LMFDB - Embedded newform 4225.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.30278 q^{2} -1.00000 q^{3} -0.302776 q^{4} -1.30278 q^{6} -1.00000 q^{7} -3.00000 q^{8} -2.00000 q^{9} +5.60555 q^{11} +0.302776 q^{12} -1.30278 q^{14} -3.30278 q^{16} -0.394449 q^{17} -2.60555 q^{18} -1.60555 q^{19} +1.00000 q^{21} +7.30278 q^{22} +3.00000 q^{23} +3.00000 q^{24} +5.00000 q^{27} +0.302776 q^{28} +8.21110 q^{29} +4.00000 q^{31} +1.69722 q^{32} -5.60555 q^{33} -0.513878 q^{34} +0.605551 q^{36} -3.60555 q^{37} -2.09167 q^{38} -3.00000 q^{41} +1.30278 q^{42} -4.21110 q^{43} -1.69722 q^{44} +3.90833 q^{46} -5.21110 q^{47} +3.30278 q^{48} -6.00000 q^{49} +0.394449 q^{51} -11.2111 q^{53} +6.51388 q^{54} +3.00000 q^{56} +1.60555 q^{57} +10.6972 q^{58} -10.8167 q^{59} -1.00000 q^{61} +5.21110 q^{62} +2.00000 q^{63} +8.81665 q^{64} -7.30278 q^{66} -7.00000 q^{67} +0.119429 q^{68} -3.00000 q^{69} +16.8167 q^{71} +6.00000 q^{72} -15.2111 q^{73} -4.69722 q^{74} +0.486122 q^{76} -5.60555 q^{77} -9.21110 q^{79} +1.00000 q^{81} -3.90833 q^{82} +5.21110 q^{83} -0.302776 q^{84} -5.48612 q^{86} -8.21110 q^{87} -16.8167 q^{88} -8.21110 q^{89} -0.908327 q^{92} -4.00000 q^{93} -6.78890 q^{94} -1.69722 q^{96} -15.6056 q^{97} -7.81665 q^{98} -11.2111 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 2 q^{3} + 3 q^{4} + q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9} + 4 q^{11} - 3 q^{12} + q^{14} - 3 q^{16} - 8 q^{17} + 2 q^{18} + 4 q^{19} + 2 q^{21} + 11 q^{22} + 6 q^{23} + 6 q^{24} + 10 q^{27}+ \cdots - 8 q^{99}+O(q^{100}) $ 2 * q - q^2 - 2 * q^3 + 3 * q^4 + q^6 - 2 * q^7 - 6 * q^8 - 4 * q^9 + 4 * q^11 - 3 * q^12 + q^14 - 3 * q^16 - 8 * q^17 + 2 * q^18 + 4 * q^19 + 2 * q^21 + 11 * q^22 + 6 * q^23 + 6 * q^24 + 10 * q^27 - 3 * q^28 + 2 * q^29 + 8 * q^31 + 7 * q^32 - 4 * q^33 + 17 * q^34 - 6 * q^36 - 15 * q^38 - 6 * q^41 - q^42 + 6 * q^43 - 7 * q^44 - 3 * q^46 + 4 * q^47 + 3 * q^48 - 12 * q^49 + 8 * q^51 - 8 * q^53 - 5 * q^54 + 6 * q^56 - 4 * q^57 + 25 * q^58 - 2 * q^61 - 4 * q^62 + 4 * q^63 - 4 * q^64 - 11 * q^66 - 14 * q^67 - 25 * q^68 - 6 * q^69 + 12 * q^71 + 12 * q^72 - 16 * q^73 - 13 * q^74 + 19 * q^76 - 4 * q^77 - 4 * q^79 + 2 * q^81 + 3 * q^82 - 4 * q^83 + 3 * q^84 - 29 * q^86 - 2 * q^87 - 12 * q^88 - 2 * q^89 + 9 * q^92 - 8 * q^93 - 28 * q^94 - 7 * q^96 - 24 * q^97 + 6 * q^98 - 8 * q^99

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 $

$2$	1.30278	0.921201	0.460601	−	0.887607i	$-0.347634\pi$
$2$	1.30278	0.921201	0.460601	+	0.887607i	$0.347634\pi$
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	−0.302776	−0.151388
$5$	0	0
$5$	0	0
$6$	−1.30278	−0.531856
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−3.00000	−1.06066
$9$	−2.00000	−0.666667
$10$	0	0
$11$	5.60555	1.69014	0.845069	−	0.534658i	$-0.179559\pi$
$11$	5.60555	1.69014	0.845069	+	0.534658i	$0.179559\pi$
$12$	0.302776	0.0874038
$13$	0	0
$13$	0	0
$14$	−1.30278	−0.348181
$15$	0	0
$16$	−3.30278	−0.825694
$17$	−0.394449	−0.0956679	−0.0478339	−	0.998855i	$-0.515232\pi$
$17$	−0.394449	−0.0956679	−0.0478339	+	0.998855i	$0.515232\pi$
$18$	−2.60555	−0.614134
$19$	−1.60555	−0.368339	−0.184169	−	0.982895i	$-0.558959\pi$
$19$	−1.60555	−0.368339	−0.184169	+	0.982895i	$0.558959\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	7.30278	1.55696
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	0	0
$27$	5.00000	0.962250
$28$	0.302776	0.0572192
$29$	8.21110	1.52476	0.762382	−	0.647128i	$-0.224030\pi$
$29$	8.21110	1.52476	0.762382	+	0.647128i	$0.224030\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.69722	0.300030
$33$	−5.60555	−0.975801
$34$	−0.513878	−0.0881294
$35$	0	0
$36$	0.605551	0.100925
$37$	−3.60555	−0.592749	−0.296374	−	0.955072i	$-0.595778\pi$
$37$	−3.60555	−0.592749	−0.296374	+	0.955072i	$0.595778\pi$
$38$	−2.09167	−0.339314
$39$	0	0
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	1.30278	0.201023
$43$	−4.21110	−0.642187	−0.321094	−	0.947047i	$-0.604050\pi$
$43$	−4.21110	−0.642187	−0.321094	+	0.947047i	$0.604050\pi$
$44$	−1.69722	−0.255866
$45$	0	0
$46$	3.90833	0.576251
$47$	−5.21110	−0.760117	−0.380059	−	0.924962i	$-0.624096\pi$
$47$	−5.21110	−0.760117	−0.380059	+	0.924962i	$0.624096\pi$
$48$	3.30278	0.476715
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0.394449	0.0552339
$52$	0	0
$53$	−11.2111	−1.53996	−0.769982	−	0.638066i	$-0.779735\pi$
$53$	−11.2111	−1.53996	−0.769982	+	0.638066i	$0.779735\pi$
$54$	6.51388	0.886427
$55$	0	0
$56$	3.00000	0.400892
$57$	1.60555	0.212660
$58$	10.6972	1.40461
$59$	−10.8167	−1.40821	−0.704104	−	0.710097i	$-0.748651\pi$
$59$	−10.8167	−1.40821	−0.704104	+	0.710097i	$0.748651\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	5.21110	0.661811
$63$	2.00000	0.251976
$64$	8.81665	1.10208
$65$	0	0
$66$	−7.30278	−0.898910
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	0.119429	0.0144829
$69$	−3.00000	−0.361158
$70$	0	0
$71$	16.8167	1.99577	0.997885	−	0.0650069i	$-0.0207069\pi$
$71$	16.8167	1.99577	0.997885	+	0.0650069i	$0.0207069\pi$
$72$	6.00000	0.707107
$73$	−15.2111	−1.78032	−0.890162	−	0.455643i	$-0.849409\pi$
$73$	−15.2111	−1.78032	−0.890162	+	0.455643i	$0.849409\pi$
$74$	−4.69722	−0.546041
$75$	0	0
$76$	0.486122	0.0557620
