LMFDB - Embedded newform 4600.2.e.w.4049.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (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:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 18x^{8} + 117x^{6} + 333x^{4} + 396x^{2} + 144 $ x^10 + 18x^8 + 117x^6 + 333x^4 + 396x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.4
Root			$-1.36629i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.w.4049.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.36629i q^{3} -3.28093i q^{7} +1.13327 q^{9} +O(q^{10})$	$q-1.36629i q^{3} -3.28093i q^{7} +1.13327 q^{9} +3.49709 q^{11} -3.41420i q^{13} -7.46023i q^{17} -3.49709 q^{19} -4.48269 q^{21} -1.00000i q^{23} -5.64722i q^{27} +3.46268 q^{29} +2.01105 q^{31} -4.77803i q^{33} +0.511497i q^{37} -4.66477 q^{39} -7.07954 q^{41} +2.76452i q^{43} -0.889198i q^{47} -3.76452 q^{49} -10.1928 q^{51} +14.2383i q^{53} +4.77803i q^{57} +4.71325 q^{59} +13.4769 q^{61} -3.71817i q^{63} -2.30025i q^{67} -1.36629 q^{69} -10.6214 q^{71} -3.70765i q^{73} -11.4737i q^{77} +7.97978 q^{79} -4.31592 q^{81} -9.42336i q^{83} -4.73101i q^{87} -0.801390 q^{89} -11.2018 q^{91} -2.74766i q^{93} +11.7722i q^{97} +3.96313 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6 q^{9}+O(q^{10}) $ 10 * q - 6 * q^9	$ 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99}+O(q^{100}) $ 10 * q - 6 * q^9 - 24 * q^29 - 36 * q^31 - 18 * q^39 - 12 * q^41 - 30 * q^49 - 12 * q^51 + 2 * q^59 + 20 * q^61 + 16 * q^71 - 54 * q^81 - 28 * q^89 - 92 * q^91 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 1.36629i	− 0.788825i	−0.918933	−	0.394413i	$-0.870948\pi$
$3$	− 1.36629i	− 0.788825i	0.918933	−	0.394413i	$-0.129052\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.28093i	− 1.24008i	−0.784572	−	0.620038i	$-0.787117\pi$
$7$	− 3.28093i	− 1.24008i	0.784572	−	0.620038i	$-0.212883\pi$
$8$	0	0
$9$	1.13327	0.377755
$10$	0	0
$11$	3.49709	1.05441	0.527207	−	0.849737i	$-0.323239\pi$
$11$	3.49709	1.05441	0.527207	+	0.849737i	$0.323239\pi$
$12$	0	0
$13$	− 3.41420i	− 0.946928i	−0.880813	−	0.473464i	$-0.843003\pi$
$13$	− 3.41420i	− 0.946928i	0.880813	−	0.473464i	$-0.156997\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.46023i	− 1.80937i	−0.426080	−	0.904685i	$-0.640106\pi$
$17$	− 7.46023i	− 1.80937i	0.426080	−	0.904685i	$-0.359894\pi$
$18$	0	0
$19$	−3.49709	−0.802288	−0.401144	−	0.916015i	$-0.631387\pi$
$19$	−3.49709	−0.802288	−0.401144	+	0.916015i	$0.631387\pi$
$20$	0	0
$21$	−4.48269	−0.978203
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.64722i	− 1.08681i
$28$	0	0
$29$	3.46268	0.643004	0.321502	−	0.946909i	$-0.395812\pi$
$29$	3.46268	0.643004	0.321502	+	0.946909i	$0.395812\pi$
$30$	0	0
$31$	2.01105	0.361195	0.180597	−	0.983557i	$-0.442197\pi$
$31$	2.01105	0.361195	0.180597	+	0.983557i	$0.442197\pi$
$32$	0	0
$33$	− 4.77803i	− 0.831748i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.511497i	0.0840896i	0.999116	+	0.0420448i	$0.0133872\pi$
$37$	0.511497i	0.0840896i	−0.999116	+	0.0420448i	$0.986613\pi$
$38$	0	0
$39$	−4.66477	−0.746961
$40$	0	0
$41$	−7.07954	−1.10564	−0.552819	−	0.833301i	$-0.686448\pi$
$41$	−7.07954	−1.10564	−0.552819	+	0.833301i	$0.686448\pi$
$42$	0	0
$43$	2.76452i	0.421586i	0.977531	+	0.210793i	$0.0676046\pi$
$43$	2.76452i	0.421586i	−0.977531	+	0.210793i	$0.932395\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 0.889198i	− 0.129703i	−0.997895	−	0.0648514i	$-0.979343\pi$
$47$	− 0.889198i	− 0.129703i	0.997895	−	0.0648514i	$-0.0206573\pi$
$48$	0	0
$49$	−3.76452	−0.537789
$50$	0	0
$51$	−10.1928	−1.42728
$52$	0	0
$53$	14.2383i	1.95577i	0.209132	+	0.977887i	$0.432936\pi$
$53$	14.2383i	1.95577i	−0.209132	+	0.977887i	$0.567064\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.77803i	0.632865i
$58$	0	0
$59$	4.71325	0.613613	0.306807	−	0.951772i	$-0.400739\pi$
$59$	4.71325	0.613613	0.306807	+	0.951772i	$0.400739\pi$
$60$	0	0
$61$	13.4769	1.72554	0.862769	−	0.505599i	$-0.168728\pi$
$61$	13.4769	1.72554	0.862769	+	0.505599i	$0.168728\pi$
$62$	0	0
$63$	− 3.71817i	− 0.468445i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2.30025i	− 0.281020i	−0.990079	−	0.140510i	$-0.955126\pi$
$67$	− 2.30025i	− 0.281020i	0.990079	−	0.140510i	$-0.0448743\pi$
$68$	0	0
$69$	−1.36629	−0.164481
$70$	0	0
$71$	−10.6214	−1.26053	−0.630264	−	0.776381i	$-0.717053\pi$
$71$	−10.6214	−1.26053	−0.630264	+	0.776381i	$0.717053\pi$
$72$	0	0
$73$	− 3.70765i	− 0.433948i	−0.976177	−	0.216974i	$-0.930381\pi$
$73$	− 3.70765i	− 0.433948i	0.976177	−	0.216974i	$-0.0696186\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 11.4737i	− 1.30755i
$78$	0	0
$79$	7.97978	0.897796	0.448898	−	0.893583i	$-0.351817\pi$
