LMFDB - Embedded newform 4536.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4536,2,Mod(1,4536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4536, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4536.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4536 = 2^{3} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4536.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$36.2201423569$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.22545.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 5x + 4 $ x^4 - x^3 - 6x^2 + 5x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 504)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.27060$ of defining polynomial
Character	$\chi$	$=$	4536.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.62393 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-3.62393 q^{5} +1.00000 q^{7} +3.91726 q^{11} +5.07288 q^{13} +1.03225 q^{17} -2.50895 q^{19} +4.95789 q^{23} +8.13288 q^{25} -9.20575 q^{29} -0.844387 q^{31} -3.62393 q^{35} +4.84439 q^{37} -4.14723 q^{41} -4.40348 q^{43} -7.87516 q^{47} +1.00000 q^{49} +12.2786 q^{53} -14.1959 q^{55} -11.2058 q^{59} +0.416697 q^{61} -18.3838 q^{65} +10.0501 q^{67} -5.05162 q^{71} +7.20723 q^{73} +3.91726 q^{77} +15.1314 q^{79} +1.86564 q^{83} -3.74080 q^{85} -0.669401 q^{89} +5.07288 q^{91} +9.09225 q^{95} +15.2703 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^5 + 4 * q^7	$ 4 q - 4 q^{5} + 4 q^{7} + 6 q^{11} + 3 q^{13} - 8 q^{17} - 2 q^{19} + 5 q^{23} + 14 q^{25} - q^{29} - 11 q^{31} - 4 q^{35} + 27 q^{37} - 2 q^{41} + 11 q^{43} - 7 q^{47} + 4 q^{49} - 4 q^{53} + 6 q^{55} - 9 q^{59} + 7 q^{61} + 9 q^{65} + 12 q^{67} - 12 q^{71} + 13 q^{73} + 6 q^{77} + 22 q^{79} + 6 q^{83} + 11 q^{85} - 14 q^{89} + 3 q^{91} + 23 q^{95} + q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 + 4 * q^7 + 6 * q^11 + 3 * q^13 - 8 * q^17 - 2 * q^19 + 5 * q^23 + 14 * q^25 - q^29 - 11 * q^31 - 4 * q^35 + 27 * q^37 - 2 * q^41 + 11 * q^43 - 7 * q^47 + 4 * q^49 - 4 * q^53 + 6 * q^55 - 9 * q^59 + 7 * q^61 + 9 * q^65 + 12 * q^67 - 12 * q^71 + 13 * q^73 + 6 * q^77 + 22 * q^79 + 6 * q^83 + 11 * q^85 - 14 * q^89 + 3 * q^91 + 23 * q^95 + q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.62393	−1.62067	−0.810336	−	0.585966i	$-0.800715\pi$
$5$	−3.62393	−1.62067	−0.810336	+	0.585966i	$0.800715\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.91726	1.18110	0.590550	−	0.807001i	$-0.298911\pi$
$11$	3.91726	1.18110	0.590550	+	0.807001i	$0.298911\pi$
$12$	0	0
$13$	5.07288	1.40696	0.703481	−	0.710714i	$-0.251628\pi$
$13$	5.07288	1.40696	0.703481	+	0.710714i	$0.251628\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.03225	0.250357	0.125178	−	0.992134i	$-0.460050\pi$
$17$	1.03225	0.250357	0.125178	+	0.992134i	$0.460050\pi$
$18$	0	0
$19$	−2.50895	−0.575592	−0.287796	−	0.957692i	$-0.592922\pi$
$19$	−2.50895	−0.575592	−0.287796	+	0.957692i	$0.592922\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.95789	1.03379	0.516896	−	0.856048i	$-0.327087\pi$
$23$	4.95789	1.03379	0.516896	+	0.856048i	$0.327087\pi$
$24$	0	0
$25$	8.13288	1.62658
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.20575	−1.70947	−0.854733	−	0.519068i	$-0.826279\pi$
$29$	−9.20575	−1.70947	−0.854733	+	0.519068i	$0.826279\pi$
$30$	0	0
$31$	−0.844387	−0.151656	−0.0758282	−	0.997121i	$-0.524160\pi$
$31$	−0.844387	−0.151656	−0.0758282	+	0.997121i	$0.524160\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.62393	−0.612556
$36$	0	0
$37$	4.84439	0.796412	0.398206	−	0.917296i	$-0.369633\pi$
$37$	4.84439	0.796412	0.398206	+	0.917296i	$0.369633\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.14723	−0.647689	−0.323844	−	0.946110i	$-0.604975\pi$
$41$	−4.14723	−0.647689	−0.323844	+	0.946110i	$0.604975\pi$
$42$	0	0
$43$	−4.40348	−0.671524	−0.335762	−	0.941947i	$-0.608994\pi$
$43$	−4.40348	−0.671524	−0.335762	+	0.941947i	$0.608994\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.87516	−1.14871	−0.574355	−	0.818607i	$-0.694747\pi$
$47$	−7.87516	−1.14871	−0.574355	+	0.818607i	$0.694747\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.2786	1.68660	0.843300	−	0.537443i	$-0.180610\pi$
$53$	12.2786	1.68660	0.843300	+	0.537443i	$0.180610\pi$
$54$	0	0
$55$	−14.1959	−1.91417
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.2058	−1.45886	−0.729432	−	0.684053i	$-0.760216\pi$
$59$	−11.2058	−1.45886	−0.729432	+	0.684053i	$0.760216\pi$
$60$	0	0
$61$	0.416697	0.0533525	0.0266763	−	0.999644i	$-0.491508\pi$
$61$	0.416697	0.0533525	0.0266763	+	0.999644i	$0.491508\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−18.3838	−2.28022
$66$	0	0
$67$	10.0501	1.22782	0.613910	−	0.789376i	$-0.289596\pi$
