LMFDB - Embedded newform 3750.2.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3750,2,Mod(1,3750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3750, 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("3750.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3750 = 2 \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3750.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:	$29.9439007580$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.33826$ of defining polynomial
Character	$\chi$	$=$	3750.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +0.511170 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.29456 q^{11} +1.00000 q^{12} +2.65418 q^{13} -0.511170 q^{14} +1.00000 q^{16} +5.57433 q^{17} -1.00000 q^{18} +0.374409 q^{19} +0.511170 q^{21} -3.29456 q^{22} +5.75638 q^{23} -1.00000 q^{24} -2.65418 q^{26} +1.00000 q^{27} +0.511170 q^{28} -3.89025 q^{29} -1.80008 q^{31} -1.00000 q^{32} +3.29456 q^{33} -5.57433 q^{34} +1.00000 q^{36} +10.3577 q^{37} -0.374409 q^{38} +2.65418 q^{39} +2.33728 q^{41} -0.511170 q^{42} -5.87802 q^{43} +3.29456 q^{44} -5.75638 q^{46} -8.19959 q^{47} +1.00000 q^{48} -6.73870 q^{49} +5.57433 q^{51} +2.65418 q^{52} +0.518727 q^{53} -1.00000 q^{54} -0.511170 q^{56} +0.374409 q^{57} +3.89025 q^{58} -7.62993 q^{59} +2.90345 q^{61} +1.80008 q^{62} +0.511170 q^{63} +1.00000 q^{64} -3.29456 q^{66} -1.78806 q^{67} +5.57433 q^{68} +5.75638 q^{69} +7.27786 q^{71} -1.00000 q^{72} -1.40995 q^{73} -10.3577 q^{74} +0.374409 q^{76} +1.68408 q^{77} -2.65418 q^{78} -12.1395 q^{79} +1.00000 q^{81} -2.33728 q^{82} +3.09306 q^{83} +0.511170 q^{84} +5.87802 q^{86} -3.89025 q^{87} -3.29456 q^{88} +3.42567 q^{89} +1.35674 q^{91} +5.75638 q^{92} -1.80008 q^{93} +8.19959 q^{94} -1.00000 q^{96} -15.1483 q^{97} +6.73870 q^{98} +3.29456 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} + 4 q^{7} - 4 q^{8} + 4 q^{9} + 4 q^{12} - 2 q^{13} - 4 q^{14} + 4 q^{16} + 9 q^{17} - 4 q^{18} - 11 q^{19} + 4 q^{21} + 15 q^{23} - 4 q^{24} + 2 q^{26} + 4 q^{27}+ \cdots + 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 + 4 * q^7 - 4 * q^8 + 4 * q^9 + 4 * q^12 - 2 * q^13 - 4 * q^14 + 4 * q^16 + 9 * q^17 - 4 * q^18 - 11 * q^19 + 4 * q^21 + 15 * q^23 - 4 * q^24 + 2 * q^26 + 4 * q^27 + 4 * q^28 + 6 * q^29 - 8 * q^31 - 4 * q^32 - 9 * q^34 + 4 * q^36 + 13 * q^37 + 11 * q^38 - 2 * q^39 + 6 * q^41 - 4 * q^42 + 4 * q^43 - 15 * q^46 + 15 * q^47 + 4 * q^48 - 4 * q^49 + 9 * q^51 - 2 * q^52 + 21 * q^53 - 4 * q^54 - 4 * q^56 - 11 * q^57 - 6 * q^58 + 15 * q^59 - 5 * q^61 + 8 * q^62 + 4 * q^63 + 4 * q^64 + 19 * q^67 + 9 * q^68 + 15 * q^69 - 4 * q^72 - 2 * q^73 - 13 * q^74 - 11 * q^76 + 15 * q^77 + 2 * q^78 - 23 * q^79 + 4 * q^81 - 6 * q^82 + 18 * q^83 + 4 * q^84 - 4 * q^86 + 6 * q^87 + 27 * q^89 - 22 * q^91 + 15 * q^92 - 8 * q^93 - 15 * q^94 - 4 * q^96 + 13 * q^97 + 4 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0.511170	0.193204	0.0966021	−	0.995323i	$-0.469203\pi$
$7$	0.511170	0.193204	0.0966021	+	0.995323i	$0.469203\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	3.29456	0.993346	0.496673	−	0.867938i	$-0.334555\pi$
$11$	3.29456	0.993346	0.496673	+	0.867938i	$0.334555\pi$
$12$	1.00000	0.288675
$13$	2.65418	0.736138	0.368069	−	0.929799i	$-0.380019\pi$
$13$	2.65418	0.736138	0.368069	+	0.929799i	$0.380019\pi$
$14$	−0.511170	−0.136616
$15$	0	0
$16$	1.00000	0.250000
$17$	5.57433	1.35197	0.675987	−	0.736914i	$-0.263718\pi$
$17$	5.57433	1.35197	0.675987	+	0.736914i	$0.263718\pi$
$18$	−1.00000	−0.235702
$19$	0.374409	0.0858953	0.0429477	−	0.999077i	$-0.486325\pi$
$19$	0.374409	0.0858953	0.0429477	+	0.999077i	$0.486325\pi$
$20$	0	0
$21$	0.511170	0.111547
$22$	−3.29456	−0.702402
$23$	5.75638	1.20029	0.600144	−	0.799892i	$-0.295110\pi$
$23$	5.75638	1.20029	0.600144	+	0.799892i	$0.295110\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−2.65418	−0.520528
$27$	1.00000	0.192450
$28$	0.511170	0.0966021
$29$	−3.89025	−0.722401	−0.361201	−	0.932488i	$-0.617633\pi$
$29$	−3.89025	−0.722401	−0.361201	+	0.932488i	$0.617633\pi$
$30$	0	0
$31$	−1.80008	−0.323304	−0.161652	−	0.986848i	$-0.551682\pi$
$31$	−1.80008	−0.323304	−0.161652	+	0.986848i	$0.551682\pi$
$32$	−1.00000	−0.176777
$33$	3.29456	0.573509
$34$	−5.57433	−0.955990
$35$	0	0
$36$	1.00000	0.166667
$37$	10.3577	1.70280	0.851399	−	0.524519i	$-0.175755\pi$
$37$	10.3577	1.70280	0.851399	+	0.524519i	$0.175755\pi$
$38$	−0.374409	−0.0607372
$39$	2.65418	0.425009
$40$	0	0
$41$	2.33728	0.365023	0.182511	−	0.983204i	$-0.441577\pi$
$41$	2.33728	0.365023	0.182511	+	0.983204i	$0.441577\pi$
$42$	−0.511170	−0.0788753
$43$	−5.87802	−0.896390	−0.448195	−	0.893936i	$-0.647933\pi$
$43$	−5.87802	−0.896390	−0.448195	+	0.893936i	$0.647933\pi$
$44$	3.29456	0.496673
$45$	0	0
$46$	−5.75638	−0.848731
$47$	−8.19959	−1.19603	−0.598017	−	0.801484i	$-0.704044\pi$
$47$	−8.19959	−1.19603	−0.598017	+	0.801484i	$0.704044\pi$
$48$	1.00000	0.144338
$49$	−6.73870	−0.962672
$50$	0	0
$51$	5.57433	0.780562
$52$	2.65418	0.368069
$53$	0.518727	0.0712527	0.0356263	−	0.999365i	$-0.488657\pi$
$53$	0.518727	0.0712527	0.0356263	+	0.999365i	$0.488657\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−0.511170	−0.0683080
$57$	0.374409	0.0495917
$58$	3.89025	0.510815
$59$	−7.62993	−0.993332	−0.496666	−	0.867942i	$-0.665443\pi$
$59$	−7.62993	−0.993332	−0.496666	+	0.867942i	$0.665443\pi$
$60$	0	0
