LMFDB - Embedded newform 3000.2.f.e.1249.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3000,2,Mod(1249,3000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3000, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3000.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3000 = 2^{3} \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3000.f (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:	$23.9551206064$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.43430560000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 25x^{6} + 208x^{4} + 705x^{2} + 841 $ x^8 + 25x^6 + 208x^4 + 705*x^2 + 841
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.5
Root			$3.50578i$ of defining polynomial
Character	$\chi$	$=$	3000.1249
Dual form			3000.2.f.e.1249.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -4.05444i q^{7} -1.00000 q^{9} -1.33909 q^{11} +7.12382i q^{13} -3.15746i q^{17} -4.78473 q^{19} +4.05444 q^{21} -0.103022i q^{23} -1.00000i q^{27} +6.22113 q^{29} -0.402761 q^{31} -1.33909i q^{33} -0.327525i q^{37} -7.12382 q^{39} -7.62389 q^{41} -9.18749i q^{43} +10.8993i q^{47} -9.43849 q^{49} +3.15746 q^{51} +0.730286i q^{53} -4.78473i q^{57} -8.39353 q^{59} -3.65376 q^{61} +4.05444i q^{63} -3.90854i q^{67} +0.103022 q^{69} -7.08680 q^{71} +2.86904i q^{73} +5.42926i q^{77} -12.4480 q^{79} +1.00000 q^{81} +3.96869i q^{83} +6.22113i q^{87} -18.0624 q^{89} +28.8831 q^{91} -0.402761i q^{93} +14.6355i q^{97} +1.33909 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9} + 2 q^{11} - 16 q^{19} + 2 q^{21} + 6 q^{29} + 28 q^{31} - 22 q^{39} + 14 q^{41} - 30 q^{49} - 24 q^{59} - 6 q^{61} + 6 q^{69} + 66 q^{71} - 26 q^{79} + 8 q^{81} - 24 q^{89} + 34 q^{91} - 2 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 + 2 * q^11 - 16 * q^19 + 2 * q^21 + 6 * q^29 + 28 * q^31 - 22 * q^39 + 14 * q^41 - 30 * q^49 - 24 * q^59 - 6 * q^61 + 6 * q^69 + 66 * q^71 - 26 * q^79 + 8 * q^81 - 24 * q^89 + 34 * q^91 - 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3000\mathbb{Z}\right)^\times$.

$n$	$751$	$1001$	$1501$	$2377$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.05444i	− 1.53243i	−0.642582	−	0.766217i	$-0.722137\pi$
$7$	− 4.05444i	− 1.53243i	0.642582	−	0.766217i	$-0.277863\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.33909	−0.403751	−0.201875	−	0.979411i	$-0.564704\pi$
$11$	−1.33909	−0.403751	−0.201875	+	0.979411i	$0.564704\pi$
$12$	0	0
$13$	7.12382i	1.97579i	0.155120	+	0.987896i	$0.450423\pi$
$13$	7.12382i	1.97579i	−0.155120	+	0.987896i	$0.549577\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.15746i	− 0.765797i	−0.923790	−	0.382899i	$-0.874926\pi$
$17$	− 3.15746i	− 0.765797i	0.923790	−	0.382899i	$-0.125074\pi$
$18$	0	0
$19$	−4.78473	−1.09769	−0.548846	−	0.835924i	$-0.684933\pi$
$19$	−4.78473	−1.09769	−0.548846	+	0.835924i	$0.684933\pi$
$20$	0	0
$21$	4.05444	0.884752
$22$	0	0
$23$	− 0.103022i	− 0.0214815i	−0.999942	−	0.0107408i	$-0.996581\pi$
$23$	− 0.103022i	− 0.0214815i	0.999942	−	0.0107408i	$-0.00341895\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	6.22113	1.15524	0.577618	−	0.816307i	$-0.303982\pi$
$29$	6.22113	1.15524	0.577618	+	0.816307i	$0.303982\pi$
$30$	0	0
$31$	−0.402761	−0.0723379	−0.0361690	−	0.999346i	$-0.511515\pi$
$31$	−0.402761	−0.0723379	−0.0361690	+	0.999346i	$0.511515\pi$
$32$	0	0
$33$	− 1.33909i	− 0.233106i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 0.327525i	− 0.0538448i	−0.999638	−	0.0269224i	$-0.991429\pi$
$37$	− 0.327525i	− 0.0538448i	0.999638	−	0.0269224i	$-0.00857071\pi$
$38$	0	0
$39$	−7.12382	−1.14072
$40$	0	0
$41$	−7.62389	−1.19065	−0.595326	−	0.803484i	$-0.702977\pi$
$41$	−7.62389	−1.19065	−0.595326	+	0.803484i	$0.702977\pi$
$42$	0	0
$43$	− 9.18749i	− 1.40108i	−0.713614	−	0.700539i	$-0.752943\pi$
$43$	− 9.18749i	− 1.40108i	0.713614	−	0.700539i	$-0.247057\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.8993i	1.58983i	0.606722	+	0.794914i	$0.292484\pi$
$47$	10.8993i	1.58983i	−0.606722	+	0.794914i	$0.707516\pi$
$48$	0	0
$49$	−9.43849	−1.34836
$50$	0	0
$51$	3.15746	0.442133
$52$	0	0
$53$	0.730286i	0.100312i	0.998741	+	0.0501562i	$0.0159719\pi$
$53$	0.730286i	0.100312i	−0.998741	+	0.0501562i	$0.984028\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 4.78473i	− 0.633753i
$58$	0	0
$59$	−8.39353	−1.09274	−0.546372	−	0.837542i	$-0.683992\pi$
$59$	−8.39353	−1.09274	−0.546372	+	0.837542i	$0.683992\pi$
$60$	0	0
$61$	−3.65376	−0.467816	−0.233908	−	0.972259i	$-0.575152\pi$
$61$	−3.65376	−0.467816	−0.233908	+	0.972259i	$0.575152\pi$
$62$	0	0
$63$	4.05444i	0.510812i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 3.90854i	− 0.477504i	−0.971081	−	0.238752i	$-0.923262\pi$
$67$	− 3.90854i	− 0.477504i	0.971081	−	0.238752i	$-0.0767384\pi$
$68$	0	0
