LMFDB - Embedded newform 8512.2.a.ca.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8512,2,Mod(1,8512)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.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:	$67.9686622005$
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			1.4
Root			$-0.365550$ 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)

$f(q)$	$=$	$q+0.365550 q^{3} +2.99343 q^{5} +1.00000 q^{7} -2.86637 q^{9} +O(q^{10})$	$q+0.365550 q^{3} +2.99343 q^{5} +1.00000 q^{7} -2.86637 q^{9} -0.171857 q^{11} -3.29234 q^{13} +1.09425 q^{15} +2.16529 q^{17} +1.00000 q^{19} +0.365550 q^{21} -5.12430 q^{23} +3.96062 q^{25} -2.14445 q^{27} -9.28679 q^{29} -0.165287 q^{31} -0.0628225 q^{33} +2.99343 q^{35} +10.4739 q^{37} -1.20352 q^{39} -10.9321 q^{41} +5.86544 q^{43} -8.58029 q^{45} -10.7520 q^{47} +1.00000 q^{49} +0.791521 q^{51} +0.616283 q^{53} -0.514443 q^{55} +0.365550 q^{57} -8.37359 q^{59} -1.85655 q^{61} -2.86637 q^{63} -9.85540 q^{65} -9.05752 q^{67} -1.87319 q^{69} +12.2076 q^{71} -6.43184 q^{73} +1.44781 q^{75} -0.171857 q^{77} +5.34636 q^{79} +7.81521 q^{81} +2.24736 q^{83} +6.48164 q^{85} -3.39479 q^{87} -5.83419 q^{89} -3.29234 q^{91} -0.0604208 q^{93} +2.99343 q^{95} +14.7989 q^{97} +0.492607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) $ 6 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

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
$3$	0.365550	0.211051	0.105525	−	0.994417i	$-0.466348\pi$
$3$	0.365550	0.211051	0.105525	+	0.994417i	$0.466348\pi$
$4$	0	0
$5$	2.99343	1.33870	0.669351	−	0.742946i	$-0.266572\pi$
$5$	2.99343	1.33870	0.669351	+	0.742946i	$0.266572\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.86637	−0.955458
$10$	0	0
$11$	−0.171857	−0.0518169	−0.0259085	−	0.999664i	$-0.508248\pi$
$11$	−0.171857	−0.0518169	−0.0259085	+	0.999664i	$0.508248\pi$
$12$	0	0
$13$	−3.29234	−0.913132	−0.456566	−	0.889690i	$-0.650921\pi$
$13$	−3.29234	−0.913132	−0.456566	+	0.889690i	$0.650921\pi$
$14$	0	0
$15$	1.09425	0.282534
$16$	0	0
$17$	2.16529	0.525159	0.262580	−	0.964910i	$-0.415427\pi$
$17$	2.16529	0.525159	0.262580	+	0.964910i	$0.415427\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.365550	0.0797696
$22$	0	0
$23$	−5.12430	−1.06849	−0.534245	−	0.845330i	$-0.679404\pi$
$23$	−5.12430	−1.06849	−0.534245	+	0.845330i	$0.679404\pi$
$24$	0	0
$25$	3.96062	0.792124
$26$	0	0
$27$	−2.14445	−0.412700
$28$	0	0
$29$	−9.28679	−1.72451	−0.862257	−	0.506471i	$-0.830950\pi$
$29$	−9.28679	−1.72451	−0.862257	+	0.506471i	$0.830950\pi$
$30$	0	0
$31$	−0.165287	−0.0296865	−0.0148432	−	0.999890i	$-0.504725\pi$
$31$	−0.165287	−0.0296865	−0.0148432	+	0.999890i	$0.504725\pi$
$32$	0	0
$33$	−0.0628225	−0.0109360
$34$	0	0
$35$	2.99343	0.505982
$36$	0	0
$37$	10.4739	1.72190	0.860948	−	0.508693i	$-0.169871\pi$
$37$	10.4739	1.72190	0.860948	+	0.508693i	$0.169871\pi$
$38$	0	0
$39$	−1.20352	−0.192717
$40$	0	0
$41$	−10.9321	−1.70730	−0.853651	−	0.520845i	$-0.825617\pi$
$41$	−10.9321	−1.70730	−0.853651	+	0.520845i	$0.825617\pi$
$42$	0	0
$43$	5.86544	0.894471	0.447235	−	0.894416i	$-0.352409\pi$
$43$	5.86544	0.894471	0.447235	+	0.894416i	$0.352409\pi$
$44$	0	0
$45$	−8.58029	−1.27907
$46$	0	0
$47$	−10.7520	−1.56834	−0.784170	−	0.620546i	$-0.786911\pi$
$47$	−10.7520	−1.56834	−0.784170	+	0.620546i	$0.786911\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.791521	0.110835
$52$	0	0
$53$	0.616283	0.0846530	0.0423265	−	0.999104i	$-0.486523\pi$
$53$	0.616283	0.0846530	0.0423265	+	0.999104i	$0.486523\pi$
$54$	0	0
$55$	−0.514443	−0.0693675
$56$	0	0
$57$	0.365550	0.0484183
$58$	0	0
$59$	−8.37359	−1.09015	−0.545074	−	0.838388i	$-0.683498\pi$
$59$	−8.37359	−1.09015	−0.545074	+	0.838388i	$0.683498\pi$
$60$	0	0
$61$	−1.85655	−0.237707	−0.118853	−	0.992912i	$-0.537922\pi$
$61$	−1.85655	−0.237707	−0.118853	+	0.992912i	$0.537922\pi$
$62$	0	0
$63$	−2.86637	−0.361129
$64$	0	0
$65$	−9.85540	−1.22241
$66$	0	0
$67$	−9.05752	−1.10655	−0.553276	−	0.832998i	$-0.686623\pi$
$67$	−9.05752	−1.10655	−0.553276	+	0.832998i	$0.686623\pi$
$68$	0	0
$69$	−1.87319	−0.225506
$70$	0	0
$71$	12.2076	1.44877	0.724386	−	0.689394i	$-0.242123\pi$
$71$	12.2076	1.44877	0.724386	+	0.689394i	$0.242123\pi$
$72$	0	0
$73$	−6.43184	−0.752790	−0.376395	−	0.926459i	$-0.622836\pi$
$73$	−6.43184	−0.752790	−0.376395	+	0.926459i	$0.622836\pi$
$74$	0	0
$75$	1.44781	0.167178
$76$	0	0
$77$	−0.171857	−0.0195850
$78$	0	0
$79$	5.34636	0.601513	0.300756	−	0.953701i	$-0.402761\pi$
$79$	5.34636	0.601513	0.300756	+	0.953701i	$0.402761\pi$
$80$	0	0
$81$	7.81521	0.868357
$82$	0	0
