LMFDB - Embedded newform 8512.2.a.ca.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8512,2,Mod(1,8512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8512.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-3,0,-3,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.9686622005$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.41027408.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 15x^{3} + 12x^{2} - 17x - 7 $ x^6 - 3x^5 - 6x^4 + 15x^3 + 12x^2 - 17*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4256)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.72183$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.72183 q^{3} -1.54252 q^{5} +1.00000 q^{7} -0.0353081 q^{9} -4.07847 q^{11} -0.958129 q^{13} -2.65595 q^{15} +1.53595 q^{17} +1.00000 q^{19} +1.72183 q^{21} +8.92159 q^{23} -2.62064 q^{25} -5.22628 q^{27} +2.51945 q^{29} +0.464046 q^{31} -7.02243 q^{33} -1.54252 q^{35} +6.79133 q^{37} -1.64973 q^{39} +1.97970 q^{41} -6.20088 q^{43} +0.0544633 q^{45} +2.22764 q^{47} +1.00000 q^{49} +2.64465 q^{51} -10.3396 q^{53} +6.29111 q^{55} +1.72183 q^{57} -12.8496 q^{59} +3.97107 q^{61} -0.0353081 q^{63} +1.47793 q^{65} +12.3944 q^{67} +15.3614 q^{69} -14.0908 q^{71} -9.47879 q^{73} -4.51229 q^{75} -4.07847 q^{77} +2.03911 q^{79} -8.89283 q^{81} +1.26012 q^{83} -2.36924 q^{85} +4.33807 q^{87} +2.48079 q^{89} -0.958129 q^{91} +0.799007 q^{93} -1.54252 q^{95} +2.57593 q^{97} +0.144003 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9} - 7 q^{11} - 4 q^{13} + 4 q^{15} - 2 q^{17} + 6 q^{19} - 3 q^{21} - 2 q^{23} + q^{25} - 18 q^{27} - 5 q^{29} + 14 q^{31} + 3 q^{33} - 3 q^{35} + q^{37} + 22 q^{39}+ \cdots - 3 q^{99}+O(q^{100}) $ 6 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9 - 7 * q^11 - 4 * q^13 + 4 * q^15 - 2 * q^17 + 6 * q^19 - 3 * q^21 - 2 * q^23 + q^25 - 18 * q^27 - 5 * q^29 + 14 * q^31 + 3 * q^33 - 3 * q^35 + q^37 + 22 * q^39 - 5 * q^41 - 15 * q^43 - 22 * q^45 - 7 * q^47 + 6 * q^49 + 4 * q^51 - 15 * q^53 + 20 * q^55 - 3 * q^57 - 23 * q^59 - 9 * q^61 + 3 * q^63 - 20 * q^65 - 2 * q^67 + 2 * q^69 + 23 * q^71 + 2 * q^73 - 4 * q^75 - 7 * q^77 + 17 * q^79 + 6 * q^81 - 22 * q^83 + 14 * q^85 - 10 * q^87 + 3 * q^89 - 4 * q^91 - 10 * q^93 - 3 * q^95 - 21 * q^97 - 3 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.72183	0.994098	0.497049	−	0.867723i	$-0.334417\pi$
$3$	1.72183	0.994098	0.497049	+	0.867723i	$0.334417\pi$
$4$	0	0
$5$	−1.54252	−0.689835	−0.344917	−	0.938633i	$-0.612093\pi$
$5$	−1.54252	−0.689835	−0.344917	+	0.938633i	$0.612093\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−0.0353081	−0.0117694
$10$	0	0
$11$	−4.07847	−1.22971	−0.614853	−	0.788642i	$-0.710785\pi$
$11$	−4.07847	−1.22971	−0.614853	+	0.788642i	$0.710785\pi$
$12$	0	0
$13$	−0.958129	−0.265737	−0.132869	−	0.991134i	$-0.542419\pi$
$13$	−0.958129	−0.265737	−0.132869	+	0.991134i	$0.542419\pi$
$14$	0	0
$15$	−2.65595	−0.685763
$16$	0	0
$17$	1.53595	0.372524	0.186262	−	0.982500i	$-0.440363\pi$
$17$	1.53595	0.372524	0.186262	+	0.982500i	$0.440363\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.72183	0.375734
$22$	0	0
$23$	8.92159	1.86028	0.930140	−	0.367206i	$-0.119686\pi$
$23$	8.92159	1.86028	0.930140	+	0.367206i	$0.119686\pi$
$24$	0	0
$25$	−2.62064	−0.524128
$26$	0	0
$27$	−5.22628	−1.00580
$28$	0	0
$29$	2.51945	0.467851	0.233925	−	0.972255i	$-0.424843\pi$
$29$	2.51945	0.467851	0.233925	+	0.972255i	$0.424843\pi$
$30$	0	0
$31$	0.464046	0.0833451	0.0416725	−	0.999131i	$-0.486731\pi$
$31$	0.464046	0.0833451	0.0416725	+	0.999131i	$0.486731\pi$
$32$	0	0
$33$	−7.02243	−1.22245
$34$	0	0
$35$	−1.54252	−0.260733
$36$	0	0
$37$	6.79133	1.11649	0.558244	−	0.829677i	$-0.311475\pi$
$37$	6.79133	1.11649	0.558244	+	0.829677i	$0.311475\pi$
$38$	0	0
$39$	−1.64973	−0.264169
$40$	0	0
$41$	1.97970	0.309176	0.154588	−	0.987979i	$-0.450595\pi$
$41$	1.97970	0.309176	0.154588	+	0.987979i	$0.450595\pi$
$42$	0	0
$43$	−6.20088	−0.945625	−0.472812	−	0.881163i	$-0.656761\pi$
$43$	−6.20088	−0.945625	−0.472812	+	0.881163i	$0.656761\pi$
$44$	0	0
$45$	0.0544633	0.00811891
$46$	0	0
$47$	2.22764	0.324935	0.162468	−	0.986714i	$-0.448055\pi$
$47$	2.22764	0.324935	0.162468	+	0.986714i	$0.448055\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.64465	0.370325
$52$	0	0
$53$	−10.3396	−1.42025	−0.710127	−	0.704073i	$-0.751363\pi$
$53$	−10.3396	−1.42025	−0.710127	+	0.704073i	$0.751363\pi$
$54$	0	0
$55$	6.29111	0.848293
$56$	0	0
$57$	1.72183	0.228062
$58$	0	0
$59$	−12.8496	−1.67287	−0.836437	−	0.548063i	$-0.815366\pi$
$59$	−12.8496	−1.67287	−0.836437	+	0.548063i	$0.815366\pi$
$60$	0	0
$61$	3.97107	0.508443	0.254222	−	0.967146i	$-0.418181\pi$
$61$	3.97107	0.508443	0.254222	+	0.967146i	$0.418181\pi$
