LMFDB - Embedded newform 3750.2.a.r.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3750,2,Mod(1,3750)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3750.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [4,4,-4,4,0,-4,6,4,4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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:	$\Z[a_1, \ldots, a_{11}]$
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)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.12920 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.05849 q^{11} -1.00000 q^{12} +5.89025 q^{13} +2.12920 q^{14} +1.00000 q^{16} -4.35772 q^{17} +1.00000 q^{18} +2.84943 q^{19} -2.12920 q^{21} -1.05849 q^{22} -0.467465 q^{23} -1.00000 q^{24} +5.89025 q^{26} -1.00000 q^{27} +2.12920 q^{28} -2.58189 q^{29} +8.20715 q^{31} +1.00000 q^{32} +1.05849 q^{33} -4.35772 q^{34} +1.00000 q^{36} +5.21470 q^{37} +2.84943 q^{38} -5.89025 q^{39} +5.13485 q^{41} -2.12920 q^{42} -6.49606 q^{43} -1.05849 q^{44} -0.467465 q^{46} +6.24520 q^{47} -1.00000 q^{48} -2.46649 q^{49} +4.35772 q^{51} +5.89025 q^{52} -7.32315 q^{53} -1.00000 q^{54} +2.12920 q^{56} -2.84943 q^{57} -2.58189 q^{58} -8.39228 q^{59} +7.13387 q^{61} +8.20715 q^{62} +2.12920 q^{63} +1.00000 q^{64} +1.05849 q^{66} +11.9579 q^{67} -4.35772 q^{68} +0.467465 q^{69} -14.6447 q^{71} +1.00000 q^{72} +12.1186 q^{73} +5.21470 q^{74} +2.84943 q^{76} -2.25374 q^{77} -5.89025 q^{78} -1.31999 q^{79} +1.00000 q^{81} +5.13485 q^{82} +12.9727 q^{83} -2.12920 q^{84} -6.49606 q^{86} +2.58189 q^{87} -1.05849 q^{88} -13.5287 q^{89} +12.5415 q^{91} -0.467465 q^{92} -8.20715 q^{93} +6.24520 q^{94} -1.00000 q^{96} -7.51742 q^{97} -2.46649 q^{98} -1.05849 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{6} + 6 q^{7} + 4 q^{8} + 4 q^{9} - 4 q^{12} + 2 q^{13} + 6 q^{14} + 4 q^{16} + 11 q^{17} + 4 q^{18} + 9 q^{19} - 6 q^{21} + 5 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 + 6 * q^7 + 4 * q^8 + 4 * q^9 - 4 * q^12 + 2 * q^13 + 6 * q^14 + 4 * q^16 + 11 * q^17 + 4 * q^18 + 9 * q^19 - 6 * q^21 + 5 * q^23 - 4 * q^24 + 2 * q^26 - 4 * q^27 + 6 * q^28 - 14 * q^29 + 2 * q^31 + 4 * q^32 + 11 * q^34 + 4 * q^36 + 7 * q^37 + 9 * q^38 - 2 * q^39 + 6 * q^41 - 6 * q^42 + 6 * q^43 + 5 * q^46 + 15 * q^47 - 4 * q^48 - 4 * q^49 - 11 * q^51 + 2 * q^52 + 9 * q^53 - 4 * q^54 + 6 * q^56 - 9 * q^57 - 14 * q^58 - 15 * q^59 + 15 * q^61 + 2 * q^62 + 6 * q^63 + 4 * q^64 + 21 * q^67 + 11 * q^68 - 5 * q^69 + 4 * q^72 + 22 * q^73 + 7 * q^74 + 9 * q^76 - 15 * q^77 - 2 * q^78 - 3 * q^79 + 4 * q^81 + 6 * q^82 + 12 * q^83 - 6 * q^84 + 6 * q^86 + 14 * q^87 - 3 * q^89 - 12 * q^91 + 5 * q^92 - 2 * q^93 + 15 * q^94 - 4 * q^96 - 3 * 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
$7$	2.12920	0.804764	0.402382	−	0.915472i	$-0.368182\pi$
$7$	2.12920	0.804764	0.402382	+	0.915472i	$0.368182\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.05849	−0.319146	−0.159573	−	0.987186i	$-0.551012\pi$
$11$	−1.05849	−0.319146	−0.159573	+	0.987186i	$0.551012\pi$
$12$	−1.00000	−0.288675
$13$	5.89025	1.63366	0.816831	−	0.576877i	$-0.195729\pi$
$13$	5.89025	1.63366	0.816831	+	0.576877i	$0.195729\pi$
$14$	2.12920	0.569054
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.35772	−1.05690	−0.528451	−	0.848964i	$-0.677227\pi$
$17$	−4.35772	−1.05690	−0.528451	+	0.848964i	$0.677227\pi$
$18$	1.00000	0.235702
$19$	2.84943	0.653704	0.326852	−	0.945075i	$-0.394012\pi$
$19$	2.84943	0.653704	0.326852	+	0.945075i	$0.394012\pi$
$20$	0	0
$21$	−2.12920	−0.464630
$22$	−1.05849	−0.225670
$23$	−0.467465	−0.0974733	−0.0487366	−	0.998812i	$-0.515520\pi$
$23$	−0.467465	−0.0974733	−0.0487366	+	0.998812i	$0.515520\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	5.89025	1.15517
$27$	−1.00000	−0.192450
$28$	2.12920	0.402382
$29$	−2.58189	−0.479444	−0.239722	−	0.970842i	$-0.577056\pi$
$29$	−2.58189	−0.479444	−0.239722	+	0.970842i	$0.577056\pi$
$30$	0	0
$31$	8.20715	1.47405	0.737024	−	0.675867i	$-0.236231\pi$
$31$	8.20715	1.47405	0.737024	+	0.675867i	$0.236231\pi$
$32$	1.00000	0.176777
$33$	1.05849	0.184259
$34$	−4.35772	−0.747342
$35$	0	0
$36$	1.00000	0.166667
$37$	5.21470	0.857292	0.428646	−	0.903473i	$-0.358991\pi$
$37$	5.21470	0.857292	0.428646	+	0.903473i	$0.358991\pi$
$38$	2.84943	0.462239
$39$	−5.89025	−0.943195
$40$	0	0
$41$	5.13485	0.801929	0.400965	−	0.916094i	$-0.368675\pi$
$41$	5.13485	0.801929	0.400965	+	0.916094i	$0.368675\pi$
$42$	−2.12920	−0.328543
$43$	−6.49606	−0.990639	−0.495320	−	0.868711i	$-0.664949\pi$
$43$	−6.49606	−0.990639	−0.495320	+	0.868711i	$0.664949\pi$
$44$	−1.05849	−0.159573
$45$	0	0
$46$	−0.467465	−0.0689240
$47$	6.24520	0.910957	0.455478	−	0.890247i	$-0.349468\pi$
$47$	6.24520	0.910957	0.455478	+	0.890247i	$0.349468\pi$
$48$	−1.00000	−0.144338
$49$	−2.46649	−0.352356
$50$	0	0
$51$	4.35772	0.610202
$52$	5.89025	0.816831
$53$	−7.32315	−1.00591	−0.502956	−	0.864312i	$-0.667754\pi$
$53$	−7.32315	−1.00591	−0.502956	+	0.864312i	$0.667754\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	2.12920	0.284527
$57$	−2.84943	−0.377416
$58$	−2.58189	−0.339018
$59$	−8.39228	−1.09258	−0.546291	−	0.837595i	$-0.683961\pi$
$59$	−8.39228	−1.09258	−0.546291	+	0.837595i	$0.683961\pi$
$60$	0	0
$61$	7.13387	0.913399	0.456700	−	0.889621i	$-0.349031\pi$
$61$	7.13387	0.913399	0.456700	+	0.889621i	$0.349031\pi$
$62$	8.20715	1.04231
