LMFDB - Embedded newform 825.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("825.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	825.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:	$6.58765816676$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.48119 q^{2} +1.00000 q^{3} +0.193937 q^{4} -1.48119 q^{6} +1.19394 q^{7} +2.67513 q^{8} +1.00000 q^{9} +1.00000 q^{11} +0.193937 q^{12} +0.806063 q^{13} -1.76845 q^{14} -4.35026 q^{16} +3.76845 q^{17} -1.48119 q^{18} -5.35026 q^{19} +1.19394 q^{21} -1.48119 q^{22} +4.00000 q^{23} +2.67513 q^{24} -1.19394 q^{26} +1.00000 q^{27} +0.231548 q^{28} -4.31265 q^{29} +0.962389 q^{31} +1.09332 q^{32} +1.00000 q^{33} -5.58181 q^{34} +0.193937 q^{36} +1.61213 q^{37} +7.92478 q^{38} +0.806063 q^{39} +9.08840 q^{41} -1.76845 q^{42} +4.41819 q^{43} +0.193937 q^{44} -5.92478 q^{46} +12.3127 q^{47} -4.35026 q^{48} -5.57452 q^{49} +3.76845 q^{51} +0.156325 q^{52} +1.42548 q^{53} -1.48119 q^{54} +3.19394 q^{56} -5.35026 q^{57} +6.38787 q^{58} +13.2750 q^{59} -0.0752228 q^{61} -1.42548 q^{62} +1.19394 q^{63} +7.08110 q^{64} -1.48119 q^{66} -2.70052 q^{67} +0.730841 q^{68} +4.00000 q^{69} -14.0508 q^{71} +2.67513 q^{72} -10.7308 q^{73} -2.38787 q^{74} -1.03761 q^{76} +1.19394 q^{77} -1.19394 q^{78} +13.9756 q^{79} +1.00000 q^{81} -13.4617 q^{82} -9.89446 q^{83} +0.231548 q^{84} -6.54420 q^{86} -4.31265 q^{87} +2.67513 q^{88} +16.8872 q^{89} +0.962389 q^{91} +0.775746 q^{92} +0.962389 q^{93} -18.2374 q^{94} +1.09332 q^{96} +11.4763 q^{97} +8.25694 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{3} + q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 3 q^{9} + 3 q^{11} + q^{12} + 2 q^{13} + 6 q^{14} - 3 q^{16} + q^{18} - 6 q^{19} + 4 q^{21} + q^{22} + 12 q^{23} + 3 q^{24} - 4 q^{26} + 3 q^{27}+ \cdots + 3 q^{99}+O(q^{100}) $ 3 * q + q^2 + 3 * q^3 + q^4 + q^6 + 4 * q^7 + 3 * q^8 + 3 * q^9 + 3 * q^11 + q^12 + 2 * q^13 + 6 * q^14 - 3 * q^16 + q^18 - 6 * q^19 + 4 * q^21 + q^22 + 12 * q^23 + 3 * q^24 - 4 * q^26 + 3 * q^27 + 12 * q^28 + 8 * q^29 - 8 * q^31 - 3 * q^32 + 3 * q^33 - 18 * q^34 + q^36 + 4 * q^37 + 2 * q^38 + 2 * q^39 + 8 * q^41 + 6 * q^42 + 12 * q^43 + q^44 + 4 * q^46 + 16 * q^47 - 3 * q^48 - 5 * q^49 - 10 * q^52 + 16 * q^53 + q^54 + 10 * q^56 - 6 * q^57 + 20 * q^58 + 8 * q^59 - 22 * q^61 - 16 * q^62 + 4 * q^63 - 11 * q^64 + q^66 + 12 * q^67 - 20 * q^68 + 12 * q^69 - 12 * q^71 + 3 * q^72 - 10 * q^73 - 8 * q^74 - 14 * q^76 + 4 * q^77 - 4 * q^78 - 10 * q^79 + 3 * q^81 + 4 * q^82 - 10 * q^83 + 12 * q^84 - 10 * q^86 + 8 * q^87 + 3 * q^88 + 18 * q^89 - 8 * q^91 + 4 * q^92 - 8 * q^93 - 12 * q^94 - 3 * q^96 + 16 * q^97 + 21 * q^98 + 3 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.48119	−1.04736	−0.523681	−	0.851914i	$-0.675442\pi$
$2$	−1.48119	−1.04736	−0.523681	+	0.851914i	$0.675442\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.193937	0.0969683
$5$	0	0
$5$	0	0
$6$	−1.48119	−0.604695
$7$	1.19394	0.451266	0.225633	−	0.974212i	$-0.427555\pi$
$7$	1.19394	0.451266	0.225633	+	0.974212i	$0.427555\pi$
$8$	2.67513	0.945802
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.193937	0.0559847
$13$	0.806063	0.223562	0.111781	−	0.993733i	$-0.464345\pi$
$13$	0.806063	0.223562	0.111781	+	0.993733i	$0.464345\pi$
$14$	−1.76845	−0.472639
$15$	0	0
$16$	−4.35026	−1.08757
$17$	3.76845	0.913984	0.456992	−	0.889471i	$-0.348927\pi$
$17$	3.76845	0.913984	0.456992	+	0.889471i	$0.348927\pi$
$18$	−1.48119	−0.349121
$19$	−5.35026	−1.22743	−0.613717	−	0.789526i	$-0.710326\pi$
$19$	−5.35026	−1.22743	−0.613717	+	0.789526i	$0.710326\pi$
$20$	0	0
$21$	1.19394	0.260538
$22$	−1.48119	−0.315792
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	2.67513	0.546059