$61$	2.90345	0.371749	0.185875	−	0.982573i	$-0.440488\pi$
$61$	2.90345	0.371749	0.185875	+	0.982573i	$0.440488\pi$
$62$	1.80008	0.228610
$63$	0.511170	0.0644014
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.29456	−0.405532
$67$	−1.78806	−0.218446	−0.109223	−	0.994017i	$-0.534836\pi$
$67$	−1.78806	−0.218446	−0.109223	+	0.994017i	$0.534836\pi$
$68$	5.57433	0.675987
$69$	5.75638	0.692986
$70$	0	0
$71$	7.27786	0.863723	0.431862	−	0.901940i	$-0.357857\pi$
$71$	7.27786	0.863723	0.431862	+	0.901940i	$0.357857\pi$
$72$	−1.00000	−0.117851
$73$	−1.40995	−0.165023	−0.0825113	−	0.996590i	$-0.526294\pi$
$73$	−1.40995	−0.165023	−0.0825113	+	0.996590i	$0.526294\pi$
$74$	−10.3577	−1.20406
$75$	0	0
$76$	0.374409	0.0429477
$77$	1.68408	0.191919
$78$	−2.65418	−0.300527
$79$	−12.1395	−1.36580	−0.682901	−	0.730511i	$-0.739282\pi$
$79$	−12.1395	−1.36580	−0.682901	+	0.730511i	$0.739282\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−2.33728	−0.258110
$83$	3.09306	0.339507	0.169754	−	0.985487i	$-0.445703\pi$
$83$	3.09306	0.339507	0.169754	+	0.985487i	$0.445703\pi$
$84$	0.511170	0.0557733
$85$	0	0
$86$	5.87802	0.633843
$87$	−3.89025	−0.417079
$88$	−3.29456	−0.351201
$89$	3.42567	0.363120	0.181560	−	0.983380i	$-0.441885\pi$
$89$	3.42567	0.363120	0.181560	+	0.983380i	$0.441885\pi$
$90$	0	0
$91$	1.35674	0.142225
$92$	5.75638	0.600144
$93$	−1.80008	−0.186660
$94$	8.19959	0.845723
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−15.1483	−1.53808	−0.769040	−	0.639201i	$-0.779265\pi$
$97$	−15.1483	−1.53808	−0.769040	+	0.639201i	$0.779265\pi$
$98$	6.73870	0.680712
$99$	3.29456	0.331115
$100$	0	0
$101$	5.58091	0.555321	0.277661	−	0.960679i	$-0.410441\pi$
$101$	5.58091	0.555321	0.277661	+	0.960679i	$0.410441\pi$
$102$	−5.57433	−0.551941
$103$	15.1228	1.49010	0.745048	−	0.667011i	$-0.232426\pi$
$103$	15.1228	1.49010	0.745048	+	0.667011i	$0.232426\pi$
$104$	−2.65418	−0.260264
$105$	0	0
$106$	−0.518727	−0.0503832
$107$	18.7290	1.81060	0.905302	−	0.424768i	$-0.139644\pi$
$107$	18.7290	1.81060	0.905302	+	0.424768i	$0.139644\pi$
$108$	1.00000	0.0962250
$109$	−18.5221	−1.77410	−0.887049	−	0.461675i	$-0.847249\pi$
$109$	−18.5221	−1.77410	−0.887049	+	0.461675i	$0.847249\pi$
$110$	0	0
$111$	10.3577	0.983111
$112$	0.511170	0.0483011
$113$	16.1273	1.51713	0.758564	−	0.651598i	$-0.225901\pi$
$113$	16.1273	1.51713	0.758564	+	0.651598i	$0.225901\pi$
$114$	−0.374409	−0.0350666
$115$	0	0
$116$	−3.89025	−0.361201
$117$	2.65418	0.245379
$118$	7.62993	0.702392
$119$	2.84943	0.261207
$120$	0	0
$121$	−0.145898	−0.0132635
$122$	−2.90345	−0.262866
$123$	2.33728	0.210746
$124$	−1.80008	−0.161652
$125$	0	0
$126$	−0.511170	−0.0455387
$127$	−3.62677	−0.321824	−0.160912	−	0.986969i	$-0.551444\pi$
$127$	−3.62677	−0.321824	−0.160912	+	0.986969i	$0.551444\pi$
$128$	−1.00000	−0.0883883
$129$	−5.87802	−0.517531
$130$	0	0
$131$	−17.4131	−1.52139	−0.760695	−	0.649109i	$-0.775142\pi$
$131$	−17.4131	−1.52139	−0.760695	+	0.649109i	$0.775142\pi$
$132$	3.29456	0.286754
$133$	0.191387	0.0165953
$134$	1.78806	0.154465
$135$	0	0
$136$	−5.57433	−0.477995
$137$	19.7230	1.68505	0.842524	−	0.538658i	$-0.181069\pi$
$137$	19.7230	1.68505	0.842524	+	0.538658i	$0.181069\pi$
$138$	−5.75638	−0.490015
$139$	11.4357	0.969960	0.484980	−	0.874525i	$-0.338827\pi$
$139$	11.4357	0.969960	0.484980	+	0.874525i	$0.338827\pi$
$140$	0	0
$141$	−8.19959	−0.690530
$142$	−7.27786	−0.610745
$143$	8.74435	0.731239
$144$	1.00000	0.0833333
$145$	0	0
$146$	1.40995	0.116689
$147$	−6.73870	−0.555799
$148$	10.3577	0.851399
$149$	20.5095	1.68021	0.840103	−	0.542427i	$-0.182494\pi$
$149$	20.5095	1.68021	0.840103	+	0.542427i	$0.182494\pi$
$150$	0	0
$151$	6.22542	0.506618	0.253309	−	0.967385i	$-0.418481\pi$
$151$	6.22542	0.506618	0.253309	+	0.967385i	$0.418481\pi$
$152$	−0.374409	−0.0303686
$153$	5.57433	0.450658
$154$	−1.68408	−0.135707
$155$	0	0
$156$	2.65418	0.212505
$157$	−23.9795	−1.91377	−0.956886	−	0.290465i	$-0.906190\pi$
$157$	−23.9795	−1.91377	−0.956886	+	0.290465i	$0.906190\pi$
$158$	12.1395	0.965768
$159$	0.518727	0.0411377
$160$	0	0
$161$	2.94249	0.231901
$162$	−1.00000	−0.0785674
$163$	24.1232	1.88948	0.944738	−	0.327825i	$-0.106316\pi$
$163$	24.1232	1.88948	0.944738	+	0.327825i	$0.106316\pi$
$164$	2.33728	0.182511
$165$	0	0
$166$	−3.09306	−0.240068
$167$	5.43441	0.420527	0.210264	−	0.977645i	$-0.432568\pi$
$167$	5.43441	0.420527	0.210264	+	0.977645i	$0.432568\pi$
$168$	−0.511170	−0.0394376
$169$	−5.95532	−0.458101
$170$	0	0
$171$	0.374409	0.0286318
$172$	−5.87802	−0.448195
$173$	1.14183	0.0868118	0.0434059	−	0.999058i	$-0.486179\pi$
$173$	1.14183	0.0868118	0.0434059	+	0.999058i	$0.486179\pi$
$174$	3.89025	0.294919
$175$	0	0
$176$	3.29456	0.248337
$177$	−7.62993	−0.573501
$178$	−3.42567	−0.256765
$179$	−15.4030	−1.15127	−0.575637	−	0.817705i	$-0.695246\pi$
$179$	−15.4030	−1.15127	−0.575637	+	0.817705i	$0.695246\pi$
$180$	0	0
$181$	20.5110	1.52457	0.762287	−	0.647239i	$-0.224077\pi$
$181$	20.5110	1.52457	0.762287	+	0.647239i	$0.224077\pi$
$182$	−1.35674	−0.100568
$183$	2.90345	0.214629
$184$	−5.75638	−0.424366
$185$	0	0
$186$	1.80008	0.131988
$187$	18.3649	1.34298
$188$	−8.19959	−0.598017
$189$	0.511170	0.0371822
$190$	0	0
$191$	7.27380	0.526313	0.263157	−	0.964753i	$-0.415236\pi$
$191$	7.27380	0.526313	0.263157	+	0.964753i	$0.415236\pi$
$192$	1.00000	0.0721688
$193$	11.4686	0.825531	0.412766	−	0.910837i	$-0.364563\pi$
