LMFDB - Embedded newform 1445.2.a.q.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1445,2,Mod(1,1445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1445, 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("1445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1445 = 5 \cdot 17^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1445.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:	$11.5383830921$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 21 x^{8} - 232 x^{7} + 44 x^{6} + 424 x^{5} - 137 x^{4} + \cdots + 17$ x^12 - 4x^11 - 10x^10 + 52x^9 + 21x^8 - 232x^7 + 44x^6 + 424x^5 - 137x^4 - 316x^3 + 46x^2 + 92*x + 17
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.962871$ of defining polynomial
Character	$\chi$	$=$	1445.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.962871 q^{2} -2.64897 q^{3} -1.07288 q^{4} -1.00000 q^{5} +2.55062 q^{6} +3.09463 q^{7} +2.95879 q^{8} +4.01706 q^{9} +0.962871 q^{10} +6.13071 q^{11} +2.84203 q^{12} +1.16017 q^{13} -2.97973 q^{14} +2.64897 q^{15} -0.703170 q^{16} -3.86791 q^{18} -5.42585 q^{19} +1.07288 q^{20} -8.19759 q^{21} -5.90308 q^{22} -3.11745 q^{23} -7.83775 q^{24} +1.00000 q^{25} -1.11710 q^{26} -2.69417 q^{27} -3.32016 q^{28} +4.99698 q^{29} -2.55062 q^{30} +3.72804 q^{31} -5.24051 q^{32} -16.2401 q^{33} -3.09463 q^{35} -4.30982 q^{36} -0.396716 q^{37} +5.22439 q^{38} -3.07327 q^{39} -2.95879 q^{40} +1.70263 q^{41} +7.89322 q^{42} -0.0268304 q^{43} -6.57751 q^{44} -4.01706 q^{45} +3.00170 q^{46} -5.43715 q^{47} +1.86268 q^{48} +2.57672 q^{49} -0.962871 q^{50} -1.24473 q^{52} +0.345087 q^{53} +2.59413 q^{54} -6.13071 q^{55} +9.15634 q^{56} +14.3729 q^{57} -4.81144 q^{58} +4.06060 q^{59} -2.84203 q^{60} +12.4424 q^{61} -3.58962 q^{62} +12.4313 q^{63} +6.45228 q^{64} -1.16017 q^{65} +15.6371 q^{66} +5.62508 q^{67} +8.25804 q^{69} +2.97973 q^{70} +10.7794 q^{71} +11.8856 q^{72} -1.65433 q^{73} +0.381986 q^{74} -2.64897 q^{75} +5.82128 q^{76} +18.9723 q^{77} +2.95916 q^{78} -5.27290 q^{79} +0.703170 q^{80} -4.91441 q^{81} -1.63941 q^{82} -12.4258 q^{83} +8.79502 q^{84} +0.0258342 q^{86} -13.2369 q^{87} +18.1394 q^{88} -3.22930 q^{89} +3.86791 q^{90} +3.59031 q^{91} +3.34465 q^{92} -9.87549 q^{93} +5.23528 q^{94} +5.42585 q^{95} +13.8820 q^{96} -12.9349 q^{97} -2.48105 q^{98} +24.6274 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 4 q^{2} + 8 q^{3} + 12 q^{4} - 12 q^{5} + 8 q^{6} + 16 q^{7} - 12 q^{8} + 12 q^{9} + 4 q^{10} + 16 q^{11} + 16 q^{12} - 8 q^{13} - 16 q^{14} - 8 q^{15} + 12 q^{16} + 4 q^{18} - 12 q^{20} + 16 q^{21}+ \cdots + 56 q^{99}+O(q^{100})$ 12 * q - 4 * q^2 + 8 * q^3 + 12 * q^4 - 12 * q^5 + 8 * q^6 + 16 * q^7 - 12 * q^8 + 12 * q^9 + 4 * q^10 + 16 * q^11 + 16 * q^12 - 8 * q^13 - 16 * q^14 - 8 * q^15 + 12 * q^16 + 4 * q^18 - 12 * q^20 + 16 * q^21 + 16 * q^22 + 16 * q^23 + 12 * q^25 + 16 * q^26 + 32 * q^27 + 40 * q^28 + 16 * q^29 - 8 * q^30 + 24 * q^31 - 28 * q^32 - 16 * q^35 + 12 * q^36 + 24 * q^37 - 24 * q^38 + 8 * q^39 + 12 * q^40 + 8 * q^41 - 16 * q^43 + 8 * q^44 - 12 * q^45 + 40 * q^46 - 32 * q^47 - 24 * q^48 + 20 * q^49 - 4 * q^50 - 24 * q^52 + 8 * q^54 - 16 * q^55 - 24 * q^56 + 32 * q^57 - 16 * q^58 + 8 * q^59 - 16 * q^60 + 24 * q^61 + 8 * q^62 + 48 * q^63 + 36 * q^64 + 8 * q^65 + 40 * q^66 - 8 * q^67 + 48 * q^69 + 16 * q^70 - 16 * q^71 + 12 * q^72 + 16 * q^73 + 8 * q^75 + 16 * q^76 + 24 * q^77 - 24 * q^78 + 40 * q^79 - 12 * q^80 + 36 * q^81 - 16 * q^82 - 40 * q^83 + 32 * q^84 - 8 * q^86 - 32 * q^87 + 48 * q^88 + 8 * q^89 - 4 * q^90 + 72 * q^91 - 8 * q^92 + 24 * q^93 - 16 * q^94 + 8 * q^96 + 32 * q^97 - 60 * q^98 + 56 * 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

)

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

$2$	−0.962871	−0.680853	−0.340426	−	0.940271i	$-0.610571\pi$
$2$	−0.962871	−0.680853	−0.340426	+	0.940271i	$0.610571\pi$
$3$	−2.64897	−1.52939	−0.764693	−	0.644395i	$-0.777109\pi$
$3$	−2.64897	−1.52939	−0.764693	+	0.644395i	$0.777109\pi$
$4$	−1.07288	−0.536440
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.55062	1.04129
$7$	3.09463	1.16966	0.584830	−	0.811156i	$-0.301161\pi$
$7$	3.09463	1.16966	0.584830	+	0.811156i	$0.301161\pi$
$8$	2.95879	1.04609
$9$	4.01706	1.33902
$10$	0.962871	0.304487
$11$	6.13071	1.84848	0.924239	−	0.381815i	$-0.124701\pi$
$11$	6.13071	1.84848	0.924239	+	0.381815i	$0.124701\pi$
$12$	2.84203	0.820423
$13$	1.16017	0.321775	0.160887	−	0.986973i	$-0.448564\pi$
$13$	1.16017	0.321775	0.160887	+	0.986973i	$0.448564\pi$
$14$	−2.97973	−0.796366
$15$	2.64897	0.683962
$16$	−0.703170	−0.175793
$17$	0	0
$17$	0	0
$18$	−3.86791	−0.911675
$19$	−5.42585	−1.24477	−0.622387	−	0.782709i	$-0.713837\pi$
$19$	−5.42585	−1.24477	−0.622387	+	0.782709i	$0.713837\pi$
$20$	1.07288	0.239903
$21$	−8.19759	−1.78886
$22$	−5.90308	−1.25854
$23$	−3.11745	−0.650033	−0.325017	−	0.945708i	$-0.605370\pi$
$23$	−3.11745	−0.650033	−0.325017	+	0.945708i	$0.605370\pi$
$24$	−7.83775	−1.59987
$25$	1.00000	0.200000
$26$	−1.11710	−0.219081
$27$	−2.69417	−0.518492
$28$	−3.32016	−0.627452
$29$	4.99698	0.927915	0.463958	−	0.885857i	$-0.346429\pi$
$29$	4.99698	0.927915	0.463958	+	0.885857i	$0.346429\pi$
$30$	−2.55062	−0.465677
$31$	3.72804	0.669576	0.334788	−	0.942293i	$-0.391335\pi$
$31$	3.72804	0.669576	0.334788	+	0.942293i	$0.391335\pi$
$32$	−5.24051	−0.926400
$33$	−16.2401	−2.82703
$34$	0	0
$35$	−3.09463	−0.523088
$36$	−4.30982	−0.718304
$37$	−0.396716	−0.0652197	−0.0326098	−	0.999468i	$-0.510382\pi$
$37$	−0.396716	−0.0652197	−0.0326098	+	0.999468i	$0.510382\pi$
$38$	5.22439	0.847508