$62$	0	0
$63$	−0.0353081	−0.00444840
$64$	0	0
$65$	1.47793	0.183315
$66$	0	0
$67$	12.3944	1.51422	0.757110	−	0.653288i	$-0.226611\pi$
$67$	12.3944	1.51422	0.757110	+	0.653288i	$0.226611\pi$
$68$	0	0
$69$	15.3614	1.84930
$70$	0	0
$71$	−14.0908	−1.67227	−0.836134	−	0.548526i	$-0.815189\pi$
$71$	−14.0908	−1.67227	−0.836134	+	0.548526i	$0.815189\pi$
$72$	0	0
$73$	−9.47879	−1.10941	−0.554704	−	0.832048i	$-0.687169\pi$
$73$	−9.47879	−1.10941	−0.554704	+	0.832048i	$0.687169\pi$
$74$	0	0
$75$	−4.51229	−0.521035
$76$	0	0
$77$	−4.07847	−0.464785
$78$	0	0
$79$	2.03911	0.229418	0.114709	−	0.993399i	$-0.463406\pi$
$79$	2.03911	0.229418	0.114709	+	0.993399i	$0.463406\pi$
$80$	0	0
$81$	−8.89283	−0.988092
$82$	0	0
$83$	1.26012	0.138316	0.0691581	−	0.997606i	$-0.477969\pi$
$83$	1.26012	0.138316	0.0691581	+	0.997606i	$0.477969\pi$
$84$	0	0
$85$	−2.36924	−0.256980
$86$	0	0
$87$	4.33807	0.465090
$88$	0	0
$89$	2.48079	0.262963	0.131482	−	0.991319i	$-0.458027\pi$
$89$	2.48079	0.262963	0.131482	+	0.991319i	$0.458027\pi$
$90$	0	0
$91$	−0.958129	−0.100439
$92$	0	0
$93$	0.799007	0.0828532
$94$	0	0
$95$	−1.54252	−0.158259
$96$	0	0
$97$	2.57593	0.261546	0.130773	−	0.991412i	$-0.458254\pi$
$97$	2.57593	0.261546	0.130773	+	0.991412i	$0.458254\pi$
$98$	0	0
$99$	0.144003	0.0144728
$100$	0	0
$101$	−2.78530	−0.277147	−0.138574	−	0.990352i	$-0.544252\pi$
$101$	−2.78530	−0.277147	−0.138574	+	0.990352i	$0.544252\pi$
$102$	0	0
$103$	−5.90034	−0.581377	−0.290689	−	0.956818i	$-0.593884\pi$
$103$	−5.90034	−0.581377	−0.290689	+	0.956818i	$0.593884\pi$
$104$	0	0
$105$	−2.65595	−0.259194
$106$	0	0
$107$	−10.8893	−1.05271	−0.526354	−	0.850266i	$-0.676441\pi$
$107$	−10.8893	−1.05271	−0.526354	+	0.850266i	$0.676441\pi$
$108$	0	0
$109$	−5.42638	−0.519753	−0.259876	−	0.965642i	$-0.583682\pi$
$109$	−5.42638	−0.519753	−0.259876	+	0.965642i	$0.583682\pi$
$110$	0	0
$111$	11.6935	1.10990
$112$	0	0
$113$	−5.91255	−0.556206	−0.278103	−	0.960551i	$-0.589706\pi$
$113$	−5.91255	−0.556206	−0.278103	+	0.960551i	$0.589706\pi$
$114$	0	0
$115$	−13.7617	−1.28329
$116$	0	0
$117$	0.0338297	0.00312756
$118$	0	0
$119$	1.53595	0.140801
$120$	0	0
$121$	5.63393	0.512175
$122$	0	0
$123$	3.40870	0.307352
$124$	0	0
$125$	11.7550	1.05140
$126$	0	0
$127$	3.86075	0.342586	0.171293	−	0.985220i	$-0.445206\pi$
$127$	3.86075	0.342586	0.171293	+	0.985220i	$0.445206\pi$
$128$	0	0
$129$	−10.6768	−0.940044
$130$	0	0
$131$	−14.9493	−1.30613	−0.653064	−	0.757303i	$-0.726517\pi$
$131$	−14.9493	−1.30613	−0.653064	+	0.757303i	$0.726517\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	8.06162	0.693834
$136$	0	0
$137$	1.40454	0.119998	0.0599991	−	0.998198i	$-0.480890\pi$
$137$	1.40454	0.119998	0.0599991	+	0.998198i	$0.480890\pi$
$138$	0	0
$139$	−22.0324	−1.86876	−0.934382	−	0.356274i	$-0.884047\pi$
$139$	−22.0324	−1.86876	−0.934382	+	0.356274i	$0.884047\pi$
$140$	0	0
$141$	3.83562	0.323017
$142$	0	0
$143$	3.90770	0.326778
$144$	0	0
$145$	−3.88630	−0.322740
$146$	0	0
$147$	1.72183	0.142014
$148$	0	0
$149$	−6.01028	−0.492381	−0.246191	−	0.969221i	$-0.579179\pi$
$149$	−6.01028	−0.492381	−0.246191	+	0.969221i	$0.579179\pi$
$150$	0	0
$151$	−5.04677	−0.410700	−0.205350	−	0.978689i	$-0.565833\pi$
$151$	−5.04677	−0.410700	−0.205350	+	0.978689i	$0.565833\pi$
$152$	0	0
$153$	−0.0542316	−0.00438436
$154$	0	0
$155$	−0.715799	−0.0574943
$156$	0	0
$157$	−6.06889	−0.484350	−0.242175	−	0.970233i	$-0.577861\pi$
$157$	−6.06889	−0.484350	−0.242175	+	0.970233i	$0.577861\pi$
$158$	0	0
$159$	−17.8030	−1.41187
$160$	0	0
$161$	8.92159	0.703120
$162$	0	0
$163$	−7.03167	−0.550763	−0.275381	−	0.961335i	$-0.588804\pi$
$163$	−7.03167	−0.550763	−0.275381	+	0.961335i	$0.588804\pi$
$164$	0	0
$165$	10.8322	0.843287
$166$	0	0
$167$	−11.4751	−0.887967	−0.443983	−	0.896035i	$-0.646435\pi$
$167$	−11.4751	−0.887967	−0.443983	+	0.896035i	$0.646435\pi$
$168$	0	0
$169$	−12.0820	−0.929384
$170$	0	0
$171$	−0.0353081	−0.00270008
$172$	0	0
$173$	−16.9653	−1.28985	−0.644925	−	0.764246i	$-0.723112\pi$
$173$	−16.9653	−1.28985	−0.644925	+	0.764246i	$0.723112\pi$
$174$	0	0
$175$	−2.62064	−0.198102
$176$	0	0
$177$	−22.1248	−1.66300
$178$	0	0
$179$	20.3181	1.51865	0.759324	−	0.650713i	$-0.225530\pi$
$179$	20.3181	1.51865	0.759324	+	0.650713i	$0.225530\pi$
$180$	0	0
$181$	17.3642	1.29067	0.645335	−	0.763900i	$-0.276718\pi$
$181$	17.3642	1.29067	0.645335	+	0.763900i	$0.276718\pi$
$182$	0	0
$183$	6.83750	0.505442
$184$	0	0
$185$	−10.4757	−0.770191
$186$	0	0
$187$	−6.26435	−0.458094
$188$	0	0
$189$	−5.22628	−0.380156
$190$	0	0
$191$	−11.0293	−0.798053	−0.399027	−	0.916939i	$-0.630652\pi$
$191$	−11.0293	−0.798053	−0.399027	+	0.916939i	$0.630652\pi$
$192$	0	0
