LMFDB - Embedded newform 3072.2.d.f.1537.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1537,3072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("3072.1537");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3072 = 2^{10} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3072.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$24.5300435009$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.18939904.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 $ x^8 - 4x^7 + 14x^6 - 28x^5 + 43x^4 - 44x^3 + 30x^2 - 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1537.2
Root			$0.500000 - 0.0297061i$ of defining polynomial
Character	$\chi$	$=$	3072.1537
Dual form			3072.2.d.f.1537.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -0.473626i q^{5} -4.55765 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -0.473626i q^{5} -4.55765 q^{7} -1.00000 q^{9} +3.49824i q^{11} +0.0840215i q^{13} -0.473626 q^{15} -3.61706 q^{17} -3.61706i q^{19} +4.55765i q^{21} -2.82843 q^{23} +4.77568 q^{25} +1.00000i q^{27} -7.30205i q^{29} -0.557647 q^{31} +3.49824 q^{33} +2.15862i q^{35} +6.20285i q^{37} +0.0840215 q^{39} +9.27391 q^{41} +2.27744i q^{43} +0.473626i q^{45} -2.82843 q^{47} +13.7721 q^{49} +3.61706i q^{51} +0.697947i q^{53} +1.65685 q^{55} -3.61706 q^{57} +5.65685i q^{59} +3.85970i q^{61} +4.55765 q^{63} +0.0397948 q^{65} +5.33962i q^{67} +2.82843i q^{69} +9.11529 q^{71} +0.541560 q^{73} -4.77568i q^{75} -15.9437i q^{77} +10.9937 q^{79} +1.00000 q^{81} +15.0496i q^{83} +1.71313i q^{85} -7.30205 q^{87} +14.6533 q^{89} -0.382941i q^{91} +0.557647i q^{93} -1.71313 q^{95} +4.31724 q^{97} -3.49824i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7} - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^7 - 8 * q^9	$ 8 q - 8 q^{7} - 8 q^{9} + 8 q^{15} - 8 q^{25} + 24 q^{31} - 16 q^{39} + 8 q^{49} - 32 q^{55} + 8 q^{63} - 16 q^{65} + 16 q^{71} + 16 q^{73} + 24 q^{79} + 8 q^{81} - 24 q^{87} + 16 q^{89} - 48 q^{95}+O(q^{100}) $ 8 * q - 8 * q^7 - 8 * q^9 + 8 * q^15 - 8 * q^25 + 24 * q^31 - 16 * q^39 + 8 * q^49 - 32 * q^55 + 8 * q^63 - 16 * q^65 + 16 * q^71 + 16 * q^73 + 24 * q^79 + 8 * q^81 - 24 * q^87 + 16 * q^89 - 48 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times$.

$n$	$1025$	$2047$	$2053$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	− 0.473626i	− 0.211812i	−0.994376	−	0.105906i	$-0.966226\pi$
$5$	− 0.473626i	− 0.211812i	0.994376	−	0.105906i	$-0.0337742\pi$
$6$	0	0
$7$	−4.55765	−1.72263	−0.861314	−	0.508072i	$-0.830358\pi$
$7$	−4.55765	−1.72263	−0.861314	+	0.508072i	$0.830358\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.49824i	1.05476i	0.849630	+	0.527379i	$0.176825\pi$
$11$	3.49824i	1.05476i	−0.849630	+	0.527379i	$0.823175\pi$
$12$	0	0
$13$	0.0840215i	0.0233034i	0.999932	+	0.0116517i	$0.00370893\pi$
$13$	0.0840215i	0.0233034i	−0.999932	+	0.0116517i	$0.996291\pi$
$14$	0	0
$15$	−0.473626	−0.122290
$16$	0	0
$17$	−3.61706	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$17$	−3.61706	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$18$	0	0
$19$	− 3.61706i	− 0.829810i	−0.909865	−	0.414905i	$-0.863815\pi$
$19$	− 3.61706i	− 0.829810i	0.909865	−	0.414905i	$-0.136185\pi$
$20$	0	0
$21$	4.55765i	0.994560i
$22$	0	0
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	4.77568	0.955136
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 7.30205i	− 1.35596i	−0.735082	−	0.677979i	$-0.762856\pi$
$29$	− 7.30205i	− 1.35596i	0.735082	−	0.677979i	$-0.237144\pi$
$30$	0	0
$31$	−0.557647	−0.100156	−0.0500782	−	0.998745i	$-0.515947\pi$
$31$	−0.557647	−0.100156	−0.0500782	+	0.998745i	$0.515947\pi$
$32$	0	0
$33$	3.49824	0.608965
$34$	0	0
$35$	2.15862i	0.364873i
$36$	0	0
$37$	6.20285i	1.01974i	0.860251	+	0.509871i	$0.170307\pi$
$37$	6.20285i	1.01974i	−0.860251	+	0.509871i	$0.829693\pi$
$38$	0	0
$39$	0.0840215	0.0134542
$40$	0	0
$41$	9.27391	1.44834	0.724171	−	0.689620i	$-0.242223\pi$
$41$	9.27391	1.44834	0.724171	+	0.689620i	$0.242223\pi$
$42$	0	0
$43$	2.27744i	0.347307i	0.984807	+	0.173653i	$0.0555573\pi$
$43$	2.27744i	0.347307i	−0.984807	+	0.173653i	$0.944443\pi$
$44$	0	0
$45$	0.473626i	0.0706040i
$46$	0	0
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	13.7721	1.96745
$50$	0	0
$51$	3.61706i	0.506490i
$52$	0	0
$53$	0.697947i	0.0958704i	0.998850	+	0.0479352i	$0.0152641\pi$
$53$	0.697947i	0.0958704i	−0.998850	+	0.0479352i	$0.984736\pi$
$54$	0	0
$55$	1.65685	0.223410
$56$	0	0
$57$	−3.61706	−0.479091
$58$	0	0
$59$	5.65685i	0.736460i	0.929735	+	0.368230i	$0.120036\pi$
$59$	5.65685i	0.736460i	−0.929735	+	0.368230i	$0.879964\pi$
$60$	0	0
$61$	3.85970i	0.494184i	0.968992	+	0.247092i	$0.0794750\pi$
$61$	3.85970i	0.494184i	−0.968992	+	0.247092i	$0.920525\pi$
$62$	0	0
$63$	4.55765	0.574210
$64$	0	0
$65$	0.0397948	0.00493593
$66$	0	0
$67$	5.33962i	0.652338i	0.945311	+	0.326169i	$0.105758\pi$
$67$	5.33962i	0.652338i	−0.945311	+	0.326169i	$0.894242\pi$
$68$	0	0
$69$	2.82843i	0.340503i
$70$	0	0
$71$	9.11529	1.08179	0.540893	−	0.841091i	$-0.318086\pi$