$25$	0	0
$26$	−1.19394	−0.234150
$27$	1.00000	0.192450
$28$	0.231548	0.0437585
$29$	−4.31265	−0.800839	−0.400420	−	0.916332i	$-0.631136\pi$
$29$	−4.31265	−0.800839	−0.400420	+	0.916332i	$0.631136\pi$
$30$	0	0
$31$	0.962389	0.172850	0.0864250	−	0.996258i	$-0.472456\pi$
$31$	0.962389	0.172850	0.0864250	+	0.996258i	$0.472456\pi$
$32$	1.09332	0.193274
$33$	1.00000	0.174078
$34$	−5.58181	−0.957272
$35$	0	0
$36$	0.193937	0.0323228
$37$	1.61213	0.265032	0.132516	−	0.991181i	$-0.457694\pi$
$37$	1.61213	0.265032	0.132516	+	0.991181i	$0.457694\pi$
$38$	7.92478	1.28557
$39$	0.806063	0.129073
$40$	0	0
$41$	9.08840	1.41937	0.709685	−	0.704520i	$-0.248837\pi$
$41$	9.08840	1.41937	0.709685	+	0.704520i	$0.248837\pi$
$42$	−1.76845	−0.272878
$43$	4.41819	0.673768	0.336884	−	0.941546i	$-0.390627\pi$
$43$	4.41819	0.673768	0.336884	+	0.941546i	$0.390627\pi$
$44$	0.193937	0.0292370
$45$	0	0
$46$	−5.92478	−0.873561
$47$	12.3127	1.79598	0.897992	−	0.440011i	$-0.145026\pi$
$47$	12.3127	1.79598	0.897992	+	0.440011i	$0.145026\pi$
$48$	−4.35026	−0.627906
$49$	−5.57452	−0.796359
$50$	0	0
$51$	3.76845	0.527689
$52$	0.156325	0.0216784
$53$	1.42548	0.195805	0.0979027	−	0.995196i	$-0.468787\pi$
$53$	1.42548	0.195805	0.0979027	+	0.995196i	$0.468787\pi$
$54$	−1.48119	−0.201565
$55$	0	0
$56$	3.19394	0.426808
$57$	−5.35026	−0.708659
$58$	6.38787	0.838769
$59$	13.2750	1.72826	0.864131	−	0.503266i	$-0.167868\pi$
$59$	13.2750	1.72826	0.864131	+	0.503266i	$0.167868\pi$
$60$	0	0
$61$	−0.0752228	−0.00963129	−0.00481565	−	0.999988i	$-0.501533\pi$
$61$	−0.0752228	−0.00963129	−0.00481565	+	0.999988i	$0.501533\pi$
$62$	−1.42548	−0.181037
$63$	1.19394	0.150422
$64$	7.08110	0.885138
$65$	0	0
$66$	−1.48119	−0.182322
$67$	−2.70052	−0.329921	−0.164961	−	0.986300i	$-0.552750\pi$
$67$	−2.70052	−0.329921	−0.164961	+	0.986300i	$0.552750\pi$
$68$	0.730841	0.0886274
$69$	4.00000	0.481543
$70$	0	0
$71$	−14.0508	−1.66752	−0.833761	−	0.552126i	$-0.813817\pi$
$71$	−14.0508	−1.66752	−0.833761	+	0.552126i	$0.813817\pi$
$72$	2.67513	0.315267
$73$	−10.7308	−1.25595	−0.627975	−	0.778234i	$-0.716116\pi$
$73$	−10.7308	−1.25595	−0.627975	+	0.778234i	$0.716116\pi$
$74$	−2.38787	−0.277585
$75$	0	0
$76$	−1.03761	−0.119022
$77$	1.19394	0.136062
$78$	−1.19394	−0.135187
$79$	13.9756	1.57237	0.786187	−	0.617989i	$-0.212052\pi$
$79$	13.9756	1.57237	0.786187	+	0.617989i	$0.212052\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−13.4617	−1.48659
$83$	−9.89446	−1.08606	−0.543029	−	0.839714i	$-0.682723\pi$
$83$	−9.89446	−1.08606	−0.543029	+	0.839714i	$0.682723\pi$
$84$	0.231548	0.0252640
$85$	0	0
$86$	−6.54420	−0.705679
$87$	−4.31265	−0.462365
$88$	2.67513	0.285170
$89$	16.8872	1.79004	0.895018	−	0.446030i	$-0.147163\pi$
$89$	16.8872	1.79004	0.895018	+	0.446030i	$0.147163\pi$
$90$	0	0
$91$	0.962389	0.100886
$92$	0.775746	0.0808771
$93$	0.962389	0.0997950
$94$	−18.2374	−1.88105
$95$	0	0
$96$	1.09332	0.111587
$97$	11.4763	1.16524	0.582619	−	0.812745i	$-0.302028\pi$
$97$	11.4763	1.16524	0.582619	+	0.812745i	$0.302028\pi$
$98$	8.25694	0.834077
$99$	1.00000	0.100504
$100$	0	0
$101$	10.7612	1.07078	0.535388	−	0.844606i	$-0.320166\pi$
$101$	10.7612	1.07078	0.535388	+	0.844606i	$0.320166\pi$
$102$	−5.58181	−0.552682
$103$	16.9380	1.66895	0.834473	−	0.551049i	$-0.185772\pi$
$103$	16.9380	1.66895	0.834473	+	0.551049i	$0.185772\pi$
$104$	2.15633	0.211445
$105$	0	0
$106$	−2.11142	−0.205079
$107$	8.28233	0.800683	0.400342	−	0.916366i	$-0.368891\pi$
$107$	8.28233	0.800683	0.400342	+	0.916366i	$0.368891\pi$
$108$	0.193937	0.0186616
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	1.61213	0.153016
$112$	−5.19394	−0.490781