$69$	0.103022	0.0124024
$70$	0	0
$71$	−7.08680	−0.841048	−0.420524	−	0.907281i	$-0.638154\pi$
$71$	−7.08680	−0.841048	−0.420524	+	0.907281i	$0.638154\pi$
$72$	0	0
$73$	2.86904i	0.335795i	0.985804	+	0.167898i	$0.0536978\pi$
$73$	2.86904i	0.335795i	−0.985804	+	0.167898i	$0.946302\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.42926i	0.618722i
$78$	0	0
$79$	−12.4480	−1.40051	−0.700253	−	0.713895i	$-0.746930\pi$
$79$	−12.4480	−1.40051	−0.700253	+	0.713895i	$0.746930\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.96869i	0.435620i	0.975991	+	0.217810i	$0.0698913\pi$
$83$	3.96869i	0.435620i	−0.975991	+	0.217810i	$0.930109\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.22113i	0.666975i
$88$	0	0
$89$	−18.0624	−1.91461	−0.957304	−	0.289082i	$-0.906650\pi$
$89$	−18.0624	−1.91461	−0.957304	+	0.289082i	$0.906650\pi$
$90$	0	0
$91$	28.8831	3.02777
$92$	0	0
$93$	− 0.402761i	− 0.0417643i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.6355i	1.48601i	0.669288	+	0.743003i	$0.266599\pi$
$97$	14.6355i	1.48601i	−0.669288	+	0.743003i	$0.733401\pi$
$98$	0	0
$99$	1.33909	0.134584
$100$	0	0
$101$	−7.35988	−0.732336	−0.366168	−	0.930549i	$-0.619330\pi$
$101$	−7.35988	−0.732336	−0.366168	+	0.930549i	$0.619330\pi$
$102$	0	0
$103$	− 3.88060i	− 0.382367i	−0.981554	−	0.191183i	$-0.938767\pi$
$103$	− 3.88060i	− 0.382367i	0.981554	−	0.191183i	$-0.0612326\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.39706i	0.908448i	0.890888	+	0.454224i	$0.150083\pi$
$107$	9.39706i	0.908448i	−0.890888	+	0.454224i	$0.849917\pi$
$108$	0	0
$109$	4.89361	0.468723	0.234361	−	0.972150i	$-0.424700\pi$
$109$	4.89361	0.468723	0.234361	+	0.972150i	$0.424700\pi$
$110$	0	0
$111$	0.327525	0.0310873
$112$	0	0
$113$	16.8436i	1.58451i	0.610189	+	0.792256i	$0.291093\pi$
$113$	16.8436i	1.58451i	−0.610189	+	0.792256i	$0.708907\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 7.12382i	− 0.658597i
$118$	0	0
$119$	−12.8017	−1.17353
$120$	0	0
$121$	−9.20684	−0.836985
$122$	0	0
$123$	− 7.62389i	− 0.687423i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 17.1711i	− 1.52369i	−0.647760	−	0.761845i	$-0.724294\pi$
$127$	− 17.1711i	− 1.52369i	0.647760	−	0.761845i	$-0.275706\pi$
$128$	0	0
$129$	9.18749	0.808913
$130$	0	0
$131$	6.34624	0.554473	0.277237	−	0.960802i	$-0.410581\pi$
$131$	6.34624	0.554473	0.277237	+	0.960802i	$0.410581\pi$
$132$	0	0
$133$	19.3994i	1.68214i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.9615i	1.61999i	0.586434	+	0.809997i	$0.300531\pi$
$137$	18.9615i	1.61999i	−0.586434	+	0.809997i	$0.699469\pi$
$138$	0	0
$139$	15.1412	1.28426	0.642132	−	0.766594i	$-0.278050\pi$
$139$	15.1412	1.28426	0.642132	+	0.766594i	$0.278050\pi$
$140$	0	0
$141$	−10.8993	−0.917888
$142$	0	0
$143$	− 9.53943i	− 0.797727i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 9.43849i	− 0.778474i
$148$	0	0
$149$	−23.8194	−1.95136	−0.975681	−	0.219193i	$-0.929657\pi$
$149$	−23.8194	−1.95136	−0.975681	+	0.219193i	$0.929657\pi$
$150$	0	0
$151$	−20.1469	−1.63954	−0.819768	−	0.572696i	$-0.805897\pi$
$151$	−20.1469	−1.63954	−0.819768	+	0.572696i	$0.805897\pi$
$152$	0	0
$153$	3.15746i	0.255266i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 6.15513i	− 0.491233i	−0.969367	−	0.245616i	$-0.921010\pi$
$157$	− 6.15513i	− 0.491233i	0.969367	−	0.245616i	$-0.0789903\pi$
$158$	0	0
$159$	−0.730286	−0.0579154
$160$	0	0
$161$	−0.417695	−0.0329190
$162$	0	0
$163$	− 8.16525i	− 0.639552i	−0.947493	−	0.319776i	$-0.896392\pi$
$163$	− 8.16525i	− 0.639552i	0.947493	−	0.319776i	$-0.103608\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.6401i	0.823357i	0.911329	+	0.411678i	$0.135057\pi$
$167$	10.6401i	0.823357i	−0.911329	+	0.411678i	$0.864943\pi$
$168$	0	0
$169$	−37.7488	−2.90375
$170$	0	0
$171$	4.78473	0.365897
$172$	0	0
$173$	− 6.66116i	− 0.506439i	−0.967409	−	0.253219i	$-0.918511\pi$
$173$	− 6.66116i	− 0.506439i	0.967409	−	0.253219i	$-0.0814894\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 8.39353i	− 0.630896i
$178$	0	0
$179$	−1.77316	−0.132532	−0.0662662	−	0.997802i	$-0.521109\pi$
$179$	−1.77316	−0.132532	−0.0662662	+	0.997802i	$0.521109\pi$
$180$	0	0
$181$	16.9027	1.25637	0.628183	−	0.778065i	$-0.283799\pi$
$181$	16.9027	1.25637	0.628183	+	0.778065i	$0.283799\pi$
$182$	0	0
$183$	− 3.65376i	− 0.270094i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.22813i	0.309191i
$188$	0	0
$189$	−4.05444	−0.294917
$190$	0	0
$191$	8.78473	0.635641	0.317820	−	0.948151i	$-0.397049\pi$
$191$	8.78473	0.635641	0.317820	+	0.948151i	$0.397049\pi$
$192$	0	0
$193$	− 12.4745i	− 0.897932i	−0.893549	−	0.448966i	$-0.851792\pi$
$193$	− 12.4745i	− 0.897932i	0.893549	−	0.448966i	$-0.148208\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 16.2500i	− 1.15776i	−0.815412	−	0.578881i	$-0.803490\pi$