$83$	2.24736	0.246680	0.123340	−	0.992364i	$-0.460639\pi$
$83$	2.24736	0.246680	0.123340	+	0.992364i	$0.460639\pi$
$84$	0	0
$85$	6.48164	0.703032
$86$	0	0
$87$	−3.39479	−0.363960
$88$	0	0
$89$	−5.83419	−0.618423	−0.309211	−	0.950993i	$-0.600065\pi$
$89$	−5.83419	−0.618423	−0.309211	+	0.950993i	$0.600065\pi$
$90$	0	0
$91$	−3.29234	−0.345131
$92$	0	0
$93$	−0.0604208	−0.00626534
$94$	0	0
$95$	2.99343	0.307119
$96$	0	0
$97$	14.7989	1.50260	0.751300	−	0.659961i	$-0.229427\pi$
$97$	14.7989	1.50260	0.751300	+	0.659961i	$0.229427\pi$
$98$	0	0
$99$	0.492607	0.0495089
$100$	0	0
$101$	−7.60311	−0.756537	−0.378269	−	0.925696i	$-0.623480\pi$
$101$	−7.60311	−0.756537	−0.378269	+	0.925696i	$0.623480\pi$
$102$	0	0
$103$	18.5152	1.82435	0.912176	−	0.409799i	$-0.134401\pi$
$103$	18.5152	1.82435	0.912176	+	0.409799i	$0.134401\pi$
$104$	0	0
$105$	1.09425	0.106788
$106$	0	0
$107$	−14.3739	−1.38957	−0.694786	−	0.719216i	$-0.744501\pi$
$107$	−14.3739	−1.38957	−0.694786	+	0.719216i	$0.744501\pi$
$108$	0	0
$109$	−14.4295	−1.38209	−0.691047	−	0.722809i	$-0.742850\pi$
$109$	−14.4295	−1.38209	−0.691047	+	0.722809i	$0.742850\pi$
$110$	0	0
$111$	3.82873	0.363407
$112$	0	0
$113$	−18.7646	−1.76523	−0.882613	−	0.470100i	$-0.844218\pi$
$113$	−18.7646	−1.76523	−0.882613	+	0.470100i	$0.844218\pi$
$114$	0	0
$115$	−15.3392	−1.43039
$116$	0	0
$117$	9.43709	0.872459
$118$	0	0
$119$	2.16529	0.198492
$120$	0	0
$121$	−10.9705	−0.997315
$122$	0	0
$123$	−3.99622	−0.360327
$124$	0	0
$125$	−3.11130	−0.278284
$126$	0	0
$127$	11.0463	0.980199	0.490099	−	0.871667i	$-0.336961\pi$
$127$	11.0463	0.980199	0.490099	+	0.871667i	$0.336961\pi$
$128$	0	0
$129$	2.14411	0.188779
$130$	0	0
$131$	−16.9363	−1.47973	−0.739865	−	0.672755i	$-0.765111\pi$
$131$	−16.9363	−1.47973	−0.739865	+	0.672755i	$0.765111\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−6.41927	−0.552483
$136$	0	0
$137$	−4.61526	−0.394308	−0.197154	−	0.980372i	$-0.563170\pi$
$137$	−4.61526	−0.394308	−0.197154	+	0.980372i	$0.563170\pi$
$138$	0	0
$139$	9.03314	0.766181	0.383090	−	0.923711i	$-0.374860\pi$
$139$	9.03314	0.766181	0.383090	+	0.923711i	$0.374860\pi$
$140$	0	0
$141$	−3.93040	−0.330999
$142$	0	0
$143$	0.565813	0.0473157
$144$	0	0
$145$	−27.7994	−2.30861
$146$	0	0
$147$	0.365550	0.0301501
$148$	0	0
$149$	−12.1497	−0.995345	−0.497672	−	0.867365i	$-0.665812\pi$
$149$	−12.1497	−0.995345	−0.497672	+	0.867365i	$0.665812\pi$
$150$	0	0
$151$	3.31192	0.269520	0.134760	−	0.990878i	$-0.456974\pi$
$151$	3.31192	0.269520	0.134760	+	0.990878i	$0.456974\pi$
$152$	0	0
$153$	−6.20652	−0.501767
$154$	0	0
$155$	−0.494776	−0.0397413
$156$	0	0
$157$	−4.00177	−0.319376	−0.159688	−	0.987168i	$-0.551049\pi$
$157$	−4.00177	−0.319376	−0.159688	+	0.987168i	$0.551049\pi$
$158$	0	0
$159$	0.225283	0.0178661
$160$	0	0
$161$	−5.12430	−0.403851
$162$	0	0
$163$	−21.3955	−1.67583	−0.837913	−	0.545803i	$-0.816225\pi$
$163$	−21.3955	−1.67583	−0.837913	+	0.545803i	$0.816225\pi$
$164$	0	0
$165$	−0.188055	−0.0146400
$166$	0	0
$167$	7.14255	0.552707	0.276354	−	0.961056i	$-0.410874\pi$
$167$	7.14255	0.552707	0.276354	+	0.961056i	$0.410874\pi$
$168$	0	0
$169$	−2.16047	−0.166190
$170$	0	0
$171$	−2.86637	−0.219197
$172$	0	0
$173$	−0.835091	−0.0634908	−0.0317454	−	0.999496i	$-0.510107\pi$
$173$	−0.835091	−0.0634908	−0.0317454	+	0.999496i	$0.510107\pi$
$174$	0	0
$175$	3.96062	0.299395
$176$	0	0
$177$	−3.06097	−0.230076
$178$	0	0
$179$	26.1441	1.95410	0.977050	−	0.213011i	$-0.0683272\pi$
$179$	26.1441	1.95410	0.977050	+	0.213011i	$0.0683272\pi$
$180$	0	0
$181$	−15.1120	−1.12327	−0.561634	−	0.827386i	$-0.689827\pi$
$181$	−15.1120	−1.12327	−0.561634	+	0.827386i	$0.689827\pi$
$182$	0	0
$183$	−0.678662	−0.0501681
$184$	0	0
$185$	31.3528	2.30511
$186$	0	0
$187$	−0.372120	−0.0272121
$188$	0	0
$189$	−2.14445	−0.155986
$190$	0	0
$191$	−11.2208	−0.811911	−0.405956	−	0.913893i	$-0.633061\pi$
$191$	−11.2208	−0.811911	−0.405956	+	0.913893i	$0.633061\pi$
$192$	0	0
$193$	9.60194	0.691163	0.345581	−	0.938389i	$-0.387682\pi$
$193$	9.60194	0.691163	0.345581	+	0.938389i	$0.387682\pi$
$194$	0	0
$195$	−3.60265	−0.257991
$196$	0	0
$197$	7.39729	0.527035	0.263518	−	0.964655i	$-0.415117\pi$
$197$	7.39729	0.527035	0.263518	+	0.964655i	$0.415117\pi$
$198$	0	0
$199$	22.4095	1.58857	0.794285	−	0.607545i	$-0.207846\pi$
$199$	22.4095	1.58857	0.794285	+	0.607545i	$0.207846\pi$
$200$	0	0
$201$	−3.31098	−0.233538
$202$	0	0
$203$	−9.28679	−0.651805
$204$	0	0
$205$	−32.7244	−2.28557
$206$	0	0
$207$	14.6882	1.02090
$208$	0	0
$209$	−0.171857	−0.0118876
$210$	0	0
$211$	−7.28952	−0.501831	−0.250915	−	0.968009i	$-0.580732\pi$