$193$	1.46871	0.105720	0.0528600	−	0.998602i	$-0.483166\pi$
$193$	1.46871	0.105720	0.0528600	+	0.998602i	$0.483166\pi$
$194$	0	0
$195$	2.54474	0.182233
$196$	0	0
$197$	−18.0687	−1.28734	−0.643670	−	0.765303i	$-0.722589\pi$
$197$	−18.0687	−1.28734	−0.643670	+	0.765303i	$0.722589\pi$
$198$	0	0
$199$	3.92999	0.278589	0.139295	−	0.990251i	$-0.455516\pi$
$199$	3.92999	0.278589	0.139295	+	0.990251i	$0.455516\pi$
$200$	0	0
$201$	21.3411	1.50528
$202$	0	0
$203$	2.51945	0.176831
$204$	0	0
$205$	−3.05371	−0.213281
$206$	0	0
$207$	−0.315004	−0.0218943
$208$	0	0
$209$	−4.07847	−0.282114
$210$	0	0
$211$	−23.0342	−1.58574	−0.792871	−	0.609389i	$-0.791415\pi$
$211$	−23.0342	−1.58574	−0.792871	+	0.609389i	$0.791415\pi$
$212$	0	0
$213$	−24.2619	−1.66240
$214$	0	0
$215$	9.56496	0.652325
$216$	0	0
$217$	0.464046	0.0315015
$218$	0	0
$219$	−16.3208	−1.10286
$220$	0	0
$221$	−1.47164	−0.0989934
$222$	0	0
$223$	−7.77688	−0.520779	−0.260389	−	0.965504i	$-0.583851\pi$
$223$	−7.77688	−0.520779	−0.260389	+	0.965504i	$0.583851\pi$
$224$	0	0
$225$	0.0925298	0.00616865
$226$	0	0
$227$	−11.7785	−0.781768	−0.390884	−	0.920440i	$-0.627831\pi$
$227$	−11.7785	−0.781768	−0.390884	+	0.920440i	$0.627831\pi$
$228$	0	0
$229$	−22.8995	−1.51324	−0.756620	−	0.653855i	$-0.773151\pi$
$229$	−22.8995	−1.51324	−0.756620	+	0.653855i	$0.773151\pi$
$230$	0	0
$231$	−7.02243	−0.462042
$232$	0	0
$233$	18.5286	1.21385	0.606926	−	0.794758i	$-0.292402\pi$
$233$	18.5286	1.21385	0.606926	+	0.794758i	$0.292402\pi$
$234$	0	0
$235$	−3.43618	−0.224152
$236$	0	0
$237$	3.51100	0.228064
$238$	0	0
$239$	0.249079	0.0161116	0.00805581	−	0.999968i	$-0.497436\pi$
$239$	0.249079	0.0161116	0.00805581	+	0.999968i	$0.497436\pi$
$240$	0	0
$241$	6.81520	0.439006	0.219503	−	0.975612i	$-0.429556\pi$
$241$	6.81520	0.439006	0.219503	+	0.975612i	$0.429556\pi$
$242$	0	0
$243$	0.366913	0.0235375
$244$	0	0
$245$	−1.54252	−0.0985478
$246$	0	0
$247$	−0.958129	−0.0609643
$248$	0	0
$249$	2.16971	0.137500
$250$	0	0
$251$	−0.414081	−0.0261366	−0.0130683	−	0.999915i	$-0.504160\pi$
$251$	−0.414081	−0.0261366	−0.0130683	+	0.999915i	$0.504160\pi$
$252$	0	0
$253$	−36.3864	−2.28760
$254$	0	0
$255$	−4.07942	−0.255463
$256$	0	0
$257$	0.762240	0.0475472	0.0237736	−	0.999717i	$-0.492432\pi$
$257$	0.762240	0.0475472	0.0237736	+	0.999717i	$0.492432\pi$
$258$	0	0
$259$	6.79133	0.421992
$260$	0	0
$261$	−0.0889571	−0.00550630
$262$	0	0
$263$	28.2110	1.73957	0.869783	−	0.493435i	$-0.164259\pi$
$263$	28.2110	1.73957	0.869783	+	0.493435i	$0.164259\pi$
$264$	0	0
$265$	15.9490	0.979741
$266$	0	0
$267$	4.27149	0.261411
$268$	0	0
$269$	−15.6272	−0.952807	−0.476403	−	0.879227i	$-0.658060\pi$
$269$	−15.6272	−0.952807	−0.476403	+	0.879227i	$0.658060\pi$
$270$	0	0
$271$	5.29871	0.321874	0.160937	−	0.986965i	$-0.448548\pi$
$271$	5.29871	0.321874	0.160937	+	0.986965i	$0.448548\pi$
$272$	0	0
$273$	−1.64973	−0.0998464
$274$	0	0
$275$	10.6882	0.644523
$276$	0	0
$277$	5.66726	0.340512	0.170256	−	0.985400i	$-0.445540\pi$
$277$	5.66726	0.340512	0.170256	+	0.985400i	$0.445540\pi$
$278$	0	0
$279$	−0.0163846	−0.000980918 0
$280$	0	0
$281$	−21.4634	−1.28040	−0.640200	−	0.768209i	$-0.721148\pi$
$281$	−21.4634	−1.28040	−0.640200	+	0.768209i	$0.721148\pi$
$282$	0	0
$283$	−8.71153	−0.517847	−0.258923	−	0.965898i	$-0.583368\pi$
$283$	−8.71153	−0.517847	−0.258923	+	0.965898i	$0.583368\pi$
$284$	0	0
$285$	−2.65595	−0.157325
$286$	0	0
$287$	1.97970	0.116858
$288$	0	0
$289$	−14.6408	−0.861226
$290$	0	0
$291$	4.43530	0.260002
$292$	0	0
$293$	−0.459648	−0.0268529	−0.0134265	−	0.999910i	$-0.504274\pi$
$293$	−0.459648	−0.0268529	−0.0134265	+	0.999910i	$0.504274\pi$
$294$	0	0
$295$	19.8207	1.15401
$296$	0	0
$297$	21.3152	1.23683
$298$	0	0
$299$	−8.54803	−0.494345
$300$	0	0
$301$	−6.20088	−0.357413
$302$	0	0
$303$	−4.79580	−0.275512
$304$	0	0
$305$	−6.12544	−0.350742
$306$	0	0
$307$	23.9441	1.36656	0.683282	−	0.730154i	$-0.260552\pi$
$307$	23.9441	1.36656	0.683282	+	0.730154i	$0.260552\pi$
$308$	0	0
$309$	−10.1594	−0.577946
$310$	0	0
$311$	24.7979	1.40616	0.703080	−	0.711111i	$-0.251808\pi$
$311$	24.7979	1.40616	0.703080	+	0.711111i	$0.251808\pi$
$312$	0	0
$313$	−8.74984	−0.494570	−0.247285	−	0.968943i	$-0.579538\pi$
$313$	−8.74984	−0.494570	−0.247285	+	0.968943i	$0.579538\pi$
$314$	0	0
$315$	0.0544633	0.00306866
$316$	0	0
$317$	2.70906	0.152156	0.0760779	−	0.997102i	$-0.475760\pi$
$317$	2.70906	0.152156	0.0760779	+	0.997102i	$0.475760\pi$
$318$	0	0
$319$	−10.2755	−0.575319
$320$	0	0
$321$	−18.7495	−1.04649
$322$	0	0
$323$	1.53595	0.0854628
$324$	0	0
$325$	2.51091	0.139280
$326$	0	0
$327$	−9.34329	−0.516685
$328$	0	0
$329$	2.22764	0.122814
$330$	0	0
$331$	8.09703	0.445053	0.222526	−	0.974927i	$-0.428570\pi$
