LMFDB - Embedded newform 163.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [163,2,Mod(1,163)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(163, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("163.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 163 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	163.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.30156155295$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 5x^{5} + 19x^{4} - 23x^{2} + 4x + 6 $ x^7 - 3x^6 - 5x^5 + 19x^4 - 23x^2 + 4*x + 6
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			1.1
Root			$-2.19784$ of defining polynomial
Character	$\chi$	$=$	163.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.19784 q^{2} +0.245778 q^{3} +2.83050 q^{4} +2.71621 q^{5} -0.540180 q^{6} -2.14321 q^{7} -1.82530 q^{8} -2.93959 q^{9} +O(q^{10})$	\(q-2.19784 q^{2} +0.245778 q^{3} +2.83050 q^{4} +2.71621 q^{5} -0.540180 q^{6} -2.14321 q^{7} -1.82530 q^{8} -2.93959 q^{9} -5.96979 q^{10} +2.88513 q^{11} +0.695674 q^{12} +6.09527 q^{13} +4.71043 q^{14} +0.667584 q^{15} -1.64927 q^{16} +6.48816 q^{17} +6.46075 q^{18} -1.94368 q^{19} +7.68822 q^{20} -0.526753 q^{21} -6.34105 q^{22} +6.50997 q^{23} -0.448619 q^{24} +2.37779 q^{25} -13.3964 q^{26} -1.45982 q^{27} -6.06635 q^{28} -4.59948 q^{29} -1.46724 q^{30} -3.00319 q^{31} +7.27545 q^{32} +0.709100 q^{33} -14.2599 q^{34} -5.82140 q^{35} -8.32051 q^{36} +0.110177 q^{37} +4.27189 q^{38} +1.49808 q^{39} -4.95790 q^{40} +10.2254 q^{41} +1.15772 q^{42} -0.881572 q^{43} +8.16635 q^{44} -7.98455 q^{45} -14.3079 q^{46} -13.1682 q^{47} -0.405355 q^{48} -2.40665 q^{49} -5.22599 q^{50} +1.59464 q^{51} +17.2527 q^{52} -1.74104 q^{53} +3.20845 q^{54} +7.83661 q^{55} +3.91201 q^{56} -0.477713 q^{57} +10.1089 q^{58} -10.8409 q^{59} +1.88959 q^{60} -8.64533 q^{61} +6.60053 q^{62} +6.30017 q^{63} -12.6917 q^{64} +16.5560 q^{65} -1.55849 q^{66} +3.01018 q^{67} +18.3647 q^{68} +1.60001 q^{69} +12.7945 q^{70} -0.761963 q^{71} +5.36565 q^{72} -3.27503 q^{73} -0.242152 q^{74} +0.584407 q^{75} -5.50158 q^{76} -6.18344 q^{77} -3.29254 q^{78} +2.69669 q^{79} -4.47977 q^{80} +8.45999 q^{81} -22.4738 q^{82} -0.451008 q^{83} -1.49097 q^{84} +17.6232 q^{85} +1.93755 q^{86} -1.13045 q^{87} -5.26623 q^{88} +6.43037 q^{89} +17.5488 q^{90} -13.0634 q^{91} +18.4265 q^{92} -0.738117 q^{93} +28.9416 q^{94} -5.27943 q^{95} +1.78814 q^{96} -13.4421 q^{97} +5.28943 q^{98} -8.48110 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{2} + q^{3} + 5 q^{4} + 11 q^{5} - 3 q^{6} + 3 q^{8} + 2 q^{9}+O(q^{10}) $ 7 * q + 3 * q^2 + q^3 + 5 * q^4 + 11 * q^5 - 3 * q^6 + 3 * q^8 + 2 * q^9	\( 7 q + 3 q^{2} + q^{3} + 5 q^{4} + 11 q^{5} - 3 q^{6} + 3 q^{8} + 2 q^{9} + q^{10} + 2 q^{11} - 4 q^{12} + 10 q^{13} - q^{14} - 4 q^{15} - 3 q^{16} + 13 q^{17} - 4 q^{18} - 5 q^{19} + 4 q^{20} - 5 q^{21} - 11 q^{22} + 2 q^{23} - 7 q^{24} + 4 q^{25} - 9 q^{26} - 11 q^{27} - 18 q^{28} + 7 q^{29} - 13 q^{30} - 11 q^{31} - 6 q^{32} - 6 q^{33} - 6 q^{34} - 9 q^{35} - 24 q^{36} + 3 q^{37} - 5 q^{38} - 13 q^{39} - 12 q^{40} + 17 q^{41} + q^{42} - 10 q^{43} + 8 q^{44} + 12 q^{45} - 24 q^{46} + 11 q^{47} - 8 q^{48} - 7 q^{49} + 13 q^{50} + 6 q^{51} + 23 q^{52} + 18 q^{53} - 4 q^{54} - 6 q^{55} - 2 q^{56} + 20 q^{57} - q^{58} + 11 q^{59} - 27 q^{60} + 4 q^{61} + 25 q^{62} + 7 q^{63} - 21 q^{64} + 34 q^{65} - 10 q^{66} - 18 q^{67} + 23 q^{68} + 8 q^{69} + 6 q^{70} - 3 q^{71} + 22 q^{72} + 2 q^{73} - 9 q^{75} - 24 q^{76} + 25 q^{77} - 10 q^{78} - 8 q^{80} - 13 q^{81} - q^{82} + 18 q^{83} + 16 q^{84} + 12 q^{85} + 15 q^{86} + 19 q^{87} + 3 q^{88} + 18 q^{89} + 37 q^{90} - 36 q^{91} + 23 q^{92} - 15 q^{93} + 51 q^{94} - 25 q^{95} + 28 q^{96} + 21 q^{97} + 6 q^{98} - 13 q^{99}+O(q^{100}) \) 7 * q + 3 * q^2 + q^3 + 5 * q^4 + 11 * q^5 - 3 * q^6 + 3 * q^8 + 2 * q^9 + q^10 + 2 * q^11 - 4 * q^12 + 10 * q^13 - q^14 - 4 * q^15 - 3 * q^16 + 13 * q^17 - 4 * q^18 - 5 * q^19 + 4 * q^20 - 5 * q^21 - 11 * q^22 + 2 * q^23 - 7 * q^24 + 4 * q^25 - 9 * q^26 - 11 * q^27 - 18 * q^28 + 7 * q^29 - 13 * q^30 - 11 * q^31 - 6 * q^32 - 6 * q^33 - 6 * q^34 - 9 * q^35 - 24 * q^36 + 3 * q^37 - 5 * q^38 - 13 * q^39 - 12 * q^40 + 17 * q^41 + q^42 - 10 * q^43 + 8 * q^44 + 12 * q^45 - 24 * q^46 + 11 * q^47 - 8 * q^48 - 7 * q^49 + 13 * q^50 + 6 * q^51 + 23 * q^52 + 18 * q^53 - 4 * q^54 - 6 * q^55 - 2 * q^56 + 20 * q^57 - q^58 + 11 * q^59 - 27 * q^60 + 4 * q^61 + 25 * q^62 + 7 * q^63 - 21 * q^64 + 34 * q^65 - 10 * q^66 - 18 * q^67 + 23 * q^68 + 8 * q^69 + 6 * q^70 - 3 * q^71 + 22 * q^72 + 2 * q^73 - 9 * q^75 - 24 * q^76 + 25 * q^77 - 10 * q^78 - 8 * q^80 - 13 * q^81 - q^82 + 18 * q^83 + 16 * q^84 + 12 * q^85 + 15 * q^86 + 19 * q^87 + 3 * q^88 + 18 * q^89 + 37 * q^90 - 36 * q^91 + 23 * q^92 - 15 * q^93 + 51 * q^94 - 25 * q^95 + 28 * q^96 + 21 * q^97 + 6 * q^98 - 13 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.19784	−1.55411	−0.777054	−	0.629434i	$-0.783287\pi$
$2$	−2.19784	−1.55411	−0.777054	+	0.629434i	$0.783287\pi$
$3$	0.245778	0.141900	0.0709499	−	0.997480i	$-0.477397\pi$
$3$	0.245778	0.141900	0.0709499	+	0.997480i	$0.477397\pi$
$4$	2.83050	1.41525
$5$	2.71621	1.21473	0.607363	−	0.794425i	$-0.292228\pi$
$5$	2.71621	1.21473	0.607363	+	0.794425i	$0.292228\pi$
$6$	−0.540180	−0.220528
$7$	−2.14321	−0.810057	−0.405029	−	0.914304i	$-0.632738\pi$
$7$	−2.14321	−0.810057	−0.405029	+	0.914304i	$0.632738\pi$
$8$	−1.82530	−0.645342
$9$	−2.93959	−0.979864
$10$	−5.96979	−1.88781
$11$	2.88513	0.869899	0.434949	−	0.900455i	$-0.356766\pi$
$11$	2.88513	0.869899	0.434949	+	0.900455i	$0.356766\pi$
$12$	0.695674	0.200824
$13$	6.09527	1.69052	0.845262	−	0.534352i	$-0.179444\pi$
$13$	6.09527	1.69052	0.845262	+	0.534352i	$0.179444\pi$
$14$	4.71043	1.25892
$15$	0.667584	0.172369
$16$	−1.64927	−0.412319
$17$	6.48816	1.57361	0.786805	−	0.617202i	$-0.211734\pi$
$17$	6.48816	1.57361	0.786805	+	0.617202i	$0.211734\pi$
$18$	6.46075	1.52281
$19$	−1.94368	−0.445910	−0.222955	−	0.974829i	$-0.571570\pi$
$19$	−1.94368	−0.445910	−0.222955	+	0.974829i	$0.571570\pi$
$20$	7.68822	1.71914
$21$	−0.526753	−0.114947
$22$	−6.34105	−1.35192
$23$	6.50997	1.35742	0.678711	−	0.734405i	$-0.262539\pi$
$23$	6.50997	1.35742	0.678711	+	0.734405i	$0.262539\pi$
$24$	−0.448619	−0.0915739
