LMFDB - Embedded newform 8512.2.a.ca.1.2

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			1.2
Root			$1.96359$ 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)

$f(q)$	$=$	$q-1.96359 q^{3} -3.24194 q^{5} +1.00000 q^{7} +0.855673 q^{9} +O(q^{10})$	$q-1.96359 q^{3} -3.24194 q^{5} +1.00000 q^{7} +0.855673 q^{9} -5.08354 q^{11} +0.544656 q^{13} +6.36582 q^{15} +0.841608 q^{17} +1.00000 q^{19} -1.96359 q^{21} -1.69291 q^{23} +5.51015 q^{25} +4.21057 q^{27} -10.0567 q^{29} +1.15839 q^{31} +9.98198 q^{33} -3.24194 q^{35} +4.21066 q^{37} -1.06948 q^{39} -3.95405 q^{41} +3.61492 q^{43} -2.77404 q^{45} +4.58725 q^{47} +1.00000 q^{49} -1.65257 q^{51} +5.97054 q^{53} +16.4805 q^{55} -1.96359 q^{57} -1.49236 q^{59} +8.97228 q^{61} +0.855673 q^{63} -1.76574 q^{65} +4.09105 q^{67} +3.32418 q^{69} +9.68015 q^{71} +6.33832 q^{73} -10.8197 q^{75} -5.08354 q^{77} -4.79994 q^{79} -10.8348 q^{81} -17.0163 q^{83} -2.72844 q^{85} +19.7472 q^{87} +3.73106 q^{89} +0.544656 q^{91} -2.27460 q^{93} -3.24194 q^{95} -17.5309 q^{97} -4.34985 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.96359	−1.13368	−0.566839	−	0.823829i	$-0.691834\pi$
$3$	−1.96359	−1.13368	−0.566839	+	0.823829i	$0.691834\pi$
$4$	0	0
$5$	−3.24194	−1.44984	−0.724919	−	0.688834i	$-0.758123\pi$
$5$	−3.24194	−1.44984	−0.724919	+	0.688834i	$0.758123\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0.855673	0.285224
$10$	0	0
$11$	−5.08354	−1.53275	−0.766373	−	0.642396i	$-0.777941\pi$
$11$	−5.08354	−1.53275	−0.766373	+	0.642396i	$0.777941\pi$
$12$	0	0
$13$	0.544656	0.151060	0.0755302	−	0.997144i	$-0.475935\pi$
$13$	0.544656	0.151060	0.0755302	+	0.997144i	$0.475935\pi$
$14$	0	0
$15$	6.36582	1.64365
$16$	0	0
$17$	0.841608	0.204120	0.102060	−	0.994778i	$-0.467457\pi$
$17$	0.841608	0.204120	0.102060	+	0.994778i	$0.467457\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.96359	−0.428490
$22$	0	0
$23$	−1.69291	−0.352997	−0.176498	−	0.984301i	$-0.556477\pi$
$23$	−1.69291	−0.352997	−0.176498	+	0.984301i	$0.556477\pi$
$24$	0	0
$25$	5.51015	1.10203
$26$	0	0
$27$	4.21057	0.810325
$28$	0	0
$29$	−10.0567	−1.86748	−0.933742	−	0.357946i	$-0.883477\pi$
$29$	−10.0567	−1.86748	−0.933742	+	0.357946i	$0.883477\pi$
$30$	0	0
$31$	1.15839	0.208053	0.104027	−	0.994575i	$-0.466827\pi$
$31$	1.15839	0.208053	0.104027	+	0.994575i	$0.466827\pi$
$32$	0	0
$33$	9.98198	1.73764
$34$	0	0
$35$	−3.24194	−0.547987
$36$	0	0
$37$	4.21066	0.692229	0.346114	−	0.938192i	$-0.387501\pi$
$37$	4.21066	0.692229	0.346114	+	0.938192i	$0.387501\pi$
$38$	0	0
$39$	−1.06948	−0.171254
$40$	0	0
$41$	−3.95405	−0.617520	−0.308760	−	0.951140i	$-0.599914\pi$
$41$	−3.95405	−0.617520	−0.308760	+	0.951140i	$0.599914\pi$
$42$	0	0
$43$	3.61492	0.551270	0.275635	−	0.961262i	$-0.411112\pi$
$43$	3.61492	0.551270	0.275635	+	0.961262i	$0.411112\pi$
$44$	0	0
$45$	−2.77404	−0.413529
$46$	0	0
$47$	4.58725	0.669120	0.334560	−	0.942375i	$-0.391412\pi$
$47$	4.58725	0.669120	0.334560	+	0.942375i	$0.391412\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.65257	−0.231406
$52$	0	0
$53$	5.97054	0.820117	0.410058	−	0.912059i	$-0.365508\pi$
$53$	5.97054	0.820117	0.410058	+	0.912059i	$0.365508\pi$
$54$	0	0
$55$	16.4805	2.22223
$56$	0	0
$57$	−1.96359	−0.260083
$58$	0	0
$59$	−1.49236	−0.194289	−0.0971444	−	0.995270i	$-0.530971\pi$
$59$	−1.49236	−0.194289	−0.0971444	+	0.995270i	$0.530971\pi$
$60$	0	0
$61$	8.97228	1.14878	0.574391	−	0.818581i	$-0.305239\pi$
$61$	8.97228	1.14878	0.574391	+	0.818581i	$0.305239\pi$
$62$	0	0
$63$	0.855673	0.107805
$64$	0	0
$65$	−1.76574	−0.219013
$66$	0	0
$67$	4.09105	0.499802	0.249901	−	0.968271i	$-0.419602\pi$
$67$	4.09105	0.499802	0.249901	+	0.968271i	$0.419602\pi$
$68$	0	0
$69$	3.32418	0.400184
$70$	0	0
$71$	9.68015	1.14882	0.574411	−	0.818567i	$-0.305231\pi$
$71$	9.68015	1.14882	0.574411	+	0.818567i	$0.305231\pi$
$72$	0	0
$73$	6.33832	0.741844	0.370922	−	0.928664i	$-0.379042\pi$
$73$	6.33832	0.741844	0.370922	+	0.928664i	$0.379042\pi$
$74$	0	0
$75$	−10.8197	−1.24935
$76$	0	0
$77$	−5.08354	−0.579324
$78$	0	0
$79$	−4.79994	−0.540035	−0.270018	−	0.962855i	$-0.587030\pi$
$79$	−4.79994	−0.540035	−0.270018	+	0.962855i	$0.587030\pi$
$80$	0	0
$81$	−10.8348	−1.20387
$82$	0	0
$83$	−17.0163	−1.86778	−0.933890	−	0.357561i	$-0.883608\pi$
$83$	−17.0163	−1.86778	−0.933890	+	0.357561i	$0.883608\pi$
$84$	0	0
$85$	−2.72844	−0.295941
$86$	0	0
$87$	19.7472	2.11713
$88$	0	0
$89$	3.73106	0.395492	0.197746	−	0.980253i	$-0.436638\pi$