$211$	−7.28952	−0.501831	−0.250915	+	0.968009i	$0.580732\pi$
$212$	0	0
$213$	4.46248	0.305764
$214$	0	0
$215$	17.5578	1.19743
$216$	0	0
$217$	−0.165287	−0.0112204
$218$	0	0
$219$	−2.35116	−0.158877
$220$	0	0
$221$	−7.12887	−0.479540
$222$	0	0
$223$	24.1967	1.62033	0.810164	−	0.586203i	$-0.199378\pi$
$223$	24.1967	1.62033	0.810164	+	0.586203i	$0.199378\pi$
$224$	0	0
$225$	−11.3526	−0.756841
$226$	0	0
$227$	5.15813	0.342357	0.171179	−	0.985240i	$-0.445243\pi$
$227$	5.15813	0.342357	0.171179	+	0.985240i	$0.445243\pi$
$228$	0	0
$229$	−2.56201	−0.169302	−0.0846510	−	0.996411i	$-0.526978\pi$
$229$	−2.56201	−0.169302	−0.0846510	+	0.996411i	$0.526978\pi$
$230$	0	0
$231$	−0.0628225	−0.00413342
$232$	0	0
$233$	−19.8681	−1.30160	−0.650802	−	0.759248i	$-0.725567\pi$
$233$	−19.8681	−1.30160	−0.650802	+	0.759248i	$0.725567\pi$
$234$	0	0
$235$	−32.1854	−2.09954
$236$	0	0
$237$	1.95436	0.126950
$238$	0	0
$239$	−4.52969	−0.293001	−0.146501	−	0.989211i	$-0.546801\pi$
$239$	−4.52969	−0.293001	−0.146501	+	0.989211i	$0.546801\pi$
$240$	0	0
$241$	−26.7883	−1.72559	−0.862794	−	0.505556i	$-0.831287\pi$
$241$	−26.7883	−1.72559	−0.862794	+	0.505556i	$0.831287\pi$
$242$	0	0
$243$	9.29022	0.595968
$244$	0	0
$245$	2.99343	0.191243
$246$	0	0
$247$	−3.29234	−0.209487
$248$	0	0
$249$	0.821524	0.0520620
$250$	0	0
$251$	−5.17309	−0.326522	−0.163261	−	0.986583i	$-0.552201\pi$
$251$	−5.17309	−0.326522	−0.163261	+	0.986583i	$0.552201\pi$
$252$	0	0
$253$	0.880649	0.0553659
$254$	0	0
$255$	2.36936	0.148375
$256$	0	0
$257$	19.6114	1.22333	0.611663	−	0.791118i	$-0.290501\pi$
$257$	19.6114	1.22333	0.611663	+	0.791118i	$0.290501\pi$
$258$	0	0
$259$	10.4739	0.650815
$260$	0	0
$261$	26.6194	1.64770
$262$	0	0
$263$	−12.9880	−0.800873	−0.400437	−	0.916324i	$-0.631142\pi$
$263$	−12.9880	−0.800873	−0.400437	+	0.916324i	$0.631142\pi$
$264$	0	0
$265$	1.84480	0.113325
$266$	0	0
$267$	−2.13269	−0.130519
$268$	0	0
$269$	−9.21484	−0.561839	−0.280919	−	0.959731i	$-0.590639\pi$
$269$	−9.21484	−0.561839	−0.280919	+	0.959731i	$0.590639\pi$
$270$	0	0
$271$	26.0814	1.58433	0.792165	−	0.610306i	$-0.208954\pi$
$271$	26.0814	1.58433	0.792165	+	0.610306i	$0.208954\pi$
$272$	0	0
$273$	−1.20352	−0.0728402
$274$	0	0
$275$	−0.680662	−0.0410455
$276$	0	0
$277$	−17.0157	−1.02238	−0.511189	−	0.859469i	$-0.670795\pi$
$277$	−17.0157	−1.02238	−0.511189	+	0.859469i	$0.670795\pi$
$278$	0	0
$279$	0.473775	0.0283641
$280$	0	0
$281$	14.6975	0.876779	0.438389	−	0.898785i	$-0.355549\pi$
$281$	14.6975	0.876779	0.438389	+	0.898785i	$0.355549\pi$
$282$	0	0
$283$	3.05012	0.181311	0.0906554	−	0.995882i	$-0.471104\pi$
$283$	3.05012	0.181311	0.0906554	+	0.995882i	$0.471104\pi$
$284$	0	0
$285$	1.09425	0.0648177
$286$	0	0
$287$	−10.9321	−0.645300
$288$	0	0
$289$	−12.3115	−0.724208
$290$	0	0
$291$	5.40974	0.317124
$292$	0	0
$293$	−11.7365	−0.685656	−0.342828	−	0.939398i	$-0.611385\pi$
$293$	−11.7365	−0.685656	−0.342828	+	0.939398i	$0.611385\pi$
$294$	0	0
$295$	−25.0657	−1.45938
$296$	0	0
$297$	0.368540	0.0213849
$298$	0	0
$299$	16.8710	0.975673
$300$	0	0
$301$	5.86544	0.338078
$302$	0	0
$303$	−2.77932	−0.159668
$304$	0	0
$305$	−5.55745	−0.318219
$306$	0	0
$307$	−20.5082	−1.17046	−0.585232	−	0.810866i	$-0.698997\pi$
$307$	−20.5082	−1.17046	−0.585232	+	0.810866i	$0.698997\pi$
$308$	0	0
$309$	6.76822	0.385031
$310$	0	0
$311$	−25.5156	−1.44686	−0.723428	−	0.690400i	$-0.757435\pi$
$311$	−25.5156	−1.44686	−0.723428	+	0.690400i	$0.757435\pi$
$312$	0	0
$313$	7.49982	0.423915	0.211957	−	0.977279i	$-0.432016\pi$
$313$	7.49982	0.423915	0.211957	+	0.977279i	$0.432016\pi$
$314$	0	0
$315$	−8.58029	−0.483444
$316$	0	0
$317$	31.1237	1.74808	0.874041	−	0.485853i	$-0.161491\pi$
$317$	31.1237	1.74808	0.874041	+	0.485853i	$0.161491\pi$
$318$	0	0
$319$	1.59600	0.0893591
$320$	0	0
$321$	−5.25437	−0.293270
$322$	0	0
$323$	2.16529	0.120480
$324$	0	0
$325$	−13.0397	−0.723314
$326$	0	0
$327$	−5.27471	−0.291692
$328$	0	0
$329$	−10.7520	−0.592777
$330$	0	0
$331$	−6.59453	−0.362468	−0.181234	−	0.983440i	$-0.558009\pi$
$331$	−6.59453	−0.362468	−0.181234	+	0.983440i	$0.558009\pi$
$332$	0	0
$333$	−30.0221	−1.64520
$334$	0	0
$335$	−27.1131	−1.48134
$336$	0	0
$337$	2.32019	0.126389	0.0631944	−	0.998001i	$-0.479871\pi$
$337$	2.32019	0.126389	0.0631944	+	0.998001i	$0.479871\pi$
$338$	0	0
$339$	−6.85941	−0.372552
$340$	0	0
$341$	0.0284058	0.00153826
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−5.60726	−0.301885
$346$	0	0
$347$	25.4846	1.36809	0.684043	−	0.729442i	$-0.260220\pi$
$347$	25.4846	1.36809	0.684043	+	0.729442i	$0.260220\pi$
$348$	0	0
