LMFDB - Embedded newform 627.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [627,2,Mod(1,627)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(627, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("627.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$627 = 3 \cdot 11 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	627.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$5.00662020673$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 4x + 1$ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.69963$ of defining polynomial
Character	$\chi$	$=$	627.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69963 q^{2} +1.00000 q^{3} +5.28799 q^{4} -0.888736 q^{5} -2.69963 q^{6} -0.411636 q^{7} -8.87636 q^{8} +1.00000 q^{9} +2.39926 q^{10} +1.00000 q^{11} +5.28799 q^{12} -2.30037 q^{13} +1.11126 q^{14} -0.888736 q^{15} +13.3869 q^{16} -8.09888 q^{17} -2.69963 q^{18} -1.00000 q^{19} -4.69963 q^{20} -0.411636 q^{21} -2.69963 q^{22} -5.00000 q^{23} -8.87636 q^{24} -4.21015 q^{25} +6.21015 q^{26} +1.00000 q^{27} -2.17673 q^{28} +7.38688 q^{29} +2.39926 q^{30} -1.98762 q^{31} -18.3869 q^{32} +1.00000 q^{33} +21.8640 q^{34} +0.365836 q^{35} +5.28799 q^{36} +9.35346 q^{37} +2.69963 q^{38} -2.30037 q^{39} +7.88874 q^{40} +1.69963 q^{41} +1.11126 q^{42} -7.28799 q^{43} +5.28799 q^{44} -0.888736 q^{45} +13.4981 q^{46} -4.47710 q^{47} +13.3869 q^{48} -6.83056 q^{49} +11.3658 q^{50} -8.09888 q^{51} -12.1643 q^{52} -0.333792 q^{53} -2.69963 q^{54} -0.888736 q^{55} +3.65383 q^{56} -1.00000 q^{57} -19.9418 q^{58} -4.76509 q^{59} -4.69963 q^{60} -10.5105 q^{61} +5.36584 q^{62} -0.411636 q^{63} +22.8640 q^{64} +2.04442 q^{65} -2.69963 q^{66} +10.0531 q^{67} -42.8268 q^{68} -5.00000 q^{69} -0.987620 q^{70} -5.30037 q^{71} -8.87636 q^{72} +2.47710 q^{73} -25.2509 q^{74} -4.21015 q^{75} -5.28799 q^{76} -0.411636 q^{77} +6.21015 q^{78} -16.0741 q^{79} -11.8974 q^{80} +1.00000 q^{81} -4.58836 q^{82} +8.73305 q^{83} -2.17673 q^{84} +7.19777 q^{85} +19.6749 q^{86} +7.38688 q^{87} -8.87636 q^{88} +1.98762 q^{89} +2.39926 q^{90} +0.946916 q^{91} -26.4400 q^{92} -1.98762 q^{93} +12.0865 q^{94} +0.888736 q^{95} -18.3869 q^{96} -15.6094 q^{97} +18.4400 q^{98} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 2 q^{2} + 3 q^{3} + 4 q^{4} - 3 q^{5} - 2 q^{6} - 7 q^{7} - 9 q^{8} + 3 q^{9} - 5 q^{10} + 3 q^{11} + 4 q^{12} - 13 q^{13} + 3 q^{14} - 3 q^{15} + 10 q^{16} - 6 q^{17} - 2 q^{18} - 3 q^{19} - 8 q^{20}+ \cdots + 3 q^{99}+O(q^{100})$ 3 * q - 2 * q^2 + 3 * q^3 + 4 * q^4 - 3 * q^5 - 2 * q^6 - 7 * q^7 - 9 * q^8 + 3 * q^9 - 5 * q^10 + 3 * q^11 + 4 * q^12 - 13 * q^13 + 3 * q^14 - 3 * q^15 + 10 * q^16 - 6 * q^17 - 2 * q^18 - 3 * q^19 - 8 * q^20 - 7 * q^21 - 2 * q^22 - 15 * q^23 - 9 * q^24 + 6 * q^25 + 3 * q^27 + 5 * q^28 - 8 * q^29 - 5 * q^30 + 12 * q^31 - 25 * q^32 + 3 * q^33 + 30 * q^34 - 4 * q^35 + 4 * q^36 + 5 * q^37 + 2 * q^38 - 13 * q^39 + 24 * q^40 - q^41 + 3 * q^42 - 10 * q^43 + 4 * q^44 - 3 * q^45 + 10 * q^46 - 8 * q^47 + 10 * q^48 + 8 * q^49 + 29 * q^50 - 6 * q^51 - 7 * q^52 - 2 * q^54 - 3 * q^55 - 6 * q^56 - 3 * q^57 - 31 * q^58 + 3 * q^59 - 8 * q^60 - 19 * q^61 + 11 * q^62 - 7 * q^63 + 33 * q^64 + 20 * q^65 - 2 * q^66 + q^67 - 39 * q^68 - 15 * q^69 + 15 * q^70 - 22 * q^71 - 9 * q^72 + 2 * q^73 - 10 * q^74 + 6 * q^75 - 4 * q^76 - 7 * q^77 + 6 * q^79 + 7 * q^80 + 3 * q^81 - 8 * q^82 + 13 * q^83 + 5 * q^84 - 15 * q^85 + 17 * q^86 - 8 * q^87 - 9 * q^88 - 12 * q^89 - 5 * q^90 + 32 * q^91 - 20 * q^92 + 12 * q^93 + 3 * q^95 - 25 * q^96 - 16 * q^97 - 4 * q^98 + 3 * 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

)

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

$2$	−2.69963	−1.90893	−0.954463	−	0.298330i	$-0.903570\pi$
$2$	−2.69963	−1.90893	−0.954463	+	0.298330i	$0.903570\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.28799	2.64400
$5$	−0.888736	−0.397455	−0.198727	−	0.980055i	$-0.563681\pi$
$5$	−0.888736	−0.397455	−0.198727	+	0.980055i	$0.563681\pi$
$6$	−2.69963	−1.10212
$7$	−0.411636	−0.155584	−0.0777919	−	0.996970i	$-0.524787\pi$
$7$	−0.411636	−0.155584	−0.0777919	+	0.996970i	$0.524787\pi$
$8$	−8.87636	−3.13827
$9$	1.00000	0.333333
$10$	2.39926	0.758711
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	5.28799	1.52651
$13$	−2.30037	−0.638008	−0.319004	−	0.947753i	$-0.603348\pi$
$13$	−2.30037	−0.638008	−0.319004	+	0.947753i	$0.603348\pi$
$14$	1.11126	0.296998
$15$	−0.888736	−0.229471
$16$	13.3869	3.34672
$17$	−8.09888	−1.96427	−0.982134	−	0.188183i	$-0.939740\pi$
$17$	−8.09888	−1.96427	−0.982134	+	0.188183i	$0.939740\pi$
$18$	−2.69963	−0.636308
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−4.69963	−1.05087
$21$	−0.411636	−0.0898263
$22$	−2.69963	−0.575563
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	−8.87636	−1.81188
$25$	−4.21015	−0.842030
$26$	6.21015	1.21791
$27$	1.00000	0.192450
$28$	−2.17673	−0.411363
$29$	7.38688	1.37171	0.685854	−	0.727739i	$-0.259429\pi$
$29$	7.38688	1.37171	0.685854	+	0.727739i	$0.259429\pi$
$30$	2.39926	0.438042
$31$	−1.98762	−0.356987	−0.178494	−	0.983941i	$-0.557122\pi$
$31$	−1.98762	−0.356987	−0.178494	+	0.983941i	$0.557122\pi$
$32$	−18.3869	−3.25037
$33$	1.00000	0.174078
$34$	21.8640	3.74964
$35$	0.365836	0.0618375
$36$	5.28799	0.881332
$37$	9.35346	1.53770	0.768849	−	0.639430i	$-0.220830\pi$
$37$	9.35346	1.53770	0.768849	+	0.639430i	$0.220830\pi$
$38$	2.69963	0.437938
$39$	−2.30037	−0.368354
$40$	7.88874	1.24732
$41$	1.69963	0.265437	0.132719	−	0.991154i	$-0.457629\pi$
$41$	1.69963	0.265437	0.132719	+	0.991154i	$0.457629\pi$