$89$	3.73106	0.395492	0.197746	+	0.980253i	$0.436638\pi$
$90$	0	0
$91$	0.544656	0.0570954
$92$	0	0
$93$	−2.27460	−0.235865
$94$	0	0
$95$	−3.24194	−0.332616
$96$	0	0
$97$	−17.5309	−1.77999	−0.889997	−	0.455966i	$-0.849294\pi$
$97$	−17.5309	−1.77999	−0.889997	+	0.455966i	$0.849294\pi$
$98$	0	0
$99$	−4.34985	−0.437176
$100$	0	0
$101$	−6.92968	−0.689529	−0.344765	−	0.938689i	$-0.612041\pi$
$101$	−6.92968	−0.689529	−0.344765	+	0.938689i	$0.612041\pi$
$102$	0	0
$103$	0.205898	0.0202877	0.0101439	−	0.999949i	$-0.496771\pi$
$103$	0.205898	0.0202877	0.0101439	+	0.999949i	$0.496771\pi$
$104$	0	0
$105$	6.36582	0.621241
$106$	0	0
$107$	19.3125	1.86701	0.933506	−	0.358562i	$-0.116733\pi$
$107$	19.3125	1.86701	0.933506	+	0.358562i	$0.116733\pi$
$108$	0	0
$109$	11.6192	1.11292	0.556458	−	0.830875i	$-0.312160\pi$
$109$	11.6192	1.11292	0.556458	+	0.830875i	$0.312160\pi$
$110$	0	0
$111$	−8.26800	−0.784764
$112$	0	0
$113$	−5.00716	−0.471034	−0.235517	−	0.971870i	$-0.575678\pi$
$113$	−5.00716	−0.471034	−0.235517	+	0.971870i	$0.575678\pi$
$114$	0	0
$115$	5.48831	0.511788
$116$	0	0
$117$	0.466047	0.0430861
$118$	0	0
$119$	0.841608	0.0771500
$120$	0	0
$121$	14.8424	1.34931
$122$	0	0
$123$	7.76413	0.700068
$124$	0	0
$125$	−1.65387	−0.147927
$126$	0	0
$127$	17.9736	1.59490	0.797449	−	0.603386i	$-0.206182\pi$
$127$	17.9736	1.59490	0.797449	+	0.603386i	$0.206182\pi$
$128$	0	0
$129$	−7.09821	−0.624963
$130$	0	0
$131$	−0.0888907	−0.00776641	−0.00388321	−	0.999992i	$-0.501236\pi$
$131$	−0.0888907	−0.00776641	−0.00388321	+	0.999992i	$0.501236\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−13.6504	−1.17484
$136$	0	0
$137$	0.872766	0.0745654	0.0372827	−	0.999305i	$-0.488130\pi$
$137$	0.872766	0.0745654	0.0372827	+	0.999305i	$0.488130\pi$
$138$	0	0
$139$	5.85137	0.496307	0.248153	−	0.968721i	$-0.420176\pi$
$139$	5.85137	0.496307	0.248153	+	0.968721i	$0.420176\pi$
$140$	0	0
$141$	−9.00747	−0.758566
$142$	0	0
$143$	−2.76878	−0.231537
$144$	0	0
$145$	32.6032	2.70755
$146$	0	0
$147$	−1.96359	−0.161954
$148$	0	0
$149$	−0.324551	−0.0265883	−0.0132941	−	0.999912i	$-0.504232\pi$
$149$	−0.324551	−0.0265883	−0.0132941	+	0.999912i	$0.504232\pi$
$150$	0	0
$151$	14.0261	1.14143	0.570713	−	0.821149i	$-0.306667\pi$
$151$	14.0261	1.14143	0.570713	+	0.821149i	$0.306667\pi$
$152$	0	0
$153$	0.720141	0.0582199
$154$	0	0
$155$	−3.75543	−0.301644
$156$	0	0
$157$	12.3655	0.986874	0.493437	−	0.869781i	$-0.335740\pi$
$157$	12.3655	0.986874	0.493437	+	0.869781i	$0.335740\pi$
$158$	0	0
$159$	−11.7237	−0.929748
$160$	0	0
$161$	−1.69291	−0.133420
$162$	0	0
$163$	22.1552	1.73533	0.867666	−	0.497147i	$-0.165619\pi$
$163$	22.1552	1.73533	0.867666	+	0.497147i	$0.165619\pi$
$164$	0	0
$165$	−32.3609	−2.51930
$166$	0	0
$167$	12.9999	1.00596	0.502980	−	0.864298i	$-0.332237\pi$
$167$	12.9999	1.00596	0.502980	+	0.864298i	$0.332237\pi$
$168$	0	0
$169$	−12.7034	−0.977181
$170$	0	0
$171$	0.855673	0.0654349
$172$	0	0
$173$	3.75864	0.285764	0.142882	−	0.989740i	$-0.454363\pi$
$173$	3.75864	0.285764	0.142882	+	0.989740i	$0.454363\pi$
$174$	0	0
$175$	5.51015	0.416528
$176$	0	0
$177$	2.93038	0.220261
$178$	0	0
$179$	−7.14981	−0.534402	−0.267201	−	0.963641i	$-0.586099\pi$
$179$	−7.14981	−0.534402	−0.267201	+	0.963641i	$0.586099\pi$
$180$	0	0
$181$	−12.3403	−0.917247	−0.458623	−	0.888631i	$-0.651657\pi$
$181$	−12.3403	−0.917247	−0.458623	+	0.888631i	$0.651657\pi$
$182$	0	0
$183$	−17.6179	−1.30235
$184$	0	0
$185$	−13.6507	−1.00362
$186$	0	0
$187$	−4.27835	−0.312864
$188$	0	0
$189$	4.21057	0.306274
$190$	0	0
$191$	25.2736	1.82873	0.914366	−	0.404889i	$-0.132690\pi$
$191$	25.2736	1.82873	0.914366	+	0.404889i	$0.132690\pi$
$192$	0	0
$193$	−16.3585	−1.17751	−0.588754	−	0.808313i	$-0.700381\pi$
$193$	−16.3585	−1.17751	−0.588754	+	0.808313i	$0.700381\pi$
$194$	0	0
$195$	3.46718	0.248290
$196$	0	0
$197$	−5.40014	−0.384744	−0.192372	−	0.981322i	$-0.561618\pi$
$197$	−5.40014	−0.384744	−0.192372	+	0.981322i	$0.561618\pi$
$198$	0	0
$199$	−12.2729	−0.870003	−0.435002	−	0.900430i	$-0.643252\pi$
$199$	−12.2729	−0.870003	−0.435002	+	0.900430i	$0.643252\pi$
$200$	0	0
$201$	−8.03314	−0.566614
$202$	0	0
$203$	−10.0567	−0.705843
$204$	0	0
$205$	12.8188	0.895303
$206$	0	0
$207$	−1.44858	−0.100683
$208$	0	0
$209$	−5.08354	−0.351636
$210$	0	0
$211$	−15.4206	−1.06159	−0.530797	−	0.847499i	$-0.678107\pi$
$211$	−15.4206	−1.06159	−0.530797	+	0.847499i	$0.678107\pi$
$212$	0	0
$213$	−19.0078	−1.30239
$214$	0	0
