LMFDB - Embedded newform 1305.2.a.r.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1305,2,Mod(1,1305)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1305.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1305, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1305.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,3,0,5,4,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$10.4204774638$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2225.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.75660$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.75660 q^{2} +5.59883 q^{4} +1.00000 q^{5} +0.393832 q^{7} +9.92054 q^{8} +2.75660 q^{10} +0.393832 q^{11} -2.56511 q^{13} +1.08564 q^{14} +16.1493 q^{16} -2.07830 q^{17} -0.958939 q^{19} +5.59883 q^{20} +1.08564 q^{22} -6.15661 q^{23} +1.00000 q^{25} -7.07097 q^{26} +2.20500 q^{28} +1.00000 q^{29} -10.1566 q^{31} +24.6760 q^{32} -5.72905 q^{34} +0.393832 q^{35} -7.34192 q^{37} -2.64341 q^{38} +9.92054 q^{40} +1.65745 q^{41} +10.3279 q^{43} +2.20500 q^{44} -16.9713 q^{46} +11.5915 q^{47} -6.84490 q^{49} +2.75660 q^{50} -14.3616 q^{52} +12.3279 q^{53} +0.393832 q^{55} +3.90703 q^{56} +2.75660 q^{58} +9.54022 q^{59} -6.25340 q^{61} -27.9977 q^{62} +35.7232 q^{64} -2.56511 q^{65} -7.42023 q^{67} -11.6361 q^{68} +1.08564 q^{70} -5.98533 q^{71} +3.34192 q^{73} -20.2387 q^{74} -5.36894 q^{76} +0.155104 q^{77} -2.06745 q^{79} +16.1493 q^{80} +4.56892 q^{82} -6.41000 q^{83} -2.07830 q^{85} +28.4698 q^{86} +3.90703 q^{88} -15.8302 q^{89} -1.01022 q^{91} -34.4698 q^{92} +31.9531 q^{94} -0.958939 q^{95} -18.4575 q^{97} -18.8686 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{2} + 5 q^{4} + 4 q^{5} + 2 q^{7} + 12 q^{8} + 3 q^{10} + 2 q^{11} - 8 q^{13} + 3 q^{14} + 11 q^{16} + 10 q^{17} - 2 q^{19} + 5 q^{20} + 3 q^{22} + 12 q^{23} + 4 q^{25} + 7 q^{26} - 9 q^{28} + 4 q^{29}+ \cdots - 34 q^{98}+O(q^{100}) $ 4 * q + 3 * q^2 + 5 * q^4 + 4 * q^5 + 2 * q^7 + 12 * q^8 + 3 * q^10 + 2 * q^11 - 8 * q^13 + 3 * q^14 + 11 * q^16 + 10 * q^17 - 2 * q^19 + 5 * q^20 + 3 * q^22 + 12 * q^23 + 4 * q^25 + 7 * q^26 - 9 * q^28 + 4 * q^29 - 4 * q^31 + 17 * q^32 - q^34 + 2 * q^35 - 16 * q^37 + 10 * q^38 + 12 * q^40 + 12 * q^41 + 2 * q^43 - 9 * q^44 - 8 * q^46 + 12 * q^47 + 6 * q^49 + 3 * q^50 - 3 * q^52 + 10 * q^53 + 2 * q^55 + 3 * q^58 - 2 * q^59 - 26 * q^61 - 20 * q^62 + 34 * q^64 - 8 * q^65 + 2 * q^67 - 9 * q^68 + 3 * q^70 + 10 * q^71 - 48 * q^74 + 16 * q^76 + 34 * q^77 + 22 * q^79 + 11 * q^80 + 38 * q^82 + 10 * q^83 + 10 * q^85 + 4 * q^86 + 4 * q^89 - 8 * q^91 - 28 * q^92 + 39 * q^94 - 2 * q^95 - 22 * q^97 - 34 * q^98

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 $

$2$	2.75660	1.94921	0.974605	−	0.223932i	$-0.0718895\pi$
$2$	2.75660	1.94921	0.974605	+	0.223932i	$0.0718895\pi$
$3$	0	0
$3$	0	0
$4$	5.59883	2.79942
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.393832	0.148855	0.0744273	−	0.997226i	$-0.476287\pi$
$7$	0.393832	0.148855	0.0744273	+	0.997226i	$0.476287\pi$
$8$	9.92054	3.50744
$9$	0	0
$10$	2.75660	0.871713
$11$	0.393832	0.118745	0.0593725	−	0.998236i	$-0.481090\pi$
$11$	0.393832	0.118745	0.0593725	+	0.998236i	$0.481090\pi$
$12$	0	0
$13$	−2.56511	−0.711433	−0.355716	−	0.934594i	$-0.615763\pi$
$13$	−2.56511	−0.711433	−0.355716	+	0.934594i	$0.615763\pi$
$14$	1.08564	0.290149
$15$	0	0
$16$	16.1493	4.03732
$17$	−2.07830	−0.504063	−0.252031	−	0.967719i	$-0.581099\pi$
$17$	−2.07830	−0.504063	−0.252031	+	0.967719i	$0.581099\pi$
$18$	0	0
$19$	−0.958939	−0.219996	−0.109998	−	0.993932i	$-0.535084\pi$
$19$	−0.958939	−0.219996	−0.109998	+	0.993932i	$0.535084\pi$
$20$	5.59883	1.25194
$21$	0	0
$22$	1.08564	0.231459
$23$	−6.15661	−1.28374	−0.641871	−	0.766813i	$-0.721841\pi$
$23$	−6.15661	−1.28374	−0.641871	+	0.766813i	$0.721841\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−7.07097	−1.38673
$27$	0	0
$28$	2.20500	0.416706
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−10.1566	−1.82418	−0.912090	−	0.409989i	$-0.865532\pi$
$31$	−10.1566	−1.82418	−0.912090	+	0.409989i	$0.865532\pi$
$32$	24.6760	4.36214
$33$	0	0
$34$	−5.72905	−0.982524
$35$	0.393832	0.0665698
$36$	0	0
$37$	−7.34192	−1.20700	−0.603502	−	0.797361i	$-0.706229\pi$
$37$	−7.34192	−1.20700	−0.603502	+	0.797361i	$0.706229\pi$
$38$	−2.64341	−0.428818
$39$	0	0
$40$	9.92054	1.56857
$41$	1.65745	0.258850	0.129425	−	0.991589i	$-0.458687\pi$
$41$	1.65745	0.258850	0.129425	+	0.991589i	$0.458687\pi$
$42$	0	0
$43$	10.3279	1.57499	0.787494	−	0.616323i	$-0.211378\pi$
$43$	10.3279	1.57499	0.787494	+	0.616323i	$0.211378\pi$
$44$	2.20500	0.332417
$45$	0	0
$46$	−16.9713	−2.50228
$47$	11.5915	1.69079	0.845397	−	0.534138i	$-0.179364\pi$
$47$	11.5915	1.69079	0.845397	+	0.534138i	$0.179364\pi$
$48$	0	0
$49$	−6.84490	−0.977842
$50$	2.75660	0.389842
$51$	0	0
$52$	−14.3616	−1.99160
$53$	12.3279	1.69336	0.846682	−	0.532099i	$-0.178596\pi$
$53$	12.3279	1.69336	0.846682	+	0.532099i	$0.178596\pi$
$54$	0	0
$55$	0.393832	0.0531043
$56$	3.90703	0.522099
$57$	0	0
$58$	2.75660	0.361959
$59$	9.54022	1.24203	0.621015	−	0.783798i	$-0.286720\pi$
