LMFDB - Embedded newform 209.2.a.d.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [209,2,Mod(1,209)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(209, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("209.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$209 = 11 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	209.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:	$1.66887340224$
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:	$x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30$ x^7 - x^6 - 14x^5 + 10x^4 + 59x^3 - 27x^2 - 66*x + 30
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.03821$ of defining polynomial
Character	$\chi$	$=$	209.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.03821 q^{2} +1.87275 q^{3} +2.15429 q^{4} -3.24760 q^{5} +3.81704 q^{6} +1.92338 q^{7} +0.314472 q^{8} +0.507178 q^{9} -6.61928 q^{10} -1.00000 q^{11} +4.03444 q^{12} +2.85122 q^{13} +3.92024 q^{14} -6.08193 q^{15} -3.66762 q^{16} -2.33033 q^{17} +1.03373 q^{18} +1.00000 q^{19} -6.99626 q^{20} +3.60199 q^{21} -2.03821 q^{22} -2.74653 q^{23} +0.588926 q^{24} +5.54689 q^{25} +5.81138 q^{26} -4.66842 q^{27} +4.14350 q^{28} -0.972965 q^{29} -12.3962 q^{30} -0.00551178 q^{31} -8.10431 q^{32} -1.87275 q^{33} -4.74970 q^{34} -6.24635 q^{35} +1.09261 q^{36} +9.67124 q^{37} +2.03821 q^{38} +5.33962 q^{39} -1.02128 q^{40} +6.65137 q^{41} +7.34161 q^{42} +7.99413 q^{43} -2.15429 q^{44} -1.64711 q^{45} -5.59800 q^{46} +3.46982 q^{47} -6.86852 q^{48} -3.30063 q^{49} +11.3057 q^{50} -4.36412 q^{51} +6.14236 q^{52} +10.5493 q^{53} -9.51521 q^{54} +3.24760 q^{55} +0.604847 q^{56} +1.87275 q^{57} -1.98311 q^{58} -13.7814 q^{59} -13.1022 q^{60} +3.74608 q^{61} -0.0112342 q^{62} +0.975494 q^{63} -9.18303 q^{64} -9.25963 q^{65} -3.81704 q^{66} -3.97172 q^{67} -5.02021 q^{68} -5.14356 q^{69} -12.7314 q^{70} +14.2688 q^{71} +0.159493 q^{72} -13.2263 q^{73} +19.7120 q^{74} +10.3879 q^{75} +2.15429 q^{76} -1.92338 q^{77} +10.8832 q^{78} -1.87656 q^{79} +11.9109 q^{80} -10.2643 q^{81} +13.5569 q^{82} -10.9619 q^{83} +7.75973 q^{84} +7.56799 q^{85} +16.2937 q^{86} -1.82212 q^{87} -0.314472 q^{88} +15.0195 q^{89} -3.35715 q^{90} +5.48397 q^{91} -5.91682 q^{92} -0.0103222 q^{93} +7.07220 q^{94} -3.24760 q^{95} -15.1773 q^{96} -7.57248 q^{97} -6.72736 q^{98} -0.507178 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$7 q - q^{2} + 2 q^{3} + 15 q^{4} + 2 q^{5} - 2 q^{6} + 10 q^{7} - 9 q^{8} + 11 q^{9} - 6 q^{10} - 7 q^{11} - 16 q^{12} - 4 q^{13} + 6 q^{14} + 12 q^{15} + 27 q^{16} + 2 q^{17} + 9 q^{18} + 7 q^{19} - 4 q^{20}+ \cdots - 11 q^{99}+O(q^{100})$ 7 * q - q^2 + 2 * q^3 + 15 * q^4 + 2 * q^5 - 2 * q^6 + 10 * q^7 - 9 * q^8 + 11 * q^9 - 6 * q^10 - 7 * q^11 - 16 * q^12 - 4 * q^13 + 6 * q^14 + 12 * q^15 + 27 * q^16 + 2 * q^17 + 9 * q^18 + 7 * q^19 - 4 * q^20 - 14 * q^21 + q^22 + 10 * q^23 - 2 * q^24 + 9 * q^25 - 8 * q^26 - 4 * q^27 + 26 * q^28 - 18 * q^29 - 42 * q^30 + 24 * q^31 - 49 * q^32 - 2 * q^33 - 6 * q^34 + 8 * q^35 + 29 * q^36 - q^38 + 24 * q^39 - 2 * q^40 - 12 * q^41 - 44 * q^42 + 2 * q^43 - 15 * q^44 - 4 * q^45 - 4 * q^46 + 8 * q^47 - 72 * q^48 + 17 * q^49 - 33 * q^50 - 24 * q^51 - 60 * q^52 + 2 * q^53 - 52 * q^54 - 2 * q^55 + 26 * q^56 + 2 * q^57 - 8 * q^58 - 10 * q^59 + 42 * q^60 + 14 * q^61 + 14 * q^62 + 55 * q^64 - 14 * q^65 + 2 * q^66 + 8 * q^67 - 18 * q^68 - 6 * q^69 - 66 * q^70 + 10 * q^71 + 53 * q^72 - 6 * q^73 + 26 * q^74 + 26 * q^75 + 15 * q^76 - 10 * q^77 + 22 * q^78 + 52 * q^79 - 12 * q^80 - q^81 + 24 * q^82 - 10 * q^83 - 6 * q^84 - 12 * q^85 + 8 * q^86 + 6 * q^87 + 9 * q^88 + 20 * q^90 + 12 * q^91 + 2 * q^93 + 24 * q^94 + 2 * q^95 + 6 * q^96 - 24 * q^97 + 19 * q^98 - 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

)

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

$2$	2.03821	1.44123	0.720615	−	0.693335i	$-0.243860\pi$
$2$	2.03821	1.44123	0.720615	+	0.693335i	$0.243860\pi$
$3$	1.87275	1.08123	0.540615	−	0.841270i	$-0.318191\pi$
$3$	1.87275	1.08123	0.540615	+	0.841270i	$0.318191\pi$
$4$	2.15429	1.07714
$5$	−3.24760	−1.45237	−0.726185	−	0.687499i	$-0.758708\pi$
$5$	−3.24760	−1.45237	−0.726185	+	0.687499i	$0.758708\pi$
$6$	3.81704	1.55830
$7$	1.92338	0.726967	0.363484	−	0.931601i	$-0.381587\pi$
$7$	1.92338	0.726967	0.363484	+	0.931601i	$0.381587\pi$
$8$	0.314472	0.111183
$9$	0.507178	0.169059
$10$	−6.61928	−2.09320
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	4.03444	1.16464
$13$	2.85122	0.790787	0.395394	−	0.918512i	$-0.370608\pi$
$13$	2.85122	0.790787	0.395394	+	0.918512i	$0.370608\pi$
$14$	3.92024	1.04773
$15$	−6.08193	−1.57035
$16$	−3.66762	−0.916905
$17$	−2.33033	−0.565189	−0.282594	−	0.959239i	$-0.591195\pi$
$17$	−2.33033	−0.565189	−0.282594	+	0.959239i	$0.591195\pi$
$18$	1.03373	0.243653
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−6.99626	−1.56441
$21$	3.60199	0.786019
$22$	−2.03821	−0.434547
$23$	−2.74653	−0.572691	−0.286346	−	0.958126i	$-0.592441\pi$
$23$	−2.74653	−0.572691	−0.286346	+	0.958126i	$0.592441\pi$
$24$	0.588926	0.120214
$25$	5.54689	1.10938
$26$	5.81138	1.13971
$27$	−4.66842	−0.898438
$28$	4.14350	0.783049
$29$	−0.972965	−0.180675	−0.0903376	−	0.995911i	$-0.528795\pi$
$29$	−0.972965	−0.180675	−0.0903376	+	0.995911i	$0.528795\pi$
$30$	−12.3962	−2.26323
$31$	−0.00551178	−0.000989945 0	−0.000494973	−	1.00000i	$-0.500158\pi$
$31$	−0.00551178	−0.000989945 0	−0.000494973		1.00000i	$0.500158\pi$
$32$	−8.10431	−1.43265
$33$	−1.87275	−0.326003
$34$	−4.74970	−0.814567
$35$	−6.24635	−1.05583
$36$	1.09261	0.182101
$37$	9.67124	1.58994	0.794971	−	0.606647i	$-0.207486\pi$