$215$	−11.7193	−0.799253
$216$	0	0
$217$	1.15839	0.0786368
$218$	0	0
$219$	−12.4458	−0.841012
$220$	0	0
$221$	0.458386	0.0308344
$222$	0	0
$223$	−0.772357	−0.0517209	−0.0258604	−	0.999666i	$-0.508233\pi$
$223$	−0.772357	−0.0517209	−0.0258604	+	0.999666i	$0.508233\pi$
$224$	0	0
$225$	4.71489	0.314326
$226$	0	0
$227$	4.78413	0.317534	0.158767	−	0.987316i	$-0.449248\pi$
$227$	4.78413	0.317534	0.158767	+	0.987316i	$0.449248\pi$
$228$	0	0
$229$	−11.6465	−0.769622	−0.384811	−	0.922995i	$-0.625733\pi$
$229$	−11.6465	−0.769622	−0.384811	+	0.922995i	$0.625733\pi$
$230$	0	0
$231$	9.98198	0.656766
$232$	0	0
$233$	8.79512	0.576188	0.288094	−	0.957602i	$-0.406979\pi$
$233$	8.79512	0.576188	0.288094	+	0.957602i	$0.406979\pi$
$234$	0	0
$235$	−14.8716	−0.970115
$236$	0	0
$237$	9.42510	0.612226
$238$	0	0
$239$	−3.61604	−0.233902	−0.116951	−	0.993138i	$-0.537312\pi$
$239$	−3.61604	−0.233902	−0.116951	+	0.993138i	$0.537312\pi$
$240$	0	0
$241$	−24.8510	−1.60080	−0.800398	−	0.599468i	$-0.795379\pi$
$241$	−24.8510	−1.60080	−0.800398	+	0.599468i	$0.795379\pi$
$242$	0	0
$243$	8.64344	0.554477
$244$	0	0
$245$	−3.24194	−0.207120
$246$	0	0
$247$	0.544656	0.0346556
$248$	0	0
$249$	33.4129	2.11746
$250$	0	0
$251$	−19.6594	−1.24089	−0.620446	−	0.784249i	$-0.713048\pi$
$251$	−19.6594	−1.24089	−0.620446	+	0.784249i	$0.713048\pi$
$252$	0	0
$253$	8.60599	0.541054
$254$	0	0
$255$	5.35753	0.335501
$256$	0	0
$257$	−30.1556	−1.88106	−0.940529	−	0.339714i	$-0.889670\pi$
$257$	−30.1556	−1.88106	−0.940529	+	0.339714i	$0.889670\pi$
$258$	0	0
$259$	4.21066	0.261638
$260$	0	0
$261$	−8.60526	−0.532652
$262$	0	0
$263$	10.7958	0.665700	0.332850	−	0.942980i	$-0.391990\pi$
$263$	10.7958	0.665700	0.332850	+	0.942980i	$0.391990\pi$
$264$	0	0
$265$	−19.3561	−1.18904
$266$	0	0
$267$	−7.32627	−0.448360
$268$	0	0
$269$	−19.1619	−1.16832	−0.584162	−	0.811637i	$-0.698577\pi$
$269$	−19.1619	−1.16832	−0.584162	+	0.811637i	$0.698577\pi$
$270$	0	0
$271$	−7.92293	−0.481284	−0.240642	−	0.970614i	$-0.577358\pi$
$271$	−7.92293	−0.481284	−0.240642	+	0.970614i	$0.577358\pi$
$272$	0	0
$273$	−1.06948	−0.0647278
$274$	0	0
$275$	−28.0111	−1.68913
$276$	0	0
$277$	29.4449	1.76917	0.884587	−	0.466375i	$-0.154440\pi$
$277$	29.4449	1.76917	0.884587	+	0.466375i	$0.154440\pi$
$278$	0	0
$279$	0.991205	0.0593419
$280$	0	0
$281$	−11.6644	−0.695842	−0.347921	−	0.937524i	$-0.613112\pi$
$281$	−11.6644	−0.695842	−0.347921	+	0.937524i	$0.613112\pi$
$282$	0	0
$283$	6.02546	0.358176	0.179088	−	0.983833i	$-0.442685\pi$
$283$	6.02546	0.358176	0.179088	+	0.983833i	$0.442685\pi$
$284$	0	0
$285$	6.36582	0.377079
$286$	0	0
$287$	−3.95405	−0.233400
$288$	0	0
$289$	−16.2917	−0.958335
$290$	0	0
$291$	34.4235	2.01794
$292$	0	0
$293$	18.4164	1.07590	0.537949	−	0.842978i	$-0.319199\pi$
$293$	18.4164	1.07590	0.537949	+	0.842978i	$0.319199\pi$
$294$	0	0
$295$	4.83814	0.281687
$296$	0	0
$297$	−21.4046	−1.24202
$298$	0	0
$299$	−0.922054	−0.0533238
$300$	0	0
$301$	3.61492	0.208361
$302$	0	0
$303$	13.6070	0.781704
$304$	0	0
$305$	−29.0876	−1.66555
$306$	0	0
$307$	−1.88100	−0.107354	−0.0536770	−	0.998558i	$-0.517094\pi$
$307$	−1.88100	−0.107354	−0.0536770	+	0.998558i	$0.517094\pi$
$308$	0	0
$309$	−0.404299	−0.0229998
$310$	0	0
$311$	−7.49727	−0.425131	−0.212566	−	0.977147i	$-0.568182\pi$
$311$	−7.49727	−0.425131	−0.212566	+	0.977147i	$0.568182\pi$
$312$	0	0
$313$	25.6524	1.44996	0.724980	−	0.688770i	$-0.241849\pi$
$313$	25.6524	1.44996	0.724980	+	0.688770i	$0.241849\pi$
$314$	0	0
$315$	−2.77404	−0.156299
$316$	0	0
$317$	−15.7153	−0.882659	−0.441329	−	0.897345i	$-0.645493\pi$
$317$	−15.7153	−0.882659	−0.441329	+	0.897345i	$0.645493\pi$
$318$	0	0
$319$	51.1237	2.86238
$320$	0	0
$321$	−37.9218	−2.11659
$322$	0	0
$323$	0.841608	0.0468283
$324$	0	0
$325$	3.00113	0.166473
$326$	0	0
$327$	−22.8153	−1.26169
$328$	0	0
$329$	4.58725	0.252903
$330$	0	0
$331$	28.4419	1.56331	0.781654	−	0.623713i	$-0.214377\pi$
$331$	28.4419	1.56331	0.781654	+	0.623713i	$0.214377\pi$
$332$	0	0
$333$	3.60295	0.197440
$334$	0	0
$335$	−13.2629	−0.724632
$336$	0	0
$337$	−11.8798	−0.647136	−0.323568	−	0.946205i	$-0.604882\pi$
$337$	−11.8798	−0.647136	−0.323568	+	0.946205i	$0.604882\pi$
$338$	0	0
$339$	9.83200	0.534001
$340$	0	0
$341$	−5.88874	−0.318893
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−10.7768	−0.580202
$346$	0	0
$347$	16.2827	0.874103	0.437051	−	0.899437i	$-0.356023\pi$
$347$	16.2827	0.874103	0.437051	+	0.899437i	$0.356023\pi$
$348$	0	0
