LMFDB - Embedded newform 6325.2.a.m.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6325,2,Mod(1,6325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6325.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6325 = 5^{2} \cdot 11 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6325.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-3,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$50.5053792785$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.8639957.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 4x^{4} + 10x^{3} + 6x^{2} - 7x - 2 $ x^6 - 3x^5 - 4x^4 + 10x^3 + 6x^2 - 7*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 253)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.901423$ of defining polynomial
Character	$\chi$	$=$	6325.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.0985770 q^{2} -3.34486 q^{3} -1.99028 q^{4} +0.329726 q^{6} +2.57601 q^{7} +0.393350 q^{8} +8.18807 q^{9} +1.00000 q^{11} +6.65721 q^{12} -4.27358 q^{13} -0.253935 q^{14} +3.94179 q^{16} +7.01179 q^{17} -0.807155 q^{18} -1.72242 q^{19} -8.61637 q^{21} -0.0985770 q^{22} +1.00000 q^{23} -1.31570 q^{24} +0.421276 q^{26} -17.3534 q^{27} -5.12698 q^{28} -8.02757 q^{29} -6.10007 q^{31} -1.17527 q^{32} -3.34486 q^{33} -0.691201 q^{34} -16.2966 q^{36} +6.30964 q^{37} +0.169791 q^{38} +14.2945 q^{39} +4.29179 q^{41} +0.849376 q^{42} -10.3776 q^{43} -1.99028 q^{44} -0.0985770 q^{46} -4.35702 q^{47} -13.1847 q^{48} -0.364192 q^{49} -23.4534 q^{51} +8.50563 q^{52} +1.74607 q^{53} +1.71064 q^{54} +1.01327 q^{56} +5.76125 q^{57} +0.791334 q^{58} +6.74400 q^{59} -1.19907 q^{61} +0.601326 q^{62} +21.0925 q^{63} -7.76773 q^{64} +0.329726 q^{66} +4.78988 q^{67} -13.9554 q^{68} -3.34486 q^{69} +8.69264 q^{71} +3.22078 q^{72} +5.16193 q^{73} -0.621985 q^{74} +3.42811 q^{76} +2.57601 q^{77} -1.40911 q^{78} -7.94736 q^{79} +33.4803 q^{81} -0.423071 q^{82} -1.58243 q^{83} +17.1490 q^{84} +1.02299 q^{86} +26.8511 q^{87} +0.393350 q^{88} +5.85263 q^{89} -11.0088 q^{91} -1.99028 q^{92} +20.4039 q^{93} +0.429502 q^{94} +3.93111 q^{96} -12.8485 q^{97} +0.0359010 q^{98} +8.18807 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{2} - 7 q^{3} + 5 q^{4} + 3 q^{6} + q^{7} - 3 q^{8} + 9 q^{9} + 6 q^{11} - 11 q^{12} + 3 q^{13} - 8 q^{14} - q^{16} - 5 q^{17} + 11 q^{18} + q^{19} - 15 q^{21} - 3 q^{22} + 6 q^{23} - 3 q^{24}+ \cdots + 9 q^{99}+O(q^{100}) $ 6 * q - 3 * q^2 - 7 * q^3 + 5 * q^4 + 3 * q^6 + q^7 - 3 * q^8 + 9 * q^9 + 6 * q^11 - 11 * q^12 + 3 * q^13 - 8 * q^14 - q^16 - 5 * q^17 + 11 * q^18 + q^19 - 15 * q^21 - 3 * q^22 + 6 * q^23 - 3 * q^24 + 15 * q^26 - 16 * q^27 + 6 * q^29 - 8 * q^31 + 6 * q^32 - 7 * q^33 - 6 * q^34 - 8 * q^36 - 2 * q^37 - 9 * q^38 + q^39 - 2 * q^41 + 31 * q^42 - 10 * q^43 + 5 * q^44 - 3 * q^46 - 14 * q^47 - 6 * q^48 - 5 * q^49 - 24 * q^51 + 11 * q^52 + 4 * q^53 - 13 * q^54 - 13 * q^56 + 19 * q^57 + 6 * q^58 + 39 * q^59 - 22 * q^61 + 35 * q^63 - 33 * q^64 + 3 * q^66 - 8 * q^67 - 17 * q^68 - 7 * q^69 + 18 * q^71 - 6 * q^72 + 9 * q^73 + 15 * q^74 - 25 * q^76 + q^77 - 20 * q^78 - 15 * q^79 + 18 * q^81 + 8 * q^82 - 43 * q^83 - 4 * q^84 - 55 * q^86 + 28 * q^87 - 3 * q^88 + 17 * q^89 - 9 * q^91 + 5 * q^92 - 27 * q^93 + 63 * q^94 + 26 * q^96 - 19 * q^97 - 34 * q^98 + 9 * 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 $

$2$	−0.0985770	−0.0697044	−0.0348522	−	0.999392i	$-0.511096\pi$
$2$	−0.0985770	−0.0697044	−0.0348522	+	0.999392i	$0.511096\pi$
$3$	−3.34486	−1.93115	−0.965577	−	0.260117i	$-0.916239\pi$
$3$	−3.34486	−1.93115	−0.965577	+	0.260117i	$0.916239\pi$
$4$	−1.99028	−0.995141
$5$	0	0
$5$	0	0
$6$	0.329726	0.134610
$7$	2.57601	0.973639	0.486819	−	0.873503i	$-0.338157\pi$
$7$	2.57601	0.973639	0.486819	+	0.873503i	$0.338157\pi$
$8$	0.393350	0.139070
$9$	8.18807	2.72936
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	6.65721	1.92177
$13$	−4.27358	−1.18528	−0.592639	−	0.805468i	$-0.701914\pi$
$13$	−4.27358	−1.18528	−0.592639	+	0.805468i	$0.701914\pi$
$14$	−0.253935	−0.0678669
$15$	0	0
$16$	3.94179	0.985447
$17$	7.01179	1.70061	0.850304	−	0.526292i	$-0.176418\pi$
$17$	7.01179	1.70061	0.850304	+	0.526292i	$0.176418\pi$
$18$	−0.807155	−0.190248
$19$	−1.72242	−0.395151	−0.197575	−	0.980288i	$-0.563307\pi$
$19$	−1.72242	−0.395151	−0.197575	+	0.980288i	$0.563307\pi$
$20$	0	0
$21$	−8.61637	−1.88025
$22$	−0.0985770	−0.0210167
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.31570	−0.268566
$25$	0	0
$26$	0.421276	0.0826191
$27$	−17.3534	−3.33965
$28$	−5.12698	−0.968908
$29$	−8.02757	−1.49068	−0.745341	−	0.666683i	$-0.767713\pi$
$29$	−8.02757	−1.49068	−0.745341	+	0.666683i	$0.767713\pi$
$30$	0	0
$31$	−6.10007	−1.09560	−0.547802	−	0.836608i	$-0.684535\pi$
$31$	−6.10007	−1.09560	−0.547802	+	0.836608i	$0.684535\pi$
$32$	−1.17527	−0.207760
$33$	−3.34486	−0.582265
$34$	−0.691201	−0.118540
$35$	0	0
$36$	−16.2966	−2.71610
$37$	6.30964	1.03730	0.518649	−	0.854987i	$-0.326435\pi$
$37$	6.30964	1.03730	0.518649	+	0.854987i	$0.326435\pi$
$38$	0.169791	0.0275437
$39$	14.2945	2.28895
$40$	0	0
$41$	4.29179	0.670265	0.335132	−	0.942171i	$-0.391219\pi$
$41$	4.29179	0.670265	0.335132	+	0.942171i	$0.391219\pi$
$42$	0.849376	0.131062
$43$	−10.3776	−1.58257	−0.791287	−	0.611445i	$-0.790588\pi$
$43$	−10.3776	−1.58257	−0.791287	+	0.611445i	$0.790588\pi$
$44$	−1.99028	−0.300046
$45$	0	0
$46$	−0.0985770	−0.0145344
$47$	−4.35702	−0.635537	−0.317769	−	0.948168i	$-0.602934\pi$
$47$	−4.35702	−0.635537	−0.317769	+	0.948168i	$0.602934\pi$