$67$	10.0501	1.22782	0.613910	+	0.789376i	$0.289596\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.05162	−0.599517	−0.299759	−	0.954015i	$-0.596906\pi$
$71$	−5.05162	−0.599517	−0.299759	+	0.954015i	$0.596906\pi$
$72$	0	0
$73$	7.20723	0.843543	0.421771	−	0.906702i	$-0.361408\pi$
$73$	7.20723	0.843543	0.421771	+	0.906702i	$0.361408\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.91726	0.446414
$78$	0	0
$79$	15.1314	1.70242	0.851208	−	0.524829i	$-0.175871\pi$
$79$	15.1314	1.70242	0.851208	+	0.524829i	$0.175871\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.86564	0.204781	0.102390	−	0.994744i	$-0.467351\pi$
$83$	1.86564	0.204781	0.102390	+	0.994744i	$0.467351\pi$
$84$	0	0
$85$	−3.74080	−0.405746
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.669401	−0.0709564	−0.0354782	−	0.999370i	$-0.511295\pi$
$89$	−0.669401	−0.0709564	−0.0354782	+	0.999370i	$0.511295\pi$
$90$	0	0
$91$	5.07288	0.531782
$92$	0	0
$93$	0	0
$94$	0	0
$95$	9.09225	0.932845
$96$	0	0
$97$	15.2703	1.55046	0.775230	−	0.631680i	$-0.217634\pi$
$97$	15.2703	1.55046	0.775230	+	0.631680i	$0.217634\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.7568	−1.16985	−0.584923	−	0.811089i	$-0.698875\pi$
$101$	−11.7568	−1.16985	−0.584923	+	0.811089i	$0.698875\pi$
$102$	0	0
$103$	11.0308	1.08689	0.543447	−	0.839444i	$-0.317119\pi$
$103$	11.0308	1.08689	0.543447	+	0.839444i	$0.317119\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.842907	−0.0814869	−0.0407434	−	0.999170i	$-0.512973\pi$
$107$	−0.842907	−0.0814869	−0.0407434	+	0.999170i	$0.512973\pi$
$108$	0	0
$109$	−17.1875	−1.64627	−0.823133	−	0.567849i	$-0.807776\pi$
$109$	−17.1875	−1.64627	−0.823133	+	0.567849i	$0.807776\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.09077	0.855188	0.427594	−	0.903971i	$-0.359361\pi$
$113$	9.09077	0.855188	0.427594	+	0.903971i	$0.359361\pi$
$114$	0	0
$115$	−17.9671	−1.67544
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.03225	0.0946260
$120$	0	0
$121$	4.34495	0.394996
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.3533	−1.01547
$126$	0	0
$127$	−14.4859	−1.28541	−0.642706	−	0.766113i	$-0.722188\pi$
$127$	−14.4859	−1.28541	−0.642706	+	0.766113i	$0.722188\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.06951	0.705037	0.352518	−	0.935805i	$-0.385325\pi$
$131$	8.06951	0.705037	0.352518	+	0.935805i	$0.385325\pi$
$132$	0	0
$133$	−2.50895	−0.217553
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.7158	1.68444	0.842219	−	0.539136i	$-0.181249\pi$
$137$	19.7158	1.68444	0.842219	+	0.539136i	$0.181249\pi$
$138$	0	0
$139$	16.7192	1.41810	0.709052	−	0.705156i	$-0.249123\pi$
$139$	16.7192	1.41810	0.709052	+	0.705156i	$0.249123\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	19.8718	1.66176
$144$	0	0
$145$	33.3610	2.77048
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.3288	1.50155	0.750776	−	0.660557i	$-0.229680\pi$
$149$	18.3288	1.50155	0.750776	+	0.660557i	$0.229680\pi$
$150$	0	0
$151$	−14.4620	−1.17690	−0.588450	−	0.808533i	$-0.700262\pi$
$151$	−14.4620	−1.17690	−0.588450	+	0.808533i	$0.700262\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.06000	0.245785
$156$	0	0
$157$	3.84775	0.307084	0.153542	−	0.988142i	$-0.450932\pi$
$157$	3.84775	0.307084	0.153542	+	0.988142i	$0.450932\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.95789	0.390737
$162$	0	0
$163$	13.0322	1.02076	0.510382	−	0.859948i	$-0.329504\pi$
$163$	13.0322	1.02076	0.510382	+	0.859948i	$0.329504\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.09077	−0.471318	−0.235659	−	0.971836i	$-0.575725\pi$
$167$	−6.09077	−0.471318	−0.235659	+	0.971836i	$0.575725\pi$
$168$	0	0
$169$	12.7341	0.979544
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.7971	0.896916	0.448458	−	0.893804i	$-0.351973\pi$
$173$	11.7971	0.896916	0.448458	+	0.893804i	$0.351973\pi$
$174$	0	0
$175$	8.13288	0.614788
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.06148	0.0793389	0.0396694	−	0.999213i	$-0.487370\pi$
$179$	1.06148	0.0793389	0.0396694	+	0.999213i	$0.487370\pi$
$180$	0	0
$181$	−16.0384	−1.19212	−0.596062	−	0.802938i	$-0.703269\pi$
$181$	−16.0384	−1.19212	−0.596062	+	0.802938i	$0.703269\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−17.5557	−1.29072
$186$	0	0
$187$	4.04359	0.295696
$188$	0	0
$189$	0	0
$190$	0	0
$191$	23.9352	1.73189	0.865944	−	0.500142i	$-0.166719\pi$
$191$	23.9352	1.73189	0.865944	+	0.500142i	$0.166719\pi$
$192$	0	0
$193$	13.2141	0.951174	0.475587	−	0.879669i	$-0.342236\pi$
$193$	13.2141	0.951174	0.475587	+	0.879669i	$0.342236\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−25.0403	−1.78405	−0.892023	−	0.451990i	$-0.850714\pi$