$349$	10.9770	0.587584	0.293792	−	0.955869i	$-0.405083\pi$
$349$	10.9770	0.587584	0.293792	+	0.955869i	$0.405083\pi$
$350$	0	0
$351$	7.06028	0.376850
$352$	0	0
$353$	13.3836	0.712337	0.356168	−	0.934422i	$-0.384083\pi$
$353$	13.3836	0.712337	0.356168	+	0.934422i	$0.384083\pi$
$354$	0	0
$355$	36.5425	1.93948
$356$	0	0
$357$	0.791521	0.0418918
$358$	0	0
$359$	−7.51270	−0.396505	−0.198253	−	0.980151i	$-0.563527\pi$
$359$	−7.51270	−0.396505	−0.198253	+	0.980151i	$0.563527\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−4.01026	−0.210484
$364$	0	0
$365$	−19.2533	−1.00776
$366$	0	0
$367$	−7.76527	−0.405344	−0.202672	−	0.979247i	$-0.564963\pi$
$367$	−7.76527	−0.405344	−0.202672	+	0.979247i	$0.564963\pi$
$368$	0	0
$369$	31.3354	1.63126
$370$	0	0
$371$	0.616283	0.0319958
$372$	0	0
$373$	12.9380	0.669907	0.334953	−	0.942235i	$-0.391279\pi$
$373$	12.9380	0.669907	0.334953	+	0.942235i	$0.391279\pi$
$374$	0	0
$375$	−1.13734	−0.0587319
$376$	0	0
$377$	30.5753	1.57471
$378$	0	0
$379$	−9.93156	−0.510150	−0.255075	−	0.966921i	$-0.582100\pi$
$379$	−9.93156	−0.510150	−0.255075	+	0.966921i	$0.582100\pi$
$380$	0	0
$381$	4.03797	0.206871
$382$	0	0
$383$	23.9838	1.22551	0.612757	−	0.790271i	$-0.290060\pi$
$383$	23.9838	1.22551	0.612757	+	0.790271i	$0.290060\pi$
$384$	0	0
$385$	−0.514443	−0.0262184
$386$	0	0
$387$	−16.8125	−0.854629
$388$	0	0
$389$	31.8477	1.61474	0.807371	−	0.590045i	$-0.200890\pi$
$389$	31.8477	1.61474	0.807371	+	0.590045i	$0.200890\pi$
$390$	0	0
$391$	−11.0956	−0.561128
$392$	0	0
$393$	−6.19107	−0.312298
$394$	0	0
$395$	16.0040	0.805247
$396$	0	0
$397$	−10.7966	−0.541864	−0.270932	−	0.962598i	$-0.587332\pi$
$397$	−10.7966	−0.541864	−0.270932	+	0.962598i	$0.587332\pi$
$398$	0	0
$399$	0.365550	0.0183004
$400$	0	0
$401$	−5.68303	−0.283797	−0.141898	−	0.989881i	$-0.545321\pi$
$401$	−5.68303	−0.283797	−0.141898	+	0.989881i	$0.545321\pi$
$402$	0	0
$403$	0.544182	0.0271076
$404$	0	0
$405$	23.3943	1.16247
$406$	0	0
$407$	−1.80001	−0.0892234
$408$	0	0
$409$	−1.51976	−0.0751472	−0.0375736	−	0.999294i	$-0.511963\pi$
$409$	−1.51976	−0.0751472	−0.0375736	+	0.999294i	$0.511963\pi$
$410$	0	0
$411$	−1.68711	−0.0832190
$412$	0	0
$413$	−8.37359	−0.412037
$414$	0	0
$415$	6.72732	0.330231
$416$	0	0
$417$	3.30207	0.161703
$418$	0	0
$419$	−23.4417	−1.14520	−0.572601	−	0.819834i	$-0.694066\pi$
$419$	−23.4417	−1.14520	−0.572601	+	0.819834i	$0.694066\pi$
$420$	0	0
$421$	4.28353	0.208766	0.104383	−	0.994537i	$-0.466713\pi$
$421$	4.28353	0.208766	0.104383	+	0.994537i	$0.466713\pi$
$422$	0	0
$423$	30.8192	1.49848
$424$	0	0
$425$	8.57588	0.415992
$426$	0	0
$427$	−1.85655	−0.0898447
$428$	0	0
$429$	0.206833	0.00998601
$430$	0	0
$431$	19.5225	0.940366	0.470183	−	0.882569i	$-0.344188\pi$
$431$	19.5225	0.940366	0.470183	+	0.882569i	$0.344188\pi$
$432$	0	0
$433$	22.9594	1.10336	0.551680	−	0.834056i	$-0.313987\pi$
$433$	22.9594	1.10336	0.551680	+	0.834056i	$0.313987\pi$
$434$	0	0
$435$	−10.1621	−0.487234
$436$	0	0
$437$	−5.12430	−0.245129
$438$	0	0
$439$	30.2467	1.44360	0.721798	−	0.692104i	$-0.243316\pi$
$439$	30.2467	1.44360	0.721798	+	0.692104i	$0.243316\pi$
$440$	0	0
$441$	−2.86637	−0.136494
$442$	0	0
$443$	−37.9752	−1.80426	−0.902128	−	0.431469i	$-0.857996\pi$
$443$	−37.9752	−1.80426	−0.902128	+	0.431469i	$0.857996\pi$
$444$	0	0
$445$	−17.4642	−0.827884
$446$	0	0
$447$	−4.44134	−0.210068
$448$	0	0
$449$	9.88074	0.466301	0.233150	−	0.972441i	$-0.425097\pi$
$449$	9.88074	0.466301	0.233150	+	0.972441i	$0.425097\pi$
$450$	0	0
$451$	1.87876	0.0884672
$452$	0	0
$453$	1.21067	0.0568824
$454$	0	0
$455$	−9.85540	−0.462028
$456$	0	0
$457$	−23.8177	−1.11414	−0.557072	−	0.830464i	$-0.688075\pi$
$457$	−23.8177	−1.11414	−0.557072	+	0.830464i	$0.688075\pi$
$458$	0	0
$459$	−4.64336	−0.216733
$460$	0	0
$461$	−26.2870	−1.22431	−0.612154	−	0.790739i	$-0.709697\pi$
$461$	−26.2870	−1.22431	−0.612154	+	0.790739i	$0.709697\pi$
$462$	0	0
$463$	−4.87019	−0.226337	−0.113168	−	0.993576i	$-0.536100\pi$
$463$	−4.87019	−0.226337	−0.113168	+	0.993576i	$0.536100\pi$
$464$	0	0
$465$	−0.180865	−0.00838743
$466$	0	0
$467$	41.7679	1.93279	0.966393	−	0.257069i	$-0.0827567\pi$
$467$	41.7679	1.93279	0.966393	+	0.257069i	$0.0827567\pi$
$468$	0	0
$469$	−9.05752	−0.418237
$470$	0	0
$471$	−1.46285	−0.0674045
$472$	0	0
$473$	−1.00802	−0.0463487
$474$	0	0
$475$	3.96062	0.181726
$476$	0	0
$477$	−1.76650	−0.0808824
$478$	0	0
$479$	−2.89757	−0.132393	−0.0661966	−	0.997807i	$-0.521086\pi$
$479$	−2.89757	−0.132393	−0.0661966	+	0.997807i	$0.521086\pi$
$480$	0	0
$481$	−34.4836	−1.57232
$482$	0	0
$483$	−1.87319	−0.0852331
$484$	0	0