$48$	−13.1847	−1.90305
$49$	−0.364192	−0.0520274
$50$	0	0
$51$	−23.4534	−3.28414
$52$	8.50563	1.17952
$53$	1.74607	0.239840	0.119920	−	0.992784i	$-0.461736\pi$
$53$	1.74607	0.239840	0.119920	+	0.992784i	$0.461736\pi$
$54$	1.71064	0.232789
$55$	0	0
$56$	1.01327	0.135404
$57$	5.76125	0.763097
$58$	0.791334	0.103907
$59$	6.74400	0.877994	0.438997	−	0.898488i	$-0.355334\pi$
$59$	6.74400	0.877994	0.438997	+	0.898488i	$0.355334\pi$
$60$	0	0
$61$	−1.19907	−0.153525	−0.0767626	−	0.997049i	$-0.524458\pi$
$61$	−1.19907	−0.153525	−0.0767626	+	0.997049i	$0.524458\pi$
$62$	0.601326	0.0763685
$63$	21.0925	2.65741
$64$	−7.76773	−0.970966
$65$	0	0
$66$	0.329726	0.0405864
$67$	4.78988	0.585177	0.292589	−	0.956238i	$-0.405483\pi$
$67$	4.78988	0.585177	0.292589	+	0.956238i	$0.405483\pi$
$68$	−13.9554	−1.69235
$69$	−3.34486	−0.402673
$70$	0	0
$71$	8.69264	1.03163	0.515813	−	0.856701i	$-0.327490\pi$
$71$	8.69264	1.03163	0.515813	+	0.856701i	$0.327490\pi$
$72$	3.22078	0.379572
$73$	5.16193	0.604159	0.302079	−	0.953283i	$-0.402319\pi$
$73$	5.16193	0.604159	0.302079	+	0.953283i	$0.402319\pi$
$74$	−0.621985	−0.0723043
$75$	0	0
$76$	3.42811	0.393231
$77$	2.57601	0.293563
$78$	−1.40911	−0.159550
$79$	−7.94736	−0.894148	−0.447074	−	0.894497i	$-0.647534\pi$
$79$	−7.94736	−0.894148	−0.447074	+	0.894497i	$0.647534\pi$
$80$	0	0
$81$	33.4803	3.72003
$82$	−0.423071	−0.0467204
$83$	−1.58243	−0.173694	−0.0868471	−	0.996222i	$-0.527679\pi$
$83$	−1.58243	−0.173694	−0.0868471	+	0.996222i	$0.527679\pi$
$84$	17.1490	1.87111
$85$	0	0
$86$	1.02299	0.110312
$87$	26.8511	2.87874
$88$	0.393350	0.0419312
$89$	5.85263	0.620377	0.310189	−	0.950675i	$-0.399608\pi$
$89$	5.85263	0.620377	0.310189	+	0.950675i	$0.399608\pi$
$90$	0	0
$91$	−11.0088	−1.15403
$92$	−1.99028	−0.207501
$93$	20.4039	2.11578
$94$	0.429502	0.0442998
$95$	0	0
$96$	3.93111	0.401217
$97$	−12.8485	−1.30456	−0.652282	−	0.757977i	$-0.726188\pi$
$97$	−12.8485	−1.30456	−0.652282	+	0.757977i	$0.726188\pi$
$98$	0.0359010	0.00362654
$99$	8.18807	0.822932
$100$	0	0
$101$	−1.05074	−0.104552	−0.0522762	−	0.998633i	$-0.516648\pi$
$101$	−1.05074	−0.104552	−0.0522762	+	0.998633i	$0.516648\pi$
$102$	2.31197	0.228919
$103$	7.29909	0.719200	0.359600	−	0.933106i	$-0.382913\pi$
$103$	7.29909	0.719200	0.359600	+	0.933106i	$0.382913\pi$
$104$	−1.68101	−0.164837
$105$	0	0
$106$	−0.172122	−0.0167179
$107$	−10.4902	−1.01413	−0.507064	−	0.861908i	$-0.669269\pi$
$107$	−10.4902	−1.01413	−0.507064	+	0.861908i	$0.669269\pi$
$108$	34.5381	3.32343
$109$	10.7094	1.02578	0.512888	−	0.858455i	$-0.328576\pi$
$109$	10.7094	1.02578	0.512888	+	0.858455i	$0.328576\pi$
$110$	0	0
$111$	−21.1048	−2.00318
$112$	10.1541	0.959470
$113$	0.487507	0.0458608	0.0229304	−	0.999737i	$-0.492700\pi$
$113$	0.487507	0.0458608	0.0229304	+	0.999737i	$0.492700\pi$
$114$	−0.567927	−0.0531912
$115$	0	0
$116$	15.9771	1.48344
$117$	−34.9924	−3.23505
$118$	−0.664803	−0.0612001
$119$	18.0624	1.65578
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.118201	0.0107014
$123$	−14.3554	−1.29438
$124$	12.1409	1.09028
$125$	0	0
$126$	−2.07924	−0.185233
$127$	−9.26452	−0.822093	−0.411046	−	0.911614i	$-0.634837\pi$
$127$	−9.26452	−0.822093	−0.411046	+	0.911614i	$0.634837\pi$
$128$	3.11626	0.275441
$129$	34.7117	3.05619
$130$	0	0
$131$	12.0582	1.05353	0.526766	−	0.850011i	$-0.323405\pi$
$131$	12.0582	1.05353	0.526766	+	0.850011i	$0.323405\pi$
$132$	6.65721	0.579436
$133$	−4.43697	−0.384734
$134$	−0.472172	−0.0407895
$135$	0	0
$136$	2.75809	0.236504
$137$	−9.54165	−0.815198	−0.407599	−	0.913161i	$-0.633634\pi$
$137$	−9.54165	−0.815198	−0.407599	+	0.913161i	$0.633634\pi$
$138$	0.329726	0.0280681
$139$	0.752437	0.0638209	0.0319104	−	0.999491i	$-0.489841\pi$
$139$	0.752437	0.0638209	0.0319104	+	0.999491i	$0.489841\pi$
$140$	0	0
$141$	14.5736	1.22732
$142$	−0.856894	−0.0719089
$143$	−4.27358	−0.357375
$144$	32.2757	2.68964
$145$	0	0
$146$	−0.508848	−0.0421125
$147$	1.21817	0.100473
$148$	−12.5580	−1.03226
$149$	−0.762211	−0.0624428	−0.0312214	−	0.999512i	$-0.509940\pi$
$149$	−0.762211	−0.0624428	−0.0312214	+	0.999512i	$0.509940\pi$
$150$	0	0
$151$	3.02363	0.246060	0.123030	−	0.992403i	$-0.460739\pi$
$151$	3.02363	0.246060	0.123030	+	0.992403i	$0.460739\pi$
$152$	−0.677514	−0.0549537
$153$	57.4130	4.64157
$154$	−0.253935	−0.0204627
$155$	0	0
$156$	−28.4501	−2.27783
$157$	4.27555	0.341226	0.170613	−	0.985338i	$-0.445425\pi$
$157$	4.27555	0.341226	0.170613	+	0.985338i	$0.445425\pi$
$158$	0.783427	0.0623261
$159$	−5.84034	−0.463169
$160$	0	0
$161$	2.57601	0.203018
$162$	−3.30038	−0.259303
$163$	−7.12165	−0.557810	−0.278905	−	0.960319i	$-0.589972\pi$
$163$	−7.12165	−0.557810	−0.278905	+	0.960319i	$0.589972\pi$
$164$	−8.54187	−0.667008
$165$	0	0
$166$	0.155991	0.0121073
$167$	−4.65914	−0.360535	−0.180268	−	0.983618i	$-0.557696\pi$
$167$	−4.65914	−0.360535	−0.180268	+	0.983618i	$0.557696\pi$
$168$	−3.38925	−0.261486
$169$	5.26347	0.404883
$170$	0	0
$171$	−14.1033	−1.07851
$172$	20.6544	1.57488
$173$	11.3203	0.860668	0.430334	−	0.902670i	$-0.358396\pi$
$173$	11.3203	0.860668	0.430334	+	0.902670i	$0.358396\pi$
$174$	−2.64690	−0.200661
$175$	0	0
$176$	3.94179	0.297124
$177$	−22.5577	−1.69554
$178$	−0.576934	−0.0432430
$179$	−0.975679	−0.0729257	−0.0364628	−	0.999335i	$-0.511609\pi$
$179$	−0.975679	−0.0729257	−0.0364628	+	0.999335i	$0.511609\pi$
$180$	0	0
$181$	−12.2845	−0.913100	−0.456550	−	0.889698i	$-0.650915\pi$
$181$	−12.2845	−0.913100	−0.456550	+	0.889698i	$0.650915\pi$
$182$	1.08521	0.0804412
$183$	4.01072	0.296481