$37$	9.67124	1.58994	0.794971	+	0.606647i	$0.207486\pi$
$38$	2.03821	0.330641
$39$	5.33962	0.855023
$40$	−1.02128	−0.161478
$41$	6.65137	1.03877	0.519385	−	0.854540i	$-0.326161\pi$
$41$	6.65137	1.03877	0.519385	+	0.854540i	$0.326161\pi$
$42$	7.34161	1.13283
$43$	7.99413	1.21909	0.609547	−	0.792750i	$-0.291352\pi$
$43$	7.99413	1.21909	0.609547	+	0.792750i	$0.291352\pi$
$44$	−2.15429	−0.324771
$45$	−1.64711	−0.245537
$46$	−5.59800	−0.825380
$47$	3.46982	0.506125	0.253062	−	0.967450i	$-0.418562\pi$
$47$	3.46982	0.506125	0.253062	+	0.967450i	$0.418562\pi$
$48$	−6.86852	−0.991385
$49$	−3.30063	−0.471518
$50$	11.3057	1.59887
$51$	−4.36412	−0.611100
$52$	6.14236	0.851792
$53$	10.5493	1.44905	0.724526	−	0.689247i	$-0.242058\pi$
$53$	10.5493	1.44905	0.724526	+	0.689247i	$0.242058\pi$
$54$	−9.51521	−1.29486
$55$	3.24760	0.437906
$56$	0.604847	0.0808261
$57$	1.87275	0.248051
$58$	−1.98311	−0.260394
$59$	−13.7814	−1.79419	−0.897096	−	0.441836i	$-0.854327\pi$
$59$	−13.7814	−1.79419	−0.897096	+	0.441836i	$0.854327\pi$
$60$	−13.1022	−1.69149
$61$	3.74608	0.479636	0.239818	−	0.970818i	$-0.422912\pi$
$61$	3.74608	0.479636	0.239818	+	0.970818i	$0.422912\pi$
$62$	−0.0112342	−0.00142674
$63$	0.975494	0.122901
$64$	−9.18303	−1.14788
$65$	−9.25963	−1.14852
$66$	−3.81704	−0.469846
$67$	−3.97172	−0.485223	−0.242612	−	0.970124i	$-0.578004\pi$
$67$	−3.97172	−0.485223	−0.242612	+	0.970124i	$0.578004\pi$
$68$	−5.02021	−0.608790
$69$	−5.14356	−0.619211
$70$	−12.7314	−1.52169
$71$	14.2688	1.69339	0.846695	−	0.532078i	$-0.178589\pi$
$71$	14.2688	1.69339	0.846695	+	0.532078i	$0.178589\pi$
$72$	0.159493	0.0187965
$73$	−13.2263	−1.54803	−0.774013	−	0.633170i	$-0.781753\pi$
$73$	−13.2263	−1.54803	−0.774013	+	0.633170i	$0.781753\pi$
$74$	19.7120	2.29147
$75$	10.3879	1.19949
$76$	2.15429	0.247114
$77$	−1.92338	−0.219189
$78$	10.8832	1.23229
$79$	−1.87656	−0.211130	−0.105565	−	0.994412i	$-0.533665\pi$
$79$	−1.87656	−0.211130	−0.105565	+	0.994412i	$0.533665\pi$
$80$	11.9109	1.33168
$81$	−10.2643	−1.14048
$82$	13.5569	1.49711
$83$	−10.9619	−1.20322	−0.601612	−	0.798789i	$-0.705474\pi$
$83$	−10.9619	−1.20322	−0.601612	+	0.798789i	$0.705474\pi$
$84$	7.75973	0.846656
$85$	7.56799	0.820863
$86$	16.2937	1.75699
$87$	−1.82212	−0.195351
$88$	−0.314472	−0.0335228
$89$	15.0195	1.59207	0.796034	−	0.605253i	$-0.206928\pi$
$89$	15.0195	1.59207	0.796034	+	0.605253i	$0.206928\pi$
$90$	−3.35715	−0.353875
$91$	5.48397	0.574876
$92$	−5.91682	−0.616871
$93$	−0.0103222	−0.00107036
$94$	7.07220	0.729442
$95$	−3.24760	−0.333196
$96$	−15.1773	−1.54903
$97$	−7.57248	−0.768869	−0.384434	−	0.923152i	$-0.625604\pi$
$97$	−7.57248	−0.768869	−0.384434	+	0.923152i	$0.625604\pi$
$98$	−6.72736	−0.679566
$99$	−0.507178	−0.0509733
$100$	11.9496	1.19496
$101$	−15.0513	−1.49766	−0.748831	−	0.662761i	$-0.769385\pi$
$101$	−15.0513	−1.49766	−0.748831	+	0.662761i	$0.769385\pi$
$102$	−8.89499	−0.880735
$103$	−0.543451	−0.0535478	−0.0267739	−	0.999642i	$-0.508523\pi$
$103$	−0.543451	−0.0535478	−0.0267739	+	0.999642i	$0.508523\pi$
$104$	0.896629	0.0879218
$105$	−11.6978	−1.14159
$106$	21.5016	2.08842
$107$	14.7371	1.42469	0.712344	−	0.701831i	$-0.247634\pi$
$107$	14.7371	1.42469	0.712344	+	0.701831i	$0.247634\pi$
$108$	−10.0571	−0.967748
$109$	−17.3711	−1.66385	−0.831925	−	0.554888i	$-0.812761\pi$
$109$	−17.3711	−1.66385	−0.831925	+	0.554888i	$0.812761\pi$
$110$	6.61928	0.631123
$111$	18.1118	1.71909
$112$	−7.05421	−0.666560
$113$	12.8865	1.21226	0.606132	−	0.795364i	$-0.292720\pi$
$113$	12.8865	1.21226	0.606132	+	0.795364i	$0.292720\pi$
$114$	3.81704	0.357499
$115$	8.91963	0.831760
$116$	−2.09605	−0.194613
$117$	1.44608	0.133690
$118$	−28.0894	−2.58584
$119$	−4.48211	−0.410874
$120$	−1.91259	−0.174595
$121$	1.00000	0.0909091
$122$	7.63529	0.691266
$123$	12.4563	1.12315
$124$	−0.0118740	−0.00106631
$125$	−1.77608	−0.158858
$126$	1.98826	0.177128
$127$	15.4342	1.36957	0.684784	−	0.728746i	$-0.259897\pi$
$127$	15.4342	1.36957	0.684784	+	0.728746i	$0.259897\pi$
$128$	−2.50829	−0.221704
$129$	14.9710	1.31812
$130$	−18.8730	−1.65527
$131$	−12.0655	−1.05417	−0.527083	−	0.849814i	$-0.676714\pi$
$131$	−12.0655	−1.05417	−0.527083	+	0.849814i	$0.676714\pi$
$132$	−4.03444	−0.351153
$133$	1.92338	0.166778
$134$	−8.09519	−0.699318
$135$	15.1612	1.30486
$136$	−0.732824	−0.0628392
$137$	−5.53253	−0.472676	−0.236338	−	0.971671i	$-0.575947\pi$
$137$	−5.53253	−0.472676	−0.236338	+	0.971671i	$0.575947\pi$
$138$	−10.4836	−0.892426
$139$	−8.66764	−0.735180	−0.367590	−	0.929988i	$-0.619817\pi$
$139$	−8.66764	−0.735180	−0.367590	+	0.929988i	$0.619817\pi$
$140$	−13.4564	−1.13728
$141$	6.49808	0.547237
$142$	29.0827	2.44057
$143$	−2.85122	−0.238431
$144$	−1.86014	−0.155011
$145$	3.15980	0.262407
$146$	−26.9580	−2.23106
$147$	−6.18124	−0.509820
$148$	20.8346	1.71260
$149$	−19.3027	−1.58134	−0.790671	−	0.612241i	$-0.790268\pi$
$149$	−19.3027	−1.58134	−0.790671	+	0.612241i	$0.790268\pi$
$150$	21.1727	1.72875
$151$	8.71384	0.709122	0.354561	−	0.935033i	$-0.384630\pi$
$151$	8.71384	0.709122	0.354561	+	0.935033i	$0.384630\pi$
$152$	0.314472	0.0255070
$153$	−1.18189	−0.0955505
$154$	−3.92024	−0.315902
$155$	0.0179000	0.00143777
$156$	11.5031	0.920983
$157$	−5.86640	−0.468189	−0.234095	−	0.972214i	$-0.575213\pi$
$157$	−5.86640	−0.468189	−0.234095	+	0.972214i	$0.575213\pi$
$158$	−3.82483	−0.304287
$159$	19.7561	1.56676
$160$	26.3195	2.08074
$161$	−5.28261	−0.416328
$162$	−20.9208	−1.64369
$163$	14.8802	1.16551	0.582753	−	0.812649i	$-0.301975\pi$
$163$	14.8802	1.16551	0.582753	+	0.812649i	$0.301975\pi$
$164$	14.3290	1.11891