$63$	2.12920	0.268255
$64$	1.00000	0.125000
$65$	0	0
$66$	1.05849	0.130291
$67$	11.9579	1.46089	0.730443	−	0.682973i	$-0.239313\pi$
$67$	11.9579	1.46089	0.730443	+	0.682973i	$0.239313\pi$
$68$	−4.35772	−0.528451
$69$	0.467465	0.0562762
$70$	0	0
$71$	−14.6447	−1.73801	−0.869004	−	0.494805i	$-0.835239\pi$
$71$	−14.6447	−1.73801	−0.869004	+	0.494805i	$0.835239\pi$
$72$	1.00000	0.117851
$73$	12.1186	1.41837	0.709185	−	0.705023i	$-0.249063\pi$
$73$	12.1186	1.41837	0.709185	+	0.705023i	$0.249063\pi$
$74$	5.21470	0.606197
$75$	0	0
$76$	2.84943	0.326852
$77$	−2.25374	−0.256837
$78$	−5.89025	−0.666939
$79$	−1.31999	−0.148510	−0.0742551	−	0.997239i	$-0.523658\pi$
$79$	−1.31999	−0.148510	−0.0742551	+	0.997239i	$0.523658\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	5.13485	0.567049
$83$	12.9727	1.42393	0.711967	−	0.702213i	$-0.247804\pi$
$83$	12.9727	1.42393	0.711967	+	0.702213i	$0.247804\pi$
$84$	−2.12920	−0.232315
$85$	0	0
$86$	−6.49606	−0.700488
$87$	2.58189	0.276807
$88$	−1.05849	−0.112835
$89$	−13.5287	−1.43404	−0.717020	−	0.697052i	$-0.754495\pi$
$89$	−13.5287	−1.43404	−0.717020	+	0.697052i	$0.754495\pi$
$90$	0	0
$91$	12.5415	1.31471
$92$	−0.467465	−0.0487366
$93$	−8.20715	−0.851041
$94$	6.24520	0.644144
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−7.51742	−0.763278	−0.381639	−	0.924311i	$-0.624640\pi$
$97$	−7.51742	−0.763278	−0.381639	+	0.924311i	$0.624640\pi$
$98$	−2.46649	−0.249153
$99$	−1.05849	−0.106382
$100$	0	0
$101$	5.95341	0.592386	0.296193	−	0.955128i	$-0.404283\pi$
$101$	5.95341	0.592386	0.296193	+	0.955128i	$0.404283\pi$
$102$	4.35772	0.431478
$103$	1.29647	0.127745	0.0638723	−	0.997958i	$-0.479655\pi$
$103$	1.29647	0.127745	0.0638723	+	0.997958i	$0.479655\pi$
$104$	5.89025	0.577587
$105$	0	0
$106$	−7.32315	−0.711287
$107$	13.0327	1.25991	0.629957	−	0.776630i	$-0.283072\pi$
$107$	13.0327	1.25991	0.629957	+	0.776630i	$0.283072\pi$
$108$	−1.00000	−0.0962250
$109$	−12.7526	−1.22147	−0.610737	−	0.791834i	$-0.709127\pi$
$109$	−12.7526	−1.22147	−0.610737	+	0.791834i	$0.709127\pi$
$110$	0	0
$111$	−5.21470	−0.494958
$112$	2.12920	0.201191
$113$	18.7173	1.76078	0.880390	−	0.474251i	$-0.157281\pi$
$113$	18.7173	1.76078	0.880390	+	0.474251i	$0.157281\pi$
$114$	−2.84943	−0.266874
$115$	0	0
$116$	−2.58189	−0.239722
$117$	5.89025	0.544554
$118$	−8.39228	−0.772572
$119$	−9.27847	−0.850556
$120$	0	0
$121$	−9.87960	−0.898146
$122$	7.13387	0.645871
$123$	−5.13485	−0.462994
$124$	8.20715	0.737024
$125$	0	0
$126$	2.12920	0.189685
$127$	8.43501	0.748486	0.374243	−	0.927331i	$-0.377903\pi$
$127$	8.43501	0.748486	0.374243	+	0.927331i	$0.377903\pi$
$128$	1.00000	0.0883883
$129$	6.49606	0.571946
$130$	0	0
$131$	−6.41653	−0.560615	−0.280308	−	0.959910i	$-0.590436\pi$
$131$	−6.41653	−0.560615	−0.280308	+	0.959910i	$0.590436\pi$
$132$	1.05849	0.0921296
$133$	6.06702	0.526078
$134$	11.9579	1.03300
$135$	0	0
$136$	−4.35772	−0.373671
$137$	−2.49508	−0.213169	−0.106585	−	0.994304i	$-0.533991\pi$
$137$	−2.49508	−0.213169	−0.106585	+	0.994304i	$0.533991\pi$
$138$	0.467465	0.0397933
$139$	1.78818	0.151672	0.0758358	−	0.997120i	$-0.475838\pi$
$139$	1.78818	0.151672	0.0758358	+	0.997120i	$0.475838\pi$
$140$	0	0
$141$	−6.24520	−0.525941
$142$	−14.6447	−1.22896
$143$	−6.23476	−0.521377
$144$	1.00000	0.0833333
$145$	0	0
$146$	12.1186	1.00294
$147$	2.46649	0.203433
$148$	5.21470	0.428646
$149$	10.7440	0.880185	0.440092	−	0.897953i	$-0.354946\pi$
$149$	10.7440	0.880185	0.440092	+	0.897953i	$0.354946\pi$
$150$	0	0
$151$	16.7444	1.36264	0.681318	−	0.731988i	$-0.261407\pi$
$151$	16.7444	1.36264	0.681318	+	0.731988i	$0.261407\pi$
$152$	2.84943	0.231119
$153$	−4.35772	−0.352300
$154$	−2.25374	−0.181611
$155$	0	0
$156$	−5.89025	−0.471597
$157$	4.04586	0.322895	0.161447	−	0.986881i	$-0.448384\pi$
$157$	4.04586	0.322895	0.161447	+	0.986881i	$0.448384\pi$
$158$	−1.31999	−0.105013
$159$	7.32315	0.580763
$160$	0	0
$161$	−0.995330	−0.0784430
$162$	1.00000	0.0785674
$163$	−12.7786	−1.00090	−0.500448	−	0.865766i	$-0.666831\pi$
$163$	−12.7786	−1.00090	−0.500448	+	0.865766i	$0.666831\pi$
$164$	5.13485	0.400965
$165$	0	0
$166$	12.9727	1.00687
$167$	22.9012	1.77215	0.886073	−	0.463546i	$-0.153423\pi$
$167$	22.9012	1.77215	0.886073	+	0.463546i	$0.153423\pi$
$168$	−2.12920	−0.164272
$169$	21.6950	1.66885
$170$	0	0
$171$	2.84943	0.217901
$172$	−6.49606	−0.495320
$173$	21.1514	1.60811	0.804056	−	0.594553i	$-0.202671\pi$
$173$	21.1514	1.60811	0.804056	+	0.594553i	$0.202671\pi$
$174$	2.58189	0.195732
$175$	0	0
$176$	−1.05849	−0.0797866
$177$	8.39228	0.630803
$178$	−13.5287	−1.01402
$179$	−10.0795	−0.753379	−0.376689	−	0.926340i	$-0.622938\pi$
$179$	−10.0795	−0.753379	−0.376689	+	0.926340i	$0.622938\pi$
$180$	0	0
$181$	−3.30520	−0.245674	−0.122837	−	0.992427i	$-0.539199\pi$
$181$	−3.30520	−0.245674	−0.122837	+	0.992427i	$0.539199\pi$
$182$	12.5415	0.929641
$183$	−7.13387	−0.527351
$184$	−0.467465	−0.0344620
$185$	0	0
$186$	−8.20715	−0.601777
$187$	4.61259	0.337306
$188$	6.24520	0.455478
$189$	−2.12920	−0.154877
$190$	0	0
$191$	9.94780	0.719798	0.359899	−	0.932991i	$-0.382811\pi$
$191$	9.94780	0.719798	0.359899	+	0.932991i	$0.382811\pi$
$192$	−1.00000	−0.0721688
$193$	−19.8949	−1.43207	−0.716034	−	0.698066i	$-0.754044\pi$
$193$	−19.8949	−1.43207	−0.716034	+	0.698066i	$0.754044\pi$
$194$	−7.51742	−0.539719
$195$	0	0
$196$	−2.46649	−0.176178