$77$	−5.60555	−0.638812
$78$	0	0
$79$	−9.21110	−1.03633	−0.518165	−	0.855281i	$-0.673385\pi$
$79$	−9.21110	−1.03633	−0.518165	+	0.855281i	$0.673385\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−3.90833	−0.431603
$83$	5.21110	0.571993	0.285996	−	0.958231i	$-0.407675\pi$
$83$	5.21110	0.571993	0.285996	+	0.958231i	$0.407675\pi$
$84$	−0.302776	−0.0330355
$85$	0	0
$86$	−5.48612	−0.591584
$87$	−8.21110	−0.880323
$88$	−16.8167	−1.79266
$89$	−8.21110	−0.870375	−0.435188	−	0.900340i	$-0.643318\pi$
$89$	−8.21110	−0.870375	−0.435188	+	0.900340i	$0.643318\pi$
$90$	0	0
$91$	0	0
$92$	−0.908327	−0.0946996
$93$	−4.00000	−0.414781
$94$	−6.78890	−0.700221
$95$	0	0
$96$	−1.69722	−0.173222
$97$	−15.6056	−1.58450	−0.792252	−	0.610194i	$-0.791091\pi$
$97$	−15.6056	−1.58450	−0.792252	+	0.610194i	$0.791091\pi$
$98$	−7.81665	−0.789601
$99$	−11.2111	−1.12676
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	0.513878	0.0508815
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	−14.6056	−1.41862
$107$	8.21110	0.793797	0.396899	−	0.917862i	$-0.370086\pi$
$107$	8.21110	0.793797	0.396899	+	0.917862i	$0.370086\pi$
$108$	−1.51388	−0.145673
$109$	4.78890	0.458693	0.229347	−	0.973345i	$-0.426341\pi$
$109$	4.78890	0.458693	0.229347	+	0.973345i	$0.426341\pi$
$110$	0	0
$111$	3.60555	0.342224
$112$	3.30278	0.312083
$113$	−5.60555	−0.527326	−0.263663	−	0.964615i	$-0.584931\pi$
$113$	−5.60555	−0.527326	−0.263663	+	0.964615i	$0.584931\pi$
$114$	2.09167	0.195903
$115$	0	0
$116$	−2.48612	−0.230831
$117$	0	0
$118$	−14.0917	−1.29724
$119$	0.394449	0.0361591
$120$	0	0
$121$	20.4222	1.85656
$122$	−1.30278	−0.117948
$123$	3.00000	0.270501
$124$	−1.21110	−0.108760
$125$	0	0
$126$	2.60555	0.232121
$127$	−10.2111	−0.906089	−0.453044	−	0.891488i	$-0.649662\pi$
$127$	−10.2111	−0.906089	−0.453044	+	0.891488i	$0.649662\pi$
$128$	8.09167	0.715210
$129$	4.21110	0.370767
$130$	0	0
$131$	−6.78890	−0.593149	−0.296574	−	0.955010i	$-0.595844\pi$
$131$	−6.78890	−0.593149	−0.296574	+	0.955010i	$0.595844\pi$
$132$	1.69722	0.147724
$133$	1.60555	0.139219
$134$	−9.11943	−0.787799
$135$	0	0
$136$	1.18335	0.101471
$137$	5.60555	0.478915	0.239457	−	0.970907i	$-0.423030\pi$
$137$	5.60555	0.478915	0.239457	+	0.970907i	$0.423030\pi$
$138$	−3.90833	−0.332699
$139$	13.6056	1.15401	0.577004	−	0.816741i	$-0.304222\pi$
$139$	13.6056	1.15401	0.577004	+	0.816741i	$0.304222\pi$
$140$	0	0
$141$	5.21110	0.438854
$142$	21.9083	1.83851
$143$	0	0
$144$	6.60555	0.550463
$145$	0	0
$146$	−19.8167	−1.64004
$147$	6.00000	0.494872
$148$	1.09167	0.0897350
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	−13.2111	−1.07510	−0.537552	−	0.843231i	$-0.680651\pi$
$151$	−13.2111	−1.07510	−0.537552	+	0.843231i	$0.680651\pi$
$152$	4.81665	0.390682
$153$	0.788897	0.0637786
$154$	−7.30278	−0.588474
$155$	0	0
$156$	0	0
$157$	3.21110	0.256274	0.128137	−	0.991756i	$-0.459100\pi$
$157$	3.21110	0.256274	0.128137	+	0.991756i	$0.459100\pi$
$158$	−12.0000	−0.954669
$159$	11.2111	0.889098
$160$	0	0
$161$	−3.00000	−0.236433
$162$	1.30278	0.102356
$163$	−18.2111	−1.42640	−0.713202	−	0.700959i	$-0.752756\pi$
$163$	−18.2111	−1.42640	−0.713202	+	0.700959i	$0.752756\pi$
$164$	0.908327	0.0709284
$165$	0	0
$166$	6.78890	0.526921
$167$	9.00000	0.696441	0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000	0.696441	0.348220	+	0.937413i	$0.386786\pi$
$168$	−3.00000	−0.231455
$169$	0	0
$170$	0	0
$171$	3.21110	0.245559
$172$	1.27502	0.0972193
$173$	16.8167	1.27855	0.639273	−	0.768980i	$-0.279235\pi$
$173$	16.8167	1.27855	0.639273	+	0.768980i	$0.279235\pi$
$174$	−10.6972	−0.810954
$175$	0	0
$176$	−18.5139	−1.39554
$177$	10.8167	0.813029
$178$	−10.6972	−0.801791
$179$	1.18335	0.0884474	0.0442237	−	0.999022i	$-0.485919\pi$
$179$	1.18335	0.0884474	0.0442237	+	0.999022i	$0.485919\pi$
$180$	0	0
$181$	−25.6333	−1.90531	−0.952654	−	0.304055i	$-0.901659\pi$
$181$	−25.6333	−1.90531	−0.952654	+	0.304055i	$0.901659\pi$
$182$	0	0
$183$	1.00000	0.0739221
$184$	−9.00000	−0.663489
$185$	0	0
$186$	−5.21110	−0.382097
$187$	−2.21110	−0.161692
$188$	1.57779	0.115073
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−4.81665	−0.348521	−0.174260	−	0.984700i	$-0.555753\pi$
$191$	−4.81665	−0.348521	−0.174260	+	0.984700i	$0.555753\pi$
$192$	−8.81665	−0.636287
$193$	8.39445	0.604246	0.302123	−	0.953269i	$-0.402305\pi$
$193$	8.39445	0.604246	0.302123	+	0.953269i	$0.402305\pi$
$194$	−20.3305	−1.45965
$195$	0	0
$196$	1.81665	0.129761
$197$	22.8167	1.62562	0.812810	−	0.582529i	$-0.197937\pi$
$197$	22.8167	1.62562	0.812810	+	0.582529i	$0.197937\pi$
$198$	−14.6056	−1.03797
$199$	−8.81665	−0.624996	−0.312498	−	0.949918i	$-0.601166\pi$
$199$	−8.81665	−0.624996	−0.312498	+	0.949918i	$0.601166\pi$
$200$	0	0
$201$	7.00000	0.493742
$202$	−11.7250	−0.824967
$203$	−8.21110	−0.576306
$204$	−0.119429	−0.00836174
$205$	0	0
$206$	5.21110	0.363075
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−9.00000	−0.622543
$210$	0	0
$211$	−16.3944	−1.12864	−0.564320	−	0.825556i	$-0.690862\pi$
$211$	−16.3944	−1.12864	−0.564320	+	0.825556i	$0.690862\pi$
$212$	3.39445	0.233132
$213$	−16.8167	−1.15226
$214$	10.6972	0.731247
$215$	0	0
$216$	−15.0000	−1.02062
$217$	−4.00000	−0.271538