$349$	−10.7314	−0.574439	−0.287220	−	0.957865i	$-0.592731\pi$
$349$	−10.7314	−0.574439	−0.287220	+	0.957865i	$0.592731\pi$
$350$	0	0
$351$	2.29331	0.122408
$352$	0	0
$353$	1.40594	0.0748305	0.0374152	−	0.999300i	$-0.488088\pi$
$353$	1.40594	0.0748305	0.0374152	+	0.999300i	$0.488088\pi$
$354$	0	0
$355$	−31.3824	−1.66561
$356$	0	0
$357$	−1.65257	−0.0874633
$358$	0	0
$359$	17.2944	0.912765	0.456383	−	0.889784i	$-0.349145\pi$
$359$	17.2944	0.912765	0.456383	+	0.889784i	$0.349145\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−29.1444	−1.52968
$364$	0	0
$365$	−20.5484	−1.07555
$366$	0	0
$367$	−21.0573	−1.09918	−0.549590	−	0.835434i	$-0.685216\pi$
$367$	−21.0573	−1.09918	−0.549590	+	0.835434i	$0.685216\pi$
$368$	0	0
$369$	−3.38338	−0.176132
$370$	0	0
$371$	5.97054	0.309975
$372$	0	0
$373$	1.64406	0.0851261	0.0425631	−	0.999094i	$-0.486448\pi$
$373$	1.64406	0.0851261	0.0425631	+	0.999094i	$0.486448\pi$
$374$	0	0
$375$	3.24752	0.167701
$376$	0	0
$377$	−5.47745	−0.282103
$378$	0	0
$379$	−24.9209	−1.28010	−0.640050	−	0.768334i	$-0.721086\pi$
$379$	−24.9209	−1.28010	−0.640050	+	0.768334i	$0.721086\pi$
$380$	0	0
$381$	−35.2927	−1.80810
$382$	0	0
$383$	10.9869	0.561406	0.280703	−	0.959795i	$-0.409432\pi$
$383$	10.9869	0.561406	0.280703	+	0.959795i	$0.409432\pi$
$384$	0	0
$385$	16.4805	0.839925
$386$	0	0
$387$	3.09319	0.157236
$388$	0	0
$389$	−25.1430	−1.27480	−0.637399	−	0.770534i	$-0.719990\pi$
$389$	−25.1430	−1.27480	−0.637399	+	0.770534i	$0.719990\pi$
$390$	0	0
$391$	−1.42477	−0.0720536
$392$	0	0
$393$	0.174545	0.00880461
$394$	0	0
$395$	15.5611	0.782964
$396$	0	0
$397$	−26.0706	−1.30845	−0.654224	−	0.756301i	$-0.727005\pi$
$397$	−26.0706	−1.30845	−0.654224	+	0.756301i	$0.727005\pi$
$398$	0	0
$399$	−1.96359	−0.0983023
$400$	0	0
$401$	−27.1975	−1.35818	−0.679090	−	0.734055i	$-0.737625\pi$
$401$	−27.1975	−1.35818	−0.679090	+	0.734055i	$0.737625\pi$
$402$	0	0
$403$	0.630925	0.0314286
$404$	0	0
$405$	35.1259	1.74542
$406$	0	0
$407$	−21.4051	−1.06101
$408$	0	0
$409$	−27.1957	−1.34474	−0.672371	−	0.740214i	$-0.734724\pi$
$409$	−27.1957	−1.34474	−0.672371	+	0.740214i	$0.734724\pi$
$410$	0	0
$411$	−1.71375	−0.0845331
$412$	0	0
$413$	−1.49236	−0.0734343
$414$	0	0
$415$	55.1657	2.70798
$416$	0	0
$417$	−11.4897	−0.562652
$418$	0	0
$419$	−21.6250	−1.05645	−0.528226	−	0.849104i	$-0.677143\pi$
$419$	−21.6250	−1.05645	−0.528226	+	0.849104i	$0.677143\pi$
$420$	0	0
$421$	−21.0179	−1.02435	−0.512175	−	0.858881i	$-0.671160\pi$
$421$	−21.0179	−1.02435	−0.512175	+	0.858881i	$0.671160\pi$
$422$	0	0
$423$	3.92519	0.190849
$424$	0	0
$425$	4.63738	0.224946
$426$	0	0
$427$	8.97228	0.434199
$428$	0	0
$429$	5.43674	0.262488
$430$	0	0
$431$	38.6530	1.86185	0.930925	−	0.365211i	$-0.119003\pi$
$431$	38.6530	1.86185	0.930925	+	0.365211i	$0.119003\pi$
$432$	0	0
$433$	−11.7136	−0.562919	−0.281460	−	0.959573i	$-0.590819\pi$
$433$	−11.7136	−0.562919	−0.281460	+	0.959573i	$0.590819\pi$
$434$	0	0
$435$	−64.0193	−3.06949
$436$	0	0
$437$	−1.69291	−0.0809830
$438$	0	0
$439$	1.02417	0.0488809	0.0244404	−	0.999701i	$-0.492220\pi$
$439$	1.02417	0.0488809	0.0244404	+	0.999701i	$0.492220\pi$
$440$	0	0
$441$	0.855673	0.0407463
$442$	0	0
$443$	12.7707	0.606754	0.303377	−	0.952871i	$-0.401886\pi$
$443$	12.7707	0.606754	0.303377	+	0.952871i	$0.401886\pi$
$444$	0	0
$445$	−12.0959	−0.573399
$446$	0	0
$447$	0.637285	0.0301425
$448$	0	0
$449$	−4.65219	−0.219551	−0.109775	−	0.993956i	$-0.535013\pi$
$449$	−4.65219	−0.219551	−0.109775	+	0.993956i	$0.535013\pi$
$450$	0	0
$451$	20.1006	0.946501
$452$	0	0
$453$	−27.5414	−1.29401
$454$	0	0
$455$	−1.76574	−0.0827791
$456$	0	0
$457$	25.1653	1.17719	0.588593	−	0.808430i	$-0.299682\pi$
$457$	25.1653	1.17719	0.588593	+	0.808430i	$0.299682\pi$
$458$	0	0
$459$	3.54365	0.165403
$460$	0	0
$461$	17.2538	0.803591	0.401796	−	0.915729i	$-0.368386\pi$
$461$	17.2538	0.803591	0.401796	+	0.915729i	$0.368386\pi$
$462$	0	0
$463$	−6.46544	−0.300474	−0.150237	−	0.988650i	$-0.548004\pi$
$463$	−6.46544	−0.300474	−0.150237	+	0.988650i	$0.548004\pi$
$464$	0	0
$465$	7.37412	0.341967
$466$	0	0
$467$	−14.1613	−0.655306	−0.327653	−	0.944798i	$-0.606258\pi$
$467$	−14.1613	−0.655306	−0.327653	+	0.944798i	$0.606258\pi$
$468$	0	0
$469$	4.09105	0.188907
$470$	0	0
$471$	−24.2807	−1.11880
$472$	0	0
$473$	−18.3766	−0.844958
$474$	0	0
$475$	5.51015	0.252823
$476$	0	0
$477$	5.10883	0.233917
$478$	0	0
$479$	−32.4923	−1.48461	−0.742306	−	0.670061i	$-0.766268\pi$