$113$	2.26187	0.212778	0.106389	−	0.994325i	$-0.466071\pi$
$113$	2.26187	0.212778	0.106389	+	0.994325i	$0.466071\pi$
$114$	7.92478	0.742223
$115$	0	0
$116$	−0.836381	−0.0776560
$117$	0.806063	0.0745206
$118$	−19.6629	−1.81012
$119$	4.49929	0.412449
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.111420	0.0100875
$123$	9.08840	0.819473
$124$	0.186642	0.0167610
$125$	0	0
$126$	−1.76845	−0.157546
$127$	−13.8192	−1.22626	−0.613130	−	0.789982i	$-0.710090\pi$
$127$	−13.8192	−1.22626	−0.613130	+	0.789982i	$0.710090\pi$
$128$	−12.6751	−1.12033
$129$	4.41819	0.389000
$130$	0	0
$131$	−5.92478	−0.517650	−0.258825	−	0.965924i	$-0.583335\pi$
$131$	−5.92478	−0.517650	−0.258825	+	0.965924i	$0.583335\pi$
$132$	0.193937	0.0168800
$133$	−6.38787	−0.553899
$134$	4.00000	0.345547
$135$	0	0
$136$	10.0811	0.864447
$137$	3.35026	0.286232	0.143116	−	0.989706i	$-0.454288\pi$
$137$	3.35026	0.286232	0.143116	+	0.989706i	$0.454288\pi$
$138$	−5.92478	−0.504351
$139$	−21.1998	−1.79814	−0.899072	−	0.437800i	$-0.855758\pi$
$139$	−21.1998	−1.79814	−0.899072	+	0.437800i	$0.855758\pi$
$140$	0	0
$141$	12.3127	1.03691
$142$	20.8119	1.74650
$143$	0.806063	0.0674064
$144$	−4.35026	−0.362522
$145$	0	0
$146$	15.8945	1.31543
$147$	−5.57452	−0.459778
$148$	0.312650	0.0256997
$149$	6.38787	0.523315	0.261657	−	0.965161i	$-0.415731\pi$
$149$	6.38787	0.523315	0.261657	+	0.965161i	$0.415731\pi$
$150$	0	0
$151$	−2.64974	−0.215633	−0.107816	−	0.994171i	$-0.534386\pi$
$151$	−2.64974	−0.215633	−0.107816	+	0.994171i	$0.534386\pi$
$152$	−14.3127	−1.16091
$153$	3.76845	0.304661
$154$	−1.76845	−0.142506
$155$	0	0
$156$	0.156325	0.0125160
$157$	1.61213	0.128662	0.0643309	−	0.997929i	$-0.479509\pi$
$157$	1.61213	0.128662	0.0643309	+	0.997929i	$0.479509\pi$
$158$	−20.7005	−1.64685
$159$	1.42548	0.113048
$160$	0	0
$161$	4.77575	0.376382
$162$	−1.48119	−0.116374
$163$	−0.312650	−0.0244887	−0.0122443	−	0.999925i	$-0.503898\pi$
$163$	−0.312650	−0.0244887	−0.0122443	+	0.999925i	$0.503898\pi$
$164$	1.76257	0.137634
$165$	0	0
$166$	14.6556	1.13750
$167$	0.493413	0.0381815	0.0190907	−	0.999818i	$-0.493923\pi$
$167$	0.493413	0.0381815	0.0190907	+	0.999818i	$0.493923\pi$
$168$	3.19394	0.246418
$169$	−12.3503	−0.950020
$170$	0	0
$171$	−5.35026	−0.409145
$172$	0.856849	0.0653341
$173$	−23.3054	−1.77187	−0.885937	−	0.463806i	$-0.846483\pi$
$173$	−23.3054	−1.77187	−0.885937	+	0.463806i	$0.846483\pi$
$174$	6.38787	0.484263
$175$	0	0
$176$	−4.35026	−0.327913
$177$	13.2750	0.997813
$178$	−25.0132	−1.87482
$179$	−10.7005	−0.799795	−0.399897	−	0.916560i	$-0.630954\pi$
$179$	−10.7005	−0.799795	−0.399897	+	0.916560i	$0.630954\pi$
$180$	0	0
$181$	−7.79877	−0.579678	−0.289839	−	0.957075i	$-0.593602\pi$
$181$	−7.79877	−0.579678	−0.289839	+	0.957075i	$0.593602\pi$
$182$	−1.42548	−0.105664
$183$	−0.0752228	−0.00556063
$184$	10.7005	0.788853
$185$	0	0
$186$	−1.42548	−0.104522
$187$	3.76845	0.275577
$188$	2.38787	0.174154
$189$	1.19394	0.0868461
$190$	0	0
$191$	1.29948	0.0940268	0.0470134	−	0.998894i	$-0.485030\pi$
$191$	1.29948	0.0940268	0.0470134	+	0.998894i	$0.485030\pi$
$192$	7.08110	0.511035
$193$	8.59498	0.618680	0.309340	−	0.950951i	$-0.399892\pi$
$193$	8.59498	0.618680	0.309340	+	0.950951i	$0.399892\pi$
$194$	−16.9986	−1.22043
$195$	0	0
$196$	−1.08110	−0.0772216
$197$	−20.7064	−1.47527	−0.737635	−	0.675200i	$-0.764057\pi$
$197$	−20.7064	−1.47527	−0.737635	+	0.675200i	$0.764057\pi$
$198$	−1.48119	−0.105264
$199$	5.55149	0.393535	0.196767	−	0.980450i	$-0.436956\pi$
$199$	5.55149	0.393535	0.196767	+	0.980450i	$0.436956\pi$
$200$	0	0
$201$	−2.70052	−0.190480
$202$	−15.9394	−1.12149
$203$	−5.14903	−0.361391
$204$	0.730841	0.0511691
$205$	0	0
$206$	−25.0884	−1.74799
$207$	4.00000	0.278019