$218$	6.23886	0.422549
$219$	15.2111	1.02787
$220$	0	0
$221$	0	0
$222$	4.69722	0.315257
$223$	10.2111	0.683786	0.341893	−	0.939739i	$-0.388932\pi$
$223$	10.2111	0.683786	0.341893	+	0.939739i	$0.388932\pi$
$224$	−1.69722	−0.113401
$225$	0	0
$226$	−7.30278	−0.485773
$227$	1.42221	0.0943951	0.0471975	−	0.998886i	$-0.484971\pi$
$227$	1.42221	0.0943951	0.0471975	+	0.998886i	$0.484971\pi$
$228$	−0.486122	−0.0321942
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	5.60555	0.368818
$232$	−24.6333	−1.61726
$233$	−0.788897	−0.0516824	−0.0258412	−	0.999666i	$-0.508226\pi$
$233$	−0.788897	−0.0516824	−0.0258412	+	0.999666i	$0.508226\pi$
$234$	0	0
$235$	0	0
$236$	3.27502	0.213186
$237$	9.21110	0.598325
$238$	0.513878	0.0333098
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−16.2111	−1.04425	−0.522124	−	0.852869i	$-0.674860\pi$
$241$	−16.2111	−1.04425	−0.522124	+	0.852869i	$0.674860\pi$
$242$	26.6056	1.71027
$243$	−16.0000	−1.02640
$244$	0.302776	0.0193832
$245$	0	0
$246$	3.90833	0.249186
$247$	0	0
$248$	−12.0000	−0.762001
$249$	−5.21110	−0.330240
$250$	0	0
$251$	−28.8167	−1.81889	−0.909446	−	0.415823i	$-0.863494\pi$
$251$	−28.8167	−1.81889	−0.909446	+	0.415823i	$0.863494\pi$
$252$	−0.605551	−0.0381461
$253$	16.8167	1.05725
$254$	−13.3028	−0.834690
$255$	0	0
$256$	−7.09167	−0.443230
$257$	23.6056	1.47247	0.736237	−	0.676724i	$-0.236601\pi$
$257$	23.6056	1.47247	0.736237	+	0.676724i	$0.236601\pi$
$258$	5.48612	0.341551
$259$	3.60555	0.224038
$260$	0	0
$261$	−16.4222	−1.01651
$262$	−8.84441	−0.546409
$263$	−26.2111	−1.61625	−0.808123	−	0.589014i	$-0.799516\pi$
$263$	−26.2111	−1.61625	−0.808123	+	0.589014i	$0.799516\pi$
$264$	16.8167	1.03499
$265$	0	0
$266$	2.09167	0.128249
$267$	8.21110	0.502511
$268$	2.11943	0.129465
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	−0.816654	−0.0496082	−0.0248041	−	0.999692i	$-0.507896\pi$
$271$	−0.816654	−0.0496082	−0.0248041	+	0.999692i	$0.507896\pi$
$272$	1.30278	0.0789924
$273$	0	0
$274$	7.30278	0.441177
$275$	0	0
$276$	0.908327	0.0546749
$277$	−20.3944	−1.22538	−0.612692	−	0.790322i	$-0.709913\pi$
$277$	−20.3944	−1.22538	−0.612692	+	0.790322i	$0.709913\pi$
$278$	17.7250	1.06307
$279$	−8.00000	−0.478947
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	6.78890	0.404273
$283$	−5.00000	−0.297219	−0.148610	−	0.988896i	$-0.547480\pi$
$283$	−5.00000	−0.297219	−0.148610	+	0.988896i	$0.547480\pi$
$284$	−5.09167	−0.302135
$285$	0	0
$286$	0	0
$287$	3.00000	0.177084
$288$	−3.39445	−0.200020
$289$	−16.8444	−0.990848
$290$	0	0
$291$	15.6056	0.914814
$292$	4.60555	0.269520
$293$	17.6056	1.02853	0.514264	−	0.857632i	$-0.328065\pi$
$293$	17.6056	1.02853	0.514264	+	0.857632i	$0.328065\pi$
$294$	7.81665	0.455877
$295$	0	0
$296$	10.8167	0.628705
$297$	28.0278	1.62634
$298$	−3.90833	−0.226403
$299$	0	0
$300$	0	0
$301$	4.21110	0.242724
$302$	−17.2111	−0.990388
$303$	9.00000	0.517036
$304$	5.30278	0.304135
$305$	0	0
$306$	1.02776	0.0587529
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	1.69722	0.0967083
$309$	−4.00000	−0.227552
$310$	0	0
$311$	5.21110	0.295495	0.147747	−	0.989025i	$-0.452798\pi$
$311$	5.21110	0.295495	0.147747	+	0.989025i	$0.452798\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	4.18335	0.236080
$315$	0	0
$316$	2.78890	0.156888
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	14.6056	0.819039
$319$	46.0278	2.57706
$320$	0	0
$321$	−8.21110	−0.458299
$322$	−3.90833	−0.217803
$323$	0.633308	0.0352382
$324$	−0.302776	−0.0168209
$325$	0	0
$326$	−23.7250	−1.31401
$327$	−4.78890	−0.264827
$328$	9.00000	0.496942
$329$	5.21110	0.287297
$330$	0	0
$331$	26.0278	1.43061	0.715307	−	0.698810i	$-0.246287\pi$
$331$	26.0278	1.43061	0.715307	+	0.698810i	$0.246287\pi$
$332$	−1.57779	−0.0865927
$333$	7.21110	0.395166
$334$	11.7250	0.641562
$335$	0	0
$336$	−3.30278	−0.180181
$337$	−17.6333	−0.960547	−0.480274	−	0.877119i	$-0.659463\pi$
$337$	−17.6333	−0.960547	−0.480274	+	0.877119i	$0.659463\pi$
$338$	0	0
$339$	5.60555	0.304452
$340$	0	0
$341$	22.4222	1.21423
$342$	4.18335	0.226209
$343$	13.0000	0.701934
$344$	12.6333	0.681142
$345$	0	0
$346$	21.9083	1.17780
$347$	−20.2111	−1.08499	−0.542494	−	0.840059i	$-0.682520\pi$
$347$	−20.2111	−1.08499	−0.542494	+	0.840059i	$0.682520\pi$
$348$	2.48612	0.133270
$349$	18.2111	0.974818	0.487409	−	0.873174i	$-0.337942\pi$
$349$	18.2111	0.974818	0.487409	+	0.873174i	$0.337942\pi$
$350$	0	0
$351$	0	0
$352$	9.51388	0.507091
$353$	−4.81665	−0.256365	−0.128182	−	0.991751i	$-0.540914\pi$
$353$	−4.81665	−0.256365	−0.128182	+	0.991751i	$0.540914\pi$
$354$	14.0917	0.748964
$355$	0	0
$356$	2.48612	0.131764
$357$	−0.394449	−0.0208764
$358$	1.54163	0.0814779
$359$	10.4222	0.550063	0.275031	−	0.961435i	$-0.411312\pi$
$359$	10.4222	0.550063	0.275031	+	0.961435i	$0.411312\pi$
$360$	0	0
$361$	−16.4222	−0.864327
$362$	−33.3944	−1.75517
$363$	−20.4222	−1.07189
$364$	0	0
$365$	0	0
$366$	1.30278	0.0680972
$367$	17.4222	0.909432	0.454716	−	0.890637i	$-0.349741\pi$
$367$	17.4222	0.909432	0.454716	+	0.890637i	$0.349741\pi$
$368$	−9.90833	−0.516507
$369$	6.00000	0.312348
$370$	0	0
$371$	11.2111	0.582051
$372$	1.21110	0.0627927
$373$	27.6056	1.42936	0.714681	−	0.699451i	$-0.246572\pi$