$42$	1.11126	0.171472
$43$	−7.28799	−1.11141	−0.555704	−	0.831380i	$-0.687551\pi$
$43$	−7.28799	−1.11141	−0.555704	+	0.831380i	$0.687551\pi$
$44$	5.28799	0.797195
$45$	−0.888736	−0.132485
$46$	13.4981	1.99019
$47$	−4.47710	−0.653052	−0.326526	−	0.945188i	$-0.605878\pi$
$47$	−4.47710	−0.653052	−0.326526	+	0.945188i	$0.605878\pi$
$48$	13.3869	1.93223
$49$	−6.83056	−0.975794
$50$	11.3658	1.60737
$51$	−8.09888	−1.13407
$52$	−12.1643	−1.68689
$53$	−0.333792	−0.0458499	−0.0229250	−	0.999737i	$-0.507298\pi$
$53$	−0.333792	−0.0458499	−0.0229250	+	0.999737i	$0.507298\pi$
$54$	−2.69963	−0.367373
$55$	−0.888736	−0.119837
$56$	3.65383	0.488263
$57$	−1.00000	−0.132453
$58$	−19.9418	−2.61849
$59$	−4.76509	−0.620362	−0.310181	−	0.950677i	$-0.600390\pi$
$59$	−4.76509	−0.620362	−0.310181	+	0.950677i	$0.600390\pi$
$60$	−4.69963	−0.606719
$61$	−10.5105	−1.34573	−0.672867	−	0.739763i	$-0.734937\pi$
$61$	−10.5105	−1.34573	−0.672867	+	0.739763i	$0.734937\pi$
$62$	5.36584	0.681462
$63$	−0.411636	−0.0518613
$64$	22.8640	2.85800
$65$	2.04442	0.253579
$66$	−2.69963	−0.332301
$67$	10.0531	1.22818	0.614090	−	0.789236i	$-0.289523\pi$
$67$	10.0531	1.22818	0.614090	+	0.789236i	$0.289523\pi$
$68$	−42.8268	−5.19352
$69$	−5.00000	−0.601929
$70$	−0.987620	−0.118043
$71$	−5.30037	−0.629038	−0.314519	−	0.949251i	$-0.601843\pi$
$71$	−5.30037	−0.629038	−0.314519	+	0.949251i	$0.601843\pi$
$72$	−8.87636	−1.04609
$73$	2.47710	0.289923	0.144961	−	0.989437i	$-0.453694\pi$
$73$	2.47710	0.289923	0.144961	+	0.989437i	$0.453694\pi$
$74$	−25.2509	−2.93535
$75$	−4.21015	−0.486146
$76$	−5.28799	−0.606574
$77$	−0.411636	−0.0469103
$78$	6.21015	0.703161
$79$	−16.0741	−1.80848	−0.904240	−	0.427024i	$-0.859562\pi$
$79$	−16.0741	−1.80848	−0.904240	+	0.427024i	$0.859562\pi$
$80$	−11.8974	−1.33017
$81$	1.00000	0.111111
$82$	−4.58836	−0.506700
$83$	8.73305	0.958577	0.479288	−	0.877658i	$-0.340895\pi$
$83$	8.73305	0.958577	0.479288	+	0.877658i	$0.340895\pi$
$84$	−2.17673	−0.237500
$85$	7.19777	0.780708
$86$	19.6749	2.12160
$87$	7.38688	0.791956
$88$	−8.87636	−0.946223
$89$	1.98762	0.210687	0.105344	−	0.994436i	$-0.466406\pi$
$89$	1.98762	0.210687	0.105344	+	0.994436i	$0.466406\pi$
$90$	2.39926	0.252904
$91$	0.946916	0.0992638
$92$	−26.4400	−2.75656
$93$	−1.98762	−0.206107
$94$	12.0865	1.24663
$95$	0.888736	0.0911824
$96$	−18.3869	−1.87660
$97$	−15.6094	−1.58489	−0.792447	−	0.609940i	$-0.791193\pi$
$97$	−15.6094	−1.58489	−0.792447	+	0.609940i	$0.791193\pi$
$98$	18.4400	1.86272
$99$	1.00000	0.100504
$100$	−22.2632	−2.22632
$101$	0.888736	0.0884325	0.0442163	−	0.999022i	$-0.485921\pi$
$101$	0.888736	0.0884325	0.0442163	+	0.999022i	$0.485921\pi$
$102$	21.8640	2.16486
$103$	−11.7193	−1.15474	−0.577368	−	0.816484i	$-0.695920\pi$
$103$	−11.7193	−1.15474	−0.577368	+	0.816484i	$0.695920\pi$
$104$	20.4189	2.00224
$105$	0.365836	0.0357019
$106$	0.901116	0.0875241
$107$	−6.18911	−0.598324	−0.299162	−	0.954202i	$-0.596707\pi$
$107$	−6.18911	−0.598324	−0.299162	+	0.954202i	$0.596707\pi$
$108$	5.28799	0.508837
$109$	−11.7651	−1.12689	−0.563446	−	0.826153i	$-0.690525\pi$
$109$	−11.7651	−1.12689	−0.563446	+	0.826153i	$0.690525\pi$
$110$	2.39926	0.228760
$111$	9.35346	0.887791
$112$	−5.51052	−0.520695
$113$	−1.84294	−0.173369	−0.0866844	−	0.996236i	$-0.527627\pi$
$113$	−1.84294	−0.173369	−0.0866844	+	0.996236i	$0.527627\pi$
$114$	2.69963	0.252843
$115$	4.44368	0.414375
$116$	39.0617	3.62679
$117$	−2.30037	−0.212669
$118$	12.8640	1.18423
$119$	3.33379	0.305608
$120$	7.88874	0.720140
$121$	1.00000	0.0909091
$122$	28.3745	2.56891
$123$	1.69963	0.153250
$124$	−10.5105	−0.943873
$125$	8.18539	0.732123
$126$	1.11126	0.0989993
$127$	9.43268	0.837015	0.418507	−	0.908213i	$-0.362553\pi$
$127$	9.43268	0.837015	0.418507	+	0.908213i	$0.362553\pi$
$128$	−24.9505	−2.20533
$129$	−7.28799	−0.641672
$130$	−5.51918	−0.484064
$131$	9.24357	0.807614	0.403807	−	0.914844i	$-0.367687\pi$
$131$	9.24357	0.807614	0.403807	+	0.914844i	$0.367687\pi$
$132$	5.28799	0.460261
$133$	0.411636	0.0356934
$134$	−27.1396	−2.34450
$135$	−0.888736	−0.0764902
$136$	71.8886	6.16440
$137$	−1.15706	−0.0988547	−0.0494273	−	0.998778i	$-0.515740\pi$
$137$	−1.15706	−0.0988547	−0.0494273	+	0.998778i	$0.515740\pi$
$138$	13.4981	1.14904
$139$	−4.25457	−0.360868	−0.180434	−	0.983587i	$-0.557750\pi$
$139$	−4.25457	−0.360868	−0.180434	+	0.983587i	$0.557750\pi$
$140$	1.93454	0.163498
$141$	−4.47710	−0.377040
$142$	14.3090	1.20079
$143$	−2.30037	−0.192367
$144$	13.3869	1.11557
$145$	−6.56498	−0.545192
$146$	−6.68725	−0.553441
$147$	−6.83056	−0.563375
$148$	49.4610	4.06567
$149$	8.30037	0.679993	0.339996	−	0.940427i	$-0.389574\pi$
$149$	8.30037	0.679993	0.339996	+	0.940427i	$0.389574\pi$
$150$	11.3658	0.928017
$151$	21.4189	1.74305	0.871523	−	0.490354i	$-0.163132\pi$
$151$	21.4189	1.74305	0.871523	+	0.490354i	$0.163132\pi$
$152$	8.87636	0.719968
$153$	−8.09888	−0.654756
$154$	1.11126	0.0895482
$155$	1.76647	0.141886
$156$	−12.1643	−0.973927
$157$	−5.85532	−0.467305	−0.233653	−	0.972320i	$-0.575068\pi$
$157$	−5.85532	−0.467305	−0.233653	+	0.972320i	$0.575068\pi$
$158$	43.3942	3.45225
$159$	−0.333792	−0.0264715
$160$	16.3411	1.29188
$161$	2.05818	0.162207
$162$	−2.69963	−0.212103
$163$	18.6094	1.45760	0.728801	−	0.684726i	$-0.240078\pi$
$163$	18.6094	1.45760	0.728801	+	0.684726i	$0.240078\pi$
$164$	8.98762	0.701815
$165$	−0.888736	−0.0691880
$166$	−23.5760	−1.82985
$167$	−1.11126	−0.0859922	−0.0429961	−	0.999075i	$-0.513690\pi$
$167$	−1.11126	−0.0859922	−0.0429961	+	0.999075i	$0.513690\pi$