$197$	− 16.2500i	− 1.15776i	0.815412	−	0.578881i	$-0.196510\pi$
$198$	0	0
$199$	15.6334	1.10822	0.554110	−	0.832443i	$-0.313059\pi$
$199$	15.6334	1.10822	0.554110	+	0.832443i	$0.313059\pi$
$200$	0	0
$201$	3.90854	0.275687
$202$	0	0
$203$	− 25.2232i	− 1.77032i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.103022i	0.00716050i
$208$	0	0
$209$	6.40718	0.443194
$210$	0	0
$211$	−23.1389	−1.59295	−0.796474	−	0.604673i	$-0.793304\pi$
$211$	−23.1389	−1.59295	−0.796474	+	0.604673i	$0.793304\pi$
$212$	0	0
$213$	− 7.08680i	− 0.485580i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.63297i	0.110853i
$218$	0	0
$219$	−2.86904	−0.193872
$220$	0	0
$221$	22.4932	1.51306
$222$	0	0
$223$	− 16.9651i	− 1.13606i	−0.823006	−	0.568032i	$-0.807705\pi$
$223$	− 16.9651i	− 1.13606i	0.823006	−	0.568032i	$-0.192295\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 10.7757i	− 0.715205i	−0.933874	−	0.357603i	$-0.883594\pi$
$227$	− 10.7757i	− 0.715205i	0.933874	−	0.357603i	$-0.116406\pi$
$228$	0	0
$229$	−18.5696	−1.22711	−0.613557	−	0.789650i	$-0.710262\pi$
$229$	−18.5696	−1.22711	−0.613557	+	0.789650i	$0.710262\pi$
$230$	0	0
$231$	−5.42926	−0.357219
$232$	0	0
$233$	1.65039i	0.108121i	0.998538	+	0.0540604i	$0.0172164\pi$
$233$	1.65039i	0.108121i	−0.998538	+	0.0540604i	$0.982784\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 12.4480i	− 0.808583i
$238$	0	0
$239$	−19.0660	−1.23328	−0.616639	−	0.787246i	$-0.711506\pi$
$239$	−19.0660	−1.23328	−0.616639	+	0.787246i	$0.711506\pi$
$240$	0	0
$241$	15.5752	1.00328	0.501642	−	0.865075i	$-0.332729\pi$
$241$	15.5752	1.00328	0.501642	+	0.865075i	$0.332729\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 34.0855i	− 2.16881i
$248$	0	0
$249$	−3.96869	−0.251505
$250$	0	0
$251$	18.2130	1.14959	0.574796	−	0.818297i	$-0.305082\pi$
$251$	18.2130	1.14959	0.574796	+	0.818297i	$0.305082\pi$
$252$	0	0
$253$	0.137955i	0.00867317i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.8505i	0.926348i	0.886267	+	0.463174i	$0.153289\pi$
$257$	14.8505i	0.926348i	−0.886267	+	0.463174i	$0.846711\pi$
$258$	0	0
$259$	−1.32793	−0.0825137
$260$	0	0
$261$	−6.22113	−0.385078
$262$	0	0
$263$	− 28.2291i	− 1.74068i	−0.492452	−	0.870340i	$-0.663899\pi$
$263$	− 28.2291i	− 1.74068i	0.492452	−	0.870340i	$-0.336101\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 18.0624i	− 1.10540i
$268$	0	0
$269$	2.33130	0.142142	0.0710710	−	0.997471i	$-0.477358\pi$
$269$	2.33130	0.142142	0.0710710	+	0.997471i	$0.477358\pi$
$270$	0	0
$271$	−9.84503	−0.598043	−0.299021	−	0.954246i	$-0.596660\pi$
$271$	−9.84503	−0.598043	−0.299021	+	0.954246i	$0.596660\pi$
$272$	0	0
$273$	28.8831i	1.74808i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 24.4616i	− 1.46976i	−0.678199	−	0.734878i	$-0.737239\pi$
$277$	− 24.4616i	− 1.46976i	0.678199	−	0.734878i	$-0.262761\pi$
$278$	0	0
$279$	0.402761	0.0241126
$280$	0	0
$281$	15.6447	0.933284	0.466642	−	0.884446i	$-0.345464\pi$
$281$	15.6447	0.933284	0.466642	+	0.884446i	$0.345464\pi$
$282$	0	0
$283$	9.82969i	0.584314i	0.956370	+	0.292157i	$0.0943730\pi$
$283$	9.82969i	0.584314i	−0.956370	+	0.292157i	$0.905627\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	30.9106i	1.82460i
$288$	0	0
$289$	7.03043	0.413555
$290$	0	0
$291$	−14.6355	−0.857946
$292$	0	0
$293$	17.3449i	1.01330i	0.862151	+	0.506651i	$0.169117\pi$
$293$	17.3449i	1.01330i	−0.862151	+	0.506651i	$0.830883\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.33909i	0.0777019i
$298$	0	0
$299$	0.733907	0.0424430
$300$	0	0
$301$	−37.2501	−2.14706
$302$	0	0
$303$	− 7.35988i	− 0.422814i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	22.1862i	1.26623i	0.774056	+	0.633117i	$0.218225\pi$
$307$	22.1862i	1.26623i	−0.774056	+	0.633117i	$0.781775\pi$
$308$	0	0
$309$	3.88060	0.220760
$310$	0	0
$311$	−19.9407	−1.13073	−0.565365	−	0.824841i	$-0.691265\pi$
$311$	−19.9407	−1.13073	−0.565365	+	0.824841i	$0.691265\pi$
$312$	0	0
$313$	− 10.9945i	− 0.621449i	−0.950500	−	0.310724i	$-0.899428\pi$
$313$	− 10.9945i	− 0.621449i	0.950500	−	0.310724i	$-0.100572\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.6249i	1.38307i	0.722341	+	0.691537i	$0.243066\pi$
$317$	24.6249i	1.38307i	−0.722341	+	0.691537i	$0.756934\pi$
$318$	0	0
$319$	−8.33066	−0.466427
$320$	0	0
$321$	−9.39706	−0.524493
$322$	0	0
$323$	15.1076i	0.840609i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.89361i	0.270617i
$328$	0	0
$329$	44.1906	2.43631
$330$	0	0
$331$	5.98481	0.328955	0.164478	−	0.986381i	$-0.447406\pi$
$331$	5.98481	0.328955	0.164478	+	0.986381i	$0.447406\pi$
$332$	0	0
$333$	0.327525i	0.0179483i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 21.3564i	− 1.16335i	−0.813420	−	0.581677i	$-0.802397\pi$
$337$	− 21.3564i	− 1.16335i	0.813420	−	0.581677i	$-0.197603\pi$
$338$	0	0