$197$	−25.0403	−1.78405	−0.892023	+	0.451990i	$0.850714\pi$
$198$	0	0
$199$	8.28159	0.587066	0.293533	−	0.955949i	$-0.405169\pi$
$199$	8.28159	0.587066	0.293533	+	0.955949i	$0.405169\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−9.20575	−0.646117
$204$	0	0
$205$	15.0293	1.04969
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−9.82820	−0.679831
$210$	0	0
$211$	3.59504	0.247493	0.123747	−	0.992314i	$-0.460509\pi$
$211$	3.59504	0.247493	0.123747	+	0.992314i	$0.460509\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	15.9579	1.08832
$216$	0	0
$217$	−0.844387	−0.0573207
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.23647	0.352243
$222$	0	0
$223$	10.1720	0.681169	0.340585	−	0.940214i	$-0.389375\pi$
$223$	10.1720	0.681169	0.340585	+	0.940214i	$0.389375\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.9011	0.922647	0.461324	−	0.887232i	$-0.347375\pi$
$227$	13.9011	0.922647	0.461324	+	0.887232i	$0.347375\pi$
$228$	0	0
$229$	−5.68695	−0.375804	−0.187902	−	0.982188i	$-0.560169\pi$
$229$	−5.68695	−0.375804	−0.187902	+	0.982188i	$0.560169\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.4775	1.27601	0.638006	−	0.770031i	$-0.279759\pi$
$233$	19.4775	1.27601	0.638006	+	0.770031i	$0.279759\pi$
$234$	0	0
$235$	28.5390	1.86168
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.00000	0.323423	0.161712	−	0.986838i	$-0.448299\pi$
$239$	5.00000	0.323423	0.161712	+	0.986838i	$0.448299\pi$
$240$	0	0
$241$	10.2816	0.662296	0.331148	−	0.943579i	$-0.392564\pi$
$241$	10.2816	0.662296	0.331148	+	0.943579i	$0.392564\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.62393	−0.231524
$246$	0	0
$247$	−12.7276	−0.809836
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−0.829685	−0.0523693	−0.0261846	−	0.999657i	$-0.508336\pi$
$251$	−0.829685	−0.0523693	−0.0261846	+	0.999657i	$0.508336\pi$
$252$	0	0
$253$	19.4214	1.22101
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.7503	−0.857721	−0.428860	−	0.903371i	$-0.641085\pi$
$257$	−13.7503	−0.857721	−0.428860	+	0.903371i	$0.641085\pi$
$258$	0	0
$259$	4.84439	0.301016
$260$	0	0
$261$	0	0
$262$	0	0
$263$	25.8571	1.59442	0.797208	−	0.603704i	$-0.206309\pi$
$263$	25.8571	1.59442	0.797208	+	0.603704i	$0.206309\pi$
$264$	0	0
$265$	−44.4969	−2.73342
$266$	0	0
$267$	0	0
$268$	0	0
$269$	9.32410	0.568500	0.284250	−	0.958750i	$-0.408255\pi$
$269$	9.32410	0.568500	0.284250	+	0.958750i	$0.408255\pi$
$270$	0	0
$271$	25.7421	1.56372	0.781861	−	0.623453i	$-0.214271\pi$
$271$	25.7421	1.56372	0.781861	+	0.623453i	$0.214271\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	31.8586	1.92115
$276$	0	0
$277$	−10.1314	−0.608737	−0.304368	−	0.952554i	$-0.598445\pi$
$277$	−10.1314	−0.608737	−0.304368	+	0.952554i	$0.598445\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6.95937	−0.415161	−0.207581	−	0.978218i	$-0.566559\pi$
$281$	−6.95937	−0.415161	−0.207581	+	0.978218i	$0.566559\pi$
$282$	0	0
$283$	−7.91840	−0.470700	−0.235350	−	0.971911i	$-0.575624\pi$
$283$	−7.91840	−0.470700	−0.235350	+	0.971911i	$0.575624\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.14723	−0.244803
$288$	0	0
$289$	−15.9345	−0.937321
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−11.2626	−0.657966	−0.328983	−	0.944336i	$-0.606706\pi$
$293$	−11.2626	−0.657966	−0.328983	+	0.944336i	$0.606706\pi$
$294$	0	0
$295$	40.6089	2.36434
$296$	0	0
$297$	0	0
$298$	0	0
$299$	25.1508	1.45451
$300$	0	0
$301$	−4.40348	−0.253812
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.51008	−0.0864669
$306$	0	0
$307$	23.0142	1.31349	0.656744	−	0.754113i	$-0.271933\pi$
$307$	23.0142	1.31349	0.656744	+	0.754113i	$0.271933\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−13.5766	−0.769860	−0.384930	−	0.922946i	$-0.625774\pi$
$311$	−13.5766	−0.769860	−0.384930	+	0.922946i	$0.625774\pi$
$312$	0	0
$313$	−17.8472	−1.00879	−0.504393	−	0.863474i	$-0.668284\pi$
$313$	−17.8472	−1.00879	−0.504393	+	0.863474i	$0.668284\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−19.3951	−1.08934	−0.544669	−	0.838651i	$-0.683345\pi$
$317$	−19.3951	−1.08934	−0.544669	+	0.838651i	$0.683345\pi$
$318$	0	0
$319$	−36.0614	−2.01905
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.58986	−0.144103
$324$	0	0
$325$	41.2571	2.28853
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7.87516	−0.434171
$330$	0	0
$331$	14.5746	0.801091	0.400546	−	0.916277i	$-0.368821\pi$
$331$	14.5746	0.801091	0.400546	+	0.916277i	$0.368821\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−36.4210	−1.98989
$336$	0	0
$337$	27.7924	1.51395	0.756975	−	0.653444i	$-0.226677\pi$