$79$	7.97978	0.897796	0.448898	+	0.893583i	$0.351817\pi$
$80$	0	0
$81$	−4.31592	−0.479546
$82$	0	0
$83$	− 9.42336i	− 1.03435i	−0.855880	−	0.517174i	$-0.826984\pi$
$83$	− 9.42336i	− 1.03435i	0.855880	−	0.517174i	$-0.173016\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 4.73101i	− 0.507218i
$88$	0	0
$89$	−0.801390	−0.0849471	−0.0424736	−	0.999098i	$-0.513524\pi$
$89$	−0.801390	−0.0849471	−0.0424736	+	0.999098i	$0.513524\pi$
$90$	0	0
$91$	−11.2018	−1.17426
$92$	0	0
$93$	− 2.74766i	− 0.284919i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.7722i	1.19529i	0.801762	+	0.597644i	$0.203896\pi$
$97$	11.7722i	1.19529i	−0.801762	+	0.597644i	$0.796104\pi$
$98$	0	0
$99$	3.96313	0.398310
$100$	0	0
$101$	11.1559	1.11006	0.555028	−	0.831831i	$-0.312707\pi$
$101$	11.1559	1.11006	0.555028	+	0.831831i	$0.312707\pi$
$102$	0	0
$103$	− 1.28585i	− 0.126698i	−0.997991	−	0.0633491i	$-0.979822\pi$
$103$	− 1.28585i	− 0.126698i	0.997991	−	0.0633491i	$-0.0201782\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 13.8975i	− 1.34352i	−0.740770	−	0.671759i	$-0.765539\pi$
$107$	− 13.8975i	− 1.34352i	0.740770	−	0.671759i	$-0.234461\pi$
$108$	0	0
$109$	3.77713	0.361783	0.180892	−	0.983503i	$-0.442102\pi$
$109$	3.77713	0.361783	0.180892	+	0.983503i	$0.442102\pi$
$110$	0	0
$111$	0.698851	0.0663320
$112$	0	0
$113$	19.3111i	1.81663i	0.418282	+	0.908317i	$0.362632\pi$
$113$	19.3111i	1.81663i	−0.418282	+	0.908317i	$0.637368\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.86919i	− 0.357707i
$118$	0	0
$119$	−24.4765	−2.24376
$120$	0	0
$121$	1.22966	0.111788
$122$	0	0
$123$	9.67267i	0.872155i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 13.7612i	− 1.22111i	−0.791975	−	0.610553i	$-0.790947\pi$
$127$	− 13.7612i	− 1.22111i	0.791975	−	0.610553i	$-0.209053\pi$
$128$	0	0
$129$	3.77713	0.332558
$130$	0	0
$131$	−7.41665	−0.647996	−0.323998	−	0.946058i	$-0.605027\pi$
$131$	−7.41665	−0.647996	−0.323998	+	0.946058i	$0.605027\pi$
$132$	0	0
$133$	11.4737i	0.994899i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.5502i	1.75573i	0.478912	+	0.877863i	$0.341031\pi$
$137$	20.5502i	1.75573i	−0.478912	+	0.877863i	$0.658969\pi$
$138$	0	0
$139$	5.49652	0.466209	0.233104	−	0.972452i	$-0.425112\pi$
$139$	5.49652	0.466209	0.233104	+	0.972452i	$0.425112\pi$
$140$	0	0
$141$	−1.21490	−0.102313
$142$	0	0
$143$	− 11.9398i	− 0.998454i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.14341i	0.424221i
$148$	0	0
$149$	−4.80139	−0.393345	−0.196673	−	0.980469i	$-0.563014\pi$
$149$	−4.80139	−0.393345	−0.196673	+	0.980469i	$0.563014\pi$
$150$	0	0
$151$	−3.26862	−0.265997	−0.132998	−	0.991116i	$-0.542461\pi$
$151$	−3.26862	−0.265997	−0.132998	+	0.991116i	$0.542461\pi$
$152$	0	0
$153$	− 8.45441i	− 0.683499i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 21.8392i	− 1.74296i	−0.490430	−	0.871480i	$-0.663160\pi$
$157$	− 21.8392i	− 1.74296i	0.490430	−	0.871480i	$-0.336840\pi$
$158$	0	0
$159$	19.4535	1.54276
$160$	0	0
$161$	−3.28093	−0.258574
$162$	0	0
$163$	23.2435i	1.82057i	0.413982	+	0.910285i	$0.364138\pi$
$163$	23.2435i	1.82057i	−0.413982	+	0.910285i	$0.635862\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.13179i	0.319728i	0.987139	+	0.159864i	$0.0511055\pi$
$167$	4.13179i	0.319728i	−0.987139	+	0.159864i	$0.948894\pi$
$168$	0	0
$169$	1.34325	0.103327
$170$	0	0
$171$	−3.96313	−0.303068
$172$	0	0
$173$	5.88548i	0.447465i	0.974651	+	0.223732i	$0.0718241\pi$
$173$	5.88548i	0.447465i	−0.974651	+	0.223732i	$0.928176\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 6.43965i	− 0.484034i
$178$	0	0
$179$	6.85789	0.512583	0.256292	−	0.966600i	$-0.417499\pi$
$179$	6.85789	0.512583	0.256292	+	0.966600i	$0.417499\pi$
$180$	0	0
$181$	−2.29042	−0.170246	−0.0851229	−	0.996370i	$-0.527128\pi$
$181$	−2.29042	−0.170246	−0.0851229	+	0.996370i	$0.527128\pi$
$182$	0	0
$183$	− 18.4133i	− 1.36115i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 26.0891i	− 1.90782i
$188$	0	0
$189$	−18.5281	−1.34772
$190$	0	0
$191$	−9.58748	−0.693725	−0.346863	−	0.937916i	$-0.612753\pi$
$191$	−9.58748	−0.693725	−0.346863	+	0.937916i	$0.612753\pi$
$192$	0	0
$193$	13.9410i	1.00350i	0.865013	+	0.501749i	$0.167310\pi$
$193$	13.9410i	1.00350i	−0.865013	+	0.501749i	$0.832690\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 15.1011i	− 1.07591i	−0.842974	−	0.537954i	$-0.819197\pi$
$197$	− 15.1011i	− 1.07591i	0.842974	−	0.537954i	$-0.180803\pi$
$198$	0	0
$199$	−14.5128	−1.02879	−0.514394	−	0.857554i	$-0.671983\pi$
$199$	−14.5128	−1.02879	−0.514394	+	0.857554i	$0.671983\pi$
$200$	0	0
$201$	−3.14280	−0.221676
$202$	0	0
$203$	− 11.3608i	− 0.797374i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.13327i	− 0.0787674i