$331$	8.09703	0.445053	0.222526	+	0.974927i	$0.428570\pi$
$332$	0	0
$333$	−0.239789	−0.0131403
$334$	0	0
$335$	−19.1186	−1.04456
$336$	0	0
$337$	4.33064	0.235905	0.117952	−	0.993019i	$-0.462367\pi$
$337$	4.33064	0.235905	0.117952	+	0.993019i	$0.462367\pi$
$338$	0	0
$339$	−10.1804	−0.552924
$340$	0	0
$341$	−1.89260	−0.102490
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−23.6953	−1.27571
$346$	0	0
$347$	−3.59896	−0.193202	−0.0966010	−	0.995323i	$-0.530797\pi$
$347$	−3.59896	−0.193202	−0.0966010	+	0.995323i	$0.530797\pi$
$348$	0	0
$349$	−23.5143	−1.25869	−0.629346	−	0.777125i	$-0.716677\pi$
$349$	−23.5143	−1.25869	−0.629346	+	0.777125i	$0.716677\pi$
$350$	0	0
$351$	5.00745	0.267278
$352$	0	0
$353$	−5.69809	−0.303279	−0.151639	−	0.988436i	$-0.548455\pi$
$353$	−5.69809	−0.303279	−0.151639	+	0.988436i	$0.548455\pi$
$354$	0	0
$355$	21.7353	1.15359
$356$	0	0
$357$	2.64465	0.139970
$358$	0	0
$359$	25.7605	1.35959	0.679793	−	0.733404i	$-0.262070\pi$
$359$	25.7605	1.35959	0.679793	+	0.733404i	$0.262070\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	9.70066	0.509152
$364$	0	0
$365$	14.6212	0.765308
$366$	0	0
$367$	36.9046	1.92640	0.963202	−	0.268778i	$-0.0866198\pi$
$367$	36.9046	1.92640	0.963202	+	0.268778i	$0.0866198\pi$
$368$	0	0
$369$	−0.0698992	−0.00363881
$370$	0	0
$371$	−10.3396	−0.536806
$372$	0	0
$373$	−16.3997	−0.849142	−0.424571	−	0.905395i	$-0.639575\pi$
$373$	−16.3997	−0.849142	−0.424571	+	0.905395i	$0.639575\pi$
$374$	0	0
$375$	20.2400	1.04519
$376$	0	0
$377$	−2.41396	−0.124325
$378$	0	0
$379$	18.5177	0.951188	0.475594	−	0.879665i	$-0.342233\pi$
$379$	18.5177	0.951188	0.475594	+	0.879665i	$0.342233\pi$
$380$	0	0
$381$	6.64755	0.340564
$382$	0	0
$383$	32.3346	1.65222	0.826110	−	0.563509i	$-0.190549\pi$
$383$	32.3346	1.65222	0.826110	+	0.563509i	$0.190549\pi$
$384$	0	0
$385$	6.29111	0.320625
$386$	0	0
$387$	0.218941	0.0111294
$388$	0	0
$389$	−25.3717	−1.28640	−0.643198	−	0.765700i	$-0.722393\pi$
$389$	−25.3717	−1.28640	−0.643198	+	0.765700i	$0.722393\pi$
$390$	0	0
$391$	13.7031	0.692998
$392$	0	0
$393$	−25.7401	−1.29842
$394$	0	0
$395$	−3.14537	−0.158261
$396$	0	0
$397$	22.2599	1.11719	0.558596	−	0.829440i	$-0.311340\pi$
$397$	22.2599	1.11719	0.558596	+	0.829440i	$0.311340\pi$
$398$	0	0
$399$	1.72183	0.0861992
$400$	0	0
$401$	−17.3432	−0.866076	−0.433038	−	0.901376i	$-0.642558\pi$
$401$	−17.3432	−0.866076	−0.433038	+	0.901376i	$0.642558\pi$
$402$	0	0
$403$	−0.444616	−0.0221479
$404$	0	0
$405$	13.7173	0.681620
$406$	0	0
$407$	−27.6982	−1.37295
$408$	0	0
$409$	35.8282	1.77159	0.885795	−	0.464077i	$-0.153614\pi$
$409$	35.8282	1.77159	0.885795	+	0.464077i	$0.153614\pi$
$410$	0	0
$411$	2.41838	0.119290
$412$	0	0
$413$	−12.8496	−0.632287
$414$	0	0
$415$	−1.94376	−0.0954153
$416$	0	0
$417$	−37.9360	−1.85773
$418$	0	0
$419$	−2.89443	−0.141402	−0.0707011	−	0.997498i	$-0.522524\pi$
$419$	−2.89443	−0.141402	−0.0707011	+	0.997498i	$0.522524\pi$
$420$	0	0
$421$	−22.2319	−1.08352	−0.541759	−	0.840534i	$-0.682242\pi$
$421$	−22.2319	−1.08352	−0.541759	+	0.840534i	$0.682242\pi$
$422$	0	0
$423$	−0.0786538	−0.00382428
$424$	0	0
$425$	−4.02518	−0.195250
$426$	0	0
$427$	3.97107	0.192173
$428$	0	0
$429$	6.72839	0.324850
$430$	0	0
$431$	−12.5425	−0.604152	−0.302076	−	0.953284i	$-0.597680\pi$
$431$	−12.5425	−0.604152	−0.302076	+	0.953284i	$0.597680\pi$
$432$	0	0
$433$	−9.76196	−0.469130	−0.234565	−	0.972100i	$-0.575367\pi$
$433$	−9.76196	−0.469130	−0.234565	+	0.972100i	$0.575367\pi$
$434$	0	0
$435$	−6.69154	−0.320835
$436$	0	0
$437$	8.92159	0.426777
$438$	0	0
$439$	36.0623	1.72116	0.860579	−	0.509317i	$-0.170102\pi$
$439$	36.0623	1.72116	0.860579	+	0.509317i	$0.170102\pi$
$440$	0	0
$441$	−0.0353081	−0.00168134
$442$	0	0
$443$	34.4276	1.63570	0.817852	−	0.575428i	$-0.195165\pi$
$443$	34.4276	1.63570	0.817852	+	0.575428i	$0.195165\pi$
$444$	0	0
$445$	−3.82666	−0.181401
$446$	0	0
$447$	−10.3487	−0.489475
$448$	0	0
$449$	14.0491	0.663017	0.331509	−	0.943452i	$-0.392442\pi$
$449$	14.0491	0.663017	0.331509	+	0.943452i	$0.392442\pi$
$450$	0	0
$451$	−8.07413	−0.380196
$452$	0	0
$453$	−8.68967	−0.408276
$454$	0	0
$455$	1.47793	0.0692864
$456$	0	0
$457$	−17.7557	−0.830578	−0.415289	−	0.909689i	$-0.636320\pi$
$457$	−17.7557	−0.830578	−0.415289	+	0.909689i	$0.636320\pi$
$458$	0	0
$459$	−8.02732	−0.374683
$460$	0	0
$461$	39.9013	1.85839	0.929194	−	0.369592i	$-0.120503\pi$
$461$	39.9013	1.85839	0.929194	+	0.369592i	$0.120503\pi$
$462$	0	0
$463$	5.76594	0.267966	0.133983	−	0.990984i	$-0.457223\pi$
$463$	5.76594	0.267966	0.133983	+	0.990984i	$0.457223\pi$
$464$	0	0
$465$	−1.23248	−0.0571550
$466$	0	0
$467$	−23.2761	−1.07709	−0.538545	−	0.842597i	$-0.681026\pi$
$467$	−23.2761	−1.07709	−0.538545	+	0.842597i	$0.681026\pi$
