LMFDB - Embedded newform 637.2.a.l.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [637,2,Mod(1,637)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("637.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(637, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 637 = 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	637.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,4,0,8,2,-5,0,9,3,-5,11,5,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.08647060876$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.746052.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 7x^{3} + 8x + 2 $ x^5 - x^4 - 7x^3 + 8x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.21332$ of defining polynomial
Character	$\chi$	$=$	637.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.21332 q^{2} +2.47443 q^{3} +2.89879 q^{4} -2.12280 q^{5} +5.47671 q^{6} +1.98932 q^{8} +3.12280 q^{9} -4.69843 q^{10} +4.78896 q^{11} +7.17286 q^{12} +1.00000 q^{13} -5.25271 q^{15} -1.39458 q^{16} -3.77828 q^{17} +6.91175 q^{18} -3.56723 q^{19} -6.15355 q^{20} +10.5995 q^{22} +4.47443 q^{23} +4.92243 q^{24} -0.493740 q^{25} +2.21332 q^{26} +0.303848 q^{27} -5.90107 q^{29} -11.6259 q^{30} -3.77116 q^{31} -7.06530 q^{32} +11.8499 q^{33} -8.36254 q^{34} +9.05234 q^{36} +5.62570 q^{37} -7.89544 q^{38} +2.47443 q^{39} -4.22292 q^{40} +10.3948 q^{41} +3.40733 q^{43} +13.8822 q^{44} -6.62906 q^{45} +9.90335 q^{46} -7.10876 q^{47} -3.45079 q^{48} -1.09280 q^{50} -9.34907 q^{51} +2.89879 q^{52} -12.3801 q^{53} +0.672514 q^{54} -10.1660 q^{55} -8.82686 q^{57} -13.0610 q^{58} +4.78896 q^{59} -15.2265 q^{60} +3.20697 q^{61} -8.34680 q^{62} -12.8486 q^{64} -2.12280 q^{65} +26.2277 q^{66} -2.89955 q^{67} -10.9524 q^{68} +11.0717 q^{69} -2.53876 q^{71} +6.21224 q^{72} +7.70071 q^{73} +12.4515 q^{74} -1.22172 q^{75} -10.3407 q^{76} +5.47671 q^{78} -5.17850 q^{79} +2.96041 q^{80} -8.61654 q^{81} +23.0071 q^{82} +3.46731 q^{83} +8.02051 q^{85} +7.54152 q^{86} -14.6018 q^{87} +9.52677 q^{88} +3.66432 q^{89} -14.6722 q^{90} +12.9704 q^{92} -9.33147 q^{93} -15.7340 q^{94} +7.57251 q^{95} -17.4826 q^{96} -5.40733 q^{97} +14.9549 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 8 q^{4} + 2 q^{5} - 5 q^{6} + 9 q^{8} + 3 q^{9} - 5 q^{10} + 11 q^{11} + 5 q^{12} + 5 q^{13} + 10 q^{16} - 5 q^{17} + 9 q^{18} + 9 q^{19} + q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} + 4 q^{26}+ \cdots + 11 q^{99}+O(q^{100}) $ 5 * q + 4 * q^2 + 8 * q^4 + 2 * q^5 - 5 * q^6 + 9 * q^8 + 3 * q^9 - 5 * q^10 + 11 * q^11 + 5 * q^12 + 5 * q^13 + 10 * q^16 - 5 * q^17 + 9 * q^18 + 9 * q^19 + q^20 + 8 * q^22 + 10 * q^23 + 9 * q^25 + 4 * q^26 - 3 * q^29 - 13 * q^30 - 6 * q^31 + 22 * q^32 + 8 * q^33 - 22 * q^34 + 7 * q^36 + 4 * q^37 - 10 * q^38 + 28 * q^40 + 14 * q^41 + 2 * q^43 - 32 * q^45 + 3 * q^46 + q^47 - 23 * q^48 + 9 * q^50 - 8 * q^51 + 8 * q^52 + 17 * q^53 + 23 * q^54 - 16 * q^57 - 27 * q^58 + 11 * q^59 - 29 * q^60 - 11 * q^61 - 23 * q^62 + 9 * q^64 + 2 * q^65 + 21 * q^66 + 13 * q^67 - 32 * q^68 + 18 * q^69 + 15 * q^71 - 19 * q^72 - 33 * q^74 - 20 * q^75 + 8 * q^76 - 5 * q^78 + 2 * q^79 + 55 * q^80 - 19 * q^81 + 34 * q^82 + 6 * q^83 - 22 * q^85 + 28 * q^86 - 8 * q^87 - 3 * q^88 - 4 * q^89 - 34 * q^90 + 21 * q^92 + 18 * q^93 + 20 * q^94 - 12 * q^95 - 37 * q^96 - 12 * q^97 + 11 * q^99

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 $

$2$	2.21332	1.56505	0.782527	−	0.622616i	$-0.213930\pi$
$2$	2.21332	1.56505	0.782527	+	0.622616i	$0.213930\pi$
$3$	2.47443	1.42861	0.714306	−	0.699834i	$-0.246743\pi$
$3$	2.47443	1.42861	0.714306	+	0.699834i	$0.246743\pi$
$4$	2.89879	1.44940
$5$	−2.12280	−0.949343	−0.474671	−	0.880163i	$-0.657433\pi$
$5$	−2.12280	−0.949343	−0.474671	+	0.880163i	$0.657433\pi$
$6$	5.47671	2.23586
$7$	0	0
$7$	0	0
$8$	1.98932	0.703331
$9$	3.12280	1.04093
$10$	−4.69843	−1.48577
$11$	4.78896	1.44392	0.721962	−	0.691932i	$-0.243240\pi$
$11$	4.78896	1.44392	0.721962	+	0.691932i	$0.243240\pi$
$12$	7.17286	2.07063
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−5.25271	−1.35624
$16$	−1.39458	−0.348645
$17$	−3.77828	−0.916367	−0.458183	−	0.888858i	$-0.651500\pi$
$17$	−3.77828	−0.916367	−0.458183	+	0.888858i	$0.651500\pi$
$18$	6.91175	1.62912
$19$	−3.56723	−0.818379	−0.409190	−	0.912449i	$-0.634189\pi$
$19$	−3.56723	−0.818379	−0.409190	+	0.912449i	$0.634189\pi$
$20$	−6.15355	−1.37597
$21$	0	0
$22$	10.5995	2.25982
$23$	4.47443	0.932983	0.466491	−	0.884526i	$-0.345518\pi$
$23$	4.47443	0.932983	0.466491	+	0.884526i	$0.345518\pi$
$24$	4.92243	1.00479
$25$	−0.493740	−0.0987479
$26$	2.21332	0.434068
$27$	0.303848	0.0584757
$28$	0	0
$29$	−5.90107	−1.09580	−0.547901	−	0.836543i	$-0.684573\pi$
$29$	−5.90107	−1.09580	−0.547901	+	0.836543i	$0.684573\pi$
$30$	−11.6259	−2.12259
$31$	−3.77116	−0.677321	−0.338660	−	0.940909i	$-0.609974\pi$
$31$	−3.77116	−0.677321	−0.338660	+	0.940909i	$0.609974\pi$
$32$	−7.06530	−1.24898
$33$	11.8499	2.06281
$34$	−8.36254	−1.43416
$35$	0	0
$36$	9.05234	1.50872
$37$	5.62570	0.924859	0.462429	−	0.886656i	$-0.346978\pi$
$37$	5.62570	0.924859	0.462429	+	0.886656i	$0.346978\pi$
$38$	−7.89544	−1.28081
$39$	2.47443	0.396226
$40$	−4.22292	−0.667702
$41$	10.3948	1.62340	0.811698	−	0.584077i	$-0.198543\pi$
$41$	10.3948	1.62340	0.811698	+	0.584077i	$0.198543\pi$
$42$	0	0
$43$	3.40733	0.519613	0.259807	−	0.965661i	$-0.416341\pi$
$43$	3.40733	0.519613	0.259807	+	0.965661i	$0.416341\pi$
$44$	13.8822	2.09282
$45$	−6.62906	−0.988201
$46$	9.90335	1.46017
$47$	−7.10876	−1.03692	−0.518459	−	0.855102i	$-0.673494\pi$
$47$	−7.10876	−1.03692	−0.518459	+	0.855102i	$0.673494\pi$