$25$	2.37779	0.475557
$26$	−13.3964	−2.62726
$27$	−1.45982	−0.280942
$28$	−6.06635	−1.14643
$29$	−4.59948	−0.854102	−0.427051	−	0.904228i	$-0.640447\pi$
$29$	−4.59948	−0.854102	−0.427051	+	0.904228i	$0.640447\pi$
$30$	−1.46724	−0.267880
$31$	−3.00319	−0.539389	−0.269694	−	0.962946i	$-0.586923\pi$
$31$	−3.00319	−0.539389	−0.269694	+	0.962946i	$0.586923\pi$
$32$	7.27545	1.28613
$33$	0.709100	0.123439
$34$	−14.2599	−2.44556
$35$	−5.82140	−0.983997
$36$	−8.32051	−1.38675
$37$	0.110177	0.0181130	0.00905652	−	0.999959i	$-0.497117\pi$
$37$	0.110177	0.0181130	0.00905652	+	0.999959i	$0.497117\pi$
$38$	4.27189	0.692992
$39$	1.49808	0.239885
$40$	−4.95790	−0.783913
$41$	10.2254	1.59694	0.798470	−	0.602035i	$-0.205643\pi$
$41$	10.2254	1.59694	0.798470	+	0.602035i	$0.205643\pi$
$42$	1.15772	0.178640
$43$	−0.881572	−0.134438	−0.0672192	−	0.997738i	$-0.521413\pi$
$43$	−0.881572	−0.134438	−0.0672192	+	0.997738i	$0.521413\pi$
$44$	8.16635	1.23112
$45$	−7.98455	−1.19027
$46$	−14.3079	−2.10958
$47$	−13.1682	−1.92078	−0.960392	−	0.278654i	$-0.910112\pi$
$47$	−13.1682	−1.92078	−0.960392	+	0.278654i	$0.910112\pi$
$48$	−0.405355	−0.0585080
$49$	−2.40665	−0.343807
$50$	−5.22599	−0.739067
$51$	1.59464	0.223295
$52$	17.2527	2.39251
$53$	−1.74104	−0.239151	−0.119575	−	0.992825i	$-0.538153\pi$
$53$	−1.74104	−0.239151	−0.119575	+	0.992825i	$0.538153\pi$
$54$	3.20845	0.436615
$55$	7.83661	1.05669
$56$	3.91201	0.522764
$57$	−0.477713	−0.0632746
$58$	10.1089	1.32737
$59$	−10.8409	−1.41137	−0.705684	−	0.708526i	$-0.749360\pi$
$59$	−10.8409	−1.41137	−0.705684	+	0.708526i	$0.749360\pi$
$60$	1.88959	0.243946
$61$	−8.64533	−1.10692	−0.553461	−	0.832875i	$-0.686693\pi$
$61$	−8.64533	−1.10692	−0.553461	+	0.832875i	$0.686693\pi$
$62$	6.60053	0.838268
$63$	6.30017	0.793746
$64$	−12.6917	−1.58646
$65$	16.5560	2.05352
$66$	−1.55849	−0.191837
$67$	3.01018	0.367752	0.183876	−	0.982949i	$-0.441135\pi$
$67$	3.01018	0.367752	0.183876	+	0.982949i	$0.441135\pi$
$68$	18.3647	2.22705
$69$	1.60001	0.192618
$70$	12.7945	1.52924
$71$	−0.761963	−0.0904283	−0.0452142	−	0.998977i	$-0.514397\pi$
$71$	−0.761963	−0.0904283	−0.0452142	+	0.998977i	$0.514397\pi$
$72$	5.36565	0.632348
$73$	−3.27503	−0.383314	−0.191657	−	0.981462i	$-0.561386\pi$
$73$	−3.27503	−0.383314	−0.191657	+	0.981462i	$0.561386\pi$
$74$	−0.242152	−0.0281496
$75$	0.584407	0.0674815
$76$	−5.50158	−0.631074
$77$	−6.18344	−0.704668
$78$	−3.29254	−0.372807
$79$	2.69669	0.303401	0.151701	−	0.988427i	$-0.451525\pi$
$79$	2.69669	0.303401	0.151701	+	0.988427i	$0.451525\pi$
$80$	−4.47977	−0.500854
$81$	8.45999	0.939999
$82$	−22.4738	−2.48182
$83$	−0.451008	−0.0495046	−0.0247523	−	0.999694i	$-0.507880\pi$
$83$	−0.451008	−0.0495046	−0.0247523	+	0.999694i	$0.507880\pi$
$84$	−1.49097	−0.162679
$85$	17.6232	1.91150
$86$	1.93755	0.208932
$87$	−1.13045	−0.121197
$88$	−5.26623	−0.561382
$89$	6.43037	0.681618	0.340809	−	0.940133i	$-0.389299\pi$
$89$	6.43037	0.681618	0.340809	+	0.940133i	$0.389299\pi$
$90$	17.5488	1.84980
$91$	−13.0634	−1.36942
$92$	18.4265	1.92109
$93$	−0.738117	−0.0765392
$94$	28.9416	2.98510
$95$	−5.27943	−0.541658
$96$	1.78814	0.182502
$97$	−13.4421	−1.36484	−0.682422	−	0.730959i	$-0.739073\pi$
$97$	−13.4421	−1.36484	−0.682422	+	0.730959i	$0.739073\pi$
$98$	5.28943	0.534314
$99$	−8.48110	−0.852383
$100$	6.73032	0.673032
$101$	2.06168	0.205145	0.102573	−	0.994726i	$-0.467293\pi$
$101$	2.06168	0.205145	0.102573	+	0.994726i	$0.467293\pi$
$102$	−3.50477	−0.347024
$103$	8.26630	0.814502	0.407251	−	0.913316i	$-0.366487\pi$
$103$	8.26630	0.814502	0.407251	+	0.913316i	$0.366487\pi$
$104$	−11.1257	−1.09097
$105$	−1.43077	−0.139629
$106$	3.82653	0.371666
$107$	−16.5037	−1.59548	−0.797738	−	0.603004i	$-0.793970\pi$
$107$	−16.5037	−1.59548	−0.797738	+	0.603004i	$0.793970\pi$
$108$	−4.13202	−0.397604
$109$	10.9214	1.04608	0.523041	−	0.852308i	$-0.324798\pi$
$109$	10.9214	1.04608	0.523041	+	0.852308i	$0.324798\pi$
$110$	−17.2236	−1.64221
$111$	0.0270791	0.00257024
$112$	3.53474	0.334002
$113$	−15.6986	−1.47680	−0.738402	−	0.674361i	$-0.764419\pi$
$113$	−15.6986	−1.47680	−0.738402	+	0.674361i	$0.764419\pi$
$114$	1.04994	0.0983355
$115$	17.6824	1.64890
$116$	−13.0188	−1.20877
$117$	−17.9176	−1.65648
$118$	23.8266	2.19342
$119$	−13.9055	−1.27471
$120$	−1.21854	−0.111237
$121$	−2.67603	−0.243276
$122$	19.0011	1.72027
$123$	2.51318	0.226605
$124$	−8.50052	−0.763369
$125$	−7.12248	−0.637054
$126$	−13.8468	−1.23357
$127$	2.74233	0.243343	0.121671	−	0.992570i	$-0.461175\pi$
$127$	2.74233	0.243343	0.121671	+	0.992570i	$0.461175\pi$
$128$	13.3435	1.17941
$129$	−0.216671	−0.0190768
$130$	−36.3875	−3.19139
$131$	−3.83619	−0.335169	−0.167585	−	0.985858i	$-0.553597\pi$
$131$	−3.83619	−0.335169	−0.167585	+	0.985858i	$0.553597\pi$
$132$	2.00711	0.174696
$133$	4.16571	0.361213
$134$	−6.61590	−0.571527
$135$	−3.96517	−0.341268
$136$	−11.8429	−1.01552
$137$	19.4432	1.66115	0.830574	−	0.556908i	$-0.188013\pi$
$137$	19.4432	1.66115	0.830574	+	0.556908i	$0.188013\pi$
$138$	−3.51656	−0.299349
$139$	−3.88818	−0.329791	−0.164895	−	0.986311i	$-0.552729\pi$
$139$	−3.88818	−0.329791	−0.164895	+	0.986311i	$0.552729\pi$
$140$	−16.4775	−1.39260
$141$	−3.23646	−0.272559
$142$	1.67467	0.140535
$143$	17.5856	1.47059
$144$	4.84820	0.404016
$145$	−12.4931	−1.03750
$146$	7.19800	0.595711
$147$	−0.591502	−0.0487862
$148$	0.311857	0.0256345
$149$	−0.180300	−0.0147707	−0.00738536	−	0.999973i	$-0.502351\pi$
$149$	−0.180300	−0.0147707	−0.00738536	+	0.999973i	$0.502351\pi$
$150$	−1.28443	−0.104874
$151$	0.333476	0.0271379	0.0135689	−	0.999908i	$-0.495681\pi$
$151$	0.333476	0.0271379	0.0135689	+	0.999908i	$0.495681\pi$
$152$	3.54780	0.287765
$153$	−19.0725	−1.54192
$154$	13.5902	1.09513
$155$	−8.15729	−0.655209
$156$	4.24032	0.339497
$157$	−1.72548	−0.137708	−0.0688540	−	0.997627i	$-0.521934\pi$
$157$	−1.72548	−0.137708	−0.0688540	+	0.997627i	$0.521934\pi$
$158$	−5.92689	−0.471518
$159$	−0.427910	−0.0339355
$160$	19.7616	1.56229
$161$	−13.9522	−1.09959
$162$	−18.5937	−1.46086
$163$	1.00000	0.0783260
$163$	1.00000	0.0783260
$164$	28.9430	2.26007
$165$	1.92606	0.149944
$166$	0.991243	0.0769354
$167$	15.5270	1.20151	0.600757	−	0.799431i	$-0.294866\pi$
$167$	15.5270	1.20151	0.600757	+	0.799431i	$0.294866\pi$