$208$	0	0
$209$	−12.2297	−0.845944
$210$	0	0
$211$	−23.4846	−1.61674	−0.808372	−	0.588672i	$-0.799651\pi$
$211$	−23.4846	−1.61674	−0.808372	+	0.588672i	$0.799651\pi$
$212$	0	0
$213$	14.5119i	0.994336i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 6.59811i	− 0.447909i
$218$	0	0
$219$	−5.06571	−0.342309
$220$	0	0
$221$	−25.4707	−1.71334
$222$	0	0
$223$	− 8.20720i	− 0.549595i	−0.961502	−	0.274797i	$-0.911389\pi$
$223$	− 8.20720i	− 0.549595i	0.961502	−	0.274797i	$-0.0886108\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 16.9942i	− 1.12794i	−0.825794	−	0.563972i	$-0.809273\pi$
$227$	− 16.9942i	− 1.12794i	0.825794	−	0.563972i	$-0.190727\pi$
$228$	0	0
$229$	−19.6818	−1.30061	−0.650306	−	0.759672i	$-0.725359\pi$
$229$	−19.6818	−1.30061	−0.650306	+	0.759672i	$0.725359\pi$
$230$	0	0
$231$	−15.6764	−1.03143
$232$	0	0
$233$	24.8799i	1.62993i	0.579507	+	0.814967i	$0.303245\pi$
$233$	24.8799i	1.62993i	−0.579507	+	0.814967i	$0.696755\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 10.9027i	− 0.708204i
$238$	0	0
$239$	0.0718483	0.00464748	0.00232374	−	0.999997i	$-0.499260\pi$
$239$	0.0718483	0.00464748	0.00232374	+	0.999997i	$0.499260\pi$
$240$	0	0
$241$	−8.50658	−0.547957	−0.273979	−	0.961736i	$-0.588340\pi$
$241$	−8.50658	−0.547957	−0.273979	+	0.961736i	$0.588340\pi$
$242$	0	0
$243$	− 11.0449i	− 0.708530i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	11.9398i	0.759709i
$248$	0	0
$249$	−12.8750	−0.815920
$250$	0	0
$251$	−19.8858	−1.25518	−0.627591	−	0.778543i	$-0.715959\pi$
$251$	−19.8858	−1.25518	−0.627591	+	0.778543i	$0.715959\pi$
$252$	0	0
$253$	− 3.49709i	− 0.219860i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.4062i	0.711498i	0.934582	+	0.355749i	$0.115774\pi$
$257$	11.4062i	0.711498i	−0.934582	+	0.355749i	$0.884226\pi$
$258$	0	0
$259$	1.67819	0.104277
$260$	0	0
$261$	3.92414	0.242898
$262$	0	0
$263$	− 13.3903i	− 0.825679i	−0.910804	−	0.412840i	$-0.864537\pi$
$263$	− 13.3903i	− 0.825679i	0.910804	−	0.412840i	$-0.135463\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.09493i	0.0670084i
$268$	0	0
$269$	−13.8103	−0.842027	−0.421014	−	0.907054i	$-0.638326\pi$
$269$	−13.8103	−0.842027	−0.421014	+	0.907054i	$0.638326\pi$
$270$	0	0
$271$	26.7574	1.62540	0.812699	−	0.582683i	$-0.197997\pi$
$271$	26.7574	1.62540	0.812699	+	0.582683i	$0.197997\pi$
$272$	0	0
$273$	15.3048i	0.926288i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.80667i	0.108552i	0.998526	+	0.0542760i	$0.0172851\pi$
$277$	1.80667i	0.108552i	−0.998526	+	0.0542760i	$0.982715\pi$
$278$	0	0
$279$	2.27905	0.136443
$280$	0	0
$281$	27.1348	1.61873	0.809364	−	0.587308i	$-0.199812\pi$
$281$	27.1348	1.61873	0.809364	+	0.587308i	$0.199812\pi$
$282$	0	0
$283$	20.9875i	1.24758i	0.781594	+	0.623788i	$0.214407\pi$
$283$	20.9875i	1.24758i	−0.781594	+	0.623788i	$0.785593\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	23.2275i	1.37108i
$288$	0	0
$289$	−38.6550	−2.27382
$290$	0	0
$291$	16.0842	0.942873
$292$	0	0
$293$	0.00671252i	0 0.000392149i	1.00000		0.000196075i	$6.24125e-5\pi$
$293$	0.00671252i	0 0.000392149i	−1.00000		0.000196075i	$0.999938\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 19.7489i	− 1.14594i
$298$	0	0
$299$	−3.41420	−0.197448
$300$	0	0
$301$	9.07022	0.522799
$302$	0	0
$303$	− 15.2422i	− 0.875641i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.9767i	1.14013i	0.821598	+	0.570067i	$0.193083\pi$
$307$	19.9767i	1.14013i	−0.821598	+	0.570067i	$0.806917\pi$
$308$	0	0
$309$	−1.75683	−0.0999427
$310$	0	0
$311$	23.8601	1.35298	0.676491	−	0.736451i	$-0.263500\pi$
$311$	23.8601	1.35298	0.676491	+	0.736451i	$0.263500\pi$
$312$	0	0
$313$	− 17.4406i	− 0.985803i	−0.870085	−	0.492901i	$-0.835936\pi$
$313$	− 17.4406i	− 0.985803i	0.870085	−	0.492901i	$-0.164064\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.35581i	− 0.0761499i	−0.999275	−	0.0380750i	$-0.987877\pi$
$317$	− 1.35581i	− 0.0761499i	0.999275	−	0.0380750i	$-0.0121226\pi$
$318$	0	0
$319$	12.1093	0.677992
$320$	0	0
$321$	−18.9879	−1.05980
$322$	0	0
$323$	26.0891i	1.45164i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 5.16063i	− 0.285384i
$328$	0	0
$329$	−2.91740	−0.160841
$330$	0	0
$331$	17.4050	0.956667	0.478333	−	0.878178i	$-0.341241\pi$
$331$	17.4050	0.956667	0.478333	+	0.878178i	$0.341241\pi$
$332$	0	0
$333$	0.579662i	0.0317653i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 22.4648i	− 1.22373i	−0.790961	−	0.611867i	$-0.790419\pi$
$337$	− 22.4648i	− 1.22373i	0.790961	−	0.611867i	$-0.209581\pi$
$338$	0	0
$339$	26.3845	1.43301
$340$	0	0
$341$	7.03282	0.380849
$342$	0	0