$71$	9.11529	1.08179	0.540893	+	0.841091i	$0.318086\pi$
$72$	0	0
$73$	0.541560	0.0633848	0.0316924	−	0.999498i	$-0.489910\pi$
$73$	0.541560	0.0633848	0.0316924	+	0.999498i	$0.489910\pi$
$74$	0	0
$75$	− 4.77568i	− 0.551448i
$76$	0	0
$77$	− 15.9437i	− 1.81696i
$78$	0	0
$79$	10.9937	1.23689	0.618445	−	0.785828i	$-0.287763\pi$
$79$	10.9937	1.23689	0.618445	+	0.785828i	$0.287763\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.0496i	1.65191i	0.563738	+	0.825954i	$0.309363\pi$
$83$	15.0496i	1.65191i	−0.563738	+	0.825954i	$0.690637\pi$
$84$	0	0
$85$	1.71313i	0.185815i
$86$	0	0
$87$	−7.30205	−0.782862
$88$	0	0
$89$	14.6533	1.55325	0.776625	−	0.629964i	$-0.216930\pi$
$89$	14.6533	1.55325	0.776625	+	0.629964i	$0.216930\pi$
$90$	0	0
$91$	− 0.382941i	− 0.0401431i
$92$	0	0
$93$	0.557647i	0.0578253i
$94$	0	0
$95$	−1.71313	−0.175764
$96$	0	0
$97$	4.31724	0.438349	0.219175	−	0.975686i	$-0.429664\pi$
$97$	4.31724	0.438349	0.219175	+	0.975686i	$0.429664\pi$
$98$	0	0
$99$	− 3.49824i	− 0.351586i
$100$	0	0
$101$	− 0.641669i	− 0.0638484i	−0.999490	−	0.0319242i	$-0.989836\pi$
$101$	− 0.641669i	− 0.0638484i	0.999490	−	0.0319242i	$-0.0101635\pi$
$102$	0	0
$103$	1.33686	0.131724	0.0658622	−	0.997829i	$-0.479020\pi$
$103$	1.33686	0.131724	0.0658622	+	0.997829i	$0.479020\pi$
$104$	0	0
$105$	2.15862	0.210660
$106$	0	0
$107$	− 8.57373i	− 0.828854i	−0.910083	−	0.414427i	$-0.863982\pi$
$107$	− 8.57373i	− 0.828854i	0.910083	−	0.414427i	$-0.136018\pi$
$108$	0	0
$109$	− 8.08402i	− 0.774309i	−0.922015	−	0.387154i	$-0.873458\pi$
$109$	− 8.08402i	− 0.774309i	0.922015	−	0.387154i	$-0.126542\pi$
$110$	0	0
$111$	6.20285	0.588748
$112$	0	0
$113$	9.55136	0.898516	0.449258	−	0.893402i	$-0.351688\pi$
$113$	9.55136	0.898516	0.449258	+	0.893402i	$0.351688\pi$
$114$	0	0
$115$	1.33962i	0.124920i
$116$	0	0
$117$	− 0.0840215i	− 0.00776779i
$118$	0	0
$119$	16.4853	1.51120
$120$	0	0
$121$	−1.23765	−0.112514
$122$	0	0
$123$	− 9.27391i	− 0.836201i
$124$	0	0
$125$	− 4.63001i	− 0.414121i
$126$	0	0
$127$	5.09921	0.452481	0.226241	−	0.974071i	$-0.427356\pi$
$127$	5.09921	0.452481	0.226241	+	0.974071i	$0.427356\pi$
$128$	0	0
$129$	2.27744	0.200518
$130$	0	0
$131$	2.99647i	0.261803i	0.991395	+	0.130901i	$0.0417871\pi$
$131$	2.99647i	0.261803i	−0.991395	+	0.130901i	$0.958213\pi$
$132$	0	0
$133$	16.4853i	1.42946i
$134$	0	0
$135$	0.473626	0.0407632
$136$	0	0
$137$	3.37941	0.288723	0.144361	−	0.989525i	$-0.453887\pi$
$137$	3.37941	0.288723	0.144361	+	0.989525i	$0.453887\pi$
$138$	0	0
$139$	− 8.31724i	− 0.705459i	−0.935725	−	0.352729i	$-0.885254\pi$
$139$	− 8.31724i	− 0.705459i	0.935725	−	0.352729i	$-0.114746\pi$
$140$	0	0
$141$	2.82843i	0.238197i
$142$	0	0
$143$	−0.293927	−0.0245794
$144$	0	0
$145$	−3.45844	−0.287208
$146$	0	0
$147$	− 13.7721i	− 1.13591i
$148$	0	0
$149$	14.1305i	1.15761i	0.815465	+	0.578807i	$0.196482\pi$
$149$	14.1305i	1.15761i	−0.815465	+	0.578807i	$0.803518\pi$
$150$	0	0
$151$	−9.97685	−0.811905	−0.405952	−	0.913894i	$-0.633060\pi$
$151$	−9.97685	−0.811905	−0.405952	+	0.913894i	$0.633060\pi$
$152$	0	0
$153$	3.61706	0.292422
$154$	0	0
$155$	0.264116i	0.0212143i
$156$	0	0
$157$	− 22.8562i	− 1.82412i	−0.410056	−	0.912060i	$-0.634491\pi$
$157$	− 22.8562i	− 1.82412i	0.410056	−	0.912060i	$-0.365509\pi$
$158$	0	0
$159$	0.697947	0.0553508
$160$	0	0
$161$	12.8910	1.01595
$162$	0	0
$163$	− 10.6135i	− 0.831316i	−0.909521	−	0.415658i	$-0.863551\pi$
$163$	− 10.6135i	− 0.831316i	0.909521	−	0.415658i	$-0.136449\pi$
$164$	0	0
$165$	− 1.65685i	− 0.128986i
$166$	0	0
$167$	−5.83822	−0.451775	−0.225888	−	0.974153i	$-0.572528\pi$
$167$	−5.83822	−0.451775	−0.225888	+	0.974153i	$0.572528\pi$
$168$	0	0
$169$	12.9929	0.999457
$170$	0	0
$171$	3.61706i	0.276603i
$172$	0	0
$173$	5.12695i	0.389795i	0.980824	+	0.194897i	$0.0624374\pi$
$173$	5.12695i	0.389795i	−0.980824	+	0.194897i	$0.937563\pi$
$174$	0	0
$175$	−21.7659	−1.64534
$176$	0	0
$177$	5.65685	0.425195
$178$	0	0
$179$	− 13.1286i	− 0.981279i	−0.871363	−	0.490640i	$-0.836763\pi$
$179$	− 13.1286i	− 0.981279i	0.871363	−	0.490640i	$-0.163237\pi$
$180$	0	0
$181$	15.3181i	1.13859i	0.822134	+	0.569294i	$0.192783\pi$
$181$	15.3181i	1.13859i	−0.822134	+	0.569294i	$0.807217\pi$
$182$	0	0
$183$	3.85970	0.285317
$184$	0	0
$185$	2.93783	0.215993
$186$	0	0
$187$	− 12.6533i	− 0.925303i
$188$	0	0
$189$	− 4.55765i	− 0.331520i
$190$	0	0
$191$	−8.63001	−0.624446	−0.312223	−	0.950009i	$-0.601074\pi$
$191$	−8.63001	−0.624446	−0.312223	+	0.950009i	$0.601074\pi$
$192$	0	0
$193$	11.4514	0.824288	0.412144	−	0.911119i	$-0.364780\pi$
$193$	11.4514	0.824288	0.412144	+	0.911119i	$0.364780\pi$
$194$	0	0
$195$	− 0.0397948i	− 0.00284976i
$196$	0	0
$197$	10.5925i	0.754681i	0.926075	+	0.377340i	$0.123161\pi$
$197$	10.5925i	0.754681i	−0.926075	+	0.377340i	$0.876839\pi$
$198$	0	0
$199$	−3.68000	−0.260868	−0.130434	−	0.991457i	$-0.541637\pi$
$199$	−3.68000	−0.260868	−0.130434	+	0.991457i	$0.541637\pi$
$200$	0	0
$201$	5.33962	0.376627
$202$	0	0
$203$	33.2802i	2.33581i
$204$	0	0