$168$	0.961484	0.0741801
$169$	24.1523	1.85787
$170$	−38.7329	−2.97068
$171$	5.71362	0.436931
$172$	−2.49529	−0.190264
$173$	11.8562	0.901408	0.450704	−	0.892673i	$-0.351173\pi$
$173$	11.8562	0.901408	0.450704	+	0.892673i	$0.351173\pi$
$174$	2.48455	0.188353
$175$	−5.09610	−0.385229
$176$	−4.75837	−0.358676
$177$	−2.66446	−0.200273
$178$	−14.1329	−1.05931
$179$	12.8131	0.957694	0.478847	−	0.877898i	$-0.341055\pi$
$179$	12.8131	0.957694	0.478847	+	0.877898i	$0.341055\pi$
$180$	−22.6002	−1.68452
$181$	3.40224	0.252887	0.126443	−	0.991974i	$-0.459644\pi$
$181$	3.40224	0.252887	0.126443	+	0.991974i	$0.459644\pi$
$182$	28.7114	2.12823
$183$	−2.12483	−0.157072
$184$	−11.8827	−0.876002
$185$	0.299265	0.0220024
$186$	1.62226	0.118950
$187$	18.7192	1.36888
$188$	−37.2726	−2.71839
$189$	3.12870	0.227579
$190$	11.6033	0.841795
$191$	−11.5859	−0.838324	−0.419162	−	0.907911i	$-0.637676\pi$
$191$	−11.5859	−0.838324	−0.419162	+	0.907911i	$0.637676\pi$
$192$	−3.11934	−0.225119
$193$	9.67127	0.696153	0.348077	−	0.937466i	$-0.386835\pi$
$193$	9.67127	0.696153	0.348077	+	0.937466i	$0.386835\pi$
$194$	29.5437	2.12111
$195$	4.06910	0.291395
$196$	−6.81202	−0.486573
$197$	20.9422	1.49207	0.746036	−	0.665905i	$-0.231954\pi$
$197$	20.9422	1.49207	0.746036	+	0.665905i	$0.231954\pi$
$198$	18.6401	1.32469
$199$	−9.55701	−0.677478	−0.338739	−	0.940880i	$-0.610000\pi$
$199$	−9.55701	−0.677478	−0.338739	+	0.940880i	$0.610000\pi$
$200$	−4.34018	−0.306897
$201$	0.739836	0.0521840
$202$	−4.53125	−0.318818
$203$	9.85765	0.691871
$204$	4.51364	0.316018
$205$	27.7743	1.93984
$206$	−18.1680	−1.26582
$207$	−19.1367	−1.33009
$208$	−10.0528	−0.697035
$209$	−5.60776	−0.387897
$210$	3.14461	0.216998
$211$	−14.7790	−1.01743	−0.508713	−	0.860936i	$-0.669878\pi$
$211$	−14.7790	−1.01743	−0.508713	+	0.860936i	$0.669878\pi$
$212$	−4.92802	−0.338458
$213$	−0.187273	−0.0128318
$214$	36.2726	2.47954
$215$	−2.39453	−0.163306
$216$	2.66461	0.181304
$217$	6.43646	0.436936
$218$	−24.0035	−1.62572
$219$	−0.804930	−0.0543922
$220$	22.1815	1.49548
$221$	39.5471	2.66022
$222$	−0.0595156	−0.00399442
$223$	−21.7059	−1.45353	−0.726767	−	0.686884i	$-0.758978\pi$
$223$	−21.7059	−1.45353	−0.726767	+	0.686884i	$0.758978\pi$
$224$	−15.5928	−1.04184
$225$	−6.98973	−0.465982
$226$	34.5031	2.29511
$227$	24.4749	1.62446	0.812228	−	0.583340i	$-0.198254\pi$
$227$	24.4749	1.62446	0.812228	+	0.583340i	$0.198254\pi$
$228$	−1.35216	−0.0895493
$229$	−17.6325	−1.16519	−0.582595	−	0.812762i	$-0.697963\pi$
$229$	−17.6325	−1.16519	−0.582595	+	0.812762i	$0.697963\pi$
$230$	−38.8631	−2.56256
$231$	−1.51975	−0.0999923
$232$	8.39544	0.551188
$233$	−25.6667	−1.68148	−0.840739	−	0.541440i	$-0.817879\pi$
$233$	−25.6667	−1.68148	−0.840739	+	0.541440i	$0.817879\pi$
$234$	39.3801	2.57435
$235$	−35.7676	−2.33322
$236$	−30.6852	−1.99744
$237$	0.662786	0.0430526
$238$	30.5620	1.98104
$239$	−17.4776	−1.13053	−0.565267	−	0.824908i	$-0.691227\pi$
$239$	−17.4776	−1.13053	−0.565267	+	0.824908i	$0.691227\pi$
$240$	−1.10103	−0.0710711
$241$	−20.2443	−1.30405	−0.652024	−	0.758198i	$-0.726080\pi$
$241$	−20.2443	−1.30405	−0.652024	+	0.758198i	$0.726080\pi$
$242$	5.88149	0.378077
$243$	6.45874	0.414328
$244$	−24.4706	−1.56657
$245$	−6.53697	−0.417632
$246$	−5.52356	−0.352169
$247$	−11.8472	−0.753822
$248$	5.48173	0.348090
$249$	−0.110848	−0.00702469
$250$	15.6541	0.990050
$251$	−30.8764	−1.94890	−0.974449	−	0.224607i	$-0.927890\pi$
$251$	−30.8764	−1.94890	−0.974449	+	0.224607i	$0.927890\pi$
$252$	17.8326	1.12335
$253$	18.7821	1.18082
$254$	−6.02720	−0.378180
$255$	4.33139	0.271242
$256$	−3.94335	−0.246460
$257$	4.45630	0.277976	0.138988	−	0.990294i	$-0.455615\pi$
$257$	4.45630	0.277976	0.138988	+	0.990294i	$0.455615\pi$
$258$	0.476208	0.0296474
$259$	−0.236133	−0.0146726
$260$	46.8618	2.90625
$261$	13.5206	0.836904
$262$	8.43132	0.520889
$263$	25.9581	1.60064	0.800322	−	0.599571i	$-0.204662\pi$
$263$	25.9581	1.60064	0.800322	+	0.599571i	$0.204662\pi$
$264$	−1.29432	−0.0796601
$265$	−4.72904	−0.290502
$266$	−9.15556	−0.561363
$267$	1.58044	0.0967215
$268$	8.52032	0.520461
$269$	14.1995	0.865760	0.432880	−	0.901451i	$-0.357497\pi$
$269$	14.1995	0.865760	0.432880	+	0.901451i	$0.357497\pi$
$270$	8.71482	0.530367
$271$	−6.85209	−0.416235	−0.208117	−	0.978104i	$-0.566734\pi$
$271$	−6.85209	−0.416235	−0.208117	+	0.978104i	$0.566734\pi$
$272$	−10.7008	−0.648829
$273$	−3.21070	−0.194321
$274$	−42.7331	−2.58160
$275$	6.86022	0.413687
$276$	4.52881	0.272603
$277$	−4.39985	−0.264361	−0.132181	−	0.991226i	$-0.542198\pi$
$277$	−4.39985	−0.264361	−0.132181	+	0.991226i	$0.542198\pi$
$278$	8.54559	0.512530
$279$	8.82815	0.528528
$280$	10.6258	0.635014
$281$	−6.43701	−0.384000	−0.192000	−	0.981395i	$-0.561497\pi$
$281$	−6.43701	−0.384000	−0.192000	+	0.981395i	$0.561497\pi$
$282$	7.11321	0.423586
$283$	−24.9521	−1.48325	−0.741625	−	0.670815i	$-0.765945\pi$
$283$	−24.9521	−1.48325	−0.741625	+	0.670815i	$0.765945\pi$
$284$	−2.15673	−0.127979
$285$	−1.29757	−0.0768612
$286$	−38.6504	−2.28545
$287$	−21.9152	−1.29361
$288$	−21.3869	−1.26023
$289$	25.0962	1.47625
$290$	27.4579	1.61238
$291$	−3.30378	−0.193671
$292$	−9.26998	−0.542484
$293$	24.2763	1.41824	0.709118	−	0.705090i	$-0.249093\pi$
$293$	24.2763	1.41824	0.709118	+	0.705090i	$0.249093\pi$
$294$	1.30003	0.0758190
$295$	−29.4462	−1.71443
$296$	−0.201107	−0.0116891
$297$	−4.21177	−0.244392
$298$	0.396269	0.0229553
$299$	39.6800	2.29476
$300$	1.65416	0.0955032
$301$	1.88939	0.108903
$302$	−0.732926	−0.0421752
$303$	0.506716	0.0291101
$304$	3.20566	0.183857
$305$	−23.4825	−1.34461
$306$	41.9184	2.39632
$307$	32.9004	1.87772	0.938862	−	0.344293i	$-0.111881\pi$
$307$	32.9004	1.87772	0.938862	+	0.344293i	$0.111881\pi$
$308$	−17.5022	−0.997281
$309$	2.03167	0.115578
$310$	17.9284	1.01827
$311$	18.4644	1.04702	0.523511	−	0.852019i	$-0.324622\pi$
$311$	18.4644	1.04702	0.523511	+	0.852019i	$0.324622\pi$
$312$	−2.73445	−0.154808
$313$	−8.94398	−0.505544	−0.252772	−	0.967526i	$-0.581342\pi$
$313$	−8.94398	−0.505544	−0.252772	+	0.967526i	$0.581342\pi$
$314$	3.79232	0.214013
$315$	17.1126	0.964183
$316$	7.63297	0.429388
$317$	−11.1833	−0.628118	−0.314059	−	0.949403i	$-0.601689\pi$
$317$	−11.1833	−0.628118	−0.314059	+	0.949403i	$0.601689\pi$