$197$	27.3096	1.94573	0.972865	−	0.231375i	$-0.0743224\pi$
$197$	27.3096	1.94573	0.972865	+	0.231375i	$0.0743224\pi$
$198$	−1.05849	−0.0752235
$199$	1.70256	0.120691	0.0603455	−	0.998178i	$-0.480780\pi$
$199$	1.70256	0.120691	0.0603455	+	0.998178i	$0.480780\pi$
$200$	0	0
$201$	−11.9579	−0.843443
$202$	5.95341	0.418880
$203$	−5.49736	−0.385839
$204$	4.35772	0.305101
$205$	0	0
$206$	1.29647	0.0903291
$207$	−0.467465	−0.0324911
$208$	5.89025	0.408415
$209$	−3.01609	−0.208627
$210$	0	0
$211$	19.8374	1.36566	0.682832	−	0.730576i	$-0.260748\pi$
$211$	19.8374	1.36566	0.682832	+	0.730576i	$0.260748\pi$
$212$	−7.32315	−0.502956
$213$	14.6447	1.00344
$214$	13.0327	0.890894
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	17.4747	1.18626
$218$	−12.7526	−0.863712
$219$	−12.1186	−0.818896
$220$	0	0
$221$	−25.6680	−1.72662
$222$	−5.21470	−0.349988
$223$	11.5590	0.774049	0.387024	−	0.922070i	$-0.373503\pi$
$223$	11.5590	0.774049	0.387024	+	0.922070i	$0.373503\pi$
$224$	2.12920	0.142263
$225$	0	0
$226$	18.7173	1.24506
$227$	−17.3898	−1.15420	−0.577101	−	0.816673i	$-0.695816\pi$
$227$	−17.3898	−1.15420	−0.577101	+	0.816673i	$0.695816\pi$
$228$	−2.84943	−0.188708
$229$	−23.5935	−1.55910	−0.779550	−	0.626340i	$-0.784552\pi$
$229$	−23.5935	−1.55910	−0.779550	+	0.626340i	$0.784552\pi$
$230$	0	0
$231$	2.25374	0.148285
$232$	−2.58189	−0.169509
$233$	−10.6944	−0.700613	−0.350307	−	0.936635i	$-0.613923\pi$
$233$	−10.6944	−0.700613	−0.350307	+	0.936635i	$0.613923\pi$
$234$	5.89025	0.385058
$235$	0	0
$236$	−8.39228	−0.546291
$237$	1.31999	0.0857424
$238$	−9.27847	−0.601434
$239$	15.7067	1.01598	0.507991	−	0.861363i	$-0.330388\pi$
$239$	15.7067	1.01598	0.507991	+	0.861363i	$0.330388\pi$
$240$	0	0
$241$	22.5174	1.45047	0.725237	−	0.688499i	$-0.241730\pi$
$241$	22.5174	1.45047	0.725237	+	0.688499i	$0.241730\pi$
$242$	−9.87960	−0.635085
$243$	−1.00000	−0.0641500
$244$	7.13387	0.456700
$245$	0	0
$246$	−5.13485	−0.327386
$247$	16.7839	1.06793
$248$	8.20715	0.521154
$249$	−12.9727	−0.822109
$250$	0	0
$251$	4.11677	0.259848	0.129924	−	0.991524i	$-0.458527\pi$
$251$	4.11677	0.259848	0.129924	+	0.991524i	$0.458527\pi$
$252$	2.12920	0.134127
$253$	0.494807	0.0311082
$254$	8.43501	0.529260
$255$	0	0
$256$	1.00000	0.0625000
$257$	−22.7141	−1.41687	−0.708434	−	0.705777i	$-0.750598\pi$
$257$	−22.7141	−1.41687	−0.708434	+	0.705777i	$0.750598\pi$
$258$	6.49606	0.404427
$259$	11.1032	0.689917
$260$	0	0
$261$	−2.58189	−0.159815
$262$	−6.41653	−0.396415
$263$	12.5881	0.776218	0.388109	−	0.921614i	$-0.373128\pi$
$263$	12.5881	0.776218	0.388109	+	0.921614i	$0.373128\pi$
$264$	1.05849	0.0651455
$265$	0	0
$266$	6.06702	0.371993
$267$	13.5287	0.827944
$268$	11.9579	0.730443
$269$	−24.4467	−1.49054	−0.745271	−	0.666762i	$-0.767680\pi$
$269$	−24.4467	−1.49054	−0.745271	+	0.666762i	$0.767680\pi$
$270$	0	0
$271$	−13.8542	−0.841581	−0.420790	−	0.907158i	$-0.638247\pi$
$271$	−13.8542	−0.841581	−0.420790	+	0.907158i	$0.638247\pi$
$272$	−4.35772	−0.264225
$273$	−12.5415	−0.759049
$274$	−2.49508	−0.150733
$275$	0	0
$276$	0.467465	0.0281381
$277$	−14.7886	−0.888563	−0.444281	−	0.895887i	$-0.646541\pi$
$277$	−14.7886	−0.888563	−0.444281	+	0.895887i	$0.646541\pi$
$278$	1.78818	0.107248
$279$	8.20715	0.491349
$280$	0	0
$281$	−5.26150	−0.313875	−0.156937	−	0.987609i	$-0.550162\pi$
$281$	−5.26150	−0.313875	−0.156937	+	0.987609i	$0.550162\pi$
$282$	−6.24520	−0.371897
$283$	−30.1332	−1.79123	−0.895617	−	0.444826i	$-0.853265\pi$
$283$	−30.1332	−1.79123	−0.895617	+	0.444826i	$0.853265\pi$
$284$	−14.6447	−0.869004
$285$	0	0
$286$	−6.23476	−0.368669
$287$	10.9331	0.645363
$288$	1.00000	0.0589256
$289$	1.98968	0.117040
$290$	0	0
$291$	7.51742	0.440679
$292$	12.1186	0.709185
$293$	27.0382	1.57959	0.789794	−	0.613373i	$-0.210188\pi$
$293$	27.0382	1.57959	0.789794	+	0.613373i	$0.210188\pi$
$294$	2.46649	0.143849
$295$	0	0
$296$	5.21470	0.303099
$297$	1.05849	0.0614197
$298$	10.7440	0.622384
$299$	−2.75349	−0.159238
$300$	0	0
$301$	−13.8314	−0.797230
$302$	16.7444	0.963529
$303$	−5.95341	−0.342014
$304$	2.84943	0.163426
$305$	0	0
$306$	−4.35772	−0.249114
$307$	−12.1714	−0.694658	−0.347329	−	0.937743i	$-0.612911\pi$
$307$	−12.1714	−0.694658	−0.347329	+	0.937743i	$0.612911\pi$
$308$	−2.25374	−0.128419
$309$	−1.29647	−0.0737534
$310$	0	0
$311$	15.2313	0.863686	0.431843	−	0.901949i	$-0.357863\pi$
$311$	15.2313	0.863686	0.431843	+	0.901949i	$0.357863\pi$
$312$	−5.89025	−0.333470
$313$	−25.5390	−1.44355	−0.721776	−	0.692127i	$-0.756674\pi$
$313$	−25.5390	−1.44355	−0.721776	+	0.692127i	$0.756674\pi$
$314$	4.04586	0.228321
$315$	0	0
$316$	−1.31999	−0.0742551
$317$	−3.77603	−0.212083	−0.106042	−	0.994362i	$-0.533818\pi$
$317$	−3.77603	−0.212083	−0.106042	+	0.994362i	$0.533818\pi$
$318$	7.32315	0.410662
$319$	2.73290	0.153013
$320$	0	0
$321$	−13.0327	−0.727412
$322$	−0.995330	−0.0554675
$323$	−12.4170	−0.690901
$324$	1.00000	0.0555556
$325$	0	0
$326$	−12.7786	−0.707741
$327$	12.7526	0.705218
$328$	5.13485	0.283525
$329$	13.2973	0.733105
$330$	0	0
$331$	15.2466	0.838028	0.419014	−	0.907980i	$-0.362376\pi$
$331$	15.2466	0.838028	0.419014	+	0.907980i	$0.362376\pi$
$332$	12.9727	0.711967
$333$	5.21470	0.285764
$334$	22.9012	1.25310
$335$	0	0
$336$	−2.12920	−0.116158
$337$	10.4906	0.571460	0.285730	−	0.958310i	$-0.407764\pi$
$337$	10.4906	0.571460	0.285730	+	0.958310i	$0.407764\pi$