$205$	− 4.39236i	− 0.306776i
$206$	0	0
$207$	2.82843	0.196589
$208$	0	0
$209$	12.6533	0.875249
$210$	0	0
$211$	− 14.3102i	− 0.985153i	−0.870269	−	0.492577i	$-0.836055\pi$
$211$	− 14.3102i	− 0.985153i	0.870269	−	0.492577i	$-0.163945\pi$
$212$	0	0
$213$	− 9.11529i	− 0.624570i
$214$	0	0
$215$	1.07866	0.0735637
$216$	0	0
$217$	2.54156	0.172532
$218$	0	0
$219$	− 0.541560i	− 0.0365952i
$220$	0	0
$221$	− 0.303911i	− 0.0204433i
$222$	0	0
$223$	−4.86156	−0.325554	−0.162777	−	0.986663i	$-0.552045\pi$
$223$	−4.86156	−0.325554	−0.162777	+	0.986663i	$0.552045\pi$
$224$	0	0
$225$	−4.77568	−0.318379
$226$	0	0
$227$	15.0496i	0.998877i	0.866349	+	0.499438i	$0.166460\pi$
$227$	15.0496i	0.998877i	−0.866349	+	0.499438i	$0.833540\pi$
$228$	0	0
$229$	− 28.5264i	− 1.88507i	−0.334101	−	0.942537i	$-0.608433\pi$
$229$	− 28.5264i	− 1.88507i	0.334101	−	0.942537i	$-0.391567\pi$
$230$	0	0
$231$	−15.9437	−1.04902
$232$	0	0
$233$	−13.5702	−0.889014	−0.444507	−	0.895775i	$-0.646621\pi$
$233$	−13.5702	−0.889014	−0.444507	+	0.895775i	$0.646621\pi$
$234$	0	0
$235$	1.33962i	0.0873869i
$236$	0	0
$237$	− 10.9937i	− 0.714118i
$238$	0	0
$239$	−29.3629	−1.89933	−0.949665	−	0.313267i	$-0.898576\pi$
$239$	−29.3629	−1.89933	−0.949665	+	0.313267i	$0.898576\pi$
$240$	0	0
$241$	−24.0063	−1.54638	−0.773190	−	0.634175i	$-0.781340\pi$
$241$	−24.0063	−1.54638	−0.773190	+	0.634175i	$0.781340\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 6.52284i	− 0.416729i
$246$	0	0
$247$	0.303911	0.0193374
$248$	0	0
$249$	15.0496	0.953729
$250$	0	0
$251$	22.2837i	1.40654i	0.710925	+	0.703268i	$0.248276\pi$
$251$	22.2837i	1.40654i	−0.710925	+	0.703268i	$0.751724\pi$
$252$	0	0
$253$	− 9.89450i	− 0.622062i
$254$	0	0
$255$	1.71313	0.107281
$256$	0	0
$257$	8.66038	0.540220	0.270110	−	0.962829i	$-0.412940\pi$
$257$	8.66038	0.540220	0.270110	+	0.962829i	$0.412940\pi$
$258$	0	0
$259$	− 28.2704i	− 1.75664i
$260$	0	0
$261$	7.30205i	0.451986i
$262$	0	0
$263$	13.3208	0.821394	0.410697	−	0.911772i	$-0.365285\pi$
$263$	13.3208	0.821394	0.410697	+	0.911772i	$0.365285\pi$
$264$	0	0
$265$	0.330566	0.0203065
$266$	0	0
$267$	− 14.6533i	− 0.896769i
$268$	0	0
$269$	16.5058i	1.00638i	0.864177	+	0.503188i	$0.167840\pi$
$269$	16.5058i	1.00638i	−0.864177	+	0.503188i	$0.832160\pi$
$270$	0	0
$271$	21.9769	1.33500	0.667499	−	0.744610i	$-0.267365\pi$
$271$	21.9769	1.33500	0.667499	+	0.744610i	$0.267365\pi$
$272$	0	0
$273$	−0.382941	−0.0231766
$274$	0	0
$275$	16.7064i	1.00744i
$276$	0	0
$277$	15.4862i	0.930475i	0.885186	+	0.465237i	$0.154031\pi$
$277$	15.4862i	0.930475i	−0.885186	+	0.465237i	$0.845969\pi$
$278$	0	0
$279$	0.557647	0.0333855
$280$	0	0
$281$	−22.8910	−1.36556	−0.682780	−	0.730624i	$-0.739229\pi$
$281$	−22.8910	−1.36556	−0.682780	+	0.730624i	$0.739229\pi$
$282$	0	0
$283$	6.34315i	0.377061i	0.982067	+	0.188530i	$0.0603724\pi$
$283$	6.34315i	0.377061i	−0.982067	+	0.188530i	$0.939628\pi$
$284$	0	0
$285$	1.71313i	0.101477i
$286$	0	0
$287$	−42.2672	−2.49496
$288$	0	0
$289$	−3.91688	−0.230405
$290$	0	0
$291$	− 4.31724i	− 0.253081i
$292$	0	0
$293$	30.5783i	1.78641i	0.449654	+	0.893203i	$0.351547\pi$
$293$	30.5783i	1.78641i	−0.449654	+	0.893203i	$0.648453\pi$
$294$	0	0
$295$	2.67923	0.155991
$296$	0	0
$297$	−3.49824	−0.202988
$298$	0	0
$299$	− 0.237649i	− 0.0137436i
$300$	0	0
$301$	− 10.3798i	− 0.598281i
$302$	0	0
$303$	−0.641669	−0.0368629
$304$	0	0
$305$	1.82805	0.104674
$306$	0	0
$307$	17.1286i	0.977582i	0.872401	+	0.488791i	$0.162562\pi$
$307$	17.1286i	0.977582i	−0.872401	+	0.488791i	$0.837438\pi$
$308$	0	0
$309$	− 1.33686i	− 0.0760511i
$310$	0	0
$311$	26.8651	1.52338	0.761689	−	0.647943i	$-0.224370\pi$
$311$	26.8651	1.52338	0.761689	+	0.647943i	$0.224370\pi$
$312$	0	0
$313$	19.6890	1.11289	0.556445	−	0.830885i	$-0.312165\pi$
$313$	19.6890	1.11289	0.556445	+	0.830885i	$0.312165\pi$
$314$	0	0
$315$	− 2.15862i	− 0.121624i
$316$	0	0
$317$	30.1860i	1.69541i	0.530466	+	0.847706i	$0.322017\pi$
$317$	30.1860i	1.69541i	−0.530466	+	0.847706i	$0.677983\pi$
$318$	0	0
$319$	25.5443	1.43021
$320$	0	0
$321$	−8.57373	−0.478539
$322$	0	0
$323$	13.0831i	0.727964i
$324$	0	0
$325$	0.401260i	0.0222579i
$326$	0	0
$327$	−8.08402	−0.447047
$328$	0	0
$329$	12.8910	0.710702
$330$	0	0
$331$	− 20.7784i	− 1.14209i	−0.820920	−	0.571043i	$-0.806539\pi$
$331$	− 20.7784i	− 1.14209i	0.820920	−	0.571043i	$-0.193461\pi$
$332$	0	0
$333$	− 6.20285i	− 0.339914i
$334$	0	0
$335$	2.52898	0.138173
$336$	0	0
$337$	23.0098	1.25342	0.626712	−	0.779251i	$-0.284400\pi$
$337$	23.0098	1.25342	0.626712	+	0.779251i	$0.284400\pi$
$338$	0	0
$339$	− 9.55136i	− 0.518759i
$340$	0	0
$341$	− 1.95078i	− 0.105641i
$342$	0	0
$343$	−30.8651	−1.66656
$344$	0	0
$345$	1.33962	0.0721225
$346$	0	0
$347$	− 15.4186i	− 0.827716i	−0.910341	−	0.413858i	$-0.864181\pi$
$347$	− 15.4186i	− 0.827716i	0.910341	−	0.413858i	$-0.135819\pi$
$348$	0	0
$349$	28.3638i	1.51828i	0.650927	+	0.759140i	$0.274380\pi$
$349$	28.3638i	1.51828i	−0.650927	+	0.759140i	$0.725620\pi$
$350$	0	0
$351$	−0.0840215	−0.00448474