$48$	−3.45079	−0.498079
$49$	0	0
$50$	−1.09280	−0.154546
$51$	−9.34907	−1.30913
$52$	2.89879	0.401990
$53$	−12.3801	−1.70053	−0.850266	−	0.526354i	$-0.823559\pi$
$53$	−12.3801	−1.70053	−0.850266	+	0.526354i	$0.823559\pi$
$54$	0.672514	0.0915176
$55$	−10.1660	−1.37078
$56$	0	0
$57$	−8.82686	−1.16915
$58$	−13.0610	−1.71499
$59$	4.78896	0.623469	0.311734	−	0.950169i	$-0.399090\pi$
$59$	4.78896	0.623469	0.311734	+	0.950169i	$0.399090\pi$
$60$	−15.2265	−1.96573
$61$	3.20697	0.410610	0.205305	−	0.978698i	$-0.434181\pi$
$61$	3.20697	0.410610	0.205305	+	0.978698i	$0.434181\pi$
$62$	−8.34680	−1.06004
$63$	0	0
$64$	−12.8486	−1.60608
$65$	−2.12280	−0.263300
$66$	26.2277	3.22841
$67$	−2.89955	−0.354237	−0.177118	−	0.984190i	$-0.556678\pi$
$67$	−2.89955	−0.354237	−0.177118	+	0.984190i	$0.556678\pi$
$68$	−10.9524	−1.32818
$69$	11.0717	1.33287
$70$	0	0
$71$	−2.53876	−0.301295	−0.150648	−	0.988588i	$-0.548136\pi$
$71$	−2.53876	−0.301295	−0.150648	+	0.988588i	$0.548136\pi$
$72$	6.21224	0.732120
$73$	7.70071	0.901300	0.450650	−	0.892701i	$-0.351192\pi$
$73$	7.70071	0.901300	0.450650	+	0.892701i	$0.351192\pi$
$74$	12.4515	1.44745
$75$	−1.22172	−0.141072
$76$	−10.3407	−1.18616
$77$	0	0
$78$	5.47671	0.620115
$79$	−5.17850	−0.582626	−0.291313	−	0.956628i	$-0.594092\pi$
$79$	−5.17850	−0.582626	−0.291313	+	0.956628i	$0.594092\pi$
$80$	2.96041	0.330984
$81$	−8.61654	−0.957393
$82$	23.0071	2.54071
$83$	3.46731	0.380587	0.190294	−	0.981727i	$-0.439056\pi$
$83$	3.46731	0.380587	0.190294	+	0.981727i	$0.439056\pi$
$84$	0	0
$85$	8.02051	0.869946
$86$	7.54152	0.813223
$87$	−14.6018	−1.56548
$88$	9.52677	1.01556
$89$	3.66432	0.388417	0.194209	−	0.980960i	$-0.437786\pi$
$89$	3.66432	0.388417	0.194209	+	0.980960i	$0.437786\pi$
$90$	−14.6722	−1.54659
$91$	0	0
$92$	12.9704	1.35226
$93$	−9.33147	−0.967629
$94$	−15.7340	−1.62283
$95$	7.57251	0.776923
$96$	−17.4826	−1.78431
$97$	−5.40733	−0.549031	−0.274516	−	0.961583i	$-0.588518\pi$
$97$	−5.40733	−0.549031	−0.274516	+	0.961583i	$0.588518\pi$
$98$	0	0
$99$	14.9549	1.50303
$100$	−1.43125	−0.143125
$101$	9.31724	0.927100	0.463550	−	0.886071i	$-0.346575\pi$
$101$	9.31724	0.927100	0.463550	+	0.886071i	$0.346575\pi$
$102$	−20.6925	−2.04886
$103$	7.30636	0.719917	0.359958	−	0.932968i	$-0.382791\pi$
$103$	7.30636	0.719917	0.359958	+	0.932968i	$0.382791\pi$
$104$	1.98932	0.195069
$105$	0	0
$106$	−27.4011	−2.66143
$107$	6.74729	0.652286	0.326143	−	0.945321i	$-0.394251\pi$
$107$	6.74729	0.652286	0.326143	+	0.945321i	$0.394251\pi$
$108$	0.880794	0.0847545
$109$	4.17645	0.400031	0.200016	−	0.979793i	$-0.435901\pi$
$109$	4.17645	0.400031	0.200016	+	0.979793i	$0.435901\pi$
$110$	−22.5006	−2.14535
$111$	13.9204	1.32126
$112$	0	0
$113$	5.90107	0.555126	0.277563	−	0.960707i	$-0.410473\pi$
$113$	5.90107	0.555126	0.277563	+	0.960707i	$0.410473\pi$
$114$	−19.5367	−1.82978
$115$	−9.49830	−0.885721
$116$	−17.1060	−1.58825
$117$	3.12280	0.288703
$118$	10.5995	0.975763
$119$	0	0
$120$	−10.4493	−0.953888
$121$	11.9341	1.08492
$122$	7.09805	0.642628
$123$	25.7212	2.31920
$124$	−10.9318	−0.981707
$125$	11.6621	1.04309
$126$	0	0
$127$	−10.5268	−0.934100	−0.467050	−	0.884231i	$-0.654683\pi$
$127$	−10.5268	−0.934100	−0.467050	+	0.884231i	$0.654683\pi$
$128$	−14.3075	−1.26462
$129$	8.43120	0.742326
$130$	−4.69843	−0.412080
$131$	5.42409	0.473905	0.236952	−	0.971521i	$-0.423851\pi$
$131$	5.42409	0.473905	0.236952	+	0.971521i	$0.423851\pi$
$132$	34.3505	2.98983
$133$	0	0
$134$	−6.41765	−0.554400
$135$	−0.645008	−0.0555135
$136$	−7.51620	−0.644509
$137$	22.2447	1.90050	0.950248	−	0.311494i	$-0.100829\pi$
$137$	22.2447	1.90050	0.950248	+	0.311494i	$0.100829\pi$
$138$	24.5051	2.08602
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−17.5901	−1.48135
$142$	−5.61909	−0.471544
$143$	4.78896	0.400473
$144$	−4.35499	−0.362916
$145$	12.5268	1.04029
$146$	17.0441	1.41058
$147$	0	0
$148$	16.3077	1.34049
$149$	2.95472	0.242060	0.121030	−	0.992649i	$-0.461380\pi$
$149$	2.95472	0.242060	0.121030	+	0.992649i	$0.461380\pi$
$150$	−2.70407	−0.220786
$151$	−18.5547	−1.50996	−0.754981	−	0.655747i	$-0.772354\pi$
$151$	−18.5547	−1.50996	−0.754981	+	0.655747i	$0.772354\pi$
$152$	−7.09637	−0.575592
$153$	−11.7988	−0.953875
$154$	0	0
$155$	8.00541	0.643010
$156$	7.17286	0.574288
$157$	−9.79964	−0.782096	−0.391048	−	0.920370i	$-0.627887\pi$
$157$	−9.79964	−0.782096	−0.391048	+	0.920370i	$0.627887\pi$
$158$	−11.4617	−0.911842
$159$	−30.6336	−2.42940
$160$	14.9982	1.18571
$161$	0	0
$162$	−19.0712	−1.49837
$163$	13.8342	1.08358	0.541788	−	0.840515i	$-0.317747\pi$
$163$	13.8342	1.08358	0.541788	+	0.840515i	$0.317747\pi$
$164$	30.1324	2.35295
$165$	−25.1550	−1.95831
$166$	7.67428	0.595640
$167$	17.3534	1.34285	0.671424	−	0.741073i	$-0.265683\pi$
$167$	17.3534	1.34285	0.671424	+	0.741073i	$0.265683\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	17.7520	1.36151
$171$	−11.1397	−0.851877
$172$	9.87716	0.753126
$173$	−2.96138	−0.225149	−0.112575	−	0.993643i	$-0.535910\pi$
$173$	−2.96138	−0.225149	−0.112575	+	0.993643i	$0.535910\pi$
$174$	−32.3184	−2.45005
$175$	0	0
$176$	−6.67859	−0.503418
$177$	11.8499	0.890695
$178$	8.11032	0.607894
$179$	−5.66888	−0.423712	−0.211856	−	0.977301i	$-0.567951\pi$
$179$	−5.66888	−0.423712	−0.211856	+	0.977301i	$0.567951\pi$
$180$	−19.2163	−1.43230
$181$	7.17645	0.533421	0.266711	−	0.963777i	$-0.414063\pi$
$181$	7.17645	0.533421	0.266711	+	0.963777i	$0.414063\pi$
$182$	0	0
$183$	7.93541	0.586603
$184$	8.90107	0.656196
$185$	−11.9422	−0.878008
$186$	−20.6536	−1.51439
$187$	−18.0940	−1.32316
$188$	−20.6068	−1.50291
$189$	0	0
$190$	16.7604	1.21593