$165$	6.08193	0.473477
$166$	−22.3426	−1.73412
$167$	4.18971	0.324209	0.162105	−	0.986774i	$-0.448172\pi$
$167$	4.18971	0.324209	0.162105	+	0.986774i	$0.448172\pi$
$168$	1.13273	0.0873917
$169$	−4.87053	−0.374656
$170$	15.4251	1.18305
$171$	0.507178	0.0387849
$172$	17.2217	1.31314
$173$	0.707136	0.0537626	0.0268813	−	0.999639i	$-0.491442\pi$
$173$	0.707136	0.0537626	0.0268813	+	0.999639i	$0.491442\pi$
$174$	−3.71385	−0.281546
$175$	10.6688	0.806482
$176$	3.66762	0.276457
$177$	−25.8091	−1.93993
$178$	30.6129	2.29454
$179$	−21.7962	−1.62913	−0.814563	−	0.580076i	$-0.803023\pi$
$179$	−21.7962	−1.62913	−0.814563	+	0.580076i	$0.803023\pi$
$180$	−3.54835	−0.264478
$181$	−2.93416	−0.218094	−0.109047	−	0.994037i	$-0.534780\pi$
$181$	−2.93416	−0.218094	−0.109047	+	0.994037i	$0.534780\pi$
$182$	11.1775	0.828529
$183$	7.01546	0.518598
$184$	−0.863707	−0.0636733
$185$	−31.4083	−2.30918
$186$	−0.0210387	−0.00154263
$187$	2.33033	0.170411
$188$	7.47498	0.545169
$189$	−8.97913	−0.653135
$190$	−6.61928	−0.480213
$191$	22.4018	1.62094	0.810468	−	0.585783i	$-0.199213\pi$
$191$	22.4018	1.62094	0.810468	+	0.585783i	$0.199213\pi$
$192$	−17.1975	−1.24112
$193$	2.55447	0.183875	0.0919375	−	0.995765i	$-0.470694\pi$
$193$	2.55447	0.183875	0.0919375	+	0.995765i	$0.470694\pi$
$194$	−15.4343	−1.10812
$195$	−17.3409	−1.24181
$196$	−7.11051	−0.507893
$197$	−12.6968	−0.904607	−0.452303	−	0.891864i	$-0.649398\pi$
$197$	−12.6968	−0.904607	−0.452303	+	0.891864i	$0.649398\pi$
$198$	−1.03373	−0.0734643
$199$	10.1553	0.719892	0.359946	−	0.932973i	$-0.382795\pi$
$199$	10.1553	0.719892	0.359946	+	0.932973i	$0.382795\pi$
$200$	1.74434	0.123344
$201$	−7.43803	−0.524638
$202$	−30.6777	−2.15848
$203$	−1.87138	−0.131345
$204$	−9.40158	−0.658242
$205$	−21.6010	−1.50868
$206$	−1.10767	−0.0771747
$207$	−1.39298	−0.0968188
$208$	−10.4572	−0.725076
$209$	−1.00000	−0.0691714
$210$	−23.8426	−1.64530
$211$	−8.25858	−0.568544	−0.284272	−	0.958744i	$-0.591752\pi$
$211$	−8.25858	−0.568544	−0.284272	+	0.958744i	$0.591752\pi$
$212$	22.7262	1.56084
$213$	26.7218	1.83095
$214$	30.0372	2.05330
$215$	−25.9617	−1.77057
$216$	−1.46809	−0.0998907
$217$	−0.0106012	−0.000719658 0
$218$	−35.4059	−2.39799
$219$	−24.7696	−1.67377
$220$	6.99626	0.471688
$221$	−6.64430	−0.446944
$222$	36.9156	2.47761
$223$	−21.2289	−1.42159	−0.710797	−	0.703398i	$-0.751665\pi$
$223$	−21.2289	−1.42159	−0.710797	+	0.703398i	$0.751665\pi$
$224$	−15.5876	−1.04149
$225$	2.81326	0.187551
$226$	26.2655	1.74715
$227$	−9.06652	−0.601766	−0.300883	−	0.953661i	$-0.597281\pi$
$227$	−9.06652	−0.601766	−0.300883	+	0.953661i	$0.597281\pi$
$228$	4.03444	0.267187
$229$	−5.25556	−0.347297	−0.173649	−	0.984808i	$-0.555556\pi$
$229$	−5.25556	−0.347297	−0.173649	+	0.984808i	$0.555556\pi$
$230$	18.1800	1.19876
$231$	−3.60199	−0.236994
$232$	−0.305970	−0.0200879
$233$	−5.68870	−0.372679	−0.186340	−	0.982485i	$-0.559662\pi$
$233$	−5.68870	−0.372679	−0.186340	+	0.982485i	$0.559662\pi$
$234$	2.94741	0.192678
$235$	−11.2686	−0.735080
$236$	−29.6892	−1.93260
$237$	−3.51433	−0.228280
$238$	−9.13546	−0.592164
$239$	−20.3787	−1.31819	−0.659094	−	0.752060i	$-0.729060\pi$
$239$	−20.3787	−1.31819	−0.659094	+	0.752060i	$0.729060\pi$
$240$	22.3062	1.43986
$241$	17.6930	1.13971	0.569854	−	0.821746i	$-0.307000\pi$
$241$	17.6930	1.13971	0.569854	+	0.821746i	$0.307000\pi$
$242$	2.03821	0.131021
$243$	−5.21717	−0.334682
$244$	8.07014	0.516638
$245$	10.7191	0.684819
$246$	25.3886	1.61872
$247$	2.85122	0.181419
$248$	−0.00173330	−0.000110065 0
$249$	−20.5288	−1.30096
$250$	−3.62002	−0.228950
$251$	−0.776543	−0.0490149	−0.0245075	−	0.999700i	$-0.507802\pi$
$251$	−0.776543	−0.0490149	−0.0245075	+	0.999700i	$0.507802\pi$
$252$	2.10149	0.132382
$253$	2.74653	0.172673
$254$	31.4582	1.97386
$255$	14.1729	0.887543
$256$	13.2536	0.828352
$257$	29.2762	1.82620	0.913100	−	0.407736i	$-0.133682\pi$
$257$	29.2762	1.82620	0.913100	+	0.407736i	$0.133682\pi$
$258$	30.5139	1.89972
$259$	18.6014	1.15584
$260$	−19.9479	−1.23712
$261$	−0.493467	−0.0305448
$262$	−24.5919	−1.51929
$263$	−5.90041	−0.363835	−0.181918	−	0.983314i	$-0.558230\pi$
$263$	−5.90041	−0.363835	−0.181918	+	0.983314i	$0.558230\pi$
$264$	−0.588926	−0.0362459
$265$	−34.2598	−2.10456
$266$	3.92024	0.240365
$267$	28.1278	1.72139
$268$	−8.55623	−0.522655
$269$	−10.7278	−0.654087	−0.327044	−	0.945009i	$-0.606052\pi$
$269$	−10.7278	−0.654087	−0.327044	+	0.945009i	$0.606052\pi$
$270$	30.9016	1.88061
$271$	16.2707	0.988375	0.494188	−	0.869355i	$-0.335466\pi$
$271$	16.2707	0.988375	0.494188	+	0.869355i	$0.335466\pi$
$272$	8.54677	0.518224
$273$	10.2701	0.621574
$274$	−11.2765	−0.681235
$275$	−5.54689	−0.334490
$276$	−11.0807	−0.666980
$277$	6.77040	0.406794	0.203397	−	0.979096i	$-0.434802\pi$
$277$	6.77040	0.406794	0.203397	+	0.979096i	$0.434802\pi$
$278$	−17.6664	−1.05956
$279$	−0.00279545	−0.000167359 0
$280$	−1.96430	−0.117389
$281$	−17.1455	−1.02281	−0.511407	−	0.859339i	$-0.670876\pi$
$281$	−17.1455	−1.02281	−0.511407	+	0.859339i	$0.670876\pi$
$282$	13.2444	0.788695
$283$	−2.94787	−0.175232	−0.0876162	−	0.996154i	$-0.527925\pi$
$283$	−2.94787	−0.175232	−0.0876162	+	0.996154i	$0.527925\pi$
$284$	30.7390	1.82403
$285$	−6.08193	−0.360262
$286$	−5.81138	−0.343634
$287$	12.7931	0.755152
$288$	−4.11033	−0.242203
$289$	−11.5695	−0.680561
$290$	6.44033	0.378189
$291$	−14.1813	−0.831325
$292$	−28.4933	−1.66745
$293$	2.57851	0.150638	0.0753192	−	0.997159i	$-0.476002\pi$
$293$	2.57851	0.150638	0.0753192	+	0.997159i	$0.476002\pi$
$294$	−12.5986	−0.734768
$295$	44.7566	2.60583
$296$	3.04133	0.176774
$297$	4.66842	0.270889