$352$	0	0
$353$	−12.2117	−0.649965	−0.324983	−	0.945720i	$-0.605358\pi$
$353$	−12.2117	−0.649965	−0.324983	+	0.945720i	$0.605358\pi$
$354$	0	0
$355$	− 4.31724i	− 0.229135i
$356$	0	0
$357$	− 16.4853i	− 0.872494i
$358$	0	0
$359$	33.4780	1.76690	0.883452	−	0.468522i	$-0.155214\pi$
$359$	33.4780	1.76690	0.883452	+	0.468522i	$0.155214\pi$
$360$	0	0
$361$	5.91688	0.311415
$362$	0	0
$363$	1.23765i	0.0649597i
$364$	0	0
$365$	− 0.256497i	− 0.0134256i
$366$	0	0
$367$	0.702379	0.0366639	0.0183319	−	0.999832i	$-0.494164\pi$
$367$	0.702379	0.0366639	0.0183319	+	0.999832i	$0.494164\pi$
$368$	0	0
$369$	−9.27391	−0.482781
$370$	0	0
$371$	− 3.18100i	− 0.165149i
$372$	0	0
$373$	− 26.8132i	− 1.38834i	−0.719813	−	0.694168i	$-0.755773\pi$
$373$	− 26.8132i	− 1.38834i	0.719813	−	0.694168i	$-0.244227\pi$
$374$	0	0
$375$	−4.63001	−0.239093
$376$	0	0
$377$	0.613530	0.0315984
$378$	0	0
$379$	− 2.51509i	− 0.129192i	−0.997912	−	0.0645958i	$-0.979424\pi$
$379$	− 2.51509i	− 0.129192i	0.997912	−	0.0645958i	$-0.0205758\pi$
$380$	0	0
$381$	− 5.09921i	− 0.261240i
$382$	0	0
$383$	25.4880	1.30238	0.651188	−	0.758916i	$-0.274271\pi$
$383$	25.4880	1.30238	0.651188	+	0.758916i	$0.274271\pi$
$384$	0	0
$385$	−7.55136	−0.384853
$386$	0	0
$387$	− 2.27744i	− 0.115769i
$388$	0	0
$389$	− 16.5532i	− 0.839281i	−0.907690	−	0.419641i	$-0.862156\pi$
$389$	− 16.5532i	− 0.839281i	0.907690	−	0.419641i	$-0.137844\pi$
$390$	0	0
$391$	10.2306	0.517383
$392$	0	0
$393$	2.99647	0.151152
$394$	0	0
$395$	− 5.20690i	− 0.261988i
$396$	0	0
$397$	12.7936i	0.642094i	0.947063	+	0.321047i	$0.104035\pi$
$397$	12.7936i	0.642094i	−0.947063	+	0.321047i	$0.895965\pi$
$398$	0	0
$399$	16.4853	0.825296
$400$	0	0
$401$	18.0853	0.903137	0.451568	−	0.892237i	$-0.350865\pi$
$401$	18.0853	0.903137	0.451568	+	0.892237i	$0.350865\pi$
$402$	0	0
$403$	− 0.0468544i	− 0.00233398i
$404$	0	0
$405$	− 0.473626i	− 0.0235347i
$406$	0	0
$407$	−21.6990	−1.07558
$408$	0	0
$409$	−25.2271	−1.24740	−0.623699	−	0.781665i	$-0.714371\pi$
$409$	−25.2271	−1.24740	−0.623699	+	0.781665i	$0.714371\pi$
$410$	0	0
$411$	− 3.37941i	− 0.166694i
$412$	0	0
$413$	− 25.7819i	− 1.26865i
$414$	0	0
$415$	7.12787	0.349894
$416$	0	0
$417$	−8.31724	−0.407297
$418$	0	0
$419$	10.2571i	0.501090i	0.968105	+	0.250545i	$0.0806098\pi$
$419$	10.2571i	0.501090i	−0.968105	+	0.250545i	$0.919390\pi$
$420$	0	0
$421$	3.38775i	0.165109i	0.996587	+	0.0825543i	$0.0263078\pi$
$421$	3.38775i	0.165109i	−0.996587	+	0.0825543i	$0.973692\pi$
$422$	0	0
$423$	2.82843	0.137523
$424$	0	0
$425$	−17.2739	−0.837908
$426$	0	0
$427$	− 17.5912i	− 0.851296i
$428$	0	0
$429$	0.293927i	0.0141909i
$430$	0	0
$431$	4.42454	0.213123	0.106561	−	0.994306i	$-0.466016\pi$
$431$	4.42454	0.213123	0.106561	+	0.994306i	$0.466016\pi$
$432$	0	0
$433$	−7.31371	−0.351474	−0.175737	−	0.984437i	$-0.556231\pi$
$433$	−7.31371	−0.351474	−0.175737	+	0.984437i	$0.556231\pi$
$434$	0	0
$435$	3.45844i	0.165820i
$436$	0	0
$437$	10.2306i	0.489395i
$438$	0	0
$439$	29.6533	1.41527	0.707637	−	0.706576i	$-0.249761\pi$
$439$	29.6533	1.41527	0.707637	+	0.706576i	$0.249761\pi$
$440$	0	0
$441$	−13.7721	−0.655817
$442$	0	0
$443$	14.5743i	0.692446i	0.938152	+	0.346223i	$0.112536\pi$
$443$	14.5743i	0.692446i	−0.938152	+	0.346223i	$0.887464\pi$
$444$	0	0
$445$	− 6.94019i	− 0.328997i
$446$	0	0
$447$	14.1305	0.668349
$448$	0	0
$449$	−6.48844	−0.306208	−0.153104	−	0.988210i	$-0.548927\pi$
$449$	−6.48844	−0.306208	−0.153104	+	0.988210i	$0.548927\pi$
$450$	0	0
$451$	32.4423i	1.52765i
$452$	0	0
$453$	9.97685i	0.468753i
$454$	0	0
$455$	−0.181370	−0.00850278
$456$	0	0
$457$	9.00353	0.421167	0.210584	−	0.977576i	$-0.432464\pi$
$457$	9.00353	0.421167	0.210584	+	0.977576i	$0.432464\pi$
$458$	0	0
$459$	− 3.61706i	− 0.168830i
$460$	0	0
$461$	− 20.6783i	− 0.963085i	−0.876423	−	0.481542i	$-0.840077\pi$
$461$	− 20.6783i	− 0.963085i	0.876423	−	0.481542i	$-0.159923\pi$
$462$	0	0
$463$	18.6435	0.866437	0.433219	−	0.901289i	$-0.357378\pi$
$463$	18.6435	0.866437	0.433219	+	0.901289i	$0.357378\pi$
$464$	0	0
$465$	0.264116	0.0122481
$466$	0	0
$467$	33.2535i	1.53879i	0.638773	+	0.769395i	$0.279442\pi$
$467$	33.2535i	1.53879i	−0.638773	+	0.769395i	$0.720558\pi$
$468$	0	0
$469$	− 24.3361i	− 1.12374i
$470$	0	0
$471$	−22.8562	−1.05316
$472$	0	0
$473$	−7.96703	−0.366325
$474$	0	0
$475$	− 17.2739i	− 0.792582i
$476$	0	0
$477$	− 0.697947i	− 0.0319568i
$478$	0	0
$479$	−1.08864	−0.0497412	−0.0248706	−	0.999691i	$-0.507917\pi$
$479$	−1.08864	−0.0497412	−0.0248706	+	0.999691i	$0.507917\pi$
$480$	0	0
$481$	−0.521173	−0.0237634
$482$	0	0
$483$	− 12.8910i	− 0.586560i
$484$	0	0
$485$	− 2.04476i	− 0.0928476i
$486$	0	0
$487$	−35.3298	−1.60095	−0.800473	−	0.599369i	$-0.795418\pi$
$487$	−35.3298	−1.60095	−0.800473	+	0.599369i	$0.795418\pi$
$488$	0	0
$489$	−10.6135	−0.479960
$490$	0	0
$491$	18.2306i	0.822735i	0.911470	+	0.411367i	$0.134949\pi$
$491$	18.2306i	0.822735i	−0.911470	+	0.411367i	$0.865051\pi$
$492$	0	0
$493$	26.4120i	1.18953i
$494$	0	0
$495$	−1.65685	−0.0744701
$496$	0	0