$168$	3.65383	0.281899
$169$	−7.70829	−0.592945
$170$	−19.4313	−1.49031
$171$	−1.00000	−0.0764719
$172$	−38.5388	−2.93856
$173$	−9.66621	−0.734908	−0.367454	−	0.930042i	$-0.619770\pi$
$173$	−9.66621	−0.734908	−0.367454	+	0.930042i	$0.619770\pi$
$174$	−19.9418	−1.51179
$175$	1.73305	0.131006
$176$	13.3869	1.00907
$177$	−4.76509	−0.358166
$178$	−5.36584	−0.402186
$179$	14.3658	1.07375	0.536876	−	0.843661i	$-0.319604\pi$
$179$	14.3658	1.07375	0.536876	+	0.843661i	$0.319604\pi$
$180$	−4.69963	−0.350290
$181$	4.14468	0.308072	0.154036	−	0.988065i	$-0.450773\pi$
$181$	4.14468	0.308072	0.154036	+	0.988065i	$0.450773\pi$
$182$	−2.55632	−0.189487
$183$	−10.5105	−0.776960
$184$	44.3818	3.27187
$185$	−8.31275	−0.611166
$186$	5.36584	0.393442
$187$	−8.09888	−0.592249
$188$	−23.6749	−1.72667
$189$	−0.411636	−0.0299421
$190$	−2.39926	−0.174060
$191$	−18.2101	−1.31764	−0.658820	−	0.752301i	$-0.728944\pi$
$191$	−18.2101	−1.31764	−0.658820	+	0.752301i	$0.728944\pi$
$192$	22.8640	1.65007
$193$	24.0989	1.73468	0.867338	−	0.497720i	$-0.165829\pi$
$193$	24.0989	1.73468	0.867338	+	0.497720i	$0.165829\pi$
$194$	42.1396	3.02545
$195$	2.04442	0.146404
$196$	−36.1199	−2.57999
$197$	−16.5512	−1.17923	−0.589613	−	0.807686i	$-0.700720\pi$
$197$	−16.5512	−1.17923	−0.589613	+	0.807686i	$0.700720\pi$
$198$	−2.69963	−0.191854
$199$	1.53528	0.108833	0.0544166	−	0.998518i	$-0.482670\pi$
$199$	1.53528	0.108833	0.0544166	+	0.998518i	$0.482670\pi$
$200$	37.3708	2.64251
$201$	10.0531	0.709090
$202$	−2.39926	−0.168811
$203$	−3.04070	−0.213416
$204$	−42.8268	−2.99848
$205$	−1.51052	−0.105499
$206$	31.6377	2.20431
$207$	−5.00000	−0.347524
$208$	−30.7948	−2.13523
$209$	−1.00000	−0.0691714
$210$	−0.987620	−0.0681523
$211$	9.01966	0.620939	0.310470	−	0.950583i	$-0.399514\pi$
$211$	9.01966	0.620939	0.310470	+	0.950583i	$0.399514\pi$
$212$	−1.76509	−0.121227
$213$	−5.30037	−0.363175
$214$	16.7083	1.14216
$215$	6.47710	0.441735
$216$	−8.87636	−0.603960
$217$	0.818176	0.0555414
$218$	31.7614	2.15115
$219$	2.47710	0.167387
$220$	−4.69963	−0.316849
$221$	18.6304	1.25322
$222$	−25.2509	−1.69473
$223$	11.3535	0.760284	0.380142	−	0.924928i	$-0.375875\pi$
$223$	11.3535	0.760284	0.380142	+	0.924928i	$0.375875\pi$
$224$	7.56870	0.505705
$225$	−4.21015	−0.280677
$226$	4.97524	0.330948
$227$	−13.7651	−0.913621	−0.456811	−	0.889564i	$-0.651008\pi$
$227$	−13.7651	−0.913621	−0.456811	+	0.889564i	$0.651008\pi$
$228$	−5.28799	−0.350206
$229$	−23.5512	−1.55631	−0.778154	−	0.628073i	$-0.783844\pi$
$229$	−23.5512	−1.55631	−0.778154	+	0.628073i	$0.783844\pi$
$230$	−11.9963	−0.791011
$231$	−0.411636	−0.0270837
$232$	−65.5685	−4.30479
$233$	27.2632	1.78607	0.893037	−	0.449983i	$-0.148570\pi$
$233$	27.2632	1.78607	0.893037	+	0.449983i	$0.148570\pi$
$234$	6.21015	0.405970
$235$	3.97896	0.259559
$236$	−25.1978	−1.64024
$237$	−16.0741	−1.04413
$238$	−9.00000	−0.583383
$239$	0.320035	0.0207014	0.0103507	−	0.999946i	$-0.496705\pi$
$239$	0.320035	0.0207014	0.0103507	+	0.999946i	$0.496705\pi$
$240$	−11.8974	−0.767974
$241$	26.8392	1.72887	0.864433	−	0.502748i	$-0.167678\pi$
$241$	26.8392	1.72887	0.864433	+	0.502748i	$0.167678\pi$
$242$	−2.69963	−0.173539
$243$	1.00000	0.0641500
$244$	−55.5795	−3.55812
$245$	6.07056	0.387834
$246$	−4.58836	−0.292543
$247$	2.30037	0.146369
$248$	17.6428	1.12032
$249$	8.73305	0.553434
$250$	−22.0975	−1.39757
$251$	27.5709	1.74026	0.870130	−	0.492823i	$-0.164035\pi$
$251$	27.5709	1.74026	0.870130	+	0.492823i	$0.164035\pi$
$252$	−2.17673	−0.137121
$253$	−5.00000	−0.314347
$254$	−25.4647	−1.59780
$255$	7.19777	0.450742
$256$	21.6291	1.35182
$257$	1.19049	0.0742604	0.0371302	−	0.999310i	$-0.488178\pi$
$257$	1.19049	0.0742604	0.0371302	+	0.999310i	$0.488178\pi$
$258$	19.6749	1.22490
$259$	−3.85022	−0.239241
$260$	10.8109	0.670463
$261$	7.38688	0.457236
$262$	−24.9542	−1.54168
$263$	−19.0334	−1.17365	−0.586825	−	0.809713i	$-0.699623\pi$
$263$	−19.0334	−1.17365	−0.586825	+	0.809713i	$0.699623\pi$
$264$	−8.87636	−0.546302
$265$	0.296653	0.0182233
$266$	−1.11126	−0.0681360
$267$	1.98762	0.121640
$268$	53.1606	3.24730
$269$	11.7069	0.713783	0.356892	−	0.934146i	$-0.383837\pi$
$269$	11.7069	0.713783	0.356892	+	0.934146i	$0.383837\pi$
$270$	2.39926	0.146014
$271$	−15.4523	−0.938663	−0.469331	−	0.883022i	$-0.655505\pi$
$271$	−15.4523	−0.938663	−0.469331	+	0.883022i	$0.655505\pi$
$272$	−108.419	−6.57385
$273$	0.946916	0.0573100
$274$	3.12364	0.188706
$275$	−4.21015	−0.253882
$276$	−26.4400	−1.59150
$277$	−20.9876	−1.26102	−0.630512	−	0.776180i	$-0.717155\pi$
$277$	−20.9876	−1.26102	−0.630512	+	0.776180i	$0.717155\pi$
$278$	11.4858	0.688870
$279$	−1.98762	−0.118996
$280$	−3.24729	−0.194063
$281$	9.68725	0.577893	0.288946	−	0.957345i	$-0.406695\pi$
$281$	9.68725	0.577893	0.288946	+	0.957345i	$0.406695\pi$
$282$	12.0865	0.719741
$283$	13.0865	0.777912	0.388956	−	0.921256i	$-0.372836\pi$
$283$	13.0865	0.777912	0.388956	+	0.921256i	$0.372836\pi$
$284$	−28.0283	−1.66318
$285$	0.888736	0.0526442
$286$	6.21015	0.367214
$287$	−0.699628	−0.0412977
$288$	−18.3869	−1.08346
$289$	48.5919	2.85835
$290$	17.7230	1.04073
$291$	−15.6094	−0.915040
$292$	13.0989	0.766554
$293$	−19.1447	−1.11844	−0.559222	−	0.829018i	$-0.688900\pi$
$293$	−19.1447	−1.11844	−0.559222	+	0.829018i	$0.688900\pi$
$294$	18.4400	1.07544
$295$	4.23491	0.246566
$296$	−83.0246	−4.82571
$297$	1.00000	0.0580259
$298$	−22.4079	−1.29806
$299$	11.5019	0.665170
$300$	−22.2632	−1.28537
$301$	3.00000	0.172917
$302$	−57.8231	−3.32735
$303$	0.888736	0.0510565
$304$	−13.3869	−0.767790