$208$	−3.50659	−0.243138
$209$	−5.35026	−0.370085
$210$	0	0
$211$	−18.4993	−1.27354	−0.636772	−	0.771052i	$-0.719731\pi$
$211$	−18.4993	−1.27354	−0.636772	+	0.771052i	$0.719731\pi$
$212$	0.276454	0.0189869
$213$	−14.0508	−0.962744
$214$	−12.2677	−0.838606
$215$	0	0
$216$	2.67513	0.182020
$217$	1.14903	0.0780013
$218$	14.8119	1.00319
$219$	−10.7308	−0.725123
$220$	0	0
$221$	3.03761	0.204332
$222$	−2.38787	−0.160264
$223$	17.6121	1.17940	0.589698	−	0.807624i	$-0.299247\pi$
$223$	17.6121	1.17940	0.589698	+	0.807624i	$0.299247\pi$
$224$	1.30536	0.0872178
$225$	0	0
$226$	−3.35026	−0.222856
$227$	−17.4314	−1.15696	−0.578480	−	0.815696i	$-0.696354\pi$
$227$	−17.4314	−1.15696	−0.578480	+	0.815696i	$0.696354\pi$
$228$	−1.03761	−0.0687175
$229$	13.0738	0.863942	0.431971	−	0.901888i	$-0.357818\pi$
$229$	13.0738	0.863942	0.431971	+	0.901888i	$0.357818\pi$
$230$	0	0
$231$	1.19394	0.0785553
$232$	−11.5369	−0.757435
$233$	−13.8437	−0.906929	−0.453465	−	0.891274i	$-0.649812\pi$
$233$	−13.8437	−0.906929	−0.453465	+	0.891274i	$0.649812\pi$
$234$	−1.19394	−0.0780501
$235$	0	0
$236$	2.57452	0.167587
$237$	13.9756	0.907810
$238$	−6.66433	−0.431984
$239$	−12.3733	−0.800361	−0.400181	−	0.916436i	$-0.631053\pi$
$239$	−12.3733	−0.800361	−0.400181	+	0.916436i	$0.631053\pi$
$240$	0	0
$241$	−24.5501	−1.58141	−0.790705	−	0.612198i	$-0.790286\pi$
$241$	−24.5501	−1.58141	−0.790705	+	0.612198i	$0.790286\pi$
$242$	−1.48119	−0.0952148
$243$	1.00000	0.0641500
$244$	−0.0145884	−0.000933930 0
$245$	0	0
$246$	−13.4617	−0.858285
$247$	−4.31265	−0.274407
$248$	2.57452	0.163482
$249$	−9.89446	−0.627036
$250$	0	0
$251$	13.9003	0.877382	0.438691	−	0.898638i	$-0.355442\pi$
$251$	13.9003	0.877382	0.438691	+	0.898638i	$0.355442\pi$
$252$	0.231548	0.0145862
$253$	4.00000	0.251478
$254$	20.4690	1.28434
$255$	0	0
$256$	4.61213	0.288258
$257$	−18.8872	−1.17815	−0.589075	−	0.808079i	$-0.700508\pi$
$257$	−18.8872	−1.17815	−0.589075	+	0.808079i	$0.700508\pi$
$258$	−6.54420	−0.407424
$259$	1.92478	0.119600
$260$	0	0
$261$	−4.31265	−0.266946
$262$	8.77575	0.542167
$263$	20.8061	1.28296	0.641478	−	0.767141i	$-0.278321\pi$
$263$	20.8061	1.28296	0.641478	+	0.767141i	$0.278321\pi$
$264$	2.67513	0.164643
$265$	0	0
$266$	9.46168	0.580133
$267$	16.8872	1.03348
$268$	−0.523730	−0.0319919
$269$	−32.3996	−1.97544	−0.987720	−	0.156233i	$-0.950065\pi$
$269$	−32.3996	−1.97544	−0.987720	+	0.156233i	$0.950065\pi$
$270$	0	0
$271$	−16.8265	−1.02214	−0.511069	−	0.859539i	$-0.670751\pi$
$271$	−16.8265	−1.02214	−0.511069	+	0.859539i	$0.670751\pi$
$272$	−16.3938	−0.994017
$273$	0.962389	0.0582464
$274$	−4.96239	−0.299789
$275$	0	0
$276$	0.775746	0.0466944
$277$	16.9076	1.01588	0.507941	−	0.861392i	$-0.330407\pi$
$277$	16.9076	1.01588	0.507941	+	0.861392i	$0.330407\pi$
$278$	31.4010	1.88331
$279$	0.962389	0.0576167
$280$	0	0
$281$	5.61213	0.334791	0.167396	−	0.985890i	$-0.446464\pi$
$281$	5.61213	0.334791	0.167396	+	0.985890i	$0.446464\pi$
$282$	−18.2374	−1.08602
$283$	5.81924	0.345918	0.172959	−	0.984929i	$-0.444667\pi$
$283$	5.81924	0.345918	0.172959	+	0.984929i	$0.444667\pi$
$284$	−2.72496	−0.161697
$285$	0	0
$286$	−1.19394	−0.0705989
$287$	10.8510	0.640512
$288$	1.09332	0.0644246
$289$	−2.79877	−0.164633
$290$	0	0
$291$	11.4763	0.672751
$292$	−2.08110	−0.121787
$293$	−8.29218	−0.484434	−0.242217	−	0.970222i	$-0.577875\pi$
$293$	−8.29218	−0.484434	−0.242217	+	0.970222i	$0.577875\pi$
$294$	8.25694	0.481555
$295$	0	0
$296$	4.31265	0.250668
$297$	1.00000	0.0580259
$298$	−9.46168	−0.548100
$299$	3.22425	0.186463
$300$	0	0
$301$	5.27504	0.304048
$302$	3.92478	0.225846
$303$	10.7612	0.618212
$304$	23.2750	1.33492
$305$	0	0
$306$	−5.58181	−0.319091