$39$	−3.07327	−0.492117
$40$	−2.95879	−0.467825
$41$	1.70263	0.265906	0.132953	−	0.991122i	$-0.457554\pi$
$41$	1.70263	0.265906	0.132953	+	0.991122i	$0.457554\pi$
$42$	7.89322	1.21795
$43$	−0.0268304	−0.00409160	−0.00204580	−	0.999998i	$-0.500651\pi$
$43$	−0.0268304	−0.00409160	−0.00204580	+	0.999998i	$0.500651\pi$
$44$	−6.57751	−0.991597
$45$	−4.01706	−0.598828
$46$	3.00170	0.442577
$47$	−5.43715	−0.793090	−0.396545	−	0.918015i	$-0.629791\pi$
$47$	−5.43715	−0.793090	−0.396545	+	0.918015i	$0.629791\pi$
$48$	1.86268	0.268855
$49$	2.57672	0.368103
$50$	−0.962871	−0.136171
$51$	0	0
$52$	−1.24473	−0.172613
$53$	0.345087	0.0474014	0.0237007	−	0.999719i	$-0.492455\pi$
$53$	0.345087	0.0474014	0.0237007	+	0.999719i	$0.492455\pi$
$54$	2.59413	0.353017
$55$	−6.13071	−0.826664
$56$	9.15634	1.22357
$57$	14.3729	1.90374
$58$	−4.81144	−0.631773
$59$	4.06060	0.528645	0.264322	−	0.964434i	$-0.414852\pi$
$59$	4.06060	0.528645	0.264322	+	0.964434i	$0.414852\pi$
$60$	−2.84203	−0.366904
$61$	12.4424	1.59308	0.796541	−	0.604584i	$-0.206661\pi$
$61$	12.4424	1.59308	0.796541	+	0.604584i	$0.206661\pi$
$62$	−3.58962	−0.455883
$63$	12.4313	1.56620
$64$	6.45228	0.806534
$65$	−1.16017	−0.143902
$66$	15.6371	1.92479
$67$	5.62508	0.687213	0.343607	−	0.939114i	$-0.388351\pi$
$67$	5.62508	0.687213	0.343607	+	0.939114i	$0.388351\pi$
$68$	0	0
$69$	8.25804	0.994152
$70$	2.97973	0.356146
$71$	10.7794	1.27928	0.639640	−	0.768674i	$-0.279083\pi$
$71$	10.7794	1.27928	0.639640	+	0.768674i	$0.279083\pi$
$72$	11.8856	1.40073
$73$	−1.65433	−0.193624	−0.0968121	−	0.995303i	$-0.530865\pi$
$73$	−1.65433	−0.193624	−0.0968121	+	0.995303i	$0.530865\pi$
$74$	0.381986	0.0444050
$75$	−2.64897	−0.305877
$76$	5.82128	0.667747
$77$	18.9723	2.16209
$78$	2.95916	0.335059
$79$	−5.27290	−0.593248	−0.296624	−	0.954994i	$-0.595861\pi$
$79$	−5.27290	−0.593248	−0.296624	+	0.954994i	$0.595861\pi$
$80$	0.703170	0.0786168
$81$	−4.91441	−0.546045
$82$	−1.63941	−0.181043
$83$	−12.4258	−1.36391	−0.681955	−	0.731394i	$-0.738870\pi$
$83$	−12.4258	−1.36391	−0.681955	+	0.731394i	$0.738870\pi$
$84$	8.79502	0.959616
$85$	0	0
$86$	0.0258342	0.00278577
$87$	−13.2369	−1.41914
$88$	18.1394	1.93367
$89$	−3.22930	−0.342305	−0.171152	−	0.985245i	$-0.554749\pi$
$89$	−3.22930	−0.342305	−0.171152	+	0.985245i	$0.554749\pi$
$90$	3.86791	0.407714
$91$	3.59031	0.376367
$92$	3.34465	0.348704
$93$	−9.87549	−1.02404
$94$	5.23528	0.539978
$95$	5.42585	0.556680
$96$	13.8820	1.41682
$97$	−12.9349	−1.31334	−0.656668	−	0.754180i	$-0.728035\pi$
$97$	−12.9349	−1.31334	−0.656668	+	0.754180i	$0.728035\pi$
$98$	−2.48105	−0.250624
$99$	24.6274	2.47515
$100$	−1.07288	−0.107288
$101$	−1.46947	−0.146218	−0.0731088	−	0.997324i	$-0.523292\pi$
$101$	−1.46947	−0.146218	−0.0731088	+	0.997324i	$0.523292\pi$
$102$	0	0
$103$	9.80978	0.966586	0.483293	−	0.875459i	$-0.339441\pi$
$103$	9.80978	0.966586	0.483293	+	0.875459i	$0.339441\pi$
$104$	3.43271	0.336605
$105$	8.19759	0.800003
$106$	−0.332275	−0.0322734
$107$	−2.88742	−0.279137	−0.139569	−	0.990212i	$-0.544572\pi$
$107$	−2.88742	−0.279137	−0.139569	+	0.990212i	$0.544572\pi$
$108$	2.89052	0.278140
$109$	5.76330	0.552024	0.276012	−	0.961154i	$-0.410987\pi$
$109$	5.76330	0.552024	0.276012	+	0.961154i	$0.410987\pi$
$110$	5.90308	0.562836
$111$	1.05089	0.0997460
$112$	−2.17605	−0.205617
$113$	−9.02043	−0.848571	−0.424285	−	0.905529i	$-0.639475\pi$
$113$	−9.02043	−0.848571	−0.424285	+	0.905529i	$0.639475\pi$
$114$	−13.8393	−1.29617
$115$	3.11745	0.290704
$116$	−5.36115	−0.497771
$117$	4.66049	0.430863
$118$	−3.90983	−0.359929
$119$	0	0
$120$	7.83775	0.715485
$121$	26.5856	2.41687
$122$	−11.9804	−1.08465
$123$	−4.51022	−0.406673
$124$	−3.99974	−0.359187
$125$	−1.00000	−0.0894427
$126$	−11.9697	−1.06635
$127$	−13.9992	−1.24223	−0.621114	−	0.783720i	$-0.713319\pi$
$127$	−13.9992	−1.24223	−0.621114	+	0.783720i	$0.713319\pi$
$128$	4.26831	0.377269
$129$	0.0710730	0.00625763
$130$	1.11710	0.0979760
$131$	−12.6704	−1.10702	−0.553509	−	0.832843i	$-0.686712\pi$
$131$	−12.6704	−1.10702	−0.553509	+	0.832843i	$0.686712\pi$
$132$	17.4236	1.51653
$133$	−16.7910	−1.45596
$134$	−5.41623	−0.467891
$135$	2.69417	0.231877
$136$	0	0
$137$	2.97888	0.254503	0.127251	−	0.991870i	$-0.459384\pi$
$137$	2.97888	0.254503	0.127251	+	0.991870i	$0.459384\pi$
$138$	−7.95143	−0.676871
$139$	19.6413	1.66595	0.832977	−	0.553308i	$-0.186635\pi$
$139$	19.6413	1.66595	0.832977	+	0.553308i	$0.186635\pi$
$140$	3.32016	0.280605
$141$	14.4029	1.21294
$142$	−10.3792	−0.871002
$143$	7.11269	0.594793
$144$	−2.82468	−0.235390
$145$	−4.99698	−0.414976
$146$	1.59290	0.131830
$147$	−6.82567	−0.562972
$148$	0.425628	0.0349864
$149$	2.95573	0.242143	0.121072	−	0.992644i	$-0.461367\pi$
$149$	2.95573	0.242143	0.121072	+	0.992644i	$0.461367\pi$
$150$	2.55062	0.208257
$151$	22.0403	1.79361	0.896807	−	0.442422i	$-0.145881\pi$
$151$	22.0403	1.79361	0.896807	+	0.442422i	$0.145881\pi$
$152$	−16.0539	−1.30214
$153$	0	0
$154$	−18.2678	−1.47206
$155$	−3.72804	−0.299444
$156$	3.29725	0.263991
$157$	12.8666	1.02686	0.513432	−	0.858130i	$-0.328374\pi$
$157$	12.8666	1.02686	0.513432	+	0.858130i	$0.328374\pi$
$158$	5.07713	0.403914
$159$	−0.914127	−0.0724950
$160$	5.24051	0.414299
$161$	−9.64735	−0.760318
$162$	4.73194	0.371776
$163$	19.2925	1.51110	0.755551	−	0.655089i	$-0.227369\pi$
$163$	19.2925	1.51110	0.755551	+	0.655089i	$0.227369\pi$
$164$	−1.82672	−0.142643
$165$	16.2401	1.26429
$166$	11.9645	0.928622
$167$	18.8591	1.45936	0.729681	−	0.683787i	$-0.239668\pi$
$167$	18.8591	1.45936	0.729681	+	0.683787i	$0.239668\pi$
$168$	−24.2549	−1.87131