$497$	−41.5443	−1.86352
$498$	0	0
$499$	− 20.3361i	− 0.910368i	−0.890397	−	0.455184i	$-0.849573\pi$
$499$	− 20.3361i	− 0.910368i	0.890397	−	0.455184i	$-0.150427\pi$
$500$	0	0
$501$	5.83822i	0.260833i
$502$	0	0
$503$	30.2969	1.35087	0.675435	−	0.737420i	$-0.263956\pi$
$503$	30.2969	1.35087	0.675435	+	0.737420i	$0.263956\pi$
$504$	0	0
$505$	−0.303911	−0.0135239
$506$	0	0
$507$	− 12.9929i	− 0.577037i
$508$	0	0
$509$	14.9660i	0.663355i	0.943393	+	0.331677i	$0.107615\pi$
$509$	14.9660i	0.663355i	−0.943393	+	0.331677i	$0.892385\pi$
$510$	0	0
$511$	−2.46824	−0.109188
$512$	0	0
$513$	3.61706	0.159697
$514$	0	0
$515$	− 0.633169i	− 0.0279008i
$516$	0	0
$517$	− 9.89450i	− 0.435160i
$518$	0	0
$519$	5.12695	0.225048
$520$	0	0
$521$	−24.9049	−1.09110	−0.545551	−	0.838078i	$-0.683680\pi$
$521$	−24.9049	−1.09110	−0.545551	+	0.838078i	$0.683680\pi$
$522$	0	0
$523$	18.2445i	0.797775i	0.917000	+	0.398888i	$0.130604\pi$
$523$	18.2445i	0.797775i	−0.917000	+	0.398888i	$0.869396\pi$
$524$	0	0
$525$	21.7659i	0.949940i
$526$	0	0
$527$	2.01704	0.0878638
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	− 5.65685i	− 0.245487i
$532$	0	0
$533$	0.779208i	0.0337513i
$534$	0	0
$535$	−4.06074	−0.175561
$536$	0	0
$537$	−13.1286	−0.566542
$538$	0	0
$539$	48.1782i	2.07518i
$540$	0	0
$541$	25.8471i	1.11125i	0.831432	+	0.555627i	$0.187522\pi$
$541$	25.8471i	1.11125i	−0.831432	+	0.555627i	$0.812478\pi$
$542$	0	0
$543$	15.3181	0.657364
$544$	0	0
$545$	−3.82880	−0.164008
$546$	0	0
$547$	19.4249i	0.830549i	0.909696	+	0.415275i	$0.136315\pi$
$547$	19.4249i	0.830549i	−0.909696	+	0.415275i	$0.863685\pi$
$548$	0	0
$549$	− 3.85970i	− 0.164728i
$550$	0	0
$551$	−26.4120	−1.12519
$552$	0	0
$553$	−50.1055	−2.13070
$554$	0	0
$555$	− 2.93783i	− 0.124704i
$556$	0	0
$557$	38.9652i	1.65101i	0.564397	+	0.825504i	$0.309109\pi$
$557$	38.9652i	1.65101i	−0.564397	+	0.825504i	$0.690891\pi$
$558$	0	0
$559$	−0.191354	−0.00809342
$560$	0	0
$561$	−12.6533	−0.534224
$562$	0	0
$563$	28.1327i	1.18565i	0.805330	+	0.592826i	$0.201988\pi$
$563$	28.1327i	1.18565i	−0.805330	+	0.592826i	$0.798012\pi$
$564$	0	0
$565$	− 4.52377i	− 0.190316i
$566$	0	0
$567$	−4.55765	−0.191403
$568$	0	0
$569$	13.4849	0.565317	0.282658	−	0.959221i	$-0.408784\pi$
$569$	13.4849	0.565317	0.282658	+	0.959221i	$0.408784\pi$
$570$	0	0
$571$	− 20.9706i	− 0.877591i	−0.898587	−	0.438795i	$-0.855405\pi$
$571$	− 20.9706i	− 0.877591i	0.898587	−	0.438795i	$-0.144595\pi$
$572$	0	0
$573$	8.63001i	0.360524i
$574$	0	0
$575$	−13.5077	−0.563308
$576$	0	0
$577$	−11.6176	−0.483648	−0.241824	−	0.970320i	$-0.577746\pi$
$577$	−11.6176	−0.483648	−0.241824	+	0.970320i	$0.577746\pi$
$578$	0	0
$579$	− 11.4514i	− 0.475903i
$580$	0	0
$581$	− 68.5907i	− 2.84562i
$582$	0	0
$583$	−2.44158	−0.101120
$584$	0	0
$585$	−0.0397948	−0.00164531
$586$	0	0
$587$	24.0796i	0.993871i	0.867787	+	0.496936i	$0.165541\pi$
$587$	24.0796i	0.993871i	−0.867787	+	0.496936i	$0.834459\pi$
$588$	0	0
$589$	2.01704i	0.0831108i
$590$	0	0
$591$	10.5925	0.435715
$592$	0	0
$593$	−41.5372	−1.70573	−0.852865	−	0.522132i	$-0.825137\pi$
$593$	−41.5372	−1.70573	−0.852865	+	0.522132i	$0.825137\pi$
$594$	0	0
$595$	− 7.80785i	− 0.320091i
$596$	0	0
$597$	3.68000i	0.150612i
$598$	0	0
$599$	6.43160	0.262788	0.131394	−	0.991330i	$-0.458055\pi$
$599$	6.43160	0.262788	0.131394	+	0.991330i	$0.458055\pi$
$600$	0	0
$601$	3.45844	0.141073	0.0705364	−	0.997509i	$-0.477529\pi$
$601$	3.45844	0.141073	0.0705364	+	0.997509i	$0.477529\pi$
$602$	0	0
$603$	− 5.33962i	− 0.217446i
$604$	0	0
$605$	0.586182i	0.0238317i
$606$	0	0
$607$	−30.1019	−1.22180	−0.610900	−	0.791708i	$-0.709192\pi$
$607$	−30.1019	−1.22180	−0.610900	+	0.791708i	$0.709192\pi$
$608$	0	0
$609$	33.2802	1.34858
$610$	0	0
$611$	− 0.237649i	− 0.00961424i
$612$	0	0
$613$	3.54246i	0.143079i	0.997438	+	0.0715393i	$0.0227911\pi$
$613$	3.54246i	0.143079i	−0.997438	+	0.0715393i	$0.977209\pi$
$614$	0	0
$615$	−4.39236	−0.177117
$616$	0	0
$617$	22.9098	0.922315	0.461157	−	0.887318i	$-0.347434\pi$
$617$	22.9098	0.922315	0.461157	+	0.887318i	$0.347434\pi$
$618$	0	0
$619$	40.4612i	1.62627i	0.582074	+	0.813136i	$0.302242\pi$
$619$	40.4612i	1.62627i	−0.582074	+	0.813136i	$0.697758\pi$
$620$	0	0
$621$	− 2.82843i	− 0.113501i
$622$	0	0
$623$	−66.7847	−2.67567
$624$	0	0
$625$	21.6855	0.867420
$626$	0	0
$627$	− 12.6533i	− 0.505325i
$628$	0	0
$629$	− 22.4361i	− 0.894584i
$630$	0	0
$631$	11.1851	0.445270	0.222635	−	0.974902i	$-0.428534\pi$
$631$	11.1851	0.445270	0.222635	+	0.974902i	$0.428534\pi$
$632$	0	0
$633$	−14.3102	−0.568779
$634$	0	0
$635$	− 2.41512i	− 0.0958409i
$636$	0	0
$637$	1.15716i	0.0458482i
$638$	0	0
$639$	−9.11529	−0.360595
$640$	0	0
$641$	−6.69312	−0.264362	−0.132181	−	0.991226i	$-0.542198\pi$
$641$	−6.69312	−0.264362	−0.132181	+	0.991226i	$0.542198\pi$
$642$	0	0
$643$	− 25.3724i	− 1.00059i	−0.865856	−	0.500294i	$-0.833225\pi$
$643$	− 25.3724i	− 1.00059i	0.865856	−	0.500294i	$-0.166775\pi$
$644$	0	0
$645$	− 1.07866i	− 0.0424720i
$646$	0	0
$647$	−6.72999	−0.264583	−0.132292	−	0.991211i	$-0.542234\pi$