$337$	27.7924	1.51395	0.756975	+	0.653444i	$0.226677\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.30769	−0.179121
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.9121	0.800522	0.400261	−	0.916401i	$-0.368919\pi$
$347$	14.9121	0.800522	0.400261	+	0.916401i	$0.368919\pi$
$348$	0	0
$349$	−21.9158	−1.17312	−0.586562	−	0.809904i	$-0.699519\pi$
$349$	−21.9158	−1.17312	−0.586562	+	0.809904i	$0.699519\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.4620	0.822959	0.411480	−	0.911419i	$-0.365012\pi$
$353$	15.4620	0.822959	0.411480	+	0.911419i	$0.365012\pi$
$354$	0	0
$355$	18.3067	0.971620
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.56506	0.399269	0.199634	−	0.979870i	$-0.436025\pi$
$359$	7.56506	0.399269	0.199634	+	0.979870i	$0.436025\pi$
$360$	0	0
$361$	−12.7052	−0.668694
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−26.1185	−1.36711
$366$	0	0
$367$	−29.0092	−1.51427	−0.757133	−	0.653261i	$-0.773401\pi$
$367$	−29.0092	−1.51427	−0.757133	+	0.653261i	$0.773401\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	12.2786	0.637475
$372$	0	0
$373$	1.31105	0.0678835	0.0339418	−	0.999424i	$-0.489194\pi$
$373$	1.31105	0.0678835	0.0339418	+	0.999424i	$0.489194\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−46.6997	−2.40515
$378$	0	0
$379$	−15.7015	−0.806531	−0.403265	−	0.915083i	$-0.632125\pi$
$379$	−15.7015	−0.806531	−0.403265	+	0.915083i	$0.632125\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−20.9607	−1.07104	−0.535522	−	0.844521i	$-0.679885\pi$
$383$	−20.9607	−1.07104	−0.535522	+	0.844521i	$0.679885\pi$
$384$	0	0
$385$	−14.1959	−0.723490
$386$	0	0
$387$	0	0
$388$	0	0
$389$	11.6187	0.589094	0.294547	−	0.955637i	$-0.404831\pi$
$389$	11.6187	0.589094	0.294547	+	0.955637i	$0.404831\pi$
$390$	0	0
$391$	5.11778	0.258817
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−54.8351	−2.75906
$396$	0	0
$397$	−35.5217	−1.78278	−0.891390	−	0.453237i	$-0.850269\pi$
$397$	−35.5217	−1.78278	−0.891390	+	0.453237i	$0.850269\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.90775	−0.245081	−0.122541	−	0.992463i	$-0.539104\pi$
$401$	−4.90775	−0.245081	−0.122541	+	0.992463i	$0.539104\pi$
$402$	0	0
$403$	−4.28347	−0.213375
$404$	0	0
$405$	0	0
$406$	0	0
$407$	18.9767	0.940642
$408$	0	0
$409$	4.43123	0.219110	0.109555	−	0.993981i	$-0.465057\pi$
$409$	4.43123	0.219110	0.109555	+	0.993981i	$0.465057\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−11.2058	−0.551399
$414$	0	0
$415$	−6.76096	−0.331882
$416$	0	0
$417$	0	0
$418$	0	0
$419$	35.5439	1.73643	0.868216	−	0.496187i	$-0.165267\pi$
$419$	35.5439	1.73643	0.868216	+	0.496187i	$0.165267\pi$
$420$	0	0
$421$	−3.63846	−0.177328	−0.0886639	−	0.996062i	$-0.528260\pi$
$421$	−3.63846	−0.177328	−0.0886639	+	0.996062i	$0.528260\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.39515	0.407225
$426$	0	0
$427$	0.416697	0.0201654
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.9303	0.526495	0.263247	−	0.964728i	$-0.415206\pi$
$431$	10.9303	0.526495	0.263247	+	0.964728i	$0.415206\pi$
$432$	0	0
$433$	15.3189	0.736180	0.368090	−	0.929790i	$-0.380012\pi$
$433$	15.3189	0.736180	0.368090	+	0.929790i	$0.380012\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12.4391	−0.595042
$438$	0	0
$439$	10.6023	0.506022	0.253011	−	0.967463i	$-0.418579\pi$
$439$	10.6023	0.506022	0.253011	+	0.967463i	$0.418579\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−20.1361	−0.956694	−0.478347	−	0.878171i	$-0.658764\pi$
$443$	−20.1361	−0.956694	−0.478347	+	0.878171i	$0.658764\pi$
$444$	0	0
$445$	2.42586	0.114997
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.74616	−0.129599	−0.0647997	−	0.997898i	$-0.520641\pi$
$449$	−2.74616	−0.129599	−0.0647997	+	0.997898i	$0.520641\pi$
$450$	0	0
$451$	−16.2458	−0.764985
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−18.3838	−0.861844
$456$	0	0
$457$	25.5429	1.19485	0.597423	−	0.801926i	$-0.296191\pi$
$457$	25.5429	1.19485	0.597423	+	0.801926i	$0.296191\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−17.6463	−0.821871	−0.410936	−	0.911664i	$-0.634798\pi$
$461$	−17.6463	−0.821871	−0.410936	+	0.911664i	$0.634798\pi$
$462$	0	0
$463$	26.5002	1.23157	0.615785	−	0.787914i	$-0.288839\pi$
$463$	26.5002	1.23157	0.615785	+	0.787914i	$0.288839\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.1735	0.933519	0.466759	−	0.884384i	$-0.345421\pi$
$467$	20.1735	0.933519	0.466759	+	0.884384i	$0.345421\pi$
$468$	0	0
$469$	10.0501	0.464072
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−17.2496	−0.793136
$474$	0	0
$475$	−20.4050	−0.936244
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.84941	−0.450031	−0.225015	−	0.974355i	$-0.572243\pi$