$373$	27.6056	1.42936	0.714681	+	0.699451i	$0.246572\pi$
$374$	−2.88057	−0.148951
$375$	0	0
$376$	15.6333	0.806226
$377$	0	0
$378$	−6.51388	−0.335038
$379$	−2.39445	−0.122995	−0.0614973	−	0.998107i	$-0.519588\pi$
$379$	−2.39445	−0.122995	−0.0614973	+	0.998107i	$0.519588\pi$
$380$	0	0
$381$	10.2111	0.523131
$382$	−6.27502	−0.321058
$383$	18.6333	0.952118	0.476059	−	0.879413i	$-0.342065\pi$
$383$	18.6333	0.952118	0.476059	+	0.879413i	$0.342065\pi$
$384$	−8.09167	−0.412926
$385$	0	0
$386$	10.9361	0.556632
$387$	8.42221	0.428125
$388$	4.72498	0.239875
$389$	−0.788897	−0.0399987	−0.0199993	−	0.999800i	$-0.506366\pi$
$389$	−0.788897	−0.0399987	−0.0199993	+	0.999800i	$0.506366\pi$
$390$	0	0
$391$	−1.18335	−0.0598444
$392$	18.0000	0.909137
$393$	6.78890	0.342455
$394$	29.7250	1.49752
$395$	0	0
$396$	3.39445	0.170577
$397$	−14.0278	−0.704033	−0.352016	−	0.935994i	$-0.614504\pi$
$397$	−14.0278	−0.704033	−0.352016	+	0.935994i	$0.614504\pi$
$398$	−11.4861	−0.575747
$399$	−1.60555	−0.0803781
$400$	0	0
$401$	2.21110	0.110417	0.0552086	−	0.998475i	$-0.482418\pi$
$401$	2.21110	0.110417	0.0552086	+	0.998475i	$0.482418\pi$
$402$	9.11943	0.454836
$403$	0	0
$404$	2.72498	0.135573
$405$	0	0
$406$	−10.6972	−0.530894
$407$	−20.2111	−1.00183
$408$	−1.18335	−0.0585844
$409$	6.21110	0.307119	0.153560	−	0.988139i	$-0.450926\pi$
$409$	6.21110	0.307119	0.153560	+	0.988139i	$0.450926\pi$
$410$	0	0
$411$	−5.60555	−0.276501
$412$	−1.21110	−0.0596667
$413$	10.8167	0.532253
$414$	−7.81665	−0.384168
$415$	0	0
$416$	0	0
$417$	−13.6056	−0.666267
$418$	−11.7250	−0.573488
$419$	−33.2389	−1.62382	−0.811912	−	0.583779i	$-0.801573\pi$
$419$	−33.2389	−1.62382	−0.811912	+	0.583779i	$0.801573\pi$
$420$	0	0
$421$	−3.57779	−0.174371	−0.0871855	−	0.996192i	$-0.527787\pi$
$421$	−3.57779	−0.174371	−0.0871855	+	0.996192i	$0.527787\pi$
$422$	−21.3583	−1.03971
$423$	10.4222	0.506745
$424$	33.6333	1.63338
$425$	0	0
$426$	−21.9083	−1.06146
$427$	1.00000	0.0483934
$428$	−2.48612	−0.120171
$429$	0	0
$430$	0	0
$431$	21.2389	1.02304	0.511520	−	0.859271i	$-0.329083\pi$
$431$	21.2389	1.02304	0.511520	+	0.859271i	$0.329083\pi$
$432$	−16.5139	−0.794524
$433$	3.60555	0.173272	0.0866359	−	0.996240i	$-0.472388\pi$
$433$	3.60555	0.173272	0.0866359	+	0.996240i	$0.472388\pi$
$434$	−5.21110	−0.250141
$435$	0	0
$436$	−1.44996	−0.0694406
$437$	−4.81665	−0.230412
$438$	19.8167	0.946876
$439$	23.2389	1.10913	0.554565	−	0.832140i	$-0.312885\pi$
$439$	23.2389	1.10913	0.554565	+	0.832140i	$0.312885\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	22.4222	1.06531	0.532656	−	0.846332i	$-0.321194\pi$
$443$	22.4222	1.06531	0.532656	+	0.846332i	$0.321194\pi$
$444$	−1.09167	−0.0518085
$445$	0	0
$446$	13.3028	0.629905
$447$	3.00000	0.141895
$448$	−8.81665	−0.416548
$449$	12.6333	0.596203	0.298101	−	0.954534i	$-0.403647\pi$
$449$	12.6333	0.596203	0.298101	+	0.954534i	$0.403647\pi$
$450$	0	0
$451$	−16.8167	−0.791865
$452$	1.69722	0.0798307
$453$	13.2111	0.620712
$454$	1.85281	0.0869569
$455$	0	0
$456$	−4.81665	−0.225560
$457$	−5.18335	−0.242467	−0.121233	−	0.992624i	$-0.538685\pi$
$457$	−5.18335	−0.242467	−0.121233	+	0.992624i	$0.538685\pi$
$458$	−18.2389	−0.852246
$459$	−1.97224	−0.0920564
$460$	0	0
$461$	−21.7889	−1.01481	−0.507405	−	0.861708i	$-0.669395\pi$
$461$	−21.7889	−1.01481	−0.507405	+	0.861708i	$0.669395\pi$
$462$	7.30278	0.339756
$463$	−5.57779	−0.259222	−0.129611	−	0.991565i	$-0.541373\pi$
$463$	−5.57779	−0.259222	−0.129611	+	0.991565i	$0.541373\pi$
$464$	−27.1194	−1.25899
$465$	0	0
$466$	−1.02776	−0.0476099
$467$	−17.2111	−0.796435	−0.398217	−	0.917291i	$-0.630371\pi$
$467$	−17.2111	−0.796435	−0.398217	+	0.917291i	$0.630371\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	0	0
$471$	−3.21110	−0.147960
$472$	32.4500	1.49363
$473$	−23.6056	−1.08538
$474$	12.0000	0.551178
$475$	0	0
$476$	−0.119429	−0.00547404
$477$	22.4222	1.02664
$478$	0	0
$479$	7.18335	0.328215	0.164108	−	0.986442i	$-0.447526\pi$
$479$	7.18335	0.328215	0.164108	+	0.986442i	$0.447526\pi$
$480$	0	0
$481$	0	0
$482$	−21.1194	−0.961964
$483$	3.00000	0.136505
$484$	−6.18335	−0.281061
$485$	0	0
$486$	−20.8444	−0.945522
$487$	−1.00000	−0.0453143	−0.0226572	−	0.999743i	$-0.507213\pi$
$487$	−1.00000	−0.0453143	−0.0226572	+	0.999743i	$0.507213\pi$
$488$	3.00000	0.135804
$489$	18.2111	0.823535
$490$	0	0
$491$	4.81665	0.217373	0.108686	−	0.994076i	$-0.465336\pi$
$491$	4.81665	0.217373	0.108686	+	0.994076i	$0.465336\pi$
$492$	−0.908327	−0.0409505
$493$	−3.23886	−0.145871
$494$	0	0
$495$	0	0
$496$	−13.2111	−0.593196
$497$	−16.8167	−0.754330
$498$	−6.78890	−0.304218
$499$	26.4222	1.18282	0.591410	−	0.806371i	$-0.298571\pi$
$499$	26.4222	1.18282	0.591410	+	0.806371i	$0.298571\pi$
$500$	0	0
$501$	−9.00000	−0.402090
$502$	−37.5416	−1.67557
$503$	−3.00000	−0.133763	−0.0668817	−	0.997761i	$-0.521305\pi$
$503$	−3.00000	−0.133763	−0.0668817	+	0.997761i	$0.521305\pi$
$504$	−6.00000	−0.267261
$505$	0	0
$506$	21.9083	0.973944
$507$	0	0
$508$	3.09167	0.137171
$509$	−3.00000	−0.132973	−0.0664863	−	0.997787i	$-0.521179\pi$
$509$	−3.00000	−0.132973	−0.0664863	+	0.997787i	$0.521179\pi$
$510$	0	0
$511$	15.2111	0.672900
$512$	−25.4222	−1.12351
$513$	−8.02776	−0.354434
$514$	30.7527	1.35645