$338$	21.6950	1.18005
$339$	−18.7173	−1.01659
$340$	0	0
$341$	−8.68717	−0.470437
$342$	2.84943	0.154080
$343$	−20.1561	−1.08833
$344$	−6.49606	−0.350244
$345$	0	0
$346$	21.1514	1.13711
$347$	−12.4717	−0.669518	−0.334759	−	0.942304i	$-0.608655\pi$
$347$	−12.4717	−0.669518	−0.334759	+	0.942304i	$0.608655\pi$
$348$	2.58189	0.138404
$349$	−23.9316	−1.28103	−0.640516	−	0.767945i	$-0.721279\pi$
$349$	−23.9316	−1.28103	−0.640516	+	0.767945i	$0.721279\pi$
$350$	0	0
$351$	−5.89025	−0.314398
$352$	−1.05849	−0.0564176
$353$	−23.4753	−1.24947	−0.624733	−	0.780838i	$-0.714792\pi$
$353$	−23.4753	−1.24947	−0.624733	+	0.780838i	$0.714792\pi$
$354$	8.39228	0.446045
$355$	0	0
$356$	−13.5287	−0.717020
$357$	9.27847	0.491069
$358$	−10.0795	−0.532719
$359$	31.5610	1.66573	0.832864	−	0.553478i	$-0.186700\pi$
$359$	31.5610	1.66573	0.832864	+	0.553478i	$0.186700\pi$
$360$	0	0
$361$	−10.8807	−0.572671
$362$	−3.30520	−0.173718
$363$	9.87960	0.518545
$364$	12.5415	0.657356
$365$	0	0
$366$	−7.13387	−0.372894
$367$	2.82202	0.147308	0.0736542	−	0.997284i	$-0.476534\pi$
$367$	2.82202	0.147308	0.0736542	+	0.997284i	$0.476534\pi$
$368$	−0.467465	−0.0243683
$369$	5.13485	0.267310
$370$	0	0
$371$	−15.5925	−0.809521
$372$	−8.20715	−0.425521
$373$	21.1247	1.09379	0.546897	−	0.837200i	$-0.315809\pi$
$373$	21.1247	1.09379	0.546897	+	0.837200i	$0.315809\pi$
$374$	4.61259	0.238511
$375$	0	0
$376$	6.24520	0.322072
$377$	−15.2080	−0.783249
$378$	−2.12920	−0.109514
$379$	−21.2311	−1.09057	−0.545285	−	0.838250i	$-0.683579\pi$
$379$	−21.2311	−1.09057	−0.545285	+	0.838250i	$0.683579\pi$
$380$	0	0
$381$	−8.43501	−0.432139
$382$	9.94780	0.508974
$383$	27.3115	1.39555	0.697776	−	0.716316i	$-0.254173\pi$
$383$	27.3115	1.39555	0.697776	+	0.716316i	$0.254173\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−19.8949	−1.01262
$387$	−6.49606	−0.330213
$388$	−7.51742	−0.381639
$389$	7.84655	0.397836	0.198918	−	0.980016i	$-0.436257\pi$
$389$	7.84655	0.397836	0.198918	+	0.980016i	$0.436257\pi$
$390$	0	0
$391$	2.03708	0.103020
$392$	−2.46649	−0.124577
$393$	6.41653	0.323671
$394$	27.3096	1.37584
$395$	0	0
$396$	−1.05849	−0.0531910
$397$	20.5604	1.03190	0.515948	−	0.856620i	$-0.327440\pi$
$397$	20.5604	1.03190	0.515948	+	0.856620i	$0.327440\pi$
$398$	1.70256	0.0853415
$399$	−6.06702	−0.303731
$400$	0	0
$401$	2.38184	0.118943	0.0594717	−	0.998230i	$-0.481058\pi$
$401$	2.38184	0.118943	0.0594717	+	0.998230i	$0.481058\pi$
$402$	−11.9579	−0.596405
$403$	48.3421	2.40809
$404$	5.95341	0.296193
$405$	0	0
$406$	−5.49736	−0.272830
$407$	−5.51970	−0.273602
$408$	4.35772	0.215739
$409$	−14.1166	−0.698023	−0.349012	−	0.937118i	$-0.613483\pi$
$409$	−14.1166	−0.698023	−0.349012	+	0.937118i	$0.613483\pi$
$410$	0	0
$411$	2.49508	0.123073
$412$	1.29647	0.0638723
$413$	−17.8689	−0.879270
$414$	−0.467465	−0.0229747
$415$	0	0
$416$	5.89025	0.288793
$417$	−1.78818	−0.0875677
$418$	−3.01609	−0.147522
$419$	−23.3766	−1.14202	−0.571012	−	0.820942i	$-0.693449\pi$
$419$	−23.3766	−1.14202	−0.571012	+	0.820942i	$0.693449\pi$
$420$	0	0
$421$	21.4696	1.04636	0.523181	−	0.852221i	$-0.324745\pi$
$421$	21.4696	1.04636	0.523181	+	0.852221i	$0.324745\pi$
$422$	19.8374	0.965670
$423$	6.24520	0.303652
$424$	−7.32315	−0.355643
$425$	0	0
$426$	14.6447	0.709539
$427$	15.1895	0.735070
$428$	13.0327	0.629957
$429$	6.23476	0.301017
$430$	0	0
$431$	−35.4598	−1.70804	−0.854019	−	0.520242i	$-0.825842\pi$
$431$	−35.4598	−1.70804	−0.854019	+	0.520242i	$0.825842\pi$
$432$	−1.00000	−0.0481125
$433$	−33.1379	−1.59251	−0.796253	−	0.604964i	$-0.793187\pi$
$433$	−33.1379	−1.59251	−0.796253	+	0.604964i	$0.793187\pi$
$434$	17.4747	0.838812
$435$	0	0
$436$	−12.7526	−0.610737
$437$	−1.33201	−0.0637187
$438$	−12.1186	−0.579047
$439$	−7.69473	−0.367249	−0.183625	−	0.982996i	$-0.558783\pi$
$439$	−7.69473	−0.367249	−0.183625	+	0.982996i	$0.558783\pi$
$440$	0	0
$441$	−2.46649	−0.117452
$442$	−25.6680	−1.22090
$443$	−8.12762	−0.386155	−0.193078	−	0.981184i	$-0.561847\pi$
$443$	−8.12762	−0.386155	−0.193078	+	0.981184i	$0.561847\pi$
$444$	−5.21470	−0.247479
$445$	0	0
$446$	11.5590	0.547335
$447$	−10.7440	−0.508175
$448$	2.12920	0.100595
$449$	1.49021	0.0703276	0.0351638	−	0.999382i	$-0.488805\pi$
$449$	1.49021	0.0703276	0.0351638	+	0.999382i	$0.488805\pi$
$450$	0	0
$451$	−5.43518	−0.255933
$452$	18.7173	0.880390
$453$	−16.7444	−0.786718
$454$	−17.3898	−0.816144
$455$	0	0
$456$	−2.84943	−0.133437
$457$	−7.94841	−0.371811	−0.185905	−	0.982568i	$-0.559522\pi$
$457$	−7.94841	−0.371811	−0.185905	+	0.982568i	$0.559522\pi$
$458$	−23.5935	−1.10245
$459$	4.35772	0.203401
$460$	0	0
$461$	−29.7969	−1.38778	−0.693889	−	0.720082i	$-0.744104\pi$
$461$	−29.7969	−1.38778	−0.693889	+	0.720082i	$0.744104\pi$
$462$	2.25374	0.104853
$463$	−12.4986	−0.580860	−0.290430	−	0.956896i	$-0.593798\pi$
$463$	−12.4986	−0.580860	−0.290430	+	0.956896i	$0.593798\pi$
$464$	−2.58189	−0.119861
$465$	0	0
$466$	−10.6944	−0.495408
$467$	−23.3113	−1.07872	−0.539358	−	0.842077i	$-0.681333\pi$
$467$	−23.3113	−1.07872	−0.539358	+	0.842077i	$0.681333\pi$
$468$	5.89025	0.272277
$469$	25.4608	1.17567
$470$	0	0
$471$	−4.04586	−0.186423
$472$	−8.39228	−0.386286
$473$	6.87600	0.316159
$474$	1.31999	0.0606290
$475$	0	0
$476$	−9.27847	−0.425278
$477$	−7.32315	−0.335304
$478$	15.7067	0.718407
$479$	36.4911	1.66732	0.833661	−	0.552276i	$-0.186241\pi$