$647$	−6.72999	−0.264583	−0.132292	+	0.991211i	$0.542234\pi$
$648$	0	0
$649$	−19.7890	−0.776786
$650$	0	0
$651$	− 2.54156i	− 0.0996116i
$652$	0	0
$653$	− 37.0144i	− 1.44849i	−0.689545	−	0.724243i	$-0.742190\pi$
$653$	− 37.0144i	− 1.44849i	0.689545	−	0.724243i	$-0.257810\pi$
$654$	0	0
$655$	1.41921	0.0554529
$656$	0	0
$657$	−0.541560	−0.0211283
$658$	0	0
$659$	− 19.7624i	− 0.769832i	−0.922952	−	0.384916i	$-0.874230\pi$
$659$	− 19.7624i	− 0.769832i	0.922952	−	0.384916i	$-0.125770\pi$
$660$	0	0
$661$	− 16.8632i	− 0.655904i	−0.944694	−	0.327952i	$-0.893642\pi$
$661$	− 16.8632i	− 0.655904i	0.944694	−	0.327952i	$-0.106358\pi$
$662$	0	0
$663$	−0.303911	−0.0118029
$664$	0	0
$665$	7.80785	0.302776
$666$	0	0
$667$	20.6533i	0.799700i
$668$	0	0
$669$	4.86156i	0.187959i
$670$	0	0
$671$	−13.5021	−0.521244
$672$	0	0
$673$	−37.3066	−1.43807	−0.719033	−	0.694976i	$-0.755415\pi$
$673$	−37.3066	−1.43807	−0.719033	+	0.694976i	$0.755415\pi$
$674$	0	0
$675$	4.77568i	0.183816i
$676$	0	0
$677$	0.632805i	0.0243207i	0.999926	+	0.0121603i	$0.00387085\pi$
$677$	0.632805i	0.0243207i	−0.999926	+	0.0121603i	$0.996129\pi$
$678$	0	0
$679$	−19.6764	−0.755113
$680$	0	0
$681$	15.0496	0.576702
$682$	0	0
$683$	6.04606i	0.231346i	0.993287	+	0.115673i	$0.0369025\pi$
$683$	6.04606i	0.231346i	−0.993287	+	0.115673i	$0.963098\pi$
$684$	0	0
$685$	− 1.60058i	− 0.0611549i
$686$	0	0
$687$	−28.5264	−1.08835
$688$	0	0
$689$	−0.0586426	−0.00223410
$690$	0	0
$691$	− 28.3955i	− 1.08021i	−0.841596	−	0.540107i	$-0.818384\pi$
$691$	− 28.3955i	− 1.08021i	0.841596	−	0.540107i	$-0.181616\pi$
$692$	0	0
$693$	15.9437i	0.605652i
$694$	0	0
$695$	−3.93926	−0.149425
$696$	0	0
$697$	−33.5443	−1.27058
$698$	0	0
$699$	13.5702i	0.513272i
$700$	0	0
$701$	− 14.7738i	− 0.558000i	−0.960291	−	0.279000i	$-0.909997\pi$
$701$	− 14.7738i	− 0.558000i	0.960291	−	0.279000i	$-0.0900029\pi$
$702$	0	0
$703$	22.4361	0.846192
$704$	0	0
$705$	1.33962	0.0504529
$706$	0	0
$707$	2.92450i	0.109987i
$708$	0	0
$709$	22.7569i	0.854655i	0.904097	+	0.427327i	$0.140545\pi$
$709$	22.7569i	0.854655i	−0.904097	+	0.427327i	$0.859455\pi$
$710$	0	0
$711$	−10.9937	−0.412296
$712$	0	0
$713$	1.57726	0.0590690
$714$	0	0
$715$	0.139211i	0.00520621i
$716$	0	0
$717$	29.3629i	1.09658i
$718$	0	0
$719$	30.9957	1.15594	0.577972	−	0.816057i	$-0.303844\pi$
$719$	30.9957	1.15594	0.577972	+	0.816057i	$0.303844\pi$
$720$	0	0
$721$	−6.09292	−0.226912
$722$	0	0
$723$	24.0063i	0.892803i
$724$	0	0
$725$	− 34.8723i	− 1.29512i
$726$	0	0
$727$	−41.1117	−1.52475	−0.762375	−	0.647135i	$-0.775967\pi$
$727$	−41.1117	−1.52475	−0.762375	+	0.647135i	$0.775967\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 8.23765i	− 0.304680i
$732$	0	0
$733$	0.206562i	0.00762954i	0.999993	+	0.00381477i	$0.00121428\pi$
$733$	0.206562i	0.00762954i	−0.999993	+	0.00381477i	$0.998786\pi$
$734$	0	0
$735$	−6.52284	−0.240599
$736$	0	0
$737$	−18.6792	−0.688058
$738$	0	0
$739$	− 2.13215i	− 0.0784325i	−0.999231	−	0.0392162i	$-0.987514\pi$
$739$	− 2.13215i	− 0.0784325i	0.999231	−	0.0392162i	$-0.0124861\pi$
$740$	0	0
$741$	− 0.303911i	− 0.0111644i
$742$	0	0
$743$	40.5175	1.48644	0.743221	−	0.669046i	$-0.233297\pi$
$743$	40.5175	1.48644	0.743221	+	0.669046i	$0.233297\pi$
$744$	0	0
$745$	6.69256	0.245196
$746$	0	0
$747$	− 15.0496i	− 0.550636i
$748$	0	0
$749$	39.0761i	1.42781i
$750$	0	0
$751$	12.5843	0.459208	0.229604	−	0.973284i	$-0.426257\pi$
$751$	12.5843	0.459208	0.229604	+	0.973284i	$0.426257\pi$
$752$	0	0
$753$	22.2837	0.812064
$754$	0	0
$755$	4.72529i	0.171971i
$756$	0	0
$757$	10.6052i	0.385452i	0.981253	+	0.192726i	$0.0617329\pi$
$757$	10.6052i	0.385452i	−0.981253	+	0.192726i	$0.938267\pi$
$758$	0	0
$759$	−9.89450	−0.359148
$760$	0	0
$761$	−42.8182	−1.55216	−0.776079	−	0.630635i	$-0.782794\pi$
$761$	−42.8182	−1.55216	−0.776079	+	0.630635i	$0.782794\pi$
$762$	0	0
$763$	36.8441i	1.33385i
$764$	0	0
$765$	− 1.71313i	− 0.0619384i
$766$	0	0
$767$	−0.475298	−0.0171620
$768$	0	0
$769$	12.7455	0.459614	0.229807	−	0.973236i	$-0.426190\pi$
$769$	12.7455	0.459614	0.229807	+	0.973236i	$0.426190\pi$
$770$	0	0
$771$	− 8.66038i	− 0.311896i
$772$	0	0
$773$	− 32.3522i	− 1.16363i	−0.813322	−	0.581814i	$-0.802343\pi$
$773$	− 32.3522i	− 1.16363i	0.813322	−	0.581814i	$-0.197657\pi$
$774$	0	0
$775$	−2.66314	−0.0956630
$776$	0	0
$777$	−28.2704	−1.01419
$778$	0	0
$779$	− 33.5443i	− 1.20185i
$780$	0	0
$781$	31.8874i	1.14102i
$782$	0	0
$783$	7.30205	0.260954
$784$	0	0
$785$	−10.8253	−0.386370
$786$	0	0
$787$	− 7.36056i	− 0.262376i	−0.991358	−	0.131188i	$-0.958121\pi$
$787$	− 7.36056i	− 0.262376i	0.991358	−	0.131188i	$-0.0418791\pi$
$788$	0	0
$789$	− 13.3208i	− 0.474232i
$790$	0	0
$791$	−43.5317	−1.54781
$792$	0	0
$793$	−0.324298	−0.0115162
$794$	0	0
$795$	− 0.330566i	− 0.0117240i
$796$	0	0
$797$	24.0627i	0.852344i	0.904642	+	0.426172i	$0.140138\pi$
$797$	24.0627i	0.852344i	−0.904642	+	0.426172i	$0.859862\pi$
$798$	0	0
$799$	10.2306	0.361932
$800$	0	0
$801$	−14.6533	−0.517750
$802$	0	0