$298$	−39.3430	−2.27908
$299$	−7.83097	−0.452877
$300$	22.3786	1.29203
$301$	15.3757	0.886241
$302$	17.7606	1.02201
$303$	−28.1873	−1.61932
$304$	−3.66762	−0.210352
$305$	−12.1658	−0.696610
$306$	−2.40895	−0.137710
$307$	−19.7888	−1.12941	−0.564704	−	0.825293i	$-0.691010\pi$
$307$	−19.7888	−1.12941	−0.564704	+	0.825293i	$0.691010\pi$
$308$	−4.14350	−0.236098
$309$	−1.01775	−0.0578975
$310$	0.0364840	0.00207215
$311$	19.7979	1.12264	0.561319	−	0.827599i	$-0.310294\pi$
$311$	19.7979	1.12264	0.561319	+	0.827599i	$0.310294\pi$
$312$	1.67916	0.0950637
$313$	10.9847	0.620890	0.310445	−	0.950591i	$-0.399522\pi$
$313$	10.9847	0.620890	0.310445	+	0.950591i	$0.399522\pi$
$314$	−11.9569	−0.674769
$315$	−3.16801	−0.178497
$316$	−4.04266	−0.227417
$317$	9.88351	0.555113	0.277557	−	0.960709i	$-0.410475\pi$
$317$	9.88351	0.555113	0.277557	+	0.960709i	$0.410475\pi$
$318$	40.2670	2.25806
$319$	0.972965	0.0544756
$320$	29.8228	1.66714
$321$	27.5988	1.54042
$322$	−10.7671	−0.600024
$323$	−2.33033	−0.129663
$324$	−22.1123	−1.22846
$325$	15.8154	0.877282
$326$	30.3289	1.67976
$327$	−32.5317	−1.79901
$328$	2.09167	0.115493
$329$	6.67376	0.367936
$330$	12.3962	0.682390
$331$	25.7597	1.41588	0.707942	−	0.706271i	$-0.249624\pi$
$331$	25.7597	1.41588	0.707942	+	0.706271i	$0.249624\pi$
$332$	−23.6151	−1.29604
$333$	4.90504	0.268795
$334$	8.53949	0.467260
$335$	12.8986	0.704723
$336$	−13.2107	−0.720705
$337$	21.8924	1.19256	0.596278	−	0.802778i	$-0.296646\pi$
$337$	21.8924	1.19256	0.596278	+	0.802778i	$0.296646\pi$
$338$	−9.92714	−0.539965
$339$	24.1332	1.31074
$340$	16.3036	0.884188
$341$	0.00551178	0.000298480 0
$342$	1.03373	0.0558979
$343$	−19.8120	−1.06975
$344$	2.51393	0.135542
$345$	16.7042	0.899324
$346$	1.44129	0.0774842
$347$	19.2857	1.03531	0.517656	−	0.855589i	$-0.326805\pi$
$347$	19.2857	1.03531	0.517656	+	0.855589i	$0.326805\pi$
$348$	−3.92537	−0.210422
$349$	15.5220	0.830872	0.415436	−	0.909622i	$-0.363629\pi$
$349$	15.5220	0.830872	0.415436	+	0.909622i	$0.363629\pi$
$350$	21.7451	1.16233
$351$	−13.3107	−0.710473
$352$	8.10431	0.431961
$353$	−8.92859	−0.475221	−0.237610	−	0.971361i	$-0.576364\pi$
$353$	−8.92859	−0.475221	−0.237610	+	0.971361i	$0.576364\pi$
$354$	−52.6044	−2.79589
$355$	−46.3392	−2.45943
$356$	32.3564	1.71489
$357$	−8.39385	−0.444249
$358$	−44.4252	−2.34794
$359$	26.2672	1.38633	0.693165	−	0.720779i	$-0.256216\pi$
$359$	26.2672	1.38633	0.693165	+	0.720779i	$0.256216\pi$
$360$	−0.517970	−0.0272994
$361$	1.00000	0.0526316
$362$	−5.98042	−0.314324
$363$	1.87275	0.0982937
$364$	11.8141	0.619225
$365$	42.9538	2.24830
$366$	14.2990	0.747418
$367$	10.2560	0.535358	0.267679	−	0.963508i	$-0.413743\pi$
$367$	10.2560	0.535358	0.267679	+	0.963508i	$0.413743\pi$
$368$	10.0732	0.525103
$369$	3.37343	0.175614
$370$	−64.0166	−3.32807
$371$	20.2902	1.05341
$372$	−0.0222369	−0.00115293
$373$	−20.2242	−1.04717	−0.523584	−	0.851974i	$-0.675405\pi$
$373$	−20.2242	−1.04717	−0.523584	+	0.851974i	$0.675405\pi$
$374$	4.74970	0.245601
$375$	−3.32615	−0.171762
$376$	1.09116	0.0562722
$377$	−2.77414	−0.142876
$378$	−18.3013	−0.941318
$379$	14.1534	0.727011	0.363505	−	0.931592i	$-0.381580\pi$
$379$	14.1534	0.727011	0.363505	+	0.931592i	$0.381580\pi$
$380$	−6.99626	−0.358901
$381$	28.9044	1.48082
$382$	45.6595	2.33614
$383$	12.3217	0.629611	0.314806	−	0.949156i	$-0.398061\pi$
$383$	12.3217	0.629611	0.314806	+	0.949156i	$0.398061\pi$
$384$	−4.69739	−0.239713
$385$	6.24635	0.318343
$386$	5.20655	0.265006
$387$	4.05445	0.206099
$388$	−16.3133	−0.828183
$389$	33.9248	1.72006	0.860029	−	0.510245i	$-0.170445\pi$
$389$	33.9248	1.72006	0.860029	+	0.510245i	$0.170445\pi$
$390$	−35.3444	−1.78973
$391$	6.40033	0.323679
$392$	−1.03795	−0.0524246
$393$	−22.5956	−1.13980
$394$	−25.8786	−1.30375
$395$	6.09432	0.306639
$396$	−1.09261	−0.0549056
$397$	−20.3357	−1.02062	−0.510309	−	0.859991i	$-0.670469\pi$
$397$	−20.3357	−1.02062	−0.510309	+	0.859991i	$0.670469\pi$
$398$	20.6987	1.03753
$399$	3.60199	0.180325
$400$	−20.3439	−1.01719
$401$	−30.6815	−1.53216	−0.766080	−	0.642746i	$-0.777795\pi$
$401$	−30.6815	−1.53216	−0.766080	+	0.642746i	$0.777795\pi$
$402$	−15.1602	−0.756124
$403$	−0.0157153	−0.000782836 0
$404$	−32.4249	−1.61320
$405$	33.3343	1.65640
$406$	−3.81425	−0.189298
$407$	−9.67124	−0.479386
$408$	−1.37239	−0.0679436
$409$	−34.8086	−1.72117	−0.860586	−	0.509305i	$-0.829903\pi$
$409$	−34.8086	−1.72117	−0.860586	+	0.509305i	$0.829903\pi$
$410$	−44.0273	−2.17435
$411$	−10.3610	−0.511072
$412$	−1.17075	−0.0576787
$413$	−26.5069	−1.30432
$414$	−2.83918	−0.139538
$415$	35.5998	1.74752
$416$	−23.1072	−1.13292
$417$	−16.2323	−0.794899
$418$	−2.03821	−0.0996920
$419$	9.04478	0.441866	0.220933	−	0.975289i	$-0.429090\pi$
$419$	9.04478	0.441866	0.220933	+	0.975289i	$0.429090\pi$
$420$	−25.2005	−1.22966
$421$	29.9089	1.45767	0.728836	−	0.684688i	$-0.240062\pi$
$421$	29.9089	1.45767	0.728836	+	0.684688i	$0.240062\pi$
$422$	−16.8327	−0.819402
$423$	1.75981	0.0855651
$424$	3.31745	0.161109
$425$	−12.9261	−0.627008
$426$	54.4645	2.63881
$427$	7.20512	0.348680
$428$	31.7479	1.53459
$429$	−5.33962	−0.257799
$430$	−52.9154	−2.55180
$431$	−13.7402	−0.661840	−0.330920	−	0.943659i	$-0.607359\pi$
$431$	−13.7402	−0.661840	−0.330920	+	0.943659i	$0.607359\pi$
$432$	17.1220	0.823782
$433$	3.28875	0.158047	0.0790236	−	0.996873i	$-0.474820\pi$
$433$	3.28875	0.158047	0.0790236	+	0.996873i	$0.474820\pi$
$434$	−0.0216075	−0.00103719
$435$	5.91750	0.283723
$436$	−37.4224	−1.79221
$437$	−2.74653	−0.131384
$438$	−50.4855	−2.41229
$439$	13.1729	0.628707	0.314353	−	0.949306i	$-0.398212\pi$