$59$	9.54022	1.24203	0.621015	+	0.783798i	$0.286720\pi$
$60$	0	0
$61$	−6.25340	−0.800665	−0.400333	−	0.916370i	$-0.631105\pi$
$61$	−6.25340	−0.800665	−0.400333	+	0.916370i	$0.631105\pi$
$62$	−27.9977	−3.55571
$63$	0	0
$64$	35.7232	4.46540
$65$	−2.56511	−0.318162
$66$	0	0
$67$	−7.42023	−0.906525	−0.453262	−	0.891377i	$-0.649740\pi$
$67$	−7.42023	−0.906525	−0.453262	+	0.891377i	$0.649740\pi$
$68$	−11.6361	−1.41108
$69$	0	0
$70$	1.08564	0.129758
$71$	−5.98533	−0.710328	−0.355164	−	0.934804i	$-0.615575\pi$
$71$	−5.98533	−0.710328	−0.355164	+	0.934804i	$0.615575\pi$
$72$	0	0
$73$	3.34192	0.391142	0.195571	−	0.980690i	$-0.437344\pi$
$73$	3.34192	0.391142	0.195571	+	0.980690i	$0.437344\pi$
$74$	−20.2387	−2.35270
$75$	0	0
$76$	−5.36894	−0.615860
$77$	0.155104	0.0176757
$78$	0	0
$79$	−2.06745	−0.232607	−0.116303	−	0.993214i	$-0.537104\pi$
$79$	−2.06745	−0.232607	−0.116303	+	0.993214i	$0.537104\pi$
$80$	16.1493	1.80554
$81$	0	0
$82$	4.56892	0.504553
$83$	−6.41000	−0.703589	−0.351795	−	0.936077i	$-0.614428\pi$
$83$	−6.41000	−0.703589	−0.351795	+	0.936077i	$0.614428\pi$
$84$	0	0
$85$	−2.07830	−0.225424
$86$	28.4698	3.06998
$87$	0	0
$88$	3.90703	0.416491
$89$	−15.8302	−1.67800	−0.839000	−	0.544131i	$-0.816860\pi$
$89$	−15.8302	−1.67800	−0.839000	+	0.544131i	$0.816860\pi$
$90$	0	0
$91$	−1.01022	−0.105900
$92$	−34.4698	−3.59373
$93$	0	0
$94$	31.9531	3.29571
$95$	−0.958939	−0.0983851
$96$	0	0
$97$	−18.4575	−1.87407	−0.937036	−	0.349233i	$-0.886442\pi$
$97$	−18.4575	−1.87407	−0.937036	+	0.349233i	$0.886442\pi$
$98$	−18.8686	−1.90602
$99$	0	0
$100$	5.59883	0.559883
$101$	12.8038	1.27403	0.637015	−	0.770852i	$-0.280169\pi$
$101$	12.8038	1.27403	0.637015	+	0.770852i	$0.280169\pi$
$102$	0	0
$103$	4.86979	0.479834	0.239917	−	0.970793i	$-0.422880\pi$
$103$	4.86979	0.479834	0.239917	+	0.970793i	$0.422880\pi$
$104$	−25.4472	−2.49531
$105$	0	0
$106$	33.9830	3.30072
$107$	2.34255	0.226463	0.113231	−	0.993569i	$-0.463880\pi$
$107$	2.34255	0.226463	0.113231	+	0.993569i	$0.463880\pi$
$108$	0	0
$109$	8.55044	0.818984	0.409492	−	0.912314i	$-0.365706\pi$
$109$	8.55044	0.818984	0.409492	+	0.912314i	$0.365706\pi$
$110$	1.08564	0.103511
$111$	0	0
$112$	6.36011	0.600973
$113$	11.2085	1.05441	0.527204	−	0.849739i	$-0.323240\pi$
$113$	11.2085	1.05441	0.527204	+	0.849739i	$0.323240\pi$
$114$	0	0
$115$	−6.15661	−0.574107
$116$	5.59883	0.519839
$117$	0	0
$118$	26.2985	2.42098
$119$	−0.818503	−0.0750321
$120$	0	0
$121$	−10.8449	−0.985900
$122$	−17.2381	−1.56066
$123$	0	0
$124$	−56.8652	−5.10664
$125$	1.00000	0.0894427
$126$	0	0
$127$	20.1566	1.78861	0.894305	−	0.447458i	$-0.147671\pi$
$127$	20.1566	1.78861	0.894305	+	0.447458i	$0.147671\pi$
$128$	49.1226	4.34187
$129$	0	0
$130$	−7.07097	−0.620165
$131$	5.50235	0.480742	0.240371	−	0.970681i	$-0.422731\pi$
$131$	5.50235	0.480742	0.240371	+	0.970681i	$0.422731\pi$
$132$	0	0
$133$	−0.377661	−0.0327474
$134$	−20.4546	−1.76701
$135$	0	0
$136$	−20.6179	−1.76797
$137$	−7.49853	−0.640643	−0.320321	−	0.947309i	$-0.603791\pi$
$137$	−7.49853	−0.640643	−0.320321	+	0.947309i	$0.603791\pi$
$138$	0	0
$139$	−9.35277	−0.793292	−0.396646	−	0.917972i	$-0.629826\pi$
$139$	−9.35277	−0.793292	−0.396646	+	0.917972i	$0.629826\pi$
$140$	2.20500	0.186357
$141$	0	0
$142$	−16.4992	−1.38458
$143$	−1.01022	−0.0844790
$144$	0	0
$145$	1.00000	0.0830455
$146$	9.21234	0.762418
$147$	0	0
$148$	−41.1062	−3.37891
$149$	11.5402	0.945411	0.472706	−	0.881220i	$-0.343277\pi$
$149$	11.5402	0.945411	0.472706	+	0.881220i	$0.343277\pi$
$150$	0	0
$151$	6.08212	0.494956	0.247478	−	0.968894i	$-0.420398\pi$
$151$	6.08212	0.494956	0.247478	+	0.968894i	$0.420398\pi$
$152$	−9.51320	−0.771622
$153$	0	0
$154$	0.427559	0.0344537
$155$	−10.1566	−0.815798
$156$	0	0
$157$	−11.5953	−0.925407	−0.462704	−	0.886513i	$-0.653121\pi$
$157$	−11.5953	−0.925407	−0.462704	+	0.886513i	$0.653121\pi$
$158$	−5.69914	−0.453399
$159$	0	0
$160$	24.6760	1.95081
$161$	−2.42467	−0.191091
$162$	0	0
$163$	−0.855118	−0.0669780	−0.0334890	−	0.999439i	$-0.510662\pi$
$163$	−0.855118	−0.0669780	−0.0334890	+	0.999439i	$0.510662\pi$
$164$	9.27979	0.724630
$165$	0	0
$166$	−17.6698	−1.37144
$167$	−7.73194	−0.598315	−0.299158	−	0.954204i	$-0.596706\pi$
$167$	−7.73194	−0.598315	−0.299158	+	0.954204i	$0.596706\pi$
$168$	0	0
$169$	−6.42023	−0.493863
$170$	−5.72905	−0.439398
$171$	0	0
$172$	57.8241	4.40905
$173$	−8.07449	−0.613892	−0.306946	−	0.951727i	$-0.599307\pi$
$173$	−8.07449	−0.613892	−0.306946	+	0.951727i	$0.599307\pi$
$174$	0	0
$175$	0.393832	0.0297709
$176$	6.36011	0.479411
$177$	0	0
$178$	−43.6376	−3.27077
$179$	−12.4100	−0.927567	−0.463784	−	0.885949i	$-0.653508\pi$
$179$	−12.4100	−0.927567	−0.463784	+	0.885949i	$0.653508\pi$
$180$	0	0
$181$	−2.49765	−0.185649	−0.0928246	−	0.995682i	$-0.529590\pi$
$181$	−2.49765	−0.185649	−0.0928246	+	0.995682i	$0.529590\pi$
$182$	−2.78478	−0.206421
$183$	0	0
$184$	−61.0769	−4.50265
$185$	−7.34192	−0.539789
$186$	0	0
$187$	−0.818503	−0.0598549
$188$	64.8989	4.73324
$189$	0	0
$190$	−2.64341	−0.191773
$191$	6.32085	0.457361	0.228680	−	0.973502i	$-0.426559\pi$