$339$	−16.8436	−0.914818
$340$	0	0
$341$	0.539332	0.0292065
$342$	0	0
$343$	9.88671i	0.533832i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.13096i	0.490176i	0.969501	+	0.245088i	$0.0788168\pi$
$347$	9.13096i	0.490176i	−0.969501	+	0.245088i	$0.921183\pi$
$348$	0	0
$349$	−14.9106	−0.798147	−0.399074	−	0.916919i	$-0.630668\pi$
$349$	−14.9106	−0.798147	−0.399074	+	0.916919i	$0.630668\pi$
$350$	0	0
$351$	7.12382	0.380241
$352$	0	0
$353$	15.7519i	0.838388i	0.907897	+	0.419194i	$0.137687\pi$
$353$	15.7519i	0.838388i	−0.907897	+	0.419194i	$0.862313\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 12.8017i	− 0.677540i
$358$	0	0
$359$	17.4164	0.919203	0.459601	−	0.888125i	$-0.347992\pi$
$359$	17.4164	0.919203	0.459601	+	0.888125i	$0.347992\pi$
$360$	0	0
$361$	3.89361	0.204927
$362$	0	0
$363$	− 9.20684i	− 0.483234i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.5432i	0.550351i	0.961394	+	0.275175i	$0.0887359\pi$
$367$	10.5432i	0.550351i	−0.961394	+	0.275175i	$0.911264\pi$
$368$	0	0
$369$	7.62389	0.396884
$370$	0	0
$371$	2.96090	0.153722
$372$	0	0
$373$	36.8381i	1.90741i	0.300750	+	0.953703i	$0.402763\pi$
$373$	36.8381i	1.90741i	−0.300750	+	0.953703i	$0.597237\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	44.3182i	2.28250i
$378$	0	0
$379$	32.4151	1.66505	0.832526	−	0.553985i	$-0.186894\pi$
$379$	32.4151	1.66505	0.832526	+	0.553985i	$0.186894\pi$
$380$	0	0
$381$	17.1711	0.879703
$382$	0	0
$383$	− 13.0618i	− 0.667429i	−0.942674	−	0.333714i	$-0.891698\pi$
$383$	− 13.0618i	− 0.667429i	0.942674	−	0.333714i	$-0.108302\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.18749i	0.467026i
$388$	0	0
$389$	8.43279	0.427559	0.213780	−	0.976882i	$-0.431423\pi$
$389$	8.43279	0.427559	0.213780	+	0.976882i	$0.431423\pi$
$390$	0	0
$391$	−0.325287	−0.0164505
$392$	0	0
$393$	6.34624i	0.320125i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 6.21309i	− 0.311826i	−0.987771	−	0.155913i	$-0.950168\pi$
$397$	− 6.21309i	− 0.311826i	0.987771	−	0.155913i	$-0.0498320\pi$
$398$	0	0
$399$	−19.3994	−0.971184
$400$	0	0
$401$	1.29845	0.0648416	0.0324208	−	0.999474i	$-0.489678\pi$
$401$	1.29845	0.0648416	0.0324208	+	0.999474i	$0.489678\pi$
$402$	0	0
$403$	− 2.86919i	− 0.142925i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.438586i	0.0217399i
$408$	0	0
$409$	13.5402	0.669521	0.334760	−	0.942303i	$-0.391345\pi$
$409$	13.5402	0.669521	0.334760	+	0.942303i	$0.391345\pi$
$410$	0	0
$411$	−18.9615	−0.935304
$412$	0	0
$413$	34.0311i	1.67456i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	15.1412i	0.741470i
$418$	0	0
$419$	−0.547371	−0.0267408	−0.0133704	−	0.999911i	$-0.504256\pi$
$419$	−0.547371	−0.0267408	−0.0133704	+	0.999911i	$0.504256\pi$
$420$	0	0
$421$	−0.319091	−0.0155515	−0.00777577	−	0.999970i	$-0.502475\pi$
$421$	−0.319091	−0.0155515	−0.00777577	+	0.999970i	$0.502475\pi$
$422$	0	0
$423$	− 10.8993i	− 0.529943i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	14.8140i	0.716898i
$428$	0	0
$429$	9.53943	0.460568
$430$	0	0
$431$	34.4929	1.66147	0.830733	−	0.556671i	$-0.187922\pi$
$431$	34.4929	1.66147	0.830733	+	0.556671i	$0.187922\pi$
$432$	0	0
$433$	− 24.6600i	− 1.18509i	−0.805539	−	0.592543i	$-0.798124\pi$
$433$	− 24.6600i	− 1.18509i	0.805539	−	0.592543i	$-0.201876\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.492930i	0.0235801i
$438$	0	0
$439$	−27.1693	−1.29672	−0.648361	−	0.761333i	$-0.724545\pi$
$439$	−27.1693	−1.29672	−0.648361	+	0.761333i	$0.724545\pi$
$440$	0	0
$441$	9.43849	0.449452
$442$	0	0
$443$	7.75124i	0.368272i	0.982901	+	0.184136i	$0.0589487\pi$
$443$	7.75124i	0.368272i	−0.982901	+	0.184136i	$0.941051\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 23.8194i	− 1.12662i
$448$	0	0
$449$	34.4432	1.62547	0.812737	−	0.582631i	$-0.197977\pi$
$449$	34.4432	1.62547	0.812737	+	0.582631i	$0.197977\pi$
$450$	0	0
$451$	10.2091	0.480727
$452$	0	0
$453$	− 20.1469i	− 0.946586i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11.0286i	0.515895i	0.966159	+	0.257948i	$0.0830462\pi$
$457$	11.0286i	0.515895i	−0.966159	+	0.257948i	$0.916954\pi$
$458$	0	0
$459$	−3.15746	−0.147378
$460$	0	0
$461$	−9.38574	−0.437138	−0.218569	−	0.975822i	$-0.570139\pi$
$461$	−9.38574	−0.437138	−0.218569	+	0.975822i	$0.570139\pi$
$462$	0	0
$463$	2.63515i	0.122466i	0.998124	+	0.0612328i	$0.0195032\pi$
$463$	2.63515i	0.122466i	−0.998124	+	0.0612328i	$0.980497\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	37.8923i	1.75345i	0.480994	+	0.876724i	$0.340276\pi$
$467$	37.8923i	1.75345i	−0.480994	+	0.876724i	$0.659724\pi$
$468$	0	0
$469$	−15.8470	−0.731744
$470$	0	0
$471$	6.15513	0.283613
$472$	0	0
$473$	12.3029i	0.565687i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 0.730286i	− 0.0334375i
$478$	0	0
$479$	−6.31492	−0.288536	−0.144268	−	0.989539i	$-0.546083\pi$