$191$	11.8818	0.859734	0.429867	−	0.902892i	$-0.358560\pi$
$191$	11.8818	0.859734	0.429867	+	0.902892i	$0.358560\pi$
$192$	−31.7930	−2.29446
$193$	22.9702	1.65343	0.826714	−	0.562622i	$-0.190207\pi$
$193$	22.9702	1.65343	0.826714	+	0.562622i	$0.190207\pi$
$194$	−11.9682	−0.859264
$195$	−5.25271	−0.376154
$196$	0	0
$197$	16.9216	1.20561	0.602806	−	0.797888i	$-0.294049\pi$
$197$	16.9216	1.20561	0.602806	+	0.797888i	$0.294049\pi$
$198$	33.1001	2.35232
$199$	10.0591	0.713068	0.356534	−	0.934282i	$-0.383958\pi$
$199$	10.0591	0.713068	0.356534	+	0.934282i	$0.383958\pi$
$200$	−0.982206	−0.0694525
$201$	−7.17474	−0.506067
$202$	20.6221	1.45096
$203$	0	0
$204$	−27.1010	−1.89745
$205$	−22.0661	−1.54116
$206$	16.1713	1.12671
$207$	13.9727	0.971171
$208$	−1.39458	−0.0966968
$209$	−17.0833	−1.18168
$210$	0	0
$211$	−24.4609	−1.68396	−0.841978	−	0.539512i	$-0.818609\pi$
$211$	−24.4609	−1.68396	−0.841978	+	0.539512i	$0.818609\pi$
$212$	−35.8872	−2.46475
$213$	−6.28198	−0.430434
$214$	14.9339	1.02086
$215$	−7.23307	−0.493291
$216$	0.604452	0.0411277
$217$	0	0
$218$	9.24382	0.626071
$219$	19.0548	1.28761
$220$	−29.4691	−1.98680
$221$	−3.77828	−0.254154
$222$	30.8103	2.06785
$223$	−29.2625	−1.95956	−0.979780	−	0.200076i	$-0.935881\pi$
$223$	−29.2625	−1.95956	−0.979780	+	0.200076i	$0.935881\pi$
$224$	0	0
$225$	−1.54185	−0.102790
$226$	13.0610	0.868803
$227$	−10.0737	−0.668615	−0.334307	−	0.942464i	$-0.608502\pi$
$227$	−10.0737	−0.668615	−0.334307	+	0.942464i	$0.608502\pi$
$228$	−25.5873	−1.69456
$229$	−11.1399	−0.736148	−0.368074	−	0.929796i	$-0.619983\pi$
$229$	−11.1399	−0.736148	−0.368074	+	0.929796i	$0.619983\pi$
$230$	−21.0228	−1.38620
$231$	0	0
$232$	−11.7391	−0.770711
$233$	−17.0833	−1.11917	−0.559583	−	0.828774i	$-0.689039\pi$
$233$	−17.0833	−1.11917	−0.559583	+	0.828774i	$0.689039\pi$
$234$	6.91175	0.451835
$235$	15.0904	0.984392
$236$	13.8822	0.903654
$237$	−12.8138	−0.832347
$238$	0	0
$239$	6.92142	0.447710	0.223855	−	0.974622i	$-0.428136\pi$
$239$	6.92142	0.447710	0.223855	+	0.974622i	$0.428136\pi$
$240$	7.32532	0.472848
$241$	6.49625	0.418460	0.209230	−	0.977866i	$-0.432904\pi$
$241$	6.49625	0.418460	0.209230	+	0.977866i	$0.432904\pi$
$242$	26.4140	1.69796
$243$	−22.2325	−1.42622
$244$	9.29634	0.595137
$245$	0	0
$246$	56.9293	3.62968
$247$	−3.56723	−0.226978
$248$	−7.50205	−0.476381
$249$	8.57962	0.543711
$250$	25.8119	1.63249
$251$	−9.86804	−0.622865	−0.311433	−	0.950268i	$-0.600809\pi$
$251$	−9.86804	−0.622865	−0.311433	+	0.950268i	$0.600809\pi$
$252$	0	0
$253$	21.4278	1.34716
$254$	−23.2991	−1.46192
$255$	19.8462	1.24282
$256$	−5.96994	−0.373121
$257$	6.86468	0.428207	0.214104	−	0.976811i	$-0.431317\pi$
$257$	6.86468	0.428207	0.214104	+	0.976811i	$0.431317\pi$
$258$	18.6610	1.16178
$259$	0	0
$260$	−6.15355	−0.381627
$261$	−18.4278	−1.14065
$262$	12.0052	0.741687
$263$	−0.126551	−0.00780345	−0.00390172	−	0.999992i	$-0.501242\pi$
$263$	−0.126551	−0.00780345	−0.00390172	+	0.999992i	$0.501242\pi$
$264$	23.5733	1.45084
$265$	26.2803	1.61439
$266$	0	0
$267$	9.06710	0.554897
$268$	−8.40521	−0.513430
$269$	−4.24308	−0.258705	−0.129353	−	0.991599i	$-0.541290\pi$
$269$	−4.24308	−0.258705	−0.129353	+	0.991599i	$0.541290\pi$
$270$	−1.42761	−0.0868816
$271$	1.56723	0.0952026	0.0476013	−	0.998866i	$-0.484842\pi$
$271$	1.56723	0.0952026	0.0476013	+	0.998866i	$0.484842\pi$
$272$	5.26911	0.319487
$273$	0	0
$274$	49.2348	2.97438
$275$	−2.36450	−0.142585
$276$	32.0944	1.93186
$277$	−12.7452	−0.765785	−0.382892	−	0.923793i	$-0.625072\pi$
$277$	−12.7452	−0.765785	−0.382892	+	0.923793i	$0.625072\pi$
$278$	−8.85329	−0.530985
$279$	−11.7766	−0.705045
$280$	0	0
$281$	4.62986	0.276194	0.138097	−	0.990419i	$-0.455901\pi$
$281$	4.62986	0.276194	0.138097	+	0.990419i	$0.455901\pi$
$282$	−38.9326	−2.31840
$283$	3.64832	0.216870	0.108435	−	0.994104i	$-0.465416\pi$
$283$	3.64832	0.216870	0.108435	+	0.994104i	$0.465416\pi$
$284$	−7.35934	−0.436697
$285$	18.7376	1.10992
$286$	10.5995	0.626762
$287$	0	0
$288$	−22.0635	−1.30010
$289$	−2.72462	−0.160272
$290$	27.7258	1.62811
$291$	−13.3801	−0.784353
$292$	22.3228	1.30634
$293$	−21.0415	−1.22926	−0.614630	−	0.788816i	$-0.710695\pi$
$293$	−21.0415	−1.22926	−0.614630	+	0.788816i	$0.710695\pi$
$294$	0	0
$295$	−10.1660	−0.591886
$296$	11.1913	0.650482
$297$	1.45512	0.0844345
$298$	6.53976	0.378838
$299$	4.47443	0.258763
$300$	−3.54152	−0.204470
$301$	0	0
$302$	−41.0676	−2.36317
$303$	23.0548	1.32447
$304$	4.97480	0.285324
$305$	−6.80774	−0.389810
$306$	−26.1145	−1.49287
$307$	−4.95861	−0.283003	−0.141502	−	0.989938i	$-0.545193\pi$
$307$	−4.95861	−0.283003	−0.141502	+	0.989938i	$0.545193\pi$
$308$	0	0
$309$	18.0791	1.02848
$310$	17.7185	1.00635
$311$	−2.42158	−0.137315	−0.0686575	−	0.997640i	$-0.521872\pi$
$311$	−2.42158	−0.137315	−0.0686575	+	0.997640i	$0.521872\pi$
$312$	4.92243	0.278678
$313$	13.9605	0.789096	0.394548	−	0.918875i	$-0.370901\pi$
$313$	13.9605	0.789096	0.394548	+	0.918875i	$0.370901\pi$
$314$	−21.6897	−1.22402
$315$	0	0
$316$	−15.0114	−0.844457
$317$	3.06862	0.172351	0.0861753	−	0.996280i	$-0.472535\pi$
$317$	3.06862	0.172351	0.0861753	+	0.996280i	$0.472535\pi$
$318$	−67.8019	−3.80214
$319$	−28.2600	−1.58225
$320$	27.2750	1.52472
$321$	16.6957	0.931863
$322$	0	0
$323$	13.4780	0.749936
$324$	−24.9776	−1.38764
$325$	−0.493740	−0.0273877
$326$	30.6195	1.69586
$327$	10.3343	0.571489
$328$	20.6786	1.14179
$329$	0	0
$330$	−55.6761	−3.06487
$331$	13.6052	0.747810	0.373905	−	0.927467i	$-0.378019\pi$
$331$	13.6052	0.747810	0.373905	+	0.927467i	$0.378019\pi$
$332$	10.0510	0.551622
$333$	17.5679	0.962715
$334$	38.4087	2.10163
$335$	6.15516	0.336292