$479$	−9.84941	−0.450031	−0.225015	+	0.974355i	$0.572243\pi$
$480$	0	0
$481$	24.5750	1.12052
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−55.3383	−2.51278
$486$	0	0
$487$	10.0278	0.454403	0.227202	−	0.973848i	$-0.427042\pi$
$487$	10.0278	0.454403	0.227202	+	0.973848i	$0.427042\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.22011	−0.0550628	−0.0275314	−	0.999621i	$-0.508765\pi$
$491$	−1.22011	−0.0550628	−0.0275314	+	0.999621i	$0.508765\pi$
$492$	0	0
$493$	−9.50262	−0.427977
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.05162	−0.226596
$498$	0	0
$499$	20.6444	0.924172	0.462086	−	0.886835i	$-0.347101\pi$
$499$	20.6444	0.924172	0.462086	+	0.886835i	$0.347101\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.85094	0.394644	0.197322	−	0.980339i	$-0.436776\pi$
$503$	8.85094	0.394644	0.197322	+	0.980339i	$0.436776\pi$
$504$	0	0
$505$	42.6059	1.89594
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−25.6920	−1.13878	−0.569388	−	0.822069i	$-0.692820\pi$
$509$	−25.6920	−1.13878	−0.569388	+	0.822069i	$0.692820\pi$
$510$	0	0
$511$	7.20723	0.318829
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−39.9747	−1.76150
$516$	0	0
$517$	−30.8491	−1.35674
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14.6797	−0.643129	−0.321565	−	0.946888i	$-0.604209\pi$
$521$	−14.6797	−0.643129	−0.321565	+	0.946888i	$0.604209\pi$
$522$	0	0
$523$	20.7922	0.909182	0.454591	−	0.890700i	$-0.349786\pi$
$523$	20.7922	0.909182	0.454591	+	0.890700i	$0.349786\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.871617	−0.0379682
$528$	0	0
$529$	1.58069	0.0687256
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−21.0384	−0.911274
$534$	0	0
$535$	3.05464	0.132063
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.91726	0.168728
$540$	0	0
$541$	−30.9593	−1.33104	−0.665521	−	0.746379i	$-0.731791\pi$
$541$	−30.9593	−1.33104	−0.665521	+	0.746379i	$0.731791\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	62.2864	2.66805
$546$	0	0
$547$	−11.6107	−0.496438	−0.248219	−	0.968704i	$-0.579845\pi$
$547$	−11.6107	−0.496438	−0.248219	+	0.968704i	$0.579845\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	23.0967	0.983954
$552$	0	0
$553$	15.1314	0.643452
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19.0116	0.805546	0.402773	−	0.915300i	$-0.368046\pi$
$557$	19.0116	0.805546	0.402773	+	0.915300i	$0.368046\pi$
$558$	0	0
$559$	−22.3383	−0.944809
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21.5917	−0.909983	−0.454992	−	0.890496i	$-0.650358\pi$
$563$	−21.5917	−0.909983	−0.454992	+	0.890496i	$0.650358\pi$
$564$	0	0
$565$	−32.9443	−1.38598
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.5772	0.778795	0.389397	−	0.921070i	$-0.372683\pi$
$569$	18.5772	0.778795	0.389397	+	0.921070i	$0.372683\pi$
$570$	0	0
$571$	−8.85805	−0.370698	−0.185349	−	0.982673i	$-0.559342\pi$
$571$	−8.85805	−0.370698	−0.185349	+	0.982673i	$0.559342\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	40.3219	1.68154
$576$	0	0
$577$	−3.76684	−0.156816	−0.0784078	−	0.996921i	$-0.524984\pi$
$577$	−3.76684	−0.156816	−0.0784078	+	0.996921i	$0.524984\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.86564	0.0773999
$582$	0	0
$583$	48.0986	1.99204
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.7523	0.691442	0.345721	−	0.938337i	$-0.387634\pi$
$587$	16.7523	0.691442	0.345721	+	0.938337i	$0.387634\pi$
$588$	0	0
$589$	2.11852	0.0872922
$590$	0	0
$591$	0	0
$592$	0	0
$593$	30.3776	1.24746	0.623729	−	0.781640i	$-0.285617\pi$
$593$	30.3776	1.24746	0.623729	+	0.781640i	$0.285617\pi$
$594$	0	0
$595$	−3.74080	−0.153358
$596$	0	0
$597$	0	0
$598$	0	0
$599$	17.9516	0.733481	0.366741	−	0.930323i	$-0.380474\pi$
$599$	17.9516	0.733481	0.366741	+	0.930323i	$0.380474\pi$
$600$	0	0
$601$	−5.33356	−0.217560	−0.108780	−	0.994066i	$-0.534694\pi$
$601$	−5.33356	−0.217560	−0.108780	+	0.994066i	$0.534694\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−15.7458	−0.640158
$606$	0	0
$607$	−18.2365	−0.740198	−0.370099	−	0.928992i	$-0.620676\pi$
$607$	−18.2365	−0.740198	−0.370099	+	0.928992i	$0.620676\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−39.9497	−1.61619
$612$	0	0
$613$	24.2030	0.977550	0.488775	−	0.872410i	$-0.337444\pi$
$613$	24.2030	0.977550	0.488775	+	0.872410i	$0.337444\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.31271	−0.254140	−0.127070	−	0.991894i	$-0.540557\pi$
$617$	−6.31271	−0.254140	−0.127070	+	0.991894i	$0.540557\pi$
$618$	0	0
$619$	14.0690	0.565481	0.282740	−	0.959197i	$-0.408757\pi$
$619$	14.0690	0.565481	0.282740	+	0.959197i	$0.408757\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.669401	−0.0268190
$624$	0	0