$485$	44.2994	2.01153
$486$	0	0
$487$	32.3162	1.46439	0.732193	−	0.681097i	$-0.238497\pi$
$487$	32.3162	1.46439	0.732193	+	0.681097i	$0.238497\pi$
$488$	0	0
$489$	−7.82114	−0.353684
$490$	0	0
$491$	32.9150	1.48543	0.742716	−	0.669607i	$-0.233537\pi$
$491$	32.9150	1.48543	0.742716	+	0.669607i	$0.233537\pi$
$492$	0	0
$493$	−20.1086	−0.905645
$494$	0	0
$495$	1.47459	0.0662777
$496$	0	0
$497$	12.2076	0.547585
$498$	0	0
$499$	−3.57773	−0.160161	−0.0800807	−	0.996788i	$-0.525518\pi$
$499$	−3.57773	−0.160161	−0.0800807	+	0.996788i	$0.525518\pi$
$500$	0	0
$501$	2.61096	0.116649
$502$	0	0
$503$	−10.5044	−0.468368	−0.234184	−	0.972192i	$-0.575242\pi$
$503$	−10.5044	−0.468368	−0.234184	+	0.972192i	$0.575242\pi$
$504$	0	0
$505$	−22.7594	−1.01278
$506$	0	0
$507$	−0.789761	−0.0350745
$508$	0	0
$509$	−29.5156	−1.30826	−0.654128	−	0.756384i	$-0.726964\pi$
$509$	−29.5156	−1.30826	−0.654128	+	0.756384i	$0.726964\pi$
$510$	0	0
$511$	−6.43184	−0.284528
$512$	0	0
$513$	−2.14445	−0.0946800
$514$	0	0
$515$	55.4238	2.44226
$516$	0	0
$517$	1.84781	0.0812666
$518$	0	0
$519$	−0.305268	−0.0133998
$520$	0	0
$521$	15.3233	0.671324	0.335662	−	0.941982i	$-0.391040\pi$
$521$	15.3233	0.671324	0.335662	+	0.941982i	$0.391040\pi$
$522$	0	0
$523$	−17.3500	−0.758660	−0.379330	−	0.925261i	$-0.623846\pi$
$523$	−17.3500	−0.758660	−0.379330	+	0.925261i	$0.623846\pi$
$524$	0	0
$525$	1.44781	0.0631875
$526$	0	0
$527$	−0.357894	−0.0155901
$528$	0	0
$529$	3.25845	0.141672
$530$	0	0
$531$	24.0018	1.04159
$532$	0	0
$533$	35.9921	1.55899
$534$	0	0
$535$	−43.0271	−1.86022
$536$	0	0
$537$	9.55697	0.412414
$538$	0	0
$539$	−0.171857	−0.00740242
$540$	0	0
$541$	27.5864	1.18603	0.593016	−	0.805191i	$-0.297937\pi$
$541$	27.5864	1.18603	0.593016	+	0.805191i	$0.297937\pi$
$542$	0	0
$543$	−5.52420	−0.237066
$544$	0	0
$545$	−43.1937	−1.85021
$546$	0	0
$547$	−6.13309	−0.262232	−0.131116	−	0.991367i	$-0.541856\pi$
$547$	−6.13309	−0.262232	−0.131116	+	0.991367i	$0.541856\pi$
$548$	0	0
$549$	5.32156	0.227119
$550$	0	0
$551$	−9.28679	−0.395631
$552$	0	0
$553$	5.34636	0.227350
$554$	0	0
$555$	11.4610	0.486494
$556$	0	0
$557$	37.7910	1.60125	0.800627	−	0.599163i	$-0.204500\pi$
$557$	37.7910	1.60125	0.800627	+	0.599163i	$0.204500\pi$
$558$	0	0
$559$	−19.3110	−0.816770
$560$	0	0
$561$	−0.136029	−0.00574314
$562$	0	0
$563$	−41.3967	−1.74466	−0.872331	−	0.488915i	$-0.837393\pi$
$563$	−41.3967	−1.74466	−0.872331	+	0.488915i	$0.837393\pi$
$564$	0	0
$565$	−56.1706	−2.36311
$566$	0	0
$567$	7.81521	0.328208
$568$	0	0
$569$	9.04847	0.379332	0.189666	−	0.981849i	$-0.439260\pi$
$569$	9.04847	0.379332	0.189666	+	0.981849i	$0.439260\pi$
$570$	0	0
$571$	10.3946	0.435001	0.217500	−	0.976060i	$-0.430210\pi$
$571$	10.3946	0.435001	0.217500	+	0.976060i	$0.430210\pi$
$572$	0	0
$573$	−4.10178	−0.171354
$574$	0	0
$575$	−20.2954	−0.846377
$576$	0	0
$577$	29.6336	1.23366	0.616831	−	0.787096i	$-0.288416\pi$
$577$	29.6336	1.23366	0.616831	+	0.787096i	$0.288416\pi$
$578$	0	0
$579$	3.50999	0.145870
$580$	0	0
$581$	2.24736	0.0932363
$582$	0	0
$583$	−0.105913	−0.00438646
$584$	0	0
$585$	28.2493	1.16796
$586$	0	0
$587$	17.8978	0.738721	0.369360	−	0.929286i	$-0.379577\pi$
$587$	17.8978	0.738721	0.369360	+	0.929286i	$0.379577\pi$
$588$	0	0
$589$	−0.165287	−0.00681054
$590$	0	0
$591$	2.70408	0.111231
$592$	0	0
$593$	6.33464	0.260133	0.130066	−	0.991505i	$-0.458481\pi$
$593$	6.33464	0.260133	0.130066	+	0.991505i	$0.458481\pi$
$594$	0	0
$595$	6.48164	0.265721
$596$	0	0
$597$	8.19181	0.335269
$598$	0	0
$599$	8.39548	0.343030	0.171515	−	0.985182i	$-0.445134\pi$
$599$	8.39548	0.343030	0.171515	+	0.985182i	$0.445134\pi$
$600$	0	0
$601$	−20.7692	−0.847195	−0.423597	−	0.905851i	$-0.639233\pi$
$601$	−20.7692	−0.847195	−0.423597	+	0.905851i	$0.639233\pi$
$602$	0	0
$603$	25.9622	1.05726
$604$	0	0
$605$	−32.8393	−1.33511
$606$	0	0
$607$	14.7345	0.598054	0.299027	−	0.954245i	$-0.403338\pi$
$607$	14.7345	0.598054	0.299027	+	0.954245i	$0.403338\pi$
$608$	0	0
$609$	−3.39479	−0.137564
$610$	0	0
$611$	35.3993	1.43210
$612$	0	0
$613$	−14.9040	−0.601966	−0.300983	−	0.953629i	$-0.597315\pi$
$613$	−14.9040	−0.601966	−0.300983	+	0.953629i	$0.597315\pi$
$614$	0	0
$615$	−11.9624	−0.482371
$616$	0	0
$617$	−42.5921	−1.71469	−0.857346	−	0.514741i	$-0.827888\pi$
$617$	−42.5921	−1.71469	−0.857346	+	0.514741i	$0.827888\pi$
$618$	0	0
$619$	−46.0020	−1.84898	−0.924488	−	0.381210i	$-0.875507\pi$
$619$	−46.0020	−1.84898	−0.924488	+	0.381210i	$0.875507\pi$
$620$	0	0
$621$	10.9888	0.440967
$622$	0	0
$623$	−5.83419	−0.233742
$624$	0	0
$625$	−29.1166	−1.16466
$626$	0	0