$307$	25.6688	1.46500	0.732498	−	0.680770i	$-0.238354\pi$
$307$	25.6688	1.46500	0.732498	+	0.680770i	$0.238354\pi$
$308$	0.231548	0.0131937
$309$	16.9380	0.963566
$310$	0	0
$311$	−15.7235	−0.891601	−0.445800	−	0.895132i	$-0.647081\pi$
$311$	−15.7235	−0.891601	−0.445800	+	0.895132i	$0.647081\pi$
$312$	2.15633	0.122078
$313$	−26.8627	−1.51837	−0.759186	−	0.650874i	$-0.774403\pi$
$313$	−26.8627	−1.51837	−0.759186	+	0.650874i	$0.774403\pi$
$314$	−2.38787	−0.134755
$315$	0	0
$316$	2.71037	0.152470
$317$	0.710373	0.0398985	0.0199492	−	0.999801i	$-0.493650\pi$
$317$	0.710373	0.0398985	0.0199492	+	0.999801i	$0.493650\pi$
$318$	−2.11142	−0.118403
$319$	−4.31265	−0.241462
$320$	0	0
$321$	8.28233	0.462275
$322$	−7.07381	−0.394208
$323$	−20.1622	−1.12186
$324$	0.193937	0.0107743
$325$	0	0
$326$	0.463096	0.0256485
$327$	−10.0000	−0.553001
$328$	24.3127	1.34244
$329$	14.7005	0.810466
$330$	0	0
$331$	0.962389	0.0528977	0.0264488	−	0.999650i	$-0.491580\pi$
$331$	0.962389	0.0528977	0.0264488	+	0.999650i	$0.491580\pi$
$332$	−1.91890	−0.105313
$333$	1.61213	0.0883440
$334$	−0.730841	−0.0399898
$335$	0	0
$336$	−5.19394	−0.283352
$337$	19.8192	1.07962	0.539811	−	0.841786i	$-0.318496\pi$
$337$	19.8192	1.07962	0.539811	+	0.841786i	$0.318496\pi$
$338$	18.2931	0.995015
$339$	2.26187	0.122848
$340$	0	0
$341$	0.962389	0.0521163
$342$	7.92478	0.428523
$343$	−15.0132	−0.810635
$344$	11.8192	0.637251
$345$	0	0
$346$	34.5198	1.85579
$347$	6.20711	0.333215	0.166608	−	0.986023i	$-0.446719\pi$
$347$	6.20711	0.333215	0.166608	+	0.986023i	$0.446719\pi$
$348$	−0.836381	−0.0448347
$349$	4.44851	0.238123	0.119062	−	0.992887i	$-0.462011\pi$
$349$	4.44851	0.238123	0.119062	+	0.992887i	$0.462011\pi$
$350$	0	0
$351$	0.806063	0.0430245
$352$	1.09332	0.0582742
$353$	6.57452	0.349926	0.174963	−	0.984575i	$-0.444019\pi$
$353$	6.57452	0.349926	0.174963	+	0.984575i	$0.444019\pi$
$354$	−19.6629	−1.04507
$355$	0	0
$356$	3.27504	0.173577
$357$	4.49929	0.238128
$358$	15.8496	0.837675
$359$	8.62530	0.455226	0.227613	−	0.973752i	$-0.426908\pi$
$359$	8.62530	0.455226	0.227613	+	0.973752i	$0.426908\pi$
$360$	0	0
$361$	9.62530	0.506595
$362$	11.5515	0.607133
$363$	1.00000	0.0524864
$364$	0.186642	0.00978272
$365$	0	0
$366$	0.111420	0.00582399
$367$	23.0132	1.20128	0.600639	−	0.799520i	$-0.294913\pi$
$367$	23.0132	1.20128	0.600639	+	0.799520i	$0.294913\pi$
$368$	−17.4010	−0.907092
$369$	9.08840	0.473123
$370$	0	0
$371$	1.70194	0.0883602
$372$	0.186642	0.00967695
$373$	28.1925	1.45975	0.729877	−	0.683579i	$-0.239577\pi$
$373$	28.1925	1.45975	0.729877	+	0.683579i	$0.239577\pi$
$374$	−5.58181	−0.288629
$375$	0	0
$376$	32.9380	1.69865
$377$	−3.47627	−0.179037
$378$	−1.76845	−0.0909594
$379$	3.74798	0.192521	0.0962605	−	0.995356i	$-0.469312\pi$
$379$	3.74798	0.192521	0.0962605	+	0.995356i	$0.469312\pi$
$380$	0	0
$381$	−13.8192	−0.707981
$382$	−1.92478	−0.0984802
$383$	1.76257	0.0900632	0.0450316	−	0.998986i	$-0.485661\pi$
$383$	1.76257	0.0900632	0.0450316	+	0.998986i	$0.485661\pi$
$384$	−12.6751	−0.646825
$385$	0	0
$386$	−12.7308	−0.647983
$387$	4.41819	0.224589
$388$	2.22567	0.112991
$389$	6.52373	0.330766	0.165383	−	0.986229i	$-0.447114\pi$
$389$	6.52373	0.330766	0.165383	+	0.986229i	$0.447114\pi$
$390$	0	0
$391$	15.0738	0.762315
$392$	−14.9126	−0.753198
$393$	−5.92478	−0.298865
$394$	30.6702	1.54514
$395$	0	0
$396$	0.193937	0.00974568
$397$	−23.6991	−1.18942	−0.594712	−	0.803939i	$-0.702734\pi$
$397$	−23.6991	−1.18942	−0.594712	+	0.803939i	$0.702734\pi$
$398$	−8.22284	−0.412174
$399$	−6.38787	−0.319794
$400$	0	0
$401$	8.88717	0.443804	0.221902	−	0.975069i	$-0.428774\pi$
$401$	8.88717	0.443804	0.221902	+	0.975069i	$0.428774\pi$
$402$	4.00000	0.199502