$479$	36.4911	1.66732	0.833661	+	0.552276i	$0.186241\pi$
$480$	0	0
$481$	30.7159	1.40052
$482$	22.5174	1.02564
$483$	0.995330	0.0452891
$484$	−9.87960	−0.449073
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	8.61685	0.390467	0.195233	−	0.980757i	$-0.437454\pi$
$487$	8.61685	0.390467	0.195233	+	0.980757i	$0.437454\pi$
$488$	7.13387	0.322935
$489$	12.7786	0.577868
$490$	0	0
$491$	−27.4118	−1.23708	−0.618539	−	0.785754i	$-0.712275\pi$
$491$	−27.4118	−1.23708	−0.618539	+	0.785754i	$0.712275\pi$
$492$	−5.13485	−0.231497
$493$	11.2511	0.506725
$494$	16.7839	0.755142
$495$	0	0
$496$	8.20715	0.368512
$497$	−31.1816	−1.39869
$498$	−12.9727	−0.581319
$499$	−9.07912	−0.406437	−0.203219	−	0.979133i	$-0.565140\pi$
$499$	−9.07912	−0.406437	−0.203219	+	0.979133i	$0.565140\pi$
$500$	0	0
$501$	−22.9012	−1.02315
$502$	4.11677	0.183741
$503$	15.2313	0.679131	0.339565	−	0.940582i	$-0.389720\pi$
$503$	15.2313	0.679131	0.339565	+	0.940582i	$0.389720\pi$
$504$	2.12920	0.0948423
$505$	0	0
$506$	0.494807	0.0219968
$507$	−21.6950	−0.963511
$508$	8.43501	0.374243
$509$	−30.6693	−1.35939	−0.679696	−	0.733494i	$-0.737888\pi$
$509$	−30.6693	−1.35939	−0.679696	+	0.733494i	$0.737888\pi$
$510$	0	0
$511$	25.8029	1.14145
$512$	1.00000	0.0441942
$513$	−2.84943	−0.125805
$514$	−22.7141	−1.00188
$515$	0	0
$516$	6.49606	0.285973
$517$	−6.61048	−0.290728
$518$	11.1032	0.487845
$519$	−21.1514	−0.928444
$520$	0	0
$521$	−18.5375	−0.812141	−0.406070	−	0.913842i	$-0.633101\pi$
$521$	−18.5375	−0.812141	−0.406070	+	0.913842i	$0.633101\pi$
$522$	−2.58189	−0.113006
$523$	20.4297	0.893328	0.446664	−	0.894702i	$-0.352612\pi$
$523$	20.4297	0.893328	0.446664	+	0.894702i	$0.352612\pi$
$524$	−6.41653	−0.280308
$525$	0	0
$526$	12.5881	0.548869
$527$	−35.7644	−1.55792
$528$	1.05849	0.0460648
$529$	−22.7815	−0.990499
$530$	0	0
$531$	−8.39228	−0.364194
$532$	6.06702	0.263039
$533$	30.2456	1.31008
$534$	13.5287	0.585445
$535$	0	0
$536$	11.9579	0.516502
$537$	10.0795	0.434963
$538$	−24.4467	−1.05397
$539$	2.61075	0.112453
$540$	0	0
$541$	−18.3399	−0.788493	−0.394247	−	0.919005i	$-0.628994\pi$
$541$	−18.3399	−0.788493	−0.394247	+	0.919005i	$0.628994\pi$
$542$	−13.8542	−0.595088
$543$	3.30520	0.141840
$544$	−4.35772	−0.186836
$545$	0	0
$546$	−12.5415	−0.536729
$547$	−5.73208	−0.245086	−0.122543	−	0.992463i	$-0.539105\pi$
$547$	−5.73208	−0.245086	−0.122543	+	0.992463i	$0.539105\pi$
$548$	−2.49508	−0.106585
$549$	7.13387	0.304466
$550$	0	0
$551$	−7.35691	−0.313415
$552$	0.467465	0.0198967
$553$	−2.81052	−0.119516
$554$	−14.7886	−0.628309
$555$	0	0
$556$	1.78818	0.0758358
$557$	−21.1606	−0.896602	−0.448301	−	0.893883i	$-0.647971\pi$
$557$	−21.1606	−0.896602	−0.448301	+	0.893883i	$0.647971\pi$
$558$	8.20715	0.347436
$559$	−38.2634	−1.61837
$560$	0	0
$561$	−4.61259	−0.194744
$562$	−5.26150	−0.221943
$563$	−21.6721	−0.913370	−0.456685	−	0.889628i	$-0.650963\pi$
$563$	−21.6721	−0.913370	−0.456685	+	0.889628i	$0.650963\pi$
$564$	−6.24520	−0.262971
$565$	0	0
$566$	−30.1332	−1.26659
$567$	2.12920	0.0894182
$568$	−14.6447	−0.614479
$569$	−7.26491	−0.304561	−0.152280	−	0.988337i	$-0.548662\pi$
$569$	−7.26491	−0.304561	−0.152280	+	0.988337i	$0.548662\pi$
$570$	0	0
$571$	37.4515	1.56730	0.783648	−	0.621205i	$-0.213357\pi$
$571$	37.4515	1.56730	0.783648	+	0.621205i	$0.213357\pi$
$572$	−6.23476	−0.260688
$573$	−9.94780	−0.415575
$574$	10.9331	0.456341
$575$	0	0
$576$	1.00000	0.0416667
$577$	7.91413	0.329469	0.164735	−	0.986338i	$-0.447323\pi$
$577$	7.91413	0.329469	0.164735	+	0.986338i	$0.447323\pi$
$578$	1.98968	0.0827599
$579$	19.8949	0.826805
$580$	0	0
$581$	27.6214	1.14593
$582$	7.51742	0.311607
$583$	7.75147	0.321033
$584$	12.1186	0.501469
$585$	0	0
$586$	27.0382	1.11694
$587$	−10.9336	−0.451276	−0.225638	−	0.974211i	$-0.572447\pi$
$587$	−10.9336	−0.451276	−0.225638	+	0.974211i	$0.572447\pi$
$588$	2.46649	0.101716
$589$	23.3857	0.963591
$590$	0	0
$591$	−27.3096	−1.12337
$592$	5.21470	0.214323
$593$	40.1889	1.65036	0.825179	−	0.564871i	$-0.191074\pi$
$593$	40.1889	1.65036	0.825179	+	0.564871i	$0.191074\pi$
$594$	1.05849	0.0434303
$595$	0	0
$596$	10.7440	0.440092
$597$	−1.70256	−0.0696810
$598$	−2.75349	−0.112599
$599$	−34.8233	−1.42284	−0.711422	−	0.702765i	$-0.751948\pi$
$599$	−34.8233	−1.42284	−0.711422	+	0.702765i	$0.751948\pi$
$600$	0	0
$601$	−35.1280	−1.43290	−0.716450	−	0.697638i	$-0.754234\pi$
$601$	−35.1280	−1.43290	−0.716450	+	0.697638i	$0.754234\pi$
$602$	−13.8314	−0.563727
$603$	11.9579	0.486962
$604$	16.7444	0.681318
$605$	0	0
$606$	−5.95341	−0.241841
$607$	−20.8402	−0.845876	−0.422938	−	0.906159i	$-0.639001\pi$
$607$	−20.8402	−0.845876	−0.422938	+	0.906159i	$0.639001\pi$
$608$	2.84943	0.115560
$609$	5.49736	0.222764
$610$	0	0
$611$	36.7858	1.48819
$612$	−4.35772	−0.176150
$613$	−4.56401	−0.184339	−0.0921694	−	0.995743i	$-0.529380\pi$
$613$	−4.56401	−0.184339	−0.0921694	+	0.995743i	$0.529380\pi$
$614$	−12.1714	−0.491198
$615$	0	0
$616$	−2.25374	−0.0908057
$617$	20.2929	0.816962	0.408481	−	0.912767i	$-0.366059\pi$
$617$	20.2929	0.816962	0.408481	+	0.912767i	$0.366059\pi$
$618$	−1.29647	−0.0521515
$619$	−34.4978	−1.38658	−0.693292	−	0.720657i	$-0.743840\pi$
$619$	−34.4978	−1.38658	−0.693292	+	0.720657i	$0.743840\pi$
$620$	0	0
$621$	0.467465	0.0187587
$622$	15.2313	0.610718
$623$	−28.8054	−1.15406
$624$	−5.89025	−0.235799
$625$	0	0
$626$	−25.5390	−1.02074
$627$	3.01609	0.120451