$479$	−32.4923	−1.48461	−0.742306	+	0.670061i	$0.766268\pi$
$480$	0	0
$481$	2.29336	0.104568
$482$	0	0
$483$	3.32418	0.151255
$484$	0	0
$485$	56.8341	2.58070
$486$	0	0
$487$	−3.77671	−0.171139	−0.0855696	−	0.996332i	$-0.527271\pi$
$487$	−3.77671	−0.171139	−0.0855696	+	0.996332i	$0.527271\pi$
$488$	0	0
$489$	−43.5037	−1.96731
$490$	0	0
$491$	−6.29576	−0.284124	−0.142062	−	0.989858i	$-0.545373\pi$
$491$	−6.29576	−0.284124	−0.142062	+	0.989858i	$0.545373\pi$
$492$	0	0
$493$	−8.46381	−0.381191
$494$	0	0
$495$	14.1019	0.633835
$496$	0	0
$497$	9.68015	0.434214
$498$	0	0
$499$	−28.3391	−1.26863	−0.634317	−	0.773073i	$-0.718719\pi$
$499$	−28.3391	−1.26863	−0.634317	+	0.773073i	$0.718719\pi$
$500$	0	0
$501$	−25.5264	−1.14043
$502$	0	0
$503$	17.5176	0.781071	0.390535	−	0.920588i	$-0.372290\pi$
$503$	17.5176	0.781071	0.390535	+	0.920588i	$0.372290\pi$
$504$	0	0
$505$	22.4656	0.999706
$506$	0	0
$507$	24.9441	1.10781
$508$	0	0
$509$	−4.95134	−0.219464	−0.109732	−	0.993961i	$-0.534999\pi$
$509$	−4.95134	−0.219464	−0.109732	+	0.993961i	$0.534999\pi$
$510$	0	0
$511$	6.33832	0.280391
$512$	0	0
$513$	4.21057	0.185901
$514$	0	0
$515$	−0.667508	−0.0294139
$516$	0	0
$517$	−23.3195	−1.02559
$518$	0	0
$519$	−7.38042	−0.323964
$520$	0	0
$521$	41.1136	1.80122	0.900610	−	0.434628i	$-0.143120\pi$
$521$	41.1136	1.80122	0.900610	+	0.434628i	$0.143120\pi$
$522$	0	0
$523$	34.4424	1.50606	0.753031	−	0.657985i	$-0.228591\pi$
$523$	34.4424	1.50606	0.753031	+	0.657985i	$0.228591\pi$
$524$	0	0
$525$	−10.8197	−0.472209
$526$	0	0
$527$	0.974912	0.0424678
$528$	0	0
$529$	−20.1340	−0.875393
$530$	0	0
$531$	−1.27697	−0.0554159
$532$	0	0
$533$	−2.15360	−0.0932827
$534$	0	0
$535$	−62.6100	−2.70686
$536$	0	0
$537$	14.0393	0.605839
$538$	0	0
$539$	−5.08354	−0.218964
$540$	0	0
$541$	4.24048	0.182313	0.0911563	−	0.995837i	$-0.470944\pi$
$541$	4.24048	0.182313	0.0911563	+	0.995837i	$0.470944\pi$
$542$	0	0
$543$	24.2312	1.03986
$544$	0	0
$545$	−37.6687	−1.61355
$546$	0	0
$547$	−20.8869	−0.893059	−0.446529	−	0.894769i	$-0.647340\pi$
$547$	−20.8869	−0.893059	−0.446529	+	0.894769i	$0.647340\pi$
$548$	0	0
$549$	7.67734	0.327661
$550$	0	0
$551$	−10.0567	−0.428430
$552$	0	0
$553$	−4.79994	−0.204114
$554$	0	0
$555$	26.8043	1.13778
$556$	0	0
$557$	30.1484	1.27743	0.638713	−	0.769445i	$-0.279467\pi$
$557$	30.1484	1.27743	0.638713	+	0.769445i	$0.279467\pi$
$558$	0	0
$559$	1.96889	0.0832751
$560$	0	0
$561$	8.40091	0.354687
$562$	0	0
$563$	−8.11592	−0.342045	−0.171023	−	0.985267i	$-0.554707\pi$
$563$	−8.11592	−0.342045	−0.171023	+	0.985267i	$0.554707\pi$
$564$	0	0
$565$	16.2329	0.682923
$566$	0	0
$567$	−10.8348	−0.455021
$568$	0	0
$569$	−12.8588	−0.539067	−0.269534	−	0.962991i	$-0.586870\pi$
$569$	−12.8588	−0.539067	−0.269534	+	0.962991i	$0.586870\pi$
$570$	0	0
$571$	−15.6641	−0.655521	−0.327761	−	0.944761i	$-0.606294\pi$
$571$	−15.6641	−0.655521	−0.327761	+	0.944761i	$0.606294\pi$
$572$	0	0
$573$	−49.6269	−2.07319
$574$	0	0
$575$	−9.32820	−0.389013
$576$	0	0
$577$	35.4583	1.47615	0.738074	−	0.674719i	$-0.235735\pi$
$577$	35.4583	1.47615	0.738074	+	0.674719i	$0.235735\pi$
$578$	0	0
$579$	32.1212	1.33491
$580$	0	0
$581$	−17.0163	−0.705954
$582$	0	0
$583$	−30.3515	−1.25703
$584$	0	0
$585$	−1.51089	−0.0624678
$586$	0	0
$587$	31.2079	1.28809	0.644045	−	0.764988i	$-0.277255\pi$
$587$	31.2079	1.28809	0.644045	+	0.764988i	$0.277255\pi$
$588$	0	0
$589$	1.15839	0.0477307
$590$	0	0
$591$	10.6037	0.436176
$592$	0	0
$593$	4.12011	0.169193	0.0845963	−	0.996415i	$-0.473040\pi$
$593$	4.12011	0.169193	0.0845963	+	0.996415i	$0.473040\pi$
$594$	0	0
$595$	−2.72844	−0.111855
$596$	0	0
$597$	24.0989	0.986303
$598$	0	0
$599$	13.3065	0.543688	0.271844	−	0.962341i	$-0.412366\pi$
$599$	13.3065	0.543688	0.271844	+	0.962341i	$0.412366\pi$
$600$	0	0
$601$	−19.2511	−0.785270	−0.392635	−	0.919694i	$-0.628436\pi$
$601$	−19.2511	−0.785270	−0.392635	+	0.919694i	$0.628436\pi$
$602$	0	0
$603$	3.50060	0.142556
$604$	0	0
$605$	−48.1182	−1.95628
$606$	0	0
$607$	−0.918010	−0.0372609	−0.0186304	−	0.999826i	$-0.505931\pi$
$607$	−0.918010	−0.0372609	−0.0186304	+	0.999826i	$0.505931\pi$
$608$	0	0
$609$	19.7472	0.800198
$610$	0	0
$611$	2.49847	0.101077
$612$	0	0
$613$	−38.0053	−1.53502	−0.767509	−	0.641038i	$-0.778504\pi$
$613$	−38.0053	−1.53502	−0.767509	+	0.641038i	$0.778504\pi$
$614$	0	0
$615$	−25.1708	−1.01499
$616$	0	0
$617$	24.3519	0.980369	0.490184	−	0.871619i	$-0.336929\pi$
$617$	24.3519	0.980369	0.490184	+	0.871619i	$0.336929\pi$