$803$	1.89450i	0.0668556i
$804$	0	0
$805$	− 6.10550i	− 0.215190i
$806$	0	0
$807$	16.5058	0.581032
$808$	0	0
$809$	7.83586	0.275494	0.137747	−	0.990467i	$-0.456014\pi$
$809$	7.83586	0.275494	0.137747	+	0.990467i	$0.456014\pi$
$810$	0	0
$811$	− 45.7351i	− 1.60598i	−0.595995	−	0.802988i	$-0.703242\pi$
$811$	− 45.7351i	− 1.60598i	0.595995	−	0.802988i	$-0.296758\pi$
$812$	0	0
$813$	− 21.9769i	− 0.770762i
$814$	0	0
$815$	−5.02684	−0.176083
$816$	0	0
$817$	8.23765	0.288199
$818$	0	0
$819$	0.382941i	0.0133810i
$820$	0	0
$821$	− 27.3709i	− 0.955250i	−0.878564	−	0.477625i	$-0.841498\pi$
$821$	− 27.3709i	− 0.955250i	0.878564	−	0.477625i	$-0.158502\pi$
$822$	0	0
$823$	28.8560	1.00586	0.502929	−	0.864328i	$-0.332256\pi$
$823$	28.8560	1.00586	0.502929	+	0.864328i	$0.332256\pi$
$824$	0	0
$825$	16.7064	0.581644
$826$	0	0
$827$	− 14.4227i	− 0.501528i	−0.968048	−	0.250764i	$-0.919318\pi$
$827$	− 14.4227i	− 0.501528i	0.968048	−	0.250764i	$-0.0806818\pi$
$828$	0	0
$829$	21.7497i	0.755400i	0.925928	+	0.377700i	$0.123285\pi$
$829$	21.7497i	0.755400i	−0.925928	+	0.377700i	$0.876715\pi$
$830$	0	0
$831$	15.4862	0.537210
$832$	0	0
$833$	−49.8147	−1.72598
$834$	0	0
$835$	2.76513i	0.0956914i
$836$	0	0
$837$	− 0.557647i	− 0.0192751i
$838$	0	0
$839$	44.4557	1.53478	0.767390	−	0.641181i	$-0.221555\pi$
$839$	44.4557	1.53478	0.767390	+	0.641181i	$0.221555\pi$
$840$	0	0
$841$	−24.3200	−0.838620
$842$	0	0
$843$	22.8910i	0.788407i
$844$	0	0
$845$	− 6.15379i	− 0.211697i
$846$	0	0
$847$	5.64077	0.193819
$848$	0	0
$849$	6.34315	0.217696
$850$	0	0
$851$	− 17.5443i	− 0.601411i
$852$	0	0
$853$	16.5648i	0.567169i	0.958947	+	0.283585i	$0.0915237\pi$
$853$	16.5648i	0.567169i	−0.958947	+	0.283585i	$0.908476\pi$
$854$	0	0
$855$	1.71313	0.0585879
$856$	0	0
$857$	19.0888	0.652062	0.326031	−	0.945359i	$-0.394289\pi$
$857$	19.0888	0.652062	0.326031	+	0.945359i	$0.394289\pi$
$858$	0	0
$859$	− 53.9272i	− 1.83997i	−0.391949	−	0.919987i	$-0.628199\pi$
$859$	− 53.9272i	− 1.83997i	0.391949	−	0.919987i	$-0.371801\pi$
$860$	0	0
$861$	42.2672i	1.44046i
$862$	0	0
$863$	3.64533	0.124089	0.0620443	−	0.998073i	$-0.480238\pi$
$863$	3.64533	0.124089	0.0620443	+	0.998073i	$0.480238\pi$
$864$	0	0
$865$	2.42826	0.0825632
$866$	0	0
$867$	3.91688i	0.133024i
$868$	0	0
$869$	38.4586i	1.30462i
$870$	0	0
$871$	−0.448643	−0.0152017
$872$	0	0
$873$	−4.31724	−0.146116
$874$	0	0
$875$	21.1020i	0.713377i
$876$	0	0
$877$	− 56.6481i	− 1.91287i	−0.291945	−	0.956435i	$-0.594302\pi$
$877$	− 56.6481i	− 1.91287i	0.291945	−	0.956435i	$-0.405698\pi$
$878$	0	0
$879$	30.5783	1.03138
$880$	0	0
$881$	20.0118	0.674214	0.337107	−	0.941466i	$-0.390552\pi$
$881$	20.0118	0.674214	0.337107	+	0.941466i	$0.390552\pi$
$882$	0	0
$883$	15.0292i	0.505773i	0.967496	+	0.252887i	$0.0813799\pi$
$883$	15.0292i	0.505773i	−0.967496	+	0.252887i	$0.918620\pi$
$884$	0	0
$885$	− 2.67923i	− 0.0900614i
$886$	0	0
$887$	26.1180	0.876958	0.438479	−	0.898742i	$-0.355517\pi$
$887$	26.1180	0.876958	0.438479	+	0.898742i	$0.355517\pi$
$888$	0	0
$889$	−23.2404	−0.779458
$890$	0	0
$891$	3.49824i	0.117195i
$892$	0	0
$893$	10.2306i	0.342354i
$894$	0	0
$895$	−6.21805	−0.207847
$896$	0	0
$897$	−0.237649	−0.00793486
$898$	0	0
$899$	4.07197i	0.135808i
$900$	0	0
$901$	− 2.52452i	− 0.0841038i
$902$	0	0
$903$	−10.3798	−0.345418
$904$	0	0
$905$	7.25507	0.241167
$906$	0	0
$907$	− 51.2480i	− 1.70166i	−0.525439	−	0.850831i	$-0.676099\pi$
$907$	− 51.2480i	− 1.70166i	0.525439	−	0.850831i	$-0.323901\pi$
$908$	0	0
$909$	0.641669i	0.0212828i
$910$	0	0
$911$	21.0535	0.697533	0.348767	−	0.937210i	$-0.386601\pi$
$911$	21.0535	0.697533	0.348767	+	0.937210i	$0.386601\pi$
$912$	0	0
$913$	−52.6470	−1.74236
$914$	0	0
$915$	− 1.82805i	− 0.0604336i
$916$	0	0
$917$	− 13.6569i	− 0.450989i
$918$	0	0
$919$	−17.8839	−0.589937	−0.294968	−	0.955507i	$-0.595309\pi$
$919$	−17.8839	−0.589937	−0.294968	+	0.955507i	$0.595309\pi$
$920$	0	0
$921$	17.1286	0.564407
$922$	0	0
$923$	0.765881i	0.0252093i
$924$	0	0
$925$	29.6228i	0.973992i
$926$	0	0
$927$	−1.33686	−0.0439081
$928$	0	0
$929$	10.2774	0.337192	0.168596	−	0.985685i	$-0.446077\pi$
$929$	10.2774	0.337192	0.168596	+	0.985685i	$0.446077\pi$
$930$	0	0
$931$	− 49.8147i	− 1.63261i
$932$	0	0
$933$	− 26.8651i	− 0.879523i
$934$	0	0
$935$	−5.99294	−0.195990
$936$	0	0
$937$	13.5780	0.443574	0.221787	−	0.975095i	$-0.428811\pi$
$937$	13.5780	0.443574	0.221787	+	0.975095i	$0.428811\pi$
$938$	0	0
$939$	− 19.6890i	− 0.642527i
$940$	0	0
$941$	5.59890i	0.182519i	0.995827	+	0.0912595i	$0.0290893\pi$
$941$	5.59890i	0.182519i	−0.995827	+	0.0912595i	$0.970911\pi$
$942$	0	0
$943$	−26.2306	−0.854186
$944$	0	0
$945$	−2.15862	−0.0702199
$946$	0	0
$947$	46.9106i	1.52439i	0.647348	+	0.762194i	$0.275878\pi$
$947$	46.9106i	1.52439i	−0.647348	+	0.762194i	$0.724122\pi$
$948$	0	0
$949$	0.0455027i	0.00147708i
$950$	0	0
$951$	30.1860	0.978847
$952$	0	0
$953$	−5.59115	−0.181115	−0.0905576	−	0.995891i	$-0.528865\pi$
$953$	−5.59115	−0.181115	−0.0905576	+	0.995891i	$0.528865\pi$
$954$	0	0
$955$	4.08740i	0.132265i