$191$	6.32085	0.457361	0.228680	+	0.973502i	$0.426559\pi$
$192$	0	0
$193$	25.2921	1.82057	0.910284	−	0.413984i	$-0.135863\pi$
$193$	25.2921	1.82057	0.910284	+	0.413984i	$0.135863\pi$
$194$	−50.8798	−3.65296
$195$	0	0
$196$	−38.3234	−2.73739
$197$	21.2047	1.51077	0.755386	−	0.655280i	$-0.227449\pi$
$197$	21.2047	1.51077	0.755386	+	0.655280i	$0.227449\pi$
$198$	0	0
$199$	−2.64723	−0.187657	−0.0938285	−	0.995588i	$-0.529911\pi$
$199$	−2.64723	−0.187657	−0.0938285	+	0.995588i	$0.529911\pi$
$200$	9.92054	0.701488
$201$	0	0
$202$	35.2950	2.48335
$203$	0.393832	0.0276416
$204$	0	0
$205$	1.65745	0.115761
$206$	13.4240	0.935297
$207$	0	0
$208$	−41.4246	−2.87228
$209$	−0.377661	−0.0261234
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	69.0218	4.74043
$213$	0	0
$214$	6.45747	0.441423
$215$	10.3279	0.704356
$216$	0	0
$217$	−4.00000	−0.271538
$218$	23.5701	1.59637
$219$	0	0
$220$	2.20500	0.148661
$221$	5.33107	0.358607
$222$	0	0
$223$	12.5504	0.840440	0.420220	−	0.907422i	$-0.361953\pi$
$223$	12.5504	0.840440	0.420220	+	0.907422i	$0.361953\pi$
$224$	9.71820	0.649324
$225$	0	0
$226$	30.8974	2.05526
$227$	−1.30149	−0.0863829	−0.0431914	−	0.999067i	$-0.513753\pi$
$227$	−1.30149	−0.0863829	−0.0431914	+	0.999067i	$0.513753\pi$
$228$	0	0
$229$	3.28682	0.217199	0.108600	−	0.994086i	$-0.465363\pi$
$229$	3.28682	0.217199	0.108600	+	0.994086i	$0.465363\pi$
$230$	−16.9713	−1.11905
$231$	0	0
$232$	9.92054	0.651315
$233$	−17.7115	−1.16032	−0.580159	−	0.814503i	$-0.697010\pi$
$233$	−17.7115	−1.16032	−0.580159	+	0.814503i	$0.697010\pi$
$234$	0	0
$235$	11.5915	0.756146
$236$	53.4141	3.47696
$237$	0	0
$238$	−2.25628	−0.146253
$239$	−21.2651	−1.37553	−0.687763	−	0.725935i	$-0.741407\pi$
$239$	−21.2651	−1.37553	−0.687763	+	0.725935i	$0.741407\pi$
$240$	0	0
$241$	15.3177	0.986697	0.493349	−	0.869832i	$-0.335773\pi$
$241$	15.3177	0.986697	0.493349	+	0.869832i	$0.335773\pi$
$242$	−29.8950	−1.92172
$243$	0	0
$244$	−35.0117	−2.24140
$245$	−6.84490	−0.437304
$246$	0	0
$247$	2.45978	0.156512
$248$	−100.759	−6.39820
$249$	0	0
$250$	2.75660	0.174343
$251$	−26.8917	−1.69739	−0.848696	−	0.528882i	$-0.822612\pi$
$251$	−26.8917	−1.69739	−0.848696	+	0.528882i	$0.822612\pi$
$252$	0	0
$253$	−2.42467	−0.152438
$254$	55.5637	3.48637
$255$	0	0
$256$	63.9648	3.99780
$257$	6.85512	0.427611	0.213805	−	0.976876i	$-0.431414\pi$
$257$	6.85512	0.427611	0.213805	+	0.976876i	$0.431414\pi$
$258$	0	0
$259$	−2.89149	−0.179668
$260$	−14.3616	−0.890669
$261$	0	0
$262$	15.1678	0.937067
$263$	13.6294	0.840423	0.420212	−	0.907426i	$-0.361956\pi$
$263$	13.6294	0.840423	0.420212	+	0.907426i	$0.361956\pi$
$264$	0	0
$265$	12.3279	0.757296
$266$	−1.04106	−0.0638315
$267$	0	0
$268$	−41.5446	−2.53774
$269$	8.46129	0.515894	0.257947	−	0.966159i	$-0.416954\pi$
$269$	8.46129	0.515894	0.257947	+	0.966159i	$0.416954\pi$
$270$	0	0
$271$	6.34255	0.385282	0.192641	−	0.981269i	$-0.438295\pi$
$271$	6.34255	0.385282	0.192641	+	0.981269i	$0.438295\pi$
$272$	−33.5631	−2.03506
$273$	0	0
$274$	−20.6704	−1.24875
$275$	0.393832	0.0237490
$276$	0	0
$277$	−17.1464	−1.03023	−0.515113	−	0.857122i	$-0.672250\pi$
$277$	−17.1464	−1.03023	−0.515113	+	0.857122i	$0.672250\pi$
$278$	−25.7818	−1.54629
$279$	0	0
$280$	3.90703	0.233490
$281$	12.2985	0.733670	0.366835	−	0.930286i	$-0.380441\pi$
$281$	12.2985	0.733670	0.366835	+	0.930286i	$0.380441\pi$
$282$	0	0
$283$	25.1830	1.49697	0.748487	−	0.663149i	$-0.230781\pi$
$283$	25.1830	1.49697	0.748487	+	0.663149i	$0.230781\pi$
$284$	−33.5109	−1.98851
$285$	0	0
$286$	−2.78478	−0.164667
$287$	0.652757	0.0385311
$288$	0	0
$289$	−12.6807	−0.745921
$290$	2.75660	0.161873
$291$	0	0
$292$	18.7109	1.09497
$293$	−12.3170	−0.719569	−0.359784	−	0.933035i	$-0.617150\pi$
$293$	−12.3170	−0.719569	−0.359784	+	0.933035i	$0.617150\pi$
$294$	0	0
$295$	9.54022	0.555453
$296$	−72.8358	−4.23350
$297$	0	0
$298$	31.8117	1.84280
$299$	15.7924	0.913296
$300$	0	0
$301$	4.06745	0.234444
$302$	16.7660	0.964773
$303$	0	0
$304$	−15.4862	−0.888193
$305$	−6.25340	−0.358068
$306$	0	0
$307$	−22.7379	−1.29772	−0.648860	−	0.760908i	$-0.724754\pi$
$307$	−22.7379	−1.29772	−0.648860	+	0.760908i	$0.724754\pi$
$308$	0.868401	0.0494817
$309$	0	0
$310$	−27.9977	−1.59016
$311$	5.33810	0.302696	0.151348	−	0.988481i	$-0.451639\pi$
$311$	5.33810	0.302696	0.151348	+	0.988481i	$0.451639\pi$
$312$	0	0
$313$	−1.96338	−0.110977	−0.0554885	−	0.998459i	$-0.517672\pi$
$313$	−1.96338	−0.110977	−0.0554885	+	0.998459i	$0.517672\pi$
$314$	−31.9636	−1.80381
$315$	0	0
$316$	−11.5753	−0.651163
$317$	−1.29064	−0.0724895	−0.0362448	−	0.999343i	$-0.511540\pi$
$317$	−1.29064	−0.0724895	−0.0362448	+	0.999343i	$0.511540\pi$
$318$	0	0
$319$	0.393832	0.0220504
$320$	35.7232	1.99699
$321$	0	0
$322$	−6.68384	−0.372476
$323$	1.99297	0.110892
$324$	0	0
$325$	−2.56511	−0.142287
$326$	−2.35722	−0.130554
$327$	0	0
$328$	16.4428	0.907902
$329$	4.56511	0.251683
$330$	0	0
$331$	28.9971	1.59382	0.796911	−	0.604096i	$-0.206466\pi$
$331$	28.9971	1.59382	0.796911	+	0.604096i	$0.206466\pi$
$332$	−35.8885	−1.96964
$333$	0	0