$625$	0.479312	0.0191725
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5.00061	0.199387
$630$	0	0
$631$	−1.75345	−0.0698036	−0.0349018	−	0.999391i	$-0.511112\pi$
$631$	−1.75345	−0.0698036	−0.0349018	+	0.999391i	$0.511112\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	52.4958	2.08323
$636$	0	0
$637$	5.07288	0.200995
$638$	0	0
$639$	0	0
$640$	0	0
$641$	27.3884	1.08178	0.540889	−	0.841094i	$-0.318088\pi$
$641$	27.3884	1.08178	0.540889	+	0.841094i	$0.318088\pi$
$642$	0	0
$643$	−42.6646	−1.68253	−0.841263	−	0.540626i	$-0.818187\pi$
$643$	−42.6646	−1.68253	−0.841263	+	0.540626i	$0.818187\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	45.7615	1.79907	0.899536	−	0.436847i	$-0.143905\pi$
$647$	45.7615	1.79907	0.899536	+	0.436847i	$0.143905\pi$
$648$	0	0
$649$	−43.8959	−1.72306
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−0.776529	−0.0303879	−0.0151940	−	0.999885i	$-0.504837\pi$
$653$	−0.776529	−0.0303879	−0.0151940	+	0.999885i	$0.504837\pi$
$654$	0	0
$655$	−29.2434	−1.14263
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−11.9587	−0.465844	−0.232922	−	0.972495i	$-0.574829\pi$
$659$	−11.9587	−0.465844	−0.232922	+	0.972495i	$0.574829\pi$
$660$	0	0
$661$	12.6275	0.491152	0.245576	−	0.969377i	$-0.421023\pi$
$661$	12.6275	0.491152	0.245576	+	0.969377i	$0.421023\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.09225	0.352582
$666$	0	0
$667$	−45.6411	−1.76723
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.63231	0.0630147
$672$	0	0
$673$	29.1471	1.12354	0.561768	−	0.827295i	$-0.310121\pi$
$673$	29.1471	1.12354	0.561768	+	0.827295i	$0.310121\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.85465	−0.301879	−0.150939	−	0.988543i	$-0.548230\pi$
$677$	−7.85465	−0.301879	−0.150939	+	0.988543i	$0.548230\pi$
$678$	0	0
$679$	15.2703	0.586018
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.87441	0.339570	0.169785	−	0.985481i	$-0.445693\pi$
$683$	8.87441	0.339570	0.169785	+	0.985481i	$0.445693\pi$
$684$	0	0
$685$	−71.4488	−2.72992
$686$	0	0
$687$	0	0
$688$	0	0
$689$	62.2880	2.37298
$690$	0	0
$691$	−3.64017	−0.138478	−0.0692392	−	0.997600i	$-0.522057\pi$
$691$	−3.64017	−0.138478	−0.0692392	+	0.997600i	$0.522057\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−60.5892	−2.29828
$696$	0	0
$697$	−4.28097	−0.162153
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.1003	0.683638	0.341819	−	0.939766i	$-0.388957\pi$
$701$	18.1003	0.683638	0.341819	+	0.939766i	$0.388957\pi$
$702$	0	0
$703$	−12.1543	−0.458408
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−11.7568	−0.442160
$708$	0	0
$709$	35.1249	1.31914	0.659572	−	0.751642i	$-0.270738\pi$
$709$	35.1249	1.31914	0.659572	+	0.751642i	$0.270738\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−4.18638	−0.156781
$714$	0	0
$715$	−72.0140	−2.69317
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−23.8092	−0.887933	−0.443966	−	0.896044i	$-0.646429\pi$
$719$	−23.8092	−0.887933	−0.443966	+	0.896044i	$0.646429\pi$
$720$	0	0
$721$	11.0308	0.410807
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−74.8693	−2.78058
$726$	0	0
$727$	2.58501	0.0958729	0.0479364	−	0.998850i	$-0.484736\pi$
$727$	2.58501	0.0958729	0.0479364	+	0.998850i	$0.484736\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.54548	−0.168121
$732$	0	0
$733$	−23.4987	−0.867944	−0.433972	−	0.900926i	$-0.642888\pi$
$733$	−23.4987	−0.867944	−0.433972	+	0.900926i	$0.642888\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	39.3691	1.45018
$738$	0	0
$739$	16.9236	0.622544	0.311272	−	0.950321i	$-0.399245\pi$
$739$	16.9236	0.622544	0.311272	+	0.950321i	$0.399245\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.78325	0.322226	0.161113	−	0.986936i	$-0.448492\pi$
$743$	8.78325	0.322226	0.161113	+	0.986936i	$0.448492\pi$
$744$	0	0
$745$	−66.4222	−2.43352
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−0.842907	−0.0307991
$750$	0	0
$751$	1.96775	0.0718043	0.0359021	−	0.999355i	$-0.488570\pi$
$751$	1.96775	0.0718043	0.0359021	+	0.999355i	$0.488570\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	52.4093	1.90737
$756$	0	0
$757$	−10.3423	−0.375899	−0.187949	−	0.982179i	$-0.560184\pi$
$757$	−10.3423	−0.375899	−0.187949	+	0.982179i	$0.560184\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−15.7641	−0.571449	−0.285724	−	0.958312i	$-0.592234\pi$
$761$	−15.7641	−0.571449	−0.285724	+	0.958312i	$0.592234\pi$
$762$	0	0
$763$	−17.1875	−0.622230
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−56.8454	−2.05257
$768$	0	0
$769$	−44.3789	−1.60034	−0.800172	−	0.599770i	$-0.795259\pi$
$769$	−44.3789	−1.60034	−0.800172	+	0.599770i	$0.795259\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−25.5222	−0.917969	−0.458984	−	0.888444i	$-0.651787\pi$