$627$	−0.0628225	−0.00250889
$628$	0	0
$629$	22.6790	0.904269
$630$	0	0
$631$	22.4418	0.893396	0.446698	−	0.894685i	$-0.352600\pi$
$631$	22.4418	0.893396	0.446698	+	0.894685i	$0.352600\pi$
$632$	0	0
$633$	−2.66468	−0.105912
$634$	0	0
$635$	33.0663	1.31219
$636$	0	0
$637$	−3.29234	−0.130447
$638$	0	0
$639$	−34.9915	−1.38424
$640$	0	0
$641$	−36.4952	−1.44147	−0.720737	−	0.693209i	$-0.756196\pi$
$641$	−36.4952	−1.44147	−0.720737	+	0.693209i	$0.756196\pi$
$642$	0	0
$643$	−16.5653	−0.653272	−0.326636	−	0.945150i	$-0.605915\pi$
$643$	−16.5653	−0.653272	−0.326636	+	0.945150i	$0.605915\pi$
$644$	0	0
$645$	6.41825	0.252718
$646$	0	0
$647$	−20.6288	−0.811001	−0.405501	−	0.914095i	$-0.632903\pi$
$647$	−20.6288	−0.811001	−0.405501	+	0.914095i	$0.632903\pi$
$648$	0	0
$649$	1.43906	0.0564881
$650$	0	0
$651$	−0.0604208	−0.00236808
$652$	0	0
$653$	19.2671	0.753980	0.376990	−	0.926217i	$-0.376959\pi$
$653$	19.2671	0.753980	0.376990	+	0.926217i	$0.376959\pi$
$654$	0	0
$655$	−50.6976	−1.98092
$656$	0	0
$657$	18.4360	0.719259
$658$	0	0
$659$	−43.9706	−1.71285	−0.856426	−	0.516270i	$-0.827320\pi$
$659$	−43.9706	−1.71285	−0.856426	+	0.516270i	$0.827320\pi$
$660$	0	0
$661$	−25.2507	−0.982139	−0.491069	−	0.871120i	$-0.663394\pi$
$661$	−25.2507	−0.982139	−0.491069	+	0.871120i	$0.663394\pi$
$662$	0	0
$663$	−2.60596	−0.101207
$664$	0	0
$665$	2.99343	0.116080
$666$	0	0
$667$	47.5883	1.84263
$668$	0	0
$669$	8.84510	0.341971
$670$	0	0
$671$	0.319061	0.0123172
$672$	0	0
$673$	−46.0886	−1.77658	−0.888292	−	0.459278i	$-0.848108\pi$
$673$	−46.0886	−1.77658	−0.888292	+	0.459278i	$0.848108\pi$
$674$	0	0
$675$	−8.49337	−0.326910
$676$	0	0
$677$	−32.3740	−1.24423	−0.622117	−	0.782924i	$-0.713727\pi$
$677$	−32.3740	−1.24423	−0.622117	+	0.782924i	$0.713727\pi$
$678$	0	0
$679$	14.7989	0.567929
$680$	0	0
$681$	1.88556	0.0722547
$682$	0	0
$683$	−25.0338	−0.957893	−0.478947	−	0.877844i	$-0.658981\pi$
$683$	−25.0338	−0.957893	−0.478947	+	0.877844i	$0.658981\pi$
$684$	0	0
$685$	−13.8155	−0.527862
$686$	0	0
$687$	−0.936542	−0.0357313
$688$	0	0
$689$	−2.02902	−0.0772994
$690$	0	0
$691$	37.2009	1.41519	0.707595	−	0.706619i	$-0.249780\pi$
$691$	37.2009	1.41519	0.707595	+	0.706619i	$0.249780\pi$
$692$	0	0
$693$	0.492607	0.0187126
$694$	0	0
$695$	27.0401	1.02569
$696$	0	0
$697$	−23.6711	−0.896606
$698$	0	0
$699$	−7.26280	−0.274704
$700$	0	0
$701$	−29.2895	−1.10625	−0.553125	−	0.833098i	$-0.686565\pi$
$701$	−29.2895	−1.10625	−0.553125	+	0.833098i	$0.686565\pi$
$702$	0	0
$703$	10.4739	0.395030
$704$	0	0
$705$	−11.7654	−0.443109
$706$	0	0
$707$	−7.60311	−0.285944
$708$	0	0
$709$	−16.0828	−0.604002	−0.302001	−	0.953308i	$-0.597655\pi$
$709$	−16.0828	−0.604002	−0.302001	+	0.953308i	$0.597655\pi$
$710$	0	0
$711$	−15.3247	−0.574720
$712$	0	0
$713$	0.846981	0.0317197
$714$	0	0
$715$	1.69372	0.0633416
$716$	0	0
$717$	−1.65583	−0.0618381
$718$	0	0
$719$	13.7032	0.511045	0.255522	−	0.966803i	$-0.417753\pi$
$719$	13.7032	0.511045	0.255522	+	0.966803i	$0.417753\pi$
$720$	0	0
$721$	18.5152	0.689540
$722$	0	0
$723$	−9.79248	−0.364186
$724$	0	0
$725$	−36.7815	−1.36603
$726$	0	0
$727$	−24.4739	−0.907688	−0.453844	−	0.891081i	$-0.649948\pi$
$727$	−24.4739	−0.907688	−0.453844	+	0.891081i	$0.649948\pi$
$728$	0	0
$729$	−20.0496	−0.742578
$730$	0	0
$731$	12.7004	0.469740
$732$	0	0
$733$	−6.90095	−0.254893	−0.127446	−	0.991845i	$-0.540678\pi$
$733$	−6.90095	−0.254893	−0.127446	+	0.991845i	$0.540678\pi$
$734$	0	0
$735$	1.09425	0.0403620
$736$	0	0
$737$	1.55660	0.0573381
$738$	0	0
$739$	−35.9002	−1.32061	−0.660306	−	0.750997i	$-0.729573\pi$
$739$	−35.9002	−1.32061	−0.660306	+	0.750997i	$0.729573\pi$
$740$	0	0
$741$	−1.20352	−0.0442123
$742$	0	0
$743$	51.0087	1.87133	0.935663	−	0.352894i	$-0.114802\pi$
$743$	51.0087	1.87133	0.935663	+	0.352894i	$0.114802\pi$
$744$	0	0
$745$	−36.3694	−1.33247
$746$	0	0
$747$	−6.44178	−0.235692
$748$	0	0
$749$	−14.3739	−0.525209
$750$	0	0
$751$	−14.4964	−0.528981	−0.264491	−	0.964388i	$-0.585204\pi$
$751$	−14.4964	−0.528981	−0.264491	+	0.964388i	$0.585204\pi$
$752$	0	0
$753$	−1.89102	−0.0689128
$754$	0	0
$755$	9.91401	0.360808
$756$	0	0
$757$	−14.2738	−0.518790	−0.259395	−	0.965771i	$-0.583523\pi$
$757$	−14.2738	−0.518790	−0.259395	+	0.965771i	$0.583523\pi$
$758$	0	0
$759$	0.321921	0.0116850
$760$	0	0
$761$	4.95752	0.179710	0.0898550	−	0.995955i	$-0.471360\pi$
$761$	4.95752	0.179710	0.0898550	+	0.995955i	$0.471360\pi$
$762$	0	0
$763$	−14.4295	−0.522383
$764$	0	0
$765$	−18.5788	−0.671717
$766$	0	0
$767$	27.5687	0.995449
$768$	0	0
$769$	−26.4011	−0.952050	−0.476025	−	0.879432i	$-0.657923\pi$