$334$	−21.3138	−1.16624
$335$	−7.42023	−0.405410
$336$	0	0
$337$	−16.7854	−0.914356	−0.457178	−	0.889375i	$-0.651140\pi$
$337$	−16.7854	−0.914356	−0.457178	+	0.889375i	$0.651140\pi$
$338$	−17.6980	−0.962643
$339$	0	0
$340$	−11.6361	−0.631055
$341$	−4.00000	−0.216612
$342$	0	0
$343$	−5.45257	−0.294411
$344$	102.458	5.52417
$345$	0	0
$346$	−22.2581	−1.19660
$347$	17.8681	0.959210	0.479605	−	0.877485i	$-0.340780\pi$
$347$	17.8681	0.959210	0.479605	+	0.877485i	$0.340780\pi$
$348$	0	0
$349$	8.73789	0.467728	0.233864	−	0.972269i	$-0.424863\pi$
$349$	8.73789	0.467728	0.233864	+	0.972269i	$0.424863\pi$
$350$	1.08564	0.0580298
$351$	0	0
$352$	9.71820	0.517982
$353$	22.6017	1.20297	0.601484	−	0.798885i	$-0.294576\pi$
$353$	22.6017	1.20297	0.601484	+	0.798885i	$0.294576\pi$
$354$	0	0
$355$	−5.98533	−0.317668
$356$	−88.6308	−4.69742
$357$	0	0
$358$	−34.2094	−1.80802
$359$	13.1830	0.695772	0.347886	−	0.937537i	$-0.386900\pi$
$359$	13.1830	0.695772	0.347886	+	0.937537i	$0.386900\pi$
$360$	0	0
$361$	−18.0804	−0.951602
$362$	−6.88503	−0.361869
$363$	0	0
$364$	−5.65607	−0.296458
$365$	3.34192	0.174924
$366$	0	0
$367$	17.4511	0.910938	0.455469	−	0.890252i	$-0.349472\pi$
$367$	17.4511	0.910938	0.455469	+	0.890252i	$0.349472\pi$
$368$	−99.4247	−5.18287
$369$	0	0
$370$	−20.2387	−1.05216
$371$	4.85512	0.252065
$372$	0	0
$373$	−32.2018	−1.66734	−0.833672	−	0.552260i	$-0.813766\pi$
$373$	−32.2018	−1.66734	−0.833672	+	0.552260i	$0.813766\pi$
$374$	−2.25628	−0.116670
$375$	0	0
$376$	114.994	5.93036
$377$	−2.56511	−0.132110
$378$	0	0
$379$	32.3660	1.66253	0.831265	−	0.555876i	$-0.187617\pi$
$379$	32.3660	1.66253	0.831265	+	0.555876i	$0.187617\pi$
$380$	−5.36894	−0.275421
$381$	0	0
$382$	17.4240	0.891492
$383$	25.8739	1.32209	0.661047	−	0.750345i	$-0.270113\pi$
$383$	25.8739	1.32209	0.661047	+	0.750345i	$0.270113\pi$
$384$	0	0
$385$	0.155104	0.00790483
$386$	69.7203	3.54867
$387$	0	0
$388$	−103.340	−5.24631
$389$	−32.6103	−1.65341	−0.826703	−	0.562639i	$-0.809786\pi$
$389$	−32.6103	−1.65341	−0.826703	+	0.562639i	$0.809786\pi$
$390$	0	0
$391$	12.7953	0.647086
$392$	−67.9051	−3.42972
$393$	0	0
$394$	58.4528	2.94481
$395$	−2.06745	−0.104025
$396$	0	0
$397$	4.39534	0.220596	0.110298	−	0.993899i	$-0.464820\pi$
$397$	4.39534	0.220596	0.110298	+	0.993899i	$0.464820\pi$
$398$	−7.29734	−0.365783
$399$	0	0
$400$	16.1493	0.807464
$401$	19.0658	0.952099	0.476049	−	0.879418i	$-0.342068\pi$
$401$	19.0658	0.952099	0.476049	+	0.879418i	$0.342068\pi$
$402$	0	0
$403$	26.0528	1.29778
$404$	71.6865	3.56654
$405$	0	0
$406$	1.08564	0.0538793
$407$	−2.89149	−0.143326
$408$	0	0
$409$	−1.38235	−0.0683530	−0.0341765	−	0.999416i	$-0.510881\pi$
$409$	−1.38235	−0.0683530	−0.0341765	+	0.999416i	$0.510881\pi$
$410$	4.56892	0.225643
$411$	0	0
$412$	27.2651	1.34326
$413$	3.75725	0.184882
$414$	0	0
$415$	−6.41000	−0.314655
$416$	−63.2965	−3.10337
$417$	0	0
$418$	−1.04106	−0.0509199
$419$	−2.86979	−0.140198	−0.0700991	−	0.997540i	$-0.522332\pi$
$419$	−2.86979	−0.140198	−0.0700991	+	0.997540i	$0.522332\pi$
$420$	0	0
$421$	13.6440	0.664970	0.332485	−	0.943109i	$-0.392113\pi$
$421$	13.6440	0.664970	0.332485	+	0.943109i	$0.392113\pi$
$422$	5.51320	0.268378
$423$	0	0
$424$	122.299	5.93938
$425$	−2.07830	−0.100813
$426$	0	0
$427$	−2.46279	−0.119183
$428$	13.1155	0.633964
$429$	0	0
$430$	28.4698	1.37294
$431$	11.9707	0.576607	0.288303	−	0.957539i	$-0.406909\pi$
$431$	11.9707	0.576607	0.288303	+	0.957539i	$0.406909\pi$
$432$	0	0
$433$	17.9907	0.864576	0.432288	−	0.901736i	$-0.357706\pi$
$433$	17.9907	0.864576	0.432288	+	0.901736i	$0.357706\pi$
$434$	−11.0264	−0.529284
$435$	0	0
$436$	47.8725	2.29268
$437$	5.90381	0.282418
$438$	0	0
$439$	−16.2226	−0.774260	−0.387130	−	0.922025i	$-0.626534\pi$
$439$	−16.2226	−0.774260	−0.387130	+	0.922025i	$0.626534\pi$
$440$	3.90703	0.186260
$441$	0	0
$442$	14.6956	0.698999
$443$	35.3762	1.68078	0.840388	−	0.541986i	$-0.182327\pi$
$443$	35.3762	1.68078	0.840388	+	0.541986i	$0.182327\pi$
$444$	0	0
$445$	−15.8302	−0.750425
$446$	34.5965	1.63819
$447$	0	0
$448$	14.0690	0.664696
$449$	−41.6971	−1.96781	−0.983903	−	0.178702i	$-0.942810\pi$
$449$	−41.6971	−1.96781	−0.983903	+	0.178702i	$0.942810\pi$
$450$	0	0
$451$	0.652757	0.0307371
$452$	62.7546	2.95173
$453$	0	0
$454$	−3.58768	−0.168378
$455$	−1.01022	−0.0473599
$456$	0	0
$457$	−16.1111	−0.753646	−0.376823	−	0.926285i	$-0.622983\pi$
$457$	−16.1111	−0.753646	−0.376823	+	0.926285i	$0.622983\pi$
$458$	9.06045	0.423367
$459$	0	0
$460$	−34.4698	−1.60716
$461$	−17.3396	−0.807586	−0.403793	−	0.914850i	$-0.632308\pi$
$461$	−17.3396	−0.807586	−0.403793	+	0.914850i	$0.632308\pi$
$462$	0	0
$463$	19.3177	0.897768	0.448884	−	0.893590i	$-0.351822\pi$
$463$	19.3177	0.897768	0.448884	+	0.893590i	$0.351822\pi$
$464$	16.1493	0.749711
$465$	0	0
$466$	−48.8235	−2.26170
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	−2.92232	−0.134940
$470$	31.9531	1.47389
$471$	0	0
$472$	94.6441	4.35635
$473$	4.06745	0.187022
$474$	0	0
$475$	−0.958939	−0.0439992
$476$	−4.58266	−0.210046
$477$	0	0
$478$	−58.6194	−2.68119