$169$	−11.6540	−0.896461
$170$	0	0
$171$	−21.7960	−1.66678
$172$	0.0287858	0.00219490
$173$	−22.5770	−1.71650	−0.858249	−	0.513233i	$-0.828448\pi$
$173$	−22.5770	−1.71650	−0.858249	+	0.513233i	$0.828448\pi$
$174$	12.7454	0.966225
$175$	3.09463	0.233932
$176$	−4.31093	−0.324949
$177$	−10.7564	−0.808502
$178$	3.10940	0.233059
$179$	1.96068	0.146548	0.0732740	−	0.997312i	$-0.476655\pi$
$179$	1.96068	0.146548	0.0732740	+	0.997312i	$0.476655\pi$
$180$	4.30982	0.321235
$181$	−0.711719	−0.0529016	−0.0264508	−	0.999650i	$-0.508421\pi$
$181$	−0.711719	−0.0529016	−0.0264508	+	0.999650i	$0.508421\pi$
$182$	−3.45700	−0.256250
$183$	−32.9595	−2.43644
$184$	−9.22387	−0.679993
$185$	0.396716	0.0291671
$186$	9.50882	0.697220
$187$	0	0
$188$	5.83341	0.425445
$189$	−8.33744	−0.606460
$190$	−5.22439	−0.379017
$191$	−5.26341	−0.380847	−0.190423	−	0.981702i	$-0.560986\pi$
$191$	−5.26341	−0.380847	−0.190423	+	0.981702i	$0.560986\pi$
$192$	−17.0919	−1.23350
$193$	14.9301	1.07469	0.537347	−	0.843361i	$-0.319426\pi$
$193$	14.9301	1.07469	0.537347	+	0.843361i	$0.319426\pi$
$194$	12.4546	0.894188
$195$	3.07327	0.220082
$196$	−2.76451	−0.197465
$197$	−6.33014	−0.451004	−0.225502	−	0.974243i	$-0.572402\pi$
$197$	−6.33014	−0.451004	−0.225502	+	0.974243i	$0.572402\pi$
$198$	−23.7130	−1.68521
$199$	14.9149	1.05729	0.528645	−	0.848843i	$-0.322700\pi$
$199$	14.9149	1.05729	0.528645	+	0.848843i	$0.322700\pi$
$200$	2.95879	0.209218
$201$	−14.9007	−1.05101
$202$	1.41491	0.0995526
$203$	15.4638	1.08534
$204$	0	0
$205$	−1.70263	−0.118917
$206$	−9.44555	−0.658103
$207$	−12.5230	−0.870408
$208$	−0.815800	−0.0565656
$209$	−33.2643	−2.30094
$210$	−7.89322	−0.544684
$211$	13.3987	0.922404	0.461202	−	0.887295i	$-0.347418\pi$
$211$	13.3987	0.922404	0.461202	+	0.887295i	$0.347418\pi$
$212$	−0.370237	−0.0254280
$213$	−28.5544	−1.95651
$214$	2.78021	0.190051
$215$	0.0268304	0.00182982
$216$	−7.97146	−0.542389
$217$	11.5369	0.783176
$218$	−5.54932	−0.375847
$219$	4.38227	0.296126
$220$	6.57751	0.443456
$221$	0	0
$222$	−1.01187	−0.0679123
$223$	−2.11298	−0.141496	−0.0707478	−	0.997494i	$-0.522539\pi$
$223$	−2.11298	−0.141496	−0.0707478	+	0.997494i	$0.522539\pi$
$224$	−16.2174	−1.08357
$225$	4.01706	0.267804
$226$	8.68551	0.577752
$227$	14.5595	0.966348	0.483174	−	0.875524i	$-0.339484\pi$
$227$	14.5595	0.966348	0.483174	+	0.875524i	$0.339484\pi$
$228$	−15.4204	−1.02124
$229$	−2.65040	−0.175143	−0.0875716	−	0.996158i	$-0.527911\pi$
$229$	−2.65040	−0.175143	−0.0875716	+	0.996158i	$0.527911\pi$
$230$	−3.00170	−0.197926
$231$	−50.2570	−3.30667
$232$	14.7850	0.970682
$233$	3.13924	0.205658	0.102829	−	0.994699i	$-0.467210\pi$
$233$	3.13924	0.205658	0.102829	+	0.994699i	$0.467210\pi$
$234$	−4.48745	−0.293354
$235$	5.43715	0.354681
$236$	−4.35653	−0.283586
$237$	13.9678	0.907305
$238$	0	0
$239$	13.7090	0.886760	0.443380	−	0.896334i	$-0.353779\pi$
$239$	13.7090	0.886760	0.443380	+	0.896334i	$0.353779\pi$
$240$	−1.86268	−0.120235
$241$	−12.4877	−0.804404	−0.402202	−	0.915551i	$-0.631755\pi$
$241$	−12.4877	−0.804404	−0.402202	+	0.915551i	$0.631755\pi$
$242$	−25.5985	−1.64553
$243$	21.1006	1.35361
$244$	−13.3492	−0.854593
$245$	−2.57672	−0.164621
$246$	4.34276	0.276884
$247$	−6.29493	−0.400537
$248$	11.0305	0.700436
$249$	32.9157	2.08595
$250$	0.962871	0.0608973
$251$	17.4413	1.10088	0.550441	−	0.834874i	$-0.314459\pi$
$251$	17.4413	1.10088	0.550441	+	0.834874i	$0.314459\pi$
$252$	−13.3373	−0.840171
$253$	−19.1122	−1.20157
$254$	13.4794	0.845774
$255$	0	0
$256$	−17.0144	−1.06340
$257$	6.88101	0.429226	0.214613	−	0.976699i	$-0.431151\pi$
$257$	6.88101	0.429226	0.214613	+	0.976699i	$0.431151\pi$
$258$	−0.0684341	−0.00426052
$259$	−1.22769	−0.0762848
$260$	1.24473	0.0771947
$261$	20.0732	1.24250
$262$	12.2000	0.753716
$263$	6.23809	0.384657	0.192329	−	0.981331i	$-0.438396\pi$
$263$	6.23809	0.384657	0.192329	+	0.981331i	$0.438396\pi$
$264$	−48.0509	−2.95733
$265$	−0.345087	−0.0211986
$266$	16.1675	0.991296
$267$	8.55432	0.523516
$268$	−6.03504	−0.368648
$269$	−1.29087	−0.0787059	−0.0393530	−	0.999225i	$-0.512530\pi$
$269$	−1.29087	−0.0787059	−0.0393530	+	0.999225i	$0.512530\pi$
$270$	−2.59413	−0.157874
$271$	10.2849	0.624764	0.312382	−	0.949957i	$-0.398873\pi$
$271$	10.2849	0.624764	0.312382	+	0.949957i	$0.398873\pi$
$272$	0	0
$273$	−9.51063	−0.575610
$274$	−2.86828	−0.173279
$275$	6.13071	0.369695
$276$	−8.85989	−0.533302
$277$	25.3802	1.52495	0.762474	−	0.647019i	$-0.223985\pi$
$277$	25.3802	1.52495	0.762474	+	0.647019i	$0.223985\pi$
$278$	−18.9120	−1.13427
$279$	14.9758	0.896576
$280$	−9.15634	−0.547196
$281$	5.89557	0.351700	0.175850	−	0.984417i	$-0.443733\pi$
$281$	5.89557	0.351700	0.175850	+	0.984417i	$0.443733\pi$
$282$	−13.8681	−0.825834
$283$	18.4663	1.09771	0.548854	−	0.835918i	$-0.315064\pi$
$283$	18.4663	1.09771	0.548854	+	0.835918i	$0.315064\pi$
$284$	−11.5650	−0.686257
$285$	−14.3729	−0.851378
$286$	−6.84860	−0.404966
$287$	5.26901	0.311020
$288$	−21.0514	−1.24047
$289$	0	0
$290$	4.81144	0.282538
$291$	34.2641	2.00860
$292$	1.77489	0.103868
$293$	8.31894	0.485998	0.242999	−	0.970027i	$-0.421869\pi$
$293$	8.31894	0.485998	0.242999	+	0.970027i	$0.421869\pi$
$294$	6.57224	0.383301
$295$	−4.06060	−0.236417
$296$	−1.17380	−0.0682256
$297$	−16.5171	−0.958422
$298$	−2.84599	−0.164864
$299$	−3.61679	−0.209164
$300$	2.84203	0.164085
$301$	−0.0830301	−0.00478577
$302$	−21.2220	−1.22119
$303$	3.89258	0.223623
$304$	3.81529	0.218822
$305$	−12.4424	−0.712448
$306$	0	0
$307$	−12.2369	−0.698398	−0.349199	−	0.937049i	$-0.613546\pi$