$439$	13.1729	0.628707	0.314353	+	0.949306i	$0.398212\pi$
$440$	1.02128	0.0486875
$441$	−1.67401	−0.0797146
$442$	−13.5425	−0.644149
$443$	−24.9484	−1.18533	−0.592666	−	0.805448i	$-0.701925\pi$
$443$	−24.9484	−1.18533	−0.592666	+	0.805448i	$0.701925\pi$
$444$	39.0180	1.85171
$445$	−48.7774	−2.31227
$446$	−43.2689	−2.04884
$447$	−36.1491	−1.70980
$448$	−17.6624	−0.834470
$449$	15.5530	0.733993	0.366996	−	0.930222i	$-0.380386\pi$
$449$	15.5530	0.733993	0.366996	+	0.930222i	$0.380386\pi$
$450$	5.73401	0.270304
$451$	−6.65137	−0.313201
$452$	27.7613	1.30578
$453$	16.3188	0.766724
$454$	−18.4794	−0.867283
$455$	−17.8097	−0.834933
$456$	0.588926	0.0275790
$457$	−19.1451	−0.895571	−0.447786	−	0.894141i	$-0.647787\pi$
$457$	−19.1451	−0.895571	−0.447786	+	0.894141i	$0.647787\pi$
$458$	−10.7119	−0.500535
$459$	10.8790	0.507787
$460$	19.2155	0.895925
$461$	33.5045	1.56046	0.780229	−	0.625493i	$-0.215102\pi$
$461$	33.5045	1.56046	0.780229	+	0.625493i	$0.215102\pi$
$462$	−7.34161	−0.341563
$463$	2.83101	0.131568	0.0657842	−	0.997834i	$-0.479045\pi$
$463$	2.83101	0.131568	0.0657842	+	0.997834i	$0.479045\pi$
$464$	3.56847	0.165662
$465$	0.0335222	0.00155456
$466$	−11.5948	−0.537117
$467$	−20.3717	−0.942689	−0.471344	−	0.881949i	$-0.656231\pi$
$467$	−20.3717	−0.942689	−0.471344	+	0.881949i	$0.656231\pi$
$468$	3.11527	0.144003
$469$	−7.63911	−0.352741
$470$	−22.9677	−1.05942
$471$	−10.9863	−0.506221
$472$	−4.33388	−0.199483
$473$	−7.99413	−0.367570
$474$	−7.16293	−0.329004
$475$	5.54689	0.254509
$476$	−9.65575	−0.442571
$477$	5.35036	0.244976
$478$	−41.5360	−1.89981
$479$	−7.81572	−0.357109	−0.178555	−	0.983930i	$-0.557142\pi$
$479$	−7.81572	−0.357109	−0.178555	+	0.983930i	$0.557142\pi$
$480$	49.2898	2.24976
$481$	27.5749	1.25731
$482$	36.0621	1.64258
$483$	−9.89299	−0.450146
$484$	2.15429	0.0979222
$485$	24.5924	1.11668
$486$	−10.6337	−0.482353
$487$	−9.10523	−0.412597	−0.206299	−	0.978489i	$-0.566142\pi$
$487$	−9.10523	−0.412597	−0.206299	+	0.978489i	$0.566142\pi$
$488$	1.17804	0.0533272
$489$	27.8668	1.26018
$490$	21.8478	0.986982
$491$	34.4175	1.55324	0.776619	−	0.629970i	$-0.216933\pi$
$491$	34.4175	1.55324	0.776619	+	0.629970i	$0.216933\pi$
$492$	26.8345	1.20979
$493$	2.26733	0.102116
$494$	5.81138	0.261467
$495$	1.64711	0.0740321
$496$	0.0202151	0.000907685 0
$497$	27.4442	1.23104
$498$	−41.8420	−1.87498
$499$	7.80798	0.349533	0.174767	−	0.984610i	$-0.444083\pi$
$499$	7.80798	0.349533	0.174767	+	0.984610i	$0.444083\pi$
$500$	−3.82619	−0.171113
$501$	7.84625	0.350545
$502$	−1.58275	−0.0706418
$503$	34.9580	1.55870	0.779350	−	0.626589i	$-0.215550\pi$
$503$	34.9580	1.55870	0.779350	+	0.626589i	$0.215550\pi$
$504$	0.306765	0.0136644
$505$	48.8806	2.17516
$506$	5.59800	0.248861
$507$	−9.12126	−0.405089
$508$	33.2498	1.47522
$509$	−11.3952	−0.505082	−0.252541	−	0.967586i	$-0.581266\pi$
$509$	−11.3952	−0.505082	−0.252541	+	0.967586i	$0.581266\pi$
$510$	28.8873	1.27915
$511$	−25.4392	−1.12536
$512$	32.0302	1.41555
$513$	−4.66842	−0.206116
$514$	59.6710	2.63197
$515$	1.76491	0.0777712
$516$	32.2518	1.41981
$517$	−3.46982	−0.152602
$518$	37.9136	1.66583
$519$	1.32429	0.0581297
$520$	−2.91189	−0.127695
$521$	−28.0648	−1.22954	−0.614770	−	0.788707i	$-0.710751\pi$
$521$	−28.0648	−1.22954	−0.614770	+	0.788707i	$0.710751\pi$
$522$	−1.00579	−0.0440221
$523$	20.5683	0.899389	0.449694	−	0.893182i	$-0.351533\pi$
$523$	20.5683	0.899389	0.449694	+	0.893182i	$0.351533\pi$
$524$	−25.9925	−1.13549
$525$	19.9799	0.871993
$526$	−12.0263	−0.524370
$527$	0.0128443	0.000559506 0
$528$	6.86852	0.298914
$529$	−15.4566	−0.672025
$530$	−69.8285	−3.03316
$531$	−6.98965	−0.303325
$532$	4.14350	0.179644
$533$	18.9645	0.821446
$534$	57.3302	2.48092
$535$	−47.8601	−2.06917
$536$	−1.24899	−0.0539484
$537$	−40.8188	−1.76146
$538$	−21.8655	−0.942690
$539$	3.30063	0.142168
$540$	32.6615	1.40553
$541$	−4.13908	−0.177953	−0.0889766	−	0.996034i	$-0.528360\pi$
$541$	−4.13908	−0.177953	−0.0889766	+	0.996034i	$0.528360\pi$
$542$	33.1631	1.42448
$543$	−5.49493	−0.235810
$544$	18.8857	0.809720
$545$	56.4144	2.41653
$546$	20.9326	0.895831
$547$	−30.7624	−1.31530	−0.657652	−	0.753322i	$-0.728450\pi$
$547$	−30.7624	−1.31530	−0.657652	+	0.753322i	$0.728450\pi$
$548$	−11.9187	−0.509141
$549$	1.89993	0.0810870
$550$	−11.3057	−0.482077
$551$	−0.972965	−0.0414497
$552$	−1.61750	−0.0688455
$553$	−3.60934	−0.153485
$554$	13.7995	0.586284
$555$	−58.8198	−2.49676
$556$	−18.6726	−0.791895
$557$	8.84004	0.374565	0.187282	−	0.982306i	$-0.440032\pi$
$557$	8.84004	0.374565	0.187282	+	0.982306i	$0.440032\pi$
$558$	−0.00569772	−0.000241204 0
$559$	22.7930	0.964043
$560$	22.9092	0.968091
$561$	4.36412	0.184253
$562$	−34.9461	−1.47411
$563$	−3.15807	−0.133097	−0.0665484	−	0.997783i	$-0.521199\pi$
$563$	−3.15807	−0.133097	−0.0665484	+	0.997783i	$0.521199\pi$
$564$	13.9987	0.589454
$565$	−41.8503	−1.76066
$566$	−6.00836	−0.252550
$567$	−19.7421	−0.829091
$568$	4.48712	0.188276
$569$	18.7192	0.784749	0.392375	−	0.919806i	$-0.371654\pi$
$569$	18.7192	0.784749	0.392375	+	0.919806i	$0.371654\pi$
$570$	−12.3962	−0.519221
$571$	37.6834	1.57700	0.788500	−	0.615034i	$-0.210858\pi$
$571$	37.6834	1.57700	0.788500	+	0.615034i	$0.210858\pi$
$572$	−6.14236	−0.256825
$573$	41.9529	1.75261
$574$	26.0750	1.08835
$575$	−15.2347	−0.635331
$576$	−4.65743	−0.194060
$577$	1.77272	0.0737993	0.0368997	−	0.999319i	$-0.488252\pi$
$577$	1.77272	0.0737993	0.0368997	+	0.999319i	$0.488252\pi$
$578$	−23.5811	−0.980846
$579$	4.78388	0.198811
$580$	6.80712	0.282650
$581$	−21.0838	−0.874704
$582$	−28.9045	−1.19813
$583$	−10.5493	−0.436906
$584$	−4.15931	−0.172113