$769$	−26.4011	−0.952050	−0.476025	+	0.879432i	$0.657923\pi$
$770$	0	0
$771$	7.16896	0.258184
$772$	0	0
$773$	5.05334	0.181756	0.0908780	−	0.995862i	$-0.471033\pi$
$773$	5.05334	0.181756	0.0908780	+	0.995862i	$0.471033\pi$
$774$	0	0
$775$	−0.654640	−0.0235154
$776$	0	0
$777$	3.82873	0.137355
$778$	0	0
$779$	−10.9321	−0.391682
$780$	0	0
$781$	−2.09796	−0.0750710
$782$	0	0
$783$	19.9151	0.711708
$784$	0	0
$785$	−11.9790	−0.427550
$786$	0	0
$787$	15.4293	0.549995	0.274998	−	0.961445i	$-0.411323\pi$
$787$	15.4293	0.549995	0.274998	+	0.961445i	$0.411323\pi$
$788$	0	0
$789$	−4.74776	−0.169025
$790$	0	0
$791$	−18.7646	−0.667193
$792$	0	0
$793$	6.11240	0.217058
$794$	0	0
$795$	0.674367	0.0239173
$796$	0	0
$797$	12.6469	0.447977	0.223989	−	0.974592i	$-0.428092\pi$
$797$	12.6469	0.447977	0.223989	+	0.974592i	$0.428092\pi$
$798$	0	0
$799$	−23.2812	−0.823628
$800$	0	0
$801$	16.7230	0.590877
$802$	0	0
$803$	1.10536	0.0390073
$804$	0	0
$805$	−15.3392	−0.540637
$806$	0	0
$807$	−3.36849	−0.118576
$808$	0	0
$809$	33.2585	1.16931	0.584654	−	0.811283i	$-0.301230\pi$
$809$	33.2585	1.16931	0.584654	+	0.811283i	$0.301230\pi$
$810$	0	0
$811$	25.4719	0.894440	0.447220	−	0.894424i	$-0.352414\pi$
$811$	25.4719	0.894440	0.447220	+	0.894424i	$0.352414\pi$
$812$	0	0
$813$	9.53406	0.334374
$814$	0	0
$815$	−64.0460	−2.24343
$816$	0	0
$817$	5.86544	0.205206
$818$	0	0
$819$	9.43709	0.329758
$820$	0	0
$821$	−18.2688	−0.637585	−0.318792	−	0.947825i	$-0.603277\pi$
$821$	−18.2688	−0.637585	−0.318792	+	0.947825i	$0.603277\pi$
$822$	0	0
$823$	−36.0233	−1.25569	−0.627846	−	0.778337i	$-0.716063\pi$
$823$	−36.0233	−1.25569	−0.627846	+	0.778337i	$0.716063\pi$
$824$	0	0
$825$	−0.248816	−0.00866267
$826$	0	0
$827$	20.8635	0.725496	0.362748	−	0.931887i	$-0.381839\pi$
$827$	20.8635	0.725496	0.362748	+	0.931887i	$0.381839\pi$
$828$	0	0
$829$	47.3818	1.64564	0.822819	−	0.568304i	$-0.192400\pi$
$829$	47.3818	1.64564	0.822819	+	0.568304i	$0.192400\pi$
$830$	0	0
$831$	−6.22011	−0.215773
$832$	0	0
$833$	2.16529	0.0750228
$834$	0	0
$835$	21.3807	0.739910
$836$	0	0
$837$	0.354451	0.0122516
$838$	0	0
$839$	−37.4381	−1.29251	−0.646254	−	0.763123i	$-0.723665\pi$
$839$	−37.4381	−1.29251	−0.646254	+	0.763123i	$0.723665\pi$
$840$	0	0
$841$	57.2446	1.97395
$842$	0	0
$843$	5.37267	0.185045
$844$	0	0
$845$	−6.46722	−0.222479
$846$	0	0
$847$	−10.9705	−0.376950
$848$	0	0
$849$	1.11497	0.0382658
$850$	0	0
$851$	−53.6713	−1.83983
$852$	0	0
$853$	−18.0157	−0.616847	−0.308423	−	0.951249i	$-0.599801\pi$
$853$	−18.0157	−0.616847	−0.308423	+	0.951249i	$0.599801\pi$
$854$	0	0
$855$	−8.58029	−0.293440
$856$	0	0
$857$	−18.3046	−0.625274	−0.312637	−	0.949873i	$-0.601212\pi$
$857$	−18.3046	−0.625274	−0.312637	+	0.949873i	$0.601212\pi$
$858$	0	0
$859$	6.52496	0.222629	0.111314	−	0.993785i	$-0.464494\pi$
$859$	6.52496	0.222629	0.111314	+	0.993785i	$0.464494\pi$
$860$	0	0
$861$	−3.99622	−0.136191
$862$	0	0
$863$	9.55638	0.325303	0.162651	−	0.986684i	$-0.447995\pi$
$863$	9.55638	0.325303	0.162651	+	0.986684i	$0.447995\pi$
$864$	0	0
$865$	−2.49979	−0.0849952
$866$	0	0
$867$	−4.50048	−0.152844
$868$	0	0
$869$	−0.918812	−0.0311686
$870$	0	0
$871$	29.8205	1.01043
$872$	0	0
$873$	−42.4191	−1.43567
$874$	0	0
$875$	−3.11130	−0.105181
$876$	0	0
$877$	3.26070	0.110106	0.0550529	−	0.998483i	$-0.482467\pi$
$877$	3.26070	0.110106	0.0550529	+	0.998483i	$0.482467\pi$
$878$	0	0
$879$	−4.29030	−0.144708
$880$	0	0
$881$	−25.0102	−0.842616	−0.421308	−	0.906918i	$-0.638429\pi$
$881$	−25.0102	−0.842616	−0.421308	+	0.906918i	$0.638429\pi$
$882$	0	0
$883$	−9.20216	−0.309677	−0.154839	−	0.987940i	$-0.549486\pi$
$883$	−9.20216	−0.309677	−0.154839	+	0.987940i	$0.549486\pi$
$884$	0	0
$885$	−9.16279	−0.308004
$886$	0	0
$887$	−28.8673	−0.969271	−0.484635	−	0.874716i	$-0.661048\pi$
$887$	−28.8673	−0.969271	−0.484635	+	0.874716i	$0.661048\pi$
$888$	0	0
$889$	11.0463	0.370480
$890$	0	0
$891$	−1.34310	−0.0449956
$892$	0	0
$893$	−10.7520	−0.359802
$894$	0	0
$895$	78.2604	2.61596
$896$	0	0
$897$	6.16718	0.205916
$898$	0	0
$899$	1.53499	0.0511947
$900$	0	0
$901$	1.33443	0.0444563
$902$	0	0
$903$	2.14411	0.0713516
$904$	0	0
$905$	−45.2367	−1.50372
$906$	0	0
$907$	24.5442	0.814978	0.407489	−	0.913210i	$-0.366404\pi$
$907$	24.5442	0.814978	0.407489	+	0.913210i	$0.366404\pi$
$908$	0	0
$909$	21.7933	0.722839
$910$	0	0
$911$	−48.3590	−1.60221	−0.801103	−	0.598526i	$-0.795753\pi$
$911$	−48.3590	−1.60221	−0.801103	+	0.598526i	$0.795753\pi$
$912$	0	0
$913$	−0.386226	−0.0127822
$914$	0	0
$915$	−2.03153	−0.0671602
$916$	0	0
$917$	−16.9363	−0.559286
$918$	0	0