$184$	0.393350	0.0289981
$185$	0	0
$186$	−2.01135	−0.147479
$187$	7.01179	0.512753
$188$	8.67171	0.632449
$189$	−44.7023	−3.25162
$190$	0	0
$191$	10.5415	0.762759	0.381380	−	0.924418i	$-0.375449\pi$
$191$	10.5415	0.762759	0.381380	+	0.924418i	$0.375449\pi$
$192$	25.9819	1.87508
$193$	2.78596	0.200537	0.100269	−	0.994960i	$-0.468030\pi$
$193$	2.78596	0.200537	0.100269	+	0.994960i	$0.468030\pi$
$194$	1.26656	0.0909339
$195$	0	0
$196$	0.724845	0.0517747
$197$	17.9830	1.28123	0.640616	−	0.767861i	$-0.278679\pi$
$197$	17.9830	1.28123	0.640616	+	0.767861i	$0.278679\pi$
$198$	−0.807155	−0.0573620
$199$	0.857794	0.0608074	0.0304037	−	0.999538i	$-0.490321\pi$
$199$	0.857794	0.0608074	0.0304037	+	0.999538i	$0.490321\pi$
$200$	0	0
$201$	−16.0215	−1.13007
$202$	0.103579	0.00728777
$203$	−20.6791	−1.45139
$204$	46.6789	3.26818
$205$	0	0
$206$	−0.719522	−0.0501315
$207$	8.18807	0.569110
$208$	−16.8455	−1.16803
$209$	−1.72242	−0.119142
$210$	0	0
$211$	−20.7330	−1.42732	−0.713658	−	0.700495i	$-0.752963\pi$
$211$	−20.7330	−1.42732	−0.713658	+	0.700495i	$0.752963\pi$
$212$	−3.47516	−0.238675
$213$	−29.0756	−1.99223
$214$	1.03410	0.0706893
$215$	0	0
$216$	−6.82594	−0.464446
$217$	−15.7138	−1.06672
$218$	−1.05570	−0.0715012
$219$	−17.2659	−1.16672
$220$	0	0
$221$	−29.9654	−2.01569
$222$	2.08045	0.139631
$223$	0.901902	0.0603958	0.0301979	−	0.999544i	$-0.490386\pi$
$223$	0.901902	0.0603958	0.0301979	+	0.999544i	$0.490386\pi$
$224$	−3.02750	−0.202283
$225$	0	0
$226$	−0.0480570	−0.00319670
$227$	−10.8182	−0.718028	−0.359014	−	0.933332i	$-0.616887\pi$
$227$	−10.8182	−0.718028	−0.359014	+	0.933332i	$0.616887\pi$
$228$	−11.4665	−0.759389
$229$	−27.6949	−1.83013	−0.915066	−	0.403304i	$-0.867862\pi$
$229$	−27.6949	−1.83013	−0.915066	+	0.403304i	$0.867862\pi$
$230$	0	0
$231$	−8.61637	−0.566916
$232$	−3.15764	−0.207310
$233$	−9.85448	−0.645589	−0.322794	−	0.946469i	$-0.604622\pi$
$233$	−9.85448	−0.645589	−0.322794	+	0.946469i	$0.604622\pi$
$234$	3.44944	0.225497
$235$	0	0
$236$	−13.4225	−0.873728
$237$	26.5828	1.72674
$238$	−1.78054	−0.115415
$239$	22.2862	1.44158	0.720788	−	0.693156i	$-0.243780\pi$
$239$	22.2862	1.44158	0.720788	+	0.693156i	$0.243780\pi$
$240$	0	0
$241$	−8.26747	−0.532555	−0.266277	−	0.963896i	$-0.585794\pi$
$241$	−8.26747	−0.532555	−0.266277	+	0.963896i	$0.585794\pi$
$242$	−0.0985770	−0.00633677
$243$	−59.9267	−3.84430
$244$	2.38649	0.152779
$245$	0	0
$246$	1.41511	0.0902243
$247$	7.36090	0.468363
$248$	−2.39946	−0.152366
$249$	5.29300	0.335430
$250$	0	0
$251$	7.07648	0.446663	0.223332	−	0.974743i	$-0.428307\pi$
$251$	7.07648	0.446663	0.223332	+	0.974743i	$0.428307\pi$
$252$	−41.9801	−2.64450
$253$	1.00000	0.0628695
$254$	0.913268	0.0573035
$255$	0	0
$256$	15.2283	0.951766
$257$	−5.69650	−0.355338	−0.177669	−	0.984090i	$-0.556856\pi$
$257$	−5.69650	−0.355338	−0.177669	+	0.984090i	$0.556856\pi$
$258$	−3.42177	−0.213030
$259$	16.2537	1.00995
$260$	0	0
$261$	−65.7303	−4.06860
$262$	−1.18866	−0.0734358
$263$	−2.50258	−0.154316	−0.0771579	−	0.997019i	$-0.524585\pi$
$263$	−2.50258	−0.154316	−0.0771579	+	0.997019i	$0.524585\pi$
$264$	−1.31570	−0.0809757
$265$	0	0
$266$	0.437383	0.0268177
$267$	−19.5762	−1.19804
$268$	−9.53322	−0.582334
$269$	−3.23678	−0.197350	−0.0986749	−	0.995120i	$-0.531460\pi$
$269$	−3.23678	−0.197350	−0.0986749	+	0.995120i	$0.531460\pi$
$270$	0	0
$271$	9.71834	0.590347	0.295173	−	0.955444i	$-0.404623\pi$
$271$	9.71834	0.590347	0.295173	+	0.955444i	$0.404623\pi$
$272$	27.6390	1.67586
$273$	36.8227	2.22861
$274$	0.940587	0.0568229
$275$	0	0
$276$	6.65721	0.400717
$277$	−20.3613	−1.22339	−0.611695	−	0.791094i	$-0.709512\pi$
$277$	−20.3613	−1.22339	−0.611695	+	0.791094i	$0.709512\pi$
$278$	−0.0741730	−0.00444860
$279$	−49.9478	−2.99030
$280$	0	0
$281$	27.8912	1.66385	0.831926	−	0.554887i	$-0.187238\pi$
$281$	27.8912	1.66385	0.831926	+	0.554887i	$0.187238\pi$
$282$	−1.43662	−0.0855497
$283$	32.2777	1.91871	0.959356	−	0.282198i	$-0.0910636\pi$
$283$	32.2777	1.91871	0.959356	+	0.282198i	$0.0910636\pi$
$284$	−17.3008	−1.02661
$285$	0	0
$286$	0.421276	0.0249106
$287$	11.0557	0.652596
$288$	−9.62319	−0.567052
$289$	32.1651	1.89207
$290$	0	0
$291$	42.9763	2.51931
$292$	−10.2737	−0.601223
$293$	−0.0837545	−0.00489299	−0.00244650	−	0.999997i	$-0.500779\pi$
$293$	−0.0837545	−0.00489299	−0.00244650	+	0.999997i	$0.500779\pi$
$294$	−0.120084	−0.00700342
$295$	0	0
$296$	2.48190	0.144257
$297$	−17.3534	−1.00694
$298$	0.0751365	0.00435254
$299$	−4.27358	−0.247147
$300$	0	0
$301$	−26.7328	−1.54085
$302$	−0.298060	−0.0171514
$303$	3.51457	0.201907
$304$	−6.78942	−0.389400
$305$	0	0
$306$	−5.65960	−0.323538
$307$	−34.7795	−1.98497	−0.992485	−	0.122367i	$-0.960952\pi$
$307$	−34.7795	−1.98497	−0.992485	+	0.122367i	$0.960952\pi$
$308$	−5.12698	−0.292137
$309$	−24.4144	−1.38889
$310$	0	0
$311$	−25.1850	−1.42811	−0.714054	−	0.700091i	$-0.753143\pi$
$311$	−25.1850	−1.42811	−0.714054	+	0.700091i	$0.753143\pi$
$312$	5.62274	0.318325
$313$	8.68839	0.491097	0.245548	−	0.969384i	$-0.421032\pi$
$313$	8.68839	0.491097	0.245548	+	0.969384i	$0.421032\pi$
$314$	−0.421471	−0.0237850
$315$	0	0
$316$	15.8175	0.889804
$317$	−8.22291	−0.461845	−0.230922	−	0.972972i	$-0.574174\pi$
$317$	−8.22291	−0.461845	−0.230922	+	0.972972i	$0.574174\pi$
$318$	0.575723	0.0322849
$319$	−8.02757	−0.449458
$320$	0	0
$321$	35.0883	1.95844
$322$	−0.253935	−0.0141512
$323$	−12.0772	−0.671996
$324$	−66.6352	−3.70196
$325$	0	0
$326$	0.702030	0.0388819
$327$	−35.8215	−1.98093