$585$	−4.69628	−0.194167
$586$	5.25554	0.217105
$587$	−32.1703	−1.32781	−0.663906	−	0.747816i	$-0.731103\pi$
$587$	−32.1703	−1.32781	−0.663906	+	0.747816i	$0.731103\pi$
$588$	−13.3162	−0.549150
$589$	−0.00551178	−0.000227109 0
$590$	91.2232	3.75560
$591$	−23.7778	−0.978088
$592$	−35.4704	−1.45783
$593$	12.2719	0.503946	0.251973	−	0.967734i	$-0.418921\pi$
$593$	12.2719	0.503946	0.251973	+	0.967734i	$0.418921\pi$
$594$	9.51521	0.390414
$595$	14.5561	0.596741
$596$	−41.5837	−1.70333
$597$	19.0183	0.778369
$598$	−15.9611	−0.652700
$599$	16.7005	0.682366	0.341183	−	0.939997i	$-0.389172\pi$
$599$	16.7005	0.682366	0.341183	+	0.939997i	$0.389172\pi$
$600$	3.26671	0.133363
$601$	14.3030	0.583432	0.291716	−	0.956505i	$-0.405774\pi$
$601$	14.3030	0.583432	0.291716	+	0.956505i	$0.405774\pi$
$602$	31.3389	1.27728
$603$	−2.01437	−0.0820315
$604$	18.7721	0.763827
$605$	−3.24760	−0.132034
$606$	−57.4516	−2.33381
$607$	−9.04370	−0.367073	−0.183536	−	0.983013i	$-0.558754\pi$
$607$	−9.04370	−0.367073	−0.183536	+	0.983013i	$0.558754\pi$
$608$	−8.10431	−0.328673
$609$	−3.50461	−0.142014
$610$	−24.7963	−1.00397
$611$	9.89322	0.400237
$612$	−2.54614	−0.102922
$613$	−24.0096	−0.969738	−0.484869	−	0.874587i	$-0.661133\pi$
$613$	−24.0096	−0.969738	−0.484869	+	0.874587i	$0.661133\pi$
$614$	−40.3337	−1.62774
$615$	−40.4532	−1.63123
$616$	−0.604847	−0.0243700
$617$	1.30852	0.0526791	0.0263395	−	0.999653i	$-0.491615\pi$
$617$	1.30852	0.0526791	0.0263395	+	0.999653i	$0.491615\pi$
$618$	−2.07438	−0.0834436
$619$	32.2747	1.29723	0.648616	−	0.761116i	$-0.275348\pi$
$619$	32.2747	1.29723	0.648616	+	0.761116i	$0.275348\pi$
$620$	0.0385619	0.00154868
$621$	12.8220	0.514528
$622$	40.3523	1.61798
$623$	28.8882	1.15738
$624$	−19.5837	−0.783975
$625$	−21.9665	−0.878658
$626$	22.3890	0.894845
$627$	−1.87275	−0.0747903
$628$	−12.6379	−0.504308
$629$	−22.5372	−0.898618
$630$	−6.45706	−0.257256
$631$	29.6689	1.18110	0.590551	−	0.807000i	$-0.298911\pi$
$631$	29.6689	1.18110	0.590551	+	0.807000i	$0.298911\pi$
$632$	−0.590127	−0.0234740
$633$	−15.4662	−0.614727
$634$	20.1446	0.800046
$635$	−50.1242	−1.98912
$636$	42.5603	1.68763
$637$	−9.41083	−0.372871
$638$	1.98311	0.0785119
$639$	7.23680	0.286283
$640$	8.14591	0.321995
$641$	22.2716	0.879675	0.439838	−	0.898077i	$-0.355036\pi$
$641$	22.2716	0.879675	0.439838	+	0.898077i	$0.355036\pi$
$642$	56.2521	2.22009
$643$	−16.6461	−0.656456	−0.328228	−	0.944599i	$-0.606451\pi$
$643$	−16.6461	−0.656456	−0.328228	+	0.944599i	$0.606451\pi$
$644$	−11.3803	−0.448445
$645$	−48.6197	−1.91440
$646$	−4.74970	−0.186875
$647$	−15.9531	−0.627180	−0.313590	−	0.949559i	$-0.601532\pi$
$647$	−15.9531	−0.627180	−0.313590	+	0.949559i	$0.601532\pi$
$648$	−3.22783	−0.126801
$649$	13.7814	0.540969
$650$	32.2351	1.26437
$651$	−0.0198534	−0.000778116 0
$652$	32.0562	1.25542
$653$	14.7613	0.577655	0.288828	−	0.957381i	$-0.406735\pi$
$653$	14.7613	0.577655	0.288828	+	0.957381i	$0.406735\pi$
$654$	−66.3063	−2.59278
$655$	39.1838	1.53104
$656$	−24.3947	−0.952453
$657$	−6.70811	−0.261708
$658$	13.6025	0.530281
$659$	−19.6143	−0.764065	−0.382032	−	0.924149i	$-0.624776\pi$
$659$	−19.6143	−0.764065	−0.382032	+	0.924149i	$0.624776\pi$
$660$	13.1022	0.510003
$661$	−31.5523	−1.22724	−0.613621	−	0.789601i	$-0.710288\pi$
$661$	−31.5523	−1.22724	−0.613621	+	0.789601i	$0.710288\pi$
$662$	52.5037	2.04061
$663$	−12.4431	−0.483250
$664$	−3.44720	−0.133777
$665$	−6.24635	−0.242223
$666$	9.99749	0.387395
$667$	2.67228	0.103471
$668$	9.02583	0.349220
$669$	−39.7564	−1.53707
$670$	26.2899	1.01567
$671$	−3.74608	−0.144616
$672$	−29.1917	−1.12609
$673$	41.3876	1.59537	0.797687	−	0.603071i	$-0.206057\pi$
$673$	41.3876	1.59537	0.797687	+	0.603071i	$0.206057\pi$
$674$	44.6213	1.71875
$675$	−25.8952	−0.996708
$676$	−10.4925	−0.403558
$677$	−20.6981	−0.795491	−0.397746	−	0.917496i	$-0.630207\pi$
$677$	−20.6981	−0.795491	−0.397746	+	0.917496i	$0.630207\pi$
$678$	49.1885	1.88907
$679$	−14.5647	−0.558943
$680$	2.37992	0.0912657
$681$	−16.9793	−0.650647
$682$	0.0112342	0.000430178 0
$683$	−0.658543	−0.0251985	−0.0125992	−	0.999921i	$-0.504011\pi$
$683$	−0.658543	−0.0251985	−0.0125992	+	0.999921i	$0.504011\pi$
$684$	1.09261	0.0417769
$685$	17.9674	0.686501
$686$	−40.3809	−1.54175
$687$	−9.84233	−0.375508
$688$	−29.3194	−1.11779
$689$	30.0783	1.14589
$690$	34.0466	1.29613
$691$	34.2462	1.30279	0.651393	−	0.758741i	$-0.274185\pi$
$691$	34.2462	1.30279	0.651393	+	0.758741i	$0.274185\pi$
$692$	1.52338	0.0579100
$693$	−0.975494	−0.0370559
$694$	39.3083	1.49212
$695$	28.1490	1.06775
$696$	−0.573005	−0.0217197
$697$	−15.4999	−0.587101
$698$	31.6370	1.19748
$699$	−10.6535	−0.402952
$700$	22.9836	0.868697
$701$	22.4638	0.848445	0.424223	−	0.905558i	$-0.360547\pi$
$701$	22.4638	0.848445	0.424223	+	0.905558i	$0.360547\pi$
$702$	−27.1300	−1.02396
$703$	9.67124	0.364758
$704$	9.18303	0.346098
$705$	−21.1032	−0.794791
$706$	−18.1983	−0.684902
$707$	−28.9493	−1.08875
$708$	−55.6003	−2.08959
$709$	−0.410520	−0.0154174	−0.00770870	−	0.999970i	$-0.502454\pi$
$709$	−0.410520	−0.0154174	−0.00770870	+	0.999970i	$0.502454\pi$
$710$	−94.4489	−3.54460
$711$	−0.951752	−0.0356935
$712$	4.72322	0.177010
$713$	0.0151383	0.000566933 0
$714$	−17.1084	−0.640266
$715$	9.25963	0.346290
$716$	−46.9553	−1.75480
$717$	−38.1641	−1.42527
$718$	53.5380	1.99802
$719$	36.5145	1.36176	0.680880	−	0.732395i	$-0.261598\pi$
$719$	36.5145	1.36176	0.680880	+	0.732395i	$0.261598\pi$
$720$	6.04097	0.225134
$721$	−1.04526	−0.0389275
$722$	2.03821	0.0758542
$723$	33.1346	1.23229
$724$	−6.32102	−0.234919
$725$	−5.39693	−0.200437