$305$	9.34108	0.534868
$306$	21.8640	1.24988
$307$	−3.28799	−0.187656	−0.0938278	−	0.995588i	$-0.529910\pi$
$307$	−3.28799	−0.187656	−0.0938278	+	0.995588i	$0.529910\pi$
$308$	−2.17673	−0.124031
$309$	−11.7193	−0.666687
$310$	−4.76881	−0.270850
$311$	9.83056	0.557440	0.278720	−	0.960372i	$-0.410090\pi$
$311$	9.83056	0.557440	0.278720	+	0.960372i	$0.410090\pi$
$312$	20.4189	1.15599
$313$	−29.8516	−1.68731	−0.843656	−	0.536884i	$-0.819601\pi$
$313$	−29.8516	−1.68731	−0.843656	+	0.536884i	$0.819601\pi$
$314$	15.8072	0.892050
$315$	0.365836	0.0206125
$316$	−84.9998	−4.78161
$317$	−9.66249	−0.542699	−0.271350	−	0.962481i	$-0.587470\pi$
$317$	−9.66249	−0.542699	−0.271350	+	0.962481i	$0.587470\pi$
$318$	0.901116	0.0505321
$319$	7.38688	0.413586
$320$	−20.3200	−1.13592
$321$	−6.18911	−0.345442
$322$	−5.55632	−0.309642
$323$	8.09888	0.450634
$324$	5.28799	0.293777
$325$	9.68491	0.537222
$326$	−50.2385	−2.78245
$327$	−11.7651	−0.650611
$328$	−15.0865	−0.833013
$329$	1.84294	0.101604
$330$	2.39926	0.132075
$331$	−0.333792	−0.0183469	−0.00917345	−	0.999958i	$-0.502920\pi$
$331$	−0.333792	−0.0183469	−0.00917345	+	0.999958i	$0.502920\pi$
$332$	46.1803	2.53447
$333$	9.35346	0.512566
$334$	3.00000	0.164153
$335$	−8.93454	−0.488146
$336$	−5.51052	−0.300624
$337$	22.8306	1.24366	0.621830	−	0.783152i	$-0.286390\pi$
$337$	22.8306	1.24366	0.621830	+	0.783152i	$0.286390\pi$
$338$	20.8095	1.13189
$339$	−1.84294	−0.100095
$340$	38.0617	2.06419
$341$	−1.98762	−0.107636
$342$	2.69963	0.145979
$343$	5.69315	0.307401
$344$	64.6908	3.48789
$345$	4.44368	0.239240
$346$	26.0952	1.40288
$347$	26.5054	1.42289	0.711443	−	0.702744i	$-0.248042\pi$
$347$	26.5054	1.42289	0.711443	+	0.702744i	$0.248042\pi$
$348$	39.0617	2.09393
$349$	−11.9469	−0.639504	−0.319752	−	0.947501i	$-0.603600\pi$
$349$	−11.9469	−0.639504	−0.319752	+	0.947501i	$0.603600\pi$
$350$	−4.67859	−0.250081
$351$	−2.30037	−0.122785
$352$	−18.3869	−0.980024
$353$	28.2312	1.50259	0.751297	−	0.659964i	$-0.229428\pi$
$353$	28.2312	1.50259	0.751297	+	0.659964i	$0.229428\pi$
$354$	12.8640	0.683713
$355$	4.71063	0.250014
$356$	10.5105	0.557056
$357$	3.33379	0.176443
$358$	−38.7824	−2.04971
$359$	18.3039	0.966045	0.483022	−	0.875608i	$-0.339539\pi$
$359$	18.3039	0.966045	0.483022	+	0.875608i	$0.339539\pi$
$360$	7.88874	0.415773
$361$	1.00000	0.0526316
$362$	−11.1891	−0.588086
$363$	1.00000	0.0524864
$364$	5.00728	0.262453
$365$	−2.20149	−0.115231
$366$	28.3745	1.48316
$367$	30.7600	1.60566	0.802829	−	0.596209i	$-0.203327\pi$
$367$	30.7600	1.60566	0.802829	+	0.596209i	$0.203327\pi$
$368$	−66.9344	−3.48920
$369$	1.69963	0.0884791
$370$	22.4413	1.16667
$371$	0.137401	0.00713350
$372$	−10.5105	−0.544945
$373$	−2.22115	−0.115007	−0.0575034	−	0.998345i	$-0.518314\pi$
$373$	−2.22115	−0.115007	−0.0575034	+	0.998345i	$0.518314\pi$
$374$	21.8640	1.13056
$375$	8.18539	0.422692
$376$	39.7403	2.04945
$377$	−16.9926	−0.875162
$378$	1.11126	0.0571573
$379$	−8.05308	−0.413659	−0.206830	−	0.978377i	$-0.566315\pi$
$379$	−8.05308	−0.413659	−0.206830	+	0.978377i	$0.566315\pi$
$380$	4.69963	0.241086
$381$	9.43268	0.483251
$382$	49.1606	2.51528
$383$	−6.59703	−0.337092	−0.168546	−	0.985694i	$-0.553907\pi$
$383$	−6.59703	−0.337092	−0.168546	+	0.985694i	$0.553907\pi$
$384$	−24.9505	−1.27325
$385$	0.365836	0.0186447
$386$	−65.0580	−3.31137
$387$	−7.28799	−0.370469
$388$	−82.5424	−4.19046
$389$	−25.1199	−1.27363	−0.636815	−	0.771016i	$-0.719749\pi$
$389$	−25.1199	−1.27363	−0.636815	+	0.771016i	$0.719749\pi$
$390$	−5.51918	−0.279475
$391$	40.4944	2.04789
$392$	60.6304	3.06230
$393$	9.24357	0.466276
$394$	44.6822	2.25105
$395$	14.2857	0.718789
$396$	5.28799	0.265732
$397$	−31.5068	−1.58128	−0.790641	−	0.612281i	$-0.790252\pi$
$397$	−31.5068	−1.58128	−0.790641	+	0.612281i	$0.790252\pi$
$398$	−4.14468	−0.207754
$399$	0.411636	0.0206076
$400$	−56.3607	−2.81804
$401$	−34.1643	−1.70609	−0.853043	−	0.521841i	$-0.825246\pi$
$401$	−34.1643	−1.70609	−0.853043	+	0.521841i	$0.825246\pi$
$402$	−27.1396	−1.35360
$403$	4.57227	0.227761
$404$	4.69963	0.233815
$405$	−0.888736	−0.0441616
$406$	8.20877	0.407394
$407$	9.35346	0.463634
$408$	71.8886	3.55902
$409$	0.642826	0.0317857	0.0158928	−	0.999874i	$-0.494941\pi$
$409$	0.642826	0.0317857	0.0158928	+	0.999874i	$0.494941\pi$
$410$	4.07784	0.201390
$411$	−1.15706	−0.0570738
$412$	−61.9715	−3.05312
$413$	1.96148	0.0965183
$414$	13.4981	0.663397
$415$	−7.76137	−0.380991
$416$	42.2967	2.07376
$417$	−4.25457	−0.208347
$418$	2.69963	0.132043
$419$	20.8502	1.01860	0.509300	−	0.860589i	$-0.329904\pi$
$419$	20.8502	1.01860	0.509300	+	0.860589i	$0.329904\pi$
$420$	1.93454	0.0943957
$421$	−10.3425	−0.504060	−0.252030	−	0.967719i	$-0.581098\pi$
$421$	−10.3425	−0.504060	−0.252030	+	0.967719i	$0.581098\pi$
$422$	−24.3497	−1.18533
$423$	−4.47710	−0.217684
$424$	2.96286	0.143889
$425$	34.0975	1.65397
$426$	14.3090	0.693275
$427$	4.32651	0.209374
$428$	−32.7280	−1.58197
$429$	−2.30037	−0.111063
$430$	−17.4858	−0.843238
$431$	−13.9615	−0.672501	−0.336250	−	0.941773i	$-0.609159\pi$
$431$	−13.9615	−0.672501	−0.336250	+	0.941773i	$0.609159\pi$
$432$	13.3869	0.644076
$433$	5.53156	0.265830	0.132915	−	0.991127i	$-0.457566\pi$
$433$	5.53156	0.265830	0.132915	+	0.991127i	$0.457566\pi$
$434$	−2.20877	−0.106024
$435$	−6.56498	−0.314767
$436$	−62.2137	−2.97950
$437$	5.00000	0.239182
$438$	−6.68725	−0.319529
$439$	−20.2509	−0.966520	−0.483260	−	0.875477i	$-0.660548\pi$
$439$	−20.2509	−0.966520	−0.483260	+	0.875477i	$0.660548\pi$
$440$	7.88874	0.376081
$441$	−6.83056	−0.325265