$193$	11.4686	0.825531	0.412766	+	0.910837i	$0.364563\pi$
$194$	15.1483	1.08759
$195$	0	0
$196$	−6.73870	−0.481336
$197$	−2.16068	−0.153942	−0.0769711	−	0.997033i	$-0.524525\pi$
$197$	−2.16068	−0.153942	−0.0769711	+	0.997033i	$0.524525\pi$
$198$	−3.29456	−0.234134
$199$	−7.27998	−0.516064	−0.258032	−	0.966136i	$-0.583074\pi$
$199$	−7.27998	−0.516064	−0.258032	+	0.966136i	$0.583074\pi$
$200$	0	0
$201$	−1.78806	−0.126120
$202$	−5.58091	−0.392671
$203$	−1.98858	−0.139571
$204$	5.57433	0.390281
$205$	0	0
$206$	−15.1228	−1.05366
$207$	5.75638	0.400096
$208$	2.65418	0.184034
$209$	1.23351	0.0853238
$210$	0	0
$211$	−28.7670	−1.98040	−0.990200	−	0.139658i	$-0.955400\pi$
$211$	−28.7670	−1.98040	−0.990200	+	0.139658i	$0.955400\pi$
$212$	0.518727	0.0356263
$213$	7.27786	0.498671
$214$	−18.7290	−1.28029
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−0.920147	−0.0624637
$218$	18.5221	1.25448
$219$	−1.40995	−0.0952758
$220$	0	0
$221$	14.7953	0.995238
$222$	−10.3577	−0.695164
$223$	−0.662342	−0.0443537	−0.0221769	−	0.999754i	$-0.507060\pi$
$223$	−0.662342	−0.0443537	−0.0221769	+	0.999754i	$0.507060\pi$
$224$	−0.511170	−0.0341540
$225$	0	0
$226$	−16.1273	−1.07277
$227$	−15.5730	−1.03362	−0.516809	−	0.856101i	$-0.672880\pi$
$227$	−15.5730	−1.03362	−0.516809	+	0.856101i	$0.672880\pi$
$228$	0.374409	0.0247958
$229$	−27.7376	−1.83296	−0.916478	−	0.400086i	$-0.868980\pi$
$229$	−27.7376	−1.83296	−0.916478	+	0.400086i	$0.868980\pi$
$230$	0	0
$231$	1.68408	0.110804
$232$	3.89025	0.255407
$233$	−23.2480	−1.52303	−0.761514	−	0.648149i	$-0.775544\pi$
$233$	−23.2480	−1.52303	−0.761514	+	0.648149i	$0.775544\pi$
$234$	−2.65418	−0.173509
$235$	0	0
$236$	−7.62993	−0.496666
$237$	−12.1395	−0.788547
$238$	−2.84943	−0.184701
$239$	5.81933	0.376421	0.188211	−	0.982129i	$-0.439731\pi$
$239$	5.81933	0.376421	0.188211	+	0.982129i	$0.439731\pi$
$240$	0	0
$241$	14.1483	0.911374	0.455687	−	0.890140i	$-0.349394\pi$
$241$	14.1483	0.911374	0.455687	+	0.890140i	$0.349394\pi$
$242$	0.145898	0.00937868
$243$	1.00000	0.0641500
$244$	2.90345	0.185875
$245$	0	0
$246$	−2.33728	−0.149020
$247$	0.993750	0.0632308
$248$	1.80008	0.114305
$249$	3.09306	0.196014
$250$	0	0
$251$	3.98246	0.251370	0.125685	−	0.992070i	$-0.459887\pi$
$251$	3.98246	0.251370	0.125685	+	0.992070i	$0.459887\pi$
$252$	0.511170	0.0322007
$253$	18.9647	1.19230
$254$	3.62677	0.227564
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.5114	0.780442	0.390221	−	0.920721i	$-0.372399\pi$
$257$	12.5114	0.780442	0.390221	+	0.920721i	$0.372399\pi$
$258$	5.87802	0.365950
$259$	5.29456	0.328988
$260$	0	0
$261$	−3.89025	−0.240800
$262$	17.4131	1.07579
$263$	−2.25374	−0.138971	−0.0694857	−	0.997583i	$-0.522136\pi$
$263$	−2.25374	−0.138971	−0.0694857	+	0.997583i	$0.522136\pi$
$264$	−3.29456	−0.202766
$265$	0	0
$266$	−0.191387	−0.0117347
$267$	3.42567	0.209648
$268$	−1.78806	−0.109223
$269$	25.4934	1.55436	0.777179	−	0.629279i	$-0.216650\pi$
$269$	25.4934	1.55436	0.777179	+	0.629279i	$0.216650\pi$
$270$	0	0
$271$	−19.2503	−1.16937	−0.584687	−	0.811259i	$-0.698783\pi$
$271$	−19.2503	−1.16937	−0.584687	+	0.811259i	$0.698783\pi$
$272$	5.57433	0.337993
$273$	1.35674	0.0821136
$274$	−19.7230	−1.19151
$275$	0	0
$276$	5.75638	0.346493
$277$	7.73948	0.465020	0.232510	−	0.972594i	$-0.425306\pi$
$277$	7.73948	0.465020	0.232510	+	0.972594i	$0.425306\pi$
$278$	−11.4357	−0.685865
$279$	−1.80008	−0.107768
$280$	0	0
$281$	−10.2724	−0.612801	−0.306401	−	0.951903i	$-0.599125\pi$
$281$	−10.2724	−0.612801	−0.306401	+	0.951903i	$0.599125\pi$
$282$	8.19959	0.488278
$283$	9.86811	0.586598	0.293299	−	0.956021i	$-0.405247\pi$
$283$	9.86811	0.586598	0.293299	+	0.956021i	$0.405247\pi$
$284$	7.27786	0.431862
$285$	0	0
$286$	−8.74435	−0.517064
$287$	1.19475	0.0705239
$288$	−1.00000	−0.0589256
$289$	14.0731	0.827832
$290$	0	0
$291$	−15.1483	−0.888011
$292$	−1.40995	−0.0825113
$293$	3.08103	0.179996	0.0899979	−	0.995942i	$-0.471314\pi$
$293$	3.08103	0.179996	0.0899979	+	0.995942i	$0.471314\pi$
$294$	6.73870	0.393009
$295$	0	0
$296$	−10.3577	−0.602030
$297$	3.29456	0.191170
$298$	−20.5095	−1.18809
$299$	15.2785	0.883577
$300$	0	0
$301$	−3.00467	−0.173186
$302$	−6.22542	−0.358233
$303$	5.58091	0.320615
$304$	0.374409	0.0214738
$305$	0	0
$306$	−5.57433	−0.318663
$307$	6.43192	0.367089	0.183545	−	0.983011i	$-0.441243\pi$
$307$	6.43192	0.367089	0.183545	+	0.983011i	$0.441243\pi$
$308$	1.68408	0.0959593
$309$	15.1228	0.860308
$310$	0	0
$311$	20.8818	1.18410	0.592048	−	0.805903i	$-0.298320\pi$
$311$	20.8818	1.18410	0.592048	+	0.805903i	$0.298320\pi$
$312$	−2.65418	−0.150263
$313$	−11.7245	−0.662708	−0.331354	−	0.943507i	$-0.607505\pi$
$313$	−11.7245	−0.662708	−0.331354	+	0.943507i	$0.607505\pi$
$314$	23.9795	1.35324
$315$	0	0
$316$	−12.1395	−0.682901
$317$	−5.92233	−0.332631	−0.166316	−	0.986073i	$-0.553187\pi$
$317$	−5.92233	−0.332631	−0.166316	+	0.986073i	$0.553187\pi$
$318$	−0.518727	−0.0290888
$319$	−12.8166	−0.717594
$320$	0	0
$321$	18.7290	1.04535
$322$	−2.94249	−0.163978
$323$	2.08708	0.116128
$324$	1.00000	0.0555556
$325$	0	0
$326$	−24.1232	−1.33606
$327$	−18.5221	−1.02428
$328$	−2.33728	−0.129055
$329$	−4.19139	−0.231079
$330$	0	0
$331$	27.3753	1.50468	0.752340	−	0.658775i	$-0.228925\pi$
$331$	27.3753	1.50468	0.752340	+	0.658775i	$0.228925\pi$
$332$	3.09306	0.169754
$333$	10.3577	0.567599
$334$	−5.43441	−0.297358
$335$	0	0
$336$	0.511170	0.0278866