$336$	0	0
$337$	−35.1646	−1.91554	−0.957769	−	0.287538i	$-0.907163\pi$
$337$	−35.1646	−1.91554	−0.957769	+	0.287538i	$0.907163\pi$
$338$	2.21332	0.120389
$339$	14.6018	0.793060
$340$	23.2498	1.26090
$341$	−18.0599	−0.978000
$342$	−24.6558	−1.33323
$343$	0	0
$344$	6.77828	0.365460
$345$	−23.5029	−1.26535
$346$	−6.55448	−0.352371
$347$	−5.47102	−0.293700	−0.146850	−	0.989159i	$-0.546913\pi$
$347$	−5.47102	−0.293700	−0.146850	+	0.989159i	$0.546913\pi$
$348$	−42.3276	−2.26899
$349$	4.34196	0.232420	0.116210	−	0.993225i	$-0.462925\pi$
$349$	4.34196	0.232420	0.116210	+	0.993225i	$0.462925\pi$
$350$	0	0
$351$	0.303848	0.0162182
$352$	−33.8354	−1.80343
$353$	27.5992	1.46896	0.734479	−	0.678631i	$-0.237427\pi$
$353$	27.5992	1.46896	0.734479	+	0.678631i	$0.237427\pi$
$354$	26.2277	1.39399
$355$	5.38927	0.286033
$356$	10.6221	0.562971
$357$	0	0
$358$	−12.5470	−0.663132
$359$	6.62855	0.349841	0.174921	−	0.984583i	$-0.444033\pi$
$359$	6.62855	0.349841	0.174921	+	0.984583i	$0.444033\pi$
$360$	−13.1873	−0.695033
$361$	−6.27485	−0.330255
$362$	15.8838	0.834833
$363$	29.5301	1.54993
$364$	0	0
$365$	−16.3470	−0.855643
$366$	17.5636	0.918065
$367$	31.2074	1.62901	0.814506	−	0.580156i	$-0.197008\pi$
$367$	31.2074	1.62901	0.814506	+	0.580156i	$0.197008\pi$
$368$	−6.23995	−0.325280
$369$	32.4609	1.68985
$370$	−26.4319	−1.37413
$371$	0	0
$372$	−27.0500	−1.40248
$373$	−15.7746	−0.816778	−0.408389	−	0.912808i	$-0.633909\pi$
$373$	−15.7746	−0.816778	−0.408389	+	0.912808i	$0.633909\pi$
$374$	−40.0479	−2.07083
$375$	28.8570	1.49017
$376$	−14.1416	−0.729297
$377$	−5.90107	−0.303921
$378$	0	0
$379$	31.6512	1.62581	0.812907	−	0.582393i	$-0.197884\pi$
$379$	31.6512	1.62581	0.812907	+	0.582393i	$0.197884\pi$
$380$	21.9511	1.12607
$381$	−26.0477	−1.33447
$382$	26.2982	1.34553
$383$	12.3935	0.633278	0.316639	−	0.948546i	$-0.397446\pi$
$383$	12.3935	0.633278	0.316639	+	0.948546i	$0.397446\pi$
$384$	−35.4030	−1.80665
$385$	0	0
$386$	50.8404	2.58771
$387$	10.6404	0.540882
$388$	−15.6747	−0.795765
$389$	−14.0741	−0.713585	−0.356792	−	0.934184i	$-0.616130\pi$
$389$	−14.0741	−0.713585	−0.356792	+	0.934184i	$0.616130\pi$
$390$	−11.6259	−0.588702
$391$	−16.9056	−0.854954
$392$	0	0
$393$	13.4215	0.677026
$394$	37.4529	1.88685
$395$	10.9929	0.553112
$396$	43.3513	2.17848
$397$	−6.97305	−0.349967	−0.174984	−	0.984571i	$-0.555987\pi$
$397$	−6.97305	−0.349967	−0.174984	+	0.984571i	$0.555987\pi$
$398$	22.2639	1.11599
$399$	0	0
$400$	0.688560	0.0344280
$401$	2.73682	0.136670	0.0683352	−	0.997662i	$-0.478231\pi$
$401$	2.73682	0.136670	0.0683352	+	0.997662i	$0.478231\pi$
$402$	−15.8800	−0.792023
$403$	−3.77116	−0.187855
$404$	27.0088	1.34374
$405$	18.2911	0.908894
$406$	0	0
$407$	26.9412	1.33543
$408$	−18.5983	−0.920753
$409$	−24.5154	−1.21221	−0.606104	−	0.795386i	$-0.707268\pi$
$409$	−24.5154	−1.21221	−0.606104	+	0.795386i	$0.707268\pi$
$410$	−48.8393	−2.41200
$411$	55.0430	2.71507
$412$	21.1796	1.04345
$413$	0	0
$414$	30.9261	1.51994
$415$	−7.36040	−0.361308
$416$	−7.06530	−0.346405
$417$	−9.89771	−0.484693
$418$	−37.8109	−1.84939
$419$	−3.01252	−0.147171	−0.0735856	−	0.997289i	$-0.523444\pi$
$419$	−3.01252	−0.147171	−0.0735856	+	0.997289i	$0.523444\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−54.1398	−2.63548
$423$	−22.1992	−1.07936
$424$	−24.6279	−1.19604
$425$	1.86548	0.0904893
$426$	−13.9040	−0.673653
$427$	0	0
$428$	19.5590	0.945421
$429$	11.8499	0.572120
$430$	−16.0091	−0.772028
$431$	−18.7942	−0.905286	−0.452643	−	0.891692i	$-0.649519\pi$
$431$	−18.7942	−0.905286	−0.452643	+	0.891692i	$0.649519\pi$
$432$	−0.423741	−0.0203873
$433$	−7.76911	−0.373360	−0.186680	−	0.982421i	$-0.559773\pi$
$433$	−7.76911	−0.373360	−0.186680	+	0.982421i	$0.559773\pi$
$434$	0	0
$435$	30.9966	1.48617
$436$	12.1067	0.579804
$437$	−15.9613	−0.763534
$438$	42.1745	2.01518
$439$	−37.9681	−1.81212	−0.906060	−	0.423150i	$-0.860924\pi$
$439$	−37.9681	−1.81212	−0.906060	+	0.423150i	$0.860924\pi$
$440$	−20.2234	−0.964112
$441$	0	0
$442$	−8.36254	−0.397766
$443$	35.6270	1.69269	0.846344	−	0.532637i	$-0.178799\pi$
$443$	35.6270	1.69269	0.846344	+	0.532637i	$0.178799\pi$
$444$	40.3523	1.91504
$445$	−7.77860	−0.368741
$446$	−64.7673	−3.06682
$447$	7.31125	0.345810
$448$	0	0
$449$	−8.05285	−0.380038	−0.190019	−	0.981780i	$-0.560855\pi$
$449$	−8.05285	−0.380038	−0.190019	+	0.981780i	$0.560855\pi$
$450$	−3.41261	−0.160872
$451$	49.7803	2.34406
$452$	17.1060	0.804598
$453$	−45.9123	−2.15715
$454$	−22.2963	−1.04642
$455$	0	0
$456$	−17.5595	−0.822297
$457$	15.5976	0.729626	0.364813	−	0.931081i	$-0.381133\pi$
$457$	15.5976	0.729626	0.364813	+	0.931081i	$0.381133\pi$
$458$	−24.6563	−1.15211
$459$	−1.14802	−0.0535852
$460$	−27.5336	−1.28376
$461$	−25.6991	−1.19692	−0.598462	−	0.801151i	$-0.704221\pi$
$461$	−25.6991	−1.19692	−0.598462	+	0.801151i	$0.704221\pi$
$462$	0	0
$463$	−20.5209	−0.953685	−0.476842	−	0.878989i	$-0.658219\pi$
$463$	−20.5209	−0.953685	−0.476842	+	0.878989i	$0.658219\pi$
$464$	8.22952	0.382046
$465$	19.8088	0.918611
$466$	−37.8109	−1.75156
$467$	11.8248	0.547187	0.273594	−	0.961845i	$-0.411788\pi$
$467$	11.8248	0.547187	0.273594	+	0.961845i	$0.411788\pi$
$468$	9.05234	0.418445
$469$	0	0
$470$	33.4000	1.54063
$471$	−24.2485	−1.11731
$472$	9.52677	0.438505
$473$	16.3176	0.750283
$474$	−28.3611	−1.30267
$475$	1.76128	0.0808133
$476$	0	0
$477$	−38.6604	−1.77014
$478$	15.3193	0.700690
$479$	22.6552	1.03514	0.517571	−	0.855640i	$-0.326836\pi$
$479$	22.6552	1.03514	0.517571	+	0.855640i	$0.326836\pi$
$480$	37.1119	1.69392
$481$	5.62570	0.256510
$482$	14.3783	0.654913
$483$	0	0
$484$	34.5945	1.57248
$485$	11.4787	0.521219