$468$	0	0
$469$	12.3944	0.572321
$470$	0	0
$471$	−10.4496	−0.481491
$472$	0	0
$473$	25.2901	1.16284
$474$	0	0
$475$	−2.62064	−0.120243
$476$	0	0
$477$	0.365072	0.0167155
$478$	0	0
$479$	3.08460	0.140939	0.0704695	−	0.997514i	$-0.477550\pi$
$479$	3.08460	0.140939	0.0704695	+	0.997514i	$0.477550\pi$
$480$	0	0
$481$	−6.50697	−0.296692
$482$	0	0
$483$	15.3614	0.698970
$484$	0	0
$485$	−3.97341	−0.180423
$486$	0	0
$487$	11.4573	0.519181	0.259590	−	0.965719i	$-0.416412\pi$
$487$	11.4573	0.519181	0.259590	+	0.965719i	$0.416412\pi$
$488$	0	0
$489$	−12.1073	−0.547512
$490$	0	0
$491$	10.4849	0.473175	0.236588	−	0.971610i	$-0.423971\pi$
$491$	10.4849	0.473175	0.236588	+	0.971610i	$0.423971\pi$
$492$	0	0
$493$	3.86977	0.174286
$494$	0	0
$495$	−0.222127	−0.00998387
$496$	0	0
$497$	−14.0908	−0.632058
$498$	0	0
$499$	−40.9058	−1.83119	−0.915597	−	0.402098i	$-0.868281\pi$
$499$	−40.9058	−1.83119	−0.915597	+	0.402098i	$0.868281\pi$
$500$	0	0
$501$	−19.7581	−0.882726
$502$	0	0
$503$	−37.6788	−1.68002	−0.840008	−	0.542573i	$-0.817450\pi$
$503$	−37.6788	−1.68002	−0.840008	+	0.542573i	$0.817450\pi$
$504$	0	0
$505$	4.29637	0.191186
$506$	0	0
$507$	−20.8031	−0.923898
$508$	0	0
$509$	−32.6186	−1.44579	−0.722897	−	0.690956i	$-0.757190\pi$
$509$	−32.6186	−1.44579	−0.722897	+	0.690956i	$0.757190\pi$
$510$	0	0
$511$	−9.47879	−0.419317
$512$	0	0
$513$	−5.22628	−0.230746
$514$	0	0
$515$	9.10137	0.401054
$516$	0	0
$517$	−9.08538	−0.399575
$518$	0	0
$519$	−29.2114	−1.28224
$520$	0	0
$521$	9.58355	0.419863	0.209931	−	0.977716i	$-0.432676\pi$
$521$	9.58355	0.419863	0.209931	+	0.977716i	$0.432676\pi$
$522$	0	0
$523$	7.43831	0.325254	0.162627	−	0.986688i	$-0.448003\pi$
$523$	7.43831	0.325254	0.162627	+	0.986688i	$0.448003\pi$
$524$	0	0
$525$	−4.51229	−0.196933
$526$	0	0
$527$	0.712753	0.0310480
$528$	0	0
$529$	56.5947	2.46064
$530$	0	0
$531$	0.453694	0.0196887
$532$	0	0
$533$	−1.89680	−0.0821597
$534$	0	0
$535$	16.7969	0.726194
$536$	0	0
$537$	34.9843	1.50968
$538$	0	0
$539$	−4.07847	−0.175672
$540$	0	0
$541$	−15.8767	−0.682591	−0.341295	−	0.939956i	$-0.610866\pi$
$541$	−15.8767	−0.682591	−0.341295	+	0.939956i	$0.610866\pi$
$542$	0	0
$543$	29.8982	1.28305
$544$	0	0
$545$	8.37028	0.358543
$546$	0	0
$547$	29.3980	1.25697	0.628484	−	0.777822i	$-0.283676\pi$
$547$	29.3980	1.25697	0.628484	+	0.777822i	$0.283676\pi$
$548$	0	0
$549$	−0.140211	−0.00598405
$550$	0	0
$551$	2.51945	0.107332
$552$	0	0
$553$	2.03911	0.0867119
$554$	0	0
$555$	−18.0374	−0.765646
$556$	0	0
$557$	−13.5091	−0.572401	−0.286200	−	0.958170i	$-0.592392\pi$
$557$	−13.5091	−0.572401	−0.286200	+	0.958170i	$0.592392\pi$
$558$	0	0
$559$	5.94124	0.251288
$560$	0	0
$561$	−10.7861	−0.455391
$562$	0	0
$563$	22.4945	0.948031	0.474016	−	0.880516i	$-0.342804\pi$
$563$	22.4945	0.948031	0.474016	+	0.880516i	$0.342804\pi$
$564$	0	0
$565$	9.12022	0.383690
$566$	0	0
$567$	−8.89283	−0.373464
$568$	0	0
$569$	−20.9924	−0.880046	−0.440023	−	0.897986i	$-0.645030\pi$
$569$	−20.9924	−0.880046	−0.440023	+	0.897986i	$0.645030\pi$
$570$	0	0
$571$	44.9996	1.88318	0.941588	−	0.336766i	$-0.109333\pi$
$571$	44.9996	1.88318	0.941588	+	0.336766i	$0.109333\pi$
$572$	0	0
$573$	−18.9906	−0.793343
$574$	0	0
$575$	−23.3803	−0.975025
$576$	0	0
$577$	16.2515	0.676560	0.338280	−	0.941045i	$-0.390155\pi$
$577$	16.2515	0.676560	0.338280	+	0.941045i	$0.390155\pi$
$578$	0	0
$579$	2.52887	0.105096
$580$	0	0
$581$	1.26012	0.0522786
$582$	0	0
$583$	42.1698	1.74649
$584$	0	0
$585$	−0.0521829	−0.00215750
$586$	0	0
$587$	−44.6567	−1.84318	−0.921590	−	0.388165i	$-0.873109\pi$
$587$	−44.6567	−1.84318	−0.921590	+	0.388165i	$0.873109\pi$
$588$	0	0
$589$	0.464046	0.0191207
$590$	0	0
$591$	−31.1112	−1.27974
$592$	0	0
$593$	−40.1075	−1.64702	−0.823509	−	0.567303i	$-0.807987\pi$
$593$	−40.1075	−1.64702	−0.823509	+	0.567303i	$0.807987\pi$
$594$	0	0
$595$	−2.36924	−0.0971292
$596$	0	0
$597$	6.76676	0.276945
$598$	0	0
$599$	−25.7938	−1.05391	−0.526954	−	0.849894i	$-0.676666\pi$
$599$	−25.7938	−1.05391	−0.526954	+	0.849894i	$0.676666\pi$
$600$	0	0
$601$	−32.1526	−1.31153	−0.655765	−	0.754965i	$-0.727654\pi$
$601$	−32.1526	−1.31153	−0.655765	+	0.754965i	$0.727654\pi$
$602$	0	0
$603$	−0.437623	−0.0178214
$604$	0	0
$605$	−8.69043	−0.353316
$606$	0	0
$607$	−22.6642	−0.919911	−0.459956	−	0.887942i	$-0.652135\pi$
$607$	−22.6642	−0.919911	−0.459956	+	0.887942i	$0.652135\pi$
$608$	0	0
$609$	4.33807	0.175787
$610$	0	0
$611$	−2.13437	−0.0863474
$612$	0	0
$613$	43.8074	1.76936	0.884682	−	0.466196i	$-0.154376\pi$
$613$	43.8074	1.76936	0.884682	+	0.466196i	$0.154376\pi$
$614$	0	0
$615$	−5.25797	−0.212022
$616$	0	0
$617$	−46.2073	−1.86024	−0.930118	−	0.367261i	$-0.880295\pi$