$307$	−12.2369	−0.698398	−0.349199	+	0.937049i	$0.613546\pi$
$308$	−20.3549	−1.15983
$309$	−25.9858	−1.47828
$310$	3.58962	0.203877
$311$	−32.6368	−1.85066	−0.925331	−	0.379160i	$-0.876213\pi$
$311$	−32.6368	−1.85066	−0.925331	+	0.379160i	$0.876213\pi$
$312$	−9.09315	−0.514799
$313$	−13.4290	−0.759053	−0.379526	−	0.925181i	$-0.623913\pi$
$313$	−13.4290	−0.759053	−0.379526	+	0.925181i	$0.623913\pi$
$314$	−12.3889	−0.699143
$315$	−12.4313	−0.700425
$316$	5.65719	0.318242
$317$	9.87357	0.554555	0.277278	−	0.960790i	$-0.410568\pi$
$317$	9.87357	0.554555	0.277278	+	0.960790i	$0.410568\pi$
$318$	0.880187	0.0493584
$319$	30.6350	1.71523
$320$	−6.45228	−0.360693
$321$	7.64869	0.426909
$322$	9.28915	0.517664
$323$	0	0
$324$	5.27257	0.292920
$325$	1.16017	0.0643549
$326$	−18.5762	−1.02884
$327$	−15.2668	−0.844258
$328$	5.03772	0.278162
$329$	−16.8260	−0.927646
$330$	−15.6371	−0.860794
$331$	−5.64305	−0.310170	−0.155085	−	0.987901i	$-0.549565\pi$
$331$	−5.64305	−0.310170	−0.155085	+	0.987901i	$0.549565\pi$
$332$	13.3314	0.731656
$333$	−1.59363	−0.0873304
$334$	−18.1589	−0.993611
$335$	−5.62508	−0.307331
$336$	5.76430	0.314468
$337$	−16.5925	−0.903853	−0.451927	−	0.892055i	$-0.649263\pi$
$337$	−16.5925	−0.903853	−0.451927	+	0.892055i	$0.649263\pi$
$338$	11.2213	0.610358
$339$	23.8949	1.29779
$340$	0	0
$341$	22.8555	1.23770
$342$	20.9867	1.13483
$343$	−13.6884	−0.739104
$344$	−0.0793854	−0.00428017
$345$	−8.25804	−0.444598
$346$	21.7387	1.16868
$347$	9.24424	0.496257	0.248129	−	0.968727i	$-0.420184\pi$
$347$	9.24424	0.496257	0.248129	+	0.968727i	$0.420184\pi$
$348$	14.2016	0.761283
$349$	26.9273	1.44139	0.720694	−	0.693254i	$-0.243823\pi$
$349$	26.9273	1.44139	0.720694	+	0.693254i	$0.243823\pi$
$350$	−2.97973	−0.159273
$351$	−3.12570	−0.166838
$352$	−32.1280	−1.71243
$353$	−22.9722	−1.22269	−0.611343	−	0.791366i	$-0.709370\pi$
$353$	−22.9722	−1.22269	−0.611343	+	0.791366i	$0.709370\pi$
$354$	10.3570	0.550470
$355$	−10.7794	−0.572112
$356$	3.46465	0.183626
$357$	0	0
$358$	−1.88788	−0.0997775
$359$	34.1089	1.80020	0.900101	−	0.435682i	$-0.143493\pi$
$359$	34.1089	1.80020	0.900101	+	0.435682i	$0.143493\pi$
$360$	−11.8856	−0.626427
$361$	10.4398	0.549463
$362$	0.685293	0.0360182
$363$	−70.4244	−3.69632
$364$	−3.85197	−0.201898
$365$	1.65433	0.0865914
$366$	31.7358	1.65885
$367$	−9.27610	−0.484209	−0.242104	−	0.970250i	$-0.577838\pi$
$367$	−9.27610	−0.484209	−0.242104	+	0.970250i	$0.577838\pi$
$368$	2.19210	0.114271
$369$	6.83957	0.356054
$370$	−0.381986	−0.0198585
$371$	1.06792	0.0554435
$372$	10.5952	0.549336
$373$	−3.06857	−0.158884	−0.0794422	−	0.996839i	$-0.525314\pi$
$373$	−3.06857	−0.158884	−0.0794422	+	0.996839i	$0.525314\pi$
$374$	0	0
$375$	2.64897	0.136792
$376$	−16.0874	−0.829643
$377$	5.79736	0.298579
$378$	8.02788	0.412910
$379$	36.5339	1.87662	0.938310	−	0.345795i	$-0.112391\pi$
$379$	36.5339	1.87662	0.938310	+	0.345795i	$0.112391\pi$
$380$	−5.82128	−0.298625
$381$	37.0835	1.89984
$382$	5.06798	0.259301
$383$	−9.32150	−0.476306	−0.238153	−	0.971228i	$-0.576542\pi$
$383$	−9.32150	−0.476306	−0.238153	+	0.971228i	$0.576542\pi$
$384$	−11.3066	−0.576990
$385$	−18.9723	−0.966916
$386$	−14.3758	−0.731709
$387$	−0.107779	−0.00547873
$388$	13.8775	0.704525
$389$	33.1623	1.68139	0.840697	−	0.541507i	$-0.182146\pi$
$389$	33.1623	1.68139	0.840697	+	0.541507i	$0.182146\pi$
$390$	−2.95916	−0.149843
$391$	0	0
$392$	7.62397	0.385069
$393$	33.5636	1.69306
$394$	6.09511	0.307067
$395$	5.27290	0.265309
$396$	−26.4223	−1.32777
$397$	−8.82540	−0.442934	−0.221467	−	0.975168i	$-0.571085\pi$
$397$	−8.82540	−0.442934	−0.221467	+	0.975168i	$0.571085\pi$
$398$	−14.3611	−0.719859
$399$	44.4789	2.22673
$400$	−0.703170	−0.0351585
$401$	−19.5190	−0.974732	−0.487366	−	0.873198i	$-0.662042\pi$
$401$	−19.5190	−0.974732	−0.487366	+	0.873198i	$0.662042\pi$
$402$	14.3474	0.715585
$403$	4.32518	0.215453
$404$	1.57656	0.0784369
$405$	4.91441	0.244199
$406$	−14.8896	−0.738960
$407$	−2.43215	−0.120557
$408$	0	0
$409$	10.4152	0.514998	0.257499	−	0.966279i	$-0.417102\pi$
$409$	10.4152	0.514998	0.257499	+	0.966279i	$0.417102\pi$
$410$	1.63941	0.0809649
$411$	−7.89097	−0.389233
$412$	−10.5247	−0.518515
$413$	12.5660	0.618334
$414$	12.0580	0.592619
$415$	12.4258	0.609959
$416$	−6.07991	−0.298092
$417$	−52.0293	−2.54789
$418$	32.0292	1.56660
$419$	−35.3206	−1.72552	−0.862762	−	0.505610i	$-0.831267\pi$
$419$	−35.3206	−1.72552	−0.862762	+	0.505610i	$0.831267\pi$
$420$	−8.79502	−0.429153
$421$	−15.0205	−0.732052	−0.366026	−	0.930605i	$-0.619282\pi$
$421$	−15.0205	−0.732052	−0.366026	+	0.930605i	$0.619282\pi$
$422$	−12.9012	−0.628021
$423$	−21.8414	−1.06196
$424$	1.02104	0.0495861
$425$	0	0
$426$	27.4942	1.33210
$427$	38.5045	1.86336
$428$	3.09785	0.149740
$429$	−18.8413	−0.909668
$430$	−0.0258342	−0.00124584
$431$	−23.3518	−1.12481	−0.562407	−	0.826860i	$-0.690125\pi$
$431$	−23.3518	−1.12481	−0.562407	+	0.826860i	$0.690125\pi$
$432$	1.89446	0.0911471
$433$	−18.3077	−0.879811	−0.439906	−	0.898044i	$-0.644988\pi$
$433$	−18.3077	−0.879811	−0.439906	+	0.898044i	$0.644988\pi$
$434$	−11.1086	−0.533228
$435$	13.2369	0.634659
$436$	−6.18333	−0.296128
$437$	16.9148	0.809145
$438$	−4.21956	−0.201618
$439$	−20.2697	−0.967421	−0.483711	−	0.875228i	$-0.660711\pi$
$439$	−20.2697	−0.967421	−0.483711	+	0.875228i	$0.660711\pi$
$440$	−18.1394	−0.864764
$441$	10.3509	0.492898
$442$	0	0
$443$	29.3849	1.39612	0.698058	−	0.716041i	$-0.254048\pi$
$443$	29.3849	1.39612	0.698058	+	0.716041i	$0.254048\pi$
$444$	−1.12748	−0.0535077
$445$	3.22930	0.153083
$446$	2.03453	0.0963376
$447$	−7.82966	−0.370330