$442$	−50.2953	−2.39230
$443$	−30.6080	−1.45423	−0.727116	−	0.686515i	$-0.759140\pi$
$443$	−30.6080	−1.45423	−0.727116	+	0.686515i	$0.759140\pi$
$444$	49.4610	2.34732
$445$	−1.76647	−0.0837387
$446$	−30.6501	−1.45132
$447$	8.30037	0.392594
$448$	−9.41164	−0.444658
$449$	−3.75643	−0.177277	−0.0886385	−	0.996064i	$-0.528252\pi$
$449$	−3.75643	−0.177277	−0.0886385	+	0.996064i	$0.528252\pi$
$450$	11.3658	0.535791
$451$	1.69963	0.0800324
$452$	−9.74543	−0.458386
$453$	21.4189	1.00635
$454$	37.1606	1.74403
$455$	−0.841558	−0.0394529
$456$	8.87636	0.415673
$457$	−2.41026	−0.112747	−0.0563736	−	0.998410i	$-0.517954\pi$
$457$	−2.41026	−0.112747	−0.0563736	+	0.998410i	$0.517954\pi$
$458$	63.5795	2.97088
$459$	−8.09888	−0.378024
$460$	23.4981	1.09561
$461$	4.35855	0.202998	0.101499	−	0.994836i	$-0.467636\pi$
$461$	4.35855	0.202998	0.101499	+	0.994836i	$0.467636\pi$
$462$	1.11126	0.0517007
$463$	38.6167	1.79467	0.897335	−	0.441350i	$-0.145500\pi$
$463$	38.6167	1.79467	0.897335	+	0.441350i	$0.145500\pi$
$464$	98.8872	4.59072
$465$	1.76647	0.0819181
$466$	−73.6006	−3.40948
$467$	−16.1767	−0.748570	−0.374285	−	0.927314i	$-0.622112\pi$
$467$	−16.1767	−0.748570	−0.374285	+	0.927314i	$0.622112\pi$
$468$	−12.1643	−0.562297
$469$	−4.13821	−0.191085
$470$	−10.7417	−0.495478
$471$	−5.85532	−0.269799
$472$	42.2967	1.94686
$473$	−7.28799	−0.335102
$474$	43.3942	1.99316
$475$	4.21015	0.193175
$476$	17.6291	0.808027
$477$	−0.333792	−0.0152833
$478$	−0.863976	−0.0395174
$479$	−28.9825	−1.32425	−0.662123	−	0.749395i	$-0.730344\pi$
$479$	−28.9825	−1.32425	−0.662123	+	0.749395i	$0.730344\pi$
$480$	16.3411	0.745865
$481$	−21.5164	−0.981065
$482$	−72.4559	−3.30028
$483$	2.05818	0.0936504
$484$	5.28799	0.240363
$485$	13.8726	0.629924
$486$	−2.69963	−0.122458
$487$	21.2670	0.963698	0.481849	−	0.876254i	$-0.339965\pi$
$487$	21.2670	0.963698	0.481849	+	0.876254i	$0.339965\pi$
$488$	93.2951	4.22327
$489$	18.6094	0.841546
$490$	−16.3883	−0.740346
$491$	−3.55494	−0.160432	−0.0802162	−	0.996777i	$-0.525561\pi$
$491$	−3.55494	−0.160432	−0.0802162	+	0.996777i	$0.525561\pi$
$492$	8.98762	0.405193
$493$	−59.8255	−2.69440
$494$	−6.21015	−0.279408
$495$	−0.888736	−0.0399457
$496$	−26.6080	−1.19474
$497$	2.18182	0.0978682
$498$	−23.5760	−1.05647
$499$	−23.0952	−1.03388	−0.516941	−	0.856021i	$-0.672929\pi$
$499$	−23.0952	−1.03388	−0.516941	+	0.856021i	$0.672929\pi$
$500$	43.2843	1.93573
$501$	−1.11126	−0.0496476
$502$	−74.4311	−3.32202
$503$	36.4930	1.62714	0.813572	−	0.581464i	$-0.197520\pi$
$503$	36.4930	1.62714	0.813572	+	0.581464i	$0.197520\pi$
$504$	3.65383	0.162754
$505$	−0.789851	−0.0351479
$506$	13.4981	0.600066
$507$	−7.70829	−0.342337
$508$	49.8799	2.21306
$509$	−25.0036	−1.10826	−0.554132	−	0.832429i	$-0.686950\pi$
$509$	−25.0036	−1.10826	−0.554132	+	0.832429i	$0.686950\pi$
$510$	−19.4313	−0.860432
$511$	−1.01966	−0.0451073
$512$	−8.48948	−0.375186
$513$	−1.00000	−0.0441511
$514$	−3.21387	−0.141758
$515$	10.4154	0.458955
$516$	−38.5388	−1.69658
$517$	−4.47710	−0.196903
$518$	10.3942	0.456693
$519$	−9.66621	−0.424299
$520$	−18.1470	−0.795800
$521$	−43.8355	−1.92047	−0.960234	−	0.279196i	$-0.909932\pi$
$521$	−43.8355	−1.92047	−0.960234	+	0.279196i	$0.909932\pi$
$522$	−19.9418	−0.872830
$523$	23.2829	1.01809	0.509045	−	0.860740i	$-0.329999\pi$
$523$	23.2829	1.01809	0.509045	+	0.860740i	$0.329999\pi$
$524$	48.8799	2.13533
$525$	1.73305	0.0756364
$526$	51.3832	2.24041
$527$	16.0975	0.701218
$528$	13.3869	0.582589
$529$	2.00000	0.0869565
$530$	−0.800854	−0.0347869
$531$	−4.76509	−0.206787
$532$	2.17673	0.0943731
$533$	−3.90978	−0.169351
$534$	−5.36584	−0.232202
$535$	5.50048	0.237807
$536$	−89.2348	−3.85435
$537$	14.3658	0.619932
$538$	−31.6043	−1.36256
$539$	−6.83056	−0.294213
$540$	−4.69963	−0.202240
$541$	29.9556	1.28789	0.643945	−	0.765071i	$-0.277296\pi$
$541$	29.9556	1.28789	0.643945	+	0.765071i	$0.277296\pi$
$542$	41.7156	1.79184
$543$	4.14468	0.177865
$544$	148.913	6.38460
$545$	10.4561	0.447888
$546$	−2.55632	−0.109400
$547$	45.7700	1.95699	0.978493	−	0.206282i	$-0.0661363\pi$
$547$	45.7700	1.95699	0.978493	+	0.206282i	$0.0661363\pi$
$548$	−6.11855	−0.261371
$549$	−10.5105	−0.448578
$550$	11.3658	0.484641
$551$	−7.38688	−0.314692
$552$	44.3818	1.88901
$553$	6.61669	0.281370
$554$	56.6588	2.40720
$555$	−8.31275	−0.352857
$556$	−22.4981	−0.954134
$557$	0.987620	0.0418468	0.0209234	−	0.999781i	$-0.493339\pi$
$557$	0.987620	0.0418468	0.0209234	+	0.999781i	$0.493339\pi$
$558$	5.36584	0.227154
$559$	16.7651	0.709088
$560$	4.89740	0.206953
$561$	−8.09888	−0.341935
$562$	−26.1520	−1.10315
$563$	5.97896	0.251983	0.125992	−	0.992031i	$-0.459789\pi$
$563$	5.97896	0.251983	0.125992	+	0.992031i	$0.459789\pi$
$564$	−23.6749	−0.996892
$565$	1.63788	0.0689062
$566$	−35.3287	−1.48498
$567$	−0.411636	−0.0172871
$568$	47.0480	1.97409
$569$	29.8799	1.25263	0.626316	−	0.779569i	$-0.284562\pi$
$569$	29.8799	1.25263	0.626316	+	0.779569i	$0.284562\pi$
$570$	−2.39926	−0.100494
$571$	−15.4785	−0.647754	−0.323877	−	0.946099i	$-0.604986\pi$
$571$	−15.4785	−0.647754	−0.323877	+	0.946099i	$0.604986\pi$
$572$	−12.1643	−0.508617
$573$	−18.2101	−0.760740
$574$	1.88874	0.0788343
$575$	21.0507	0.877877
$576$	22.8640	0.952666
$577$	−4.14840	−0.172700	−0.0863501	−	0.996265i	$-0.527520\pi$
$577$	−4.14840	−0.172700	−0.0863501	+	0.996265i	$0.527520\pi$
$578$	−131.180	−5.45637
$579$	24.0989	1.00152
$580$	−34.7156	−1.44149
$581$	−3.59484	−0.149139
$582$	42.1396	1.74674
$583$	−0.333792	−0.0138243
$584$	−21.9876	−0.909854
$585$	2.04442	0.0845265