$403$	0.775746	0.0386427
$404$	2.08698	0.103831
$405$	0	0
$406$	7.62672	0.378508
$407$	1.61213	0.0799102
$408$	10.0811	0.499089
$409$	−4.85097	−0.239865	−0.119932	−	0.992782i	$-0.538268\pi$
$409$	−4.85097	−0.239865	−0.119932	+	0.992782i	$0.538268\pi$
$410$	0	0
$411$	3.35026	0.165256
$412$	3.28489	0.161835
$413$	15.8496	0.779906
$414$	−5.92478	−0.291187
$415$	0	0
$416$	0.881286	0.0432086
$417$	−21.1998	−1.03816
$418$	7.92478	0.387614
$419$	−10.7005	−0.522755	−0.261377	−	0.965237i	$-0.584177\pi$
$419$	−10.7005	−0.522755	−0.261377	+	0.965237i	$0.584177\pi$
$420$	0	0
$421$	−30.6009	−1.49139	−0.745697	−	0.666285i	$-0.767884\pi$
$421$	−30.6009	−1.49139	−0.745697	+	0.666285i	$0.767884\pi$
$422$	27.4010	1.33386
$423$	12.3127	0.598662
$424$	3.81336	0.185193
$425$	0	0
$426$	20.8119	1.00834
$427$	−0.0898112	−0.00434627
$428$	1.60625	0.0776409
$429$	0.806063	0.0389171
$430$	0	0
$431$	−5.92478	−0.285386	−0.142693	−	0.989767i	$-0.545576\pi$
$431$	−5.92478	−0.285386	−0.142693	+	0.989767i	$0.545576\pi$
$432$	−4.35026	−0.209302
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	−1.70194	−0.0816956
$435$	0	0
$436$	−1.93937	−0.0928788
$437$	−21.4010	−1.02375
$438$	15.8945	0.759467
$439$	5.35026	0.255354	0.127677	−	0.991816i	$-0.459248\pi$
$439$	5.35026	0.255354	0.127677	+	0.991816i	$0.459248\pi$
$440$	0	0
$441$	−5.57452	−0.265453
$442$	−4.49929	−0.214010
$443$	19.6873	0.935374	0.467687	−	0.883894i	$-0.345087\pi$
$443$	19.6873	0.935374	0.467687	+	0.883894i	$0.345087\pi$
$444$	0.312650	0.0148377
$445$	0	0
$446$	−26.0870	−1.23525
$447$	6.38787	0.302136
$448$	8.45439	0.399432
$449$	−31.3357	−1.47882	−0.739411	−	0.673254i	$-0.764896\pi$
$449$	−31.3357	−1.47882	−0.739411	+	0.673254i	$0.764896\pi$
$450$	0	0
$451$	9.08840	0.427956
$452$	0.438658	0.0206328
$453$	−2.64974	−0.124496
$454$	25.8192	1.21176
$455$	0	0
$456$	−14.3127	−0.670251
$457$	−37.5936	−1.75855	−0.879276	−	0.476312i	$-0.841973\pi$
$457$	−37.5936	−1.75855	−0.879276	+	0.476312i	$0.841973\pi$
$458$	−19.3649	−0.904860
$459$	3.76845	0.175896
$460$	0	0
$461$	17.9854	0.837664	0.418832	−	0.908064i	$-0.362440\pi$
$461$	17.9854	0.837664	0.418832	+	0.908064i	$0.362440\pi$
$462$	−1.76845	−0.0822758
$463$	39.0132	1.81310	0.906548	−	0.422103i	$-0.138708\pi$
$463$	39.0132	1.81310	0.906548	+	0.422103i	$0.138708\pi$
$464$	18.7612	0.870965
$465$	0	0
$466$	20.5052	0.949884
$467$	14.5501	0.673297	0.336649	−	0.941630i	$-0.390707\pi$
$467$	14.5501	0.673297	0.336649	+	0.941630i	$0.390707\pi$
$468$	0.156325	0.00722613
$469$	−3.22425	−0.148882
$470$	0	0
$471$	1.61213	0.0742829
$472$	35.5125	1.63459
$473$	4.41819	0.203149
$474$	−20.7005	−0.950807
$475$	0	0
$476$	0.872577	0.0399945
$477$	1.42548	0.0652685
$478$	18.3272	0.838268
$479$	28.6253	1.30792	0.653962	−	0.756528i	$-0.273106\pi$
$479$	28.6253	1.30792	0.653962	+	0.756528i	$0.273106\pi$
$480$	0	0
$481$	1.29948	0.0592510
$482$	36.3634	1.65631
$483$	4.77575	0.217304
$484$	0.193937	0.00881530
$485$	0	0
$486$	−1.48119	−0.0671883
$487$	1.44992	0.0657022	0.0328511	−	0.999460i	$-0.489541\pi$
$487$	1.44992	0.0657022	0.0328511	+	0.999460i	$0.489541\pi$
$488$	−0.201231	−0.00910929
$489$	−0.312650	−0.0141385
$490$	0	0
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	1.76257	0.0794629
$493$	−16.2520	−0.731954
$494$	6.38787	0.287404
$495$	0	0
$496$	−4.18664	−0.187986
$497$	−16.7757	−0.752495
$498$	14.6556	0.656734
$499$	30.7005	1.37434	0.687172	−	0.726495i	$-0.258852\pi$
$499$	30.7005	1.37434	0.687172	+	0.726495i	$0.258852\pi$
$500$	0	0
$501$	0.493413	0.0220441
$502$	−20.5891	−0.918937
$503$	−19.7586	−0.880993	−0.440496	−	0.897754i	$-0.645198\pi$
$503$	−19.7586	−0.880993	−0.440496	+	0.897754i	$0.645198\pi$
$504$	3.19394	0.142269