$448$	19.9674	0.943371
$449$	−32.6419	−1.54047	−0.770234	−	0.637761i	$-0.779861\pi$
$449$	−32.6419	−1.54047	−0.770234	+	0.637761i	$0.779861\pi$
$450$	−3.86791	−0.182335
$451$	10.4383	0.491522
$452$	9.67784	0.455207
$453$	−58.3842	−2.74313
$454$	−14.0189	−0.657940
$455$	−3.59031	−0.168316
$456$	42.5264	1.99148
$457$	−8.45112	−0.395327	−0.197663	−	0.980270i	$-0.563335\pi$
$457$	−8.45112	−0.395327	−0.197663	+	0.980270i	$0.563335\pi$
$458$	2.55199	0.119247
$459$	0	0
$460$	−3.34465	−0.155945
$461$	−38.2583	−1.78187	−0.890933	−	0.454134i	$-0.849949\pi$
$461$	−38.2583	−1.78187	−0.890933	+	0.454134i	$0.849949\pi$
$462$	48.3910	2.25135
$463$	27.1761	1.26298	0.631491	−	0.775383i	$-0.282443\pi$
$463$	27.1761	1.26298	0.631491	+	0.775383i	$0.282443\pi$
$464$	−3.51373	−0.163121
$465$	9.87549	0.457965
$466$	−3.02268	−0.140023
$467$	3.44525	0.159427	0.0797136	−	0.996818i	$-0.474599\pi$
$467$	3.44525	0.159427	0.0797136	+	0.996818i	$0.474599\pi$
$468$	−5.00015	−0.231132
$469$	17.4075	0.803805
$470$	−5.23528	−0.241485
$471$	−34.0832	−1.57047
$472$	12.0144	0.553009
$473$	−0.164489	−0.00756322
$474$	−13.4492	−0.617741
$475$	−5.42585	−0.248955
$476$	0	0
$477$	1.38624	0.0634714
$478$	−13.2000	−0.603753
$479$	15.3593	0.701785	0.350893	−	0.936416i	$-0.385878\pi$
$479$	15.3593	0.701785	0.350893	+	0.936416i	$0.385878\pi$
$480$	−13.8820	−0.633622
$481$	−0.460260	−0.0209860
$482$	12.0241	0.547681
$483$	25.5556	1.16282
$484$	−28.5231	−1.29650
$485$	12.9349	0.587341
$486$	−20.3172	−0.921606
$487$	26.3095	1.19220	0.596099	−	0.802911i	$-0.296717\pi$
$487$	26.3095	1.19220	0.596099	+	0.802911i	$0.296717\pi$
$488$	36.8143	1.66651
$489$	−51.1052	−2.31106
$490$	2.48105	0.112083
$491$	−29.0366	−1.31040	−0.655202	−	0.755454i	$-0.727417\pi$
$491$	−29.0366	−1.31040	−0.655202	+	0.755454i	$0.727417\pi$
$492$	4.83893	0.218156
$493$	0	0
$494$	6.06120	0.272706
$495$	−24.6274	−1.10692
$496$	−2.62145	−0.117707
$497$	33.3583	1.49632
$498$	−31.6935	−1.42022
$499$	32.5905	1.45895	0.729475	−	0.684007i	$-0.239764\pi$
$499$	32.5905	1.45895	0.729475	+	0.684007i	$0.239764\pi$
$500$	1.07288	0.0479806
$501$	−49.9573	−2.23193
$502$	−16.7937	−0.749539
$503$	−23.5391	−1.04956	−0.524779	−	0.851239i	$-0.675852\pi$
$503$	−23.5391	−1.04956	−0.524779	+	0.851239i	$0.675852\pi$
$504$	36.7816	1.63838
$505$	1.46947	0.0653905
$506$	18.4026	0.818093
$507$	30.8711	1.37103
$508$	15.0194	0.666380
$509$	−17.5818	−0.779300	−0.389650	−	0.920963i	$-0.627404\pi$
$509$	−17.5818	−0.779300	−0.389650	+	0.920963i	$0.627404\pi$
$510$	0	0
$511$	−5.11953	−0.226474
$512$	7.84603	0.346749
$513$	14.6181	0.645406
$514$	−6.62553	−0.292239
$515$	−9.80978	−0.432271
$516$	−0.0762528	−0.00335684
$517$	−33.3336	−1.46601
$518$	1.18210	0.0519387
$519$	59.8059	2.62519
$520$	−3.43271	−0.150534
$521$	20.5061	0.898388	0.449194	−	0.893434i	$-0.351711\pi$
$521$	20.5061	0.898388	0.449194	+	0.893434i	$0.351711\pi$
$522$	−19.3279	−0.845957
$523$	9.19853	0.402224	0.201112	−	0.979568i	$-0.435545\pi$
$523$	9.19853	0.402224	0.201112	+	0.979568i	$0.435545\pi$
$524$	13.5938	0.593849
$525$	−8.19759	−0.357772
$526$	−6.00648	−0.261895
$527$	0	0
$528$	11.4195	0.496972
$529$	−13.2815	−0.577457
$530$	0.332275	0.0144331
$531$	16.3117	0.707866
$532$	18.0147	0.781036
$533$	1.97535	0.0855618
$534$	−8.23671	−0.356437
$535$	2.88742	0.124834
$536$	16.6434	0.718886
$537$	−5.19378	−0.224128
$538$	1.24294	0.0535871
$539$	15.7971	0.680431
$540$	−2.89052	−0.124388
$541$	14.2880	0.614291	0.307145	−	0.951663i	$-0.400626\pi$
$541$	14.2880	0.614291	0.307145	+	0.951663i	$0.400626\pi$
$542$	−9.90305	−0.425372
$543$	1.88532	0.0809070
$544$	0	0
$545$	−5.76330	−0.246873
$546$	9.15751	0.391905
$547$	23.7010	1.01338	0.506691	−	0.862128i	$-0.330869\pi$
$547$	23.7010	1.01338	0.506691	+	0.862128i	$0.330869\pi$
$548$	−3.19598	−0.136525
$549$	49.9818	2.13317
$550$	−5.90308	−0.251708
$551$	−27.1128	−1.15505
$552$	24.4338	1.03997
$553$	−16.3177	−0.693898
$554$	−24.4378	−1.03826
$555$	−1.05089	−0.0446078
$556$	−21.0728	−0.893684
$557$	0.399812	0.0169406	0.00847028	−	0.999964i	$-0.497304\pi$
$557$	0.399812	0.0169406	0.00847028	+	0.999964i	$0.497304\pi$
$558$	−14.4197	−0.610436
$559$	−0.0311279	−0.00131657
$560$	2.17605	0.0919549
$561$	0	0
$562$	−5.67668	−0.239456
$563$	9.26358	0.390413	0.195207	−	0.980762i	$-0.437462\pi$
$563$	9.26358	0.390413	0.195207	+	0.980762i	$0.437462\pi$
$564$	−15.4525	−0.650670
$565$	9.02043	0.379492
$566$	−17.7807	−0.747377
$567$	−15.2083	−0.638687
$568$	31.8940	1.33824
$569$	5.75650	0.241325	0.120662	−	0.992694i	$-0.461498\pi$
$569$	5.75650	0.241325	0.120662	+	0.992694i	$0.461498\pi$
$570$	13.8393	0.579663
$571$	13.9987	0.585825	0.292913	−	0.956139i	$-0.405375\pi$
$571$	13.9987	0.585825	0.292913	+	0.956139i	$0.405375\pi$
$572$	−7.63106	−0.319071
$573$	13.9426	0.582462
$574$	−5.07337	−0.211759
$575$	−3.11745	−0.130007
$576$	25.9192	1.07997
$577$	18.4417	0.767736	0.383868	−	0.923388i	$-0.374592\pi$
$577$	18.4417	0.767736	0.383868	+	0.923388i	$0.374592\pi$
$578$	0	0
$579$	−39.5495	−1.64362
$580$	5.36115	0.222610
$581$	−38.4533	−1.59531
$582$	−32.9919	−1.36756
$583$	2.11563	0.0876204
$584$	−4.89480	−0.202548
$585$	−4.66049	−0.192688
$586$	−8.01007	−0.330893
$587$	4.03167	0.166405	0.0832024	−	0.996533i	$-0.473485\pi$
$587$	4.03167	0.166405	0.0832024	+	0.996533i	$0.473485\pi$
$588$	7.32312	0.302001
$589$	−20.2278	−0.833471
$590$	3.90983	0.160965
$591$	16.7684	0.689758
$592$	0.278959	0.0114651
$593$	−6.52416	−0.267915	−0.133958	−	0.990987i	$-0.542769\pi$
$593$	−6.52416	−0.267915	−0.133958	+	0.990987i	$0.542769\pi$
$594$	15.9039	0.652544