$343$	− 10.6154i	− 0.573177i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.97296i	− 0.159597i	−0.996811	−	0.0797984i	$-0.974572\pi$
$347$	− 2.97296i	− 0.159597i	0.996811	−	0.0797984i	$-0.0254277\pi$
$348$	0	0
$349$	4.36975	0.233908	0.116954	−	0.993137i	$-0.462687\pi$
$349$	4.36975	0.233908	0.116954	+	0.993137i	$0.462687\pi$
$350$	0	0
$351$	−19.2807	−1.02913
$352$	0	0
$353$	31.4419i	1.67348i	0.547598	+	0.836742i	$0.315542\pi$
$353$	31.4419i	1.67348i	−0.547598	+	0.836742i	$0.684458\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	33.4419i	1.76993i
$358$	0	0
$359$	−27.7912	−1.46676	−0.733381	−	0.679818i	$-0.762059\pi$
$359$	−27.7912	−1.46676	−0.733381	+	0.679818i	$0.762059\pi$
$360$	0	0
$361$	−6.77034	−0.356333
$362$	0	0
$363$	− 1.68007i	− 0.0881809i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 13.9061i	− 0.725890i	−0.931811	−	0.362945i	$-0.881771\pi$
$367$	− 13.9061i	− 0.725890i	0.931811	−	0.362945i	$-0.118229\pi$
$368$	0	0
$369$	−8.02299	−0.417660
$370$	0	0
$371$	46.7148	2.42531
$372$	0	0
$373$	− 26.2799i	− 1.36072i	−0.732876	−	0.680362i	$-0.761823\pi$
$373$	− 26.2799i	− 1.36072i	0.732876	−	0.680362i	$-0.238177\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 11.8223i	− 0.608879i
$378$	0	0
$379$	3.06792	0.157588	0.0787942	−	0.996891i	$-0.474893\pi$
$379$	3.06792	0.157588	0.0787942	+	0.996891i	$0.474893\pi$
$380$	0	0
$381$	−18.8017	−0.963239
$382$	0	0
$383$	28.8872i	1.47607i	0.674763	+	0.738034i	$0.264246\pi$
$383$	28.8872i	1.47607i	−0.674763	+	0.738034i	$0.735754\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.13294i	0.159256i
$388$	0	0
$389$	−24.4697	−1.24066	−0.620331	−	0.784340i	$-0.713002\pi$
$389$	−24.4697	−1.24066	−0.620331	+	0.784340i	$0.713002\pi$
$390$	0	0
$391$	−7.46023	−0.377280
$392$	0	0
$393$	10.1333i	0.511156i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.44828i	0.424007i	0.977269	+	0.212004i	$0.0679988\pi$
$397$	8.44828i	0.424007i	−0.977269	+	0.212004i	$0.932001\pi$
$398$	0	0
$399$	15.6764	0.784801
$400$	0	0
$401$	20.3352	1.01549	0.507746	−	0.861507i	$-0.330479\pi$
$401$	20.3352	1.01549	0.507746	+	0.861507i	$0.330479\pi$
$402$	0	0
$403$	− 6.86611i	− 0.342025i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.78875i	0.0886652i
$408$	0	0
$409$	−6.05655	−0.299477	−0.149738	−	0.988726i	$-0.547843\pi$
$409$	−6.05655	−0.299477	−0.149738	+	0.988726i	$0.547843\pi$
$410$	0	0
$411$	28.0775	1.38496
$412$	0	0
$413$	− 15.4639i	− 0.760927i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 7.50982i	− 0.367757i
$418$	0	0
$419$	−7.02434	−0.343162	−0.171581	−	0.985170i	$-0.554887\pi$
$419$	−7.02434	−0.343162	−0.171581	+	0.985170i	$0.554887\pi$
$420$	0	0
$421$	7.17652	0.349762	0.174881	−	0.984590i	$-0.444046\pi$
$421$	7.17652	0.349762	0.174881	+	0.984590i	$0.444046\pi$
$422$	0	0
$423$	− 1.00770i	− 0.0489959i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 44.2167i	− 2.13980i
$428$	0	0
$429$	−16.3131	−0.787605
$430$	0	0
$431$	11.2944	0.544034	0.272017	−	0.962292i	$-0.412309\pi$
$431$	11.2944	0.544034	0.272017	+	0.962292i	$0.412309\pi$
$432$	0	0
$433$	− 22.2266i	− 1.06814i	−0.845439	−	0.534072i	$-0.820661\pi$
$433$	− 22.2266i	− 1.06814i	0.845439	−	0.534072i	$-0.179339\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.49709i	0.167289i
$438$	0	0
$439$	30.2573	1.44410	0.722052	−	0.691839i	$-0.243199\pi$
$439$	30.2573	1.44410	0.722052	+	0.691839i	$0.243199\pi$
$440$	0	0
$441$	−4.26620	−0.203153
$442$	0	0
$443$	14.0307i	0.666620i	0.942817	+	0.333310i	$0.108166\pi$
$443$	14.0307i	0.666620i	−0.942817	+	0.333310i	$0.891834\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.56007i	0.310281i
$448$	0	0
$449$	−8.70110	−0.410630	−0.205315	−	0.978696i	$-0.565822\pi$
$449$	−8.70110	−0.410630	−0.205315	+	0.978696i	$0.565822\pi$
$450$	0	0
$451$	−24.7578	−1.16580
$452$	0	0
$453$	4.46587i	0.209825i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	33.8379i	1.58287i	0.611255	+	0.791434i	$0.290665\pi$
$457$	33.8379i	1.58287i	−0.611255	+	0.791434i	$0.709335\pi$
$458$	0	0
$459$	−42.1295	−1.96644
$460$	0	0
$461$	39.6512	1.84674	0.923370	−	0.383912i	$-0.125423\pi$
$461$	39.6512	1.84674	0.923370	+	0.383912i	$0.125423\pi$
$462$	0	0
$463$	− 34.8490i	− 1.61957i	−0.586727	−	0.809784i	$-0.699584\pi$
$463$	− 34.8490i	− 1.61957i	0.586727	−	0.809784i	$-0.300416\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 13.9442i	− 0.645258i	−0.946525	−	0.322629i	$-0.895433\pi$
$467$	− 13.9442i	− 0.645258i	0.946525	−	0.322629i	$-0.104567\pi$
$468$	0	0
$469$	−7.54697	−0.348486
$470$	0	0
$471$	−29.8386	−1.37489
$472$	0	0
$473$	9.66780i	0.444526i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	16.1357i	0.738804i