$337$	5.86677	0.319583	0.159792	−	0.987151i	$-0.448918\pi$
$337$	5.86677	0.319583	0.159792	+	0.987151i	$0.448918\pi$
$338$	5.95532	0.323927
$339$	16.1273	0.875914
$340$	0	0
$341$	−5.93046	−0.321153
$342$	−0.374409	−0.0202457
$343$	−7.02282	−0.379197
$344$	5.87802	0.316922
$345$	0	0
$346$	−1.14183	−0.0613852
$347$	−11.8634	−0.636863	−0.318431	−	0.947946i	$-0.603156\pi$
$347$	−11.8634	−0.636863	−0.318431	+	0.947946i	$0.603156\pi$
$348$	−3.89025	−0.208539
$349$	28.3391	1.51696	0.758479	−	0.651698i	$-0.225943\pi$
$349$	28.3391	1.51696	0.758479	+	0.651698i	$0.225943\pi$
$350$	0	0
$351$	2.65418	0.141670
$352$	−3.29456	−0.175600
$353$	−5.83404	−0.310515	−0.155257	−	0.987874i	$-0.549621\pi$
$353$	−5.83404	−0.310515	−0.155257	+	0.987874i	$0.549621\pi$
$354$	7.62993	0.405526
$355$	0	0
$356$	3.42567	0.181560
$357$	2.84943	0.150808
$358$	15.4030	0.814074
$359$	9.94932	0.525105	0.262552	−	0.964918i	$-0.415436\pi$
$359$	9.94932	0.525105	0.262552	+	0.964918i	$0.415436\pi$
$360$	0	0
$361$	−18.8598	−0.992622
$362$	−20.5110	−1.07804
$363$	−0.145898	−0.00765766
$364$	1.35674	0.0711124
$365$	0	0
$366$	−2.90345	−0.151766
$367$	−16.2990	−0.850802	−0.425401	−	0.905005i	$-0.639867\pi$
$367$	−16.2990	−0.850802	−0.425401	+	0.905005i	$0.639867\pi$
$368$	5.75638	0.300072
$369$	2.33728	0.121674
$370$	0	0
$371$	0.265158	0.0137663
$372$	−1.80008	−0.0933298
$373$	8.85626	0.458560	0.229280	−	0.973361i	$-0.426363\pi$
$373$	8.85626	0.458560	0.229280	+	0.973361i	$0.426363\pi$
$374$	−18.3649	−0.949629
$375$	0	0
$376$	8.19959	0.422862
$377$	−10.3254	−0.531787
$378$	−0.511170	−0.0262918
$379$	−14.1431	−0.726480	−0.363240	−	0.931696i	$-0.618329\pi$
$379$	−14.1431	−0.726480	−0.363240	+	0.931696i	$0.618329\pi$
$380$	0	0
$381$	−3.62677	−0.185805
$382$	−7.27380	−0.372160
$383$	−32.9848	−1.68544	−0.842721	−	0.538350i	$-0.819048\pi$
$383$	−32.9848	−1.68544	−0.842721	+	0.538350i	$0.819048\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−11.4686	−0.583739
$387$	−5.87802	−0.298797
$388$	−15.1483	−0.769040
$389$	11.4919	0.582663	0.291332	−	0.956622i	$-0.405902\pi$
$389$	11.4919	0.582663	0.291332	+	0.956622i	$0.405902\pi$
$390$	0	0
$391$	32.0879	1.62276
$392$	6.73870	0.340356
$393$	−17.4131	−0.878375
$394$	2.16068	0.108854
$395$	0	0
$396$	3.29456	0.165558
$397$	4.82958	0.242390	0.121195	−	0.992629i	$-0.461327\pi$
$397$	4.82958	0.242390	0.121195	+	0.992629i	$0.461327\pi$
$398$	7.27998	0.364912
$399$	0.191387	0.00958132
$400$	0	0
$401$	−21.5779	−1.07755	−0.538775	−	0.842449i	$-0.681113\pi$
$401$	−21.5779	−1.07755	−0.538775	+	0.842449i	$0.681113\pi$
$402$	1.78806	0.0891802
$403$	−4.77774	−0.237996
$404$	5.58091	0.277661
$405$	0	0
$406$	1.98858	0.0986916
$407$	34.1241	1.69147
$408$	−5.57433	−0.275970
$409$	−1.57998	−0.0781248	−0.0390624	−	0.999237i	$-0.512437\pi$
$409$	−1.57998	−0.0781248	−0.0390624	+	0.999237i	$0.512437\pi$
$410$	0	0
$411$	19.7230	0.972863
$412$	15.1228	0.745048
$413$	−3.90019	−0.191916
$414$	−5.75638	−0.282910
$415$	0	0
$416$	−2.65418	−0.130132
$417$	11.4357	0.560007
$418$	−1.23351	−0.0603330
$419$	−36.7568	−1.79569	−0.897843	−	0.440316i	$-0.854866\pi$
$419$	−36.7568	−1.79569	−0.897843	+	0.440316i	$0.854866\pi$
$420$	0	0
$421$	6.96946	0.339671	0.169835	−	0.985472i	$-0.445676\pi$
$421$	6.96946	0.339671	0.169835	+	0.985472i	$0.445676\pi$
$422$	28.7670	1.40035
$423$	−8.19959	−0.398678
$424$	−0.518727	−0.0251916
$425$	0	0
$426$	−7.27786	−0.352614
$427$	1.48416	0.0718235
$428$	18.7290	0.905302
$429$	8.74435	0.422181
$430$	0	0
$431$	25.4098	1.22395	0.611974	−	0.790878i	$-0.290376\pi$
$431$	25.4098	1.22395	0.611974	+	0.790878i	$0.290376\pi$
$432$	1.00000	0.0481125
$433$	−13.4595	−0.646823	−0.323411	−	0.946258i	$-0.604830\pi$
$433$	−13.4595	−0.646823	−0.323411	+	0.946258i	$0.604830\pi$
$434$	0.920147	0.0441685
$435$	0	0
$436$	−18.5221	−0.887049
$437$	2.15524	0.103099
$438$	1.40995	0.0673702
$439$	−37.8392	−1.80597	−0.902983	−	0.429676i	$-0.858628\pi$
$439$	−37.8392	−1.80597	−0.902983	+	0.429676i	$0.858628\pi$
$440$	0	0
$441$	−6.73870	−0.320891
$442$	−14.7953	−0.703740
$443$	2.70138	0.128346	0.0641731	−	0.997939i	$-0.479559\pi$
$443$	2.70138	0.128346	0.0641731	+	0.997939i	$0.479559\pi$
$444$	10.3577	0.491555
$445$	0	0
$446$	0.662342	0.0313628
$447$	20.5095	0.970068
$448$	0.511170	0.0241505
$449$	18.9478	0.894200	0.447100	−	0.894484i	$-0.352457\pi$
$449$	18.9478	0.894200	0.447100	+	0.894484i	$0.352457\pi$
$450$	0	0
$451$	7.70032	0.362594
$452$	16.1273	0.758564
$453$	6.22542	0.292496
$454$	15.5730	0.730878
$455$	0	0
$456$	−0.374409	−0.0175333
$457$	−38.2879	−1.79103	−0.895517	−	0.445028i	$-0.853194\pi$
$457$	−38.2879	−1.79103	−0.895517	+	0.445028i	$0.853194\pi$
$458$	27.7376	1.29610
$459$	5.57433	0.260187
$460$	0	0
$461$	17.3427	0.807731	0.403866	−	0.914818i	$-0.367666\pi$
$461$	17.3427	0.807731	0.403866	+	0.914818i	$0.367666\pi$
$462$	−1.68408	−0.0783505
$463$	17.6641	0.820922	0.410461	−	0.911878i	$-0.365368\pi$
$463$	17.6641	0.820922	0.410461	+	0.911878i	$0.365368\pi$
$464$	−3.89025	−0.180600
$465$	0	0
$466$	23.2480	1.07694
$467$	−2.88972	−0.133720	−0.0668600	−	0.997762i	$-0.521298\pi$
$467$	−2.88972	−0.133720	−0.0668600	+	0.997762i	$0.521298\pi$
$468$	2.65418	0.122690
$469$	−0.914001	−0.0422047
$470$	0	0
$471$	−23.9795	−1.10492
$472$	7.62993	0.351196
$473$	−19.3655	−0.890426
$474$	12.1395	0.557587
$475$	0	0
$476$	2.84943	0.130603
$477$	0.518727	0.0237509
$478$	−5.81933	−0.266170