$515$	0	0
$516$	−1.27502	−0.0561296
$517$	−29.2111	−1.28470
$518$	4.69722	0.206384
$519$	−16.8167	−0.738169
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	−21.3944	−0.936410
$523$	−27.4222	−1.19909	−0.599545	−	0.800341i	$-0.704652\pi$
$523$	−27.4222	−1.19909	−0.599545	+	0.800341i	$0.704652\pi$
$524$	2.05551	0.0897955
$525$	0	0
$526$	−34.1472	−1.48889
$527$	−1.57779	−0.0687298
$528$	18.5139	0.805713
$529$	−14.0000	−0.608696
$530$	0	0
$531$	21.6333	0.938806
$532$	−0.486122	−0.0210761
$533$	0	0
$534$	10.6972	0.462914
$535$	0	0
$536$	21.0000	0.907062
$537$	−1.18335	−0.0510652
$538$	−11.7250	−0.505500
$539$	−33.6333	−1.44869
$540$	0	0
$541$	−17.6333	−0.758115	−0.379058	−	0.925373i	$-0.623752\pi$
$541$	−17.6333	−0.758115	−0.379058	+	0.925373i	$0.623752\pi$
$542$	−1.06392	−0.0456991
$543$	25.6333	1.10003
$544$	−0.669468	−0.0287032
$545$	0	0
$546$	0	0
$547$	24.8444	1.06227	0.531135	−	0.847287i	$-0.321766\pi$
$547$	24.8444	1.06227	0.531135	+	0.847287i	$0.321766\pi$
$548$	−1.69722	−0.0725018
$549$	2.00000	0.0853579
$550$	0	0
$551$	−13.1833	−0.561629
$552$	9.00000	0.383065
$553$	9.21110	0.391696
$554$	−26.5694	−1.12883
$555$	0	0
$556$	−4.11943	−0.174703
$557$	5.60555	0.237515	0.118757	−	0.992923i	$-0.462109\pi$
$557$	5.60555	0.237515	0.118757	+	0.992923i	$0.462109\pi$
$558$	−10.4222	−0.441207
$559$	0	0
$560$	0	0
$561$	2.21110	0.0933528
$562$	7.81665	0.329726
$563$	19.4222	0.818548	0.409274	−	0.912411i	$-0.365782\pi$
$563$	19.4222	0.818548	0.409274	+	0.912411i	$0.365782\pi$
$564$	−1.57779	−0.0664372
$565$	0	0
$566$	−6.51388	−0.273799
$567$	−1.00000	−0.0419961
$568$	−50.4500	−2.11683
$569$	1.42221	0.0596219	0.0298110	−	0.999556i	$-0.490509\pi$
$569$	1.42221	0.0596219	0.0298110	+	0.999556i	$0.490509\pi$
$570$	0	0
$571$	−36.8444	−1.54189	−0.770945	−	0.636901i	$-0.780216\pi$
$571$	−36.8444	−1.54189	−0.770945	+	0.636901i	$0.780216\pi$
$572$	0	0
$573$	4.81665	0.201219
$574$	3.90833	0.163130
$575$	0	0
$576$	−17.6333	−0.734721
$577$	29.6333	1.23365	0.616825	−	0.787100i	$-0.288418\pi$
$577$	29.6333	1.23365	0.616825	+	0.787100i	$0.288418\pi$
$578$	−21.9445	−0.912770
$579$	−8.39445	−0.348861
$580$	0	0
$581$	−5.21110	−0.216193
$582$	20.3305	0.842728
$583$	−62.8444	−2.60275
$584$	45.6333	1.88832
$585$	0	0
$586$	22.9361	0.947481
$587$	−4.57779	−0.188946	−0.0944729	−	0.995527i	$-0.530117\pi$
$587$	−4.57779	−0.188946	−0.0944729	+	0.995527i	$0.530117\pi$
$588$	−1.81665	−0.0749175
$589$	−6.42221	−0.264622
$590$	0	0
$591$	−22.8167	−0.938552
$592$	11.9083	0.489429
$593$	35.2111	1.44595	0.722973	−	0.690876i	$-0.242775\pi$
$593$	35.2111	1.44595	0.722973	+	0.690876i	$0.242775\pi$
$594$	36.5139	1.49818
$595$	0	0
$596$	0.908327	0.0372065
$597$	8.81665	0.360842
$598$	0	0
$599$	−6.78890	−0.277387	−0.138693	−	0.990335i	$-0.544290\pi$
$599$	−6.78890	−0.277387	−0.138693	+	0.990335i	$0.544290\pi$
$600$	0	0
$601$	28.2111	1.15075	0.575377	−	0.817888i	$-0.304855\pi$
$601$	28.2111	1.15075	0.575377	+	0.817888i	$0.304855\pi$
$602$	5.48612	0.223598
$603$	14.0000	0.570124
$604$	4.00000	0.162758
$605$	0	0
$606$	11.7250	0.476295
$607$	19.7889	0.803207	0.401603	−	0.915814i	$-0.368453\pi$
$607$	19.7889	0.803207	0.401603	+	0.915814i	$0.368453\pi$
$608$	−2.72498	−0.110513
$609$	8.21110	0.332731
$610$	0	0
$611$	0	0
$612$	−0.238859	−0.00965530
$613$	1.60555	0.0648476	0.0324238	−	0.999474i	$-0.489677\pi$
$613$	1.60555	0.0648476	0.0324238	+	0.999474i	$0.489677\pi$
$614$	−20.8444	−0.841212
$615$	0	0
$616$	16.8167	0.677562
$617$	26.4500	1.06484	0.532418	−	0.846482i	$-0.321284\pi$
$617$	26.4500	1.06484	0.532418	+	0.846482i	$0.321284\pi$
$618$	−5.21110	−0.209621
$619$	14.4222	0.579677	0.289839	−	0.957076i	$-0.406398\pi$
$619$	14.4222	0.579677	0.289839	+	0.957076i	$0.406398\pi$
$620$	0	0
$621$	15.0000	0.601929
$622$	6.78890	0.272210
$623$	8.21110	0.328971
$624$	0	0
$625$	0	0
$626$	−18.2389	−0.728971
$627$	9.00000	0.359425
$628$	−0.972244	−0.0387967
$629$	1.42221	0.0567070
$630$	0	0
$631$	−0.0277564	−0.00110496	−0.000552482	−	1.00000i	$-0.500176\pi$
$631$	−0.0277564	−0.00110496	−0.000552482		1.00000i	$0.500176\pi$
$632$	27.6333	1.09919
$633$	16.3944	0.651621
$634$	7.81665	0.310439
$635$	0	0
$636$	−3.39445	−0.134599
$637$	0	0
$638$	59.9638	2.37399
$639$	−33.6333	−1.33051
$640$	0	0
$641$	−19.4222	−0.767131	−0.383565	−	0.923514i	$-0.625304\pi$
$641$	−19.4222	−0.767131	−0.383565	+	0.923514i	$0.625304\pi$
$642$	−10.6972	−0.422186
$643$	−40.6333	−1.60242	−0.801211	−	0.598382i	$-0.795810\pi$
$643$	−40.6333	−1.60242	−0.801211	+	0.598382i	$0.795810\pi$
$644$	0.908327	0.0357931
$645$	0	0
$646$	0.825058	0.0324615
$647$	−10.5778	−0.415856	−0.207928	−	0.978144i	$-0.566672\pi$
$647$	−10.5778	−0.415856	−0.207928	+	0.978144i	$0.566672\pi$
$648$	−3.00000	−0.117851
$649$	−60.6333	−2.38007
$650$	0	0
$651$	4.00000	0.156772
$652$	5.51388	0.215940
$653$	28.8167	1.12768	0.563841	−	0.825883i	$-0.309323\pi$
$653$	28.8167	1.12768	0.563841	+	0.825883i	$0.309323\pi$
$654$	−6.23886	−0.243959
$655$	0	0
$656$	9.90833	0.386855
$657$	30.4222	1.18688
$658$	6.78890	0.264659
$659$	−13.1833	−0.513550	−0.256775	−	0.966471i	$-0.582660\pi$
$659$	−13.1833	−0.513550	−0.256775	+	0.966471i	$0.582660\pi$
$660$	0	0