$478$	0	0
$479$	32.4103	1.48086	0.740432	−	0.672131i	$-0.234621\pi$
$479$	32.4103	1.48086	0.740432	+	0.672131i	$0.234621\pi$
$480$	0	0
$481$	1.74635	0.0796268
$482$	0	0
$483$	4.48269i	0.203969i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 25.1999i	− 1.14192i	−0.820979	−	0.570959i	$-0.806572\pi$
$487$	− 25.1999i	− 1.14192i	0.820979	−	0.570959i	$-0.193428\pi$
$488$	0	0
$489$	31.7572	1.43611
$490$	0	0
$491$	−32.9247	−1.48587	−0.742936	−	0.669363i	$-0.766567\pi$
$491$	−32.9247	−1.48587	−0.742936	+	0.669363i	$0.766567\pi$
$492$	0	0
$493$	− 25.8324i	− 1.16343i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	34.8481i	1.56315i
$498$	0	0
$499$	−12.4047	−0.555309	−0.277654	−	0.960681i	$-0.589557\pi$
$499$	−12.4047	−0.555309	−0.277654	+	0.960681i	$0.589557\pi$
$500$	0	0
$501$	5.64521	0.252209
$502$	0	0
$503$	− 3.74517i	− 0.166989i	−0.996508	−	0.0834945i	$-0.973392\pi$
$503$	− 3.74517i	− 0.166989i	0.996508	−	0.0834945i	$-0.0266081\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1.83526i	− 0.0815069i
$508$	0	0
$509$	5.84783	0.259201	0.129600	−	0.991566i	$-0.458631\pi$
$509$	5.84783	0.259201	0.129600	+	0.991566i	$0.458631\pi$
$510$	0	0
$511$	−12.1646	−0.538128
$512$	0	0
$513$	19.7489i	0.871933i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 3.10961i	− 0.136760i
$518$	0	0
$519$	8.04124	0.352971
$520$	0	0
$521$	18.3808	0.805278	0.402639	−	0.915359i	$-0.368093\pi$
$521$	18.3808	0.805278	0.402639	+	0.915359i	$0.368093\pi$
$522$	0	0
$523$	23.4541i	1.02558i	0.858515	+	0.512789i	$0.171388\pi$
$523$	23.4541i	1.02558i	−0.858515	+	0.512789i	$0.828612\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 15.0029i	− 0.653535i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	5.34137	0.231796
$532$	0	0
$533$	24.1710i	1.04696i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 9.36984i	− 0.404338i
$538$	0	0
$539$	−13.1649	−0.567052
$540$	0	0
$541$	3.03474	0.130474	0.0652368	−	0.997870i	$-0.479220\pi$
$541$	3.03474	0.130474	0.0652368	+	0.997870i	$0.479220\pi$
$542$	0	0
$543$	3.12937i	0.134294i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 16.4383i	− 0.702851i	−0.936216	−	0.351426i	$-0.885697\pi$
$547$	− 16.4383i	− 0.702851i	0.936216	−	0.351426i	$-0.114303\pi$
$548$	0	0
$549$	15.2729	0.651830
$550$	0	0
$551$	−12.1093	−0.515875
$552$	0	0
$553$	− 26.1811i	− 1.11334i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.72395i	0.284903i	0.989802	+	0.142451i	$0.0454985\pi$
$557$	6.72395i	0.284903i	−0.989802	+	0.142451i	$0.954502\pi$
$558$	0	0
$559$	9.43863	0.399212
$560$	0	0
$561$	−35.6452	−1.50494
$562$	0	0
$563$	− 42.0044i	− 1.77027i	−0.465331	−	0.885137i	$-0.654065\pi$
$563$	− 42.0044i	− 1.77027i	0.465331	−	0.885137i	$-0.345935\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	14.1602i	0.594674i
$568$	0	0
$569$	39.1152	1.63980	0.819898	−	0.572510i	$-0.194030\pi$
$569$	39.1152	1.63980	0.819898	+	0.572510i	$0.194030\pi$
$570$	0	0
$571$	−13.2607	−0.554944	−0.277472	−	0.960734i	$-0.589497\pi$
$571$	−13.2607	−0.554944	−0.277472	+	0.960734i	$0.589497\pi$
$572$	0	0
$573$	13.0992i	0.547228i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 9.31588i	− 0.387825i	−0.981019	−	0.193913i	$-0.937882\pi$
$577$	− 9.31588i	− 0.387825i	0.981019	−	0.193913i	$-0.0621178\pi$
$578$	0	0
$579$	19.0474	0.791584
$580$	0	0
$581$	−30.9174	−1.28267
$582$	0	0
$583$	49.7925i	2.06220i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 7.80336i	− 0.322079i	−0.986948	−	0.161040i	$-0.948515\pi$
$587$	− 7.80336i	− 0.322079i	0.986948	−	0.161040i	$-0.0514847\pi$
$588$	0	0
$589$	−7.03282	−0.289782
$590$	0	0
$591$	−20.6324	−0.848704
$592$	0	0
$593$	12.8383i	0.527204i	0.964632	+	0.263602i	$0.0849106\pi$
$593$	12.8383i	0.527204i	−0.964632	+	0.263602i	$0.915089\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	19.8287i	0.811534i
$598$	0	0
$599$	10.4957	0.428843	0.214422	−	0.976741i	$-0.431213\pi$
$599$	10.4957	0.428843	0.214422	+	0.976741i	$0.431213\pi$
$600$	0	0
$601$	−28.1978	−1.15021	−0.575106	−	0.818079i	$-0.695039\pi$
$601$	−28.1978	−1.15021	−0.575106	+	0.818079i	$0.695039\pi$
$602$	0	0
$603$	− 2.60679i	− 0.106157i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	46.5662i	1.89007i	0.326976	+	0.945033i	$0.393970\pi$
$607$	46.5662i	1.89007i	−0.326976	+	0.945033i	$0.606030\pi$
$608$	0	0
$609$	−15.5221	−0.628989
$610$	0	0
$611$	−3.03590	−0.122819
$612$	0	0
$613$	− 30.3120i	− 1.22429i	−0.790746	−	0.612145i	$-0.790307\pi$
$613$	− 30.3120i	− 1.22429i	0.790746	−	0.612145i	$-0.209693\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.1879i	0.973768i	0.873467	+	0.486884i	$0.161866\pi$
$617$	24.1879i	0.973768i	−0.873467	+	0.486884i	$0.838134\pi$