$773$	−25.5222	−0.917969	−0.458984	+	0.888444i	$0.651787\pi$
$774$	0	0
$775$	−6.86730	−0.246681
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.4052	0.372804
$780$	0	0
$781$	−19.7885	−0.708089
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−13.9440	−0.497682
$786$	0	0
$787$	24.5683	0.875764	0.437882	−	0.899033i	$-0.355729\pi$
$787$	24.5683	0.875764	0.437882	+	0.899033i	$0.355729\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9.09077	0.323231
$792$	0	0
$793$	2.11385	0.0750650
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−37.3966	−1.32466	−0.662328	−	0.749214i	$-0.730432\pi$
$797$	−37.3966	−1.32466	−0.662328	+	0.749214i	$0.730432\pi$
$798$	0	0
$799$	−8.12912	−0.287587
$800$	0	0
$801$	0	0
$802$	0	0
$803$	28.2326	0.996308
$804$	0	0
$805$	−17.9671	−0.633256
$806$	0	0
$807$	0	0
$808$	0	0
$809$	12.6565	0.444980	0.222490	−	0.974935i	$-0.428582\pi$
$809$	12.6565	0.444980	0.222490	+	0.974935i	$0.428582\pi$
$810$	0	0
$811$	−42.2499	−1.48360	−0.741798	−	0.670624i	$-0.766027\pi$
$811$	−42.2499	−1.48360	−0.741798	+	0.670624i	$0.766027\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−47.2280	−1.65432
$816$	0	0
$817$	11.0481	0.386524
$818$	0	0
$819$	0	0
$820$	0	0
$821$	39.4436	1.37659	0.688295	−	0.725431i	$-0.258359\pi$
$821$	39.4436	1.37659	0.688295	+	0.725431i	$0.258359\pi$
$822$	0	0
$823$	−2.66416	−0.0928667	−0.0464334	−	0.998921i	$-0.514786\pi$
$823$	−2.66416	−0.0928667	−0.0464334	+	0.998921i	$0.514786\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.13009	−0.0740704	−0.0370352	−	0.999314i	$-0.511791\pi$
$827$	−2.13009	−0.0740704	−0.0370352	+	0.999314i	$0.511791\pi$
$828$	0	0
$829$	−7.61167	−0.264364	−0.132182	−	0.991225i	$-0.542198\pi$
$829$	−7.61167	−0.264364	−0.132182	+	0.991225i	$0.542198\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.03225	0.0357653
$834$	0	0
$835$	22.0725	0.763851
$836$	0	0
$837$	0	0
$838$	0	0
$839$	39.9085	1.37780	0.688898	−	0.724858i	$-0.258095\pi$
$839$	39.9085	1.37780	0.688898	+	0.724858i	$0.258095\pi$
$840$	0	0
$841$	55.7459	1.92227
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−46.1474	−1.58752
$846$	0	0
$847$	4.34495	0.149294
$848$	0	0
$849$	0	0
$850$	0	0
$851$	24.0179	0.823325
$852$	0	0
$853$	−48.2008	−1.65036	−0.825182	−	0.564867i	$-0.808927\pi$
$853$	−48.2008	−1.65036	−0.825182	+	0.564867i	$0.808927\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	40.6734	1.38938	0.694688	−	0.719311i	$-0.255542\pi$
$857$	40.6734	1.38938	0.694688	+	0.719311i	$0.255542\pi$
$858$	0	0
$859$	−10.1165	−0.345171	−0.172586	−	0.984995i	$-0.555212\pi$
$859$	−10.1165	−0.345171	−0.172586	+	0.984995i	$0.555212\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	16.5930	0.564832	0.282416	−	0.959292i	$-0.408864\pi$
$863$	16.5930	0.564832	0.282416	+	0.959292i	$0.408864\pi$
$864$	0	0
$865$	−42.7518	−1.45361
$866$	0	0
$867$	0	0
$868$	0	0
$869$	59.2737	2.01072
$870$	0	0
$871$	50.9831	1.72750
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−11.3533	−0.383813
$876$	0	0
$877$	37.9298	1.28080	0.640399	−	0.768042i	$-0.278769\pi$
$877$	37.9298	1.28080	0.640399	+	0.768042i	$0.278769\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20.9737	0.706623	0.353312	−	0.935506i	$-0.385056\pi$
$881$	20.9737	0.706623	0.353312	+	0.935506i	$0.385056\pi$
$882$	0	0
$883$	39.6536	1.33445	0.667225	−	0.744856i	$-0.267482\pi$
$883$	39.6536	1.33445	0.667225	+	0.744856i	$0.267482\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.2320	0.780055	0.390028	−	0.920803i	$-0.372465\pi$
$887$	23.2320	0.780055	0.390028	+	0.920803i	$0.372465\pi$
$888$	0	0
$889$	−14.4859	−0.485840
$890$	0	0
$891$	0	0
$892$	0	0
$893$	19.7583	0.661188
$894$	0	0
$895$	−3.84674	−0.128582
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.77322	0.259251
$900$	0	0
$901$	12.6746	0.422252
$902$	0	0
$903$	0	0
$904$	0	0
$905$	58.1221	1.93204
$906$	0	0
$907$	−42.3823	−1.40728	−0.703640	−	0.710556i	$-0.748443\pi$
$907$	−42.3823	−1.40728	−0.703640	+	0.710556i	$0.748443\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	19.3364	0.640644	0.320322	−	0.947309i	$-0.396209\pi$
$911$	19.3364	0.640644	0.320322	+	0.947309i	$0.396209\pi$
$912$	0	0
$913$	7.30821	0.241866
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.06951	0.266479
$918$	0	0
$919$	28.4866	0.939685	0.469842	−	0.882750i	$-0.344311\pi$
$919$	28.4866	0.939685	0.469842	+	0.882750i	$0.344311\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−25.6263	−0.843498
$924$	0	0
$925$	39.3988	1.29542
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−25.1860	−0.826327	−0.413163	−	0.910657i	$-0.635576\pi$