$586$	51.6835	2.13503
$587$	−23.2522	−0.959722	−0.479861	−	0.877344i	$-0.659313\pi$
$587$	−23.2522	−0.959722	−0.479861	+	0.877344i	$0.659313\pi$
$588$	−36.1199	−1.48956
$589$	1.98762	0.0818985
$590$	−11.4327	−0.470676
$591$	−16.5512	−0.680826
$592$	125.214	5.14625
$593$	−17.2225	−0.707244	−0.353622	−	0.935388i	$-0.615050\pi$
$593$	−17.2225	−0.707244	−0.353622	+	0.935388i	$0.615050\pi$
$594$	−2.69963	−0.110767
$595$	−2.96286	−0.121465
$596$	43.8923	1.79790
$597$	1.53528	0.0628348
$598$	−31.0507	−1.26976
$599$	−15.8268	−0.646667	−0.323334	−	0.946285i	$-0.604804\pi$
$599$	−15.8268	−0.646667	−0.323334	+	0.946285i	$0.604804\pi$
$600$	37.3708	1.52566
$601$	−37.5599	−1.53210	−0.766050	−	0.642781i	$-0.777780\pi$
$601$	−37.5599	−1.53210	−0.766050	+	0.642781i	$0.777780\pi$
$602$	−8.09888	−0.330086
$603$	10.0531	0.409393
$604$	113.263	4.60861
$605$	−0.888736	−0.0361323
$606$	−2.39926	−0.0974631
$607$	1.07413	0.0435974	0.0217987	−	0.999762i	$-0.493061\pi$
$607$	1.07413	0.0435974	0.0217987	+	0.999762i	$0.493061\pi$
$608$	18.3869	0.745686
$609$	−3.04070	−0.123216
$610$	−25.2174	−1.02102
$611$	10.2990	0.416653
$612$	−42.8268	−1.73117
$613$	19.4771	0.786673	0.393336	−	0.919395i	$-0.371321\pi$
$613$	19.4771	0.786673	0.393336	+	0.919395i	$0.371321\pi$
$614$	8.87636	0.358221
$615$	−1.51052	−0.0609101
$616$	3.65383	0.147217
$617$	−44.3845	−1.78685	−0.893427	−	0.449208i	$-0.851706\pi$
$617$	−44.3845	−1.78685	−0.893427	+	0.449208i	$0.851706\pi$
$618$	31.6377	1.27266
$619$	39.5636	1.59020	0.795098	−	0.606481i	$-0.207419\pi$
$619$	39.5636	1.59020	0.795098	+	0.606481i	$0.207419\pi$
$620$	9.34108	0.375147
$621$	−5.00000	−0.200643
$622$	−26.5388	−1.06411
$623$	−0.818176	−0.0327795
$624$	−30.7948	−1.23278
$625$	13.7761	0.551044
$626$	80.5882	3.22095
$627$	−1.00000	−0.0399362
$628$	−30.9629	−1.23555
$629$	−75.7526	−3.02045
$630$	−0.987620	−0.0393477
$631$	13.4203	0.534254	0.267127	−	0.963661i	$-0.413926\pi$
$631$	13.4203	0.534254	0.267127	+	0.963661i	$0.413926\pi$
$632$	142.680	5.67549
$633$	9.01966	0.358499
$634$	26.0851	1.03597
$635$	−8.38316	−0.332675
$636$	−1.76509	−0.0699904
$637$	15.7128	0.622565
$638$	−19.9418	−0.789504
$639$	−5.30037	−0.209679
$640$	22.1744	0.876520
$641$	−23.9381	−0.945498	−0.472749	−	0.881197i	$-0.656738\pi$
$641$	−23.9381	−0.945498	−0.472749	+	0.881197i	$0.656738\pi$
$642$	16.7083	0.659424
$643$	−11.5956	−0.457288	−0.228644	−	0.973510i	$-0.573429\pi$
$643$	−11.5956	−0.457288	−0.228644	+	0.973510i	$0.573429\pi$
$644$	10.8836	0.428875
$645$	6.47710	0.255036
$646$	−21.8640	−0.860227
$647$	−48.9998	−1.92638	−0.963191	−	0.268817i	$-0.913367\pi$
$647$	−48.9998	−1.92638	−0.963191	+	0.268817i	$0.913367\pi$
$648$	−8.87636	−0.348696
$649$	−4.76509	−0.187046
$650$	−26.1456	−1.02552
$651$	0.818176	0.0320668
$652$	98.4064	3.85389
$653$	−31.6996	−1.24050	−0.620251	−	0.784403i	$-0.712969\pi$
$653$	−31.6996	−1.24050	−0.620251	+	0.784403i	$0.712969\pi$
$654$	31.7614	1.24197
$655$	−8.21509	−0.320990
$656$	22.7527	0.888344
$657$	2.47710	0.0966409
$658$	−4.97524	−0.193955
$659$	8.70101	0.338943	0.169472	−	0.985535i	$-0.445794\pi$
$659$	8.70101	0.338943	0.169472	+	0.985535i	$0.445794\pi$
$660$	−4.69963	−0.182933
$661$	23.4647	0.912672	0.456336	−	0.889808i	$-0.349162\pi$
$661$	23.4647	0.912672	0.456336	+	0.889808i	$0.349162\pi$
$662$	0.901116	0.0350229
$663$	18.6304	0.723547
$664$	−77.5177	−3.00827
$665$	−0.365836	−0.0141865
$666$	−25.2509	−0.978451
$667$	−36.9344	−1.43011
$668$	−5.87636	−0.227363
$669$	11.3535	0.438950
$670$	24.1199	0.931834
$671$	−10.5105	−0.405754
$672$	7.56870	0.291969
$673$	−37.7861	−1.45655	−0.728274	−	0.685286i	$-0.759677\pi$
$673$	−37.7861	−1.45655	−0.728274	+	0.685286i	$0.759677\pi$
$674$	−61.6340	−2.37405
$675$	−4.21015	−0.162049
$676$	−40.7614	−1.56775
$677$	−41.2705	−1.58615	−0.793077	−	0.609121i	$-0.791522\pi$
$677$	−41.2705	−1.58615	−0.793077	+	0.609121i	$0.791522\pi$
$678$	4.97524	0.191073
$679$	6.42539	0.246584
$680$	−63.8900	−2.45007
$681$	−13.7651	−0.527479
$682$	5.36584	0.205468
$683$	36.3374	1.39041	0.695205	−	0.718811i	$-0.255314\pi$
$683$	36.3374	1.39041	0.695205	+	0.718811i	$0.255314\pi$
$684$	−5.28799	−0.202191
$685$	1.02832	0.0392903
$686$	−15.3694	−0.586806
$687$	−23.5512	−0.898535
$688$	−97.5635	−3.71957
$689$	0.767847	0.0292526
$690$	−11.9963	−0.456691
$691$	35.8777	1.36485	0.682427	−	0.730954i	$-0.260925\pi$
$691$	35.8777	1.36485	0.682427	+	0.730954i	$0.260925\pi$
$692$	−51.1148	−1.94309
$693$	−0.411636	−0.0156368
$694$	−71.5548	−2.71618
$695$	3.78119	0.143429
$696$	−65.5685	−2.48537
$697$	−13.7651	−0.521390
$698$	32.2522	1.22076
$699$	27.2632	1.03119
$700$	9.16435	0.346380
$701$	26.2632	0.991949	0.495974	−	0.868337i	$-0.334811\pi$
$701$	26.2632	0.991949	0.495974	+	0.868337i	$0.334811\pi$
$702$	6.21015	0.234387
$703$	−9.35346	−0.352772
$704$	22.8640	0.861719
$705$	3.97896	0.149856
$706$	−76.2137	−2.86834
$707$	−0.365836	−0.0137587
$708$	−25.1978	−0.946990
$709$	33.7824	1.26873	0.634363	−	0.773036i	$-0.281263\pi$
$709$	33.7824	1.26873	0.634363	+	0.773036i	$0.281263\pi$
$710$	−12.7170	−0.477259
$711$	−16.0741	−0.602827
$712$	−17.6428	−0.661193
$713$	9.93810	0.372185
$714$	−9.00000	−0.336817
$715$	2.04442	0.0764571
$716$	75.9664	2.83900
$717$	0.320035	0.0119519
$718$	−49.4138	−1.84411
$719$	−17.8777	−0.666727	−0.333363	−	0.942798i	$-0.608184\pi$
$719$	−17.8777	−0.666727	−0.333363	+	0.942798i	$0.608184\pi$
$720$	−11.8974	−0.443390
$721$	4.82408	0.179658
$722$	−2.69963	−0.100470
$723$	26.8392	0.998161
$724$	21.9171	0.814541
$725$	−31.0998	−1.15502
$726$	−2.69963	−0.100193