$505$	0	0
$506$	−5.92478	−0.263388
$507$	−12.3503	−0.548494
$508$	−2.68006	−0.118908
$509$	22.1016	0.979635	0.489817	−	0.871825i	$-0.337063\pi$
$509$	22.1016	0.979635	0.489817	+	0.871825i	$0.337063\pi$
$510$	0	0
$511$	−12.8119	−0.566767
$512$	18.5188	0.818423
$513$	−5.35026	−0.236220
$514$	27.9756	1.23395
$515$	0	0
$516$	0.856849	0.0377207
$517$	12.3127	0.541510
$518$	−2.85097	−0.125264
$519$	−23.3054	−1.02299
$520$	0	0
$521$	−22.8119	−0.999409	−0.499705	−	0.866196i	$-0.666558\pi$
$521$	−22.8119	−0.999409	−0.499705	+	0.866196i	$0.666558\pi$
$522$	6.38787	0.279590
$523$	−12.2677	−0.536431	−0.268216	−	0.963359i	$-0.586434\pi$
$523$	−12.2677	−0.536431	−0.268216	+	0.963359i	$0.586434\pi$
$524$	−1.14903	−0.0501957
$525$	0	0
$526$	−30.8178	−1.34372
$527$	3.62672	0.157982
$528$	−4.35026	−0.189321
$529$	−7.00000	−0.304348
$530$	0	0
$531$	13.2750	0.576088
$532$	−1.23884	−0.0537106
$533$	7.32582	0.317317
$534$	−25.0132	−1.08243
$535$	0	0
$536$	−7.22425	−0.312040
$537$	−10.7005	−0.461762
$538$	47.9902	2.06900
$539$	−5.57452	−0.240111
$540$	0	0
$541$	−5.22425	−0.224608	−0.112304	−	0.993674i	$-0.535823\pi$
$541$	−5.22425	−0.224608	−0.112304	+	0.993674i	$0.535823\pi$
$542$	24.9234	1.07055
$543$	−7.79877	−0.334677
$544$	4.12013	0.176649
$545$	0	0
$546$	−1.42548	−0.0610051
$547$	−17.9697	−0.768328	−0.384164	−	0.923265i	$-0.625510\pi$
$547$	−17.9697	−0.768328	−0.384164	+	0.923265i	$0.625510\pi$
$548$	0.649738	0.0277554
$549$	−0.0752228	−0.00321043
$550$	0	0
$551$	23.0738	0.982977
$552$	10.7005	0.455445
$553$	16.6859	0.709558
$554$	−25.0435	−1.06400
$555$	0	0
$556$	−4.11142	−0.174363
$557$	15.8700	0.672434	0.336217	−	0.941784i	$-0.390852\pi$
$557$	15.8700	0.672434	0.336217	+	0.941784i	$0.390852\pi$
$558$	−1.42548	−0.0603456
$559$	3.56134	0.150629
$560$	0	0
$561$	3.76845	0.159104
$562$	−8.31265	−0.350648
$563$	31.6688	1.33468	0.667340	−	0.744753i	$-0.267433\pi$
$563$	31.6688	1.33468	0.667340	+	0.744753i	$0.267433\pi$
$564$	2.38787	0.100548
$565$	0	0
$566$	−8.61942	−0.362301
$567$	1.19394	0.0501406
$568$	−37.5877	−1.57714
$569$	−24.3127	−1.01924	−0.509620	−	0.860400i	$-0.670214\pi$
$569$	−24.3127	−1.01924	−0.509620	+	0.860400i	$0.670214\pi$
$570$	0	0
$571$	8.05079	0.336915	0.168457	−	0.985709i	$-0.446121\pi$
$571$	8.05079	0.336915	0.168457	+	0.985709i	$0.446121\pi$
$572$	0.156325	0.00653628
$573$	1.29948	0.0542864
$574$	−16.0724	−0.670849
$575$	0	0
$576$	7.08110	0.295046
$577$	−44.5355	−1.85404	−0.927018	−	0.375016i	$-0.877637\pi$
$577$	−44.5355	−1.85404	−0.927018	+	0.375016i	$0.877637\pi$
$578$	4.14552	0.172431
$579$	8.59498	0.357195
$580$	0	0
$581$	−11.8134	−0.490101
$582$	−16.9986	−0.704614
$583$	1.42548	0.0590375
$584$	−28.7064	−1.18788
$585$	0	0
$586$	12.2823	0.507379
$587$	−15.4763	−0.638774	−0.319387	−	0.947624i	$-0.603477\pi$
$587$	−15.4763	−0.638774	−0.319387	+	0.947624i	$0.603477\pi$
$588$	−1.08110	−0.0445839
$589$	−5.14903	−0.212162
$590$	0	0
$591$	−20.7064	−0.851748
$592$	−7.01317	−0.288240
$593$	−9.53102	−0.391392	−0.195696	−	0.980665i	$-0.562697\pi$
$593$	−9.53102	−0.391392	−0.195696	+	0.980665i	$0.562697\pi$
$594$	−1.48119	−0.0607741
$595$	0	0
$596$	1.23884	0.0507450
$597$	5.55149	0.227207
$598$	−4.77575	−0.195295
$599$	−25.5515	−1.04401	−0.522003	−	0.852944i	$-0.674815\pi$
$599$	−25.5515	−1.04401	−0.522003	+	0.852944i	$0.674815\pi$
$600$	0	0
$601$	12.0263	0.490565	0.245282	−	0.969452i	$-0.421119\pi$
$601$	12.0263	0.490565	0.245282	+	0.969452i	$0.421119\pi$
$602$	−7.81336	−0.318449
$603$	−2.70052	−0.109974
$604$	−0.513881	−0.0209095
$605$	0	0
$606$	−15.9394	−0.647492
$607$	−6.86670	−0.278711	−0.139355	−	0.990242i	$-0.544503\pi$
$607$	−6.86670	−0.278711	−0.139355	+	0.990242i	$0.544503\pi$