$661$	−38.6333	−1.50266	−0.751331	−	0.659926i	$-0.770588\pi$
$661$	−38.6333	−1.50266	−0.751331	+	0.659926i	$0.770588\pi$
$662$	33.9083	1.31788
$663$	0	0
$664$	−15.6333	−0.606690
$665$	0	0
$666$	9.39445	0.364027
$667$	24.6333	0.953805
$668$	−2.72498	−0.105433
$669$	−10.2111	−0.394784
$670$	0	0
$671$	−5.60555	−0.216400
$672$	1.69722	0.0654719
$673$	10.3944	0.400677	0.200338	−	0.979727i	$-0.435796\pi$
$673$	10.3944	0.400677	0.200338	+	0.979727i	$0.435796\pi$
$674$	−22.9722	−0.884858
$675$	0	0
$676$	0	0
$677$	−33.6333	−1.29263	−0.646317	−	0.763069i	$-0.723691\pi$
$677$	−33.6333	−1.29263	−0.646317	+	0.763069i	$0.723691\pi$
$678$	7.30278	0.280461
$679$	15.6056	0.598886
$680$	0	0
$681$	−1.42221	−0.0544990
$682$	29.2111	1.11855
$683$	21.7889	0.833729	0.416864	−	0.908969i	$-0.363129\pi$
$683$	21.7889	0.833729	0.416864	+	0.908969i	$0.363129\pi$
$684$	−0.972244	−0.0371747
$685$	0	0
$686$	16.9361	0.646623
$687$	14.0000	0.534133
$688$	13.9083	0.530250
$689$	0	0
$690$	0	0
$691$	−6.02776	−0.229307	−0.114653	−	0.993406i	$-0.536576\pi$
$691$	−6.02776	−0.229307	−0.114653	+	0.993406i	$0.536576\pi$
$692$	−5.09167	−0.193556
$693$	11.2111	0.425875
$694$	−26.3305	−0.999493
$695$	0	0
$696$	24.6333	0.933723
$697$	1.18335	0.0448224
$698$	23.7250	0.898004
$699$	0.788897	0.0298388
$700$	0	0
$701$	−7.57779	−0.286209	−0.143105	−	0.989708i	$-0.545709\pi$
$701$	−7.57779	−0.286209	−0.143105	+	0.989708i	$0.545709\pi$
$702$	0	0
$703$	5.78890	0.218332
$704$	49.4222	1.86267
$705$	0	0
$706$	−6.27502	−0.236163
$707$	9.00000	0.338480
$708$	−3.27502	−0.123083
$709$	−43.8444	−1.64661	−0.823306	−	0.567598i	$-0.807873\pi$
$709$	−43.8444	−1.64661	−0.823306	+	0.567598i	$0.807873\pi$
$710$	0	0
$711$	18.4222	0.690887
$712$	24.6333	0.923172
$713$	12.0000	0.449404
$714$	−0.513878	−0.0192314
$715$	0	0
$716$	−0.358288	−0.0133899
$717$	0	0
$718$	13.5778	0.506719
$719$	−18.3944	−0.685997	−0.342999	−	0.939336i	$-0.611443\pi$
$719$	−18.3944	−0.685997	−0.342999	+	0.939336i	$0.611443\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	−21.3944	−0.796219
$723$	16.2111	0.602897
$724$	7.76114	0.288441
$725$	0	0
$726$	−26.6056	−0.987425
$727$	−42.4222	−1.57335	−0.786676	−	0.617366i	$-0.788200\pi$
$727$	−42.4222	−1.57335	−0.786676	+	0.617366i	$0.788200\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	1.66106	0.0614367
$732$	−0.302776	−0.0111909
$733$	10.8444	0.400547	0.200274	−	0.979740i	$-0.435817\pi$
$733$	10.8444	0.400547	0.200274	+	0.979740i	$0.435817\pi$
$734$	22.6972	0.837770
$735$	0	0
$736$	5.09167	0.187682
$737$	−39.2389	−1.44538
$738$	7.81665	0.287735
$739$	28.3944	1.04451	0.522253	−	0.852790i	$-0.325092\pi$
$739$	28.3944	1.04451	0.522253	+	0.852790i	$0.325092\pi$
$740$	0	0
$741$	0	0
$742$	14.6056	0.536187
$743$	−6.63331	−0.243352	−0.121676	−	0.992570i	$-0.538827\pi$
$743$	−6.63331	−0.243352	−0.121676	+	0.992570i	$0.538827\pi$
$744$	12.0000	0.439941
$745$	0	0
$746$	35.9638	1.31673
$747$	−10.4222	−0.381329
$748$	0.669468	0.0244782
$749$	−8.21110	−0.300027
$750$	0	0
$751$	−18.4500	−0.673249	−0.336624	−	0.941639i	$-0.609285\pi$
$751$	−18.4500	−0.673249	−0.336624	+	0.941639i	$0.609285\pi$
$752$	17.2111	0.627624
$753$	28.8167	1.05014
$754$	0	0
$755$	0	0
$756$	1.51388	0.0550592
$757$	20.8167	0.756594	0.378297	−	0.925684i	$-0.376510\pi$
$757$	20.8167	0.756594	0.378297	+	0.925684i	$0.376510\pi$
$758$	−3.11943	−0.113303
$759$	−16.8167	−0.610406
$760$	0	0
$761$	24.6333	0.892957	0.446478	−	0.894794i	$-0.352678\pi$
$761$	24.6333	0.892957	0.446478	+	0.894794i	$0.352678\pi$
$762$	13.3028	0.481909
$763$	−4.78890	−0.173370
$764$	1.45837	0.0527618
$765$	0	0
$766$	24.2750	0.877092
$767$	0	0
$768$	7.09167	0.255899
$769$	−11.0000	−0.396670	−0.198335	−	0.980134i	$-0.563553\pi$
$769$	−11.0000	−0.396670	−0.198335	+	0.980134i	$0.563553\pi$
$770$	0	0
$771$	−23.6056	−0.850133
$772$	−2.54163	−0.0914754
$773$	29.6056	1.06484	0.532419	−	0.846481i	$-0.321283\pi$
$773$	29.6056	1.06484	0.532419	+	0.846481i	$0.321283\pi$
$774$	10.9722	0.394389
$775$	0	0
$776$	46.8167	1.68062
$777$	−3.60555	−0.129348
$778$	−1.02776	−0.0368469
$779$	4.81665	0.172575
$780$	0	0
$781$	94.2666	3.37312
$782$	−1.54163	−0.0551287
$783$	41.0555	1.46720
$784$	19.8167	0.707738
$785$	0	0
$786$	8.84441	0.315470
$787$	−28.6333	−1.02067	−0.510334	−	0.859977i	$-0.670478\pi$
$787$	−28.6333	−1.02067	−0.510334	+	0.859977i	$0.670478\pi$
$788$	−6.90833	−0.246099
$789$	26.2111	0.933140
$790$	0	0
$791$	5.60555	0.199310
$792$	33.6333	1.19511
$793$	0	0
$794$	−18.2750	−0.648556
$795$	0	0
$796$	2.66947	0.0946168
$797$	−50.4500	−1.78703	−0.893515	−	0.449034i	$-0.851768\pi$
$797$	−50.4500	−1.78703	−0.893515	+	0.449034i	$0.851768\pi$
$798$	−2.09167	−0.0740444
$799$	2.05551	0.0727188
$800$	0	0
$801$	16.4222	0.580250
$802$	2.88057	0.101716
$803$	−85.2666	−3.00899
$804$	−2.11943	−0.0747465
$805$	0	0
$806$	0	0
$807$	9.00000	0.316815
$808$	27.0000	0.949857
$809$	17.0555	0.599640	0.299820	−	0.953996i	$-0.403073\pi$
$809$	17.0555	0.599640	0.299820	+	0.953996i	$0.403073\pi$
$810$	0	0
$811$	17.5778	0.617240	0.308620	−	0.951185i	$-0.400133\pi$
$811$	17.5778	0.617240	0.308620	+	0.951185i	$0.400133\pi$
$812$	2.48612	0.0872458
$813$	0.816654	0.0286413
$814$	−26.3305	−0.922885
$815$	0	0
$816$	−1.30278	−0.0456063