$595$	0	0
$596$	−3.17114	−0.129895
$597$	−39.5092	−1.61700
$598$	3.48250	0.142410
$599$	6.81189	0.278326	0.139163	−	0.990269i	$-0.455559\pi$
$599$	6.81189	0.278326	0.139163	+	0.990269i	$0.455559\pi$
$600$	−7.83775	−0.319975
$601$	38.1591	1.55654	0.778272	−	0.627927i	$-0.216096\pi$
$601$	38.1591	1.55654	0.778272	+	0.627927i	$0.216096\pi$
$602$	0.0799473	0.00325841
$603$	22.5963	0.920192
$604$	−23.6466	−0.962166
$605$	−26.5856	−1.08086
$606$	−3.74805	−0.152254
$607$	27.1567	1.10226	0.551129	−	0.834420i	$-0.314197\pi$
$607$	27.1567	1.10226	0.551129	+	0.834420i	$0.314197\pi$
$608$	28.4342	1.15316
$609$	−40.9632	−1.65991
$610$	11.9804	0.485072
$611$	−6.30805	−0.255196
$612$	0	0
$613$	2.48591	0.100405	0.0502025	−	0.998739i	$-0.484013\pi$
$613$	2.48591	0.100405	0.0502025	+	0.998739i	$0.484013\pi$
$614$	11.7826	0.475506
$615$	4.51022	0.181870
$616$	56.1349	2.26174
$617$	43.1941	1.73893	0.869465	−	0.493994i	$-0.164464\pi$
$617$	43.1941	1.73893	0.869465	+	0.493994i	$0.164464\pi$
$618$	25.0210	1.00649
$619$	−1.37692	−0.0553430	−0.0276715	−	0.999617i	$-0.508809\pi$
$619$	−1.37692	−0.0553430	−0.0276715	+	0.999617i	$0.508809\pi$
$620$	3.99974	0.160633
$621$	8.39893	0.337037
$622$	31.4250	1.26003
$623$	−9.99347	−0.400380
$624$	2.16103	0.0865106
$625$	1.00000	0.0400000
$626$	12.9304	0.516803
$627$	88.1162	3.51902
$628$	−13.8043	−0.550851
$629$	0	0
$630$	11.9697	0.476886
$631$	−32.4351	−1.29122	−0.645611	−	0.763667i	$-0.723397\pi$
$631$	−32.4351	−1.29122	−0.645611	+	0.763667i	$0.723397\pi$
$632$	−15.6014	−0.620590
$633$	−35.4928	−1.41071
$634$	−9.50698	−0.377570
$635$	13.9992	0.555541
$636$	0.980749	0.0388892
$637$	2.98945	0.118446
$638$	−29.4975	−1.16782
$639$	43.3015	1.71298
$640$	−4.26831	−0.168720
$641$	−22.2992	−0.880764	−0.440382	−	0.897811i	$-0.645157\pi$
$641$	−22.2992	−0.880764	−0.440382	+	0.897811i	$0.645157\pi$
$642$	−7.36471	−0.290662
$643$	11.7260	0.462430	0.231215	−	0.972903i	$-0.425730\pi$
$643$	11.7260	0.462430	0.231215	+	0.972903i	$0.425730\pi$
$644$	10.3504	0.407865
$645$	−0.0710730	−0.00279850
$646$	0	0
$647$	−37.3217	−1.46727	−0.733634	−	0.679544i	$-0.762177\pi$
$647$	−37.3217	−1.46727	−0.733634	+	0.679544i	$0.762177\pi$
$648$	−14.5407	−0.571212
$649$	24.8943	0.977188
$650$	−1.11710	−0.0438162
$651$	−30.5610	−1.19778
$652$	−20.6985	−0.810616
$653$	−1.75573	−0.0687071	−0.0343535	−	0.999410i	$-0.510937\pi$
$653$	−1.75573	−0.0687071	−0.0343535	+	0.999410i	$0.510937\pi$
$654$	14.7000	0.574815
$655$	12.6704	0.495074
$656$	−1.19724	−0.0467443
$657$	−6.64553	−0.259267
$658$	16.2012	0.631590
$659$	6.02834	0.234831	0.117415	−	0.993083i	$-0.462539\pi$
$659$	6.02834	0.234831	0.117415	+	0.993083i	$0.462539\pi$
$660$	−17.4236	−0.678215
$661$	32.0199	1.24543	0.622715	−	0.782449i	$-0.286030\pi$
$661$	32.0199	1.24543	0.622715	+	0.782449i	$0.286030\pi$
$662$	5.43353	0.211180
$663$	0	0
$664$	−36.7653	−1.42677
$665$	16.7910	0.651126
$666$	1.53446	0.0594592
$667$	−15.5778	−0.603176
$668$	−20.2336	−0.782860
$669$	5.59723	0.216401
$670$	5.41623	0.209247
$671$	76.2805	2.94478
$672$	42.9595	1.65720
$673$	39.5902	1.52609	0.763044	−	0.646346i	$-0.223704\pi$
$673$	39.5902	1.52609	0.763044	+	0.646346i	$0.223704\pi$
$674$	15.9765	0.615391
$675$	−2.69417	−0.103698
$676$	12.5033	0.480897
$677$	−6.99467	−0.268827	−0.134413	−	0.990925i	$-0.542915\pi$
$677$	−6.99467	−0.268827	−0.134413	+	0.990925i	$0.542915\pi$
$678$	−23.0077	−0.883605
$679$	−40.0286	−1.53616
$680$	0	0
$681$	−38.5677	−1.47792
$682$	−22.0069	−0.842689
$683$	37.2155	1.42401	0.712005	−	0.702174i	$-0.247787\pi$
$683$	37.2155	1.42401	0.712005	+	0.702174i	$0.247787\pi$
$684$	23.3844	0.894126
$685$	−2.97888	−0.113817
$686$	13.1802	0.503221
$687$	7.02084	0.267862
$688$	0.0188663	0.000719272 0
$689$	0.400362	0.0152526
$690$	7.95143	0.302706
$691$	15.6482	0.595286	0.297643	−	0.954677i	$-0.403800\pi$
$691$	15.6482	0.595286	0.297643	+	0.954677i	$0.403800\pi$
$692$	24.2224	0.920798
$693$	76.2127	2.89508
$694$	−8.90101	−0.337878
$695$	−19.6413	−0.745037
$696$	−39.1650	−1.48455
$697$	0	0
$698$	−25.9276	−0.981372
$699$	−8.31576	−0.314531
$700$	−3.32016	−0.125490
$701$	−6.08551	−0.229847	−0.114923	−	0.993374i	$-0.536662\pi$
$701$	−6.08551	−0.229847	−0.114923	+	0.993374i	$0.536662\pi$
$702$	3.00965	0.113592
$703$	2.15252	0.0811838
$704$	39.5570	1.49086
$705$	−14.4029	−0.542444
$706$	22.1193	0.832469
$707$	−4.54746	−0.171025
$708$	11.5403	0.433712
$709$	−19.7081	−0.740154	−0.370077	−	0.929001i	$-0.620669\pi$
$709$	−19.7081	−0.740154	−0.370077	+	0.929001i	$0.620669\pi$
$710$	10.3792	0.389524
$711$	−21.1816	−0.794371
$712$	−9.55480	−0.358081
$713$	−11.6220	−0.435247
$714$	0	0
$715$	−7.11269	−0.266000
$716$	−2.10357	−0.0786141
$717$	−36.3147	−1.35620
$718$	−32.8425	−1.22567
$719$	−15.9164	−0.593583	−0.296791	−	0.954942i	$-0.595917\pi$
$719$	−15.9164	−0.593583	−0.296791	+	0.954942i	$0.595917\pi$
$720$	2.82468	0.105270
$721$	30.3576	1.13058
$722$	−10.0522	−0.374104
$723$	33.0796	1.23024
$724$	0.763588	0.0283785
$725$	4.99698	0.185583
$726$	67.8096	2.51665
$727$	22.2643	0.825736	0.412868	−	0.910791i	$-0.364527\pi$
$727$	22.2643	0.825736	0.412868	+	0.910791i	$0.364527\pi$
$728$	10.6230	0.393713
$729$	−41.1518	−1.52414
$730$	−1.59290	−0.0589560
$731$	0	0
$732$	35.3616	1.30700
$733$	−5.67351	−0.209556	−0.104778	−	0.994496i	$-0.533413\pi$
$733$	−5.67351	−0.209556	−0.104778	+	0.994496i	$0.533413\pi$
$734$	8.93169	0.329675
$735$	6.82567	0.251769
$736$	16.3370	0.602191
$737$	34.4857	1.27030
$738$	−6.58562	−0.242420
$739$	−39.7975	−1.46397	−0.731987	−	0.681319i	$-0.761407\pi$