$628$	4.04586	0.161447
$629$	−22.7242	−0.906073
$630$	0	0
$631$	20.8998	0.832009	0.416005	−	0.909362i	$-0.363430\pi$
$631$	20.8998	0.832009	0.416005	+	0.909362i	$0.363430\pi$
$632$	−1.31999	−0.0525063
$633$	−19.8374	−0.788466
$634$	−3.77603	−0.149965
$635$	0	0
$636$	7.32315	0.290382
$637$	−14.5282	−0.575630
$638$	2.73290	0.108196
$639$	−14.6447	−0.579336
$640$	0	0
$641$	−11.5946	−0.457960	−0.228980	−	0.973431i	$-0.573539\pi$
$641$	−11.5946	−0.457960	−0.228980	+	0.973431i	$0.573539\pi$
$642$	−13.0327	−0.514358
$643$	9.98993	0.393964	0.196982	−	0.980407i	$-0.436886\pi$
$643$	9.98993	0.393964	0.196982	+	0.980407i	$0.436886\pi$
$644$	−0.995330	−0.0392215
$645$	0	0
$646$	−12.4170	−0.488541
$647$	11.8474	0.465767	0.232884	−	0.972505i	$-0.425184\pi$
$647$	11.8474	0.465767	0.232884	+	0.972505i	$0.425184\pi$
$648$	1.00000	0.0392837
$649$	8.88313	0.348694
$650$	0	0
$651$	−17.4747	−0.684887
$652$	−12.7786	−0.500448
$653$	−9.60772	−0.375979	−0.187990	−	0.982171i	$-0.560197\pi$
$653$	−9.60772	−0.375979	−0.187990	+	0.982171i	$0.560197\pi$
$654$	12.7526	0.498664
$655$	0	0
$656$	5.13485	0.200482
$657$	12.1186	0.472790
$658$	13.2973	0.518383
$659$	15.5196	0.604557	0.302278	−	0.953220i	$-0.402253\pi$
$659$	15.5196	0.604557	0.302278	+	0.953220i	$0.402253\pi$
$660$	0	0
$661$	−3.68034	−0.143149	−0.0715744	−	0.997435i	$-0.522802\pi$
$661$	−3.68034	−0.143149	−0.0715744	+	0.997435i	$0.522802\pi$
$662$	15.2466	0.592576
$663$	25.6680	0.996864
$664$	12.9727	0.503437
$665$	0	0
$666$	5.21470	0.202066
$667$	1.20694	0.0467330
$668$	22.9012	0.886073
$669$	−11.5590	−0.446897
$670$	0	0
$671$	−7.55112	−0.291508
$672$	−2.12920	−0.0821358
$673$	−15.9336	−0.614196	−0.307098	−	0.951678i	$-0.599358\pi$
$673$	−15.9336	−0.614196	−0.307098	+	0.951678i	$0.599358\pi$
$674$	10.4906	0.404083
$675$	0	0
$676$	21.6950	0.834425
$677$	10.0691	0.386986	0.193493	−	0.981102i	$-0.438018\pi$
$677$	10.0691	0.386986	0.193493	+	0.981102i	$0.438018\pi$
$678$	−18.7173	−0.718835
$679$	−16.0061	−0.614259
$680$	0	0
$681$	17.3898	0.666378
$682$	−8.68717	−0.332649
$683$	4.75947	0.182116	0.0910579	−	0.995846i	$-0.470975\pi$
$683$	4.75947	0.182116	0.0910579	+	0.995846i	$0.470975\pi$
$684$	2.84943	0.108951
$685$	0	0
$686$	−20.1561	−0.769563
$687$	23.5935	0.900146
$688$	−6.49606	−0.247660
$689$	−43.1352	−1.64332
$690$	0	0
$691$	−20.9548	−0.797159	−0.398580	−	0.917134i	$-0.630497\pi$
$691$	−20.9548	−0.797159	−0.398580	+	0.917134i	$0.630497\pi$
$692$	21.1514	0.804056
$693$	−2.25374	−0.0856124
$694$	−12.4717	−0.473421
$695$	0	0
$696$	2.58189	0.0978661
$697$	−22.3762	−0.847560
$698$	−23.9316	−0.905826
$699$	10.6944	0.404499
$700$	0	0
$701$	−13.5605	−0.512172	−0.256086	−	0.966654i	$-0.582433\pi$
$701$	−13.5605	−0.512172	−0.256086	+	0.966654i	$0.582433\pi$
$702$	−5.89025	−0.222313
$703$	14.8589	0.560416
$704$	−1.05849	−0.0398933
$705$	0	0
$706$	−23.4753	−0.883506
$707$	12.6760	0.476731
$708$	8.39228	0.315401
$709$	−26.2036	−0.984095	−0.492048	−	0.870568i	$-0.663751\pi$
$709$	−26.2036	−0.984095	−0.492048	+	0.870568i	$0.663751\pi$
$710$	0	0
$711$	−1.31999	−0.0495034
$712$	−13.5287	−0.507010
$713$	−3.83656	−0.143680
$714$	9.27847	0.347238
$715$	0	0
$716$	−10.0795	−0.376689
$717$	−15.7067	−0.586577
$718$	31.5610	1.17785
$719$	−2.51142	−0.0936601	−0.0468300	−	0.998903i	$-0.514912\pi$
$719$	−2.51142	−0.0936601	−0.0468300	+	0.998903i	$0.514912\pi$
$720$	0	0
$721$	2.76044	0.102804
$722$	−10.8807	−0.404939
$723$	−22.5174	−0.837432
$724$	−3.30520	−0.122837
$725$	0	0
$726$	9.87960	0.366666
$727$	−16.4323	−0.609441	−0.304721	−	0.952442i	$-0.598563\pi$
$727$	−16.4323	−0.609441	−0.304721	+	0.952442i	$0.598563\pi$
$728$	12.5415	0.464821
$729$	1.00000	0.0370370
$730$	0	0
$731$	28.3080	1.04701
$732$	−7.13387	−0.263676
$733$	−23.5466	−0.869713	−0.434857	−	0.900500i	$-0.643201\pi$
$733$	−23.5466	−0.869713	−0.434857	+	0.900500i	$0.643201\pi$
$734$	2.82202	0.104163
$735$	0	0
$736$	−0.467465	−0.0172310
$737$	−12.6573	−0.466237
$738$	5.13485	0.189016
$739$	29.2458	1.07582	0.537912	−	0.843001i	$-0.319213\pi$
$739$	29.2458	1.07582	0.537912	+	0.843001i	$0.319213\pi$
$740$	0	0
$741$	−16.7839	−0.616571
$742$	−15.5925	−0.572418
$743$	−18.8806	−0.692661	−0.346331	−	0.938113i	$-0.612572\pi$
$743$	−18.8806	−0.692661	−0.346331	+	0.938113i	$0.612572\pi$
$744$	−8.20715	−0.300889
$745$	0	0
$746$	21.1247	0.773430
$747$	12.9727	0.474645
$748$	4.61259	0.168653
$749$	27.7492	1.01393
$750$	0	0
$751$	48.7324	1.77827	0.889135	−	0.457645i	$-0.151307\pi$
$751$	48.7324	1.77827	0.889135	+	0.457645i	$0.151307\pi$
$752$	6.24520	0.227739
$753$	−4.11677	−0.150024
$754$	−15.2080	−0.553841
$755$	0	0
$756$	−2.12920	−0.0774384
$757$	2.52286	0.0916950	0.0458475	−	0.998948i	$-0.485401\pi$
$757$	2.52286	0.0916950	0.0458475	+	0.998948i	$0.485401\pi$
$758$	−21.2311	−0.771150
$759$	−0.494807	−0.0179603
$760$	0	0
$761$	27.6782	1.00334	0.501668	−	0.865060i	$-0.332720\pi$
$761$	27.6782	1.00334	0.501668	+	0.865060i	$0.332720\pi$
$762$	−8.43501	−0.305568
$763$	−27.1528	−0.982997
$764$	9.94780	0.359899
$765$	0	0
$766$	27.3115	0.986804
$767$	−49.4326	−1.78491
$768$	−1.00000	−0.0360844
$769$	46.5952	1.68027	0.840134	−	0.542379i	$-0.182476\pi$
$769$	46.5952	1.68027	0.840134	+	0.542379i	$0.182476\pi$
$770$	0	0
$771$	22.7141	0.818029
$772$	−19.8949	−0.716034
$773$	−27.5113	−0.989512	−0.494756	−	0.869032i	$-0.664743\pi$