$486$	−49.2078	−2.23211
$487$	−32.7167	−1.48254	−0.741268	−	0.671209i	$-0.765775\pi$
$487$	−32.7167	−1.48254	−0.741268	+	0.671209i	$0.765775\pi$
$488$	6.37969	0.288795
$489$	34.2317	1.54801
$490$	0	0
$491$	6.17281	0.278575	0.139288	−	0.990252i	$-0.455519\pi$
$491$	6.17281	0.278575	0.139288	+	0.990252i	$0.455519\pi$
$492$	74.5605	3.36145
$493$	22.2959	1.00416
$494$	−7.89544	−0.355232
$495$	−31.7463	−1.42689
$496$	5.25919	0.236145
$497$	0	0
$498$	18.9895	0.850938
$499$	−14.6387	−0.655317	−0.327659	−	0.944796i	$-0.606260\pi$
$499$	−14.6387	−0.655317	−0.327659	+	0.944796i	$0.606260\pi$
$500$	33.8060	1.51185
$501$	42.9398	1.91841
$502$	−21.8412	−0.974818
$503$	12.7787	0.569774	0.284887	−	0.958561i	$-0.408044\pi$
$503$	12.7787	0.569774	0.284887	+	0.958561i	$0.408044\pi$
$504$	0	0
$505$	−19.7786	−0.880136
$506$	47.4267	2.10837
$507$	2.47443	0.109893
$508$	−30.5149	−1.35388
$509$	11.6853	0.517940	0.258970	−	0.965885i	$-0.416617\pi$
$509$	11.6853	0.517940	0.258970	+	0.965885i	$0.416617\pi$
$510$	43.9260	1.94507
$511$	0	0
$512$	15.4017	0.680664
$513$	−1.08390	−0.0478553
$514$	15.1938	0.670168
$515$	−15.5099	−0.683448
$516$	24.4403	1.07592
$517$	−34.0435	−1.49723
$518$	0	0
$519$	−7.32772	−0.321651
$520$	−4.22292	−0.185187
$521$	8.47675	0.371373	0.185687	−	0.982609i	$-0.440549\pi$
$521$	8.47675	0.371373	0.185687	+	0.982609i	$0.440549\pi$
$522$	−40.7867	−1.78519
$523$	32.7108	1.43034	0.715172	−	0.698949i	$-0.246348\pi$
$523$	32.7108	1.43034	0.715172	+	0.698949i	$0.246348\pi$
$524$	15.7233	0.686876
$525$	0	0
$526$	−0.280097	−0.0122128
$527$	14.2485	0.620674
$528$	−16.5257	−0.719188
$529$	−2.97949	−0.129543
$530$	58.1668	2.52661
$531$	14.9549	0.648989
$532$	0	0
$533$	10.3948	0.450249
$534$	20.0684	0.868445
$535$	−14.3231	−0.619243
$536$	−5.76814	−0.249146
$537$	−14.0272	−0.605320
$538$	−9.39131	−0.404888
$539$	0	0
$540$	−1.86975	−0.0804610
$541$	−28.1705	−1.21115	−0.605573	−	0.795790i	$-0.707056\pi$
$541$	−28.1705	−1.21115	−0.605573	+	0.795790i	$0.707056\pi$
$542$	3.46879	0.148997
$543$	17.7576	0.762052
$544$	26.6947	1.14452
$545$	−8.86574	−0.379767
$546$	0	0
$547$	−18.5377	−0.792615	−0.396307	−	0.918118i	$-0.629709\pi$
$547$	−18.5377	−0.792615	−0.396307	+	0.918118i	$0.629709\pi$
$548$	64.4829	2.75457
$549$	10.0147	0.427417
$550$	−5.23339	−0.223153
$551$	21.0505	0.896781
$552$	22.0251	0.937449
$553$	0	0
$554$	−28.2092	−1.19850
$555$	−29.5501	−1.25433
$556$	−11.5952	−0.491745
$557$	4.00283	0.169605	0.0848027	−	0.996398i	$-0.472974\pi$
$557$	4.00283	0.169605	0.0848027	+	0.996398i	$0.472974\pi$
$558$	−26.0653	−1.10343
$559$	3.40733	0.144115
$560$	0	0
$561$	−44.7723	−1.89029
$562$	10.2474	0.432259
$563$	−17.8620	−0.752794	−0.376397	−	0.926459i	$-0.622837\pi$
$563$	−17.8620	−0.752794	−0.376397	+	0.926459i	$0.622837\pi$
$564$	−50.9901	−2.14707
$565$	−12.5268	−0.527005
$566$	8.07490	0.339413
$567$	0	0
$568$	−5.05041	−0.211910
$569$	−37.4672	−1.57071	−0.785353	−	0.619048i	$-0.787519\pi$
$569$	−37.4672	−1.57071	−0.785353	+	0.619048i	$0.787519\pi$
$570$	41.4724	1.73709
$571$	17.5703	0.735293	0.367646	−	0.929966i	$-0.380164\pi$
$571$	17.5703	0.735293	0.367646	+	0.929966i	$0.380164\pi$
$572$	13.8822	0.580444
$573$	29.4006	1.22823
$574$	0	0
$575$	−2.20920	−0.0921301
$576$	−40.1236	−1.67182
$577$	−34.2494	−1.42582	−0.712910	−	0.701256i	$-0.752623\pi$
$577$	−34.2494	−1.42582	−0.712910	+	0.701256i	$0.752623\pi$
$578$	−6.03047	−0.250835
$579$	56.8380	2.36211
$580$	36.3125	1.50780
$581$	0	0
$582$	−29.6144	−1.22756
$583$	−59.2876	−2.45544
$584$	15.3192	0.633912
$585$	−6.62906	−0.274078
$586$	−46.5717	−1.92386
$587$	−29.4494	−1.21551	−0.607754	−	0.794126i	$-0.707929\pi$
$587$	−29.4494	−1.21551	−0.607754	+	0.794126i	$0.707929\pi$
$588$	0	0
$589$	13.4526	0.554305
$590$	−22.5006	−0.926334
$591$	41.8712	1.72235
$592$	−7.84549	−0.322448
$593$	−34.0001	−1.39622	−0.698109	−	0.715992i	$-0.745975\pi$
$593$	−34.0001	−1.39622	−0.698109	+	0.715992i	$0.745975\pi$
$594$	3.22064	0.132145
$595$	0	0
$596$	8.56514	0.350842
$597$	24.8904	1.01870
$598$	9.90335	0.404978
$599$	21.4418	0.876087	0.438043	−	0.898954i	$-0.355672\pi$
$599$	21.4418	0.876087	0.438043	+	0.898954i	$0.355672\pi$
$600$	−2.43040	−0.0992206
$601$	40.4039	1.64811	0.824054	−	0.566511i	$-0.191707\pi$
$601$	40.4039	1.64811	0.824054	+	0.566511i	$0.191707\pi$
$602$	0	0
$603$	−9.05472	−0.368737
$604$	−53.7863	−2.18853
$605$	−25.3337	−1.02996
$606$	51.0278	2.07286
$607$	−43.8913	−1.78149	−0.890746	−	0.454501i	$-0.849818\pi$
$607$	−43.8913	−1.78149	−0.890746	+	0.454501i	$0.849818\pi$
$608$	25.2036	1.02214
$609$	0	0
$610$	−15.0677	−0.610074
$611$	−7.10876	−0.287590
$612$	−34.2022	−1.38254
$613$	−14.3155	−0.578199	−0.289100	−	0.957299i	$-0.593356\pi$
$613$	−14.3155	−0.578199	−0.289100	+	0.957299i	$0.593356\pi$
$614$	−10.9750	−0.442915
$615$	−54.6009	−2.20172
$616$	0	0
$617$	−36.9097	−1.48593	−0.742965	−	0.669330i	$-0.766581\pi$
$617$	−36.9097	−1.48593	−0.742965	+	0.669330i	$0.766581\pi$
$618$	40.0148	1.60963
$619$	14.2929	0.574481	0.287240	−	0.957859i	$-0.407262\pi$
$619$	14.2929	0.574481	0.287240	+	0.957859i	$0.407262\pi$
$620$	23.2060	0.931976
$621$	1.35955	0.0545568
$622$	−5.35973	−0.214906
$623$	0	0
$624$	−3.45079	−0.138142
$625$	−22.2875	−0.891501
$626$	30.8991	1.23498
$627$	−42.2715	−1.68816
$628$	−28.4071	−1.13357
$629$	−21.2554	−0.847510
$630$	0	0
$631$	−0.0431064	−0.00171604	−0.000858019	−	1.00000i	$-0.500273\pi$
$631$	−0.0431064	−0.00171604	−0.000858019		1.00000i	$0.500273\pi$
$632$	−10.3017	−0.409779
$633$	−60.5267	−2.40572
$634$	6.79184	0.269738
$635$	22.3462	0.886781
$636$	−88.8004	−3.52116
$637$	0	0
$638$	−62.5484	−2.47632