$929$	−25.1860	−0.826327	−0.413163	+	0.910657i	$0.635576\pi$
$930$	0	0
$931$	−2.50895	−0.0822274
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−14.6537	−0.479227
$936$	0	0
$937$	−17.4922	−0.571445	−0.285722	−	0.958312i	$-0.592234\pi$
$937$	−17.4922	−0.571445	−0.285722	+	0.958312i	$0.592234\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−31.9154	−1.04041	−0.520207	−	0.854040i	$-0.674145\pi$
$941$	−31.9154	−1.04041	−0.520207	+	0.854040i	$0.674145\pi$
$942$	0	0
$943$	−20.5615	−0.669576
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16.0964	−0.523063	−0.261531	−	0.965195i	$-0.584227\pi$
$947$	−16.0964	−0.523063	−0.261531	+	0.965195i	$0.584227\pi$
$948$	0	0
$949$	36.5614	1.18683
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24.0358	0.778595	0.389298	−	0.921112i	$-0.372718\pi$
$953$	24.0358	0.778595	0.389298	+	0.921112i	$0.372718\pi$
$954$	0	0
$955$	−86.7394	−2.80682
$956$	0	0
$957$	0	0
$958$	0	0
$959$	19.7158	0.636657
$960$	0	0
$961$	−30.2870	−0.977000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−47.8871	−1.54154
$966$	0	0
$967$	−12.5897	−0.404857	−0.202428	−	0.979297i	$-0.564883\pi$
$967$	−12.5897	−0.404857	−0.202428	+	0.979297i	$0.564883\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	30.7939	0.988223	0.494112	−	0.869398i	$-0.335493\pi$
$971$	30.7939	0.988223	0.494112	+	0.869398i	$0.335493\pi$
$972$	0	0
$973$	16.7192	0.535993
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.56564	−0.242046	−0.121023	−	0.992650i	$-0.538618\pi$
$977$	−7.56564	−0.242046	−0.121023	+	0.992650i	$0.538618\pi$
$978$	0	0
$979$	−2.62222	−0.0838065
$980$	0	0
$981$	0	0
$982$	0	0
$983$	26.7647	0.853660	0.426830	−	0.904332i	$-0.359630\pi$
$983$	26.7647	0.853660	0.426830	+	0.904332i	$0.359630\pi$
$984$	0	0
$985$	90.7443	2.89135
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−21.8320	−0.694216
$990$	0	0
$991$	−48.6851	−1.54653	−0.773266	−	0.634082i	$-0.781378\pi$
$991$	−48.6851	−1.54653	−0.773266	+	0.634082i	$0.781378\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−30.0119	−0.951442
$996$	0	0
$997$	34.1270	1.08081	0.540406	−	0.841404i	$-0.318271\pi$
$997$	34.1270	1.08081	0.540406	+	0.841404i	$0.318271\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4536.2.a.x.1.2		4
3.2	odd	2		4536.2.a.ba.1.3		4
4.3	odd	2		9072.2.a.ce.1.2		4
9.2	odd	6		1512.2.r.d.1009.2		8
9.4	even	3		504.2.r.d.169.4	✓	8
9.5	odd	6		1512.2.r.d.505.2		8
9.7	even	3		504.2.r.d.337.4	yes	8
12.11	even	2		9072.2.a.cl.1.3		4
36.7	odd	6		1008.2.r.m.337.1		8
36.11	even	6		3024.2.r.l.1009.2		8
36.23	even	6		3024.2.r.l.2017.2		8
36.31	odd	6		1008.2.r.m.673.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.r.d.169.4	✓	8	9.4	even	3
504.2.r.d.337.4	yes	8	9.7	even	3
1008.2.r.m.337.1		8	36.7	odd	6
1008.2.r.m.673.1		8	36.31	odd	6
1512.2.r.d.505.2		8	9.5	odd	6
1512.2.r.d.1009.2		8	9.2	odd	6
3024.2.r.l.1009.2		8	36.11	even	6
3024.2.r.l.2017.2		8	36.23	even	6
4536.2.a.x.1.2		4	1.1	even	1	trivial
4536.2.a.ba.1.3		4	3.2	odd	2
9072.2.a.ce.1.2		4	4.3	odd	2
9072.2.a.cl.1.3		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4536 = 2^{3} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4536.a (trivial)

\(f(q)\)	\(=\)	\(q-3.62393 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-3.62393 q^{5} +1.00000 q^{7} +3.91726 q^{11} +5.07288 q^{13} +1.03225 q^{17} -2.50895 q^{19} +4.95789 q^{23} +8.13288 q^{25} -9.20575 q^{29} -0.844387 q^{31} -3.62393 q^{35} +4.84439 q^{37} -4.14723 q^{41} -4.40348 q^{43} -7.87516 q^{47} +1.00000 q^{49} +12.2786 q^{53} -14.1959 q^{55} -11.2058 q^{59} +0.416697 q^{61} -18.3838 q^{65} +10.0501 q^{67} -5.05162 q^{71} +7.20723 q^{73} +3.91726 q^{77} +15.1314 q^{79} +1.86564 q^{83} -3.74080 q^{85} -0.669401 q^{89} +5.07288 q^{91} +9.09225 q^{95} +15.2703 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^5 + 4 * q^7	\( 4 q - 4 q^{5} + 4 q^{7} + 6 q^{11} + 3 q^{13} - 8 q^{17} - 2 q^{19} + 5 q^{23} + 14 q^{25} - q^{29} - 11 q^{31} - 4 q^{35} + 27 q^{37} - 2 q^{41} + 11 q^{43} - 7 q^{47} + 4 q^{49} - 4 q^{53} + 6 q^{55} - 9 q^{59} + 7 q^{61} + 9 q^{65} + 12 q^{67} - 12 q^{71} + 13 q^{73} + 6 q^{77} + 22 q^{79} + 6 q^{83} + 11 q^{85} - 14 q^{89} + 3 q^{91} + 23 q^{95} + q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 + 4 * q^7 + 6 * q^11 + 3 * q^13 - 8 * q^17 - 2 * q^19 + 5 * q^23 + 14 * q^25 - q^29 - 11 * q^31 - 4 * q^35 + 27 * q^37 - 2 * q^41 + 11 * q^43 - 7 * q^47 + 4 * q^49 - 4 * q^53 + 6 * q^55 - 9 * q^59 + 7 * q^61 + 9 * q^65 + 12 * q^67 - 12 * q^71 + 13 * q^73 + 6 * q^77 + 22 * q^79 + 6 * q^83 + 11 * q^85 - 14 * q^89 + 3 * q^91 + 23 * q^95 + q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more