$817$	6.76114	0.236542
$818$	8.09167	0.282919
$819$	0	0
$820$	0	0
$821$	7.42221	0.259037	0.129518	−	0.991577i	$-0.458657\pi$
$821$	7.42221	0.259037	0.129518	+	0.991577i	$0.458657\pi$
$822$	−7.30278	−0.254714
$823$	−26.6333	−0.928379	−0.464189	−	0.885736i	$-0.653654\pi$
$823$	−26.6333	−0.928379	−0.464189	+	0.885736i	$0.653654\pi$
$824$	−12.0000	−0.418040
$825$	0	0
$826$	14.0917	0.490312
$827$	−13.5778	−0.472146	−0.236073	−	0.971735i	$-0.575860\pi$
$827$	−13.5778	−0.472146	−0.236073	+	0.971735i	$0.575860\pi$
$828$	1.81665	0.0631331
$829$	0.577795	0.0200676	0.0100338	−	0.999950i	$-0.496806\pi$
$829$	0.577795	0.0200676	0.0100338	+	0.999950i	$0.496806\pi$
$830$	0	0
$831$	20.3944	0.707476
$832$	0	0
$833$	2.36669	0.0820010
$834$	−17.7250	−0.613766
$835$	0	0
$836$	2.72498	0.0942454
$837$	20.0000	0.691301
$838$	−43.3028	−1.49587
$839$	−16.0278	−0.553340	−0.276670	−	0.960965i	$-0.589231\pi$
$839$	−16.0278	−0.553340	−0.276670	+	0.960965i	$0.589231\pi$
$840$	0	0
$841$	38.4222	1.32490
$842$	−4.66106	−0.160631
$843$	−6.00000	−0.206651
$844$	4.96384	0.170862
$845$	0	0
$846$	13.5778	0.466814
$847$	−20.4222	−0.701715
$848$	37.0278	1.27154
$849$	5.00000	0.171600
$850$	0	0
$851$	−10.8167	−0.370790
$852$	5.09167	0.174438
$853$	32.7889	1.12267	0.561335	−	0.827589i	$-0.310288\pi$
$853$	32.7889	1.12267	0.561335	+	0.827589i	$0.310288\pi$
$854$	1.30278	0.0445801
$855$	0	0
$856$	−24.6333	−0.841949
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	25.2111	0.860192	0.430096	−	0.902783i	$-0.358480\pi$
$859$	25.2111	0.860192	0.430096	+	0.902783i	$0.358480\pi$
$860$	0	0
$861$	−3.00000	−0.102240
$862$	27.6695	0.942426
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	8.48612	0.288704
$865$	0	0
$866$	4.69722	0.159618
$867$	16.8444	0.572066
$868$	1.21110	0.0411075
$869$	−51.6333	−1.75154
$870$	0	0
$871$	0	0
$872$	−14.3667	−0.486518
$873$	31.2111	1.05634
$874$	−6.27502	−0.212256
$875$	0	0
$876$	−4.60555	−0.155607
$877$	−38.0278	−1.28411	−0.642053	−	0.766660i	$-0.721917\pi$
$877$	−38.0278	−1.28411	−0.642053	+	0.766660i	$0.721917\pi$
$878$	30.2750	1.02173
$879$	−17.6056	−0.593821
$880$	0	0
$881$	35.8444	1.20763	0.603814	−	0.797125i	$-0.293647\pi$
$881$	35.8444	1.20763	0.603814	+	0.797125i	$0.293647\pi$
$882$	15.6333	0.526401
$883$	31.6333	1.06455	0.532273	−	0.846573i	$-0.321338\pi$
$883$	31.6333	1.06455	0.532273	+	0.846573i	$0.321338\pi$
$884$	0	0
$885$	0	0
$886$	29.2111	0.981366
$887$	−35.0555	−1.17705	−0.588524	−	0.808479i	$-0.700291\pi$
$887$	−35.0555	−1.17705	−0.588524	+	0.808479i	$0.700291\pi$
$888$	−10.8167	−0.362983
$889$	10.2111	0.342469
$890$	0	0
$891$	5.60555	0.187793
$892$	−3.09167	−0.103517
$893$	8.36669	0.279981
$894$	3.90833	0.130714
$895$	0	0
$896$	−8.09167	−0.270324
$897$	0	0
$898$	16.4584	0.549223
$899$	32.8444	1.09542
$900$	0	0
$901$	4.42221	0.147325
$902$	−21.9083	−0.729467
$903$	−4.21110	−0.140137
$904$	16.8167	0.559314
$905$	0	0
$906$	17.2111	0.571801
$907$	−48.2666	−1.60267	−0.801333	−	0.598218i	$-0.795876\pi$
$907$	−48.2666	−1.60267	−0.801333	+	0.598218i	$0.795876\pi$
$908$	−0.430609	−0.0142903
$909$	18.0000	0.597022
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	−5.30278	−0.175592
$913$	29.2111	0.966746
$914$	−6.75274	−0.223361
$915$	0	0
$916$	4.23886	0.140056
$917$	6.78890	0.224189
$918$	−2.56939	−0.0848025
$919$	−17.1833	−0.566826	−0.283413	−	0.958998i	$-0.591467\pi$
$919$	−17.1833	−0.566826	−0.283413	+	0.958998i	$0.591467\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	−28.3860	−0.934845
$923$	0	0
$924$	−1.69722	−0.0558346
$925$	0	0
$926$	−7.26662	−0.238796
$927$	−8.00000	−0.262754
$928$	13.9361	0.457474
$929$	−13.4222	−0.440368	−0.220184	−	0.975458i	$-0.570666\pi$
$929$	−13.4222	−0.440368	−0.220184	+	0.975458i	$0.570666\pi$
$930$	0	0
$931$	9.63331	0.315719
$932$	0.238859	0.00782408
$933$	−5.21110	−0.170604
$934$	−22.4222	−0.733677
$935$	0	0
$936$	0	0
$937$	46.4777	1.51836	0.759180	−	0.650880i	$-0.225600\pi$
$937$	46.4777	1.51836	0.759180	+	0.650880i	$0.225600\pi$
$938$	9.11943	0.297760
$939$	14.0000	0.456873
$940$	0	0
$941$	−33.6333	−1.09641	−0.548207	−	0.836343i	$-0.684689\pi$
$941$	−33.6333	−1.09641	−0.548207	+	0.836343i	$0.684689\pi$
$942$	−4.18335	−0.136301
$943$	−9.00000	−0.293080
$944$	35.7250	1.16275
$945$	0	0
$946$	−30.7527	−0.999858
$947$	24.6333	0.800475	0.400237	−	0.916411i	$-0.368928\pi$
$947$	24.6333	0.800475	0.400237	+	0.916411i	$0.368928\pi$
$948$	−2.78890	−0.0905792
$949$	0	0
$950$	0	0
$951$	−6.00000	−0.194563
$952$	−1.18335	−0.0383525
$953$	−50.4500	−1.63423	−0.817117	−	0.576471i	$-0.804429\pi$
$953$	−50.4500	−1.63423	−0.817117	+	0.576471i	$0.804429\pi$
$954$	29.2111	0.945744
$955$	0	0
$956$	0	0
$957$	−46.0278	−1.48787
$958$	9.35829	0.302353
$959$	−5.60555	−0.181013
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−16.4222	−0.529198
$964$	4.90833	0.158087
$965$	0	0
$966$	3.90833	0.125748
$967$	56.4777	1.81620	0.908100	−	0.418752i	$-0.137532\pi$
$967$	56.4777	1.81620	0.908100	+	0.418752i	$0.137532\pi$
$968$	−61.2666	−1.96918
$969$	−0.633308	−0.0203448
$970$	0	0
$971$	−7.97224	−0.255841	−0.127921	−	0.991784i	$-0.540830\pi$
$971$	−7.97224	−0.255841	−0.127921	+	0.991784i	$0.540830\pi$
$972$	4.84441	0.155385