$479$	27.5859	1.26043	0.630216	−	0.776420i	$-0.282966\pi$
$479$	27.5859	1.26043	0.630216	+	0.776420i	$0.282966\pi$
$480$	0	0
$481$	27.4913	1.25349
$482$	−14.1483	−0.644439
$483$	2.94249	0.133888
$484$	−0.145898	−0.00663173
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	14.9338	0.676713	0.338356	−	0.941018i	$-0.390129\pi$
$487$	14.9338	0.676713	0.338356	+	0.941018i	$0.390129\pi$
$488$	−2.90345	−0.131433
$489$	24.1232	1.09089
$490$	0	0
$491$	26.4452	1.19346	0.596728	−	0.802444i	$-0.296467\pi$
$491$	26.4452	1.19346	0.596728	+	0.802444i	$0.296467\pi$
$492$	2.33728	0.105373
$493$	−21.6855	−0.976667
$494$	−0.993750	−0.0447109
$495$	0	0
$496$	−1.80008	−0.0808260
$497$	3.72023	0.166875
$498$	−3.09306	−0.138603
$499$	−10.3375	−0.462771	−0.231386	−	0.972862i	$-0.574326\pi$
$499$	−10.3375	−0.462771	−0.231386	+	0.972862i	$0.574326\pi$
$500$	0	0
$501$	5.43441	0.242791
$502$	−3.98246	−0.177746
$503$	16.9084	0.753908	0.376954	−	0.926232i	$-0.376971\pi$
$503$	16.9084	0.753908	0.376954	+	0.926232i	$0.376971\pi$
$504$	−0.511170	−0.0227693
$505$	0	0
$506$	−18.9647	−0.843084
$507$	−5.95532	−0.264485
$508$	−3.62677	−0.160912
$509$	6.48318	0.287362	0.143681	−	0.989624i	$-0.454106\pi$
$509$	6.48318	0.287362	0.143681	+	0.989624i	$0.454106\pi$
$510$	0	0
$511$	−0.720726	−0.0318831
$512$	−1.00000	−0.0441942
$513$	0.374409	0.0165306
$514$	−12.5114	−0.551856
$515$	0	0
$516$	−5.87802	−0.258766
$517$	−27.0140	−1.18807
$518$	−5.29456	−0.232629
$519$	1.14183	0.0501208
$520$	0	0
$521$	21.6371	0.947939	0.473970	−	0.880541i	$-0.342821\pi$
$521$	21.6371	0.947939	0.473970	+	0.880541i	$0.342821\pi$
$522$	3.89025	0.170272
$523$	15.6560	0.684589	0.342295	−	0.939593i	$-0.388796\pi$
$523$	15.6560	0.684589	0.342295	+	0.939593i	$0.388796\pi$
$524$	−17.4131	−0.760695
$525$	0	0
$526$	2.25374	0.0982677
$527$	−10.0342	−0.437098
$528$	3.29456	0.143377
$529$	10.1359	0.440689
$530$	0	0
$531$	−7.62993	−0.331111
$532$	0.191387	0.00829767
$533$	6.20358	0.268707
$534$	−3.42567	−0.148243
$535$	0	0
$536$	1.78806	0.0772323
$537$	−15.4030	−0.664689
$538$	−25.4934	−1.09910
$539$	−22.2010	−0.956267
$540$	0	0
$541$	14.2839	0.614113	0.307057	−	0.951691i	$-0.400656\pi$
$541$	14.2839	0.614113	0.307057	+	0.951691i	$0.400656\pi$
$542$	19.2503	0.826872
$543$	20.5110	0.880213
$544$	−5.57433	−0.238997
$545$	0	0
$546$	−1.35674	−0.0580631
$547$	42.8226	1.83096	0.915481	−	0.402361i	$-0.131810\pi$
$547$	42.8226	1.83096	0.915481	+	0.402361i	$0.131810\pi$
$548$	19.7230	0.842524
$549$	2.90345	0.123916
$550$	0	0
$551$	−1.45654	−0.0620509
$552$	−5.75638	−0.245008
$553$	−6.20536	−0.263879
$554$	−7.73948	−0.328819
$555$	0	0
$556$	11.4357	0.484980
$557$	14.5543	0.616688	0.308344	−	0.951275i	$-0.400225\pi$
$557$	14.5543	0.616688	0.308344	+	0.951275i	$0.400225\pi$
$558$	1.80008	0.0762035
$559$	−15.6013	−0.659866
$560$	0	0
$561$	18.3649	0.775368
$562$	10.2724	0.433316
$563$	4.52778	0.190823	0.0954116	−	0.995438i	$-0.469583\pi$
$563$	4.52778	0.190823	0.0954116	+	0.995438i	$0.469583\pi$
$564$	−8.19959	−0.345265
$565$	0	0
$566$	−9.86811	−0.414788
$567$	0.511170	0.0214671
$568$	−7.27786	−0.305372
$569$	10.0904	0.423010	0.211505	−	0.977377i	$-0.432163\pi$
$569$	10.0904	0.423010	0.211505	+	0.977377i	$0.432163\pi$
$570$	0	0
$571$	−20.3024	−0.849630	−0.424815	−	0.905280i	$-0.639661\pi$
$571$	−20.3024	−0.849630	−0.424815	+	0.905280i	$0.639661\pi$
$572$	8.74435	0.365620
$573$	7.27380	0.303867
$574$	−1.19475	−0.0498679
$575$	0	0
$576$	1.00000	0.0416667
$577$	19.3900	0.807218	0.403609	−	0.914932i	$-0.367756\pi$
$577$	19.3900	0.807218	0.403609	+	0.914932i	$0.367756\pi$
$578$	−14.0731	−0.585366
$579$	11.4686	0.476621
$580$	0	0
$581$	1.58108	0.0655942
$582$	15.1483	0.627918
$583$	1.70898	0.0707786
$584$	1.40995	0.0583443
$585$	0	0
$586$	−3.08103	−0.127276
$587$	−43.8857	−1.81136	−0.905679	−	0.423964i	$-0.860638\pi$
$587$	−43.8857	−1.81136	−0.905679	+	0.423964i	$0.860638\pi$
$588$	−6.73870	−0.277900
$589$	−0.673966	−0.0277703
$590$	0	0
$591$	−2.16068	−0.0888786
$592$	10.3577	0.425699
$593$	−27.5651	−1.13196	−0.565981	−	0.824418i	$-0.691502\pi$
$593$	−27.5651	−1.13196	−0.565981	+	0.824418i	$0.691502\pi$
$594$	−3.29456	−0.135177
$595$	0	0
$596$	20.5095	0.840103
$597$	−7.27998	−0.297950
$598$	−15.2785	−0.624783
$599$	13.3970	0.547385	0.273692	−	0.961817i	$-0.411755\pi$
$599$	13.3970	0.547385	0.273692	+	0.961817i	$0.411755\pi$
$600$	0	0
$601$	−32.3401	−1.31918	−0.659590	−	0.751625i	$-0.729270\pi$
$601$	−32.3401	−1.31918	−0.659590	+	0.751625i	$0.729270\pi$
$602$	3.00467	0.122461
$603$	−1.78806	−0.0728153
$604$	6.22542	0.253309
$605$	0	0
$606$	−5.58091	−0.226709
$607$	−18.7755	−0.762074	−0.381037	−	0.924560i	$-0.624433\pi$
$607$	−18.7755	−0.762074	−0.381037	+	0.924560i	$0.624433\pi$
$608$	−0.374409	−0.0151843
$609$	−1.98858	−0.0805813
$610$	0	0
$611$	−21.7632	−0.880445
$612$	5.57433	0.225329
$613$	−5.12696	−0.207076	−0.103538	−	0.994625i	$-0.533016\pi$
$613$	−5.12696	−0.207076	−0.103538	+	0.994625i	$0.533016\pi$
$614$	−6.43192	−0.259571
$615$	0	0
$616$	−1.68408	−0.0678535
$617$	27.9234	1.12415	0.562077	−	0.827085i	$-0.310002\pi$
$617$	27.9234	1.12415	0.562077	+	0.827085i	$0.310002\pi$
$618$	−15.1228	−0.608329
$619$	−36.9317	−1.48441	−0.742205	−	0.670172i	$-0.766220\pi$
$619$	−36.9317	−1.48441	−0.742205	+	0.670172i	$0.766220\pi$
$620$	0	0
$621$	5.75638	0.230995
$622$	−20.8818	−0.837282
$623$	1.75110	0.0701564
$624$	2.65418	0.106252
$625$	0	0