$328$	1.68817	0.0932138
$329$	−11.2237	−0.618784
$330$	0	0
$331$	−10.9564	−0.602217	−0.301108	−	0.953590i	$-0.597357\pi$
$331$	−10.9564	−0.602217	−0.301108	+	0.953590i	$0.597357\pi$
$332$	3.14948	0.172850
$333$	51.6637	2.83116
$334$	0.459284	0.0251309
$335$	0	0
$336$	−33.9639	−1.85288
$337$	32.2285	1.75560	0.877798	−	0.479031i	$-0.159012\pi$
$337$	32.2285	1.75560	0.877798	+	0.479031i	$0.159012\pi$
$338$	−0.518857	−0.0282221
$339$	−1.63064	−0.0885643
$340$	0	0
$341$	−6.10007	−0.330337
$342$	1.39026	0.0751767
$343$	−18.9702	−1.02429
$344$	−4.08204	−0.220089
$345$	0	0
$346$	−1.11592	−0.0599924
$347$	27.4718	1.47476	0.737382	−	0.675476i	$-0.236062\pi$
$347$	27.4718	1.47476	0.737382	+	0.675476i	$0.236062\pi$
$348$	−53.4412	−2.86475
$349$	13.8793	0.742941	0.371470	−	0.928445i	$-0.378854\pi$
$349$	13.8793	0.742941	0.371470	+	0.928445i	$0.378854\pi$
$350$	0	0
$351$	74.1609	3.95842
$352$	−1.17527	−0.0626421
$353$	−3.45719	−0.184008	−0.0920039	−	0.995759i	$-0.529327\pi$
$353$	−3.45719	−0.184008	−0.0920039	+	0.995759i	$0.529327\pi$
$354$	2.22367	0.118187
$355$	0	0
$356$	−11.6484	−0.617363
$357$	−60.4162	−3.19756
$358$	0.0961795	0.00508324
$359$	−19.2010	−1.01339	−0.506696	−	0.862125i	$-0.669133\pi$
$359$	−19.2010	−1.01339	−0.506696	+	0.862125i	$0.669133\pi$
$360$	0	0
$361$	−16.0333	−0.843856
$362$	1.21097	0.0636471
$363$	−3.34486	−0.175559
$364$	21.9106	1.14843
$365$	0	0
$366$	−0.395365	−0.0206660
$367$	4.36951	0.228087	0.114043	−	0.993476i	$-0.463620\pi$
$367$	4.36951	0.228087	0.114043	+	0.993476i	$0.463620\pi$
$368$	3.94179	0.205480
$369$	35.1415	1.82939
$370$	0	0
$371$	4.49787	0.233518
$372$	−40.6094	−2.10550
$373$	18.1184	0.938135	0.469068	−	0.883162i	$-0.344590\pi$
$373$	18.1184	0.938135	0.469068	+	0.883162i	$0.344590\pi$
$374$	−0.691201	−0.0357411
$375$	0	0
$376$	−1.71384	−0.0883843
$377$	34.3065	1.76687
$378$	4.40662	0.226652
$379$	−18.3481	−0.942480	−0.471240	−	0.882005i	$-0.656194\pi$
$379$	−18.3481	−0.942480	−0.471240	+	0.882005i	$0.656194\pi$
$380$	0	0
$381$	30.9885	1.58759
$382$	−1.03915	−0.0531677
$383$	−34.2931	−1.75230	−0.876148	−	0.482042i	$-0.839895\pi$
$383$	−34.2931	−1.75230	−0.876148	+	0.482042i	$0.839895\pi$
$384$	−10.4234	−0.531919
$385$	0	0
$386$	−0.274631	−0.0139784
$387$	−84.9727	−4.31941
$388$	25.5721	1.29822
$389$	18.2975	0.927720	0.463860	−	0.885908i	$-0.346464\pi$
$389$	18.2975	0.927720	0.463860	+	0.885908i	$0.346464\pi$
$390$	0	0
$391$	7.01179	0.354601
$392$	−0.143255	−0.00723547
$393$	−40.3330	−2.03453
$394$	−1.77271	−0.0893076
$395$	0	0
$396$	−16.2966	−0.818934
$397$	16.7978	0.843057	0.421528	−	0.906815i	$-0.361494\pi$
$397$	16.7978	0.843057	0.421528	+	0.906815i	$0.361494\pi$
$398$	−0.0845587	−0.00423855
$399$	14.8410	0.742981
$400$	0	0
$401$	33.6597	1.68088	0.840442	−	0.541901i	$-0.182295\pi$
$401$	33.6597	1.68088	0.840442	+	0.541901i	$0.182295\pi$
$402$	1.57935	0.0787707
$403$	26.0691	1.29860
$404$	2.09127	0.104044
$405$	0	0
$406$	2.03848	0.101168
$407$	6.30964	0.312757
$408$	−9.22540	−0.456725
$409$	−23.8012	−1.17689	−0.588447	−	0.808536i	$-0.700260\pi$
$409$	−23.8012	−1.17689	−0.588447	+	0.808536i	$0.700260\pi$
$410$	0	0
$411$	31.9155	1.57427
$412$	−14.5272	−0.715706
$413$	17.3726	0.854849
$414$	−0.807155	−0.0396695
$415$	0	0
$416$	5.02261	0.246254
$417$	−2.51679	−0.123248
$418$	0.169791	0.00830475
$419$	−32.1373	−1.57001	−0.785006	−	0.619489i	$-0.787340\pi$
$419$	−32.1373	−1.57001	−0.785006	+	0.619489i	$0.787340\pi$
$420$	0	0
$421$	−33.1371	−1.61501	−0.807503	−	0.589864i	$-0.799181\pi$
$421$	−33.1371	−1.61501	−0.807503	+	0.589864i	$0.799181\pi$
$422$	2.04379	0.0994902
$423$	−35.6756	−1.73461
$424$	0.686815	0.0333547
$425$	0	0
$426$	2.86619	0.138867
$427$	−3.08881	−0.149478
$428$	20.8785	1.00920
$429$	14.2945	0.690145
$430$	0	0
$431$	9.06476	0.436634	0.218317	−	0.975878i	$-0.429943\pi$
$431$	9.06476	0.436634	0.218317	+	0.975878i	$0.429943\pi$
$432$	−68.4033	−3.29105
$433$	−29.3934	−1.41256	−0.706279	−	0.707934i	$-0.749627\pi$
$433$	−29.3934	−1.41256	−0.706279	+	0.707934i	$0.749627\pi$
$434$	1.54902	0.0743553
$435$	0	0
$436$	−21.3148	−1.02079
$437$	−1.72242	−0.0823946
$438$	1.70202	0.0813258
$439$	7.67129	0.366131	0.183065	−	0.983101i	$-0.441398\pi$
$439$	7.67129	0.366131	0.183065	+	0.983101i	$0.441398\pi$
$440$	0	0
$441$	−2.98203	−0.142001
$442$	2.95390	0.140503
$443$	18.8452	0.895362	0.447681	−	0.894193i	$-0.352250\pi$
$443$	18.8452	0.895362	0.447681	+	0.894193i	$0.352250\pi$
$444$	42.0046	1.99345
$445$	0	0
$446$	−0.0889068	−0.00420986
$447$	2.54949	0.120587
$448$	−20.0097	−0.945370
$449$	−19.2501	−0.908467	−0.454234	−	0.890883i	$-0.650087\pi$
$449$	−19.2501	−0.908467	−0.454234	+	0.890883i	$0.650087\pi$
$450$	0	0
$451$	4.29179	0.202092
$452$	−0.970276	−0.0456380
$453$	−10.1136	−0.475179
$454$	1.06642	0.0500498
$455$	0	0
$456$	2.26619	0.106124
$457$	−14.2597	−0.667041	−0.333520	−	0.942743i	$-0.608237\pi$
$457$	−14.2597	−0.667041	−0.333520	+	0.942743i	$0.608237\pi$
$458$	2.73008	0.127568
$459$	−121.678	−5.67944
$460$	0	0
$461$	−6.79786	−0.316608	−0.158304	−	0.987390i	$-0.550603\pi$
$461$	−6.79786	−0.316608	−0.158304	+	0.987390i	$0.550603\pi$
$462$	0.849376	0.0395165
$463$	23.3464	1.08500	0.542499	−	0.840056i	$-0.317478\pi$
$463$	23.3464	1.08500	0.542499	+	0.840056i	$0.317478\pi$
$464$	−31.6430	−1.46899
$465$	0	0
$466$	0.971425	0.0450004
$467$	−20.0130	−0.926093	−0.463047	−	0.886334i	$-0.653244\pi$
$467$	−20.0130	−0.926093	−0.463047	+	0.886334i	$0.653244\pi$
$468$	69.6447	3.21933
$469$	12.3388	0.569751
$470$	0	0