$618$	0	0
$619$	−10.8816	−0.437370	−0.218685	−	0.975795i	$-0.570177\pi$
$619$	−10.8816	−0.437370	−0.218685	+	0.975795i	$0.570177\pi$
$620$	0	0
$621$	−7.12813	−0.286042
$622$	0	0
$623$	3.73106	0.149482
$624$	0	0
$625$	−22.1890	−0.887560
$626$	0	0
$627$	9.98198	0.398642
$628$	0	0
$629$	3.54373	0.141298
$630$	0	0
$631$	−0.526629	−0.0209648	−0.0104824	−	0.999945i	$-0.503337\pi$
$631$	−0.526629	−0.0209648	−0.0104824	+	0.999945i	$0.503337\pi$
$632$	0	0
$633$	30.2796	1.20351
$634$	0	0
$635$	−58.2692	−2.31234
$636$	0	0
$637$	0.544656	0.0215800
$638$	0	0
$639$	8.28304	0.327672
$640$	0	0
$641$	−4.36431	−0.172380	−0.0861899	−	0.996279i	$-0.527469\pi$
$641$	−4.36431	−0.172380	−0.0861899	+	0.996279i	$0.527469\pi$
$642$	0	0
$643$	−25.2830	−0.997062	−0.498531	−	0.866872i	$-0.666127\pi$
$643$	−25.2830	−0.997062	−0.498531	+	0.866872i	$0.666127\pi$
$644$	0	0
$645$	23.0120	0.906095
$646$	0	0
$647$	−1.48922	−0.0585474	−0.0292737	−	0.999571i	$-0.509319\pi$
$647$	−1.48922	−0.0585474	−0.0292737	+	0.999571i	$0.509319\pi$
$648$	0	0
$649$	7.58648	0.297796
$650$	0	0
$651$	−2.27460	−0.0891488
$652$	0	0
$653$	−20.5898	−0.805740	−0.402870	−	0.915257i	$-0.631987\pi$
$653$	−20.5898	−0.805740	−0.402870	+	0.915257i	$0.631987\pi$
$654$	0	0
$655$	0.288178	0.0112600
$656$	0	0
$657$	5.42353	0.211592
$658$	0	0
$659$	5.44563	0.212132	0.106066	−	0.994359i	$-0.466175\pi$
$659$	5.44563	0.212132	0.106066	+	0.994359i	$0.466175\pi$
$660$	0	0
$661$	25.1221	0.977136	0.488568	−	0.872526i	$-0.337519\pi$
$661$	25.1221	0.977136	0.488568	+	0.872526i	$0.337519\pi$
$662$	0	0
$663$	−0.900081	−0.0349563
$664$	0	0
$665$	−3.24194	−0.125717
$666$	0	0
$667$	17.0251	0.659216
$668$	0	0
$669$	1.51659	0.0586348
$670$	0	0
$671$	−45.6110	−1.76079
$672$	0	0
$673$	−27.4230	−1.05708	−0.528540	−	0.848909i	$-0.677260\pi$
$673$	−27.4230	−1.05708	−0.528540	+	0.848909i	$0.677260\pi$
$674$	0	0
$675$	23.2009	0.893002
$676$	0	0
$677$	7.07254	0.271820	0.135910	−	0.990721i	$-0.456604\pi$
$677$	7.07254	0.271820	0.135910	+	0.990721i	$0.456604\pi$
$678$	0	0
$679$	−17.5309	−0.672775
$680$	0	0
$681$	−9.39406	−0.359981
$682$	0	0
$683$	−15.3499	−0.587348	−0.293674	−	0.955906i	$-0.594878\pi$
$683$	−15.3499	−0.587348	−0.293674	+	0.955906i	$0.594878\pi$
$684$	0	0
$685$	−2.82945	−0.108108
$686$	0	0
$687$	22.8689	0.872503
$688$	0	0
$689$	3.25189	0.123887
$690$	0	0
$691$	−15.0161	−0.571241	−0.285621	−	0.958343i	$-0.592200\pi$
$691$	−15.0161	−0.571241	−0.285621	+	0.958343i	$0.592200\pi$
$692$	0	0
$693$	−4.34985	−0.165237
$694$	0	0
$695$	−18.9698	−0.719564
$696$	0	0
$697$	−3.32776	−0.126048
$698$	0	0
$699$	−17.2700	−0.653211
$700$	0	0
$701$	7.98423	0.301560	0.150780	−	0.988567i	$-0.451821\pi$
$701$	7.98423	0.301560	0.150780	+	0.988567i	$0.451821\pi$
$702$	0	0
$703$	4.21066	0.158808
$704$	0	0
$705$	29.2016	1.09980
$706$	0	0
$707$	−6.92968	−0.260618
$708$	0	0
$709$	30.8399	1.15822	0.579108	−	0.815251i	$-0.303401\pi$
$709$	30.8399	1.15822	0.579108	+	0.815251i	$0.303401\pi$
$710$	0	0
$711$	−4.10718	−0.154031
$712$	0	0
$713$	−1.96106	−0.0734422
$714$	0	0
$715$	8.97621	0.335691
$716$	0	0
$717$	7.10041	0.265170
$718$	0	0
$719$	−25.9968	−0.969516	−0.484758	−	0.874648i	$-0.661092\pi$
$719$	−25.9968	−0.969516	−0.484758	+	0.874648i	$0.661092\pi$
$720$	0	0
$721$	0.205898	0.00766805
$722$	0	0
$723$	48.7972	1.81479
$724$	0	0
$725$	−55.4140	−2.05802
$726$	0	0
$727$	−39.7938	−1.47587	−0.737935	−	0.674871i	$-0.764199\pi$
$727$	−39.7938	−1.47587	−0.737935	+	0.674871i	$0.764199\pi$
$728$	0	0
$729$	15.5324	0.575274
$730$	0	0
$731$	3.04235	0.112525
$732$	0	0
$733$	18.6116	0.687435	0.343718	−	0.939073i	$-0.388314\pi$
$733$	18.6116	0.687435	0.343718	+	0.939073i	$0.388314\pi$
$734$	0	0
$735$	6.36582	0.234807
$736$	0	0
$737$	−20.7971	−0.766069
$738$	0	0
$739$	−6.22602	−0.229028	−0.114514	−	0.993422i	$-0.536531\pi$
$739$	−6.22602	−0.229028	−0.114514	+	0.993422i	$0.536531\pi$
$740$	0	0
$741$	−1.06948	−0.0392883
$742$	0	0
$743$	−51.8756	−1.90313	−0.951566	−	0.307446i	$-0.900526\pi$
$743$	−51.8756	−1.90313	−0.951566	+	0.307446i	$0.900526\pi$
$744$	0	0
$745$	1.05217	0.0385487
$746$	0	0
$747$	−14.5604	−0.532736
$748$	0	0
$749$	19.3125	0.705664
$750$	0	0
$751$	−14.3254	−0.522742	−0.261371	−	0.965238i	$-0.584175\pi$
$751$	−14.3254	−0.522742	−0.261371	+	0.965238i	$0.584175\pi$
$752$	0	0
$753$	38.6030	1.40677
$754$	0	0
$755$	−45.4717	−1.65488
$756$	0	0
$757$	45.4777	1.65291	0.826457	−	0.563000i	$-0.190353\pi$