$956$	0	0
$957$	− 25.5443i	− 0.825730i
$958$	0	0
$959$	−15.4022	−0.497362
$960$	0	0
$961$	−30.6890	−0.989969
$962$	0	0
$963$	8.57373i	0.276285i
$964$	0	0
$965$	− 5.42367i	− 0.174594i
$966$	0	0
$967$	30.7561	0.989048	0.494524	−	0.869164i	$-0.335342\pi$
$967$	30.7561	0.989048	0.494524	+	0.869164i	$0.335342\pi$
$968$	0	0
$969$	13.0831	0.420290
$970$	0	0
$971$	− 11.3668i	− 0.364779i	−0.983226	−	0.182389i	$-0.941617\pi$
$971$	− 11.3668i	− 0.364779i	0.983226	−	0.182389i	$-0.0583832\pi$
$972$	0	0
$973$	37.9070i	1.21524i
$974$	0	0
$975$	0.401260	0.0128506
$976$	0	0
$977$	−22.8323	−0.730471	−0.365235	−	0.930915i	$-0.619012\pi$
$977$	−22.8323	−0.730471	−0.365235	+	0.930915i	$0.619012\pi$
$978$	0	0
$979$	51.2608i	1.63830i
$980$	0	0
$981$	8.08402i	0.258103i
$982$	0	0
$983$	46.3557	1.47852	0.739258	−	0.673422i	$-0.235176\pi$
$983$	46.3557	1.47852	0.739258	+	0.673422i	$0.235176\pi$
$984$	0	0
$985$	5.01686	0.159850
$986$	0	0
$987$	− 12.8910i	− 0.410324i
$988$	0	0
$989$	− 6.44158i	− 0.204830i
$990$	0	0
$991$	3.43683	0.109175	0.0545873	−	0.998509i	$-0.482616\pi$
$991$	3.43683	0.109175	0.0545873	+	0.998509i	$0.482616\pi$
$992$	0	0
$993$	−20.7784	−0.659383
$994$	0	0
$995$	1.74294i	0.0552550i
$996$	0	0
$997$	− 31.0320i	− 0.982794i	−0.870936	−	0.491397i	$-0.836486\pi$
$997$	− 31.0320i	− 0.982794i	0.870936	−	0.491397i	$-0.163514\pi$
$998$	0	0
$999$	−6.20285	−0.196249

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.d.f.1537.2		8
4.3	odd	2		3072.2.d.i.1537.6		8
8.3	odd	2		3072.2.d.i.1537.3		8
8.5	even	2	inner	3072.2.d.f.1537.7		8
16.3	odd	4		3072.2.a.n.1.2		4
16.5	even	4		3072.2.a.i.1.3		4
16.11	odd	4		3072.2.a.o.1.3		4
16.13	even	4		3072.2.a.t.1.2		4
32.3	odd	8		384.2.j.a.97.2		8
32.5	even	8		384.2.j.b.289.4		8
32.11	odd	8		192.2.j.a.145.3		8
32.13	even	8		48.2.j.a.37.4	yes	8
32.19	odd	8		192.2.j.a.49.3		8
32.21	even	8		48.2.j.a.13.4	✓	8
32.27	odd	8		384.2.j.a.289.2		8
32.29	even	8		384.2.j.b.97.4		8
48.5	odd	4		9216.2.a.bo.1.2		4
48.11	even	4		9216.2.a.bn.1.2		4
48.29	odd	4		9216.2.a.y.1.3		4
48.35	even	4		9216.2.a.x.1.3		4
96.5	odd	8		1152.2.k.c.289.2		8
96.11	even	8		576.2.k.b.145.3		8
96.29	odd	8		1152.2.k.c.865.2		8
96.35	even	8		1152.2.k.f.865.2		8
96.53	odd	8		144.2.k.b.109.1		8
96.59	even	8		1152.2.k.f.289.2		8
96.77	odd	8		144.2.k.b.37.1		8
96.83	even	8		576.2.k.b.433.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.2.j.a.13.4	✓	8	32.21	even	8
48.2.j.a.37.4	yes	8	32.13	even	8
144.2.k.b.37.1		8	96.77	odd	8
144.2.k.b.109.1		8	96.53	odd	8
192.2.j.a.49.3		8	32.19	odd	8
192.2.j.a.145.3		8	32.11	odd	8
384.2.j.a.97.2		8	32.3	odd	8
384.2.j.a.289.2		8	32.27	odd	8
384.2.j.b.97.4		8	32.29	even	8
384.2.j.b.289.4		8	32.5	even	8
576.2.k.b.145.3		8	96.11	even	8
576.2.k.b.433.3		8	96.83	even	8
1152.2.k.c.289.2		8	96.5	odd	8
1152.2.k.c.865.2		8	96.29	odd	8
1152.2.k.f.289.2		8	96.59	even	8
1152.2.k.f.865.2		8	96.35	even	8
3072.2.a.i.1.3		4	16.5	even	4
3072.2.a.n.1.2		4	16.3	odd	4
3072.2.a.o.1.3		4	16.11	odd	4
3072.2.a.t.1.2		4	16.13	even	4
3072.2.d.f.1537.2		8	1.1	even	1	trivial
3072.2.d.f.1537.7		8	8.5	even	2	inner
3072.2.d.i.1537.3		8	8.3	odd	2
3072.2.d.i.1537.6		8	4.3	odd	2
9216.2.a.x.1.3		4	48.35	even	4
9216.2.a.y.1.3		4	48.29	odd	4
9216.2.a.bn.1.2		4	48.11	even	4
9216.2.a.bo.1.2		4	48.5	odd	4

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:	\( N \)	\(=\)	\( 3072 = 2^{10} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3072.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -0.473626i q^{5} -4.55765 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -0.473626i q^{5} -4.55765 q^{7} -1.00000 q^{9} +3.49824i q^{11} +0.0840215i q^{13} -0.473626 q^{15} -3.61706 q^{17} -3.61706i q^{19} +4.55765i q^{21} -2.82843 q^{23} +4.77568 q^{25} +1.00000i q^{27} -7.30205i q^{29} -0.557647 q^{31} +3.49824 q^{33} +2.15862i q^{35} +6.20285i q^{37} +0.0840215 q^{39} +9.27391 q^{41} +2.27744i q^{43} +0.473626i q^{45} -2.82843 q^{47} +13.7721 q^{49} +3.61706i q^{51} +0.697947i q^{53} +1.65685 q^{55} -3.61706 q^{57} +5.65685i q^{59} +3.85970i q^{61} +4.55765 q^{63} +0.0397948 q^{65} +5.33962i q^{67} +2.82843i q^{69} +9.11529 q^{71} +0.541560 q^{73} -4.77568i q^{75} -15.9437i q^{77} +10.9937 q^{79} +1.00000 q^{81} +15.0496i q^{83} +1.71313i q^{85} -7.30205 q^{87} +14.6533 q^{89} -0.382941i q^{91} +0.557647i q^{93} -1.71313 q^{95} +4.31724 q^{97} -3.49824i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7} - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^7 - 8 * q^9	\( 8 q - 8 q^{7} - 8 q^{9} + 8 q^{15} - 8 q^{25} + 24 q^{31} - 16 q^{39} + 8 q^{49} - 32 q^{55} + 8 q^{63} - 16 q^{65} + 16 q^{71} + 16 q^{73} + 24 q^{79} + 8 q^{81} - 24 q^{87} + 16 q^{89} - 48 q^{95}+O(q^{100}) \) 8 * q - 8 * q^7 - 8 * q^9 + 8 * q^15 - 8 * q^25 + 24 * q^31 - 16 * q^39 + 8 * q^49 - 32 * q^55 + 8 * q^63 - 16 * q^65 + 16 * q^71 + 16 * q^73 + 24 * q^79 + 8 * q^81 - 24 * q^87 + 16 * q^89 - 48 * q^95

Introduction

L-functions

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