$479$	40.3877	1.84536	0.922681	−	0.385565i	$-0.125994\pi$
$479$	40.3877	1.84536	0.922681	+	0.385565i	$0.125994\pi$
$480$	0	0
$481$	18.8328	0.858702
$482$	42.2246	1.92328
$483$	0	0
$484$	−60.7188	−2.75994
$485$	−18.4575	−0.838110
$486$	0	0
$487$	2.84171	0.128770	0.0643850	−	0.997925i	$-0.479491\pi$
$487$	2.84171	0.128770	0.0643850	+	0.997925i	$0.479491\pi$
$488$	−62.0371	−2.80829
$489$	0	0
$490$	−18.8686	−0.852398
$491$	−0.157863	−0.00712428	−0.00356214	−	0.999994i	$-0.501134\pi$
$491$	−0.157863	−0.00712428	−0.00356214	+	0.999994i	$0.501134\pi$
$492$	0	0
$493$	−2.07830	−0.0936021
$494$	6.78063	0.305075
$495$	0	0
$496$	−164.022	−7.36480
$497$	−2.35722	−0.105736
$498$	0	0
$499$	2.64723	0.118506	0.0592531	−	0.998243i	$-0.481128\pi$
$499$	2.64723	0.118506	0.0592531	+	0.998243i	$0.481128\pi$
$500$	5.59883	0.250387
$501$	0	0
$502$	−74.1297	−3.30857
$503$	−13.9545	−0.622200	−0.311100	−	0.950377i	$-0.600697\pi$
$503$	−13.9545	−0.622200	−0.311100	+	0.950377i	$0.600697\pi$
$504$	0	0
$505$	12.8038	0.569763
$506$	−6.68384	−0.297133
$507$	0	0
$508$	112.853	5.00706
$509$	16.4921	0.731001	0.365500	−	0.930811i	$-0.380898\pi$
$509$	16.4921	0.731001	0.365500	+	0.930811i	$0.380898\pi$
$510$	0	0
$511$	1.31616	0.0582233
$512$	78.0802	3.45069
$513$	0	0
$514$	18.8968	0.833502
$515$	4.86979	0.214588
$516$	0	0
$517$	4.56511	0.200773
$518$	−7.97066	−0.350211
$519$	0	0
$520$	−25.4472	−1.11594
$521$	12.2253	0.535601	0.267800	−	0.963474i	$-0.413703\pi$
$521$	12.2253	0.535601	0.267800	+	0.963474i	$0.413703\pi$
$522$	0	0
$523$	1.66659	0.0728748	0.0364374	−	0.999336i	$-0.488399\pi$
$523$	1.66659	0.0728748	0.0364374	+	0.999336i	$0.488399\pi$
$524$	30.8067	1.34580
$525$	0	0
$526$	37.5707	1.63816
$527$	21.1085	0.919501
$528$	0	0
$529$	14.9038	0.647992
$530$	33.9830	1.47613
$531$	0	0
$532$	−2.11446	−0.0916736
$533$	−4.25154	−0.184155
$534$	0	0
$535$	2.34255	0.101277
$536$	−73.6126	−3.17958
$537$	0	0
$538$	23.3244	1.00558
$539$	−2.69574	−0.116114
$540$	0	0
$541$	−34.6558	−1.48997	−0.744984	−	0.667083i	$-0.767543\pi$
$541$	−34.6558	−1.48997	−0.744984	+	0.667083i	$0.767543\pi$
$542$	17.4839	0.750996
$543$	0	0
$544$	−51.2842	−2.19879
$545$	8.55044	0.366261
$546$	0	0
$547$	38.2530	1.63558	0.817791	−	0.575515i	$-0.195199\pi$
$547$	38.2530	1.63558	0.817791	+	0.575515i	$0.195199\pi$
$548$	−41.9830	−1.79343
$549$	0	0
$550$	1.08564	0.0462917
$551$	−0.958939	−0.0408522
$552$	0	0
$553$	−0.814230	−0.0346246
$554$	−47.2657	−2.00813
$555$	0	0
$556$	−52.3646	−2.22075
$557$	28.7760	1.21928	0.609639	−	0.792679i	$-0.291314\pi$
$557$	28.7760	1.21928	0.609639	+	0.792679i	$0.291314\pi$
$558$	0	0
$559$	−26.4921	−1.12050
$560$	6.36011	0.268764
$561$	0	0
$562$	33.9022	1.43008
$563$	32.9809	1.38998	0.694989	−	0.719020i	$-0.255409\pi$
$563$	32.9809	1.38998	0.694989	+	0.719020i	$0.255409\pi$
$564$	0	0
$565$	11.2085	0.471546
$566$	69.4194	2.91792
$567$	0	0
$568$	−59.3777	−2.49143
$569$	−16.8038	−0.704453	−0.352227	−	0.935915i	$-0.614575\pi$
$569$	−16.8038	−0.704453	−0.352227	+	0.935915i	$0.614575\pi$
$570$	0	0
$571$	−6.21703	−0.260175	−0.130087	−	0.991503i	$-0.541526\pi$
$571$	−6.21703	−0.260175	−0.130087	+	0.991503i	$0.541526\pi$
$572$	−5.65607	−0.236492
$573$	0	0
$574$	1.79939	0.0751051
$575$	−6.15661	−0.256748
$576$	0	0
$577$	11.9977	0.499470	0.249735	−	0.968314i	$-0.419656\pi$
$577$	11.9977	0.499470	0.249735	+	0.968314i	$0.419656\pi$
$578$	−34.9555	−1.45396
$579$	0	0
$580$	5.59883	0.232479
$581$	−2.52447	−0.104733
$582$	0	0
$583$	4.85512	0.201078
$584$	33.1537	1.37191
$585$	0	0
$586$	−33.9531	−1.40259
$587$	11.2194	0.463073	0.231536	−	0.972826i	$-0.425625\pi$
$587$	11.2194	0.463073	0.231536	+	0.972826i	$0.425625\pi$
$588$	0	0
$589$	9.73957	0.401312
$590$	26.2985	1.08269
$591$	0	0
$592$	−118.567	−4.87306
$593$	7.69682	0.316071	0.158035	−	0.987433i	$-0.449484\pi$
$593$	7.69682	0.316071	0.158035	+	0.987433i	$0.449484\pi$
$594$	0	0
$595$	−0.818503	−0.0335554
$596$	64.6118	2.64660
$597$	0	0
$598$	43.5332	1.78020
$599$	−20.0543	−0.819396	−0.409698	−	0.912221i	$-0.634366\pi$
$599$	−20.0543	−0.819396	−0.409698	+	0.912221i	$0.634366\pi$
$600$	0	0
$601$	3.10851	0.126799	0.0633995	−	0.997988i	$-0.479806\pi$
$601$	3.10851	0.126799	0.0633995	+	0.997988i	$0.479806\pi$
$602$	11.2123	0.456981
$603$	0	0
$604$	34.0528	1.38559
$605$	−10.8449	−0.440908
$606$	0	0
$607$	37.0441	1.50357	0.751786	−	0.659407i	$-0.229193\pi$
$607$	37.0441	1.50357	0.751786	+	0.659407i	$0.229193\pi$
$608$	−23.6628	−0.959652
$609$	0	0
$610$	−17.2381	−0.697950
$611$	−29.7334	−1.20289
$612$	0	0
$613$	−13.3839	−0.540569	−0.270284	−	0.962781i	$-0.587118\pi$
$613$	−13.3839	−0.540569	−0.270284	+	0.962781i	$0.587118\pi$
$614$	−62.6792	−2.52953
$615$	0	0
$616$	1.53871	0.0619966
$617$	12.6364	0.508721	0.254361	−	0.967109i	$-0.418135\pi$
$617$	12.6364	0.508721	0.254361	+	0.967109i	$0.418135\pi$
$618$	0	0
$619$	−41.2447	−1.65776	−0.828882	−	0.559424i	$-0.811022\pi$
$619$	−41.2447	−1.65776	−0.828882	+	0.559424i	$0.811022\pi$
$620$	−56.8652	−2.28376
$621$	0	0
$622$	14.7150	0.590018
$623$	−6.23446	−0.249778
$624$	0	0
$625$	1.00000	0.0400000
$626$	−5.41226	−0.216318