$318$	0.940477	0.0527393
$319$	−13.2701	−0.742982
$320$	−34.4733	−1.92712
$321$	−4.05625	−0.226398
$322$	30.6648	1.70888
$323$	−12.6109	−0.701688
$324$	23.9460	1.33033
$325$	14.4933	0.803941
$326$	−2.19784	−0.121727
$327$	2.68424	0.148439
$328$	−18.6645	−1.03057
$329$	28.2223	1.55594
$330$	−4.23318	−0.233029
$331$	20.9091	1.14927	0.574635	−	0.818410i	$-0.305144\pi$
$331$	20.9091	1.14927	0.574635	+	0.818410i	$0.305144\pi$
$332$	−1.27658	−0.0700613
$333$	−0.323877	−0.0177483
$334$	−34.1258	−1.86728
$335$	8.17628	0.446718
$336$	0.868761	0.0473948
$337$	−16.1160	−0.877893	−0.438946	−	0.898513i	$-0.644648\pi$
$337$	−16.1160	−0.877893	−0.438946	+	0.898513i	$0.644648\pi$
$338$	−53.0830	−2.88733
$339$	−3.85838	−0.209558
$340$	49.8824	2.70525
$341$	−8.66459	−0.469214
$342$	−12.5576	−0.679038
$343$	20.1604	1.08856
$344$	1.60914	0.0867588
$345$	4.34595	0.233978
$346$	−26.0580	−1.40089
$347$	−33.5149	−1.79918	−0.899588	−	0.436740i	$-0.856133\pi$
$347$	−33.5149	−1.79918	−0.899588	+	0.436740i	$0.856133\pi$
$348$	−3.19974	−0.171524
$349$	−9.37984	−0.502091	−0.251046	−	0.967975i	$-0.580774\pi$
$349$	−9.37984	−0.502091	−0.251046	+	0.967975i	$0.580774\pi$
$350$	11.2004	0.598687
$351$	−8.89800	−0.474940
$352$	20.9906	1.11880
$353$	0.206320	0.0109813	0.00549066	−	0.999985i	$-0.498252\pi$
$353$	0.206320	0.0109813	0.00549066	+	0.999985i	$0.498252\pi$
$354$	5.85605	0.311246
$355$	−2.06965	−0.109846
$356$	18.2012	0.964660
$357$	−3.41766	−0.180882
$358$	−28.1611	−1.48836
$359$	30.5736	1.61361	0.806807	−	0.590815i	$-0.201194\pi$
$359$	30.5736	1.61361	0.806807	+	0.590815i	$0.201194\pi$
$360$	14.5742	0.768129
$361$	−15.2221	−0.801164
$362$	−7.47758	−0.393013
$363$	−0.657710	−0.0345208
$364$	−36.9761	−1.93807
$365$	−8.89567	−0.465621
$366$	4.67004	0.244107
$367$	7.41255	0.386932	0.193466	−	0.981107i	$-0.438027\pi$
$367$	7.41255	0.386932	0.193466	+	0.981107i	$0.438027\pi$
$368$	−10.7367	−0.559691
$369$	−30.0585	−1.56478
$370$	−0.657735	−0.0341940
$371$	3.73142	0.193726
$372$	−2.08924	−0.108322
$373$	−24.0495	−1.24524	−0.622618	−	0.782526i	$-0.713931\pi$
$373$	−24.0495	−1.24524	−0.622618	+	0.782526i	$0.713931\pi$
$374$	−41.1417	−2.12739
$375$	−1.75055	−0.0903978
$376$	24.0360	1.23956
$377$	−28.0351	−1.44388
$378$	−6.87638	−0.353683
$379$	25.3208	1.30064	0.650321	−	0.759659i	$-0.274634\pi$
$379$	25.3208	1.30064	0.650321	+	0.759659i	$0.274634\pi$
$380$	−14.9434	−0.766581
$381$	0.674004	0.0345303
$382$	25.4639	1.30285
$383$	−7.20976	−0.368402	−0.184201	−	0.982889i	$-0.558970\pi$
$383$	−7.20976	−0.368402	−0.184201	+	0.982889i	$0.558970\pi$
$384$	3.27953	0.167358
$385$	−16.7955	−0.855978
$386$	−21.2559	−1.08190
$387$	2.59146	0.131731
$388$	−38.0480	−1.93159
$389$	−0.807742	−0.0409542	−0.0204771	−	0.999790i	$-0.506519\pi$
$389$	−0.807742	−0.0409542	−0.0204771	+	0.999790i	$0.506519\pi$
$390$	−8.94324	−0.452858
$391$	42.2377	2.13605
$392$	4.39287	0.221873
$393$	−0.942849	−0.0475605
$394$	−46.0277	−2.31884
$395$	7.32476	0.368549
$396$	−24.0058	−1.20633
$397$	15.4479	0.775308	0.387654	−	0.921805i	$-0.373286\pi$
$397$	15.4479	0.775308	0.387654	+	0.921805i	$0.373286\pi$
$398$	21.0048	1.05287
$399$	1.02384	0.0512560
$400$	−3.92162	−0.196081
$401$	13.5395	0.676128	0.338064	−	0.941123i	$-0.390228\pi$
$401$	13.5395	0.676128	0.338064	+	0.941123i	$0.390228\pi$
$402$	−1.62604	−0.0810996
$403$	−18.3053	−0.911850
$404$	5.83559	0.290332
$405$	22.9791	1.14184
$406$	−21.6655	−1.07524
$407$	0.317876	0.0157565
$408$	−2.91071	−0.144102
$409$	−22.5873	−1.11687	−0.558435	−	0.829548i	$-0.688598\pi$
$409$	−22.5873	−1.11687	−0.558435	+	0.829548i	$0.688598\pi$
$410$	−61.0435	−3.01472
$411$	4.77872	0.235717
$412$	23.3977	1.15272
$413$	23.2344	1.14329
$414$	42.0593	2.06710
$415$	−1.22503	−0.0601344
$416$	44.3458	2.17423
$417$	−0.955627	−0.0467973
$418$	12.3250	0.602833
$419$	−37.3228	−1.82334	−0.911670	−	0.410923i	$-0.865206\pi$
$419$	−37.3228	−1.82334	−0.911670	+	0.410923i	$0.865206\pi$
$420$	−4.04980	−0.197610
$421$	2.42793	0.118330	0.0591650	−	0.998248i	$-0.481156\pi$
$421$	2.42793	0.118330	0.0591650	+	0.998248i	$0.481156\pi$
$422$	32.4818	1.58119
$423$	38.7092	1.88211
$424$	3.17793	0.154334
$425$	15.4275	0.748341
$426$	0.411597	0.0199419
$427$	18.5288	0.896670
$428$	−46.7138	−2.25800
$429$	4.32216	0.208676
$430$	5.26280	0.253795
$431$	9.01962	0.434460	0.217230	−	0.976120i	$-0.430298\pi$
$431$	9.01962	0.434460	0.217230	+	0.976120i	$0.430298\pi$
$432$	2.40764	0.115838
$433$	−5.33736	−0.256497	−0.128249	−	0.991742i	$-0.540936\pi$
$433$	−5.33736	−0.256497	−0.128249	+	0.991742i	$0.540936\pi$
$434$	−14.1463	−0.679045
$435$	−3.07054	−0.147221
$436$	30.9130	1.48047
$437$	−12.6533	−0.605288
$438$	1.76911	0.0845312
$439$	−2.16539	−0.103348	−0.0516742	−	0.998664i	$-0.516456\pi$
$439$	−2.16539	−0.103348	−0.0516742	+	0.998664i	$0.516456\pi$
$440$	−14.3042	−0.681925
$441$	7.07458	0.336885
$442$	−86.9181	−4.13427
$443$	19.8841	0.944724	0.472362	−	0.881405i	$-0.343402\pi$
$443$	19.8841	0.944724	0.472362	+	0.881405i	$0.343402\pi$
$444$	0.0766475	0.00363753
$445$	17.4662	0.827979
$446$	47.7060	2.25895
$447$	−0.0443136	−0.00209596
$448$	27.2010	1.28513
$449$	−10.8265	−0.510934	−0.255467	−	0.966818i	$-0.582229\pi$
$449$	−10.8265	−0.510934	−0.255467	+	0.966818i	$0.582229\pi$
$450$	15.3623	0.724185
$451$	29.5016	1.38918
$452$	−44.4350	−2.09004
$453$	0.0819609	0.00385086
$454$	−53.7919	−2.52458
$455$	−35.4830	−1.66347
$456$	0.871970	0.0408337
$457$	−22.4676	−1.05099	−0.525496	−	0.850796i	$-0.676120\pi$
$457$	−22.4676	−1.05099	−0.525496	+	0.850796i	$0.676120\pi$
$458$	38.7535	1.81083
$459$	−9.47154	−0.442094
$460$	50.0501	2.33360
$461$	−8.78854	−0.409323	−0.204662	−	0.978833i	$-0.565609\pi$
$461$	−8.78854	−0.409323	−0.204662	+	0.978833i	$0.565609\pi$
$462$	3.34017	0.155399
$463$	32.9434	1.53101	0.765505	−	0.643430i	$-0.222489\pi$
$463$	32.9434	1.53101	0.765505	+	0.643430i	$0.222489\pi$
$464$	7.58580	0.352162
$465$	−2.00488	−0.0929741
$466$	56.4112	2.61320
$467$	−1.87407	−0.0867217	−0.0433608	−	0.999059i	$-0.513807\pi$
$467$	−1.87407	−0.0867217	−0.0433608	+	0.999059i	$0.513807\pi$
$468$	−50.7158	−2.34434
$469$	−6.45145	−0.297900
$470$	78.6115	3.62608
$471$	−0.424084	−0.0195407
$472$	19.7880	0.910816
$473$	−2.54345	−0.116948
$474$	−1.45670	−0.0669083