$639$	−7.92803	−0.313628
$640$	30.3720	1.20056
$641$	42.6655	1.68519	0.842594	−	0.538550i	$-0.181028\pi$
$641$	42.6655	1.68519	0.842594	+	0.538550i	$0.181028\pi$
$642$	36.9530	1.45842
$643$	−5.49737	−0.216795	−0.108398	−	0.994108i	$-0.534572\pi$
$643$	−5.49737	−0.216795	−0.108398	+	0.994108i	$0.534572\pi$
$644$	0	0
$645$	−17.8977	−0.704722
$646$	29.8311	1.17369
$647$	−38.1867	−1.50127	−0.750637	−	0.660715i	$-0.770253\pi$
$647$	−38.1867	−1.50127	−0.750637	+	0.660715i	$0.770253\pi$
$648$	−17.1411	−0.673364
$649$	22.9341	0.900242
$650$	−1.09280	−0.0428633
$651$	0	0
$652$	40.1024	1.57053
$653$	38.5019	1.50670	0.753349	−	0.657621i	$-0.228437\pi$
$653$	38.5019	1.50670	0.753349	+	0.657621i	$0.228437\pi$
$654$	22.8732	0.894412
$655$	−11.5142	−0.449898
$656$	−14.4964	−0.565990
$657$	24.0477	0.938192
$658$	0	0
$659$	19.4843	0.759002	0.379501	−	0.925191i	$-0.376096\pi$
$659$	19.4843	0.759002	0.379501	+	0.925191i	$0.376096\pi$
$660$	−72.9191	−2.83837
$661$	−41.6667	−1.62065	−0.810324	−	0.585983i	$-0.800709\pi$
$661$	−41.6667	−1.62065	−0.810324	+	0.585983i	$0.800709\pi$
$662$	30.1127	1.17036
$663$	−9.34907	−0.363088
$664$	6.89760	0.267679
$665$	0	0
$666$	38.8834	1.50670
$667$	−26.4039	−1.02236
$668$	50.3040	1.94632
$669$	−72.4079	−2.79945
$670$	13.6234	0.526316
$671$	15.3580	0.592890
$672$	0	0
$673$	−14.3157	−0.551830	−0.275915	−	0.961182i	$-0.588981\pi$
$673$	−14.3157	−0.551830	−0.275915	+	0.961182i	$0.588981\pi$
$674$	−77.8306	−2.99792
$675$	−0.150022	−0.00577435
$676$	2.89879	0.111492
$677$	−29.5281	−1.13486	−0.567429	−	0.823423i	$-0.692062\pi$
$677$	−29.5281	−1.13486	−0.567429	+	0.823423i	$0.692062\pi$
$678$	32.3184	1.24118
$679$	0	0
$680$	15.9554	0.611860
$681$	−24.9266	−0.955191
$682$	−39.9724	−1.53062
$683$	47.0699	1.80108	0.900539	−	0.434774i	$-0.143172\pi$
$683$	47.0699	1.80108	0.900539	+	0.434774i	$0.143172\pi$
$684$	−32.2918	−1.23471
$685$	−47.2210	−1.80422
$686$	0	0
$687$	−27.5650	−1.05167
$688$	−4.75180	−0.181161
$689$	−12.3801	−0.471643
$690$	−52.0194	−1.98034
$691$	−30.8668	−1.17423	−0.587113	−	0.809505i	$-0.699736\pi$
$691$	−30.8668	−1.17423	−0.587113	+	0.809505i	$0.699736\pi$
$692$	−8.58442	−0.326331
$693$	0	0
$694$	−12.1091	−0.459656
$695$	8.49118	0.322089
$696$	−29.0476	−1.10105
$697$	−39.2745	−1.48763
$698$	9.61016	0.363750
$699$	−42.2715	−1.59885
$700$	0	0
$701$	6.48958	0.245108	0.122554	−	0.992462i	$-0.460892\pi$
$701$	6.48958	0.245108	0.122554	+	0.992462i	$0.460892\pi$
$702$	0.672514	0.0253824
$703$	−20.0682	−0.756885
$704$	−61.5315	−2.31905
$705$	37.3402	1.40631
$706$	61.0860	2.29900
$707$	0	0
$708$	34.3505	1.29097
$709$	−13.3738	−0.502263	−0.251131	−	0.967953i	$-0.580803\pi$
$709$	−13.3738	−0.502263	−0.251131	+	0.967953i	$0.580803\pi$
$710$	11.9282	0.447657
$711$	−16.1714	−0.606474
$712$	7.28951	0.273186
$713$	−16.8738	−0.631929
$714$	0	0
$715$	−10.1660	−0.380186
$716$	−16.4329	−0.614126
$717$	17.1266	0.639603
$718$	14.6711	0.547521
$719$	−16.7410	−0.624333	−0.312166	−	0.950027i	$-0.601055\pi$
$719$	−16.7410	−0.624333	−0.312166	+	0.950027i	$0.601055\pi$
$720$	9.24476	0.344532
$721$	0	0
$722$	−13.8883	−0.516868
$723$	16.0745	0.597817
$724$	20.8030	0.773139
$725$	2.91359	0.108208
$726$	65.3596	2.42572
$727$	38.8138	1.43952	0.719761	−	0.694221i	$-0.244251\pi$
$727$	38.8138	1.43952	0.719761	+	0.694221i	$0.244251\pi$
$728$	0	0
$729$	−29.1632	−1.08012
$730$	−36.1812	−1.33913
$731$	−12.8738	−0.476156
$732$	23.0031	0.850220
$733$	37.7279	1.39351	0.696756	−	0.717309i	$-0.254626\pi$
$733$	37.7279	1.39351	0.696756	+	0.717309i	$0.254626\pi$
$734$	69.0719	2.54949
$735$	0	0
$736$	−31.6132	−1.16528
$737$	−13.8858	−0.511491
$738$	71.8464	2.64470
$739$	−9.22952	−0.339514	−0.169757	−	0.985486i	$-0.554298\pi$
$739$	−9.22952	−0.339514	−0.169757	+	0.985486i	$0.554298\pi$
$740$	−34.6180	−1.27258
$741$	−8.82686	−0.324263
$742$	0	0
$743$	−3.56327	−0.130724	−0.0653619	−	0.997862i	$-0.520820\pi$
$743$	−3.56327	−0.130724	−0.0653619	+	0.997862i	$0.520820\pi$
$744$	−18.5633	−0.680563
$745$	−6.27228	−0.229798
$746$	−34.9143	−1.27830
$747$	10.8277	0.396165
$748$	−52.4508	−1.91779
$749$	0	0
$750$	63.8698	2.33220
$751$	51.2106	1.86870	0.934350	−	0.356357i	$-0.115981\pi$
$751$	51.2106	1.86870	0.934350	+	0.356357i	$0.115981\pi$
$752$	9.91374	0.361517
$753$	−24.4178	−0.889833
$754$	−13.0610	−0.475653
$755$	39.3879	1.43347
$756$	0	0
$757$	25.2305	0.917019	0.458509	−	0.888690i	$-0.348384\pi$
$757$	25.2305	0.917019	0.458509	+	0.888690i	$0.348384\pi$
$758$	70.0543	2.54449
$759$	53.0217	1.92456
$760$	15.0641	0.546434
$761$	3.64744	0.132220	0.0661099	−	0.997812i	$-0.478941\pi$
$761$	3.64744	0.132220	0.0661099	+	0.997812i	$0.478941\pi$
$762$	−57.6520	−2.08851
$763$	0	0
$764$	34.4428	1.24610
$765$	25.0464	0.905555
$766$	27.4308	0.991115
$767$	4.78896	0.172919
$768$	−14.7722	−0.533045
$769$	21.9882	0.792914	0.396457	−	0.918053i	$-0.370240\pi$
$769$	21.9882	0.792914	0.396457	+	0.918053i	$0.370240\pi$
$770$	0	0
$771$	16.9862	0.611742
$772$	66.5858	2.39647
$773$	21.8590	0.786215	0.393108	−	0.919493i	$-0.371400\pi$
$773$	21.8590	0.786215	0.393108	+	0.919493i	$0.371400\pi$
$774$	23.5506	0.846510
$775$	1.86197	0.0668840
$776$	−10.7569	−0.386151
$777$	0	0
$778$	−31.1505	−1.11680
$779$	−37.0807	−1.32855
$780$	−15.2265	−0.545197
$781$	−12.1580	−0.435048
$782$	−37.4176	−1.33805
$783$	−1.79303	−0.0640777
$784$	0	0
$785$	20.8026	0.742477
$786$	29.7061	1.05958
$787$	39.8673	1.42111	0.710557	−	0.703639i	$-0.248443\pi$
$787$	39.8673	1.42111	0.710557	+	0.703639i	$0.248443\pi$
$788$	49.0522	1.74741
$789$	−0.313141	−0.0111481
$790$	24.3308	0.865651
$791$	0	0
$792$	29.7502	1.05713