$618$	0	0
$619$	4.29534	0.172644	0.0863221	−	0.996267i	$-0.472489\pi$
$619$	4.29534	0.172644	0.0863221	+	0.996267i	$0.472489\pi$
$620$	0	0
$621$	−5.64722	−0.226615
$622$	0	0
$623$	2.62931i	0.105341i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	16.7092i	0.667301i
$628$	0	0
$629$	3.81588	0.152149
$630$	0	0
$631$	12.3857	0.493066	0.246533	−	0.969134i	$-0.420709\pi$
$631$	12.3857	0.493066	0.246533	+	0.969134i	$0.420709\pi$
$632$	0	0
$633$	32.0866i	1.27533i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12.8528i	0.509248i
$638$	0	0
$639$	−12.0369	−0.476171
$640$	0	0
$641$	−5.21653	−0.206041	−0.103020	−	0.994679i	$-0.532851\pi$
$641$	−5.21653	−0.206041	−0.103020	+	0.994679i	$0.532851\pi$
$642$	0	0
$643$	− 12.5639i	− 0.495471i	−0.968828	−	0.247735i	$-0.920314\pi$
$643$	− 12.5639i	− 0.495471i	0.968828	−	0.247735i	$-0.0796863\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 22.7423i	− 0.894094i	−0.894510	−	0.447047i	$-0.852476\pi$
$647$	− 22.7423i	− 0.894094i	0.894510	−	0.447047i	$-0.147524\pi$
$648$	0	0
$649$	16.4827	0.647002
$650$	0	0
$651$	−9.01490	−0.353322
$652$	0	0
$653$	− 9.36997i	− 0.366675i	−0.983050	−	0.183338i	$-0.941310\pi$
$653$	− 9.36997i	− 0.366675i	0.983050	−	0.183338i	$-0.0586902\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 4.20175i	− 0.163926i
$658$	0	0
$659$	15.4086	0.600235	0.300117	−	0.953902i	$-0.402974\pi$
$659$	15.4086	0.600235	0.300117	+	0.953902i	$0.402974\pi$
$660$	0	0
$661$	43.1386	1.67790	0.838948	−	0.544211i	$-0.183171\pi$
$661$	43.1386	1.67790	0.838948	+	0.544211i	$0.183171\pi$
$662$	0	0
$663$	34.8002i	1.35153i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.46268i	− 0.134076i
$668$	0	0
$669$	−11.2134	−0.433534
$670$	0	0
$671$	47.1299	1.81943
$672$	0	0
$673$	12.6356i	0.487066i	0.969893	+	0.243533i	$0.0783065\pi$
$673$	12.6356i	0.487066i	−0.969893	+	0.243533i	$0.921694\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.9139i	0.419454i	0.977760	+	0.209727i	$0.0672575\pi$
$677$	10.9139i	0.419454i	−0.977760	+	0.209727i	$0.932742\pi$
$678$	0	0
$679$	38.6238	1.48225
$680$	0	0
$681$	−23.2189	−0.889750
$682$	0	0
$683$	− 17.5334i	− 0.670896i	−0.942059	−	0.335448i	$-0.891112\pi$
$683$	− 17.5334i	− 0.670896i	0.942059	−	0.335448i	$-0.108888\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	26.8910i	1.02596i
$688$	0	0
$689$	48.6122	1.85198
$690$	0	0
$691$	−52.2318	−1.98699	−0.993496	−	0.113871i	$-0.963675\pi$
$691$	−52.2318	−1.98699	−0.993496	+	0.113871i	$0.963675\pi$
$692$	0	0
$693$	− 13.0028i	− 0.493935i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	52.8150i	2.00051i
$698$	0	0
$699$	33.9930	1.28573
$700$	0	0
$701$	26.0270	0.983026	0.491513	−	0.870870i	$-0.336444\pi$
$701$	26.0270	0.983026	0.491513	+	0.870870i	$0.336444\pi$
$702$	0	0
$703$	− 1.78875i	− 0.0674641i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 36.6019i	− 1.37655i
$708$	0	0
$709$	34.7971	1.30683	0.653415	−	0.757000i	$-0.273336\pi$
$709$	34.7971	1.30683	0.653415	+	0.757000i	$0.273336\pi$
$710$	0	0
$711$	9.04321	0.339147
$712$	0	0
$713$	− 2.01105i	− 0.0753143i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 0.0981652i	− 0.00366605i
$718$	0	0
$719$	−4.40110	−0.164133	−0.0820667	−	0.996627i	$-0.526152\pi$
$719$	−4.40110	−0.164133	−0.0820667	+	0.996627i	$0.526152\pi$
$720$	0	0
$721$	−4.21878	−0.157115
$722$	0	0
$723$	11.6224i	0.432242i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 0.744921i	− 0.0276276i	−0.999905	−	0.0138138i	$-0.995603\pi$
$727$	− 0.744921i	− 0.0276276i	0.999905	−	0.0138138i	$-0.00439721\pi$
$728$	0	0
$729$	−28.0382	−1.03845
$730$	0	0
$731$	20.6240	0.762805
$732$	0	0
$733$	− 7.35631i	− 0.271711i	−0.990729	−	0.135856i	$-0.956622\pi$
$733$	− 7.35631i	− 0.271711i	0.990729	−	0.135856i	$-0.0433784\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.04419i	− 0.296311i
$738$	0	0
$739$	−13.2496	−0.487394	−0.243697	−	0.969851i	$-0.578360\pi$
$739$	−13.2496	−0.487394	−0.243697	+	0.969851i	$0.578360\pi$
$740$	0	0
$741$	16.3131	0.599278
$742$	0	0
$743$	− 28.8314i	− 1.05772i	−0.848708	−	0.528861i	$-0.822619\pi$
$743$	− 28.8314i	− 1.05772i	0.848708	−	0.528861i	$-0.177381\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 10.6792i	− 0.390730i
$748$	0	0
$749$	−45.5966	−1.66607
$750$	0	0
$751$	−24.3962	−0.890228	−0.445114	−	0.895474i	$-0.646837\pi$
$751$	−24.3962	−0.890228	−0.445114	+	0.895474i	$0.646837\pi$
$752$	0	0
$753$	27.1697i	0.990120i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	9.49825i	0.345220i	0.984990	+	0.172610i	$0.0552200\pi$
$757$	9.49825i	0.345220i	−0.984990	+	0.172610i	$0.944780\pi$
$758$	0	0
$759$	−4.77803	−0.173431
$760$	0	0