$475$	−4.62165	−0.212056
$476$	−39.3595	−1.80404
$477$	5.11796	0.234335
$478$	38.4130	1.75697
$479$	28.9833	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$479$	28.9833	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$480$	4.85697	0.221689
$481$	0.671561	0.0306205
$482$	44.4937	2.02663
$483$	−3.42915	−0.156032
$484$	−7.57451	−0.344296
$485$	−36.5117	−1.65791
$486$	−14.1953	−0.643910
$487$	4.86203	0.220320	0.110160	−	0.993914i	$-0.464864\pi$
$487$	4.86203	0.220320	0.110160	+	0.993914i	$0.464864\pi$
$488$	15.7803	0.714343
$489$	0.245778	0.0111145
$490$	14.3672	0.649044
$491$	24.5481	1.10784	0.553920	−	0.832570i	$-0.313131\pi$
$491$	24.5481	1.10784	0.553920	+	0.832570i	$0.313131\pi$
$492$	7.11354	0.320703
$493$	−29.8421	−1.34402
$494$	26.0383	1.17152
$495$	−23.0364	−1.03541
$496$	4.95308	0.222400
$497$	1.63305	0.0732521
$498$	0.243626	0.0109171
$499$	−13.0306	−0.583332	−0.291666	−	0.956520i	$-0.594210\pi$
$499$	−13.0306	−0.583332	−0.291666	+	0.956520i	$0.594210\pi$
$500$	−20.1602	−0.901590
$501$	3.81619	0.170495
$502$	67.8613	3.02880
$503$	9.54192	0.425453	0.212727	−	0.977112i	$-0.431766\pi$
$503$	9.54192	0.425453	0.212727	+	0.977112i	$0.431766\pi$
$504$	−11.4997	−0.512238
$505$	5.59996	0.249195
$506$	−41.2800	−1.83512
$507$	5.93611	0.263632
$508$	7.76217	0.344390
$509$	12.3680	0.548202	0.274101	−	0.961701i	$-0.411620\pi$
$509$	12.3680	0.548202	0.274101	+	0.961701i	$0.411620\pi$
$510$	−9.51969	−0.421539
$511$	7.01908	0.310506
$512$	−18.0201	−0.796382
$513$	2.83742	0.125275
$514$	−9.79424	−0.432005
$515$	22.4530	0.989397
$516$	−0.613286	−0.0269984
$517$	−37.9920	−1.67089
$518$	0.518983	0.0228028
$519$	2.91398	0.127910
$520$	−30.2198	−1.32522
$521$	−7.75832	−0.339898	−0.169949	−	0.985453i	$-0.554360\pi$
$521$	−7.75832	−0.339898	−0.169949	+	0.985453i	$0.554360\pi$
$522$	−29.7161	−1.30064
$523$	10.0211	0.438194	0.219097	−	0.975703i	$-0.429689\pi$
$523$	10.0211	0.438194	0.219097	+	0.975703i	$0.429689\pi$
$524$	−10.8583	−0.474348
$525$	−1.25251	−0.0546639
$526$	−57.0517	−2.48757
$527$	−19.4852	−0.848787
$528$	−1.16950	−0.0508960
$529$	19.3797	0.842596
$530$	10.3937	0.451472
$531$	31.8679	1.38295
$532$	11.7910	0.511206
$533$	62.3266	2.69966
$534$	−3.47356	−0.150316
$535$	−44.8276	−1.93806
$536$	−5.49450	−0.237326
$537$	3.14917	0.135897
$538$	−31.2083	−1.34548
$539$	−6.94350	−0.299078
$540$	−11.2234	−0.482979
$541$	7.74918	0.333163	0.166582	−	0.986028i	$-0.446727\pi$
$541$	7.74918	0.333163	0.166582	+	0.986028i	$0.446727\pi$
$542$	15.0598	0.646874
$543$	0.836196	0.0358846
$544$	47.2042	2.02387
$545$	29.6648	1.27070
$546$	7.05661	0.301995
$547$	−17.7057	−0.757042	−0.378521	−	0.925593i	$-0.623567\pi$
$547$	−17.7057	−0.757042	−0.378521	+	0.925593i	$0.623567\pi$
$548$	55.0341	2.35094
$549$	25.4138	1.08463
$550$	−15.0777	−0.642914
$551$	8.93990	0.380853
$552$	−2.92049	−0.124305
$553$	−5.77957	−0.245772
$554$	9.67016	0.410846
$555$	0.0735526	0.00312213
$556$	−11.0055	−0.466736
$557$	−16.3373	−0.692235	−0.346117	−	0.938191i	$-0.612500\pi$
$557$	−16.3373	−0.692235	−0.346117	+	0.938191i	$0.612500\pi$
$558$	−19.4029	−0.821389
$559$	−5.37342	−0.227271
$560$	9.60109	0.405720
$561$	4.60076	0.194244
$562$	14.1475	0.596777
$563$	9.02667	0.380429	0.190214	−	0.981743i	$-0.439082\pi$
$563$	9.02667	0.380429	0.190214	+	0.981743i	$0.439082\pi$
$564$	−9.16079	−0.385739
$565$	−42.6408	−1.79391
$566$	54.8408	2.30513
$567$	−18.1315	−0.761453
$568$	1.39081	0.0583572
$569$	−30.5989	−1.28277	−0.641387	−	0.767218i	$-0.721640\pi$
$569$	−30.5989	−1.28277	−0.641387	+	0.767218i	$0.721640\pi$
$570$	2.85184	0.119451
$571$	−4.84495	−0.202755	−0.101377	−	0.994848i	$-0.532325\pi$
$571$	−4.84495	−0.202755	−0.101377	+	0.994848i	$0.532325\pi$
$572$	49.7761	2.08124
$573$	−2.84755	−0.118958
$574$	48.1661	2.01041
$575$	15.4793	0.645532
$576$	37.3085	1.55452
$577$	−7.39726	−0.307952	−0.153976	−	0.988075i	$-0.549208\pi$
$577$	−7.39726	−0.307952	−0.153976	+	0.988075i	$0.549208\pi$
$578$	−55.1574	−2.29425
$579$	2.37698	0.0987840
$580$	−35.3618	−1.46832
$581$	0.966605	0.0401015
$582$	7.26118	0.300986
$583$	−5.02313	−0.208037
$584$	5.97793	0.247368
$585$	−48.6680	−2.01217
$586$	−53.3554	−2.20409
$587$	36.3661	1.50099	0.750494	−	0.660877i	$-0.229816\pi$
$587$	36.3661	1.50099	0.750494	+	0.660877i	$0.229816\pi$
$588$	−1.67424	−0.0690447
$589$	5.83723	0.240519
$590$	64.7181	2.66440
$591$	5.14714	0.211725
$592$	−0.181713	−0.00746834
$593$	−12.6250	−0.518448	−0.259224	−	0.965817i	$-0.583467\pi$
$593$	−12.6250	−0.518448	−0.259224	+	0.965817i	$0.583467\pi$
$594$	9.25679	0.379811
$595$	−37.7702	−1.54843
$596$	−0.510338	−0.0209042
$597$	−2.34890	−0.0961341
$598$	−87.2103	−3.56630
$599$	11.5871	0.473437	0.236719	−	0.971578i	$-0.423928\pi$
$599$	11.5871	0.473437	0.236719	+	0.971578i	$0.423928\pi$
$600$	−1.06672	−0.0435487
$601$	40.6684	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$601$	40.6684	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$602$	−4.15258	−0.169247
$603$	−8.84871	−0.360348
$604$	0.943903	0.0384069
$605$	−7.26867	−0.295513
$606$	−1.11368	−0.0452402
$607$	48.0450	1.95009	0.975044	−	0.222012i	$-0.0712626\pi$
$607$	48.0450	1.95009	0.975044	+	0.222012i	$0.0712626\pi$
$608$	−14.1411	−0.573498
$609$	2.42279	0.0981764
$610$	51.6108	2.08966
$611$	−80.2639	−3.24713
$612$	−53.9848	−2.18221
$613$	7.59812	0.306885	0.153443	−	0.988158i	$-0.450964\pi$
$613$	7.59812	0.306885	0.153443	+	0.988158i	$0.450964\pi$
$614$	−72.3098	−2.91819
$615$	6.82631	0.275263
$616$	11.2866	0.454752
$617$	−18.7103	−0.753250	−0.376625	−	0.926366i	$-0.622915\pi$
$617$	−18.7103	−0.753250	−0.376625	+	0.926366i	$0.622915\pi$
$618$	−4.46529	−0.179620
$619$	−12.3125	−0.494880	−0.247440	−	0.968903i	$-0.579589\pi$
$619$	−12.3125	−0.494880	−0.247440	+	0.968903i	$0.579589\pi$
$620$	−23.0892	−0.927284
$621$	−9.50338	−0.381358
$622$	−40.5818	−1.62718
$623$	−13.7816	−0.552150
$624$	−2.47075	−0.0989091
$625$	−31.2351	−1.24940
$626$	19.6574	0.785669
$627$	−1.37826	−0.0550425
$628$	−4.88396	−0.194891
$629$	0.714848	0.0285029
$630$	−37.6107	−1.49844
$631$	13.8693	0.552128	0.276064	−	0.961139i	$-0.410970\pi$
$631$	13.8693	0.552128	0.276064	+	0.961139i	$0.410970\pi$
$632$	−4.92227	−0.195797
$633$	−3.63234	−0.144373
$634$	24.5791	0.976163