$726$	3.81704	0.141664
$727$	−39.2587	−1.45602	−0.728012	−	0.685564i	$-0.759556\pi$
$727$	−39.2587	−1.45602	−0.728012	+	0.685564i	$0.759556\pi$
$728$	1.72455	0.0639163
$729$	21.0225	0.778610
$730$	87.5488	3.24032
$731$	−18.6290	−0.689018
$732$	15.1133	0.558604
$733$	−34.3259	−1.26786	−0.633928	−	0.773392i	$-0.718558\pi$
$733$	−34.3259	−1.26786	−0.633928	+	0.773392i	$0.718558\pi$
$734$	20.9038	0.771575
$735$	20.0742	0.740447
$736$	22.2587	0.820468
$737$	3.97172	0.146300
$738$	6.87575	0.253100
$739$	−26.7732	−0.984870	−0.492435	−	0.870349i	$-0.663893\pi$
$739$	−26.7732	−0.984870	−0.492435	+	0.870349i	$0.663893\pi$
$740$	−67.6625	−2.48732
$741$	5.33962	0.196156
$742$	41.3556	1.51821
$743$	−9.38431	−0.344277	−0.172139	−	0.985073i	$-0.555068\pi$
$743$	−9.38431	−0.344277	−0.172139	+	0.985073i	$0.555068\pi$
$744$	−0.00324603	−0.000119005 0
$745$	62.6876	2.29669
$746$	−41.2210	−1.50921
$747$	−5.55963	−0.203416
$748$	5.02021	0.183557
$749$	28.3449	1.03570
$750$	−6.77939	−0.247548
$751$	7.66846	0.279826	0.139913	−	0.990164i	$-0.455318\pi$
$751$	7.66846	0.279826	0.139913	+	0.990164i	$0.455318\pi$
$752$	−12.7260	−0.464068
$753$	−1.45427	−0.0529965
$754$	−5.65428	−0.205917
$755$	−28.2990	−1.02991
$756$	−19.3436	−0.703521
$757$	−6.75931	−0.245671	−0.122836	−	0.992427i	$-0.539199\pi$
$757$	−6.75931	−0.245671	−0.122836	+	0.992427i	$0.539199\pi$
$758$	28.8475	1.04779
$759$	5.14356	0.186699
$760$	−1.02128	−0.0370456
$761$	−34.8774	−1.26430	−0.632152	−	0.774844i	$-0.717828\pi$
$761$	−34.8774	−1.26430	−0.632152	+	0.774844i	$0.717828\pi$
$762$	58.9132	2.13420
$763$	−33.4111	−1.20956
$764$	48.2599	1.74598
$765$	3.83832	0.138775
$766$	25.1142	0.907415
$767$	−39.2940	−1.41882
$768$	24.8207	0.895640
$769$	−0.00622027	−0.000224309 0	−0.000112154	−	1.00000i	$-0.500036\pi$
$769$	−0.00622027	−0.000224309 0	−0.000112154		1.00000i	$0.500036\pi$
$770$	12.7314	0.458806
$771$	54.8269	1.97454
$772$	5.50307	0.198060
$773$	10.8548	0.390418	0.195209	−	0.980762i	$-0.437461\pi$
$773$	10.8548	0.390418	0.195209	+	0.980762i	$0.437461\pi$
$774$	8.26380	0.297036
$775$	−0.0305733	−0.00109822
$776$	−2.38133	−0.0854848
$777$	34.8357	1.24973
$778$	69.1459	2.47900
$779$	6.65137	0.238310
$780$	−37.3574	−1.33761
$781$	−14.2688	−0.510576
$782$	13.0452	0.466496
$783$	4.54221	0.162325
$784$	12.1054	0.432337
$785$	19.0517	0.679984
$786$	−46.0544	−1.64271
$787$	37.8221	1.34821	0.674106	−	0.738635i	$-0.264529\pi$
$787$	37.8221	1.34821	0.674106	+	0.738635i	$0.264529\pi$
$788$	−27.3525	−0.974392
$789$	−11.0500	−0.393390
$790$	12.4215	0.441937
$791$	24.7857	0.881277
$792$	−0.159493	−0.00566735
$793$	10.6809	0.379290
$794$	−41.4483	−1.47095
$795$	−64.1598	−2.27552
$796$	21.8775	0.775428
$797$	−49.2853	−1.74577	−0.872887	−	0.487922i	$-0.837755\pi$
$797$	−49.2853	−1.74577	−0.872887	+	0.487922i	$0.837755\pi$
$798$	7.34161	0.259890
$799$	−8.08583	−0.286056
$800$	−44.9537	−1.58935
$801$	7.61758	0.269154
$802$	−62.5352	−2.20819
$803$	13.2263	0.466747
$804$	−16.0237	−0.565111
$805$	17.1558	0.604662
$806$	−0.0320311	−0.00112825
$807$	−20.0905	−0.707219
$808$	−4.73322	−0.166514
$809$	34.0416	1.19684	0.598420	−	0.801183i	$-0.295796\pi$
$809$	34.0416	1.19684	0.598420	+	0.801183i	$0.295796\pi$
$810$	67.9423	2.38725
$811$	−44.7037	−1.56976	−0.784880	−	0.619648i	$-0.787275\pi$
$811$	−44.7037	−1.56976	−0.784880	+	0.619648i	$0.787275\pi$
$812$	−4.03149	−0.141477
$813$	30.4709	1.06866
$814$	−19.7120	−0.690905
$815$	−48.3249	−1.69275
$816$	16.0059	0.560320
$817$	7.99413	0.279679
$818$	−70.9470	−2.48061
$819$	2.78135	0.0971882
$820$	−46.5348	−1.62506
$821$	−48.0778	−1.67793	−0.838963	−	0.544188i	$-0.816838\pi$
$821$	−48.0778	−1.67793	−0.838963	+	0.544188i	$0.816838\pi$
$822$	−21.1179	−0.736572
$823$	18.6246	0.649211	0.324606	−	0.945849i	$-0.394768\pi$
$823$	18.6246	0.649211	0.324606	+	0.945849i	$0.394768\pi$
$824$	−0.170900	−0.00595358
$825$	−10.3879	−0.361661
$826$	−54.0265	−1.87982
$827$	6.99744	0.243325	0.121662	−	0.992572i	$-0.461177\pi$
$827$	6.99744	0.243325	0.121662	+	0.992572i	$0.461177\pi$
$828$	−3.00088	−0.104288
$829$	24.9441	0.866344	0.433172	−	0.901311i	$-0.357394\pi$
$829$	24.9441	0.866344	0.433172	+	0.901311i	$0.357394\pi$
$830$	72.5597	2.51859
$831$	12.6792	0.439838
$832$	−26.1829	−0.907727
$833$	7.69157	0.266497
$834$	−33.0848	−1.14563
$835$	−13.6065	−0.470872
$836$	−2.15429	−0.0745076
$837$	0.0257313	0.000889405 0
$838$	18.4351	0.636831
$839$	10.2122	0.352566	0.176283	−	0.984340i	$-0.443593\pi$
$839$	10.2122	0.352566	0.176283	+	0.984340i	$0.443593\pi$
$840$	−3.67864	−0.126925
$841$	−28.0533	−0.967356
$842$	60.9606	2.10084
$843$	−32.1091	−1.10590
$844$	−17.7914	−0.612404
$845$	15.8175	0.544139
$846$	3.58687	0.123319
$847$	1.92338	0.0660879
$848$	−38.6907	−1.32864
$849$	−5.52060	−0.189467
$850$	−26.3461	−0.903663
$851$	−26.5624	−0.910546
$852$	57.5664	1.97219
$853$	33.5562	1.14894	0.574472	−	0.818524i	$-0.305207\pi$
$853$	33.5562	1.14894	0.574472	+	0.818524i	$0.305207\pi$
$854$	14.6855	0.502528
$855$	−1.64711	−0.0563300
$856$	4.63440	0.158400
$857$	34.8775	1.19139	0.595696	−	0.803210i	$-0.296876\pi$
$857$	34.8775	1.19139	0.595696	+	0.803210i	$0.296876\pi$
$858$	−10.8832	−0.371548
$859$	13.1320	0.448059	0.224029	−	0.974582i	$-0.428079\pi$
$859$	13.1320	0.448059	0.224029	+	0.974582i	$0.428079\pi$
$860$	−55.9290	−1.90716
$861$	23.9582	0.816493
$862$	−28.0053	−0.953863
$863$	−27.5223	−0.936871	−0.468435	−	0.883498i	$-0.655182\pi$
$863$	−27.5223	−0.936871	−0.468435	+	0.883498i	$0.655182\pi$
$864$	37.8343	1.28715
$865$	−2.29649	−0.0780831
$866$	6.70315	0.227782
$867$	−21.6668	−0.735844