$479$	−6.31492	−0.288536	−0.144268	+	0.989539i	$0.546083\pi$
$480$	0	0
$481$	2.33323	0.106386
$482$	0	0
$483$	− 0.417695i	− 0.0190058i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 14.0946i	− 0.638687i	−0.947639	−	0.319343i	$-0.896538\pi$
$487$	− 14.0946i	− 0.638687i	0.947639	−	0.319343i	$-0.103462\pi$
$488$	0	0
$489$	8.16525	0.369245
$490$	0	0
$491$	−13.2922	−0.599869	−0.299934	−	0.953960i	$-0.596965\pi$
$491$	−13.2922	−0.599869	−0.299934	+	0.953960i	$0.596965\pi$
$492$	0	0
$493$	− 19.6430i	− 0.884676i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	28.7330i	1.28885i
$498$	0	0
$499$	37.6503	1.68546	0.842729	−	0.538337i	$-0.180947\pi$
$499$	37.6503	1.68546	0.842729	+	0.538337i	$0.180947\pi$
$500$	0	0
$501$	−10.6401	−0.475365
$502$	0	0
$503$	− 0.913200i	− 0.0407176i	−0.999793	−	0.0203588i	$-0.993519\pi$
$503$	− 0.913200i	− 0.0407176i	0.999793	−	0.0203588i	$-0.00648085\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 37.7488i	− 1.67648i
$508$	0	0
$509$	34.9232	1.54795	0.773973	−	0.633219i	$-0.218267\pi$
$509$	34.9232	1.54795	0.773973	+	0.633219i	$0.218267\pi$
$510$	0	0
$511$	11.6323	0.514584
$512$	0	0
$513$	4.78473i	0.211251i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 14.5952i	− 0.641894i
$518$	0	0
$519$	6.66116	0.292393
$520$	0	0
$521$	−28.4317	−1.24562	−0.622809	−	0.782374i	$-0.714009\pi$
$521$	−28.4317	−1.24562	−0.622809	+	0.782374i	$0.714009\pi$
$522$	0	0
$523$	− 16.4521i	− 0.719402i	−0.933068	−	0.359701i	$-0.882879\pi$
$523$	− 16.4521i	− 0.719402i	0.933068	−	0.359701i	$-0.117121\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.27170i	0.0553962i
$528$	0	0
$529$	22.9894	0.999539
$530$	0	0
$531$	8.39353	0.364248
$532$	0	0
$533$	− 54.3112i	− 2.35248i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1.77316i	− 0.0765176i
$538$	0	0
$539$	12.6390	0.544400
$540$	0	0
$541$	28.4455	1.22297	0.611484	−	0.791257i	$-0.290573\pi$
$541$	28.4455	1.22297	0.611484	+	0.791257i	$0.290573\pi$
$542$	0	0
$543$	16.9027i	0.725364i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 29.8588i	− 1.27667i	−0.769759	−	0.638334i	$-0.779624\pi$
$547$	− 29.8588i	− 1.27667i	0.769759	−	0.638334i	$-0.220376\pi$
$548$	0	0
$549$	3.65376	0.155939
$550$	0	0
$551$	−29.7664	−1.26809
$552$	0	0
$553$	50.4696i	2.14618i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 38.0727i	− 1.61319i	−0.591104	−	0.806595i	$-0.701308\pi$
$557$	− 38.0727i	− 1.61319i	0.591104	−	0.806595i	$-0.298692\pi$
$558$	0	0
$559$	65.4500	2.76824
$560$	0	0
$561$	−4.22813	−0.178512
$562$	0	0
$563$	− 29.4249i	− 1.24011i	−0.784558	−	0.620055i	$-0.787110\pi$
$563$	− 29.4249i	− 1.24011i	0.784558	−	0.620055i	$-0.212890\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 4.05444i	− 0.170271i
$568$	0	0
$569$	−26.8910	−1.12733	−0.563665	−	0.826003i	$-0.690609\pi$
$569$	−26.8910	−1.12733	−0.563665	+	0.826003i	$0.690609\pi$
$570$	0	0
$571$	23.4059	0.979506	0.489753	−	0.871861i	$-0.337087\pi$
$571$	23.4059	0.979506	0.489753	+	0.871861i	$0.337087\pi$
$572$	0	0
$573$	8.78473i	0.366987i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 14.8910i	− 0.619922i	−0.950749	−	0.309961i	$-0.899684\pi$
$577$	− 14.8910i	− 0.619922i	0.950749	−	0.309961i	$-0.100316\pi$
$578$	0	0
$579$	12.4745	0.518421
$580$	0	0
$581$	16.0908	0.667559
$582$	0	0
$583$	− 0.977918i	− 0.0405012i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 23.7896i	− 0.981900i	−0.871188	−	0.490950i	$-0.836650\pi$
$587$	− 23.7896i	− 0.981900i	0.871188	−	0.490950i	$-0.163350\pi$
$588$	0	0
$589$	1.92710	0.0794047
$590$	0	0
$591$	16.2500	0.668434
$592$	0	0
$593$	7.93152i	0.325708i	0.986650	+	0.162854i	$0.0520700\pi$
$593$	7.93152i	0.325708i	−0.986650	+	0.162854i	$0.947930\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	15.6334i	0.639831i
$598$	0	0
$599$	−46.7716	−1.91104	−0.955519	−	0.294931i	$-0.904703\pi$
$599$	−46.7716	−1.91104	−0.955519	+	0.294931i	$0.904703\pi$
$600$	0	0
$601$	−2.18877	−0.0892820	−0.0446410	−	0.999003i	$-0.514214\pi$
$601$	−2.18877	−0.0892820	−0.0446410	+	0.999003i	$0.514214\pi$
$602$	0	0
$603$	3.90854i	0.159168i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20.6382i	0.837678i	0.908061	+	0.418839i	$0.137563\pi$
$607$	20.6382i	0.837678i	−0.908061	+	0.418839i	$0.862437\pi$
$608$	0	0
$609$	25.2232	1.02210
$610$	0	0
$611$	−77.6447	−3.14117
$612$	0	0
$613$	45.4674i	1.83641i	0.396105	+	0.918205i	$0.370362\pi$
$613$	45.4674i	1.83641i	−0.396105	+	0.918205i	$0.629638\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.97648i	0.280862i	0.990090	+	0.140431i	$0.0448489\pi$
$617$	6.97648i	0.280862i	−0.990090	+	0.140431i	$0.955151\pi$
$618$	0	0
$619$	−11.1090	−0.446510	−0.223255	−	0.974760i	$-0.571668\pi$
$619$	−11.1090	−0.446510	−0.223255	+	0.974760i	$0.571668\pi$
$620$	0	0