$627$	0	0
$628$	−64.9203	−2.59060
$629$	15.2587	0.608406
$630$	0	0
$631$	29.8009	1.18635	0.593177	−	0.805072i	$-0.297873\pi$
$631$	29.8009	1.18635	0.593177	+	0.805072i	$0.297873\pi$
$632$	−20.5103	−0.815854
$633$	0	0
$634$	−3.55777	−0.141297
$635$	20.1566	0.799891
$636$	0	0
$637$	17.5579	0.695669
$638$	1.08564	0.0429808
$639$	0	0
$640$	49.1226	1.94174
$641$	−17.7774	−0.702167	−0.351083	−	0.936344i	$-0.614187\pi$
$641$	−17.7774	−0.702167	−0.351083	+	0.936344i	$0.614187\pi$
$642$	0	0
$643$	−44.5534	−1.75702	−0.878508	−	0.477727i	$-0.841461\pi$
$643$	−44.5534	−1.75702	−0.878508	+	0.477727i	$0.841461\pi$
$644$	−13.5753	−0.534943
$645$	0	0
$646$	5.49381	0.216151
$647$	−42.9339	−1.68790	−0.843952	−	0.536418i	$-0.819777\pi$
$647$	−42.9339	−1.68790	−0.843952	+	0.536418i	$0.819777\pi$
$648$	0	0
$649$	3.75725	0.147485
$650$	−7.07097	−0.277346
$651$	0	0
$652$	−4.78766	−0.187499
$653$	19.8426	0.776500	0.388250	−	0.921554i	$-0.373080\pi$
$653$	19.8426	0.776500	0.388250	+	0.921554i	$0.373080\pi$
$654$	0	0
$655$	5.50235	0.214994
$656$	26.7666	1.04506
$657$	0	0
$658$	12.5842	0.490582
$659$	−23.0483	−0.897836	−0.448918	−	0.893573i	$-0.648190\pi$
$659$	−23.0483	−0.897836	−0.448918	+	0.893573i	$0.648190\pi$
$660$	0	0
$661$	−36.2079	−1.40832	−0.704162	−	0.710040i	$-0.748677\pi$
$661$	−36.2079	−1.40832	−0.704162	+	0.710040i	$0.748677\pi$
$662$	79.9332	3.10669
$663$	0	0
$664$	−63.5907	−2.46780
$665$	−0.377661	−0.0146451
$666$	0	0
$667$	−6.15661	−0.238385
$668$	−43.2898	−1.67493
$669$	0	0
$670$	−20.4546	−0.790229
$671$	−2.46279	−0.0950749
$672$	0	0
$673$	22.9000	0.882731	0.441365	−	0.897327i	$-0.354494\pi$
$673$	22.9000	0.882731	0.441365	+	0.897327i	$0.354494\pi$
$674$	−46.2705	−1.78227
$675$	0	0
$676$	−35.9458	−1.38253
$677$	−36.0068	−1.38385	−0.691927	−	0.721967i	$-0.743238\pi$
$677$	−36.0068	−1.38385	−0.691927	+	0.721967i	$0.743238\pi$
$678$	0	0
$679$	−7.26915	−0.278964
$680$	−20.6179	−0.790660
$681$	0	0
$682$	−11.0264	−0.422222
$683$	−9.12896	−0.349310	−0.174655	−	0.984630i	$-0.555881\pi$
$683$	−9.12896	−0.349310	−0.174655	+	0.984630i	$0.555881\pi$
$684$	0	0
$685$	−7.49853	−0.286504
$686$	−15.0305	−0.573869
$687$	0	0
$688$	166.788	6.35873
$689$	−31.6223	−1.20472
$690$	0	0
$691$	5.85956	0.222908	0.111454	−	0.993770i	$-0.464449\pi$
$691$	5.85956	0.222908	0.111454	+	0.993770i	$0.464449\pi$
$692$	−45.2077	−1.71854
$693$	0	0
$694$	49.2552	1.86970
$695$	−9.35277	−0.354771
$696$	0	0
$697$	−3.44469	−0.130477
$698$	24.0868	0.911700
$699$	0	0
$700$	2.20500	0.0833412
$701$	7.56955	0.285898	0.142949	−	0.989730i	$-0.454342\pi$
$701$	7.56955	0.285898	0.142949	+	0.989730i	$0.454342\pi$
$702$	0	0
$703$	7.04046	0.265536
$704$	14.0690	0.530244
$705$	0	0
$706$	62.3039	2.34484
$707$	5.04256	0.189645
$708$	0	0
$709$	14.5209	0.545342	0.272671	−	0.962107i	$-0.412093\pi$
$709$	14.5209	0.545342	0.272671	+	0.962107i	$0.412093\pi$
$710$	−16.4992	−0.619202
$711$	0	0
$712$	−157.044	−5.88549
$713$	62.5302	2.34178
$714$	0	0
$715$	−1.01022	−0.0377802
$716$	−69.4815	−2.59665
$717$	0	0
$718$	36.3402	1.35621
$719$	−12.2164	−0.455596	−0.227798	−	0.973708i	$-0.573153\pi$
$719$	−12.2164	−0.455596	−0.227798	+	0.973708i	$0.573153\pi$
$720$	0	0
$721$	1.91788	0.0714255
$722$	−49.8405	−1.85487
$723$	0	0
$724$	−13.9839	−0.519709
$725$	1.00000	0.0371391
$726$	0	0
$727$	−40.8856	−1.51636	−0.758182	−	0.652044i	$-0.773912\pi$
$727$	−40.8856	−1.51636	−0.758182	+	0.652044i	$0.773912\pi$
$728$	−10.0219	−0.371438
$729$	0	0
$730$	9.21234	0.340964
$731$	−21.4645	−0.793892
$732$	0	0
$733$	25.4915	0.941550	0.470775	−	0.882253i	$-0.343974\pi$
$733$	25.4915	0.941550	0.470775	+	0.882253i	$0.343974\pi$
$734$	48.1056	1.77561
$735$	0	0
$736$	−151.920	−5.59986
$737$	−2.92232	−0.107645
$738$	0	0
$739$	9.56830	0.351975	0.175988	−	0.984392i	$-0.443688\pi$
$739$	9.56830	0.351975	0.175988	+	0.984392i	$0.443688\pi$
$740$	−41.1062	−1.51109
$741$	0	0
$742$	13.3836	0.491328
$743$	18.0455	0.662025	0.331013	−	0.943626i	$-0.392610\pi$
$743$	18.0455	0.662025	0.331013	+	0.943626i	$0.392610\pi$
$744$	0	0
$745$	11.5402	0.422801
$746$	−88.7673	−3.25000
$747$	0	0
$748$	−4.58266	−0.167559
$749$	0.922572	0.0337100
$750$	0	0
$751$	11.9255	0.435168	0.217584	−	0.976042i	$-0.430182\pi$
$751$	11.9255	0.435168	0.217584	+	0.976042i	$0.430182\pi$
$752$	187.194	6.82627
$753$	0	0
$754$	−7.07097	−0.257510
$755$	6.08212	0.221351
$756$	0	0
$757$	5.86152	0.213041	0.106520	−	0.994311i	$-0.466029\pi$
$757$	5.86152	0.213041	0.106520	+	0.994311i	$0.466029\pi$
$758$	89.2201	3.24062
$759$	0	0
$760$	−9.51320	−0.345080
$761$	−49.8739	−1.80793	−0.903963	−	0.427610i	$-0.859356\pi$
$761$	−49.8739	−1.80793	−0.903963	+	0.427610i	$0.859356\pi$
$762$	0	0
$763$	3.36744	0.121909
$764$	35.3894	1.28034
$765$	0	0
$766$	71.3239	2.57704
$767$	−24.4717	−0.883621
$768$	0	0
$769$	29.5121	1.06423	0.532117	−	0.846671i	$-0.321396\pi$
$769$	29.5121	1.06423	0.532117	+	0.846671i	$0.321396\pi$
$770$	0.427559	0.0154082
$771$	0	0
$772$	141.607	5.09653
$773$	2.41935	0.0870180	0.0435090	−	0.999053i	$-0.486146\pi$
$773$	2.41935	0.0870180	0.0435090	+	0.999053i	$0.486146\pi$