$617$	−46.2073	−1.86024	−0.930118	+	0.367261i	$0.880295\pi$
$618$	0	0
$619$	18.9085	0.759996	0.379998	−	0.924987i	$-0.375925\pi$
$619$	18.9085	0.759996	0.379998	+	0.924987i	$0.375925\pi$
$620$	0	0
$621$	−46.6267	−1.87106
$622$	0	0
$623$	2.48079	0.0993907
$624$	0	0
$625$	−5.02903	−0.201161
$626$	0	0
$627$	−7.02243	−0.280449
$628$	0	0
$629$	10.4312	0.415918
$630$	0	0
$631$	−19.9351	−0.793604	−0.396802	−	0.917904i	$-0.629880\pi$
$631$	−19.9351	−0.793604	−0.396802	+	0.917904i	$0.629880\pi$
$632$	0	0
$633$	−39.6610	−1.57638
$634$	0	0
$635$	−5.95527	−0.236328
$636$	0	0
$637$	−0.958129	−0.0379625
$638$	0	0
$639$	0.497518	0.0196815
$640$	0	0
$641$	−35.8503	−1.41600	−0.708000	−	0.706212i	$-0.750403\pi$
$641$	−35.8503	−1.41600	−0.708000	+	0.706212i	$0.750403\pi$
$642$	0	0
$643$	26.6068	1.04927	0.524635	−	0.851327i	$-0.324202\pi$
$643$	26.6068	1.04927	0.524635	+	0.851327i	$0.324202\pi$
$644$	0	0
$645$	16.4692	0.648475
$646$	0	0
$647$	−19.5860	−0.770005	−0.385003	−	0.922915i	$-0.625799\pi$
$647$	−19.5860	−0.770005	−0.385003	+	0.922915i	$0.625799\pi$
$648$	0	0
$649$	52.4067	2.05714
$650$	0	0
$651$	0.799007	0.0313156
$652$	0	0
$653$	23.2709	0.910659	0.455330	−	0.890323i	$-0.349521\pi$
$653$	23.2709	0.910659	0.455330	+	0.890323i	$0.349521\pi$
$654$	0	0
$655$	23.0596	0.901012
$656$	0	0
$657$	0.334678	0.0130570
$658$	0	0
$659$	18.3819	0.716058	0.358029	−	0.933710i	$-0.383449\pi$
$659$	18.3819	0.716058	0.358029	+	0.933710i	$0.383449\pi$
$660$	0	0
$661$	−3.28306	−0.127696	−0.0638482	−	0.997960i	$-0.520337\pi$
$661$	−3.28306	−0.127696	−0.0638482	+	0.997960i	$0.520337\pi$
$662$	0	0
$663$	−2.53392	−0.0984091
$664$	0	0
$665$	−1.54252	−0.0598162
$666$	0	0
$667$	22.4775	0.870333
$668$	0	0
$669$	−13.3905	−0.517705
$670$	0	0
$671$	−16.1959	−0.625235
$672$	0	0
$673$	−32.2466	−1.24302	−0.621508	−	0.783408i	$-0.713480\pi$
$673$	−32.2466	−1.24302	−0.621508	+	0.783408i	$0.713480\pi$
$674$	0	0
$675$	13.6962	0.527167
$676$	0	0
$677$	−17.5452	−0.674316	−0.337158	−	0.941448i	$-0.609466\pi$
$677$	−17.5452	−0.674316	−0.337158	+	0.941448i	$0.609466\pi$
$678$	0	0
$679$	2.57593	0.0988550
$680$	0	0
$681$	−20.2806	−0.777154
$682$	0	0
$683$	−9.60539	−0.367540	−0.183770	−	0.982969i	$-0.558830\pi$
$683$	−9.60539	−0.367540	−0.183770	+	0.982969i	$0.558830\pi$
$684$	0	0
$685$	−2.16653	−0.0827790
$686$	0	0
$687$	−39.4289	−1.50431
$688$	0	0
$689$	9.90668	0.377414
$690$	0	0
$691$	−21.8826	−0.832451	−0.416226	−	0.909261i	$-0.636647\pi$
$691$	−21.8826	−0.832451	−0.416226	+	0.909261i	$0.636647\pi$
$692$	0	0
$693$	0.144003	0.00547022
$694$	0	0
$695$	33.9853	1.28914
$696$	0	0
$697$	3.04072	0.115176
$698$	0	0
$699$	31.9031	1.20669
$700$	0	0
$701$	−9.18985	−0.347096	−0.173548	−	0.984825i	$-0.555523\pi$
$701$	−9.18985	−0.347096	−0.173548	+	0.984825i	$0.555523\pi$
$702$	0	0
$703$	6.79133	0.256140
$704$	0	0
$705$	−5.91651	−0.222829
$706$	0	0
$707$	−2.78530	−0.104752
$708$	0	0
$709$	28.5389	1.07180	0.535901	−	0.844281i	$-0.319972\pi$
$709$	28.5389	1.07180	0.535901	+	0.844281i	$0.319972\pi$
$710$	0	0
$711$	−0.0719971	−0.00270010
$712$	0	0
$713$	4.14002	0.155045
$714$	0	0
$715$	−6.02770	−0.225423
$716$	0	0
$717$	0.428872	0.0160165
$718$	0	0
$719$	−23.7733	−0.886593	−0.443296	−	0.896375i	$-0.646191\pi$
$719$	−23.7733	−0.886593	−0.443296	+	0.896375i	$0.646191\pi$
$720$	0	0
$721$	−5.90034	−0.219740
$722$	0	0
$723$	11.7346	0.436415
$724$	0	0
$725$	−6.60259	−0.245214
$726$	0	0
$727$	−3.39111	−0.125769	−0.0628846	−	0.998021i	$-0.520030\pi$
$727$	−3.39111	−0.125769	−0.0628846	+	0.998021i	$0.520030\pi$
$728$	0	0
$729$	27.3102	1.01149
$730$	0	0
$731$	−9.52426	−0.352268
$732$	0	0
$733$	6.62579	0.244729	0.122364	−	0.992485i	$-0.460952\pi$
$733$	6.62579	0.244729	0.122364	+	0.992485i	$0.460952\pi$
$734$	0	0
$735$	−2.65595	−0.0979662
$736$	0	0
$737$	−50.5503	−1.86204
$738$	0	0
$739$	−10.1808	−0.374505	−0.187253	−	0.982312i	$-0.559958\pi$
$739$	−10.1808	−0.374505	−0.187253	+	0.982312i	$0.559958\pi$
$740$	0	0
$741$	−1.64973	−0.0606045
$742$	0	0
$743$	6.22858	0.228504	0.114252	−	0.993452i	$-0.463553\pi$
$743$	6.22858	0.228504	0.114252	+	0.993452i	$0.463553\pi$
$744$	0	0
$745$	9.27096	0.339662
$746$	0	0
$747$	−0.0444924	−0.00162789
$748$	0	0
$749$	−10.8893	−0.397886
$750$	0	0
$751$	27.8628	1.01673	0.508364	−	0.861143i	$-0.330251\pi$
$751$	27.8628	1.01673	0.508364	+	0.861143i	$0.330251\pi$
$752$	0	0
$753$	−0.712977	−0.0259823
$754$	0	0
$755$	7.78472	0.283315
$756$	0	0
$757$	9.18222	0.333734	0.166867	−	0.985979i	$-0.446635\pi$
$757$	9.18222	0.333734	0.166867	+	0.985979i	$0.446635\pi$
$758$	0	0
$759$	−62.6512	−2.27409
$760$	0	0
$761$	28.3945	1.02930	0.514650	−	0.857400i	$-0.327922\pi$
$761$	28.3945	1.02930	0.514650	+	0.857400i	$0.327922\pi$