$626$	11.7245	0.468605
$627$	1.23351	0.0492617
$628$	−23.9795	−0.956886
$629$	57.7373	2.30214
$630$	0	0
$631$	−7.04589	−0.280492	−0.140246	−	0.990117i	$-0.544789\pi$
$631$	−7.04589	−0.280492	−0.140246	+	0.990117i	$0.544789\pi$
$632$	12.1395	0.482884
$633$	−28.7670	−1.14338
$634$	5.92233	0.235206
$635$	0	0
$636$	0.518727	0.0205689
$637$	−17.8857	−0.708659
$638$	12.8166	0.507416
$639$	7.27786	0.287908
$640$	0	0
$641$	−41.2092	−1.62767	−0.813834	−	0.581098i	$-0.802623\pi$
$641$	−41.2092	−1.62767	−0.813834	+	0.581098i	$0.802623\pi$
$642$	−18.7290	−0.739176
$643$	4.40938	0.173889	0.0869444	−	0.996213i	$-0.472290\pi$
$643$	4.40938	0.173889	0.0869444	+	0.996213i	$0.472290\pi$
$644$	2.94249	0.115950
$645$	0	0
$646$	−2.08708	−0.0821150
$647$	12.8764	0.506223	0.253111	−	0.967437i	$-0.418546\pi$
$647$	12.8764	0.506223	0.253111	+	0.967437i	$0.418546\pi$
$648$	−1.00000	−0.0392837
$649$	−25.1372	−0.986723
$650$	0	0
$651$	−0.920147	−0.0360634
$652$	24.1232	0.944738
$653$	−33.9475	−1.32847	−0.664234	−	0.747525i	$-0.731242\pi$
$653$	−33.9475	−1.32847	−0.664234	+	0.747525i	$0.731242\pi$
$654$	18.5221	0.724273
$655$	0	0
$656$	2.33728	0.0912556
$657$	−1.40995	−0.0550075
$658$	4.19139	0.163397
$659$	−12.6089	−0.491171	−0.245586	−	0.969375i	$-0.578980\pi$
$659$	−12.6089	−0.491171	−0.245586	+	0.969375i	$0.578980\pi$
$660$	0	0
$661$	14.6188	0.568607	0.284304	−	0.958734i	$-0.408238\pi$
$661$	14.6188	0.568607	0.284304	+	0.958734i	$0.408238\pi$
$662$	−27.3753	−1.06397
$663$	14.7953	0.574601
$664$	−3.09306	−0.120034
$665$	0	0
$666$	−10.3577	−0.401353
$667$	−22.3937	−0.867089
$668$	5.43441	0.210264
$669$	−0.662342	−0.0256076
$670$	0	0
$671$	9.56559	0.369276
$672$	−0.511170	−0.0197188
$673$	13.4341	0.517847	0.258924	−	0.965898i	$-0.416632\pi$
$673$	13.4341	0.517847	0.258924	+	0.965898i	$0.416632\pi$
$674$	−5.86677	−0.225980
$675$	0	0
$676$	−5.95532	−0.229051
$677$	31.9199	1.22678	0.613391	−	0.789780i	$-0.289805\pi$
$677$	31.9199	1.22678	0.613391	+	0.789780i	$0.289805\pi$
$678$	−16.1273	−0.619365
$679$	−7.74338	−0.297163
$680$	0	0
$681$	−15.5730	−0.596760
$682$	5.93046	0.227089
$683$	−19.0698	−0.729685	−0.364842	−	0.931069i	$-0.618877\pi$
$683$	−19.0698	−0.729685	−0.364842	+	0.931069i	$0.618877\pi$
$684$	0.374409	0.0143159
$685$	0	0
$686$	7.02282	0.268132
$687$	−27.7376	−1.05826
$688$	−5.87802	−0.224098
$689$	1.37680	0.0524518
$690$	0	0
$691$	21.5074	0.818181	0.409091	−	0.912494i	$-0.365846\pi$
$691$	21.5074	0.818181	0.409091	+	0.912494i	$0.365846\pi$
$692$	1.14183	0.0434059
$693$	1.68408	0.0639729
$694$	11.8634	0.450330
$695$	0	0
$696$	3.89025	0.147460
$697$	13.0288	0.493501
$698$	−28.3391	−1.07265
$699$	−23.2480	−0.879320
$700$	0	0
$701$	−5.65381	−0.213541	−0.106771	−	0.994284i	$-0.534051\pi$
$701$	−5.65381	−0.213541	−0.106771	+	0.994284i	$0.534051\pi$
$702$	−2.65418	−0.100176
$703$	3.87802	0.146262
$704$	3.29456	0.124168
$705$	0	0
$706$	5.83404	0.219567
$707$	2.85280	0.107290
$708$	−7.62993	−0.286750
$709$	43.7198	1.64193	0.820967	−	0.570976i	$-0.193435\pi$
$709$	43.7198	1.64193	0.820967	+	0.570976i	$0.193435\pi$
$710$	0	0
$711$	−12.1395	−0.455268
$712$	−3.42567	−0.128382
$713$	−10.3619	−0.388058
$714$	−2.84943	−0.106637
$715$	0	0
$716$	−15.4030	−0.575637
$717$	5.81933	0.217327
$718$	−9.94932	−0.371305
$719$	−8.09200	−0.301781	−0.150890	−	0.988551i	$-0.548214\pi$
$719$	−8.09200	−0.301781	−0.150890	+	0.988551i	$0.548214\pi$
$720$	0	0
$721$	7.73034	0.287893
$722$	18.8598	0.701890
$723$	14.1483	0.526182
$724$	20.5110	0.762287
$725$	0	0
$726$	0.145898	0.00541478
$727$	47.3889	1.75756	0.878779	−	0.477228i	$-0.158359\pi$
$727$	47.3889	1.75756	0.878779	+	0.477228i	$0.158359\pi$
$728$	−1.35674	−0.0502841
$729$	1.00000	0.0370370
$730$	0	0
$731$	−32.7660	−1.21190
$732$	2.90345	0.107315
$733$	−14.6145	−0.539801	−0.269900	−	0.962888i	$-0.586991\pi$
$733$	−14.6145	−0.539801	−0.269900	+	0.962888i	$0.586991\pi$
$734$	16.2990	0.601608
$735$	0	0
$736$	−5.75638	−0.212183
$737$	−5.89085	−0.216992
$738$	−2.33728	−0.0860366
$739$	−12.1592	−0.447283	−0.223641	−	0.974672i	$-0.571794\pi$
$739$	−12.1592	−0.447283	−0.223641	+	0.974672i	$0.571794\pi$
$740$	0	0
$741$	0.993750	0.0365063
$742$	−0.265158	−0.00973426
$743$	−50.5368	−1.85402	−0.927008	−	0.375041i	$-0.877629\pi$
$743$	−50.5368	−1.85402	−0.927008	+	0.375041i	$0.877629\pi$
$744$	1.80008	0.0659941
$745$	0	0
$746$	−8.85626	−0.324251
$747$	3.09306	0.113169
$748$	18.3649	0.671489
$749$	9.57373	0.349816
$750$	0	0
$751$	33.9598	1.23921	0.619605	−	0.784914i	$-0.287293\pi$
$751$	33.9598	1.23921	0.619605	+	0.784914i	$0.287293\pi$
$752$	−8.19959	−0.299008
$753$	3.98246	0.145129
$754$	10.3254	0.376030
$755$	0	0
$756$	0.511170	0.0185911
$757$	−47.9954	−1.74442	−0.872211	−	0.489130i	$-0.837315\pi$
$757$	−47.9954	−1.74442	−0.872211	+	0.489130i	$0.837315\pi$
$758$	14.1431	0.513699
$759$	18.9647	0.688375
$760$	0	0
$761$	27.6622	1.00275	0.501377	−	0.865229i	$-0.332827\pi$
$761$	27.6622	1.00275	0.501377	+	0.865229i	$0.332827\pi$
$762$	3.62677	0.131384
$763$	−9.46796	−0.342763
$764$	7.27380	0.263157
$765$	0	0
$766$	32.9848	1.19179
$767$	−20.2512	−0.731229
$768$	1.00000	0.0360844
$769$	−15.0664	−0.543310	−0.271655	−	0.962395i	$-0.587571\pi$
$769$	−15.0664	−0.543310	−0.271655	+	0.962395i	$0.587571\pi$
$770$	0	0
$771$	12.5114	0.450589
$772$	11.4686	0.412766
$773$	−29.4391	−1.05885	−0.529426	−	0.848356i	$-0.677593\pi$
$773$	−29.4391	−1.05885	−0.529426	+	0.848356i	$0.677593\pi$