$773$	−27.5113	−0.989512	−0.494756	+	0.869032i	$0.664743\pi$
$774$	−6.49606	−0.233496
$775$	0	0
$776$	−7.51742	−0.269860
$777$	−11.1032	−0.398324
$778$	7.84655	0.281312
$779$	14.6314	0.524225
$780$	0	0
$781$	15.5013	0.554679
$782$	2.03708	0.0728459
$783$	2.58189	0.0922691
$784$	−2.46649	−0.0880889
$785$	0	0
$786$	6.41653	0.228870
$787$	−35.1736	−1.25380	−0.626901	−	0.779099i	$-0.715677\pi$
$787$	−35.1736	−1.25380	−0.626901	+	0.779099i	$0.715677\pi$
$788$	27.3096	0.972865
$789$	−12.5881	−0.448150
$790$	0	0
$791$	39.8530	1.41701
$792$	−1.05849	−0.0376117
$793$	42.0203	1.49218
$794$	20.5604	0.729661
$795$	0	0
$796$	1.70256	0.0603455
$797$	−16.4851	−0.583932	−0.291966	−	0.956429i	$-0.594309\pi$
$797$	−16.4851	−0.583932	−0.291966	+	0.956429i	$0.594309\pi$
$798$	−6.06702	−0.214770
$799$	−27.2148	−0.962791
$800$	0	0
$801$	−13.5287	−0.478014
$802$	2.38184	0.0841057
$803$	−12.8274	−0.452667
$804$	−11.9579	−0.421722
$805$	0	0
$806$	48.3421	1.70278
$807$	24.4467	0.860565
$808$	5.95341	0.209440
$809$	22.8814	0.804466	0.402233	−	0.915537i	$-0.368234\pi$
$809$	22.8814	0.804466	0.402233	+	0.915537i	$0.368234\pi$
$810$	0	0
$811$	5.01376	0.176057	0.0880285	−	0.996118i	$-0.471943\pi$
$811$	5.01376	0.176057	0.0880285	+	0.996118i	$0.471943\pi$
$812$	−5.49736	−0.192920
$813$	13.8542	0.485887
$814$	−5.51970	−0.193466
$815$	0	0
$816$	4.35772	0.152551
$817$	−18.5101	−0.647585
$818$	−14.1166	−0.493577
$819$	12.5415	0.438237
$820$	0	0
$821$	42.5592	1.48532	0.742662	−	0.669666i	$-0.233563\pi$
$821$	42.5592	1.48532	0.742662	+	0.669666i	$0.233563\pi$
$822$	2.49508	0.0870259
$823$	−26.4978	−0.923657	−0.461828	−	0.886969i	$-0.652806\pi$
$823$	−26.4978	−0.923657	−0.461828	+	0.886969i	$0.652806\pi$
$824$	1.29647	0.0451646
$825$	0	0
$826$	−17.8689	−0.621738
$827$	11.8921	0.413530	0.206765	−	0.978391i	$-0.433706\pi$
$827$	11.8921	0.413530	0.206765	+	0.978391i	$0.433706\pi$
$828$	−0.467465	−0.0162455
$829$	−24.4042	−0.847592	−0.423796	−	0.905758i	$-0.639303\pi$
$829$	−24.4042	−0.847592	−0.423796	+	0.905758i	$0.639303\pi$
$830$	0	0
$831$	14.7886	0.513012
$832$	5.89025	0.204208
$833$	10.7483	0.372405
$834$	−1.78818	−0.0619197
$835$	0	0
$836$	−3.01609	−0.104314
$837$	−8.20715	−0.283680
$838$	−23.3766	−0.807532
$839$	−14.0138	−0.483810	−0.241905	−	0.970300i	$-0.577772\pi$
$839$	−14.0138	−0.483810	−0.241905	+	0.970300i	$0.577772\pi$
$840$	0	0
$841$	−22.3339	−0.770133
$842$	21.4696	0.739890
$843$	5.26150	0.181216
$844$	19.8374	0.682832
$845$	0	0
$846$	6.24520	0.214715
$847$	−21.0357	−0.722795
$848$	−7.32315	−0.251478
$849$	30.1332	1.03417
$850$	0	0
$851$	−2.43769	−0.0835631
$852$	14.6447	0.501720
$853$	13.7943	0.472310	0.236155	−	0.971715i	$-0.424113\pi$
$853$	13.7943	0.472310	0.236155	+	0.971715i	$0.424113\pi$
$854$	15.1895	0.519773
$855$	0	0
$856$	13.0327	0.445447
$857$	−11.5927	−0.396000	−0.198000	−	0.980202i	$-0.563445\pi$
$857$	−11.5927	−0.396000	−0.198000	+	0.980202i	$0.563445\pi$
$858$	6.23476	0.212851
$859$	24.8369	0.847423	0.423712	−	0.905797i	$-0.360727\pi$
$859$	24.8369	0.847423	0.423712	+	0.905797i	$0.360727\pi$
$860$	0	0
$861$	−10.9331	−0.372601
$862$	−35.4598	−1.20776
$863$	−35.9787	−1.22473	−0.612365	−	0.790575i	$-0.709782\pi$
$863$	−35.9787	−1.22473	−0.612365	+	0.790575i	$0.709782\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−33.1379	−1.12607
$867$	−1.98968	−0.0675732
$868$	17.4747	0.593130
$869$	1.39719	0.0473965
$870$	0	0
$871$	70.4349	2.38659
$872$	−12.7526	−0.431856
$873$	−7.51742	−0.254426
$874$	−1.33201	−0.0450559
$875$	0	0
$876$	−12.1186	−0.409448
$877$	−22.2472	−0.751236	−0.375618	−	0.926775i	$-0.622569\pi$
$877$	−22.2472	−0.751236	−0.375618	+	0.926775i	$0.622569\pi$
$878$	−7.69473	−0.259684
$879$	−27.0382	−0.911975
$880$	0	0
$881$	−56.7564	−1.91217	−0.956085	−	0.293088i	$-0.905317\pi$
$881$	−56.7564	−1.91217	−0.956085	+	0.293088i	$0.905317\pi$
$882$	−2.46649	−0.0830510
$883$	38.7543	1.30419	0.652093	−	0.758139i	$-0.273891\pi$
$883$	38.7543	1.30419	0.652093	+	0.758139i	$0.273891\pi$
$884$	−25.6680	−0.863309
$885$	0	0
$886$	−8.12762	−0.273053
$887$	−7.38317	−0.247903	−0.123951	−	0.992288i	$-0.539557\pi$
$887$	−7.38317	−0.247903	−0.123951	+	0.992288i	$0.539557\pi$
$888$	−5.21470	−0.174994
$889$	17.9599	0.602354
$890$	0	0
$891$	−1.05849	−0.0354607
$892$	11.5590	0.387024
$893$	17.7953	0.595496
$894$	−10.7440	−0.359334
$895$	0	0
$896$	2.12920	0.0711317
$897$	2.75349	0.0919363
$898$	1.49021	0.0497291
$899$	−21.1899	−0.706723
$900$	0	0
$901$	31.9122	1.06315
$902$	−5.43518	−0.180972
$903$	13.8314	0.460281
$904$	18.7173	0.622529
$905$	0	0
$906$	−16.7444	−0.556294
$907$	−20.2132	−0.671168	−0.335584	−	0.942010i	$-0.608934\pi$
$907$	−20.2132	−0.671168	−0.335584	+	0.942010i	$0.608934\pi$
$908$	−17.3898	−0.577101
$909$	5.95341	0.197462
$910$	0	0
$911$	−7.50406	−0.248621	−0.124310	−	0.992243i	$-0.539672\pi$
$911$	−7.50406	−0.248621	−0.124310	+	0.992243i	$0.539672\pi$
$912$	−2.84943	−0.0943541
$913$	−13.7314	−0.454443
$914$	−7.94841	−0.262910
$915$	0	0
$916$	−23.5935	−0.779550
$917$	−13.6621	−0.451163
$918$	4.35772	0.143826
$919$	17.3423	0.572071	0.286035	−	0.958219i	$-0.407663\pi$
$919$	17.3423	0.572071	0.286035	+	0.958219i	$0.407663\pi$
$920$	0	0
$921$	12.1714	0.401061
$922$	−29.7969	−0.981307
$923$	−86.2610	−2.83932
$924$	2.25374	0.0741425
$925$	0	0
$926$	−12.4986	−0.410730
$927$	1.29647	0.0425815
$928$	−2.58189	−0.0847546