$635$	7.44874	0.295594
$636$	−1.21120	−0.0480271
$637$	−14.6692	−0.581215
$638$	29.1655	1.15467
$639$	2.23986	0.0886075
$640$	36.2436	1.43265
$641$	−7.01727	−0.277165	−0.138583	−	0.990351i	$-0.544255\pi$
$641$	−7.01727	−0.277165	−0.138583	+	0.990351i	$0.544255\pi$
$642$	8.91499	0.351846
$643$	−36.8663	−1.45387	−0.726933	−	0.686708i	$-0.759055\pi$
$643$	−36.8663	−1.45387	−0.726933	+	0.686708i	$0.759055\pi$
$644$	−39.4918	−1.55619
$645$	−0.588523	−0.0231731
$646$	27.7167	1.09050
$647$	−3.36324	−0.132223	−0.0661114	−	0.997812i	$-0.521059\pi$
$647$	−3.36324	−0.132223	−0.0661114	+	0.997812i	$0.521059\pi$
$648$	−15.4420	−0.606621
$649$	−31.2775	−1.22775
$650$	−31.8538	−1.24941
$651$	1.58194	0.0620011
$652$	2.83050	0.110851
$653$	−23.3417	−0.913429	−0.456715	−	0.889613i	$-0.650974\pi$
$653$	−23.3417	−0.913429	−0.456715	+	0.889613i	$0.650974\pi$
$654$	−5.89953	−0.230690
$655$	−10.4199	−0.407138
$656$	−16.8645	−0.658448
$657$	9.62727	0.375595
$658$	−62.0280	−2.41810
$659$	3.06174	0.119268	0.0596342	−	0.998220i	$-0.481007\pi$
$659$	3.06174	0.119268	0.0596342	+	0.998220i	$0.481007\pi$
$660$	5.45172	0.212208
$661$	15.0883	0.586868	0.293434	−	0.955979i	$-0.405202\pi$
$661$	15.0883	0.586868	0.293434	+	0.955979i	$0.405202\pi$
$662$	−45.9549	−1.78609
$663$	9.71979	0.377486
$664$	0.823226	0.0319474
$665$	11.3149	0.438774
$666$	0.711829	0.0275828
$667$	−29.9425	−1.15938
$668$	43.9491	1.70044
$669$	−5.33482	−0.206256
$670$	−17.9702	−0.694248
$671$	−24.9429	−0.962910
$672$	−3.83237	−0.147837
$673$	0.00828583	0.000319395 0	0.000159698	−	1.00000i	$-0.499949\pi$
$673$	0.00828583	0.000319395 0	0.000159698		1.00000i	$0.499949\pi$
$674$	35.4203	1.36434
$675$	−3.47114	−0.133604
$676$	68.3632	2.62935
$677$	30.6154	1.17664	0.588322	−	0.808627i	$-0.299789\pi$
$677$	30.6154	1.17664	0.588322	+	0.808627i	$0.299789\pi$
$678$	8.48009	0.325676
$679$	28.8093	1.10560
$680$	−32.1677	−1.23357
$681$	6.01539	0.230510
$682$	19.0434	0.729208
$683$	43.2939	1.65659	0.828297	−	0.560289i	$-0.189310\pi$
$683$	43.2939	1.65659	0.828297	+	0.560289i	$0.189310\pi$
$684$	16.1724	0.618367
$685$	52.8119	2.01784
$686$	−44.3094	−1.69174
$687$	−4.33368	−0.165340
$688$	1.45395	0.0554315
$689$	−10.6121	−0.404290
$690$	−9.55170	−0.363627
$691$	8.54858	0.325203	0.162602	−	0.986692i	$-0.448011\pi$
$691$	8.54858	0.325203	0.162602	+	0.986692i	$0.448011\pi$
$692$	33.5589	1.27572
$693$	18.1768	0.690479
$694$	73.6604	2.79611
$695$	−10.5611	−0.400605
$696$	2.06341	0.0782135
$697$	66.3440	2.51296
$698$	20.6154	0.780304
$699$	−6.30829	−0.238602
$700$	−14.4245	−0.545194
$701$	−24.4201	−0.922333	−0.461167	−	0.887314i	$-0.652569\pi$
$701$	−24.4201	−0.922333	−0.461167	+	0.887314i	$0.652569\pi$
$702$	19.5564	0.738108
$703$	−0.214149	−0.00807679
$704$	−36.6172	−1.38006
$705$	−8.79089	−0.331084
$706$	−0.453459	−0.0170662
$707$	−4.41862	−0.166179
$708$	−7.54175	−0.283436
$709$	−11.9755	−0.449749	−0.224875	−	0.974388i	$-0.572197\pi$
$709$	−11.9755	−0.449749	−0.224875	+	0.974388i	$0.572197\pi$
$710$	4.54876	0.170712
$711$	−7.92716	−0.297292
$712$	−11.7374	−0.439877
$713$	−19.5507	−0.732178
$714$	7.51147	0.281110
$715$	47.7663	1.78636
$716$	36.2674	1.35538
$717$	−4.29561	−0.160423
$718$	−67.1959	−2.50773
$719$	−34.3069	−1.27943	−0.639715	−	0.768612i	$-0.720948\pi$
$719$	−34.3069	−1.27943	−0.639715	+	0.768612i	$0.720948\pi$
$720$	13.1687	0.490769
$721$	−17.7164	−0.659793
$722$	33.4558	1.24510
$723$	−4.97559	−0.185044
$724$	9.63004	0.357898
$725$	−10.9366	−0.406174
$726$	1.44554	0.0536490
$727$	16.7489	0.621182	0.310591	−	0.950544i	$-0.399473\pi$
$727$	16.7489	0.621182	0.310591	+	0.950544i	$0.399473\pi$
$728$	23.8447	0.883745
$729$	−23.7926	−0.881206
$730$	19.5513	0.723625
$731$	−5.71978	−0.211554
$732$	−6.01433	−0.222296
$733$	−10.9853	−0.405751	−0.202876	−	0.979205i	$-0.565029\pi$
$733$	−10.9853	−0.405751	−0.202876	+	0.979205i	$0.565029\pi$
$734$	−16.2916	−0.601334
$735$	−1.60664	−0.0592619
$736$	47.3629	1.74582
$737$	8.68477	0.319907
$738$	66.0638	2.43184
$739$	−0.0267848	−0.000985293 0	−0.000492647	−	1.00000i	$-0.500157\pi$
$739$	−0.0267848	−0.000985293 0	−0.000492647		1.00000i	$0.500157\pi$
$740$	0.847068	0.0311388
$741$	−2.91179	−0.106967
$742$	−8.20107	−0.301071
$743$	31.2029	1.14472	0.572362	−	0.820001i	$-0.306027\pi$
$743$	31.2029	1.14472	0.572362	+	0.820001i	$0.306027\pi$
$744$	1.34729	0.0493939
$745$	−0.489731	−0.0179424
$746$	52.8569	1.93523
$747$	1.32578	0.0485078
$748$	52.9846	1.93731
$749$	35.3710	1.29243
$750$	3.84742	0.140488
$751$	28.3831	1.03571	0.517856	−	0.855468i	$-0.326730\pi$
$751$	28.3831	1.03571	0.517856	+	0.855468i	$0.326730\pi$
$752$	21.7180	0.791975
$753$	−7.58872	−0.276548
$754$	61.6166	2.24394
$755$	0.905789	0.0329651
$756$	8.85578	0.322082
$757$	−1.50263	−0.0546140	−0.0273070	−	0.999627i	$-0.508693\pi$
$757$	−1.50263	−0.0546140	−0.0273070	+	0.999627i	$0.508693\pi$
$758$	−55.6511	−2.02134
$759$	4.61622	0.167558
$760$	9.63656	0.349555
$761$	−11.9292	−0.432433	−0.216217	−	0.976345i	$-0.569372\pi$
$761$	−11.9292	−0.432433	−0.216217	+	0.976345i	$0.569372\pi$
$762$	−1.48135	−0.0536638
$763$	−23.4069	−0.847386
$764$	−32.7938	−1.18644
$765$	−51.8050	−1.87301
$766$	15.8459	0.572536
$767$	−66.0784	−2.38595
$768$	−0.969188	−0.0349726
$769$	8.03072	0.289595	0.144797	−	0.989461i	$-0.453747\pi$
$769$	8.03072	0.289595	0.144797	+	0.989461i	$0.453747\pi$
$770$	36.9138	1.33028
$771$	1.09526	0.0394448
$772$	27.3745	0.985230
$773$	43.4736	1.56364	0.781819	−	0.623505i	$-0.214292\pi$
$773$	43.4736	1.56364	0.781819	+	0.623505i	$0.214292\pi$
$774$	−5.69562	−0.204725
$775$	−7.14094	−0.256510
$776$	24.5360	0.880791
$777$	−0.0580363	−0.00208204
$778$	1.77529	0.0636472
$779$	−19.8749	−0.712091
$780$	11.5176	0.412396
$781$	−2.19836	−0.0786635
$782$	−92.8317	−3.31966
$783$	6.71441	0.239953
$784$	3.96923	0.141758
$785$	−4.68675	−0.167277
$786$	2.07223	0.0739141
$787$	−9.89211	−0.352616	−0.176308	−	0.984335i	$-0.556415\pi$
$787$	−9.89211	−0.352616	−0.176308	+	0.984335i	$0.556415\pi$
$788$	59.2770	2.11165
$789$	6.37992	0.227131
$790$	−16.0987	−0.572765
$791$	33.6455	1.19630
$792$	15.4806	0.550079
$793$	−52.6956	−1.87128
$794$	−33.9520	−1.20491
$795$	−1.16229	−0.0412222
$796$	−27.0511	−0.958801
$797$	39.8797	1.41261	0.706305	−	0.707907i	$-0.250361\pi$