$471$	−14.3011	−0.658960
$472$	2.65275	0.122103
$473$	−10.3776	−0.477164
$474$	−2.62045	−0.120361
$475$	0	0
$476$	−35.9493	−1.64773
$477$	14.2969	0.654610
$478$	−2.19691	−0.100484
$479$	−32.3385	−1.47758	−0.738791	−	0.673934i	$-0.764603\pi$
$479$	−32.3385	−1.47758	−0.738791	+	0.673934i	$0.764603\pi$
$480$	0	0
$481$	−26.9647	−1.22949
$482$	0.814983	0.0371214
$483$	−8.61637	−0.392059
$484$	−1.99028	−0.0904674
$485$	0	0
$486$	5.90739	0.267965
$487$	−10.2937	−0.466450	−0.233225	−	0.972423i	$-0.574928\pi$
$487$	−10.2937	−0.466450	−0.233225	+	0.972423i	$0.574928\pi$
$488$	−0.471654	−0.0213508
$489$	23.8209	1.07722
$490$	0	0
$491$	−2.44398	−0.110295	−0.0551477	−	0.998478i	$-0.517563\pi$
$491$	−2.44398	−0.110295	−0.0551477	+	0.998478i	$0.517563\pi$
$492$	28.5713	1.28810
$493$	−56.2876	−2.53507
$494$	−0.725615	−0.0326470
$495$	0	0
$496$	−24.0452	−1.07966
$497$	22.3923	1.00443
$498$	−0.521768	−0.0233810
$499$	−15.7764	−0.706247	−0.353124	−	0.935577i	$-0.614880\pi$
$499$	−15.7764	−0.706247	−0.353124	+	0.935577i	$0.614880\pi$
$500$	0	0
$501$	15.5842	0.696249
$502$	−0.697578	−0.0311344
$503$	−37.7052	−1.68119	−0.840595	−	0.541664i	$-0.817795\pi$
$503$	−37.7052	−1.68119	−0.840595	+	0.541664i	$0.817795\pi$
$504$	8.29674	0.369566
$505$	0	0
$506$	−0.0985770	−0.00438228
$507$	−17.6056	−0.781891
$508$	18.4390	0.818099
$509$	−15.7070	−0.696201	−0.348101	−	0.937457i	$-0.613173\pi$
$509$	−15.7070	−0.696201	−0.348101	+	0.937457i	$0.613173\pi$
$510$	0	0
$511$	13.2972	0.588232
$512$	−7.73367	−0.341783
$513$	29.8898	1.31967
$514$	0.561543	0.0247686
$515$	0	0
$516$	−69.0861	−3.04134
$517$	−4.35702	−0.191622
$518$	−1.60224	−0.0703982
$519$	−37.8648	−1.66208
$520$	0	0
$521$	−29.6576	−1.29932	−0.649661	−	0.760224i	$-0.725089\pi$
$521$	−29.6576	−1.29932	−0.649661	+	0.760224i	$0.725089\pi$
$522$	6.47950	0.283600
$523$	1.12098	0.0490169	0.0245084	−	0.999700i	$-0.492198\pi$
$523$	1.12098	0.0490169	0.0245084	+	0.999700i	$0.492198\pi$
$524$	−23.9993	−1.04841
$525$	0	0
$526$	0.246697	0.0107565
$527$	−42.7724	−1.86319
$528$	−13.1847	−0.573791
$529$	1.00000	0.0434783
$530$	0	0
$531$	55.2204	2.39636
$532$	8.83082	0.382865
$533$	−18.3413	−0.794450
$534$	1.92976	0.0835090
$535$	0	0
$536$	1.88410	0.0813807
$537$	3.26351	0.140831
$538$	0.319072	0.0137562
$539$	−0.364192	−0.0156869
$540$	0	0
$541$	41.1739	1.77020	0.885102	−	0.465397i	$-0.154088\pi$
$541$	41.1739	1.77020	0.885102	+	0.465397i	$0.154088\pi$
$542$	−0.958004	−0.0411498
$543$	41.0899	1.76334
$544$	−8.24074	−0.353319
$545$	0	0
$546$	−3.62987	−0.155344
$547$	17.2965	0.739546	0.369773	−	0.929122i	$-0.379435\pi$
$547$	17.2965	0.739546	0.369773	+	0.929122i	$0.379435\pi$
$548$	18.9906	0.811237
$549$	−9.81807	−0.419025
$550$	0	0
$551$	13.8269	0.589044
$552$	−1.31570	−0.0559999
$553$	−20.4725	−0.870577
$554$	2.00715	0.0852757
$555$	0	0
$556$	−1.49756	−0.0635108
$557$	−44.5087	−1.88589	−0.942947	−	0.332944i	$-0.891958\pi$
$557$	−44.5087	−1.88589	−0.942947	+	0.332944i	$0.891958\pi$
$558$	4.92370	0.208437
$559$	44.3496	1.87579
$560$	0	0
$561$	−23.4534	−0.990204
$562$	−2.74943	−0.115978
$563$	−20.7078	−0.872732	−0.436366	−	0.899769i	$-0.643735\pi$
$563$	−20.7078	−0.872732	−0.436366	+	0.899769i	$0.643735\pi$
$564$	−29.0056	−1.22136
$565$	0	0
$566$	−3.18184	−0.133743
$567$	86.2454	3.62197
$568$	3.41925	0.143468
$569$	−8.30654	−0.348228	−0.174114	−	0.984725i	$-0.555706\pi$
$569$	−8.30654	−0.348228	−0.174114	+	0.984725i	$0.555706\pi$
$570$	0	0
$571$	12.6846	0.530836	0.265418	−	0.964133i	$-0.414490\pi$
$571$	12.6846	0.530836	0.265418	+	0.964133i	$0.414490\pi$
$572$	8.50563	0.355638
$573$	−35.2600	−1.47301
$574$	−1.08983	−0.0454888
$575$	0	0
$576$	−63.6027	−2.65011
$577$	35.3453	1.47145	0.735723	−	0.677283i	$-0.236843\pi$
$577$	35.3453	1.47145	0.735723	+	0.677283i	$0.236843\pi$
$578$	−3.17074	−0.131885
$579$	−9.31862	−0.387269
$580$	0	0
$581$	−4.07635	−0.169115
$582$	−4.23647	−0.175607
$583$	1.74607	0.0723146
$584$	2.03045	0.0840205
$585$	0	0
$586$	0.00825627	0.000341063 0
$587$	−15.9955	−0.660205	−0.330103	−	0.943945i	$-0.607083\pi$
$587$	−15.9955	−0.660205	−0.330103	+	0.943945i	$0.607083\pi$
$588$	−2.42450	−0.0999848
$589$	10.5069	0.432929
$590$	0	0
$591$	−60.1504	−2.47426
$592$	24.8713	1.02220
$593$	33.5080	1.37601	0.688005	−	0.725706i	$-0.258487\pi$
$593$	33.5080	1.37601	0.688005	+	0.725706i	$0.258487\pi$
$594$	1.71064	0.0701885
$595$	0	0
$596$	1.51702	0.0621394
$597$	−2.86920	−0.117428
$598$	0.421276	0.0172273
$599$	−5.35026	−0.218606	−0.109303	−	0.994008i	$-0.534862\pi$
$599$	−5.35026	−0.218606	−0.109303	+	0.994008i	$0.534862\pi$
$600$	0	0
$601$	−36.3268	−1.48180	−0.740901	−	0.671614i	$-0.765601\pi$
$601$	−36.3268	−1.48180	−0.740901	+	0.671614i	$0.765601\pi$
$602$	2.63524	0.107404
$603$	39.2199	1.59716
$604$	−6.01788	−0.244864
$605$	0	0
$606$	−0.346456	−0.0140738
$607$	−36.7889	−1.49321	−0.746607	−	0.665266i	$-0.768318\pi$
$607$	−36.7889	−1.49321	−0.746607	+	0.665266i	$0.768318\pi$
$608$	2.02431	0.0820966
$609$	69.1686	2.80285
$610$	0	0
$611$	18.6201	0.753288
$612$	−114.268	−4.61901
$613$	14.8696	0.600577	0.300288	−	0.953848i	$-0.402917\pi$
$613$	14.8696	0.600577	0.300288	+	0.953848i	$0.402917\pi$
$614$	3.42846	0.138361
$615$	0	0
$616$	1.01327	0.0408259
$617$	23.7451	0.955944	0.477972	−	0.878375i	$-0.341372\pi$
$617$	23.7451	0.955944	0.477972	+	0.878375i	$0.341372\pi$
$618$	2.40670	0.0968116
$619$	43.3459	1.74222	0.871110	−	0.491088i	$-0.163401\pi$
$619$	43.3459	1.74222	0.871110	+	0.491088i	$0.163401\pi$
$620$	0	0
$621$	−17.3534	−0.696366