$739$	−39.7975	−1.46397	−0.731987	+	0.681319i	$0.761407\pi$
$740$	−0.425628	−0.0156464
$741$	16.6751	0.612575
$742$	−1.02827	−0.0377488
$743$	10.6500	0.390709	0.195355	−	0.980733i	$-0.437414\pi$
$743$	10.6500	0.390709	0.195355	+	0.980733i	$0.437414\pi$
$744$	−29.2195	−1.07124
$745$	−2.95573	−0.108290
$746$	2.95464	0.108177
$747$	−49.9153	−1.82630
$748$	0	0
$749$	−8.93549	−0.326496
$750$	−2.55062	−0.0931355
$751$	18.1195	0.661190	0.330595	−	0.943773i	$-0.392751\pi$
$751$	18.1195	0.661190	0.330595	+	0.943773i	$0.392751\pi$
$752$	3.82324	0.139419
$753$	−46.2014	−1.68367
$754$	−5.58211	−0.203289
$755$	−22.0403	−0.802128
$756$	8.94507	0.325329
$757$	−17.0771	−0.620676	−0.310338	−	0.950626i	$-0.600442\pi$
$757$	−17.0771	−0.620676	−0.310338	+	0.950626i	$0.600442\pi$
$758$	−35.1774	−1.27770
$759$	50.6276	1.83767
$760$	16.0539	0.582337
$761$	−31.9719	−1.15898	−0.579491	−	0.814979i	$-0.696749\pi$
$761$	−31.9719	−1.15898	−0.579491	+	0.814979i	$0.696749\pi$
$762$	−35.7066	−1.29351
$763$	17.8353	0.645681
$764$	5.64701	0.204301
$765$	0	0
$766$	8.97540	0.324294
$767$	4.71100	0.170104
$768$	45.0707	1.62635
$769$	−34.9147	−1.25906	−0.629529	−	0.776977i	$-0.716752\pi$
$769$	−34.9147	−1.25906	−0.629529	+	0.776977i	$0.716752\pi$
$770$	18.2678	0.658327
$771$	−18.2276	−0.656452
$772$	−16.0182	−0.576509
$773$	4.49160	0.161552	0.0807758	−	0.996732i	$-0.474260\pi$
$773$	4.49160	0.161552	0.0807758	+	0.996732i	$0.474260\pi$
$774$	0.103778	0.00373021
$775$	3.72804	0.133915
$776$	−38.2715	−1.37387
$777$	3.25211	0.116669
$778$	−31.9310	−1.14478
$779$	−9.23821	−0.330993
$780$	−3.29725	−0.118061
$781$	66.0854	2.36472
$782$	0	0
$783$	−13.4627	−0.481117
$784$	−1.81188	−0.0647098
$785$	−12.8666	−0.459228
$786$	−32.3174	−1.15272
$787$	−8.65141	−0.308390	−0.154195	−	0.988040i	$-0.549278\pi$
$787$	−8.65141	−0.308390	−0.154195	+	0.988040i	$0.549278\pi$
$788$	6.79148	0.241936
$789$	−16.5245	−0.588289
$790$	−5.07713	−0.180636
$791$	−27.9149	−0.992539
$792$	72.8673	2.58923
$793$	14.4353	0.512613
$794$	8.49772	0.301573
$795$	0.914127	0.0324208
$796$	−16.0019	−0.567173
$797$	19.0137	0.673500	0.336750	−	0.941594i	$-0.390672\pi$
$797$	19.0137	0.673500	0.336750	+	0.941594i	$0.390672\pi$
$798$	−42.8274	−1.51607
$799$	0	0
$800$	−5.24051	−0.185280
$801$	−12.9723	−0.458353
$802$	18.7943	0.663649
$803$	−10.1422	−0.357910
$804$	15.9866	0.563806
$805$	9.64735	0.340024
$806$	−4.16459	−0.146691
$807$	3.41949	0.120372
$808$	−4.34784	−0.152957
$809$	22.6712	0.797077	0.398538	−	0.917152i	$-0.369518\pi$
$809$	22.6712	0.797077	0.398538	+	0.917152i	$0.369518\pi$
$810$	−4.73194	−0.166263
$811$	24.1167	0.846850	0.423425	−	0.905931i	$-0.360828\pi$
$811$	24.1167	0.846850	0.423425	+	0.905931i	$0.360828\pi$
$812$	−16.5908	−0.582222
$813$	−27.2445	−0.955505
$814$	2.34184	0.0820816
$815$	−19.2925	−0.675786
$816$	0	0
$817$	0.145578	0.00509311
$818$	−10.0285	−0.350638
$819$	14.4225	0.503963
$820$	1.82672	0.0637917
$821$	−0.114687	−0.00400260	−0.00200130	−	0.999998i	$-0.500637\pi$
$821$	−0.114687	−0.00400260	−0.00200130	+	0.999998i	$0.500637\pi$
$822$	7.59799	0.265010
$823$	−33.8610	−1.18032	−0.590161	−	0.807286i	$-0.700936\pi$
$823$	−33.8610	−1.18032	−0.590161	+	0.807286i	$0.700936\pi$
$824$	29.0250	1.01114
$825$	−16.2401	−0.565407
$826$	−12.0995	−0.420995
$827$	20.5646	0.715102	0.357551	−	0.933894i	$-0.383612\pi$
$827$	20.5646	0.715102	0.357551	+	0.933894i	$0.383612\pi$
$828$	13.4357	0.466921
$829$	26.7935	0.930576	0.465288	−	0.885159i	$-0.345951\pi$
$829$	26.7935	0.930576	0.465288	+	0.885159i	$0.345951\pi$
$830$	−11.9645	−0.415292
$831$	−67.2314	−2.33223
$832$	7.48577	0.259522
$833$	0	0
$834$	50.0975	1.73473
$835$	−18.8591	−0.652647
$836$	35.6886	1.23431
$837$	−10.0440	−0.347170
$838$	34.0092	1.17483
$839$	13.8332	0.477575	0.238788	−	0.971072i	$-0.423250\pi$
$839$	13.8332	0.477575	0.238788	+	0.971072i	$0.423250\pi$
$840$	24.2549	0.836874
$841$	−4.03023	−0.138973
$842$	14.4628	0.498419
$843$	−15.6172	−0.537886
$844$	−14.3752	−0.494814
$845$	11.6540	0.400910
$846$	21.0304	0.723041
$847$	82.2724	2.82691
$848$	−0.242655	−0.00833281
$849$	−48.9168	−1.67882
$850$	0	0
$851$	1.23674	0.0423950
$852$	30.6354	1.04955
$853$	12.7815	0.437631	0.218816	−	0.975766i	$-0.429781\pi$
$853$	12.7815	0.437631	0.218816	+	0.975766i	$0.429781\pi$
$854$	−37.0749	−1.26868
$855$	21.7960	0.745406
$856$	−8.54325	−0.292002
$857$	6.31557	0.215736	0.107868	−	0.994165i	$-0.465598\pi$
$857$	6.31557	0.215736	0.107868	+	0.994165i	$0.465598\pi$
$858$	18.1418	0.619350
$859$	−35.1334	−1.19874	−0.599369	−	0.800473i	$-0.704582\pi$
$859$	−35.1334	−1.19874	−0.599369	+	0.800473i	$0.704582\pi$
$860$	−0.0287858	−0.000981587 0
$861$	−13.9575	−0.475669
$862$	22.4847	0.765833
$863$	−44.2102	−1.50493	−0.752466	−	0.658632i	$-0.771136\pi$
$863$	−44.2102	−1.50493	−0.752466	+	0.658632i	$0.771136\pi$
$864$	14.1188	0.480331
$865$	22.5770	0.767641
$866$	17.6279	0.599022
$867$	0	0
$868$	−12.3777	−0.420127
$869$	−32.3266	−1.09661
$870$	−12.7454	−0.432109
$871$	6.52608	0.221128
$872$	17.0524	0.577467
$873$	−51.9601	−1.75858
$874$	−16.2868	−0.550908
$875$	−3.09463	−0.104618
$876$	−4.70164	−0.158854
$877$	−23.6073	−0.797164	−0.398582	−	0.917133i	$-0.630497\pi$
$877$	−23.6073	−0.797164	−0.398582	+	0.917133i	$0.630497\pi$
$878$	19.5171	0.658671
$879$	−22.0367	−0.743278
$880$	4.31093	0.145321
$881$	33.0624	1.11390	0.556950	−	0.830546i	$-0.311971\pi$
$881$	33.0624	1.11390	0.556950	+	0.830546i	$0.311971\pi$
$882$	−9.96654	−0.335591
$883$	58.1141	1.95569	0.977847	−	0.209320i	$-0.0671251\pi$
$883$	58.1141	1.95569	0.977847	+	0.209320i	$0.0671251\pi$