$973$	−13.6056	−0.436174
$974$	−1.30278	−0.0417436
$975$	0	0
$976$	3.30278	0.105719
$977$	7.18335	0.229816	0.114908	−	0.993376i	$-0.463343\pi$
$977$	7.18335	0.229816	0.114908	+	0.993376i	$0.463343\pi$
$978$	23.7250	0.758641
$979$	−46.0278	−1.47105
$980$	0	0
$981$	−9.57779	−0.305795
$982$	6.27502	0.200244
$983$	−10.4222	−0.332417	−0.166208	−	0.986091i	$-0.553152\pi$
$983$	−10.4222	−0.332417	−0.166208	+	0.986091i	$0.553152\pi$
$984$	−9.00000	−0.286910
$985$	0	0
$986$	−4.21951	−0.134376
$987$	−5.21110	−0.165871
$988$	0	0
$989$	−12.6333	−0.401716
$990$	0	0
$991$	3.97224	0.126182	0.0630912	−	0.998008i	$-0.479904\pi$
$991$	3.97224	0.126182	0.0630912	+	0.998008i	$0.479904\pi$
$992$	6.78890	0.215548
$993$	−26.0278	−0.825966
$994$	−21.9083	−0.694890
$995$	0	0
$996$	1.57779	0.0499943
$997$	−46.4500	−1.47109	−0.735543	−	0.677479i	$-0.763073\pi$
$997$	−46.4500	−1.47109	−0.735543	+	0.677479i	$0.763073\pi$
$998$	34.4222	1.08962
$999$	−18.0278	−0.570373

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.t.1.2		2
5.4	even	2		845.2.a.f.1.1		2
13.4	even	6		325.2.e.a.276.2		4
13.10	even	6		325.2.e.a.126.2		4
13.12	even	2		4225.2.a.x.1.1		2
15.14	odd	2		7605.2.a.bb.1.2		2
65.4	even	6		65.2.e.b.16.1	✓	4
65.9	even	6		845.2.e.d.146.2		4
65.17	odd	12		325.2.o.b.224.2		8
65.19	odd	12		845.2.m.d.361.3		8
65.23	odd	12		325.2.o.b.74.2		8
65.24	odd	12		845.2.m.d.316.3		8
65.29	even	6		845.2.e.d.191.2		4
65.34	odd	4		845.2.c.d.506.2		4
65.43	odd	12		325.2.o.b.224.3		8
65.44	odd	4		845.2.c.d.506.3		4
65.49	even	6		65.2.e.b.61.1	yes	4
65.54	odd	12		845.2.m.d.316.2		8
65.59	odd	12		845.2.m.d.361.2		8
65.62	odd	12		325.2.o.b.74.3		8
65.64	even	2		845.2.a.c.1.2		2
195.134	odd	6		585.2.j.d.406.2		4
195.179	odd	6		585.2.j.d.451.2		4
195.194	odd	2		7605.2.a.bg.1.1		2
260.179	odd	6		1040.2.q.o.321.1		4
260.199	odd	6		1040.2.q.o.81.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.e.b.16.1	✓	4	65.4	even	6
65.2.e.b.61.1	yes	4	65.49	even	6
325.2.e.a.126.2		4	13.10	even	6
325.2.e.a.276.2		4	13.4	even	6
325.2.o.b.74.2		8	65.23	odd	12
325.2.o.b.74.3		8	65.62	odd	12
325.2.o.b.224.2		8	65.17	odd	12
325.2.o.b.224.3		8	65.43	odd	12
585.2.j.d.406.2		4	195.134	odd	6
585.2.j.d.451.2		4	195.179	odd	6
845.2.a.c.1.2		2	65.64	even	2
845.2.a.f.1.1		2	5.4	even	2
845.2.c.d.506.2		4	65.34	odd	4
845.2.c.d.506.3		4	65.44	odd	4
845.2.e.d.146.2		4	65.9	even	6
845.2.e.d.191.2		4	65.29	even	6
845.2.m.d.316.2		8	65.54	odd	12
845.2.m.d.316.3		8	65.24	odd	12
845.2.m.d.361.2		8	65.59	odd	12
845.2.m.d.361.3		8	65.19	odd	12
1040.2.q.o.81.1		4	260.199	odd	6
1040.2.q.o.321.1		4	260.179	odd	6
4225.2.a.t.1.2		2	1.1	even	1	trivial
4225.2.a.x.1.1		2	13.12	even	2
7605.2.a.bb.1.2		2	15.14	odd	2
7605.2.a.bg.1.1		2	195.194	odd	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:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q+1.30278 q^{2} -1.00000 q^{3} -0.302776 q^{4} -1.30278 q^{6} -1.00000 q^{7} -3.00000 q^{8} -2.00000 q^{9} +5.60555 q^{11} +0.302776 q^{12} -1.30278 q^{14} -3.30278 q^{16} -0.394449 q^{17} -2.60555 q^{18} -1.60555 q^{19} +1.00000 q^{21} +7.30278 q^{22} +3.00000 q^{23} +3.00000 q^{24} +5.00000 q^{27} +0.302776 q^{28} +8.21110 q^{29} +4.00000 q^{31} +1.69722 q^{32} -5.60555 q^{33} -0.513878 q^{34} +0.605551 q^{36} -3.60555 q^{37} -2.09167 q^{38} -3.00000 q^{41} +1.30278 q^{42} -4.21110 q^{43} -1.69722 q^{44} +3.90833 q^{46} -5.21110 q^{47} +3.30278 q^{48} -6.00000 q^{49} +0.394449 q^{51} -11.2111 q^{53} +6.51388 q^{54} +3.00000 q^{56} +1.60555 q^{57} +10.6972 q^{58} -10.8167 q^{59} -1.00000 q^{61} +5.21110 q^{62} +2.00000 q^{63} +8.81665 q^{64} -7.30278 q^{66} -7.00000 q^{67} +0.119429 q^{68} -3.00000 q^{69} +16.8167 q^{71} +6.00000 q^{72} -15.2111 q^{73} -4.69722 q^{74} +0.486122 q^{76} -5.60555 q^{77} -9.21110 q^{79} +1.00000 q^{81} -3.90833 q^{82} +5.21110 q^{83} -0.302776 q^{84} -5.48612 q^{86} -8.21110 q^{87} -16.8167 q^{88} -8.21110 q^{89} -0.908327 q^{92} -4.00000 q^{93} -6.78890 q^{94} -1.69722 q^{96} -15.6056 q^{97} -7.81665 q^{98} -11.2111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 2 q^{3} + 3 q^{4} + q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9} + 4 q^{11} - 3 q^{12} + q^{14} - 3 q^{16} - 8 q^{17} + 2 q^{18} + 4 q^{19} + 2 q^{21} + 11 q^{22} + 6 q^{23} + 6 q^{24} + 10 q^{27}+ \cdots - 8 q^{99}+O(q^{100}) \) 2 * q - q^2 - 2 * q^3 + 3 * q^4 + q^6 - 2 * q^7 - 6 * q^8 - 4 * q^9 + 4 * q^11 - 3 * q^12 + q^14 - 3 * q^16 - 8 * q^17 + 2 * q^18 + 4 * q^19 + 2 * q^21 + 11 * q^22 + 6 * q^23 + 6 * q^24 + 10 * q^27 - 3 * q^28 + 2 * q^29 + 8 * q^31 + 7 * q^32 - 4 * q^33 + 17 * q^34 - 6 * q^36 - 15 * q^38 - 6 * q^41 - q^42 + 6 * q^43 - 7 * q^44 - 3 * q^46 + 4 * q^47 + 3 * q^48 - 12 * q^49 + 8 * q^51 - 8 * q^53 - 5 * q^54 + 6 * q^56 - 4 * q^57 + 25 * q^58 - 2 * q^61 - 4 * q^62 + 4 * q^63 - 4 * q^64 - 11 * q^66 - 14 * q^67 - 25 * q^68 - 6 * q^69 + 12 * q^71 + 12 * q^72 - 16 * q^73 - 13 * q^74 + 19 * q^76 - 4 * q^77 - 4 * q^79 + 2 * q^81 + 3 * q^82 - 4 * q^83 + 3 * q^84 - 29 * q^86 - 2 * q^87 - 12 * q^88 - 2 * q^89 + 9 * q^92 - 8 * q^93 - 28 * q^94 - 7 * q^96 - 24 * q^97 + 6 * q^98 - 8 * q^99

Introduction

L-functions

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