$929$	−26.8437	−0.880712	−0.440356	−	0.897823i	$-0.645148\pi$
$929$	−26.8437	−0.880712	−0.440356	+	0.897823i	$0.645148\pi$
$930$	0	0
$931$	−7.02809	−0.230336
$932$	−10.6944	−0.350307
$933$	−15.2313	−0.498649
$934$	−23.3113	−0.762767
$935$	0	0
$936$	5.89025	0.192529
$937$	−8.90963	−0.291065	−0.145532	−	0.989353i	$-0.546490\pi$
$937$	−8.90963	−0.291065	−0.145532	+	0.989353i	$0.546490\pi$
$938$	25.4608	0.831323
$939$	25.5390	0.833435
$940$	0	0
$941$	−4.44814	−0.145005	−0.0725026	−	0.997368i	$-0.523099\pi$
$941$	−4.44814	−0.145005	−0.0725026	+	0.997368i	$0.523099\pi$
$942$	−4.04586	−0.131821
$943$	−2.40037	−0.0781667
$944$	−8.39228	−0.273146
$945$	0	0
$946$	6.87600	0.223558
$947$	−26.3275	−0.855528	−0.427764	−	0.903890i	$-0.640699\pi$
$947$	−26.3275	−0.855528	−0.427764	+	0.903890i	$0.640699\pi$
$948$	1.31999	0.0428712
$949$	71.3813	2.31714
$950$	0	0
$951$	3.77603	0.122446
$952$	−9.27847	−0.300717
$953$	−17.1473	−0.555455	−0.277727	−	0.960660i	$-0.589581\pi$
$953$	−17.1473	−0.555455	−0.277727	+	0.960660i	$0.589581\pi$
$954$	−7.32315	−0.237096
$955$	0	0
$956$	15.7067	0.507991
$957$	−2.73290	−0.0883420
$958$	36.4911	1.17898
$959$	−5.31254	−0.171551
$960$	0	0
$961$	36.3573	1.17281
$962$	30.7159	0.990321
$963$	13.0327	0.419972
$964$	22.5174	0.725237
$965$	0	0
$966$	0.995330	0.0320242
$967$	−1.12375	−0.0361374	−0.0180687	−	0.999837i	$-0.505752\pi$
$967$	−1.12375	−0.0361374	−0.0180687	+	0.999837i	$0.505752\pi$
$968$	−9.87960	−0.317542
$969$	12.4170	0.398892
$970$	0	0
$971$	9.39797	0.301595	0.150798	−	0.988565i	$-0.451816\pi$
$971$	9.39797	0.301595	0.150798	+	0.988565i	$0.451816\pi$
$972$	−1.00000	−0.0320750
$973$	3.80741	0.122060
$974$	8.61685	0.276102
$975$	0	0
$976$	7.13387	0.228350
$977$	7.30876	0.233828	0.116914	−	0.993142i	$-0.462700\pi$
$977$	7.30876	0.233828	0.116914	+	0.993142i	$0.462700\pi$
$978$	12.7786	0.408614
$979$	14.3200	0.457669
$980$	0	0
$981$	−12.7526	−0.407158
$982$	−27.4118	−0.874746
$983$	−9.70301	−0.309478	−0.154739	−	0.987955i	$-0.549454\pi$
$983$	−9.70301	−0.309478	−0.154739	+	0.987955i	$0.549454\pi$
$984$	−5.13485	−0.163693
$985$	0	0
$986$	11.2511	0.358309
$987$	−13.2973	−0.423258
$988$	16.7839	0.533966
$989$	3.03668	0.0965609
$990$	0	0
$991$	32.0364	1.01767	0.508835	−	0.860864i	$-0.330076\pi$
$991$	32.0364	1.01767	0.508835	+	0.860864i	$0.330076\pi$
$992$	8.20715	0.260577
$993$	−15.2466	−0.483836
$994$	−31.1816	−0.989020
$995$	0	0
$996$	−12.9727	−0.411054
$997$	−4.00845	−0.126949	−0.0634744	−	0.997983i	$-0.520218\pi$
$997$	−4.00845	−0.126949	−0.0634744	+	0.997983i	$0.520218\pi$
$998$	−9.07912	−0.287395
$999$	−5.21470	−0.164986

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3750.2.a.r.1.3	yes	4
5.2	odd	4		3750.2.c.f.1249.7		8
5.3	odd	4		3750.2.c.f.1249.2		8
5.4	even	2		3750.2.a.k.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3750.2.a.k.1.2	✓	4	5.4	even	2
3750.2.a.r.1.3	yes	4	1.1	even	1	trivial
3750.2.c.f.1249.2		8	5.3	odd	4
3750.2.c.f.1249.7		8	5.2	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.12920 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.05849 q^{11} -1.00000 q^{12} +5.89025 q^{13} +2.12920 q^{14} +1.00000 q^{16} -4.35772 q^{17} +1.00000 q^{18} +2.84943 q^{19} -2.12920 q^{21} -1.05849 q^{22} -0.467465 q^{23} -1.00000 q^{24} +5.89025 q^{26} -1.00000 q^{27} +2.12920 q^{28} -2.58189 q^{29} +8.20715 q^{31} +1.00000 q^{32} +1.05849 q^{33} -4.35772 q^{34} +1.00000 q^{36} +5.21470 q^{37} +2.84943 q^{38} -5.89025 q^{39} +5.13485 q^{41} -2.12920 q^{42} -6.49606 q^{43} -1.05849 q^{44} -0.467465 q^{46} +6.24520 q^{47} -1.00000 q^{48} -2.46649 q^{49} +4.35772 q^{51} +5.89025 q^{52} -7.32315 q^{53} -1.00000 q^{54} +2.12920 q^{56} -2.84943 q^{57} -2.58189 q^{58} -8.39228 q^{59} +7.13387 q^{61} +8.20715 q^{62} +2.12920 q^{63} +1.00000 q^{64} +1.05849 q^{66} +11.9579 q^{67} -4.35772 q^{68} +0.467465 q^{69} -14.6447 q^{71} +1.00000 q^{72} +12.1186 q^{73} +5.21470 q^{74} +2.84943 q^{76} -2.25374 q^{77} -5.89025 q^{78} -1.31999 q^{79} +1.00000 q^{81} +5.13485 q^{82} +12.9727 q^{83} -2.12920 q^{84} -6.49606 q^{86} +2.58189 q^{87} -1.05849 q^{88} -13.5287 q^{89} +12.5415 q^{91} -0.467465 q^{92} -8.20715 q^{93} +6.24520 q^{94} -1.00000 q^{96} -7.51742 q^{97} -2.46649 q^{98} -1.05849 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{6} + 6 q^{7} + 4 q^{8} + 4 q^{9} - 4 q^{12} + 2 q^{13} + 6 q^{14} + 4 q^{16} + 11 q^{17} + 4 q^{18} + 9 q^{19} - 6 q^{21} + 5 q^{23} - 4 q^{24} + 2 q^{26} - 4 q^{27}+ \cdots - 4 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^6 + 6 * q^7 + 4 * q^8 + 4 * q^9 - 4 * q^12 + 2 * q^13 + 6 * q^14 + 4 * q^16 + 11 * q^17 + 4 * q^18 + 9 * q^19 - 6 * q^21 + 5 * q^23 - 4 * q^24 + 2 * q^26 - 4 * q^27 + 6 * q^28 - 14 * q^29 + 2 * q^31 + 4 * q^32 + 11 * q^34 + 4 * q^36 + 7 * q^37 + 9 * q^38 - 2 * q^39 + 6 * q^41 - 6 * q^42 + 6 * q^43 + 5 * q^46 + 15 * q^47 - 4 * q^48 - 4 * q^49 - 11 * q^51 + 2 * q^52 + 9 * q^53 - 4 * q^54 + 6 * q^56 - 9 * q^57 - 14 * q^58 - 15 * q^59 + 15 * q^61 + 2 * q^62 + 6 * q^63 + 4 * q^64 + 21 * q^67 + 11 * q^68 - 5 * q^69 + 4 * q^72 + 22 * q^73 + 7 * q^74 + 9 * q^76 - 15 * q^77 - 2 * q^78 - 3 * q^79 + 4 * q^81 + 6 * q^82 + 12 * q^83 - 6 * q^84 + 6 * q^86 + 14 * q^87 - 3 * q^89 - 12 * q^91 + 5 * q^92 - 2 * q^93 + 15 * q^94 - 4 * q^96 - 3 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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