$727$	20.8502	0.773292	0.386646	−	0.922228i	$-0.373634\pi$
$727$	20.8502	0.773292	0.386646	+	0.922228i	$0.373634\pi$
$728$	−8.40516	−0.311516
$729$	1.00000	0.0370370
$730$	5.94320	0.219968
$731$	59.0246	2.18310
$732$	−55.5795	−2.05428
$733$	−39.2756	−1.45068	−0.725339	−	0.688392i	$-0.758317\pi$
$733$	−39.2756	−1.45068	−0.725339	+	0.688392i	$0.758317\pi$
$734$	−83.0406	−3.06508
$735$	6.07056	0.223916
$736$	91.9344	3.38875
$737$	10.0531	0.370310
$738$	−4.58836	−0.168900
$739$	−43.7366	−1.60888	−0.804439	−	0.594036i	$-0.797534\pi$
$739$	−43.7366	−1.60888	−0.804439	+	0.594036i	$0.797534\pi$
$740$	−43.9578	−1.61592
$741$	2.30037	0.0845063
$742$	−0.370932	−0.0136173
$743$	−11.2312	−0.412032	−0.206016	−	0.978549i	$-0.566050\pi$
$743$	−11.2312	−0.412032	−0.206016	+	0.978549i	$0.566050\pi$
$744$	17.6428	0.646817
$745$	−7.37684	−0.270266
$746$	5.99628	0.219539
$747$	8.73305	0.319526
$748$	−42.8268	−1.56590
$749$	2.54766	0.0930895
$750$	−22.0975	−0.806887
$751$	−11.2742	−0.411403	−0.205701	−	0.978615i	$-0.565948\pi$
$751$	−11.2742	−0.411403	−0.205701	+	0.978615i	$0.565948\pi$
$752$	−59.9344	−2.18558
$753$	27.5709	1.00474
$754$	45.8736	1.67062
$755$	−19.0358	−0.692782
$756$	−2.17673	−0.0791668
$757$	13.9257	0.506139	0.253069	−	0.967448i	$-0.418560\pi$
$757$	13.9257	0.506139	0.253069	+	0.967448i	$0.418560\pi$
$758$	21.7403	0.789644
$759$	−5.00000	−0.181489
$760$	−7.88874	−0.286155
$761$	20.0952	0.728449	0.364225	−	0.931311i	$-0.381334\pi$
$761$	20.0952	0.728449	0.364225	+	0.931311i	$0.381334\pi$
$762$	−25.4647	−0.922489
$763$	4.84294	0.175326
$764$	−96.2951	−3.48384
$765$	7.19777	0.260236
$766$	17.8095	0.643484
$767$	10.9615	0.395796
$768$	21.6291	0.780472
$769$	−39.9098	−1.43918	−0.719592	−	0.694397i	$-0.755671\pi$
$769$	−39.9098	−1.43918	−0.719592	+	0.694397i	$0.755671\pi$
$770$	−0.987620	−0.0355914
$771$	1.19049	0.0428743
$772$	127.435	4.58648
$773$	38.6464	1.39001	0.695007	−	0.719003i	$-0.255401\pi$
$773$	38.6464	1.39001	0.695007	+	0.719003i	$0.255401\pi$
$774$	19.6749	0.707199
$775$	8.36818	0.300594
$776$	138.555	4.97382
$777$	−3.85022	−0.138126
$778$	67.8145	2.43127
$779$	−1.69963	−0.0608955
$780$	10.8109	0.387092
$781$	−5.30037	−0.189662
$782$	−109.320	−3.90927
$783$	7.38688	0.263985
$784$	−91.4398	−3.26571
$785$	5.20383	0.185733
$786$	−24.9542	−0.890087
$787$	28.8988	1.03013	0.515065	−	0.857151i	$-0.327768\pi$
$787$	28.8988	1.03013	0.515065	+	0.857151i	$0.327768\pi$
$788$	−87.5227	−3.11787
$789$	−19.0334	−0.677608
$790$	−38.5659	−1.37211
$791$	0.758619	0.0269734
$792$	−8.87636	−0.315408
$793$	24.1781	0.858590
$794$	85.0566	3.01855
$795$	0.296653	0.0105212
$796$	8.11855	0.287754
$797$	8.62907	0.305657	0.152829	−	0.988253i	$-0.451162\pi$
$797$	8.62907	0.305657	0.152829	+	0.988253i	$0.451162\pi$
$798$	−1.11126	−0.0393383
$799$	36.2595	1.28277
$800$	77.4115	2.73691
$801$	1.98762	0.0702291
$802$	92.2310	3.25679
$803$	2.47710	0.0874150
$804$	53.1606	1.87483
$805$	−1.82918	−0.0644701
$806$	−12.3434	−0.434778
$807$	11.7069	0.412103
$808$	−7.88874	−0.277525
$809$	−21.1162	−0.742406	−0.371203	−	0.928552i	$-0.621055\pi$
$809$	−21.1162	−0.742406	−0.371203	+	0.928552i	$0.621055\pi$
$810$	2.39926	0.0843013
$811$	−0.634164	−0.0222685	−0.0111343	−	0.999938i	$-0.503544\pi$
$811$	−0.634164	−0.0222685	−0.0111343	+	0.999938i	$0.503544\pi$
$812$	−16.0792	−0.564270
$813$	−15.4523	−0.541937
$814$	−25.2509	−0.885042
$815$	−16.5388	−0.579330
$816$	−108.419	−3.79542
$817$	7.28799	0.254975
$818$	−1.73539	−0.0606765
$819$	0.946916	0.0330879
$820$	−7.98762	−0.278940
$821$	5.01238	0.174933	0.0874666	−	0.996167i	$-0.472123\pi$
$821$	5.01238	0.174933	0.0874666	+	0.996167i	$0.472123\pi$
$822$	3.12364	0.108950
$823$	−5.65383	−0.197080	−0.0985400	−	0.995133i	$-0.531417\pi$
$823$	−5.65383	−0.197080	−0.0985400	+	0.995133i	$0.531417\pi$
$824$	104.025	3.62387
$825$	−4.21015	−0.146579
$826$	−5.29528	−0.184246
$827$	−10.8726	−0.378079	−0.189039	−	0.981970i	$-0.560537\pi$
$827$	−10.8726	−0.378079	−0.189039	+	0.981970i	$0.560537\pi$
$828$	−26.4400	−0.918852
$829$	8.45234	0.293562	0.146781	−	0.989169i	$-0.453109\pi$
$829$	8.45234	0.293562	0.146781	+	0.989169i	$0.453109\pi$
$830$	20.9528	0.727283
$831$	−20.9876	−0.728052
$832$	−52.5956	−1.82343
$833$	55.3199	1.91672
$834$	11.4858	0.397719
$835$	0.987620	0.0341780
$836$	−5.28799	−0.182889
$837$	−1.98762	−0.0687022
$838$	−56.2878	−1.94443
$839$	−29.6800	−1.02467	−0.512333	−	0.858787i	$-0.671219\pi$
$839$	−29.6800	−1.02467	−0.512333	+	0.858787i	$0.671219\pi$
$840$	−3.24729	−0.112042
$841$	25.5659	0.881584
$842$	27.9208	0.962214
$843$	9.68725	0.333647
$844$	47.6959	1.64176
$845$	6.85063	0.235669
$846$	12.0865	0.415543
$847$	−0.411636	−0.0141440
$848$	−4.46844	−0.153447
$849$	13.0865	0.449128
$850$	−92.0506	−3.15731
$851$	−46.7673	−1.60316
$852$	−28.0283	−0.960235
$853$	39.3584	1.34761	0.673803	−	0.738911i	$-0.264660\pi$
$853$	39.3584	1.34761	0.673803	+	0.738911i	$0.264660\pi$
$854$	−11.6800	−0.399680
$855$	0.888736	0.0303941
$856$	54.9367	1.87770
$857$	−34.4079	−1.17535	−0.587676	−	0.809096i	$-0.699957\pi$
$857$	−34.4079	−1.17535	−0.587676	+	0.809096i	$0.699957\pi$
$858$	6.21015	0.212011
$859$	−21.7293	−0.741395	−0.370698	−	0.928754i	$-0.620881\pi$
$859$	−21.7293	−0.741395	−0.370698	+	0.928754i	$0.620881\pi$
$860$	34.2509	1.16794
$861$	−0.699628	−0.0238433
$862$	37.6908	1.28375
$863$	−4.26461	−0.145169	−0.0725845	−	0.997362i	$-0.523125\pi$
$863$	−4.26461	−0.145169	−0.0725845	+	0.997362i	$0.523125\pi$
$864$	−18.3869	−0.625534
$865$	8.59071	0.292093
$866$	−14.9332	−0.507449