$797$	39.8797	1.41261	0.706305	+	0.707907i	$0.250361\pi$
$798$	−2.25023	−0.0796574
$799$	−85.4375	−3.02256
$800$	17.2995	0.611628
$801$	−18.9027	−0.667893
$802$	−29.7576	−1.05078
$803$	−9.44889	−0.333444
$804$	2.09411	0.0738534
$805$	−37.8972	−1.33570
$806$	40.2320	1.41711
$807$	3.48993	0.122851
$808$	−3.76320	−0.132389
$809$	−38.6530	−1.35897	−0.679483	−	0.733691i	$-0.737796\pi$
$809$	−38.6530	−1.35897	−0.679483	+	0.733691i	$0.737796\pi$
$810$	−50.5044	−1.77454
$811$	52.7138	1.85103	0.925516	−	0.378709i	$-0.123632\pi$
$811$	52.7138	1.85103	0.925516	+	0.378709i	$0.123632\pi$
$812$	27.9021	0.979170
$813$	−1.68409	−0.0590637
$814$	−0.698640	−0.0244873
$815$	2.71621	0.0951446
$816$	−2.63001	−0.0920687
$817$	1.71349	0.0599475
$818$	49.6432	1.73573
$819$	38.4012	1.34185
$820$	78.6152	2.74536
$821$	8.30174	0.289733	0.144866	−	0.989451i	$-0.453725\pi$
$821$	8.30174	0.289733	0.144866	+	0.989451i	$0.453725\pi$
$822$	−10.5028	−0.366329
$823$	−38.3106	−1.33542	−0.667712	−	0.744420i	$-0.732726\pi$
$823$	−38.3106	−1.33542	−0.667712	+	0.744420i	$0.732726\pi$
$824$	−15.0885	−0.525633
$825$	1.68609	0.0587021
$826$	−51.0655	−1.77679
$827$	−23.7299	−0.825170	−0.412585	−	0.910919i	$-0.635374\pi$
$827$	−23.7299	−0.825170	−0.412585	+	0.910919i	$0.635374\pi$
$828$	−54.1663	−1.88241
$829$	19.8199	0.688373	0.344187	−	0.938901i	$-0.388155\pi$
$829$	19.8199	0.688373	0.344187	+	0.938901i	$0.388155\pi$
$830$	2.69242	0.0934554
$831$	−1.08138	−0.0375128
$832$	−77.3594	−2.68196
$833$	−15.6147	−0.541019
$834$	2.10032	0.0727280
$835$	42.1746	1.45951
$836$	−15.8728	−0.548971
$837$	4.38411	0.151537
$838$	82.0296	2.83367
$839$	4.40599	0.152112	0.0760558	−	0.997104i	$-0.475767\pi$
$839$	4.40599	0.152112	0.0760558	+	0.997104i	$0.475767\pi$
$840$	2.61159	0.0901085
$841$	−7.84480	−0.270510
$842$	−5.33619	−0.183897
$843$	−1.58207	−0.0544895
$844$	−41.8318	−1.43991
$845$	65.6028	2.25680
$846$	−85.0767	−2.92500
$847$	5.73530	0.197067
$848$	2.87146	0.0986063
$849$	−6.13268	−0.210473
$850$	−33.9071	−1.16300
$851$	0.717251	0.0245870
$852$	−0.530077	−0.0181601
$853$	−33.8448	−1.15882	−0.579412	−	0.815035i	$-0.696718\pi$
$853$	−33.8448	−1.15882	−0.579412	+	0.815035i	$0.696718\pi$
$854$	−40.7232	−1.39352
$855$	15.5194	0.530752
$856$	30.1243	1.02963
$857$	−28.4936	−0.973322	−0.486661	−	0.873591i	$-0.661785\pi$
$857$	−28.4936	−0.973322	−0.486661	+	0.873591i	$0.661785\pi$
$858$	−9.49941	−0.324305
$859$	18.7414	0.639447	0.319723	−	0.947511i	$-0.396410\pi$
$859$	18.7414	0.639447	0.319723	+	0.947511i	$0.396410\pi$
$860$	−6.77772	−0.231118
$861$	−5.38626	−0.183563
$862$	−19.8237	−0.675197
$863$	−14.6361	−0.498220	−0.249110	−	0.968475i	$-0.580138\pi$
$863$	−14.6361	−0.498220	−0.249110	+	0.968475i	$0.580138\pi$
$864$	−10.6208	−0.361328
$865$	32.2038	1.09496
$866$	11.7307	0.398624
$867$	6.16809	0.209479
$868$	18.2184	0.618373
$869$	7.78029	0.263928
$870$	6.74855	0.228797
$871$	18.3479	0.621694
$872$	−19.9349	−0.675080
$873$	39.5144	1.33736
$874$	27.8099	0.940683
$875$	15.2650	0.516050
$876$	−2.27835	−0.0769785
$877$	−12.8754	−0.434770	−0.217385	−	0.976086i	$-0.569753\pi$
$877$	−12.8754	−0.434770	−0.217385	+	0.976086i	$0.569753\pi$
$878$	4.75918	0.160614
$879$	5.96657	0.201247
$880$	−12.9247	−0.435692
$881$	7.54592	0.254229	0.127114	−	0.991888i	$-0.459428\pi$
$881$	7.54592	0.254229	0.127114	+	0.991888i	$0.459428\pi$
$882$	−15.5488	−0.523555
$883$	41.3456	1.39139	0.695696	−	0.718336i	$-0.255096\pi$
$883$	41.3456	1.39139	0.695696	+	0.718336i	$0.255096\pi$
$884$	111.938	3.76488
$885$	−7.23723	−0.243277
$886$	−43.7021	−1.46820
$887$	21.3096	0.715508	0.357754	−	0.933816i	$-0.383543\pi$
$887$	21.3096	0.715508	0.357754	+	0.933816i	$0.383543\pi$
$888$	−0.0494276	−0.00165868
$889$	−5.87739	−0.197121
$890$	−38.3880	−1.28677
$891$	24.4082	0.817704
$892$	−61.4385	−2.05711
$893$	25.5948	0.856497
$894$	0.0973942	0.00325735
$895$	34.8030	1.16333
$896$	−28.5978	−0.955387
$897$	9.75247	0.325625
$898$	23.7949	0.794046
$899$	13.8131	0.460693
$900$	−19.7844	−0.659480
$901$	−11.2962	−0.376330
$902$	−64.8398	−2.15893
$903$	0.464371	0.0154533
$904$	28.6548	0.953043
$905$	9.24120	0.307188
$906$	−0.180137	−0.00598465
$907$	−41.8268	−1.38884	−0.694419	−	0.719571i	$-0.744338\pi$
$907$	−41.8268	−1.38884	−0.694419	+	0.719571i	$0.744338\pi$
$908$	69.2762	2.29901
$909$	−6.06051	−0.201014
$910$	77.9860	2.58521
$911$	9.56353	0.316854	0.158427	−	0.987371i	$-0.449358\pi$
$911$	9.56353	0.316854	0.158427	+	0.987371i	$0.449358\pi$
$912$	0.787879	0.0260893
$913$	−1.30122	−0.0430640
$914$	49.3803	1.63335
$915$	−5.77148	−0.190799
$916$	−49.9089	−1.64904
$917$	8.22175	0.271506
$918$	20.8169	0.687061
$919$	−21.1841	−0.698799	−0.349400	−	0.936974i	$-0.613614\pi$
$919$	−21.1841	−0.698799	−0.349400	+	0.936974i	$0.613614\pi$
$920$	−32.2758	−1.06410
$921$	8.08619	0.266449
$922$	19.3158	0.636132
$923$	−4.64437	−0.152871
$924$	−4.30165	−0.141514
$925$	0.261978	0.00861379
$926$	−72.4043	−2.37935
$927$	−24.2995	−0.798102
$928$	−33.4633	−1.09849
$929$	27.5801	0.904872	0.452436	−	0.891797i	$-0.350555\pi$
$929$	27.5801	0.904872	0.452436	+	0.891797i	$0.350555\pi$
$930$	4.40640	0.144492
$931$	4.67775	0.153307
$932$	−72.6495	−2.37971
$933$	4.53814	0.148572
$934$	4.11891	0.134775
$935$	50.8452	1.66281
$936$	32.7051	1.06900
$937$	43.8622	1.43292	0.716458	−	0.697630i	$-0.245762\pi$
$937$	43.8622	1.43292	0.716458	+	0.697630i	$0.245762\pi$
$938$	14.1793	0.462969
$939$	−2.19823	−0.0717366
$940$	−101.240	−3.30209
$941$	10.2759	0.334983	0.167492	−	0.985873i	$-0.446433\pi$
$941$	10.2759	0.334983	0.167492	+	0.985873i	$0.446433\pi$
$942$	0.932068	0.0303684
$943$	66.5671	2.16772
$944$	17.8797	0.581934
$945$	8.49820	0.276447
$946$	5.59009	0.181750
$947$	−24.0333	−0.780977	−0.390488	−	0.920608i	$-0.627694\pi$
$947$	−24.0333	−0.780977	−0.390488	+	0.920608i	$0.627694\pi$
$948$	1.87601	0.0609301
$949$	−19.9622	−0.648001
$950$	10.1576	0.329557
$951$	−2.74861	−0.0891298
$952$	25.3817	0.822626
$953$	2.36407	0.0765799	0.0382899	−	0.999267i	$-0.487809\pi$
$953$	2.36407	0.0765799	0.0382899	+	0.999267i	$0.487809\pi$
$954$	−11.2485	−0.364182
$955$	−31.4696	−1.01833
$956$	−49.4704	−1.59999
$957$	−3.26149	−0.105429
$958$	−63.7007	−2.05807
$959$	−41.6709	−1.34562
$960$	−8.47278	−0.273458