$762$	0	0
$763$	−5.42638	−0.196448
$764$	0	0
$765$	0.0836531	0.00302449
$766$	0	0
$767$	12.3116	0.444545
$768$	0	0
$769$	1.85252	0.0668035	0.0334018	−	0.999442i	$-0.489366\pi$
$769$	1.85252	0.0668035	0.0334018	+	0.999442i	$0.489366\pi$
$770$	0	0
$771$	1.31245	0.0472666
$772$	0	0
$773$	13.0734	0.470217	0.235109	−	0.971969i	$-0.424455\pi$
$773$	13.0734	0.470217	0.235109	+	0.971969i	$0.424455\pi$
$774$	0	0
$775$	−1.21610	−0.0436835
$776$	0	0
$777$	11.6935	0.419502
$778$	0	0
$779$	1.97970	0.0709300
$780$	0	0
$781$	57.4688	2.05640
$782$	0	0
$783$	−13.1674	−0.470563
$784$	0	0
$785$	9.36136	0.334121
$786$	0	0
$787$	15.5749	0.555187	0.277593	−	0.960699i	$-0.410463\pi$
$787$	15.5749	0.555187	0.277593	+	0.960699i	$0.410463\pi$
$788$	0	0
$789$	48.5745	1.72930
$790$	0	0
$791$	−5.91255	−0.210226
$792$	0	0
$793$	−3.80480	−0.135112
$794$	0	0
$795$	27.4615	0.973958
$796$	0	0
$797$	16.3360	0.578649	0.289325	−	0.957231i	$-0.406569\pi$
$797$	16.3360	0.578649	0.289325	+	0.957231i	$0.406569\pi$
$798$	0	0
$799$	3.42156	0.121046
$800$	0	0
$801$	−0.0875919	−0.00309491
$802$	0	0
$803$	38.6590	1.36425
$804$	0	0
$805$	−13.7617	−0.485036
$806$	0	0
$807$	−26.9073	−0.947183
$808$	0	0
$809$	−4.41795	−0.155327	−0.0776635	−	0.996980i	$-0.524746\pi$
$809$	−4.41795	−0.155327	−0.0776635	+	0.996980i	$0.524746\pi$
$810$	0	0
$811$	−22.3329	−0.784215	−0.392107	−	0.919919i	$-0.628254\pi$
$811$	−22.3329	−0.784215	−0.392107	+	0.919919i	$0.628254\pi$
$812$	0	0
$813$	9.12348	0.319974
$814$	0	0
$815$	10.8465	0.379935
$816$	0	0
$817$	−6.20088	−0.216941
$818$	0	0
$819$	0.0338297	0.00118210
$820$	0	0
$821$	−17.1268	−0.597729	−0.298864	−	0.954296i	$-0.596608\pi$
$821$	−17.1268	−0.597729	−0.298864	+	0.954296i	$0.596608\pi$
$822$	0	0
$823$	23.4609	0.817795	0.408898	−	0.912580i	$-0.365913\pi$
$823$	23.4609	0.817795	0.408898	+	0.912580i	$0.365913\pi$
$824$	0	0
$825$	18.4033	0.640719
$826$	0	0
$827$	18.0744	0.628507	0.314253	−	0.949339i	$-0.398246\pi$
$827$	18.0744	0.628507	0.314253	+	0.949339i	$0.398246\pi$
$828$	0	0
$829$	−3.59196	−0.124754	−0.0623770	−	0.998053i	$-0.519868\pi$
$829$	−3.59196	−0.124754	−0.0623770	+	0.998053i	$0.519868\pi$
$830$	0	0
$831$	9.75804	0.338503
$832$	0	0
$833$	1.53595	0.0532177
$834$	0	0
$835$	17.7005	0.612550
$836$	0	0
$837$	−2.42523	−0.0838283
$838$	0	0
$839$	31.8764	1.10050	0.550248	−	0.835001i	$-0.314533\pi$
$839$	31.8764	1.10050	0.550248	+	0.835001i	$0.314533\pi$
$840$	0	0
$841$	−22.6524	−0.781116
$842$	0	0
$843$	−36.9563	−1.27284
$844$	0	0
$845$	18.6367	0.641121
$846$	0	0
$847$	5.63393	0.193584
$848$	0	0
$849$	−14.9998	−0.514790
$850$	0	0
$851$	60.5894	2.07698
$852$	0	0
$853$	52.0420	1.78189	0.890943	−	0.454116i	$-0.150045\pi$
$853$	52.0420	1.78189	0.890943	+	0.454116i	$0.150045\pi$
$854$	0	0
$855$	0.0544633	0.00186261
$856$	0	0
$857$	47.3779	1.61840	0.809199	−	0.587534i	$-0.199901\pi$
$857$	47.3779	1.61840	0.809199	+	0.587534i	$0.199901\pi$
$858$	0	0
$859$	−1.87430	−0.0639502	−0.0319751	−	0.999489i	$-0.510180\pi$
$859$	−1.87430	−0.0639502	−0.0319751	+	0.999489i	$0.510180\pi$
$860$	0	0
$861$	3.40870	0.116168
$862$	0	0
$863$	−12.6321	−0.430000	−0.215000	−	0.976614i	$-0.568975\pi$
$863$	−12.6321	−0.430000	−0.215000	+	0.976614i	$0.568975\pi$
$864$	0	0
$865$	26.1693	0.889784
$866$	0	0
$867$	−25.2090	−0.856143
$868$	0	0
$869$	−8.31646	−0.282117
$870$	0	0
$871$	−11.8755	−0.402384
$872$	0	0
$873$	−0.0909510	−0.00307822
$874$	0	0
$875$	11.7550	0.397390
$876$	0	0
$877$	20.6355	0.696811	0.348406	−	0.937344i	$-0.386723\pi$
$877$	20.6355	0.696811	0.348406	+	0.937344i	$0.386723\pi$
$878$	0	0
$879$	−0.791435	−0.0266944
$880$	0	0
$881$	−1.70171	−0.0573321	−0.0286660	−	0.999589i	$-0.509126\pi$
$881$	−1.70171	−0.0573321	−0.0286660	+	0.999589i	$0.509126\pi$
$882$	0	0
$883$	32.9265	1.10806	0.554032	−	0.832495i	$-0.313089\pi$
$883$	32.9265	1.10806	0.554032	+	0.832495i	$0.313089\pi$
$884$	0	0
$885$	34.1279	1.14720
$886$	0	0
$887$	−29.7341	−0.998372	−0.499186	−	0.866495i	$-0.666368\pi$
$887$	−29.7341	−0.998372	−0.499186	+	0.866495i	$0.666368\pi$
$888$	0	0
$889$	3.86075	0.129485
$890$	0	0
$891$	36.2691	1.21506
$892$	0	0
$893$	2.22764	0.0745453
$894$	0	0
$895$	−31.3410	−1.04762
$896$	0	0
$897$	−14.7182	−0.491428
$898$	0	0
$899$	1.16914	0.0389931
$900$	0	0
$901$	−15.8812	−0.529078
$902$	0	0
$903$	−10.6768	−0.355303
$904$	0	0
$905$	−26.7846	−0.890349
$906$	0	0
$907$	17.1920	0.570851	0.285425	−	0.958401i	$-0.407865\pi$
$907$	17.1920	0.570851	0.285425	+	0.958401i	$0.407865\pi$
$908$	0	0
$909$	0.0983434	0.00326185
$910$	0	0
$911$	−32.7334	−1.08450	−0.542252	−	0.840216i	$-0.682428\pi$
$911$	−32.7334	−1.08450	−0.542252	+	0.840216i	$0.682428\pi$
$912$	0	0
$913$	−5.13937	−0.170088
$914$	0	0
$915$	−10.5470	−0.348672