$774$	5.87802	0.211281
$775$	0	0
$776$	15.1483	0.543793
$777$	5.29456	0.189941
$778$	−11.4919	−0.412005
$779$	0.875101	0.0313537
$780$	0	0
$781$	23.9773	0.857976
$782$	−32.0879	−1.14746
$783$	−3.89025	−0.139026
$784$	−6.73870	−0.240668
$785$	0	0
$786$	17.4131	0.621105
$787$	−17.7230	−0.631756	−0.315878	−	0.948800i	$-0.602299\pi$
$787$	−17.7230	−0.631756	−0.315878	+	0.948800i	$0.602299\pi$
$788$	−2.16068	−0.0769711
$789$	−2.25374	−0.0802352
$790$	0	0
$791$	8.24379	0.293116
$792$	−3.29456	−0.117067
$793$	7.70629	0.273659
$794$	−4.82958	−0.171395
$795$	0	0
$796$	−7.27998	−0.258032
$797$	−24.6949	−0.874737	−0.437368	−	0.899282i	$-0.644089\pi$
$797$	−24.6949	−0.874737	−0.437368	+	0.899282i	$0.644089\pi$
$798$	−0.191387	−0.00677502
$799$	−45.7072	−1.61700
$800$	0	0
$801$	3.42567	0.121040
$802$	21.5779	0.761944
$803$	−4.64517	−0.163925
$804$	−1.78806	−0.0630599
$805$	0	0
$806$	4.77774	0.168289
$807$	25.4934	0.897409
$808$	−5.58091	−0.196336
$809$	−16.4366	−0.577879	−0.288939	−	0.957347i	$-0.593303\pi$
$809$	−16.4366	−0.577879	−0.288939	+	0.957347i	$0.593303\pi$
$810$	0	0
$811$	−12.9773	−0.455696	−0.227848	−	0.973697i	$-0.573169\pi$
$811$	−12.9773	−0.455696	−0.227848	+	0.973697i	$0.573169\pi$
$812$	−1.98858	−0.0697855
$813$	−19.2503	−0.675138
$814$	−34.1241	−1.19605
$815$	0	0
$816$	5.57433	0.195141
$817$	−2.20078	−0.0769957
$818$	1.57998	0.0552426
$819$	1.35674	0.0474083
$820$	0	0
$821$	−31.7784	−1.10907	−0.554537	−	0.832159i	$-0.687105\pi$
$821$	−31.7784	−1.10907	−0.554537	+	0.832159i	$0.687105\pi$
$822$	−19.7230	−0.687918
$823$	−38.3742	−1.33764	−0.668820	−	0.743425i	$-0.733200\pi$
$823$	−38.3742	−1.33764	−0.668820	+	0.743425i	$0.733200\pi$
$824$	−15.1228	−0.526829
$825$	0	0
$826$	3.90019	0.135705
$827$	−23.0395	−0.801162	−0.400581	−	0.916261i	$-0.631192\pi$
$827$	−23.0395	−0.801162	−0.400581	+	0.916261i	$0.631192\pi$
$828$	5.75638	0.200048
$829$	−17.5852	−0.610758	−0.305379	−	0.952231i	$-0.598783\pi$
$829$	−17.5852	−0.610758	−0.305379	+	0.952231i	$0.598783\pi$
$830$	0	0
$831$	7.73948	0.268479
$832$	2.65418	0.0920172
$833$	−37.5638	−1.30151
$834$	−11.4357	−0.395984
$835$	0	0
$836$	1.23351	0.0426619
$837$	−1.80008	−0.0622199
$838$	36.7568	1.26974
$839$	−44.0619	−1.52119	−0.760593	−	0.649229i	$-0.775092\pi$
$839$	−44.0619	−1.52119	−0.760593	+	0.649229i	$0.775092\pi$
$840$	0	0
$841$	−13.8660	−0.478136
$842$	−6.96946	−0.240183
$843$	−10.2724	−0.353801
$844$	−28.7670	−0.990200
$845$	0	0
$846$	8.19959	0.281908
$847$	−0.0745787	−0.00256256
$848$	0.518727	0.0178132
$849$	9.86811	0.338673
$850$	0	0
$851$	59.6229	2.04385
$852$	7.27786	0.249335
$853$	−38.1231	−1.30531	−0.652655	−	0.757656i	$-0.726345\pi$
$853$	−38.1231	−1.30531	−0.652655	+	0.757656i	$0.726345\pi$
$854$	−1.48416	−0.0507869
$855$	0	0
$856$	−18.7290	−0.640145
$857$	−40.2837	−1.37607	−0.688033	−	0.725679i	$-0.741526\pi$
$857$	−40.2837	−1.37607	−0.688033	+	0.725679i	$0.741526\pi$
$858$	−8.74435	−0.298527
$859$	−16.9314	−0.577691	−0.288845	−	0.957376i	$-0.593271\pi$
$859$	−16.9314	−0.577691	−0.288845	+	0.957376i	$0.593271\pi$
$860$	0	0
$861$	1.19475	0.0407170
$862$	−25.4098	−0.865462
$863$	−38.3696	−1.30612	−0.653058	−	0.757308i	$-0.726514\pi$
$863$	−38.3696	−1.30612	−0.653058	+	0.757308i	$0.726514\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	13.4595	0.457373
$867$	14.0731	0.477949
$868$	−0.920147	−0.0312318
$869$	−39.9943	−1.35672
$870$	0	0
$871$	−4.74583	−0.160806
$872$	18.5221	0.627239
$873$	−15.1483	−0.512693
$874$	−2.15524	−0.0729020
$875$	0	0
$876$	−1.40995	−0.0476379
$877$	9.59661	0.324055	0.162027	−	0.986786i	$-0.448197\pi$
$877$	9.59661	0.324055	0.162027	+	0.986786i	$0.448197\pi$
$878$	37.8392	1.27701
$879$	3.08103	0.103921
$880$	0	0
$881$	21.0921	0.710610	0.355305	−	0.934750i	$-0.384377\pi$
$881$	21.0921	0.710610	0.355305	+	0.934750i	$0.384377\pi$
$882$	6.73870	0.226904
$883$	−1.26807	−0.0426740	−0.0213370	−	0.999772i	$-0.506792\pi$
$883$	−1.26807	−0.0426740	−0.0213370	+	0.999772i	$0.506792\pi$
$884$	14.7953	0.497619
$885$	0	0
$886$	−2.70138	−0.0907545
$887$	−3.50065	−0.117540	−0.0587701	−	0.998272i	$-0.518718\pi$
$887$	−3.50065	−0.117540	−0.0587701	+	0.998272i	$0.518718\pi$
$888$	−10.3577	−0.347582
$889$	−1.85390	−0.0621777
$890$	0	0
$891$	3.29456	0.110372
$892$	−0.662342	−0.0221769
$893$	−3.07000	−0.102734
$894$	−20.5095	−0.685941
$895$	0	0
$896$	−0.511170	−0.0170770
$897$	15.2785	0.510133
$898$	−18.9478	−0.632295
$899$	7.00276	0.233555
$900$	0	0
$901$	2.89156	0.0963317
$902$	−7.70032	−0.256393
$903$	−3.00467	−0.0999892
$904$	−16.1273	−0.536386
$905$	0	0
$906$	−6.22542	−0.206826
$907$	−20.2610	−0.672757	−0.336378	−	0.941727i	$-0.609202\pi$
$907$	−20.2610	−0.672757	−0.336378	+	0.941727i	$0.609202\pi$
$908$	−15.5730	−0.516809
$909$	5.58091	0.185107
$910$	0	0
$911$	−5.90220	−0.195549	−0.0977744	−	0.995209i	$-0.531172\pi$
$911$	−5.90220	−0.195549	−0.0977744	+	0.995209i	$0.531172\pi$
$912$	0.374409	0.0123979
$913$	10.1902	0.337248
$914$	38.2879	1.26645
$915$	0	0
$916$	−27.7376	−0.916478
$917$	−8.90107	−0.293939
$918$	−5.57433	−0.183980
$919$	−45.3645	−1.49644	−0.748219	−	0.663452i	$-0.769091\pi$
$919$	−45.3645	−1.49644	−0.748219	+	0.663452i	$0.769091\pi$
$920$	0	0
$921$	6.43192	0.211939
$922$	−17.3427	−0.571152
$923$	19.3168	0.635819
$924$	1.68408	0.0554021
$925$	0	0
$926$	−17.6641	−0.580479
$927$	15.1228	0.496699
$928$	3.89025	0.127704