$621$	−0.103022	−0.00413412
$622$	0	0
$623$	73.2329i	2.93401i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.40718i	0.255878i
$628$	0	0
$629$	−1.03415	−0.0412342
$630$	0	0
$631$	0.589056	0.0234500	0.0117250	−	0.999931i	$-0.496268\pi$
$631$	0.589056	0.0234500	0.0117250	+	0.999931i	$0.496268\pi$
$632$	0	0
$633$	− 23.1389i	− 0.919689i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 67.2381i	− 2.66407i
$638$	0	0
$639$	7.08680	0.280349
$640$	0	0
$641$	30.0504	1.18692	0.593460	−	0.804863i	$-0.297761\pi$
$641$	30.0504	1.18692	0.593460	+	0.804863i	$0.297761\pi$
$642$	0	0
$643$	− 9.47601i	− 0.373697i	−0.982389	−	0.186849i	$-0.940173\pi$
$643$	− 9.47601i	− 0.373697i	0.982389	−	0.186849i	$-0.0598274\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11.8150i	0.464496i	0.972657	+	0.232248i	$0.0746080\pi$
$647$	11.8150i	0.464496i	−0.972657	+	0.232248i	$0.925392\pi$
$648$	0	0
$649$	11.2397	0.441196
$650$	0	0
$651$	−1.63297	−0.0640011
$652$	0	0
$653$	0.0672925i	0.00263336i	0.999999	+	0.00131668i	$0.000419112\pi$
$653$	0.0672925i	0.00263336i	−0.999999	+	0.00131668i	$0.999581\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.86904i	− 0.111932i
$658$	0	0
$659$	21.0565	0.820246	0.410123	−	0.912030i	$-0.365486\pi$
$659$	21.0565	0.820246	0.410123	+	0.912030i	$0.365486\pi$
$660$	0	0
$661$	10.7212	0.417007	0.208503	−	0.978022i	$-0.433141\pi$
$661$	10.7212	0.417007	0.208503	+	0.978022i	$0.433141\pi$
$662$	0	0
$663$	22.4932i	0.873563i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 0.640911i	− 0.0248162i
$668$	0	0
$669$	16.9651	0.655907
$670$	0	0
$671$	4.89272	0.188881
$672$	0	0
$673$	− 30.2822i	− 1.16729i	−0.812008	−	0.583646i	$-0.801626\pi$
$673$	− 30.2822i	− 1.16729i	0.812008	−	0.583646i	$-0.198374\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.9028i	1.14926i	0.818414	+	0.574630i	$0.194854\pi$
$677$	29.9028i	1.14926i	−0.818414	+	0.574630i	$0.805146\pi$
$678$	0	0
$679$	59.3386	2.27721
$680$	0	0
$681$	10.7757	0.412924
$682$	0	0
$683$	8.49293i	0.324973i	0.986711	+	0.162486i	$0.0519514\pi$
$683$	8.49293i	0.324973i	−0.986711	+	0.162486i	$0.948049\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 18.5696i	− 0.708475i
$688$	0	0
$689$	−5.20242	−0.198196
$690$	0	0
$691$	−22.9983	−0.874897	−0.437449	−	0.899243i	$-0.644118\pi$
$691$	−22.9983	−0.874897	−0.437449	+	0.899243i	$0.644118\pi$
$692$	0	0
$693$	− 5.42926i	− 0.206241i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	24.0722i	0.911798i
$698$	0	0
$699$	−1.65039	−0.0624236
$700$	0	0
$701$	−10.5497	−0.398456	−0.199228	−	0.979953i	$-0.563844\pi$
$701$	−10.5497	−0.398456	−0.199228	+	0.979953i	$0.563844\pi$
$702$	0	0
$703$	1.56712i	0.0591050i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	29.8402i	1.12226i
$708$	0	0
$709$	−38.7411	−1.45495	−0.727476	−	0.686134i	$-0.759307\pi$
$709$	−38.7411	−1.45495	−0.727476	+	0.686134i	$0.759307\pi$
$710$	0	0
$711$	12.4480	0.466835
$712$	0	0
$713$	0.0414931i	0.00155393i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 19.0660i	− 0.712033i
$718$	0	0
$719$	−19.9535	−0.744140	−0.372070	−	0.928205i	$-0.621352\pi$
$719$	−19.9535	−0.744140	−0.372070	+	0.928205i	$0.621352\pi$
$720$	0	0
$721$	−15.7337	−0.585952
$722$	0	0
$723$	15.5752i	0.579246i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 4.98353i	− 0.184829i	−0.995721	−	0.0924144i	$-0.970542\pi$
$727$	− 4.98353i	− 0.184829i	0.995721	−	0.0924144i	$-0.0294584\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−29.0091	−1.07294
$732$	0	0
$733$	24.6102i	0.909000i	0.890747	+	0.454500i	$0.150182\pi$
$733$	24.6102i	0.909000i	−0.890747	+	0.454500i	$0.849818\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.23389i	0.192793i
$738$	0	0
$739$	28.1007	1.03370	0.516850	−	0.856076i	$-0.327104\pi$
$739$	28.1007	1.03370	0.516850	+	0.856076i	$0.327104\pi$
$740$	0	0
$741$	34.0855	1.25216
$742$	0	0
$743$	7.88253i	0.289182i	0.989492	+	0.144591i	$0.0461866\pi$
$743$	7.88253i	0.289182i	−0.989492	+	0.144591i	$0.953813\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 3.96869i	− 0.145207i
$748$	0	0
$749$	38.0998	1.39214
$750$	0	0
$751$	−21.3763	−0.780031	−0.390015	−	0.920808i	$-0.627530\pi$
$751$	−21.3763	−0.780031	−0.390015	+	0.920808i	$0.627530\pi$
$752$	0	0
$753$	18.2130i	0.663717i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 17.6845i	− 0.642755i	−0.946951	−	0.321377i	$-0.895854\pi$
$757$	− 17.6845i	− 0.642755i	0.946951	−	0.321377i	$-0.104146\pi$
$758$	0	0
$759$	−0.137955	−0.00500746
$760$	0	0
$761$	−30.5834	−1.10865	−0.554323	−	0.832302i	$-0.687023\pi$
$761$	−30.5834	−1.10865	−0.554323	+	0.832302i	$0.687023\pi$
$762$	0	0
$763$	− 19.8408i	− 0.718287i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 59.7940i	− 2.15904i
$768$	0	0
$769$	−8.39447	−0.302712	−0.151356	−	0.988479i	$-0.548364\pi$