$793$	3.20697	0.113883
$794$	−15.4336	−0.547718
$795$	65.0288	2.30633
$796$	29.1591	1.03352
$797$	−40.1971	−1.42385	−0.711927	−	0.702253i	$-0.752178\pi$
$797$	−40.1971	−1.42385	−0.711927	+	0.702253i	$0.752178\pi$
$798$	0	0
$799$	26.8589	0.950198
$800$	3.48842	0.123334
$801$	11.4429	0.404316
$802$	6.05747	0.213897
$803$	36.8784	1.30141
$804$	−20.7981	−0.733492
$805$	0	0
$806$	−8.34680	−0.294003
$807$	−10.4992	−0.369589
$808$	18.5350	0.652058
$809$	2.53849	0.0892485	0.0446243	−	0.999004i	$-0.485791\pi$
$809$	2.53849	0.0892485	0.0446243	+	0.999004i	$0.485791\pi$
$810$	40.4842	1.42247
$811$	−41.7062	−1.46450	−0.732251	−	0.681035i	$-0.761530\pi$
$811$	−41.7062	−1.46450	−0.732251	+	0.681035i	$0.761530\pi$
$812$	0	0
$813$	3.87801	0.136008
$814$	59.6296	2.09002
$815$	−29.3671	−1.02869
$816$	13.0380	0.456423
$817$	−12.1547	−0.425241
$818$	−54.2604	−1.89717
$819$	0	0
$820$	−63.9650	−2.23375
$821$	31.9304	1.11438	0.557189	−	0.830386i	$-0.311880\pi$
$821$	31.9304	1.11438	0.557189	+	0.830386i	$0.311880\pi$
$822$	121.828	4.24924
$823$	−34.2531	−1.19399	−0.596995	−	0.802245i	$-0.703639\pi$
$823$	−34.2531	−1.19399	−0.596995	+	0.802245i	$0.703639\pi$
$824$	14.5347	0.506340
$825$	−5.85078	−0.203698
$826$	0	0
$827$	36.9755	1.28576	0.642882	−	0.765965i	$-0.277739\pi$
$827$	36.9755	1.28576	0.642882	+	0.765965i	$0.277739\pi$
$828$	40.5040	1.40761
$829$	−19.9895	−0.694263	−0.347131	−	0.937817i	$-0.612844\pi$
$829$	−19.9895	−0.694263	−0.347131	+	0.937817i	$0.612844\pi$
$830$	−16.2909	−0.565466
$831$	−31.5371	−1.09401
$832$	−12.8486	−0.445446
$833$	0	0
$834$	−21.9068	−0.758571
$835$	−36.8378	−1.27482
$836$	−49.5210	−1.71272
$837$	−1.14586	−0.0396068
$838$	−6.66768	−0.230331
$839$	12.8147	0.442411	0.221206	−	0.975227i	$-0.429001\pi$
$839$	12.8147	0.442411	0.221206	+	0.975227i	$0.429001\pi$
$840$	0	0
$841$	5.82265	0.200781
$842$	−22.1332	−0.762761
$843$	11.4562	0.394574
$844$	−70.9070	−2.44072
$845$	−2.12280	−0.0730264
$846$	−49.1340	−1.68926
$847$	0	0
$848$	17.2650	0.592882
$849$	9.02750	0.309823
$850$	4.12892	0.141621
$851$	25.1718	0.862877
$852$	−18.2102	−0.623870
$853$	−30.1839	−1.03348	−0.516739	−	0.856143i	$-0.672854\pi$
$853$	−30.1839	−1.03348	−0.516739	+	0.856143i	$0.672854\pi$
$854$	0	0
$855$	23.6474	0.808724
$856$	13.4225	0.458773
$857$	−53.2327	−1.81839	−0.909197	−	0.416366i	$-0.863304\pi$
$857$	−53.2327	−1.81839	−0.909197	+	0.416366i	$0.863304\pi$
$858$	26.2277	0.895399
$859$	−12.2719	−0.418713	−0.209357	−	0.977839i	$-0.567137\pi$
$859$	−12.2719	−0.418713	−0.209357	+	0.977839i	$0.567137\pi$
$860$	−20.9672	−0.714975
$861$	0	0
$862$	−41.5977	−1.41682
$863$	24.4453	0.832127	0.416064	−	0.909335i	$-0.363409\pi$
$863$	24.4453	0.832127	0.416064	+	0.909335i	$0.363409\pi$
$864$	−2.14678	−0.0730349
$865$	6.28640	0.213744
$866$	−17.1956	−0.584329
$867$	−6.74189	−0.228967
$868$	0	0
$869$	−24.7996	−0.841268
$870$	68.6054	2.32594
$871$	−2.89955	−0.0982477
$872$	8.30829	0.281354
$873$	−16.8860	−0.571504
$874$	−35.3276	−1.19497
$875$	0	0
$876$	55.2361	1.86625
$877$	−52.8753	−1.78547	−0.892736	−	0.450580i	$-0.851217\pi$
$877$	−52.8753	−1.78547	−0.892736	+	0.450580i	$0.851217\pi$
$878$	−84.0357	−2.83607
$879$	−52.0658	−1.75614
$880$	14.1773	0.477916
$881$	55.0118	1.85339	0.926697	−	0.375809i	$-0.122635\pi$
$881$	55.0118	1.85339	0.926697	+	0.375809i	$0.122635\pi$
$882$	0	0
$883$	44.1730	1.48654	0.743269	−	0.668992i	$-0.233274\pi$
$883$	44.1730	1.48654	0.743269	+	0.668992i	$0.233274\pi$
$884$	−10.9524	−0.368371
$885$	−25.1550	−0.845575
$886$	78.8539	2.64915
$887$	−5.08659	−0.170791	−0.0853955	−	0.996347i	$-0.527215\pi$
$887$	−5.08659	−0.170791	−0.0853955	+	0.996347i	$0.527215\pi$
$888$	27.6921	0.929286
$889$	0	0
$890$	−17.2165	−0.577100
$891$	−41.2642	−1.38240
$892$	−84.8259	−2.84018
$893$	25.3586	0.848593
$894$	16.1822	0.541212
$895$	12.0339	0.402248
$896$	0	0
$897$	11.0717	0.369672
$898$	−17.8236	−0.594780
$899$	22.2539	0.742209
$900$	−4.46950	−0.148983
$901$	46.7753	1.55831
$902$	110.180	3.66859
$903$	0	0
$904$	11.7391	0.390437
$905$	−15.2341	−0.506400
$906$	−101.619	−3.37606
$907$	−18.1253	−0.601840	−0.300920	−	0.953649i	$-0.597294\pi$
$907$	−18.1253	−0.601840	−0.300920	+	0.953649i	$0.597294\pi$
$908$	−29.2016	−0.969088
$909$	29.0958	0.965048
$910$	0	0
$911$	−9.65804	−0.319985	−0.159993	−	0.987118i	$-0.551147\pi$
$911$	−9.65804	−0.319985	−0.159993	+	0.987118i	$0.551147\pi$
$912$	12.3098	0.407617
$913$	16.6048	0.549539
$914$	34.5226	1.14190
$915$	−16.8453	−0.556887
$916$	−32.2924	−1.06697
$917$	0	0
$918$	−2.54095	−0.0838637
$919$	47.7603	1.57547	0.787733	−	0.616017i	$-0.211255\pi$
$919$	47.7603	1.57547	0.787733	+	0.616017i	$0.211255\pi$
$920$	−18.8952	−0.622955
$921$	−12.2697	−0.404301
$922$	−56.8803	−1.87325
$923$	−2.53876	−0.0835643
$924$	0	0
$925$	−2.77763	−0.0913279
$926$	−45.4193	−1.49257
$927$	22.8163	0.749384
$928$	41.6928	1.36863
$929$	33.9811	1.11488	0.557442	−	0.830216i	$-0.311783\pi$
$929$	33.9811	1.11488	0.557442	+	0.830216i	$0.311783\pi$
$930$	43.8433	1.43768
$931$	0	0
$932$	−49.5210	−1.62212
$933$	−5.99202	−0.196170
$934$	26.1721	0.856378
$935$	38.4099	1.25614
$936$	6.21224	0.203053
$937$	−24.7948	−0.810012	−0.405006	−	0.914314i	$-0.632731\pi$
$937$	−24.7948	−0.810012	−0.405006	+	0.914314i	$0.632731\pi$
$938$	0	0
$939$	34.5443	1.12731
$940$	43.7441	1.42677
$941$	−8.24196	−0.268680	−0.134340	−	0.990935i	$-0.542891\pi$
$941$	−8.24196	−0.268680	−0.134340	+	0.990935i	$0.542891\pi$
$942$	−53.6697	−1.74865
$943$	46.5108	1.51460
$944$	−6.67859	−0.217370
$945$	0	0
$946$	36.1160	1.17423
$947$	19.9729	0.649031	0.324515	−	0.945880i	$-0.394799\pi$