$774$	0	0
$775$	−10.1566	−0.364836
$776$	−183.108	−6.57320
$777$	0	0
$778$	−89.8934	−3.22283
$779$	−1.58939	−0.0569460
$780$	0	0
$781$	−2.35722	−0.0843479
$782$	35.2715	1.26131
$783$	0	0
$784$	−110.540	−3.94786
$785$	−11.5953	−0.413855
$786$	0	0
$787$	−51.1549	−1.82348	−0.911738	−	0.410772i	$-0.865259\pi$
$787$	−51.1549	−1.82348	−0.911738	+	0.410772i	$0.865259\pi$
$788$	118.722	4.22928
$789$	0	0
$790$	−5.69914	−0.202766
$791$	4.41428	0.156954
$792$	0	0
$793$	16.0406	0.569619
$794$	12.1162	0.429987
$795$	0	0
$796$	−14.8214	−0.525330
$797$	−28.8662	−1.02249	−0.511247	−	0.859434i	$-0.670816\pi$
$797$	−28.8662	−1.02249	−0.511247	+	0.859434i	$0.670816\pi$
$798$	0	0
$799$	−24.0907	−0.852266
$800$	24.6760	0.872428
$801$	0	0
$802$	52.5567	1.85584
$803$	1.31616	0.0464462
$804$	0	0
$805$	−2.42467	−0.0854584
$806$	71.8171	2.52965
$807$	0	0
$808$	127.021	4.46858
$809$	−19.0426	−0.669501	−0.334750	−	0.942307i	$-0.608652\pi$
$809$	−19.0426	−0.669501	−0.334750	+	0.942307i	$0.608652\pi$
$810$	0	0
$811$	−20.8783	−0.733137	−0.366569	−	0.930391i	$-0.619467\pi$
$811$	−20.8783	−0.733137	−0.366569	+	0.930391i	$0.619467\pi$
$812$	2.20500	0.0773804
$813$	0	0
$814$	−7.97066	−0.279372
$815$	−0.855118	−0.0299535
$816$	0	0
$817$	−9.90381	−0.346491
$818$	−3.81060	−0.133234
$819$	0	0
$820$	9.27979	0.324064
$821$	−3.60466	−0.125804	−0.0629018	−	0.998020i	$-0.520035\pi$
$821$	−3.60466	−0.125804	−0.0629018	+	0.998020i	$0.520035\pi$
$822$	0	0
$823$	27.2288	0.949135	0.474567	−	0.880219i	$-0.342605\pi$
$823$	27.2288	0.949135	0.474567	+	0.880219i	$0.342605\pi$
$824$	48.3109	1.68299
$825$	0	0
$826$	10.3572	0.360374
$827$	−47.3936	−1.64804	−0.824019	−	0.566562i	$-0.808273\pi$
$827$	−47.3936	−1.64804	−0.824019	+	0.566562i	$0.808273\pi$
$828$	0	0
$829$	−41.8388	−1.45312	−0.726560	−	0.687103i	$-0.758882\pi$
$829$	−41.8388	−1.45312	−0.726560	+	0.687103i	$0.758882\pi$
$830$	−17.6698	−0.613328
$831$	0	0
$832$	−91.6339	−3.17683
$833$	14.2258	0.492894
$834$	0	0
$835$	−7.73194	−0.267575
$836$	−2.11446	−0.0731302
$837$	0	0
$838$	−7.91085	−0.273276
$839$	31.6385	1.09228	0.546141	−	0.837693i	$-0.316096\pi$
$839$	31.6385	1.09228	0.546141	+	0.837693i	$0.316096\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	37.6111	1.29617
$843$	0	0
$844$	11.1977	0.385440
$845$	−6.42023	−0.220862
$846$	0	0
$847$	−4.27107	−0.146756
$848$	199.086	6.83665
$849$	0	0
$850$	−5.72905	−0.196505
$851$	45.2013	1.54948
$852$	0	0
$853$	44.2000	1.51338	0.756690	−	0.653773i	$-0.226815\pi$
$853$	44.2000	1.51338	0.756690	+	0.653773i	$0.226815\pi$
$854$	−6.78892	−0.232312
$855$	0	0
$856$	23.2394	0.794305
$857$	−14.4775	−0.494541	−0.247270	−	0.968947i	$-0.579534\pi$
$857$	−14.4775	−0.494541	−0.247270	+	0.968947i	$0.579534\pi$
$858$	0	0
$859$	22.5883	0.770703	0.385352	−	0.922770i	$-0.374080\pi$
$859$	22.5883	0.770703	0.385352	+	0.922770i	$0.374080\pi$
$860$	57.8241	1.97179
$861$	0	0
$862$	32.9983	1.12393
$863$	22.9900	0.782590	0.391295	−	0.920265i	$-0.372027\pi$
$863$	22.9900	0.782590	0.391295	+	0.920265i	$0.372027\pi$
$864$	0	0
$865$	−8.07449	−0.274541
$866$	49.5930	1.68524
$867$	0	0
$868$	−22.3953	−0.760147
$869$	−0.814230	−0.0276209
$870$	0	0
$871$	19.0337	0.644931
$872$	84.8250	2.87254
$873$	0	0
$874$	16.2744	0.550491
$875$	0.393832	0.0133140
$876$	0	0
$877$	13.5741	0.458364	0.229182	−	0.973384i	$-0.426395\pi$
$877$	13.5741	0.458364	0.229182	+	0.973384i	$0.426395\pi$
$878$	−44.7191	−1.50920
$879$	0	0
$880$	6.36011	0.214399
$881$	−9.91235	−0.333956	−0.166978	−	0.985961i	$-0.553401\pi$
$881$	−9.91235	−0.333956	−0.166978	+	0.985961i	$0.553401\pi$
$882$	0	0
$883$	39.3192	1.32320	0.661598	−	0.749859i	$-0.269879\pi$
$883$	39.3192	1.32320	0.661598	+	0.749859i	$0.269879\pi$
$884$	29.8478	1.00389
$885$	0	0
$886$	97.5180	3.27618
$887$	59.2520	1.98949	0.994743	−	0.102403i	$-0.0326531\pi$
$887$	59.2520	1.98949	0.994743	+	0.102403i	$0.0326531\pi$
$888$	0	0
$889$	7.93832	0.266243
$890$	−43.6376	−1.46274
$891$	0	0
$892$	70.2678	2.35274
$893$	−11.1155	−0.371968
$894$	0	0
$895$	−12.4100	−0.414821
$896$	19.3461	0.646307
$897$	0	0
$898$	−114.942	−3.83567
$899$	−10.1566	−0.338742
$900$	0	0
$901$	−25.6211	−0.853562
$902$	1.79939	0.0599131
$903$	0	0
$904$	111.195	3.69828
$905$	−2.49765	−0.0830248
$906$	0	0
$907$	−5.91210	−0.196308	−0.0981541	−	0.995171i	$-0.531294\pi$
$907$	−5.91210	−0.196308	−0.0981541	+	0.995171i	$0.531294\pi$
$908$	−7.28682	−0.241822
$909$	0	0
$910$	−2.78478	−0.0923144
$911$	12.4185	0.411445	0.205722	−	0.978610i	$-0.434046\pi$
$911$	12.4185	0.411445	0.205722	+	0.978610i	$0.434046\pi$
$912$	0	0
$913$	−2.52447	−0.0835476
$914$	−44.4118	−1.46901
$915$	0	0
$916$	18.4024	0.608031
$917$	2.16700	0.0715607
$918$	0	0
$919$	16.4332	0.542081	0.271041	−	0.962568i	$-0.412632\pi$
$919$	16.4332	0.542081	0.271041	+	0.962568i	$0.412632\pi$
$920$	−61.0769	−2.01364
$921$	0	0
$922$	−47.7983	−1.57415
$923$	15.3530	0.505351
$924$	0	0
$925$	−7.34192	−0.241401
$926$	53.2510	1.74994
$927$	0	0
$928$	24.6760	0.810029
$929$	−13.8358	−0.453936	−0.226968	−	0.973902i	$-0.572881\pi$
$929$	−13.8358	−0.453936	−0.226968	+	0.973902i	$0.572881\pi$