$916$	0	0
$917$	−14.9493	−0.493670
$918$	0	0
$919$	3.93326	0.129746	0.0648732	−	0.997894i	$-0.479336\pi$
$919$	3.93326	0.129746	0.0648732	+	0.997894i	$0.479336\pi$
$920$	0	0
$921$	41.2277	1.35850
$922$	0	0
$923$	13.5008	0.444384
$924$	0	0
$925$	−17.7976	−0.585182
$926$	0	0
$927$	0.208330	0.00684244
$928$	0	0
$929$	−52.8729	−1.73470	−0.867352	−	0.497694i	$-0.834180\pi$
$929$	−52.8729	−1.73470	−0.867352	+	0.497694i	$0.834180\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	42.6977	1.39786
$934$	0	0
$935$	9.66286	0.316009
$936$	0	0
$937$	−8.33464	−0.272281	−0.136140	−	0.990690i	$-0.543470\pi$
$937$	−8.33464	−0.272281	−0.136140	+	0.990690i	$0.543470\pi$
$938$	0	0
$939$	−15.0657	−0.491651
$940$	0	0
$941$	−59.1436	−1.92803	−0.964013	−	0.265854i	$-0.914346\pi$
$941$	−59.1436	−1.92803	−0.964013	+	0.265854i	$0.914346\pi$
$942$	0	0
$943$	17.6620	0.575155
$944$	0	0
$945$	8.06162	0.262245
$946$	0	0
$947$	51.7202	1.68068	0.840340	−	0.542060i	$-0.182355\pi$
$947$	51.7202	1.68068	0.840340	+	0.542060i	$0.182355\pi$
$948$	0	0
$949$	9.08190	0.294811
$950$	0	0
$951$	4.66453	0.151258
$952$	0	0
$953$	−26.6904	−0.864588	−0.432294	−	0.901733i	$-0.642296\pi$
$953$	−26.6904	−0.864588	−0.432294	+	0.901733i	$0.642296\pi$
$954$	0	0
$955$	17.0129	0.550525
$956$	0	0
$957$	−17.6927	−0.571923
$958$	0	0
$959$	1.40454	0.0453551
$960$	0	0
$961$	−30.7847	−0.993054
$962$	0	0
$963$	0.384480	0.0123897
$964$	0	0
$965$	−2.26551	−0.0729294
$966$	0	0
$967$	18.3963	0.591584	0.295792	−	0.955252i	$-0.404417\pi$
$967$	18.3963	0.591584	0.295792	+	0.955252i	$0.404417\pi$
$968$	0	0
$969$	2.64465	0.0849584
$970$	0	0
$971$	2.87813	0.0923637	0.0461818	−	0.998933i	$-0.485295\pi$
$971$	2.87813	0.0923637	0.0461818	+	0.998933i	$0.485295\pi$
$972$	0	0
$973$	−22.0324	−0.706326
$974$	0	0
$975$	4.32336	0.138458
$976$	0	0
$977$	36.4521	1.16621	0.583103	−	0.812398i	$-0.301838\pi$
$977$	36.4521	1.16621	0.583103	+	0.812398i	$0.301838\pi$
$978$	0	0
$979$	−10.1178	−0.323367
$980$	0	0
$981$	0.191595	0.00611716
$982$	0	0
$983$	41.9792	1.33893	0.669465	−	0.742844i	$-0.266524\pi$
$983$	41.9792	1.33893	0.669465	+	0.742844i	$0.266524\pi$
$984$	0	0
$985$	27.8712	0.888052
$986$	0	0
$987$	3.83562	0.122089
$988$	0	0
$989$	−55.3217	−1.75913
$990$	0	0
$991$	−5.34878	−0.169910	−0.0849548	−	0.996385i	$-0.527075\pi$
$991$	−5.34878	−0.169910	−0.0849548	+	0.996385i	$0.527075\pi$
$992$	0	0
$993$	13.9417	0.442426
$994$	0	0
$995$	−6.06207	−0.192181
$996$	0	0
$997$	9.13068	0.289171	0.144586	−	0.989492i	$-0.453815\pi$
$997$	9.13068	0.289171	0.144586	+	0.989492i	$0.453815\pi$
$998$	0	0
$999$	−35.4934	−1.12296

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.ca.1.6		6
4.3	odd	2		8512.2.a.cf.1.1		6
8.3	odd	2		4256.2.a.i.1.6	✓	6
8.5	even	2		4256.2.a.n.1.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4256.2.a.i.1.6	✓	6	8.3	odd	2
4256.2.a.n.1.1	yes	6	8.5	even	2
8512.2.a.ca.1.6		6	1.1	even	1	trivial
8512.2.a.cf.1.1		6	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q+1.72183 q^{3} -1.54252 q^{5} +1.00000 q^{7} -0.0353081 q^{9} -4.07847 q^{11} -0.958129 q^{13} -2.65595 q^{15} +1.53595 q^{17} +1.00000 q^{19} +1.72183 q^{21} +8.92159 q^{23} -2.62064 q^{25} -5.22628 q^{27} +2.51945 q^{29} +0.464046 q^{31} -7.02243 q^{33} -1.54252 q^{35} +6.79133 q^{37} -1.64973 q^{39} +1.97970 q^{41} -6.20088 q^{43} +0.0544633 q^{45} +2.22764 q^{47} +1.00000 q^{49} +2.64465 q^{51} -10.3396 q^{53} +6.29111 q^{55} +1.72183 q^{57} -12.8496 q^{59} +3.97107 q^{61} -0.0353081 q^{63} +1.47793 q^{65} +12.3944 q^{67} +15.3614 q^{69} -14.0908 q^{71} -9.47879 q^{73} -4.51229 q^{75} -4.07847 q^{77} +2.03911 q^{79} -8.89283 q^{81} +1.26012 q^{83} -2.36924 q^{85} +4.33807 q^{87} +2.48079 q^{89} -0.958129 q^{91} +0.799007 q^{93} -1.54252 q^{95} +2.57593 q^{97} +0.144003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9} - 7 q^{11} - 4 q^{13} + 4 q^{15} - 2 q^{17} + 6 q^{19} - 3 q^{21} - 2 q^{23} + q^{25} - 18 q^{27} - 5 q^{29} + 14 q^{31} + 3 q^{33} - 3 q^{35} + q^{37} + 22 q^{39}+ \cdots - 3 q^{99}+O(q^{100}) \) 6 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9 - 7 * q^11 - 4 * q^13 + 4 * q^15 - 2 * q^17 + 6 * q^19 - 3 * q^21 - 2 * q^23 + q^25 - 18 * q^27 - 5 * q^29 + 14 * q^31 + 3 * q^33 - 3 * q^35 + q^37 + 22 * q^39 - 5 * q^41 - 15 * q^43 - 22 * q^45 - 7 * q^47 + 6 * q^49 + 4 * q^51 - 15 * q^53 + 20 * q^55 - 3 * q^57 - 23 * q^59 - 9 * q^61 + 3 * q^63 - 20 * q^65 - 2 * q^67 + 2 * q^69 + 23 * q^71 + 2 * q^73 - 4 * q^75 - 7 * q^77 + 17 * q^79 + 6 * q^81 - 22 * q^83 + 14 * q^85 - 10 * q^87 + 3 * q^89 - 4 * q^91 - 10 * q^93 - 3 * q^95 - 21 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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