$757$	45.4777	1.65291	0.826457	+	0.563000i	$0.190353\pi$
$758$	0	0
$759$	−16.8986	−0.613381
$760$	0	0
$761$	−19.5767	−0.709657	−0.354828	−	0.934931i	$-0.615461\pi$
$761$	−19.5767	−0.709657	−0.354828	+	0.934931i	$0.615461\pi$
$762$	0	0
$763$	11.6192	0.420643
$764$	0	0
$765$	−2.33465	−0.0844095
$766$	0	0
$767$	−0.812823	−0.0293493
$768$	0	0
$769$	27.1791	0.980104	0.490052	−	0.871693i	$-0.336978\pi$
$769$	27.1791	0.980104	0.490052	+	0.871693i	$0.336978\pi$
$770$	0	0
$771$	59.2132	2.13251
$772$	0	0
$773$	−39.2410	−1.41140	−0.705700	−	0.708511i	$-0.749367\pi$
$773$	−39.2410	−1.41140	−0.705700	+	0.708511i	$0.749367\pi$
$774$	0	0
$775$	6.38292	0.229281
$776$	0	0
$777$	−8.26800	−0.296613
$778$	0	0
$779$	−3.95405	−0.141669
$780$	0	0
$781$	−49.2095	−1.76085
$782$	0	0
$783$	−42.3445	−1.51327
$784$	0	0
$785$	−40.0882	−1.43081
$786$	0	0
$787$	−46.6472	−1.66279	−0.831396	−	0.555680i	$-0.812458\pi$
$787$	−46.6472	−1.66279	−0.831396	+	0.555680i	$0.812458\pi$
$788$	0	0
$789$	−21.1986	−0.754689
$790$	0	0
$791$	−5.00716	−0.178034
$792$	0	0
$793$	4.88680	0.173536
$794$	0	0
$795$	38.0074	1.34798
$796$	0	0
$797$	−45.0074	−1.59424	−0.797122	−	0.603818i	$-0.793646\pi$
$797$	−45.0074	−1.59424	−0.797122	+	0.603818i	$0.793646\pi$
$798$	0	0
$799$	3.86067	0.136581
$800$	0	0
$801$	3.19257	0.112804
$802$	0	0
$803$	−32.2211	−1.13706
$804$	0	0
$805$	5.48831	0.193438
$806$	0	0
$807$	37.6261	1.32450
$808$	0	0
$809$	−8.21185	−0.288713	−0.144357	−	0.989526i	$-0.546111\pi$
$809$	−8.21185	−0.288713	−0.144357	+	0.989526i	$0.546111\pi$
$810$	0	0
$811$	29.2902	1.02852	0.514258	−	0.857635i	$-0.328067\pi$
$811$	29.2902	1.02852	0.514258	+	0.857635i	$0.328067\pi$
$812$	0	0
$813$	15.5574	0.545621
$814$	0	0
$815$	−71.8259	−2.51595
$816$	0	0
$817$	3.61492	0.126470
$818$	0	0
$819$	0.466047	0.0162850
$820$	0	0
$821$	33.6432	1.17415	0.587077	−	0.809531i	$-0.300278\pi$
$821$	33.6432	1.17415	0.587077	+	0.809531i	$0.300278\pi$
$822$	0	0
$823$	35.7935	1.24768	0.623842	−	0.781551i	$-0.285571\pi$
$823$	35.7935	1.24768	0.623842	+	0.781551i	$0.285571\pi$
$824$	0	0
$825$	55.0022	1.91493
$826$	0	0
$827$	40.7404	1.41668	0.708342	−	0.705870i	$-0.249444\pi$
$827$	40.7404	1.41668	0.708342	+	0.705870i	$0.249444\pi$
$828$	0	0
$829$	−4.92672	−0.171112	−0.0855561	−	0.996333i	$-0.527267\pi$
$829$	−4.92672	−0.171112	−0.0855561	+	0.996333i	$0.527267\pi$
$830$	0	0
$831$	−57.8177	−2.00567
$832$	0	0
$833$	0.841608	0.0291600
$834$	0	0
$835$	−42.1447	−1.45848
$836$	0	0
$837$	4.87749	0.168591
$838$	0	0
$839$	−14.0195	−0.484008	−0.242004	−	0.970275i	$-0.577805\pi$
$839$	−14.0195	−0.484008	−0.242004	+	0.970275i	$0.577805\pi$
$840$	0	0
$841$	72.1375	2.48750
$842$	0	0
$843$	22.9041	0.788861
$844$	0	0
$845$	41.1835	1.41675
$846$	0	0
$847$	14.8424	0.509992
$848$	0	0
$849$	−11.8315	−0.406056
$850$	0	0
$851$	−7.12829	−0.244354
$852$	0	0
$853$	−18.0298	−0.617330	−0.308665	−	0.951171i	$-0.599882\pi$
$853$	−18.0298	−0.617330	−0.308665	+	0.951171i	$0.599882\pi$
$854$	0	0
$855$	−2.77404	−0.0948700
$856$	0	0
$857$	−42.4395	−1.44971	−0.724853	−	0.688903i	$-0.758092\pi$
$857$	−42.4395	−1.44971	−0.724853	+	0.688903i	$0.758092\pi$
$858$	0	0
$859$	29.7917	1.01648	0.508240	−	0.861216i	$-0.330296\pi$
$859$	29.7917	1.01648	0.508240	+	0.861216i	$0.330296\pi$
$860$	0	0
$861$	7.76413	0.264601
$862$	0	0
$863$	−24.1810	−0.823132	−0.411566	−	0.911380i	$-0.635018\pi$
$863$	−24.1810	−0.823132	−0.411566	+	0.911380i	$0.635018\pi$
$864$	0	0
$865$	−12.1853	−0.414312
$866$	0	0
$867$	31.9902	1.08644
$868$	0	0
$869$	24.4007	0.827737
$870$	0	0
$871$	2.22822	0.0755002
$872$	0	0
$873$	−15.0007	−0.507698
$874$	0	0
$875$	−1.65387	−0.0559111
$876$	0	0
$877$	29.6693	1.00186	0.500931	−	0.865487i	$-0.332991\pi$
$877$	29.6693	1.00186	0.500931	+	0.865487i	$0.332991\pi$
$878$	0	0
$879$	−36.1622	−1.21972
$880$	0	0
$881$	46.2108	1.55688	0.778441	−	0.627717i	$-0.216011\pi$
$881$	46.2108	1.55688	0.778441	+	0.627717i	$0.216011\pi$
$882$	0	0
$883$	−45.2854	−1.52397	−0.761987	−	0.647593i	$-0.775776\pi$
$883$	−45.2854	−1.52397	−0.761987	+	0.647593i	$0.775776\pi$
$884$	0	0
$885$	−9.50011	−0.319343
$886$	0	0
$887$	−56.6950	−1.90363	−0.951816	−	0.306671i	$-0.900785\pi$
$887$	−56.6950	−1.90363	−0.951816	+	0.306671i	$0.900785\pi$
$888$	0	0
$889$	17.9736	0.602815
$890$	0	0
$891$	55.0794	1.84523
$892$	0	0
$893$	4.58725	0.153507
$894$	0	0
$895$	23.1792	0.774796
$896$	0	0
$897$	1.81053	0.0604520
$898$	0	0
$899$	−11.6496	−0.388537
$900$	0	0
$901$	5.02485	0.167402
$902$	0	0
$903$	−7.09821	−0.236214