$884$	0	0
$885$	10.7564	0.361573
$886$	−28.2938	−0.950550
$887$	−28.0097	−0.940474	−0.470237	−	0.882540i	$-0.655832\pi$
$887$	−28.0097	−0.940474	−0.470237	+	0.882540i	$0.655832\pi$
$888$	3.10936	0.104343
$889$	−43.3223	−1.45298
$890$	−3.10940	−0.104227
$891$	−30.1288	−1.00935
$892$	2.26697	0.0759039
$893$	29.5012	0.987219
$894$	7.53895	0.252140
$895$	−1.96068	−0.0655382
$896$	13.2088	0.441276
$897$	9.58077	0.319893
$898$	31.4300	1.04883
$899$	18.6289	0.621310
$900$	−4.30982	−0.143661
$901$	0	0
$902$	−10.0508	−0.334654
$903$	0.219944	0.00731929
$904$	−26.6895	−0.887680
$905$	0.711719	0.0236583
$906$	56.2164	1.86766
$907$	49.9204	1.65758	0.828790	−	0.559560i	$-0.189030\pi$
$907$	49.9204	1.65758	0.828790	+	0.559560i	$0.189030\pi$
$908$	−15.6206	−0.518387
$909$	−5.90294	−0.195788
$910$	3.45700	0.114599
$911$	21.8797	0.724908	0.362454	−	0.932002i	$-0.381939\pi$
$911$	21.8797	0.724908	0.362454	+	0.932002i	$0.381939\pi$
$912$	−10.1066	−0.334663
$913$	−76.1790	−2.52116
$914$	8.13734	0.269159
$915$	32.9595	1.08961
$916$	2.84356	0.0939538
$917$	−39.2102	−1.29483
$918$	0	0
$919$	−11.0529	−0.364600	−0.182300	−	0.983243i	$-0.558354\pi$
$919$	−11.0529	−0.364600	−0.182300	+	0.983243i	$0.558354\pi$
$920$	9.22387	0.304102
$921$	32.4153	1.06812
$922$	36.8378	1.21319
$923$	12.5060	0.411640
$924$	53.9197	1.77383
$925$	−0.396716	−0.0130439
$926$	−26.1671	−0.859905
$927$	39.4065	1.29428
$928$	−26.1867	−0.859621
$929$	−33.1333	−1.08707	−0.543534	−	0.839387i	$-0.682914\pi$
$929$	−33.1333	−1.08707	−0.543534	+	0.839387i	$0.682914\pi$
$930$	−9.50882	−0.311806
$931$	−13.9809	−0.458206
$932$	−3.36803	−0.110323
$933$	86.4540	2.83038
$934$	−3.31733	−0.108546
$935$	0	0
$936$	13.7894	0.450721
$937$	−58.3815	−1.90724	−0.953621	−	0.301011i	$-0.902676\pi$
$937$	−58.3815	−1.90724	−0.953621	+	0.301011i	$0.902676\pi$
$938$	−16.7612	−0.547273
$939$	35.5731	1.16088
$940$	−5.83341	−0.190265
$941$	−6.09945	−0.198836	−0.0994181	−	0.995046i	$-0.531698\pi$
$941$	−6.09945	−0.198836	−0.0994181	+	0.995046i	$0.531698\pi$
$942$	32.8177	1.06926
$943$	−5.30787	−0.172848
$944$	−2.85529	−0.0929318
$945$	8.33744	0.271217
$946$	0.158382	0.00514944
$947$	−47.1485	−1.53212	−0.766060	−	0.642769i	$-0.777785\pi$
$947$	−47.1485	−1.53212	−0.766060	+	0.642769i	$0.777785\pi$
$948$	−14.9857	−0.486714
$949$	−1.91931	−0.0623034
$950$	5.22439	0.169502
$951$	−26.1548	−0.848129
$952$	0	0
$953$	26.8459	0.869625	0.434812	−	0.900521i	$-0.356815\pi$
$953$	26.8459	0.869625	0.434812	+	0.900521i	$0.356815\pi$
$954$	−1.33477	−0.0432147
$955$	5.26341	0.170320
$956$	−14.7081	−0.475694
$957$	−81.1513	−2.62325
$958$	−14.7890	−0.477812
$959$	9.21852	0.297682
$960$	17.0919	0.551639
$961$	−17.1017	−0.551668
$962$	0.443171	0.0142884
$963$	−11.5989	−0.373770
$964$	13.3978	0.431514
$965$	−14.9301	−0.480618
$966$	−24.6067	−0.791708
$967$	−34.1142	−1.09704	−0.548519	−	0.836138i	$-0.684808\pi$
$967$	−34.1142	−1.09704	−0.548519	+	0.836138i	$0.684808\pi$
$968$	78.6610	2.52826
$969$	0	0
$970$	−12.4546	−0.399893
$971$	22.9516	0.736551	0.368276	−	0.929717i	$-0.379948\pi$
$971$	22.9516	0.736551	0.368276	+	0.929717i	$0.379948\pi$
$972$	−22.6384	−0.726128
$973$	60.7825	1.94860
$974$	−25.3327	−0.811711
$975$	−3.07327	−0.0984235
$976$	−8.74911	−0.280052
$977$	19.8739	0.635823	0.317912	−	0.948120i	$-0.397018\pi$
$977$	19.8739	0.635823	0.317912	+	0.948120i	$0.397018\pi$
$978$	49.2077	1.57349
$979$	−19.7979	−0.632743
$980$	2.76451	0.0883092
$981$	23.1515	0.739172
$982$	27.9585	0.892192
$983$	59.0708	1.88407	0.942034	−	0.335518i	$-0.108911\pi$
$983$	59.0708	1.88407	0.942034	+	0.335518i	$0.108911\pi$
$984$	−13.3448	−0.425416
$985$	6.33014	0.201695
$986$	0	0
$987$	44.5715	1.41873
$988$	6.75370	0.214864
$989$	0.0836424	0.00265967
$990$	23.7130	0.753649
$991$	−10.5941	−0.336534	−0.168267	−	0.985741i	$-0.553817\pi$
$991$	−10.5941	−0.336534	−0.168267	+	0.985741i	$0.553817\pi$
$992$	−19.5368	−0.620296
$993$	14.9483	0.474370
$994$	−32.1197	−1.01878
$995$	−14.9149	−0.472835
$996$	−35.3145	−1.11898
$997$	−40.7864	−1.29172	−0.645860	−	0.763456i	$-0.723501\pi$
$997$	−40.7864	−1.29172	−0.645860	+	0.763456i	$0.723501\pi$
$998$	−31.3804	−0.993331
$999$	1.06882	0.0338159

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	1445.2.a.q.1.6		12
5.4	even	2		7225.2.a.bq.1.7		12
17.4	even	4		1445.2.d.j.866.14		24
17.10	odd	16		85.2.l.a.66.4	✓	24
17.12	odd	16		85.2.l.a.76.4	yes	24
17.13	even	4		1445.2.d.j.866.13		24
17.16	even	2		1445.2.a.p.1.6		12
51.29	even	16		765.2.be.b.586.3		24
51.44	even	16		765.2.be.b.406.3		24
85.12	even	16		425.2.n.f.399.4		24
85.27	even	16		425.2.n.c.49.3		24
85.29	odd	16		425.2.m.b.76.3		24
85.44	odd	16		425.2.m.b.151.3		24
85.63	even	16		425.2.n.c.399.3		24
85.78	even	16		425.2.n.f.49.4		24
85.84	even	2		7225.2.a.bs.1.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.l.a.66.4	✓	24	17.10	odd	16
85.2.l.a.76.4	yes	24	17.12	odd	16
425.2.m.b.76.3		24	85.29	odd	16
425.2.m.b.151.3		24	85.44	odd	16
425.2.n.c.49.3		24	85.27	even	16
425.2.n.c.399.3		24	85.63	even	16
425.2.n.f.49.4		24	85.78	even	16
425.2.n.f.399.4		24	85.12	even	16
765.2.be.b.406.3		24	51.44	even	16
765.2.be.b.586.3		24	51.29	even	16
1445.2.a.p.1.6		12	17.16	even	2
1445.2.a.q.1.6		12	1.1	even	1	trivial
1445.2.d.j.866.13		24	17.13	even	4
1445.2.d.j.866.14		24	17.4	even	4
7225.2.a.bq.1.7		12	5.4	even	2
7225.2.a.bs.1.7		12	85.84	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	1445.2.a.q.1.6

Level	$1445$
Weight	$2$
Character	1445.1
Self dual	yes
Analytic conductor	$11.538$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$1$