$769$	−8.39447	−0.302712	−0.151356	+	0.988479i	$0.548364\pi$
$770$	0	0
$771$	−14.8505	−0.534827
$772$	0	0
$773$	− 42.8082i	− 1.53970i	−0.638223	−	0.769851i	$-0.720330\pi$
$773$	− 42.8082i	− 1.53970i	0.638223	−	0.769851i	$-0.279670\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 1.32793i	− 0.0476393i
$778$	0	0
$779$	36.4782	1.30697
$780$	0	0
$781$	9.48986	0.339574
$782$	0	0
$783$	− 6.22113i	− 0.222325i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	10.3286i	0.368174i	0.982910	+	0.184087i	$0.0589328\pi$
$787$	10.3286i	0.368174i	−0.982910	+	0.184087i	$0.941067\pi$
$788$	0	0
$789$	28.2291	1.00498
$790$	0	0
$791$	68.2913	2.42816
$792$	0	0
$793$	− 26.0287i	− 0.924308i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 2.84333i	− 0.100716i	−0.998731	−	0.0503580i	$-0.983964\pi$
$797$	− 2.84333i	− 0.100716i	0.998731	−	0.0503580i	$-0.0160362\pi$
$798$	0	0
$799$	34.4142	1.21749
$800$	0	0
$801$	18.0624	0.638203
$802$	0	0
$803$	− 3.84190i	− 0.135578i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.33130i	0.0820657i
$808$	0	0
$809$	−17.3440	−0.609783	−0.304891	−	0.952387i	$-0.598620\pi$
$809$	−17.3440	−0.609783	−0.304891	+	0.952387i	$0.598620\pi$
$810$	0	0
$811$	−11.8281	−0.415341	−0.207670	−	0.978199i	$-0.566588\pi$
$811$	−11.8281	−0.415341	−0.207670	+	0.978199i	$0.566588\pi$
$812$	0	0
$813$	− 9.84503i	− 0.345280i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	43.9596i	1.53795i
$818$	0	0
$819$	−28.8831	−1.00926
$820$	0	0
$821$	−19.9201	−0.695217	−0.347608	−	0.937640i	$-0.613006\pi$
$821$	−19.9201	−0.695217	−0.347608	+	0.937640i	$0.613006\pi$
$822$	0	0
$823$	− 37.0022i	− 1.28982i	−0.764260	−	0.644909i	$-0.776895\pi$
$823$	− 37.0022i	− 1.28982i	0.764260	−	0.644909i	$-0.223105\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.61973i	− 0.125870i	−0.998018	−	0.0629351i	$-0.979954\pi$
$827$	− 3.61973i	− 0.125870i	0.998018	−	0.0629351i	$-0.0200461\pi$
$828$	0	0
$829$	42.3796	1.47191	0.735953	−	0.677033i	$-0.236734\pi$
$829$	42.3796	1.47191	0.735953	+	0.677033i	$0.236734\pi$
$830$	0	0
$831$	24.4616	0.848564
$832$	0	0
$833$	29.8017i	1.03257i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0.402761i	0.0139214i
$838$	0	0
$839$	31.6705	1.09339	0.546694	−	0.837332i	$-0.315886\pi$
$839$	31.6705	1.09339	0.546694	+	0.837332i	$0.315886\pi$
$840$	0	0
$841$	9.70250	0.334569
$842$	0	0
$843$	15.6447i	0.538832i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	37.3286i	1.28263i
$848$	0	0
$849$	−9.82969	−0.337354
$850$	0	0
$851$	−0.0337422	−0.00115667
$852$	0	0
$853$	10.7240i	0.367184i	0.983003	+	0.183592i	$0.0587725\pi$
$853$	10.7240i	0.367184i	−0.983003	+	0.183592i	$0.941227\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.8384i	0.404391i	0.979345	+	0.202196i	$0.0648077\pi$
$857$	11.8384i	0.404391i	−0.979345	+	0.202196i	$0.935192\pi$
$858$	0	0
$859$	8.53452	0.291194	0.145597	−	0.989344i	$-0.453490\pi$
$859$	8.53452	0.291194	0.145597	+	0.989344i	$0.453490\pi$
$860$	0	0
$861$	−30.9106	−1.05343
$862$	0	0
$863$	− 38.1498i	− 1.29864i	−0.760517	−	0.649318i	$-0.775055\pi$
$863$	− 38.1498i	− 1.29864i	0.760517	−	0.649318i	$-0.224945\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	7.03043i	0.238766i
$868$	0	0
$869$	16.6689	0.565455
$870$	0	0
$871$	27.8437	0.943449
$872$	0	0
$873$	− 14.6355i	− 0.495335i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.01003i	− 0.0678737i	−0.999424	−	0.0339369i	$-0.989195\pi$
$877$	− 2.01003i	− 0.0678737i	0.999424	−	0.0339369i	$-0.0108045\pi$
$878$	0	0
$879$	−17.3449	−0.585031
$880$	0	0
$881$	2.16862	0.0730627	0.0365313	−	0.999333i	$-0.488369\pi$
$881$	2.16862	0.0730627	0.0365313	+	0.999333i	$0.488369\pi$
$882$	0	0
$883$	55.0263i	1.85178i	0.377792	+	0.925891i	$0.376683\pi$
$883$	55.0263i	1.85178i	−0.377792	+	0.925891i	$0.623317\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	50.3100i	1.68925i	0.535362	+	0.844623i	$0.320175\pi$
$887$	50.3100i	1.68925i	−0.535362	+	0.844623i	$0.679825\pi$
$888$	0	0
$889$	−69.6192	−2.33495
$890$	0	0
$891$	−1.33909	−0.0448612
$892$	0	0
$893$	− 52.1502i	− 1.74514i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0.733907i	0.0245045i
$898$	0	0
$899$	−2.50563	−0.0835673
$900$	0	0
$901$	2.30585	0.0768190
$902$	0	0
$903$	− 37.2501i	− 1.23961i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	13.1238i	0.435769i	0.975975	+	0.217885i	$0.0699156\pi$
$907$	13.1238i	0.435769i	−0.975975	+	0.217885i	$0.930084\pi$
$908$	0	0
$909$	7.35988	0.244112
$910$	0	0
$911$	33.8534	1.12161	0.560806	−	0.827947i	$-0.310491\pi$
$911$	33.8534	1.12161	0.560806	+	0.827947i	$0.310491\pi$
$912$	0	0
$913$	− 5.31443i	− 0.175882i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 25.7304i	− 0.849694i
$918$	0	0
$919$	−0.0908103	−0.00299556	−0.00149778	−	0.999999i	$-0.500477\pi$