$961$	−21.9809	−0.709060
$962$	−1.47598	−0.0475876
$963$	48.5143	1.56335
$964$	−57.3014	−1.84555
$965$	26.2692	0.845635
$966$	7.53672	0.242490
$967$	−14.0103	−0.450540	−0.225270	−	0.974296i	$-0.572326\pi$
$967$	−14.0103	−0.450540	−0.225270	+	0.974296i	$0.572326\pi$
$968$	4.88457	0.156996
$969$	−3.09947	−0.0995695
$970$	80.2468	2.57657
$971$	27.7449	0.890375	0.445188	−	0.895437i	$-0.353137\pi$
$971$	27.7449	0.890375	0.445188	+	0.895437i	$0.353137\pi$
$972$	18.2814	0.586378
$973$	8.33318	0.267149
$974$	−10.6860	−0.342401
$975$	3.56212	0.114079
$976$	14.2585	0.456404
$977$	38.1195	1.21955	0.609776	−	0.792574i	$-0.291259\pi$
$977$	38.1195	1.21955	0.609776	+	0.792574i	$0.291259\pi$
$978$	−0.540180	−0.0172731
$979$	18.5525	0.592939
$980$	−18.5029	−0.591053
$981$	−32.1045	−1.02502
$982$	−53.9528	−1.72170
$983$	−2.52478	−0.0805278	−0.0402639	−	0.999189i	$-0.512820\pi$
$983$	−2.52478	−0.0805278	−0.0402639	+	0.999189i	$0.512820\pi$
$984$	−4.58731	−0.146238
$985$	56.8835	1.81246
$986$	65.5882	2.08876
$987$	6.93641	0.220788
$988$	−33.5336	−1.06685
$989$	−5.73901	−0.182490
$990$	50.6304	1.60914
$991$	−41.8708	−1.33007	−0.665034	−	0.746813i	$-0.731583\pi$
$991$	−41.8708	−1.33007	−0.665034	+	0.746813i	$0.731583\pi$
$992$	−21.8495	−0.693724
$993$	5.13900	0.163081
$994$	−3.58917	−0.113842
$995$	−25.9588	−0.822950
$996$	−0.313754	−0.00994169
$997$	19.6250	0.621532	0.310766	−	0.950487i	$-0.399415\pi$
$997$	19.6250	0.621532	0.310766	+	0.950487i	$0.399415\pi$
$998$	28.6393	0.906560
$999$	−0.160839	−0.00508872

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	163.2.a.c.1.1	✓	7
3.2	odd	2		1467.2.a.f.1.7		7
4.3	odd	2		2608.2.a.n.1.5		7
5.4	even	2		4075.2.a.f.1.7		7
7.6	odd	2		7987.2.a.h.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
163.2.a.c.1.1	✓	7	1.1	even	1	trivial
1467.2.a.f.1.7		7	3.2	odd	2
2608.2.a.n.1.5		7	4.3	odd	2
4075.2.a.f.1.7		7	5.4	even	2
7987.2.a.h.1.1		7	7.6	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:	\( N \)	\(=\)	\( 163 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	163.a (trivial)

\(f(q)\)	\(=\)	\(q-2.19784 q^{2} +0.245778 q^{3} +2.83050 q^{4} +2.71621 q^{5} -0.540180 q^{6} -2.14321 q^{7} -1.82530 q^{8} -2.93959 q^{9} +O(q^{10})\)	\(q-2.19784 q^{2} +0.245778 q^{3} +2.83050 q^{4} +2.71621 q^{5} -0.540180 q^{6} -2.14321 q^{7} -1.82530 q^{8} -2.93959 q^{9} -5.96979 q^{10} +2.88513 q^{11} +0.695674 q^{12} +6.09527 q^{13} +4.71043 q^{14} +0.667584 q^{15} -1.64927 q^{16} +6.48816 q^{17} +6.46075 q^{18} -1.94368 q^{19} +7.68822 q^{20} -0.526753 q^{21} -6.34105 q^{22} +6.50997 q^{23} -0.448619 q^{24} +2.37779 q^{25} -13.3964 q^{26} -1.45982 q^{27} -6.06635 q^{28} -4.59948 q^{29} -1.46724 q^{30} -3.00319 q^{31} +7.27545 q^{32} +0.709100 q^{33} -14.2599 q^{34} -5.82140 q^{35} -8.32051 q^{36} +0.110177 q^{37} +4.27189 q^{38} +1.49808 q^{39} -4.95790 q^{40} +10.2254 q^{41} +1.15772 q^{42} -0.881572 q^{43} +8.16635 q^{44} -7.98455 q^{45} -14.3079 q^{46} -13.1682 q^{47} -0.405355 q^{48} -2.40665 q^{49} -5.22599 q^{50} +1.59464 q^{51} +17.2527 q^{52} -1.74104 q^{53} +3.20845 q^{54} +7.83661 q^{55} +3.91201 q^{56} -0.477713 q^{57} +10.1089 q^{58} -10.8409 q^{59} +1.88959 q^{60} -8.64533 q^{61} +6.60053 q^{62} +6.30017 q^{63} -12.6917 q^{64} +16.5560 q^{65} -1.55849 q^{66} +3.01018 q^{67} +18.3647 q^{68} +1.60001 q^{69} +12.7945 q^{70} -0.761963 q^{71} +5.36565 q^{72} -3.27503 q^{73} -0.242152 q^{74} +0.584407 q^{75} -5.50158 q^{76} -6.18344 q^{77} -3.29254 q^{78} +2.69669 q^{79} -4.47977 q^{80} +8.45999 q^{81} -22.4738 q^{82} -0.451008 q^{83} -1.49097 q^{84} +17.6232 q^{85} +1.93755 q^{86} -1.13045 q^{87} -5.26623 q^{88} +6.43037 q^{89} +17.5488 q^{90} -13.0634 q^{91} +18.4265 q^{92} -0.738117 q^{93} +28.9416 q^{94} -5.27943 q^{95} +1.78814 q^{96} -13.4421 q^{97} +5.28943 q^{98} -8.48110 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{2} + q^{3} + 5 q^{4} + 11 q^{5} - 3 q^{6} + 3 q^{8} + 2 q^{9}+O(q^{10}) \) 7 * q + 3 * q^2 + q^3 + 5 * q^4 + 11 * q^5 - 3 * q^6 + 3 * q^8 + 2 * q^9	\( 7 q + 3 q^{2} + q^{3} + 5 q^{4} + 11 q^{5} - 3 q^{6} + 3 q^{8} + 2 q^{9} + q^{10} + 2 q^{11} - 4 q^{12} + 10 q^{13} - q^{14} - 4 q^{15} - 3 q^{16} + 13 q^{17} - 4 q^{18} - 5 q^{19} + 4 q^{20} - 5 q^{21} - 11 q^{22} + 2 q^{23} - 7 q^{24} + 4 q^{25} - 9 q^{26} - 11 q^{27} - 18 q^{28} + 7 q^{29} - 13 q^{30} - 11 q^{31} - 6 q^{32} - 6 q^{33} - 6 q^{34} - 9 q^{35} - 24 q^{36} + 3 q^{37} - 5 q^{38} - 13 q^{39} - 12 q^{40} + 17 q^{41} + q^{42} - 10 q^{43} + 8 q^{44} + 12 q^{45} - 24 q^{46} + 11 q^{47} - 8 q^{48} - 7 q^{49} + 13 q^{50} + 6 q^{51} + 23 q^{52} + 18 q^{53} - 4 q^{54} - 6 q^{55} - 2 q^{56} + 20 q^{57} - q^{58} + 11 q^{59} - 27 q^{60} + 4 q^{61} + 25 q^{62} + 7 q^{63} - 21 q^{64} + 34 q^{65} - 10 q^{66} - 18 q^{67} + 23 q^{68} + 8 q^{69} + 6 q^{70} - 3 q^{71} + 22 q^{72} + 2 q^{73} - 9 q^{75} - 24 q^{76} + 25 q^{77} - 10 q^{78} - 8 q^{80} - 13 q^{81} - q^{82} + 18 q^{83} + 16 q^{84} + 12 q^{85} + 15 q^{86} + 19 q^{87} + 3 q^{88} + 18 q^{89} + 37 q^{90} - 36 q^{91} + 23 q^{92} - 15 q^{93} + 51 q^{94} - 25 q^{95} + 28 q^{96} + 21 q^{97} + 6 q^{98} - 13 q^{99}+O(q^{100}) \) 7 * q + 3 * q^2 + q^3 + 5 * q^4 + 11 * q^5 - 3 * q^6 + 3 * q^8 + 2 * q^9 + q^10 + 2 * q^11 - 4 * q^12 + 10 * q^13 - q^14 - 4 * q^15 - 3 * q^16 + 13 * q^17 - 4 * q^18 - 5 * q^19 + 4 * q^20 - 5 * q^21 - 11 * q^22 + 2 * q^23 - 7 * q^24 + 4 * q^25 - 9 * q^26 - 11 * q^27 - 18 * q^28 + 7 * q^29 - 13 * q^30 - 11 * q^31 - 6 * q^32 - 6 * q^33 - 6 * q^34 - 9 * q^35 - 24 * q^36 + 3 * q^37 - 5 * q^38 - 13 * q^39 - 12 * q^40 + 17 * q^41 + q^42 - 10 * q^43 + 8 * q^44 + 12 * q^45 - 24 * q^46 + 11 * q^47 - 8 * q^48 - 7 * q^49 + 13 * q^50 + 6 * q^51 + 23 * q^52 + 18 * q^53 - 4 * q^54 - 6 * q^55 - 2 * q^56 + 20 * q^57 - q^58 + 11 * q^59 - 27 * q^60 + 4 * q^61 + 25 * q^62 + 7 * q^63 - 21 * q^64 + 34 * q^65 - 10 * q^66 - 18 * q^67 + 23 * q^68 + 8 * q^69 + 6 * q^70 - 3 * q^71 + 22 * q^72 + 2 * q^73 - 9 * q^75 - 24 * q^76 + 25 * q^77 - 10 * q^78 - 8 * q^80 - 13 * q^81 - q^82 + 18 * q^83 + 16 * q^84 + 12 * q^85 + 15 * q^86 + 19 * q^87 + 3 * q^88 + 18 * q^89 + 37 * q^90 - 36 * q^91 + 23 * q^92 - 15 * q^93 + 51 * q^94 - 25 * q^95 + 28 * q^96 + 21 * q^97 + 6 * q^98 - 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more