$622$	2.48266	0.0995455
$623$	15.0764	0.604023
$624$	56.3460	2.25564
$625$	0	0
$626$	−0.856475	−0.0342316
$627$	5.76125	0.230082
$628$	−8.50955	−0.339568
$629$	44.2418	1.76404
$630$	0	0
$631$	19.0544	0.758544	0.379272	−	0.925285i	$-0.376174\pi$
$631$	19.0544	0.758544	0.379272	+	0.925285i	$0.376174\pi$
$632$	−3.12609	−0.124349
$633$	69.3488	2.75637
$634$	0.810590	0.0321926
$635$	0	0
$636$	11.6239	0.460919
$637$	1.55640	0.0616670
$638$	0.791334	0.0313292
$639$	71.1759	2.81568
$640$	0	0
$641$	−10.6471	−0.420535	−0.210268	−	0.977644i	$-0.567434\pi$
$641$	−10.6471	−0.420535	−0.210268	+	0.977644i	$0.567434\pi$
$642$	−3.45890	−0.136512
$643$	−19.0246	−0.750258	−0.375129	−	0.926973i	$-0.622402\pi$
$643$	−19.0246	−0.750258	−0.375129	+	0.926973i	$0.622402\pi$
$644$	−5.12698	−0.202031
$645$	0	0
$646$	1.19054	0.0468411
$647$	24.8112	0.975430	0.487715	−	0.873003i	$-0.337831\pi$
$647$	24.8112	0.975430	0.487715	+	0.873003i	$0.337831\pi$
$648$	13.1695	0.517346
$649$	6.74400	0.264725
$650$	0	0
$651$	52.5605	2.06001
$652$	14.1741	0.555100
$653$	22.1826	0.868072	0.434036	−	0.900895i	$-0.357089\pi$
$653$	22.1826	0.868072	0.434036	+	0.900895i	$0.357089\pi$
$654$	3.53117	0.138080
$655$	0	0
$656$	16.9173	0.660511
$657$	42.2663	1.64896
$658$	1.10640	0.0431320
$659$	17.1147	0.666692	0.333346	−	0.942804i	$-0.391822\pi$
$659$	17.1147	0.666692	0.333346	+	0.942804i	$0.391822\pi$
$660$	0	0
$661$	−25.6537	−0.997815	−0.498908	−	0.866655i	$-0.666265\pi$
$661$	−25.6537	−0.997815	−0.498908	+	0.866655i	$0.666265\pi$
$662$	1.08005	0.0419772
$663$	100.230	3.89261
$664$	−0.622448	−0.0241557
$665$	0	0
$666$	−5.09286	−0.197344
$667$	−8.02757	−0.310829
$668$	9.27301	0.358784
$669$	−3.01673	−0.116634
$670$	0	0
$671$	−1.19907	−0.0462896
$672$	10.1266	0.390641
$673$	−19.0359	−0.733780	−0.366890	−	0.930264i	$-0.619578\pi$
$673$	−19.0359	−0.733780	−0.366890	+	0.930264i	$0.619578\pi$
$674$	−3.17698	−0.122373
$675$	0	0
$676$	−10.4758	−0.402915
$677$	12.5375	0.481856	0.240928	−	0.970543i	$-0.422548\pi$
$677$	12.5375	0.481856	0.240928	+	0.970543i	$0.422548\pi$
$678$	0.160744	0.00617332
$679$	−33.0977	−1.27017
$680$	0	0
$681$	36.1853	1.38662
$682$	0.601326	0.0230260
$683$	5.31521	0.203381	0.101690	−	0.994816i	$-0.467575\pi$
$683$	5.31521	0.203381	0.101690	+	0.994816i	$0.467575\pi$
$684$	28.0696	1.07327
$685$	0	0
$686$	1.87003	0.0713979
$687$	92.6356	3.53427
$688$	−40.9064	−1.55954
$689$	−7.46195	−0.284278
$690$	0	0
$691$	−51.1052	−1.94413	−0.972066	−	0.234706i	$-0.924587\pi$
$691$	−51.1052	−1.94413	−0.972066	+	0.234706i	$0.924587\pi$
$692$	−22.5306	−0.856486
$693$	21.0925	0.801239
$694$	−2.70809	−0.102798
$695$	0	0
$696$	10.5619	0.400347
$697$	30.0931	1.13986
$698$	−1.36818	−0.0517863
$699$	32.9618	1.24673
$700$	0	0
$701$	16.8643	0.636957	0.318479	−	0.947930i	$-0.396828\pi$
$701$	16.8643	0.636957	0.318479	+	0.947930i	$0.396828\pi$
$702$	−7.31056	−0.275919
$703$	−10.8679	−0.409889
$704$	−7.76773	−0.292757
$705$	0	0
$706$	0.340800	0.0128262
$707$	−2.70671	−0.101796
$708$	44.8962	1.68730
$709$	23.2166	0.871917	0.435958	−	0.899967i	$-0.356410\pi$
$709$	23.2166	0.871917	0.435958	+	0.899967i	$0.356410\pi$
$710$	0	0
$711$	−65.0735	−2.44045
$712$	2.30213	0.0862760
$713$	−6.10007	−0.228449
$714$	5.95564	0.222884
$715$	0	0
$716$	1.94188	0.0725714
$717$	−74.5442	−2.78390
$718$	1.89278	0.0706379
$719$	−22.5357	−0.840439	−0.420219	−	0.907423i	$-0.638047\pi$
$719$	−22.5357	−0.840439	−0.420219	+	0.907423i	$0.638047\pi$
$720$	0	0
$721$	18.8025	0.700242
$722$	1.58051	0.0588205
$723$	27.6535	1.02845
$724$	24.4496	0.908664
$725$	0	0
$726$	0.329726	0.0122373
$727$	−3.69172	−0.136918	−0.0684592	−	0.997654i	$-0.521808\pi$
$727$	−3.69172	−0.136918	−0.0684592	+	0.997654i	$0.521808\pi$
$728$	−4.33030	−0.160491
$729$	100.005	3.70390
$730$	0	0
$731$	−72.7657	−2.69134
$732$	−7.98246	−0.295040
$733$	6.78702	0.250684	0.125342	−	0.992114i	$-0.459997\pi$
$733$	6.78702	0.250684	0.125342	+	0.992114i	$0.459997\pi$
$734$	−0.430733	−0.0158987
$735$	0	0
$736$	−1.17527	−0.0433210
$737$	4.78988	0.176438
$738$	−3.46414	−0.127517
$739$	−30.2153	−1.11149	−0.555743	−	0.831354i	$-0.687566\pi$
$739$	−30.2153	−1.11149	−0.555743	+	0.831354i	$0.687566\pi$
$740$	0	0
$741$	−24.6212	−0.904481
$742$	−0.443387	−0.0162772
$743$	−20.0638	−0.736070	−0.368035	−	0.929812i	$-0.619969\pi$
$743$	−20.0638	−0.736070	−0.368035	+	0.929812i	$0.619969\pi$
$744$	8.02586	0.294242
$745$	0	0
$746$	−1.78606	−0.0653922
$747$	−12.9570	−0.474073
$748$	−13.9554	−0.510261
$749$	−27.0229	−0.987395
$750$	0	0
$751$	27.7596	1.01296	0.506481	−	0.862251i	$-0.330946\pi$
$751$	27.7596	1.01296	0.506481	+	0.862251i	$0.330946\pi$
$752$	−17.1745	−0.626289
$753$	−23.6698	−0.862576
$754$	−3.38183	−0.123159
$755$	0	0
$756$	88.9703	3.23582
$757$	5.91530	0.214995	0.107498	−	0.994205i	$-0.465716\pi$
$757$	5.91530	0.214995	0.107498	+	0.994205i	$0.465716\pi$
$758$	1.80870	0.0656951
$759$	−3.34486	−0.121411
$760$	0	0
$761$	−8.67915	−0.314619	−0.157310	−	0.987549i	$-0.550282\pi$
$761$	−8.67915	−0.314619	−0.157310	+	0.987549i	$0.550282\pi$
$762$	−3.05475	−0.110662
$763$	27.5875	0.998736
$764$	−20.9807	−0.759053
$765$	0	0
$766$	3.38051	0.122143
$767$	−28.8210	−1.04067
$768$	−50.9364	−1.83801
$769$	52.7197	1.90112	0.950561	−	0.310538i	$-0.100509\pi$
$769$	52.7197	1.90112	0.950561	+	0.310538i	$0.100509\pi$
$770$	0	0
$771$	19.0540	0.686212
$772$	−5.54484	−0.199563
$773$	−46.4791	−1.67174	−0.835869	−	0.548928i	$-0.815036\pi$
$773$	−46.4791	−1.67174	−0.835869	+	0.548928i	$0.815036\pi$