$947$	19.9729	0.649031	0.324515	+	0.945880i	$0.394799\pi$
$948$	−37.1446	−1.20640
$949$	7.70071	0.249976
$950$	3.89829	0.126477
$951$	7.59307	0.246222
$952$	0	0
$953$	−21.5341	−0.697557	−0.348778	−	0.937205i	$-0.613403\pi$
$953$	−21.5341	−0.697557	−0.348778	+	0.937205i	$0.613403\pi$
$954$	−85.5679	−2.77036
$955$	−25.2225	−0.816182
$956$	20.0638	0.648909
$957$	−69.9273	−2.26043
$958$	50.1432	1.62005
$959$	0	0
$960$	67.4900	2.17823
$961$	−16.7783	−0.541237
$962$	12.4515	0.401452
$963$	21.0704	0.678985
$964$	18.8313	0.606515
$965$	−48.7610	−1.56967
$966$	0	0
$967$	43.2887	1.39207	0.696036	−	0.718007i	$-0.254945\pi$
$967$	43.2887	1.39207	0.696036	+	0.718007i	$0.254945\pi$
$968$	23.7408	0.763057
$969$	33.3503	1.07137
$970$	25.4060	0.815737
$971$	−52.6713	−1.69030	−0.845151	−	0.534528i	$-0.820490\pi$
$971$	−52.6713	−1.69030	−0.845151	+	0.534528i	$0.820490\pi$
$972$	−64.4476	−2.06716
$973$	0	0
$974$	−72.4127	−2.32025
$975$	−1.22172	−0.0391265
$976$	−4.47238	−0.143157
$977$	15.4061	0.492885	0.246442	−	0.969157i	$-0.420738\pi$
$977$	15.4061	0.492885	0.246442	+	0.969157i	$0.420738\pi$
$978$	75.7658	2.42272
$979$	17.5483	0.560845
$980$	0	0
$981$	13.0422	0.416405
$982$	13.6624	0.435985
$983$	7.58146	0.241811	0.120906	−	0.992664i	$-0.461420\pi$
$983$	7.58146	0.241811	0.120906	+	0.992664i	$0.461420\pi$
$984$	51.1677	1.63117
$985$	−35.9211	−1.14454
$986$	49.3480	1.57156
$987$	0	0
$988$	−10.3407	−0.328981
$989$	15.2459	0.484790
$990$	−70.2647	−2.23316
$991$	−19.0185	−0.604141	−0.302071	−	0.953286i	$-0.597678\pi$
$991$	−19.0185	−0.604141	−0.302071	+	0.953286i	$0.597678\pi$
$992$	26.6444	0.845960
$993$	33.6651	1.06833
$994$	0	0
$995$	−21.3533	−0.676946
$996$	24.8706	0.788054
$997$	46.0998	1.46000	0.729998	−	0.683449i	$-0.239521\pi$
$997$	46.0998	1.46000	0.729998	+	0.683449i	$0.239521\pi$
$998$	−32.4001	−1.02561
$999$	1.70936	0.0540817

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	637.2.a.l.1.4		5
3.2	odd	2		5733.2.a.bl.1.2		5
7.2	even	3		91.2.e.c.53.2	✓	10
7.3	odd	6		637.2.e.m.79.2		10
7.4	even	3		91.2.e.c.79.2	yes	10
7.5	odd	6		637.2.e.m.508.2		10
7.6	odd	2		637.2.a.k.1.4		5
13.12	even	2		8281.2.a.bw.1.2		5
21.2	odd	6		819.2.j.h.235.4		10
21.11	odd	6		819.2.j.h.352.4		10
21.20	even	2		5733.2.a.bm.1.2		5
28.11	odd	6		1456.2.r.p.625.5		10
28.23	odd	6		1456.2.r.p.417.5		10
91.25	even	6		1183.2.e.f.170.4		10
91.51	even	6		1183.2.e.f.508.4		10
91.90	odd	2		8281.2.a.bx.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.e.c.53.2	✓	10	7.2	even	3
91.2.e.c.79.2	yes	10	7.4	even	3
637.2.a.k.1.4		5	7.6	odd	2
637.2.a.l.1.4		5	1.1	even	1	trivial
637.2.e.m.79.2		10	7.3	odd	6
637.2.e.m.508.2		10	7.5	odd	6
819.2.j.h.235.4		10	21.2	odd	6
819.2.j.h.352.4		10	21.11	odd	6
1183.2.e.f.170.4		10	91.25	even	6
1183.2.e.f.508.4		10	91.51	even	6
1456.2.r.p.417.5		10	28.23	odd	6
1456.2.r.p.625.5		10	28.11	odd	6
5733.2.a.bl.1.2		5	3.2	odd	2
5733.2.a.bm.1.2		5	21.20	even	2
8281.2.a.bw.1.2		5	13.12	even	2
8281.2.a.bx.1.2		5	91.90	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.a (trivial)

\(f(q)\)	\(=\)	\(q+2.21332 q^{2} +2.47443 q^{3} +2.89879 q^{4} -2.12280 q^{5} +5.47671 q^{6} +1.98932 q^{8} +3.12280 q^{9} -4.69843 q^{10} +4.78896 q^{11} +7.17286 q^{12} +1.00000 q^{13} -5.25271 q^{15} -1.39458 q^{16} -3.77828 q^{17} +6.91175 q^{18} -3.56723 q^{19} -6.15355 q^{20} +10.5995 q^{22} +4.47443 q^{23} +4.92243 q^{24} -0.493740 q^{25} +2.21332 q^{26} +0.303848 q^{27} -5.90107 q^{29} -11.6259 q^{30} -3.77116 q^{31} -7.06530 q^{32} +11.8499 q^{33} -8.36254 q^{34} +9.05234 q^{36} +5.62570 q^{37} -7.89544 q^{38} +2.47443 q^{39} -4.22292 q^{40} +10.3948 q^{41} +3.40733 q^{43} +13.8822 q^{44} -6.62906 q^{45} +9.90335 q^{46} -7.10876 q^{47} -3.45079 q^{48} -1.09280 q^{50} -9.34907 q^{51} +2.89879 q^{52} -12.3801 q^{53} +0.672514 q^{54} -10.1660 q^{55} -8.82686 q^{57} -13.0610 q^{58} +4.78896 q^{59} -15.2265 q^{60} +3.20697 q^{61} -8.34680 q^{62} -12.8486 q^{64} -2.12280 q^{65} +26.2277 q^{66} -2.89955 q^{67} -10.9524 q^{68} +11.0717 q^{69} -2.53876 q^{71} +6.21224 q^{72} +7.70071 q^{73} +12.4515 q^{74} -1.22172 q^{75} -10.3407 q^{76} +5.47671 q^{78} -5.17850 q^{79} +2.96041 q^{80} -8.61654 q^{81} +23.0071 q^{82} +3.46731 q^{83} +8.02051 q^{85} +7.54152 q^{86} -14.6018 q^{87} +9.52677 q^{88} +3.66432 q^{89} -14.6722 q^{90} +12.9704 q^{92} -9.33147 q^{93} -15.7340 q^{94} +7.57251 q^{95} -17.4826 q^{96} -5.40733 q^{97} +14.9549 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 8 q^{4} + 2 q^{5} - 5 q^{6} + 9 q^{8} + 3 q^{9} - 5 q^{10} + 11 q^{11} + 5 q^{12} + 5 q^{13} + 10 q^{16} - 5 q^{17} + 9 q^{18} + 9 q^{19} + q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} + 4 q^{26}+ \cdots + 11 q^{99}+O(q^{100}) \) 5 * q + 4 * q^2 + 8 * q^4 + 2 * q^5 - 5 * q^6 + 9 * q^8 + 3 * q^9 - 5 * q^10 + 11 * q^11 + 5 * q^12 + 5 * q^13 + 10 * q^16 - 5 * q^17 + 9 * q^18 + 9 * q^19 + q^20 + 8 * q^22 + 10 * q^23 + 9 * q^25 + 4 * q^26 - 3 * q^29 - 13 * q^30 - 6 * q^31 + 22 * q^32 + 8 * q^33 - 22 * q^34 + 7 * q^36 + 4 * q^37 - 10 * q^38 + 28 * q^40 + 14 * q^41 + 2 * q^43 - 32 * q^45 + 3 * q^46 + q^47 - 23 * q^48 + 9 * q^50 - 8 * q^51 + 8 * q^52 + 17 * q^53 + 23 * q^54 - 16 * q^57 - 27 * q^58 + 11 * q^59 - 29 * q^60 - 11 * q^61 - 23 * q^62 + 9 * q^64 + 2 * q^65 + 21 * q^66 + 13 * q^67 - 32 * q^68 + 18 * q^69 + 15 * q^71 - 19 * q^72 - 33 * q^74 - 20 * q^75 + 8 * q^76 - 5 * q^78 + 2 * q^79 + 55 * q^80 - 19 * q^81 + 34 * q^82 + 6 * q^83 - 22 * q^85 + 28 * q^86 - 8 * q^87 - 3 * q^88 - 4 * q^89 - 34 * q^90 + 21 * q^92 + 18 * q^93 + 20 * q^94 - 12 * q^95 - 37 * q^96 - 12 * q^97 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more