$930$	0	0
$931$	6.56384	0.215121
$932$	−99.1637	−3.24822
$933$	0	0
$934$	22.0528	0.721589
$935$	−0.818503	−0.0267679
$936$	0	0
$937$	20.7528	0.677964	0.338982	−	0.940793i	$-0.389917\pi$
$937$	20.7528	0.677964	0.338982	+	0.940793i	$0.389917\pi$
$938$	−8.05567	−0.263027
$939$	0	0
$940$	64.8989	2.11677
$941$	−40.1666	−1.30940	−0.654698	−	0.755891i	$-0.727204\pi$
$941$	−40.1666	−1.30940	−0.654698	+	0.755891i	$0.727204\pi$
$942$	0	0
$943$	−10.2043	−0.332297
$944$	154.068	5.01447
$945$	0	0
$946$	11.2123	0.364544
$947$	17.4144	0.565894	0.282947	−	0.959136i	$-0.408688\pi$
$947$	17.4144	0.565894	0.282947	+	0.959136i	$0.408688\pi$
$948$	0	0
$949$	−8.57239	−0.278271
$950$	−2.64341	−0.0857636
$951$	0	0
$952$	−8.11999	−0.263170
$953$	−13.5121	−0.437701	−0.218851	−	0.975758i	$-0.570231\pi$
$953$	−13.5121	−0.437701	−0.218851	+	0.975758i	$0.570231\pi$
$954$	0	0
$955$	6.32085	0.204538
$956$	−119.060	−3.85067
$957$	0	0
$958$	111.333	3.59700
$959$	−2.95316	−0.0953626
$960$	0	0
$961$	72.1567	2.32763
$962$	51.9145	1.67379
$963$	0	0
$964$	85.7610	2.76218
$965$	25.2921	0.814183
$966$	0	0
$967$	−24.0598	−0.773712	−0.386856	−	0.922140i	$-0.626439\pi$
$967$	−24.0598	−0.773712	−0.386856	+	0.922140i	$0.626439\pi$
$968$	−107.587	−3.45798
$969$	0	0
$970$	−50.8798	−1.63365
$971$	−34.7877	−1.11639	−0.558195	−	0.829710i	$-0.688506\pi$
$971$	−34.7877	−1.11639	−0.558195	+	0.829710i	$0.688506\pi$
$972$	0	0
$973$	−3.68342	−0.118085
$974$	7.83344	0.251000
$975$	0	0
$976$	−100.988	−3.23254
$977$	−35.7566	−1.14396	−0.571978	−	0.820269i	$-0.693824\pi$
$977$	−35.7566	−1.14396	−0.571978	+	0.820269i	$0.693824\pi$
$978$	0	0
$979$	−6.23446	−0.199254
$980$	−38.3234	−1.22420
$981$	0	0
$982$	−0.435166	−0.0138867
$983$	7.19064	0.229346	0.114673	−	0.993403i	$-0.463418\pi$
$983$	7.19064	0.229346	0.114673	+	0.993403i	$0.463418\pi$
$984$	0	0
$985$	21.2047	0.675638
$986$	−5.72905	−0.182450
$987$	0	0
$988$	13.7719	0.438143
$989$	−63.5847	−2.02188
$990$	0	0
$991$	−19.9123	−0.632537	−0.316268	−	0.948670i	$-0.602430\pi$
$991$	−19.9123	−0.632537	−0.316268	+	0.948670i	$0.602430\pi$
$992$	−250.624	−7.95733
$993$	0	0
$994$	−6.49790	−0.206101
$995$	−2.64723	−0.0839228
$996$	0	0
$997$	−1.57302	−0.0498179	−0.0249089	−	0.999690i	$-0.507930\pi$
$997$	−1.57302	−0.0498179	−0.0249089	+	0.999690i	$0.507930\pi$
$998$	7.29734	0.230993
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.a.r.1.4		4
3.2	odd	2		435.2.a.j.1.1	✓	4
5.4	even	2		6525.2.a.bi.1.1		4
12.11	even	2		6960.2.a.co.1.2		4
15.2	even	4		2175.2.c.n.349.1		8
15.8	even	4		2175.2.c.n.349.8		8
15.14	odd	2		2175.2.a.v.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.j.1.1	✓	4	3.2	odd	2
1305.2.a.r.1.4		4	1.1	even	1	trivial
2175.2.a.v.1.4		4	15.14	odd	2
2175.2.c.n.349.1		8	15.2	even	4
2175.2.c.n.349.8		8	15.8	even	4
6525.2.a.bi.1.1		4	5.4	even	2
6960.2.a.co.1.2		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q+2.75660 q^{2} +5.59883 q^{4} +1.00000 q^{5} +0.393832 q^{7} +9.92054 q^{8} +2.75660 q^{10} +0.393832 q^{11} -2.56511 q^{13} +1.08564 q^{14} +16.1493 q^{16} -2.07830 q^{17} -0.958939 q^{19} +5.59883 q^{20} +1.08564 q^{22} -6.15661 q^{23} +1.00000 q^{25} -7.07097 q^{26} +2.20500 q^{28} +1.00000 q^{29} -10.1566 q^{31} +24.6760 q^{32} -5.72905 q^{34} +0.393832 q^{35} -7.34192 q^{37} -2.64341 q^{38} +9.92054 q^{40} +1.65745 q^{41} +10.3279 q^{43} +2.20500 q^{44} -16.9713 q^{46} +11.5915 q^{47} -6.84490 q^{49} +2.75660 q^{50} -14.3616 q^{52} +12.3279 q^{53} +0.393832 q^{55} +3.90703 q^{56} +2.75660 q^{58} +9.54022 q^{59} -6.25340 q^{61} -27.9977 q^{62} +35.7232 q^{64} -2.56511 q^{65} -7.42023 q^{67} -11.6361 q^{68} +1.08564 q^{70} -5.98533 q^{71} +3.34192 q^{73} -20.2387 q^{74} -5.36894 q^{76} +0.155104 q^{77} -2.06745 q^{79} +16.1493 q^{80} +4.56892 q^{82} -6.41000 q^{83} -2.07830 q^{85} +28.4698 q^{86} +3.90703 q^{88} -15.8302 q^{89} -1.01022 q^{91} -34.4698 q^{92} +31.9531 q^{94} -0.958939 q^{95} -18.4575 q^{97} -18.8686 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{2} + 5 q^{4} + 4 q^{5} + 2 q^{7} + 12 q^{8} + 3 q^{10} + 2 q^{11} - 8 q^{13} + 3 q^{14} + 11 q^{16} + 10 q^{17} - 2 q^{19} + 5 q^{20} + 3 q^{22} + 12 q^{23} + 4 q^{25} + 7 q^{26} - 9 q^{28} + 4 q^{29}+ \cdots - 34 q^{98}+O(q^{100}) \) 4 * q + 3 * q^2 + 5 * q^4 + 4 * q^5 + 2 * q^7 + 12 * q^8 + 3 * q^10 + 2 * q^11 - 8 * q^13 + 3 * q^14 + 11 * q^16 + 10 * q^17 - 2 * q^19 + 5 * q^20 + 3 * q^22 + 12 * q^23 + 4 * q^25 + 7 * q^26 - 9 * q^28 + 4 * q^29 - 4 * q^31 + 17 * q^32 - q^34 + 2 * q^35 - 16 * q^37 + 10 * q^38 + 12 * q^40 + 12 * q^41 + 2 * q^43 - 9 * q^44 - 8 * q^46 + 12 * q^47 + 6 * q^49 + 3 * q^50 - 3 * q^52 + 10 * q^53 + 2 * q^55 + 3 * q^58 - 2 * q^59 - 26 * q^61 - 20 * q^62 + 34 * q^64 - 8 * q^65 + 2 * q^67 - 9 * q^68 + 3 * q^70 + 10 * q^71 - 48 * q^74 + 16 * q^76 + 34 * q^77 + 22 * q^79 + 11 * q^80 + 38 * q^82 + 10 * q^83 + 10 * q^85 + 4 * q^86 + 4 * q^89 - 8 * q^91 - 28 * q^92 + 39 * q^94 - 2 * q^95 - 22 * q^97 - 34 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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