$761$	−8.43957	−0.305934	−0.152967	−	0.988231i	$-0.548883\pi$
$761$	−8.43957	−0.305934	−0.152967	+	0.988231i	$0.548883\pi$
$762$	0	0
$763$	− 12.3925i	− 0.448639i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 16.0920i	− 0.581048i
$768$	0	0
$769$	−8.85911	−0.319468	−0.159734	−	0.987160i	$-0.551064\pi$
$769$	−8.85911	−0.319468	−0.159734	+	0.987160i	$0.551064\pi$
$770$	0	0
$771$	15.5841	0.561247
$772$	0	0
$773$	− 32.7351i	− 1.17740i	−0.808352	−	0.588699i	$-0.799640\pi$
$773$	− 32.7351i	− 1.17740i	0.808352	−	0.588699i	$-0.200360\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2.29288i	− 0.0822567i
$778$	0	0
$779$	24.7578	0.887041
$780$	0	0
$781$	−37.1440	−1.32912
$782$	0	0
$783$	− 19.5545i	− 0.698822i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0.736862i	0.0262663i	0.999914	+	0.0131331i	$0.00418053\pi$
$787$	0.736862i	0.0262663i	−0.999914	+	0.0131331i	$0.995819\pi$
$788$	0	0
$789$	−18.2949	−0.651316
$790$	0	0
$791$	63.3584	2.25276
$792$	0	0
$793$	− 46.0127i	− 1.63396i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 41.7353i	− 1.47834i	−0.673519	−	0.739170i	$-0.735218\pi$
$797$	− 41.7353i	− 1.47834i	0.673519	−	0.739170i	$-0.264782\pi$
$798$	0	0
$799$	−6.63362	−0.234681
$800$	0	0
$801$	−0.908187	−0.0320892
$802$	0	0
$803$	− 12.9660i	− 0.457560i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18.8688i	0.664212i
$808$	0	0
$809$	−21.3182	−0.749506	−0.374753	−	0.927125i	$-0.622273\pi$
$809$	−21.3182	−0.749506	−0.374753	+	0.927125i	$0.622273\pi$
$810$	0	0
$811$	24.3126	0.853732	0.426866	−	0.904315i	$-0.359618\pi$
$811$	24.3126	0.853732	0.426866	+	0.904315i	$0.359618\pi$
$812$	0	0
$813$	− 36.5583i	− 1.28216i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 9.66780i	− 0.338233i
$818$	0	0
$819$	−12.6946	−0.443584
$820$	0	0
$821$	49.3157	1.72113	0.860566	−	0.509340i	$-0.170110\pi$
$821$	49.3157	1.72113	0.860566	+	0.509340i	$0.170110\pi$
$822$	0	0
$823$	− 41.5427i	− 1.44809i	−0.689755	−	0.724043i	$-0.742282\pi$
$823$	− 41.5427i	− 1.44809i	0.689755	−	0.724043i	$-0.257718\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 0.306510i	− 0.0106584i	−0.999986	−	0.00532919i	$-0.998304\pi$
$827$	− 0.306510i	− 0.0106584i	0.999986	−	0.00532919i	$-0.00169634\pi$
$828$	0	0
$829$	−17.5423	−0.609268	−0.304634	−	0.952469i	$-0.598534\pi$
$829$	−17.5423	−0.609268	−0.304634	+	0.952469i	$0.598534\pi$
$830$	0	0
$831$	2.46842	0.0856285
$832$	0	0
$833$	28.0842i	0.973060i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 11.3568i	− 0.392549i
$838$	0	0
$839$	44.1480	1.52416	0.762078	−	0.647485i	$-0.224179\pi$
$839$	44.1480	1.52416	0.762078	+	0.647485i	$0.224179\pi$
$840$	0	0
$841$	−17.0098	−0.586546
$842$	0	0
$843$	− 37.0739i	− 1.27689i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 4.03445i	− 0.138625i
$848$	0	0
$849$	28.6749	0.984119
$850$	0	0
$851$	0.511497	0.0175339
$852$	0	0
$853$	41.2098i	1.41100i	0.708711	+	0.705499i	$0.249277\pi$
$853$	41.2098i	1.41100i	−0.708711	+	0.705499i	$0.750723\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.3245i	0.660114i	0.943961	+	0.330057i	$0.107068\pi$
$857$	19.3245i	0.660114i	−0.943961	+	0.330057i	$0.892932\pi$
$858$	0	0
$859$	16.2174	0.553330	0.276665	−	0.960966i	$-0.410771\pi$
$859$	16.2174	0.553330	0.276665	+	0.960966i	$0.410771\pi$
$860$	0	0
$861$	31.7354	1.08154
$862$	0	0
$863$	56.9036i	1.93702i	0.248974	+	0.968510i	$0.419907\pi$
$863$	56.9036i	1.93702i	−0.248974	+	0.968510i	$0.580093\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	52.8137i	1.79365i
$868$	0	0
$869$	27.9061	0.946648
$870$	0	0
$871$	−7.85351	−0.266106
$872$	0	0
$873$	13.3410i	0.451526i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	33.4569i	1.12976i	0.825173	+	0.564880i	$0.191077\pi$
$877$	33.4569i	1.12976i	−0.825173	+	0.564880i	$0.808923\pi$
$878$	0	0
$879$	0.00917121	0.000309337 0
$880$	0	0
$881$	−5.99700	−0.202044	−0.101022	−	0.994884i	$-0.532211\pi$
$881$	−5.99700	−0.202044	−0.101022	+	0.994884i	$0.532211\pi$
$882$	0	0
$883$	− 51.3210i	− 1.72709i	−0.504273	−	0.863544i	$-0.668239\pi$
$883$	− 51.3210i	− 1.72709i	0.504273	−	0.863544i	$-0.331761\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	27.1121i	0.910336i	0.890405	+	0.455168i	$0.150421\pi$
$887$	27.1121i	0.910336i	−0.890405	+	0.455168i	$0.849579\pi$
$888$	0	0
$889$	−45.1495	−1.51426
$890$	0	0
$891$	−15.0932	−0.505640
$892$	0	0
$893$	3.10961i	0.104059i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.66477i	0.155752i
$898$	0	0
$899$	6.96362	0.232250
$900$	0	0
$901$	106.221	3.53872
$902$	0	0
$903$	− 12.3925i	− 0.412397i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	40.6194i	1.34875i	0.738391	+	0.674373i	$0.235586\pi$