$867$	48.5919	1.65027
$868$	4.32651	0.146851
$869$	−16.0741	−0.545277
$870$	17.7230	0.600866
$871$	−23.1258	−0.783589
$872$	104.431	3.53648
$873$	−15.6094	−0.528298
$874$	−13.4981	−0.456581
$875$	−3.36940	−0.113907
$876$	13.0989	0.442570
$877$	−23.6908	−0.799982	−0.399991	−	0.916519i	$-0.630987\pi$
$877$	−23.6908	−0.799982	−0.399991	+	0.916519i	$0.630987\pi$
$878$	54.6698	1.84502
$879$	−19.1447	−0.645734
$880$	−11.8974	−0.401061
$881$	−10.4276	−0.351314	−0.175657	−	0.984451i	$-0.556205\pi$
$881$	−10.4276	−0.351314	−0.175657	+	0.984451i	$0.556205\pi$
$882$	18.4400	0.620906
$883$	−16.2349	−0.546348	−0.273174	−	0.961965i	$-0.588074\pi$
$883$	−16.2349	−0.546348	−0.273174	+	0.961965i	$0.588074\pi$
$884$	98.5177	3.31351
$885$	4.23491	0.142355
$886$	82.6303	2.77602
$887$	31.1991	1.04756	0.523782	−	0.851852i	$-0.324520\pi$
$887$	31.1991	1.04756	0.523782	+	0.851852i	$0.324520\pi$
$888$	−83.0246	−2.78612
$889$	−3.88283	−0.130226
$890$	4.76881	0.159851
$891$	1.00000	0.0335013
$892$	60.0370	2.01019
$893$	4.47710	0.149820
$894$	−22.4079	−0.749433
$895$	−12.7674	−0.426768
$896$	10.2705	0.343114
$897$	11.5019	0.384036
$898$	10.1410	0.338408
$899$	−14.6823	−0.489682
$900$	−22.2632	−0.742108
$901$	2.70335	0.0900615
$902$	−4.58836	−0.152776
$903$	3.00000	0.0998337
$904$	16.3586	0.544077
$905$	−3.68353	−0.122445
$906$	−57.8231	−1.92104
$907$	9.56498	0.317600	0.158800	−	0.987311i	$-0.449237\pi$
$907$	9.56498	0.317600	0.158800	+	0.987311i	$0.449237\pi$
$908$	−72.7897	−2.41561
$909$	0.888736	0.0294775
$910$	2.27189	0.0753125
$911$	32.8589	1.08866	0.544332	−	0.838870i	$-0.316783\pi$
$911$	32.8589	1.08866	0.544332	+	0.838870i	$0.316783\pi$
$912$	−13.3869	−0.443284
$913$	8.73305	0.289022
$914$	6.50680	0.215226
$915$	9.34108	0.308806
$916$	−124.539	−4.11487
$917$	−3.80499	−0.125652
$918$	21.8640	0.721619
$919$	52.0755	1.71781	0.858906	−	0.512133i	$-0.171145\pi$
$919$	52.0755	1.71781	0.858906	+	0.512133i	$0.171145\pi$
$920$	−39.4437	−1.30042
$921$	−3.28799	−0.108343
$922$	−11.7665	−0.387508
$923$	12.1928	0.401332
$924$	−2.17673	−0.0716091
$925$	−39.3794	−1.29479
$926$	−104.251	−3.42589
$927$	−11.7193	−0.384912
$928$	−135.822	−4.45856
$929$	−6.73305	−0.220904	−0.110452	−	0.993881i	$-0.535230\pi$
$929$	−6.73305	−0.220904	−0.110452	+	0.993881i	$0.535230\pi$
$930$	−4.76881	−0.156375
$931$	6.83056	0.223862
$932$	144.168	4.72237
$933$	9.83056	0.321838
$934$	43.6712	1.42896
$935$	7.19777	0.235392
$936$	20.4189	0.667413
$937$	−13.1557	−0.429778	−0.214889	−	0.976639i	$-0.568939\pi$
$937$	−13.1557	−0.429778	−0.214889	+	0.976639i	$0.568939\pi$
$938$	11.1716	0.364767
$939$	−29.8516	−0.974170
$940$	21.0407	0.686272
$941$	−18.5105	−0.603426	−0.301713	−	0.953399i	$-0.597558\pi$
$941$	−18.5105	−0.603426	−0.301713	+	0.953399i	$0.597558\pi$
$942$	15.8072	0.515026
$943$	−8.49814	−0.276738
$944$	−63.7897	−2.07618
$945$	0.365836	0.0119006
$946$	19.6749	0.639685
$947$	2.67349	0.0868768	0.0434384	−	0.999056i	$-0.486169\pi$
$947$	2.67349	0.0868768	0.0434384	+	0.999056i	$0.486169\pi$
$948$	−84.9998	−2.76067
$949$	−5.69825	−0.184973
$950$	−11.3658	−0.368756
$951$	−9.66249	−0.313328
$952$	−29.5919	−0.959080
$953$	45.3336	1.46850	0.734250	−	0.678879i	$-0.237534\pi$
$953$	45.3336	1.46850	0.734250	+	0.678879i	$0.237534\pi$
$954$	0.901116	0.0291747
$955$	16.1840	0.523702
$956$	1.69234	0.0547343
$957$	7.38688	0.238784
$958$	78.2420	2.52789
$959$	0.476289	0.0153802
$960$	−20.3200	−0.655826
$961$	−27.0494	−0.872560
$962$	58.0864	1.87278
$963$	−6.18911	−0.199441
$964$	141.926	4.57111
$965$	−21.4175	−0.689455
$966$	−5.55632	−0.178772
$967$	−49.7797	−1.60081	−0.800403	−	0.599462i	$-0.795381\pi$
$967$	−49.7797	−1.60081	−0.800403	+	0.599462i	$0.795381\pi$
$968$	−8.87636	−0.285297
$969$	8.09888	0.260174
$970$	−37.4510	−1.20248
$971$	−5.46982	−0.175535	−0.0877674	−	0.996141i	$-0.527973\pi$
$971$	−5.46982	−0.175535	−0.0877674	+	0.996141i	$0.527973\pi$
$972$	5.28799	0.169612
$973$	1.75133	0.0561452
$974$	−57.4129	−1.83963
$975$	9.68491	0.310165
$976$	−140.703	−4.50379
$977$	33.5933	1.07475	0.537373	−	0.843345i	$-0.319417\pi$
$977$	33.5933	1.07475	0.537373	+	0.843345i	$0.319417\pi$
$978$	−50.2385	−1.60645
$979$	1.98762	0.0635246
$980$	32.1011	1.02543
$981$	−11.7651	−0.375630
$982$	9.59703	0.306253
$983$	49.0122	1.56325	0.781624	−	0.623750i	$-0.214392\pi$
$983$	49.0122	1.56325	0.781624	+	0.623750i	$0.214392\pi$
$984$	−15.0865	−0.480940
$985$	14.7097	0.468689
$986$	161.506	5.14341
$987$	1.84294	0.0586613
$988$	12.1643	0.386999
$989$	36.4400	1.15872
$990$	2.39926	0.0762534
$991$	27.0159	0.858190	0.429095	−	0.903259i	$-0.358833\pi$
$991$	27.0159	0.858190	0.429095	+	0.903259i	$0.358833\pi$
$992$	36.5461	1.16034
$993$	−0.333792	−0.0105926
$994$	−5.89011	−0.186823
$995$	−1.36446	−0.0432562
$996$	46.1803	1.46328
$997$	−16.4981	−0.522501	−0.261251	−	0.965271i	$-0.584135\pi$
$997$	−16.4981	−0.522501	−0.261251	+	0.965271i	$0.584135\pi$
$998$	62.3484	1.97360
$999$	9.35346	0.295930

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	627.2.a.e.1.1	✓	3
3.2	odd	2		1881.2.a.i.1.3		3
11.10	odd	2		6897.2.a.p.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
627.2.a.e.1.1	✓	3	1.1	even	1	trivial
1881.2.a.i.1.3		3	3.2	odd	2
6897.2.a.p.1.3		3	11.10	odd	2

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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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	627.2.a.e.1.1

Level	$627$
Weight	$2$
Character	627.1
Self dual	yes
Analytic conductor	$5.007$
Analytic rank	$1$
Dimension	$3$
CM	no
Inner twists	$1$