$868$	−0.0228381	−0.000775175 0
$869$	1.87656	0.0636581
$870$	12.0611	0.408910
$871$	−11.3243	−0.383708
$872$	−5.46272	−0.184991
$873$	−3.84060	−0.129985
$874$	−5.59800	−0.189355
$875$	−3.41607	−0.115484
$876$	−53.3608	−1.80289
$877$	−18.2679	−0.616865	−0.308432	−	0.951246i	$-0.599804\pi$
$877$	−18.2679	−0.616865	−0.308432	+	0.951246i	$0.599804\pi$
$878$	26.8490	0.906111
$879$	4.82890	0.162875
$880$	−11.9109	−0.401518
$881$	−47.0363	−1.58469	−0.792346	−	0.610072i	$-0.791141\pi$
$881$	−47.0363	−1.58469	−0.792346	+	0.610072i	$0.791141\pi$
$882$	−3.41197	−0.114887
$883$	−24.7005	−0.831238	−0.415619	−	0.909539i	$-0.636435\pi$
$883$	−24.7005	−0.831238	−0.415619	+	0.909539i	$0.636435\pi$
$884$	−14.3137	−0.481423
$885$	83.8177	2.81750
$886$	−50.8500	−1.70834
$887$	23.3946	0.785515	0.392758	−	0.919642i	$-0.371521\pi$
$887$	23.3946	0.785515	0.392758	+	0.919642i	$0.371521\pi$
$888$	5.69564	0.191133
$889$	29.6858	0.995631
$890$	−99.4184	−3.33251
$891$	10.2643	0.343867
$892$	−45.7332	−1.53126
$893$	3.46982	0.116113
$894$	−73.6794	−2.46421
$895$	70.7853	2.36609
$896$	−4.82438	−0.161171
$897$	−14.6654	−0.489664
$898$	31.7003	1.05785
$899$	0.00536277	0.000178858 0
$900$	6.06058	0.202019
$901$	−24.5833	−0.818989
$902$	−13.5569	−0.451395
$903$	28.7948	0.958231
$904$	4.05246	0.134783
$905$	9.52897	0.316754
$906$	33.2611	1.10503
$907$	19.2411	0.638892	0.319446	−	0.947605i	$-0.396503\pi$
$907$	19.2411	0.638892	0.319446	+	0.947605i	$0.396503\pi$
$908$	−19.5319	−0.648189
$909$	−7.63370	−0.253194
$910$	−36.2999	−1.20333
$911$	48.5854	1.60971	0.804853	−	0.593475i	$-0.202244\pi$
$911$	48.5854	1.60971	0.804853	+	0.593475i	$0.202244\pi$
$912$	−6.86852	−0.227439
$913$	10.9619	0.362785
$914$	−39.0217	−1.29072
$915$	−22.7834	−0.753195
$916$	−11.3220	−0.374089
$917$	−23.2064	−0.766344
$918$	22.1736	0.731839
$919$	21.2754	0.701812	0.350906	−	0.936411i	$-0.385874\pi$
$919$	21.2754	0.701812	0.350906	+	0.936411i	$0.385874\pi$
$920$	2.80497	0.0924772
$921$	−37.0595	−1.22115
$922$	68.2891	2.24898
$923$	40.6834	1.33911
$924$	−7.75973	−0.255276
$925$	53.6453	1.76385
$926$	5.77019	0.189620
$927$	−0.275626	−0.00905276
$928$	7.88521	0.258845
$929$	49.9319	1.63821	0.819106	−	0.573643i	$-0.194470\pi$
$929$	49.9319	1.63821	0.819106	+	0.573643i	$0.194470\pi$
$930$	0.0683253	0.00224047
$931$	−3.30063	−0.108174
$932$	−12.2551	−0.401429
$933$	37.0765	1.21383
$934$	−41.5217	−1.35863
$935$	−7.56799	−0.247500
$936$	0.454751	0.0148640
$937$	11.2142	0.366353	0.183176	−	0.983080i	$-0.441362\pi$
$937$	11.2142	0.366353	0.183176	+	0.983080i	$0.441362\pi$
$938$	−15.5701	−0.508381
$939$	20.5715	0.671325
$940$	−24.2757	−0.791787
$941$	−47.9390	−1.56277	−0.781383	−	0.624052i	$-0.785485\pi$
$941$	−47.9390	−1.56277	−0.781383	+	0.624052i	$0.785485\pi$
$942$	−22.3923	−0.729580
$943$	−18.2682	−0.594894
$944$	50.5451	1.64510
$945$	29.1606	0.948594
$946$	−16.2937	−0.529754
$947$	14.4633	0.469993	0.234997	−	0.971996i	$-0.424492\pi$
$947$	14.4633	0.469993	0.234997	+	0.971996i	$0.424492\pi$
$948$	−7.57088	−0.245891
$949$	−37.7112	−1.22416
$950$	11.3057	0.366806
$951$	18.5093	0.600206
$952$	−1.40950	−0.0456820
$953$	19.3671	0.627363	0.313682	−	0.949528i	$-0.398438\pi$
$953$	19.3671	0.627363	0.313682	+	0.949528i	$0.398438\pi$
$954$	10.9051	0.353067
$955$	−72.7520	−2.35420
$956$	−43.9016	−1.41988
$957$	1.82212	0.0589007
$958$	−15.9300	−0.514677
$959$	−10.6411	−0.343620
$960$	55.8505	1.80257
$961$	−31.0000	−0.999999
$962$	56.2033	1.81207
$963$	7.47432	0.240857
$964$	38.1159	1.22763
$965$	−8.29591	−0.267055
$966$	−20.1640	−0.648765
$967$	−29.6452	−0.953326	−0.476663	−	0.879086i	$-0.658154\pi$
$967$	−29.6452	−0.953326	−0.476663	+	0.879086i	$0.658154\pi$
$968$	0.314472	0.0101075
$969$	−4.36412	−0.140196
$970$	50.1244	1.60940
$971$	−27.9572	−0.897190	−0.448595	−	0.893735i	$-0.648075\pi$
$971$	−27.9572	−0.897190	−0.448595	+	0.893735i	$0.648075\pi$
$972$	−11.2393	−0.360500
$973$	−16.6711	−0.534452
$974$	−18.5583	−0.594648
$975$	29.6183	0.948544
$976$	−13.7392	−0.439781
$977$	−25.4860	−0.815369	−0.407684	−	0.913123i	$-0.633664\pi$
$977$	−25.4860	−0.815369	−0.407684	+	0.913123i	$0.633664\pi$
$978$	56.7983	1.81621
$979$	−15.0195	−0.480026
$980$	23.0921	0.737649
$981$	−8.81024	−0.281289
$982$	70.1499	2.23857
$983$	−23.7523	−0.757582	−0.378791	−	0.925482i	$-0.623660\pi$
$983$	−23.7523	−0.757582	−0.378791	+	0.925482i	$0.623660\pi$
$984$	3.91717	0.124875
$985$	41.2340	1.31382
$986$	4.62130	0.147172
$987$	12.4983	0.397824
$988$	6.14236	0.195414
$989$	−21.9561	−0.698164
$990$	3.35715	0.106697
$991$	10.8187	0.343668	0.171834	−	0.985126i	$-0.445031\pi$
$991$	10.8187	0.343668	0.171834	+	0.985126i	$0.445031\pi$
$992$	0.0446692	0.00141825
$993$	48.2415	1.53090
$994$	55.9369	1.77421
$995$	−32.9804	−1.04555
$996$	−44.2250	−1.40132
$997$	−55.0261	−1.74270	−0.871348	−	0.490666i	$-0.836753\pi$
$997$	−55.0261	−1.74270	−0.871348	+	0.490666i	$0.836753\pi$
$998$	15.9143	0.503758
$999$	−45.1494	−1.42847

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	209.2.a.d.1.6	✓	7
3.2	odd	2		1881.2.a.p.1.2		7
4.3	odd	2		3344.2.a.ba.1.2		7
5.4	even	2		5225.2.a.n.1.2		7
11.10	odd	2		2299.2.a.q.1.2		7
19.18	odd	2		3971.2.a.i.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.d.1.6	✓	7	1.1	even	1	trivial
1881.2.a.p.1.2		7	3.2	odd	2
2299.2.a.q.1.2		7	11.10	odd	2
3344.2.a.ba.1.2		7	4.3	odd	2
3971.2.a.i.1.2		7	19.18	odd	2
5225.2.a.n.1.2		7	5.4	even	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	209.2.a.d.1.6

Level	$209$
Weight	$2$
Character	209.1
Self dual	yes
Analytic conductor	$1.669$
Analytic rank	$0$
Dimension	$7$
CM	no
Inner twists	$1$