$774$	8.37635	0.301082
$775$	0	0
$776$	−5.05394	−0.181426
$777$	−54.3662	−1.95038
$778$	−1.80371	−0.0646662
$779$	−7.39227	−0.264855
$780$	0	0
$781$	8.69264	0.311047
$782$	−0.691201	−0.0247173
$783$	139.305	4.97837
$784$	−1.43557	−0.0512703
$785$	0	0
$786$	3.97591	0.141816
$787$	4.85692	0.173131	0.0865653	−	0.996246i	$-0.472411\pi$
$787$	4.85692	0.173131	0.0865653	+	0.996246i	$0.472411\pi$
$788$	−35.7912	−1.27501
$789$	8.37078	0.298008
$790$	0	0
$791$	1.25582	0.0446518
$792$	3.22078	0.114445
$793$	5.12432	0.181970
$794$	−1.65588	−0.0587648
$795$	0	0
$796$	−1.70725	−0.0605120
$797$	6.47756	0.229447	0.114724	−	0.993397i	$-0.463402\pi$
$797$	6.47756	0.229447	0.114724	+	0.993397i	$0.463402\pi$
$798$	−1.46298	−0.0517890
$799$	−30.5505	−1.08080
$800$	0	0
$801$	47.9217	1.69323
$802$	−3.31807	−0.117165
$803$	5.16193	0.182161
$804$	31.8873	1.12458
$805$	0	0
$806$	−2.56981	−0.0905179
$807$	10.8266	0.381113
$808$	−0.413308	−0.0145401
$809$	−34.9544	−1.22893	−0.614466	−	0.788943i	$-0.710629\pi$
$809$	−34.9544	−1.22893	−0.614466	+	0.788943i	$0.710629\pi$
$810$	0	0
$811$	−27.6077	−0.969439	−0.484719	−	0.874670i	$-0.661078\pi$
$811$	−27.6077	−0.969439	−0.484719	+	0.874670i	$0.661078\pi$
$812$	41.1572	1.44433
$813$	−32.5065	−1.14005
$814$	−0.621985	−0.0218006
$815$	0	0
$816$	−92.4485	−3.23634
$817$	17.8746	0.625355
$818$	2.34625	0.0820347
$819$	−90.1405	−3.14977
$820$	0	0
$821$	18.5526	0.647491	0.323745	−	0.946144i	$-0.395058\pi$
$821$	18.5526	0.647491	0.323745	+	0.946144i	$0.395058\pi$
$822$	−3.14613	−0.109734
$823$	−0.337866	−0.0117773	−0.00588863	−	0.999983i	$-0.501874\pi$
$823$	−0.337866	−0.0117773	−0.00588863	+	0.999983i	$0.501874\pi$
$824$	2.87110	0.100019
$825$	0	0
$826$	−1.71254	−0.0595868
$827$	−43.1299	−1.49977	−0.749887	−	0.661566i	$-0.769892\pi$
$827$	−43.1299	−1.49977	−0.749887	+	0.661566i	$0.769892\pi$
$828$	−16.2966	−0.566345
$829$	0.536280	0.0186258	0.00931289	−	0.999957i	$-0.497036\pi$
$829$	0.536280	0.0186258	0.00931289	+	0.999957i	$0.497036\pi$
$830$	0	0
$831$	68.1056	2.36256
$832$	33.1960	1.15086
$833$	−2.55364	−0.0884783
$834$	0.248098	0.00859093
$835$	0	0
$836$	3.42811	0.118563
$837$	105.857	3.65894
$838$	3.16800	0.109437
$839$	12.8678	0.444246	0.222123	−	0.975019i	$-0.428701\pi$
$839$	12.8678	0.444246	0.222123	+	0.975019i	$0.428701\pi$
$840$	0	0
$841$	35.4419	1.22213
$842$	3.26656	0.112573
$843$	−93.2922	−3.21315
$844$	41.2644	1.42038
$845$	0	0
$846$	3.51679	0.120910
$847$	2.57601	0.0885126
$848$	6.88262	0.236350
$849$	−107.964	−3.70533
$850$	0	0
$851$	6.30964	0.216292
$852$	57.8687	1.98255
$853$	−48.4726	−1.65967	−0.829835	−	0.558009i	$-0.811566\pi$
$853$	−48.4726	−1.65967	−0.829835	+	0.558009i	$0.811566\pi$
$854$	0.304486	0.0104193
$855$	0	0
$856$	−4.12633	−0.141035
$857$	12.2020	0.416811	0.208406	−	0.978042i	$-0.433173\pi$
$857$	12.2020	0.416811	0.208406	+	0.978042i	$0.433173\pi$
$858$	−1.40911	−0.0481062
$859$	24.5683	0.838258	0.419129	−	0.907927i	$-0.362335\pi$
$859$	24.5683	0.838258	0.419129	+	0.907927i	$0.362335\pi$
$860$	0	0
$861$	−36.9796	−1.26026
$862$	−0.893577	−0.0304354
$863$	29.4190	1.00143	0.500717	−	0.865611i	$-0.333070\pi$
$863$	29.4190	1.00143	0.500717	+	0.865611i	$0.333070\pi$
$864$	20.3949	0.693848
$865$	0	0
$866$	2.89751	0.0984615
$867$	−107.588	−3.65387
$868$	31.2749	1.06154
$869$	−7.94736	−0.269596
$870$	0	0
$871$	−20.4699	−0.693598
$872$	4.21255	0.142655
$873$	−105.204	−3.56062
$874$	0.169791	0.00574327
$875$	0	0
$876$	34.3641	1.16105
$877$	3.38594	0.114335	0.0571676	−	0.998365i	$-0.481793\pi$
$877$	3.38594	0.114335	0.0571676	+	0.998365i	$0.481793\pi$
$878$	−0.756212	−0.0255209
$879$	0.280147	0.00944912
$880$	0	0
$881$	20.3444	0.685420	0.342710	−	0.939441i	$-0.388655\pi$
$881$	20.3444	0.685420	0.342710	+	0.939441i	$0.388655\pi$
$882$	0.293960	0.00989813
$883$	−48.8737	−1.64473	−0.822365	−	0.568960i	$-0.807346\pi$
$883$	−48.8737	−1.64473	−0.822365	+	0.568960i	$0.807346\pi$
$884$	59.6397	2.00590
$885$	0	0
$886$	−1.85770	−0.0624107
$887$	17.1414	0.575552	0.287776	−	0.957698i	$-0.407084\pi$
$887$	17.1414	0.575552	0.287776	+	0.957698i	$0.407084\pi$
$888$	−8.30159	−0.278583
$889$	−23.8655	−0.800422
$890$	0	0
$891$	33.4803	1.12163
$892$	−1.79504	−0.0601024
$893$	7.50463	0.251133
$894$	−0.251321	−0.00840542
$895$	0	0
$896$	8.02750	0.268180
$897$	14.2945	0.477280
$898$	1.89761	0.0633242
$899$	48.9687	1.63320
$900$	0	0
$901$	12.2430	0.407875
$902$	−0.423071	−0.0140867
$903$	89.4175	2.97563
$904$	0.191761	0.00637787
$905$	0	0
$906$	0.996969	0.0331221
$907$	11.1499	0.370228	0.185114	−	0.982717i	$-0.440735\pi$
$907$	11.1499	0.370228	0.185114	+	0.982717i	$0.440735\pi$
$908$	21.5313	0.714540
$909$	−8.60352	−0.285361
$910$	0	0
$911$	6.30489	0.208890	0.104445	−	0.994531i	$-0.466693\pi$
$911$	6.30489	0.208890	0.104445	+	0.994531i	$0.466693\pi$
$912$	22.7097	0.751992
$913$	−1.58243	−0.0523708
$914$	1.40568	0.0464957
$915$	0	0
$916$	55.1207	1.82124
$917$	31.0620	1.02576
$918$	11.9946	0.395882
$919$	−55.3838	−1.82694	−0.913472	−	0.406902i	$-0.866609\pi$
$919$	−55.3838	−1.82694	−0.913472	+	0.406902i	$0.866609\pi$
$920$	0	0
$921$	116.332	3.83328
$922$	0.670112	0.0220690
$923$	−37.1487	−1.22276
$924$	17.1490	0.564161
$925$	0	0
$926$	−2.30142	−0.0756292
$927$	59.7654	1.96295
$928$	9.43456	0.309705
$929$	22.6620	0.743517	0.371758	−	0.928330i	$-0.378755\pi$
$929$	22.6620	0.743517	0.371758	+	0.928330i	$0.378755\pi$
$930$	0	0
$931$	0.627292	0.0205587
$932$	19.6132	0.642452
$933$	84.2401	2.75790
$934$	1.97283	0.0645528
$935$	0	0