$919$	−0.0908103	−0.00299556	−0.00149778	+	0.999999i	$0.500477\pi$
$920$	0	0
$921$	−22.1862	−0.731060
$922$	0	0
$923$	− 50.4851i	− 1.66174i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.88060i	0.127456i
$928$	0	0
$929$	22.6510	0.743156	0.371578	−	0.928402i	$-0.378817\pi$
$929$	22.6510	0.743156	0.371578	+	0.928402i	$0.378817\pi$
$930$	0	0
$931$	45.1606	1.48008
$932$	0	0
$933$	− 19.9407i	− 0.652828i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 9.99276i	− 0.326449i	−0.986589	−	0.163225i	$-0.947810\pi$
$937$	− 9.99276i	− 0.326449i	0.986589	−	0.163225i	$-0.0521895\pi$
$938$	0	0
$939$	10.9945	0.358793
$940$	0	0
$941$	−28.6412	−0.933675	−0.466838	−	0.884343i	$-0.654607\pi$
$941$	−28.6412	−0.933675	−0.466838	+	0.884343i	$0.654607\pi$
$942$	0	0
$943$	0.785426i	0.0255770i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 8.22217i	− 0.267185i	−0.991036	−	0.133592i	$-0.957349\pi$
$947$	− 8.22217i	− 0.267185i	0.991036	−	0.133592i	$-0.0426513\pi$
$948$	0	0
$949$	−20.4385	−0.663461
$950$	0	0
$951$	−24.6249	−0.798518
$952$	0	0
$953$	− 48.8140i	− 1.58124i	−0.612307	−	0.790620i	$-0.709759\pi$
$953$	− 48.8140i	− 1.58124i	0.612307	−	0.790620i	$-0.290241\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 8.33066i	− 0.269292i
$958$	0	0
$959$	76.8784	2.48253
$960$	0	0
$961$	−30.8378	−0.994767
$962$	0	0
$963$	− 9.39706i	− 0.302816i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	17.4141i	0.559999i	0.960000	+	0.279999i	$0.0903343\pi$
$967$	17.4141i	0.559999i	−0.960000	+	0.279999i	$0.909666\pi$
$968$	0	0
$969$	−15.1076	−0.485326
$970$	0	0
$971$	0.795340	0.0255237	0.0127618	−	0.999919i	$-0.495938\pi$
$971$	0.795340	0.0255237	0.0127618	+	0.999919i	$0.495938\pi$
$972$	0	0
$973$	− 61.3893i	− 1.96805i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.2594i	1.51196i	0.654593	+	0.755982i	$0.272840\pi$
$977$	47.2594i	1.51196i	−0.654593	+	0.755982i	$0.727160\pi$
$978$	0	0
$979$	24.1872	0.773025
$980$	0	0
$981$	−4.89361	−0.156241
$982$	0	0
$983$	18.3638i	0.585715i	0.956156	+	0.292857i	$0.0946061\pi$
$983$	18.3638i	0.585715i	−0.956156	+	0.292857i	$0.905394\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	44.1906i	1.40660i
$988$	0	0
$989$	−0.946510	−0.0300973
$990$	0	0
$991$	−55.5141	−1.76346	−0.881732	−	0.471750i	$-0.843622\pi$
$991$	−55.5141	−1.76346	−0.881732	+	0.471750i	$0.843622\pi$
$992$	0	0
$993$	5.98481i	0.189922i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	13.2769i	0.420482i	0.977650	+	0.210241i	$0.0674249\pi$
$997$	13.2769i	0.420482i	−0.977650	+	0.210241i	$0.932575\pi$
$998$	0	0
$999$	−0.327525	−0.0103624

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3000.2.f.e.1249.5		8
4.3	odd	2		6000.2.f.s.1249.4		8
5.2	odd	4		3000.2.a.n.1.4	yes	4
5.3	odd	4		3000.2.a.k.1.1	✓	4
5.4	even	2	inner	3000.2.f.e.1249.4		8
15.2	even	4		9000.2.a.x.1.4		4
15.8	even	4		9000.2.a.u.1.1		4
20.3	even	4		6000.2.a.bi.1.4		4
20.7	even	4		6000.2.a.bf.1.1		4
20.19	odd	2		6000.2.f.s.1249.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3000.2.a.k.1.1	✓	4	5.3	odd	4
3000.2.a.n.1.4	yes	4	5.2	odd	4
3000.2.f.e.1249.4		8	5.4	even	2	inner
3000.2.f.e.1249.5		8	1.1	even	1	trivial
6000.2.a.bf.1.1		4	20.7	even	4
6000.2.a.bi.1.4		4	20.3	even	4
6000.2.f.s.1249.4		8	4.3	odd	2
6000.2.f.s.1249.5		8	20.19	odd	2
9000.2.a.u.1.1		4	15.8	even	4
9000.2.a.x.1.4		4	15.2	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:	\( N \)	\(=\)	\( 3000 = 2^{3} \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3000.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -4.05444i q^{7} -1.00000 q^{9} -1.33909 q^{11} +7.12382i q^{13} -3.15746i q^{17} -4.78473 q^{19} +4.05444 q^{21} -0.103022i q^{23} -1.00000i q^{27} +6.22113 q^{29} -0.402761 q^{31} -1.33909i q^{33} -0.327525i q^{37} -7.12382 q^{39} -7.62389 q^{41} -9.18749i q^{43} +10.8993i q^{47} -9.43849 q^{49} +3.15746 q^{51} +0.730286i q^{53} -4.78473i q^{57} -8.39353 q^{59} -3.65376 q^{61} +4.05444i q^{63} -3.90854i q^{67} +0.103022 q^{69} -7.08680 q^{71} +2.86904i q^{73} +5.42926i q^{77} -12.4480 q^{79} +1.00000 q^{81} +3.96869i q^{83} +6.22113i q^{87} -18.0624 q^{89} +28.8831 q^{91} -0.402761i q^{93} +14.6355i q^{97} +1.33909 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} + 2 q^{11} - 16 q^{19} + 2 q^{21} + 6 q^{29} + 28 q^{31} - 22 q^{39} + 14 q^{41} - 30 q^{49} - 24 q^{59} - 6 q^{61} + 6 q^{69} + 66 q^{71} - 26 q^{79} + 8 q^{81} - 24 q^{89} + 34 q^{91} - 2 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 + 2 * q^11 - 16 * q^19 + 2 * q^21 + 6 * q^29 + 28 * q^31 - 22 * q^39 + 14 * q^41 - 30 * q^49 - 24 * q^59 - 6 * q^61 + 6 * q^69 + 66 * q^71 - 26 * q^79 + 8 * q^81 - 24 * q^89 + 34 * q^91 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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