$904$	0	0
$905$	40.0065	1.32986
$906$	0	0
$907$	−11.1794	−0.371205	−0.185603	−	0.982625i	$-0.559424\pi$
$907$	−11.1794	−0.371205	−0.185603	+	0.982625i	$0.559424\pi$
$908$	0	0
$909$	−5.92954	−0.196671
$910$	0	0
$911$	16.2376	0.537976	0.268988	−	0.963144i	$-0.413311\pi$
$911$	16.2376	0.537976	0.268988	+	0.963144i	$0.413311\pi$
$912$	0	0
$913$	86.5030	2.86283
$914$	0	0
$915$	57.1160	1.88820
$916$	0	0
$917$	−0.0888907	−0.00293543
$918$	0	0
$919$	−48.1232	−1.58744	−0.793719	−	0.608285i	$-0.791858\pi$
$919$	−48.1232	−1.58744	−0.793719	+	0.608285i	$0.791858\pi$
$920$	0	0
$921$	3.69350	0.121705
$922$	0	0
$923$	5.27235	0.173541
$924$	0	0
$925$	23.2014	0.762857
$926$	0	0
$927$	0.176181	0.00578656
$928$	0	0
$929$	−8.72703	−0.286325	−0.143162	−	0.989699i	$-0.545727\pi$
$929$	−8.72703	−0.286325	−0.143162	+	0.989699i	$0.545727\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	14.7215	0.481962
$934$	0	0
$935$	13.8701	0.453602
$936$	0	0
$937$	26.6360	0.870160	0.435080	−	0.900392i	$-0.356720\pi$
$937$	26.6360	0.870160	0.435080	+	0.900392i	$0.356720\pi$
$938$	0	0
$939$	−50.3708	−1.64379
$940$	0	0
$941$	31.5793	1.02946	0.514728	−	0.857353i	$-0.327893\pi$
$941$	31.5793	1.02946	0.514728	+	0.857353i	$0.327893\pi$
$942$	0	0
$943$	6.69387	0.217982
$944$	0	0
$945$	−13.6504	−0.444048
$946$	0	0
$947$	12.9868	0.422016	0.211008	−	0.977484i	$-0.432325\pi$
$947$	12.9868	0.422016	0.211008	+	0.977484i	$0.432325\pi$
$948$	0	0
$949$	3.45220	0.112063
$950$	0	0
$951$	30.8583	1.00065
$952$	0	0
$953$	1.27092	0.0411691	0.0205845	−	0.999788i	$-0.493447\pi$
$953$	1.27092	0.0411691	0.0205845	+	0.999788i	$0.493447\pi$
$954$	0	0
$955$	−81.9353	−2.65137
$956$	0	0
$957$	−100.386	−3.24502
$958$	0	0
$959$	0.872766	0.0281831
$960$	0	0
$961$	−29.6581	−0.956714
$962$	0	0
$963$	16.5252	0.532517
$964$	0	0
$965$	53.0331	1.70719
$966$	0	0
$967$	−50.8903	−1.63652	−0.818260	−	0.574848i	$-0.805061\pi$
$967$	−50.8903	−1.63652	−0.818260	+	0.574848i	$0.805061\pi$
$968$	0	0
$969$	−1.65257	−0.0530882
$970$	0	0
$971$	24.7478	0.794193	0.397097	−	0.917777i	$-0.370018\pi$
$971$	24.7478	0.794193	0.397097	+	0.917777i	$0.370018\pi$
$972$	0	0
$973$	5.85137	0.187586
$974$	0	0
$975$	−5.89299	−0.188727
$976$	0	0
$977$	−0.504613	−0.0161440	−0.00807199	−	0.999967i	$-0.502569\pi$
$977$	−0.504613	−0.0161440	−0.00807199	+	0.999967i	$0.502569\pi$
$978$	0	0
$979$	−18.9670	−0.606189
$980$	0	0
$981$	9.94223	0.317431
$982$	0	0
$983$	−21.1896	−0.675844	−0.337922	−	0.941174i	$-0.609724\pi$
$983$	−21.1896	−0.675844	−0.337922	+	0.941174i	$0.609724\pi$
$984$	0	0
$985$	17.5069	0.557817
$986$	0	0
$987$	−9.00747	−0.286711
$988$	0	0
$989$	−6.11975	−0.194597
$990$	0	0
$991$	−23.7636	−0.754876	−0.377438	−	0.926035i	$-0.623195\pi$
$991$	−23.7636	−0.754876	−0.377438	+	0.926035i	$0.623195\pi$
$992$	0	0
$993$	−55.8481	−1.77229
$994$	0	0
$995$	39.7880	1.26136
$996$	0	0
$997$	12.6614	0.400991	0.200495	−	0.979695i	$-0.435745\pi$
$997$	12.6614	0.400991	0.200495	+	0.979695i	$0.435745\pi$
$998$	0	0
$999$	17.7293	0.560930

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.ca.1.2		6
4.3	odd	2		8512.2.a.cf.1.5		6
8.3	odd	2		4256.2.a.i.1.2	✓	6
8.5	even	2		4256.2.a.n.1.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4256.2.a.i.1.2	✓	6	8.3	odd	2
4256.2.a.n.1.5	yes	6	8.5	even	2
8512.2.a.ca.1.2		6	1.1	even	1	trivial
8512.2.a.cf.1.5		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)

\(f(q)\)	\(=\)	\(q-1.96359 q^{3} -3.24194 q^{5} +1.00000 q^{7} +0.855673 q^{9} +O(q^{10})\)	\(q-1.96359 q^{3} -3.24194 q^{5} +1.00000 q^{7} +0.855673 q^{9} -5.08354 q^{11} +0.544656 q^{13} +6.36582 q^{15} +0.841608 q^{17} +1.00000 q^{19} -1.96359 q^{21} -1.69291 q^{23} +5.51015 q^{25} +4.21057 q^{27} -10.0567 q^{29} +1.15839 q^{31} +9.98198 q^{33} -3.24194 q^{35} +4.21066 q^{37} -1.06948 q^{39} -3.95405 q^{41} +3.61492 q^{43} -2.77404 q^{45} +4.58725 q^{47} +1.00000 q^{49} -1.65257 q^{51} +5.97054 q^{53} +16.4805 q^{55} -1.96359 q^{57} -1.49236 q^{59} +8.97228 q^{61} +0.855673 q^{63} -1.76574 q^{65} +4.09105 q^{67} +3.32418 q^{69} +9.68015 q^{71} +6.33832 q^{73} -10.8197 q^{75} -5.08354 q^{77} -4.79994 q^{79} -10.8348 q^{81} -17.0163 q^{83} -2.72844 q^{85} +19.7472 q^{87} +3.73106 q^{89} +0.544656 q^{91} -2.27460 q^{93} -3.24194 q^{95} -17.5309 q^{97} -4.34985 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) \) 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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