$936$	−13.7642	−0.449898
$937$	12.4128	0.405507	0.202754	−	0.979230i	$-0.435011\pi$
$937$	12.4128	0.405507	0.202754	+	0.979230i	$0.435011\pi$
$938$	−1.21632	−0.0397142
$939$	−29.0614	−0.948384
$940$	0	0
$941$	−6.03145	−0.196620	−0.0983098	−	0.995156i	$-0.531344\pi$
$941$	−6.03145	−0.196620	−0.0983098	+	0.995156i	$0.531344\pi$
$942$	1.40976	0.0459324
$943$	4.29179	0.139760
$944$	26.5834	0.865217
$945$	0	0
$946$	1.02299	0.0332604
$947$	−41.3238	−1.34284	−0.671422	−	0.741075i	$-0.734316\pi$
$947$	−41.3238	−1.34284	−0.671422	+	0.741075i	$0.734316\pi$
$948$	−52.9073	−1.71835
$949$	−22.0599	−0.716096
$950$	0	0
$951$	27.5045	0.891893
$952$	7.10485	0.230269
$953$	31.9472	1.03487	0.517435	−	0.855722i	$-0.326887\pi$
$953$	31.9472	1.03487	0.517435	+	0.855722i	$0.326887\pi$
$954$	−1.40935	−0.0456292
$955$	0	0
$956$	−44.3559	−1.43457
$957$	26.8511	0.867972
$958$	3.18783	0.102994
$959$	−24.5794	−0.793709
$960$	0	0
$961$	6.21084	0.200350
$962$	2.65810	0.0857006
$963$	−85.8947	−2.76792
$964$	16.4546	0.529967
$965$	0	0
$966$	0.849376	0.0273282
$967$	38.5020	1.23814	0.619071	−	0.785335i	$-0.287509\pi$
$967$	38.5020	1.23814	0.619071	+	0.785335i	$0.287509\pi$
$968$	0.393350	0.0126427
$969$	40.3967	1.29773
$970$	0	0
$971$	46.5026	1.49234	0.746169	−	0.665756i	$-0.231891\pi$
$971$	46.5026	1.49234	0.746169	+	0.665756i	$0.231891\pi$
$972$	119.271	3.82562
$973$	1.93828	0.0621385
$974$	1.01472	0.0325136
$975$	0	0
$976$	−4.72648	−0.151291
$977$	32.3962	1.03645	0.518224	−	0.855245i	$-0.326593\pi$
$977$	32.3962	1.03645	0.518224	+	0.855245i	$0.326593\pi$
$978$	−2.34819	−0.0750869
$979$	5.85263	0.187051
$980$	0	0
$981$	87.6895	2.79971
$982$	0.240921	0.00768809
$983$	−41.4597	−1.32236	−0.661179	−	0.750228i	$-0.729944\pi$
$983$	−41.4597	−1.32236	−0.661179	+	0.750228i	$0.729944\pi$
$984$	−5.64670	−0.180010
$985$	0	0
$986$	5.54866	0.176705
$987$	37.5417	1.19497
$988$	−14.6503	−0.466087
$989$	−10.3776	−0.329989
$990$	0	0
$991$	5.48208	0.174144	0.0870720	−	0.996202i	$-0.472249\pi$
$991$	5.48208	0.174144	0.0870720	+	0.996202i	$0.472249\pi$
$992$	7.16922	0.227623
$993$	36.6475	1.16297
$994$	−2.20736	−0.0700133
$995$	0	0
$996$	−10.5346	−0.333801
$997$	−34.1456	−1.08140	−0.540700	−	0.841215i	$-0.681841\pi$
$997$	−34.1456	−1.08140	−0.540700	+	0.841215i	$0.681841\pi$
$998$	1.55519	0.0492286
$999$	−109.493	−3.46422

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6325.2.a.m.1.4		6
5.4	even	2		253.2.a.d.1.3	✓	6
15.14	odd	2		2277.2.a.m.1.4		6
20.19	odd	2		4048.2.a.bc.1.1		6
55.54	odd	2		2783.2.a.h.1.4		6
115.114	odd	2		5819.2.a.e.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
253.2.a.d.1.3	✓	6	5.4	even	2
2277.2.a.m.1.4		6	15.14	odd	2
2783.2.a.h.1.4		6	55.54	odd	2
4048.2.a.bc.1.1		6	20.19	odd	2
5819.2.a.e.1.3		6	115.114	odd	2
6325.2.a.m.1.4		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6325 = 5^{2} \cdot 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6325.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0985770 q^{2} -3.34486 q^{3} -1.99028 q^{4} +0.329726 q^{6} +2.57601 q^{7} +0.393350 q^{8} +8.18807 q^{9} +1.00000 q^{11} +6.65721 q^{12} -4.27358 q^{13} -0.253935 q^{14} +3.94179 q^{16} +7.01179 q^{17} -0.807155 q^{18} -1.72242 q^{19} -8.61637 q^{21} -0.0985770 q^{22} +1.00000 q^{23} -1.31570 q^{24} +0.421276 q^{26} -17.3534 q^{27} -5.12698 q^{28} -8.02757 q^{29} -6.10007 q^{31} -1.17527 q^{32} -3.34486 q^{33} -0.691201 q^{34} -16.2966 q^{36} +6.30964 q^{37} +0.169791 q^{38} +14.2945 q^{39} +4.29179 q^{41} +0.849376 q^{42} -10.3776 q^{43} -1.99028 q^{44} -0.0985770 q^{46} -4.35702 q^{47} -13.1847 q^{48} -0.364192 q^{49} -23.4534 q^{51} +8.50563 q^{52} +1.74607 q^{53} +1.71064 q^{54} +1.01327 q^{56} +5.76125 q^{57} +0.791334 q^{58} +6.74400 q^{59} -1.19907 q^{61} +0.601326 q^{62} +21.0925 q^{63} -7.76773 q^{64} +0.329726 q^{66} +4.78988 q^{67} -13.9554 q^{68} -3.34486 q^{69} +8.69264 q^{71} +3.22078 q^{72} +5.16193 q^{73} -0.621985 q^{74} +3.42811 q^{76} +2.57601 q^{77} -1.40911 q^{78} -7.94736 q^{79} +33.4803 q^{81} -0.423071 q^{82} -1.58243 q^{83} +17.1490 q^{84} +1.02299 q^{86} +26.8511 q^{87} +0.393350 q^{88} +5.85263 q^{89} -11.0088 q^{91} -1.99028 q^{92} +20.4039 q^{93} +0.429502 q^{94} +3.93111 q^{96} -12.8485 q^{97} +0.0359010 q^{98} +8.18807 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} - 7 q^{3} + 5 q^{4} + 3 q^{6} + q^{7} - 3 q^{8} + 9 q^{9} + 6 q^{11} - 11 q^{12} + 3 q^{13} - 8 q^{14} - q^{16} - 5 q^{17} + 11 q^{18} + q^{19} - 15 q^{21} - 3 q^{22} + 6 q^{23} - 3 q^{24}+ \cdots + 9 q^{99}+O(q^{100}) \) 6 * q - 3 * q^2 - 7 * q^3 + 5 * q^4 + 3 * q^6 + q^7 - 3 * q^8 + 9 * q^9 + 6 * q^11 - 11 * q^12 + 3 * q^13 - 8 * q^14 - q^16 - 5 * q^17 + 11 * q^18 + q^19 - 15 * q^21 - 3 * q^22 + 6 * q^23 - 3 * q^24 + 15 * q^26 - 16 * q^27 + 6 * q^29 - 8 * q^31 + 6 * q^32 - 7 * q^33 - 6 * q^34 - 8 * q^36 - 2 * q^37 - 9 * q^38 + q^39 - 2 * q^41 + 31 * q^42 - 10 * q^43 + 5 * q^44 - 3 * q^46 - 14 * q^47 - 6 * q^48 - 5 * q^49 - 24 * q^51 + 11 * q^52 + 4 * q^53 - 13 * q^54 - 13 * q^56 + 19 * q^57 + 6 * q^58 + 39 * q^59 - 22 * q^61 + 35 * q^63 - 33 * q^64 + 3 * q^66 - 8 * q^67 - 17 * q^68 - 7 * q^69 + 18 * q^71 - 6 * q^72 + 9 * q^73 + 15 * q^74 - 25 * q^76 + q^77 - 20 * q^78 - 15 * q^79 + 18 * q^81 + 8 * q^82 - 43 * q^83 - 4 * q^84 - 55 * q^86 + 28 * q^87 - 3 * q^88 + 17 * q^89 - 9 * q^91 + 5 * q^92 - 27 * q^93 + 63 * q^94 + 26 * q^96 - 19 * q^97 - 34 * q^98 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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