$907$	40.6194i	1.34875i	−0.738391	+	0.674373i	$0.764414\pi$
$908$	0	0
$909$	12.6426	0.419329
$910$	0	0
$911$	32.2475	1.06841	0.534203	−	0.845356i	$-0.320612\pi$
$911$	32.2475	1.06841	0.534203	+	0.845356i	$0.320612\pi$
$912$	0	0
$913$	− 32.9544i	− 1.09063i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	24.3335i	0.803565i
$918$	0	0
$919$	30.4042	1.00294	0.501470	−	0.865175i	$-0.332793\pi$
$919$	30.4042	1.00294	0.501470	+	0.865175i	$0.332793\pi$
$920$	0	0
$921$	27.2939	0.899366
$922$	0	0
$923$	36.2636i	1.19363i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1.45720i	− 0.0478609i
$928$	0	0
$929$	25.2834	0.829520	0.414760	−	0.909931i	$-0.363866\pi$
$929$	25.2834	0.829520	0.414760	+	0.909931i	$0.363866\pi$
$930$	0	0
$931$	13.1649	0.431462
$932$	0	0
$933$	− 32.5997i	− 1.06727i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.1769i	0.724487i	0.932084	+	0.362243i	$0.117989\pi$
$937$	22.1769i	0.724487i	−0.932084	+	0.362243i	$0.882011\pi$
$938$	0	0
$939$	−23.8289	−0.777626
$940$	0	0
$941$	4.23228	0.137968	0.0689842	−	0.997618i	$-0.478024\pi$
$941$	4.23228	0.137968	0.0689842	+	0.997618i	$0.478024\pi$
$942$	0	0
$943$	7.07954i	0.230542i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 45.2374i	− 1.47002i	−0.678058	−	0.735009i	$-0.737178\pi$
$947$	− 45.2374i	− 1.47002i	0.678058	−	0.735009i	$-0.262822\pi$
$948$	0	0
$949$	−12.6587	−0.410917
$950$	0	0
$951$	−1.85242	−0.0600690
$952$	0	0
$953$	− 43.7170i	− 1.41613i	−0.706146	−	0.708066i	$-0.749568\pi$
$953$	− 43.7170i	− 1.41613i	0.706146	−	0.708066i	$-0.250432\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 16.5448i	− 0.534817i
$958$	0	0
$959$	67.4240	2.17723
$960$	0	0
$961$	−26.9557	−0.869538
$962$	0	0
$963$	− 15.7495i	− 0.507521i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	5.58482i	0.179596i	0.995960	+	0.0897979i	$0.0286221\pi$
$967$	5.58482i	0.179596i	−0.995960	+	0.0897979i	$0.971378\pi$
$968$	0	0
$969$	35.6452	1.14509
$970$	0	0
$971$	39.1206	1.25544	0.627720	−	0.778439i	$-0.283988\pi$
$971$	39.1206	1.25544	0.627720	+	0.778439i	$0.283988\pi$
$972$	0	0
$973$	− 18.0337i	− 0.578135i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	52.3050i	1.67339i	0.547672	+	0.836693i	$0.315514\pi$
$977$	52.3050i	1.67339i	−0.547672	+	0.836693i	$0.684486\pi$
$978$	0	0
$979$	−2.80253	−0.0895694
$980$	0	0
$981$	4.28049	0.136665
$982$	0	0
$983$	22.4395i	0.715710i	0.933777	+	0.357855i	$0.116492\pi$
$983$	22.4395i	0.715710i	−0.933777	+	0.357855i	$0.883508\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.98600i	0.126876i
$988$	0	0
$989$	2.76452	0.0879067
$990$	0	0
$991$	−0.851383	−0.0270451	−0.0135225	−	0.999909i	$-0.504304\pi$
$991$	−0.851383	−0.0270451	−0.0135225	+	0.999909i	$0.504304\pi$
$992$	0	0
$993$	− 23.7802i	− 0.754643i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 12.1784i	− 0.385693i	−0.981229	−	0.192846i	$-0.938228\pi$
$997$	− 12.1784i	− 0.385693i	0.981229	−	0.192846i	$-0.0617719\pi$
$998$	0	0
$999$	2.88853	0.0913892

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.w.4049.4		10
5.2	odd	4		4600.2.a.bf.1.2	yes	5
5.3	odd	4		4600.2.a.bd.1.4	✓	5
5.4	even	2	inner	4600.2.e.w.4049.7		10
20.3	even	4		9200.2.a.cv.1.2		5
20.7	even	4		9200.2.a.ct.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bd.1.4	✓	5	5.3	odd	4
4600.2.a.bf.1.2	yes	5	5.2	odd	4
4600.2.e.w.4049.4		10	1.1	even	1	trivial
4600.2.e.w.4049.7		10	5.4	even	2	inner
9200.2.a.ct.1.4		5	20.7	even	4
9200.2.a.cv.1.2		5	20.3	even	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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.36629i q^{3} -3.28093i q^{7} +1.13327 q^{9} +O(q^{10})\)	\(q-1.36629i q^{3} -3.28093i q^{7} +1.13327 q^{9} +3.49709 q^{11} -3.41420i q^{13} -7.46023i q^{17} -3.49709 q^{19} -4.48269 q^{21} -1.00000i q^{23} -5.64722i q^{27} +3.46268 q^{29} +2.01105 q^{31} -4.77803i q^{33} +0.511497i q^{37} -4.66477 q^{39} -7.07954 q^{41} +2.76452i q^{43} -0.889198i q^{47} -3.76452 q^{49} -10.1928 q^{51} +14.2383i q^{53} +4.77803i q^{57} +4.71325 q^{59} +13.4769 q^{61} -3.71817i q^{63} -2.30025i q^{67} -1.36629 q^{69} -10.6214 q^{71} -3.70765i q^{73} -11.4737i q^{77} +7.97978 q^{79} -4.31592 q^{81} -9.42336i q^{83} -4.73101i q^{87} -0.801390 q^{89} -11.2018 q^{91} -2.74766i q^{93} +11.7722i q^{97} +3.96313 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{9}+O(q^{10}) \) 10 * q - 6 * q^9	\( 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99}+O(q^{100}) \) 10 * q - 6 * q^9 - 24 * q^29 - 36 * q^31 - 18 * q^39 - 12 * q^41 - 30 * q^49 - 12 * q^51 + 2 * q^59 + 20 * q^61 + 16 * q^71 - 54 * q^81 - 28 * q^89 - 92 * q^91 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more