$919$	48.2207	1.59065	0.795327	−	0.606181i	$-0.207299\pi$
$919$	48.2207	1.59065	0.795327	+	0.606181i	$0.207299\pi$
$920$	0	0
$921$	−7.49678	−0.247027
$922$	0	0
$923$	−40.1915	−1.32292
$924$	0	0
$925$	41.4831	1.36396
$926$	0	0
$927$	−53.0713	−1.74309
$928$	0	0
$929$	50.7256	1.66425	0.832126	−	0.554586i	$-0.187123\pi$
$929$	50.7256	1.66425	0.832126	+	0.554586i	$0.187123\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−9.32723	−0.305360
$934$	0	0
$935$	−1.11392	−0.0364290
$936$	0	0
$937$	31.7409	1.03693	0.518466	−	0.855098i	$-0.326503\pi$
$937$	31.7409	1.03693	0.518466	+	0.855098i	$0.326503\pi$
$938$	0	0
$939$	2.74156	0.0894675
$940$	0	0
$941$	−2.94618	−0.0960427	−0.0480213	−	0.998846i	$-0.515292\pi$
$941$	−2.94618	−0.0960427	−0.0480213	+	0.998846i	$0.515292\pi$
$942$	0	0
$943$	56.0192	1.82424
$944$	0	0
$945$	−6.41927	−0.208819
$946$	0	0
$947$	29.7624	0.967149	0.483575	−	0.875303i	$-0.339338\pi$
$947$	29.7624	0.967149	0.483575	+	0.875303i	$0.339338\pi$
$948$	0	0
$949$	21.1758	0.687396
$950$	0	0
$951$	11.3773	0.368934
$952$	0	0
$953$	12.9257	0.418705	0.209353	−	0.977840i	$-0.432864\pi$
$953$	12.9257	0.418705	0.209353	+	0.977840i	$0.432864\pi$
$954$	0	0
$955$	−33.5888	−1.08691
$956$	0	0
$957$	0.583420	0.0188593
$958$	0	0
$959$	−4.61526	−0.149035
$960$	0	0
$961$	−30.9727	−0.999119
$962$	0	0
$963$	41.2008	1.32768
$964$	0	0
$965$	28.7427	0.925261
$966$	0	0
$967$	32.0651	1.03115	0.515573	−	0.856846i	$-0.327579\pi$
$967$	32.0651	1.03115	0.515573	+	0.856846i	$0.327579\pi$
$968$	0	0
$969$	0.791521	0.0254273
$970$	0	0
$971$	13.0870	0.419983	0.209991	−	0.977703i	$-0.432656\pi$
$971$	13.0870	0.419983	0.209991	+	0.977703i	$0.432656\pi$
$972$	0	0
$973$	9.03314	0.289589
$974$	0	0
$975$	−4.76668	−0.152656
$976$	0	0
$977$	−12.8826	−0.412152	−0.206076	−	0.978536i	$-0.566069\pi$
$977$	−12.8826	−0.412152	−0.206076	+	0.978536i	$0.566069\pi$
$978$	0	0
$979$	1.00265	0.0320448
$980$	0	0
$981$	41.3603	1.32053
$982$	0	0
$983$	48.8110	1.55683	0.778415	−	0.627750i	$-0.216024\pi$
$983$	48.8110	1.55683	0.778415	+	0.627750i	$0.216024\pi$
$984$	0	0
$985$	22.1433	0.705543
$986$	0	0
$987$	−3.93040	−0.125106
$988$	0	0
$989$	−30.0563	−0.955733
$990$	0	0
$991$	17.1267	0.544047	0.272024	−	0.962291i	$-0.412307\pi$
$991$	17.1267	0.544047	0.272024	+	0.962291i	$0.412307\pi$
$992$	0	0
$993$	−2.41063	−0.0764991
$994$	0	0
$995$	67.0814	2.12662
$996$	0	0
$997$	58.3512	1.84800	0.924000	−	0.382393i	$-0.124900\pi$
$997$	58.3512	1.84800	0.924000	+	0.382393i	$0.124900\pi$
$998$	0	0
$999$	−22.4608	−0.710627

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.ca.1.4		6
4.3	odd	2		8512.2.a.cf.1.3		6
8.3	odd	2		4256.2.a.i.1.4	✓	6
8.5	even	2		4256.2.a.n.1.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4256.2.a.i.1.4	✓	6	8.3	odd	2
4256.2.a.n.1.3	yes	6	8.5	even	2
8512.2.a.ca.1.4		6	1.1	even	1	trivial
8512.2.a.cf.1.3		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.365550 q^{3} +2.99343 q^{5} +1.00000 q^{7} -2.86637 q^{9} +O(q^{10})\)	\(q+0.365550 q^{3} +2.99343 q^{5} +1.00000 q^{7} -2.86637 q^{9} -0.171857 q^{11} -3.29234 q^{13} +1.09425 q^{15} +2.16529 q^{17} +1.00000 q^{19} +0.365550 q^{21} -5.12430 q^{23} +3.96062 q^{25} -2.14445 q^{27} -9.28679 q^{29} -0.165287 q^{31} -0.0628225 q^{33} +2.99343 q^{35} +10.4739 q^{37} -1.20352 q^{39} -10.9321 q^{41} +5.86544 q^{43} -8.58029 q^{45} -10.7520 q^{47} +1.00000 q^{49} +0.791521 q^{51} +0.616283 q^{53} -0.514443 q^{55} +0.365550 q^{57} -8.37359 q^{59} -1.85655 q^{61} -2.86637 q^{63} -9.85540 q^{65} -9.05752 q^{67} -1.87319 q^{69} +12.2076 q^{71} -6.43184 q^{73} +1.44781 q^{75} -0.171857 q^{77} +5.34636 q^{79} +7.81521 q^{81} +2.24736 q^{83} +6.48164 q^{85} -3.39479 q^{87} -5.83419 q^{89} -3.29234 q^{91} -0.0604208 q^{93} +2.99343 q^{95} +14.7989 q^{97} +0.492607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) \) 6 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9	\( 6 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9} - 7 q^{11} - 4 q^{13} + 4 q^{15} - 2 q^{17} + 6 q^{19} - 3 q^{21} - 2 q^{23} + q^{25} - 18 q^{27} - 5 q^{29} + 14 q^{31} + 3 q^{33} - 3 q^{35} + q^{37} + 22 q^{39} - 5 q^{41} - 15 q^{43} - 22 q^{45} - 7 q^{47} + 6 q^{49} + 4 q^{51} - 15 q^{53} + 20 q^{55} - 3 q^{57} - 23 q^{59} - 9 q^{61} + 3 q^{63} - 20 q^{65} - 2 q^{67} + 2 q^{69} + 23 q^{71} + 2 q^{73} - 4 q^{75} - 7 q^{77} + 17 q^{79} + 6 q^{81} - 22 q^{83} + 14 q^{85} - 10 q^{87} + 3 q^{89} - 4 q^{91} - 10 q^{93} - 3 q^{95} - 21 q^{97} - 3 q^{99}+O(q^{100}) \) 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

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