$929$	46.2829	1.51849	0.759247	−	0.650803i	$-0.225568\pi$
$929$	46.2829	1.51849	0.759247	+	0.650803i	$0.225568\pi$
$930$	0	0
$931$	−2.52303	−0.0826890
$932$	−23.2480	−0.761514
$933$	20.8818	0.683638
$934$	2.88972	0.0945544
$935$	0	0
$936$	−2.65418	−0.0867546
$937$	21.2179	0.693158	0.346579	−	0.938021i	$-0.387343\pi$
$937$	21.2179	0.693158	0.346579	+	0.938021i	$0.387343\pi$
$938$	0.914001	0.0298432
$939$	−11.7245	−0.382615
$940$	0	0
$941$	−43.4850	−1.41757	−0.708784	−	0.705425i	$-0.750756\pi$
$941$	−43.4850	−1.41757	−0.708784	+	0.705425i	$0.750756\pi$
$942$	23.9795	0.781294
$943$	13.4543	0.438132
$944$	−7.62993	−0.248333
$945$	0	0
$946$	19.3655	0.629626
$947$	−34.6211	−1.12504	−0.562518	−	0.826785i	$-0.690167\pi$
$947$	−34.6211	−1.12504	−0.562518	+	0.826785i	$0.690167\pi$
$948$	−12.1395	−0.394273
$949$	−3.74227	−0.121479
$950$	0	0
$951$	−5.92233	−0.192045
$952$	−2.84943	−0.0923506
$953$	−21.1695	−0.685747	−0.342874	−	0.939382i	$-0.611400\pi$
$953$	−21.1695	−0.685747	−0.342874	+	0.939382i	$0.611400\pi$
$954$	−0.518727	−0.0167944
$955$	0	0
$956$	5.81933	0.188211
$957$	−12.8166	−0.414303
$958$	−27.5859	−0.891260
$959$	10.0818	0.325558
$960$	0	0
$961$	−27.7597	−0.895475
$962$	−27.4913	−0.886354
$963$	18.7290	0.603535
$964$	14.1483	0.455687
$965$	0	0
$966$	−2.94249	−0.0946730
$967$	43.1011	1.38604	0.693020	−	0.720919i	$-0.256280\pi$
$967$	43.1011	1.38604	0.693020	+	0.720919i	$0.256280\pi$
$968$	0.145898	0.00468934
$969$	2.08708	0.0670466
$970$	0	0
$971$	−32.5128	−1.04338	−0.521692	−	0.853134i	$-0.674699\pi$
$971$	−32.5128	−1.04338	−0.521692	+	0.853134i	$0.674699\pi$
$972$	1.00000	0.0320750
$973$	5.84557	0.187400
$974$	−14.9338	−0.478508
$975$	0	0
$976$	2.90345	0.0929373
$977$	56.2425	1.79936	0.899679	−	0.436552i	$-0.143800\pi$
$977$	56.2425	1.79936	0.899679	+	0.436552i	$0.143800\pi$
$978$	−24.1232	−0.771376
$979$	11.2861	0.360704
$980$	0	0
$981$	−18.5221	−0.591366
$982$	−26.4452	−0.843900
$983$	31.7227	1.01180	0.505898	−	0.862593i	$-0.331161\pi$
$983$	31.7227	1.01180	0.505898	+	0.862593i	$0.331161\pi$
$984$	−2.33728	−0.0745099
$985$	0	0
$986$	21.6855	0.690608
$987$	−4.19139	−0.133413
$988$	0.993750	0.0316154
$989$	−33.8361	−1.07593
$990$	0	0
$991$	15.6911	0.498445	0.249222	−	0.968446i	$-0.419825\pi$
$991$	15.6911	0.498445	0.249222	+	0.968446i	$0.419825\pi$
$992$	1.80008	0.0571526
$993$	27.3753	0.868728
$994$	−3.72023	−0.117998
$995$	0	0
$996$	3.09306	0.0980072
$997$	34.2809	1.08569	0.542843	−	0.839834i	$-0.317348\pi$
$997$	34.2809	1.08569	0.542843	+	0.839834i	$0.317348\pi$
$998$	10.3375	0.327229
$999$	10.3577	0.327704

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3750.2.a.n.1.3	✓	4
5.2	odd	4		3750.2.c.g.1249.3		8
5.3	odd	4		3750.2.c.g.1249.6		8
5.4	even	2		3750.2.a.p.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3750.2.a.n.1.3	✓	4	1.1	even	1	trivial
3750.2.a.p.1.2	yes	4	5.4	even	2
3750.2.c.g.1249.3		8	5.2	odd	4
3750.2.c.g.1249.6		8	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3750.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +0.511170 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.29456 q^{11} +1.00000 q^{12} +2.65418 q^{13} -0.511170 q^{14} +1.00000 q^{16} +5.57433 q^{17} -1.00000 q^{18} +0.374409 q^{19} +0.511170 q^{21} -3.29456 q^{22} +5.75638 q^{23} -1.00000 q^{24} -2.65418 q^{26} +1.00000 q^{27} +0.511170 q^{28} -3.89025 q^{29} -1.80008 q^{31} -1.00000 q^{32} +3.29456 q^{33} -5.57433 q^{34} +1.00000 q^{36} +10.3577 q^{37} -0.374409 q^{38} +2.65418 q^{39} +2.33728 q^{41} -0.511170 q^{42} -5.87802 q^{43} +3.29456 q^{44} -5.75638 q^{46} -8.19959 q^{47} +1.00000 q^{48} -6.73870 q^{49} +5.57433 q^{51} +2.65418 q^{52} +0.518727 q^{53} -1.00000 q^{54} -0.511170 q^{56} +0.374409 q^{57} +3.89025 q^{58} -7.62993 q^{59} +2.90345 q^{61} +1.80008 q^{62} +0.511170 q^{63} +1.00000 q^{64} -3.29456 q^{66} -1.78806 q^{67} +5.57433 q^{68} +5.75638 q^{69} +7.27786 q^{71} -1.00000 q^{72} -1.40995 q^{73} -10.3577 q^{74} +0.374409 q^{76} +1.68408 q^{77} -2.65418 q^{78} -12.1395 q^{79} +1.00000 q^{81} -2.33728 q^{82} +3.09306 q^{83} +0.511170 q^{84} +5.87802 q^{86} -3.89025 q^{87} -3.29456 q^{88} +3.42567 q^{89} +1.35674 q^{91} +5.75638 q^{92} -1.80008 q^{93} +8.19959 q^{94} -1.00000 q^{96} -15.1483 q^{97} +6.73870 q^{98} +3.29456 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} + 4 q^{7} - 4 q^{8} + 4 q^{9} + 4 q^{12} - 2 q^{13} - 4 q^{14} + 4 q^{16} + 9 q^{17} - 4 q^{18} - 11 q^{19} + 4 q^{21} + 15 q^{23} - 4 q^{24} + 2 q^{26} + 4 q^{27}+ \cdots + 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 + 4 * q^7 - 4 * q^8 + 4 * q^9 + 4 * q^12 - 2 * q^13 - 4 * q^14 + 4 * q^16 + 9 * q^17 - 4 * q^18 - 11 * q^19 + 4 * q^21 + 15 * q^23 - 4 * q^24 + 2 * q^26 + 4 * q^27 + 4 * q^28 + 6 * q^29 - 8 * q^31 - 4 * q^32 - 9 * q^34 + 4 * q^36 + 13 * q^37 + 11 * q^38 - 2 * q^39 + 6 * q^41 - 4 * q^42 + 4 * q^43 - 15 * q^46 + 15 * q^47 + 4 * q^48 - 4 * q^49 + 9 * q^51 - 2 * q^52 + 21 * q^53 - 4 * q^54 - 4 * q^56 - 11 * q^57 - 6 * q^58 + 15 * q^59 - 5 * q^61 + 8 * q^62 + 4 * q^63 + 4 * q^64 + 19 * q^67 + 9 * q^68 + 15 * q^69 - 4 * q^72 - 2 * q^73 - 13 * q^74 - 11 * q^76 + 15 * q^77 + 2 * q^78 - 23 * q^79 + 4 * q^81 - 6 * q^82 + 18 * q^83 + 4 * q^84 - 4 * q^86 + 6 * q^87 + 27 * q^89 - 22 * q^91 + 15 * q^92 - 8 * q^93 - 15 * q^94 - 4 * q^96 + 13 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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