$608$	−5.84955	−0.237231
$609$	−5.14903	−0.208649
$610$	0	0
$611$	9.92478	0.401514
$612$	0.730841	0.0295425
$613$	−7.25457	−0.293009	−0.146505	−	0.989210i	$-0.546802\pi$
$613$	−7.25457	−0.293009	−0.146505	+	0.989210i	$0.546802\pi$
$614$	−38.0205	−1.53438
$615$	0	0
$616$	3.19394	0.128687
$617$	38.3634	1.54445	0.772227	−	0.635347i	$-0.219143\pi$
$617$	38.3634	1.54445	0.772227	+	0.635347i	$0.219143\pi$
$618$	−25.0884	−1.00920
$619$	−29.6893	−1.19331	−0.596656	−	0.802497i	$-0.703504\pi$
$619$	−29.6893	−1.19331	−0.596656	+	0.802497i	$0.703504\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	23.2896	0.933829
$623$	20.1622	0.807782
$624$	−3.50659	−0.140376
$625$	0	0
$626$	39.7889	1.59029
$627$	−5.35026	−0.213669
$628$	0.312650	0.0124761
$629$	6.07522	0.242235
$630$	0	0
$631$	−19.6991	−0.784209	−0.392105	−	0.919921i	$-0.628253\pi$
$631$	−19.6991	−0.784209	−0.392105	+	0.919921i	$0.628253\pi$
$632$	37.3865	1.48715
$633$	−18.4993	−0.735281
$634$	−1.05220	−0.0417882
$635$	0	0
$636$	0.276454	0.0109621
$637$	−4.49341	−0.178036
$638$	6.38787	0.252898
$639$	−14.0508	−0.555840
$640$	0	0
$641$	−31.4372	−1.24170	−0.620848	−	0.783931i	$-0.713212\pi$
$641$	−31.4372	−1.24170	−0.620848	+	0.783931i	$0.713212\pi$
$642$	−12.2677	−0.484169
$643$	−34.4894	−1.36013	−0.680065	−	0.733151i	$-0.738049\pi$
$643$	−34.4894	−1.36013	−0.680065	+	0.733151i	$0.738049\pi$
$644$	0.926192	0.0364971
$645$	0	0
$646$	29.8641	1.17499
$647$	−5.61213	−0.220635	−0.110318	−	0.993896i	$-0.535187\pi$
$647$	−5.61213	−0.220635	−0.110318	+	0.993896i	$0.535187\pi$
$648$	2.67513	0.105089
$649$	13.2750	0.521091
$650$	0	0
$651$	1.14903	0.0450341
$652$	−0.0606343	−0.00237462
$653$	4.06537	0.159090	0.0795452	−	0.996831i	$-0.474653\pi$
$653$	4.06537	0.159090	0.0795452	+	0.996831i	$0.474653\pi$
$654$	14.8119	0.579193
$655$	0	0
$656$	−39.5369	−1.54366
$657$	−10.7308	−0.418650
$658$	−21.7743	−0.848852
$659$	11.3747	0.443095	0.221548	−	0.975150i	$-0.428889\pi$
$659$	11.3747	0.443095	0.221548	+	0.975150i	$0.428889\pi$
$660$	0	0
$661$	−42.4749	−1.65208	−0.826040	−	0.563611i	$-0.809412\pi$
$661$	−42.4749	−1.65208	−0.826040	+	0.563611i	$0.809412\pi$
$662$	−1.42548	−0.0554030
$663$	3.03761	0.117971
$664$	−26.4690	−1.02720
$665$	0	0
$666$	−2.38787	−0.0925282
$667$	−17.2506	−0.667946
$668$	0.0956908	0.00370239
$669$	17.6121	0.680924
$670$	0	0
$671$	−0.0752228	−0.00290394
$672$	1.30536	0.0503552
$673$	−14.8813	−0.573631	−0.286816	−	0.957986i	$-0.592597\pi$
$673$	−14.8813	−0.573631	−0.286816	+	0.957986i	$0.592597\pi$
$674$	−29.3561	−1.13076
$675$	0	0
$676$	−2.39517	−0.0921218
$677$	−27.7685	−1.06723	−0.533614	−	0.845728i	$-0.679167\pi$
$677$	−27.7685	−1.06723	−0.533614	+	0.845728i	$0.679167\pi$
$678$	−3.35026	−0.128666
$679$	13.7019	0.525832
$680$	0	0
$681$	−17.4314	−0.667971
$682$	−1.42548	−0.0545846
$683$	−25.4617	−0.974264	−0.487132	−	0.873328i	$-0.661957\pi$
$683$	−25.4617	−0.974264	−0.487132	+	0.873328i	$0.661957\pi$
$684$	−1.03761	−0.0396741
$685$	0	0
$686$	22.2374	0.849029
$687$	13.0738	0.498797
$688$	−19.2203	−0.732766
$689$	1.14903	0.0437746
$690$	0	0
$691$	43.6991	1.66239	0.831196	−	0.555979i	$-0.187657\pi$
$691$	43.6991	1.66239	0.831196	+	0.555979i	$0.187657\pi$
$692$	−4.51976	−0.171816
$693$	1.19394	0.0453539
$694$	−9.19394	−0.348997
$695$	0	0
$696$	−11.5369	−0.437305
$697$	34.2492	1.29728
$698$	−6.58910	−0.249401
$699$	−13.8437	−0.523616
$700$	0	0
$701$	7.01317	0.264884	0.132442	−	0.991191i	$-0.457718\pi$
$701$	7.01317	0.264884	0.132442	+	0.991191i	$0.457718\pi$
$702$	−1.19394	−0.0450622
$703$	−8.62530	−0.325309
$704$	7.08110	0.266879
$705$	0	0
$706$	−9.73813	−0.366500
$707$	12.8481	0.483204
$708$	2.57452	0.0967562
$709$	45.6747	1.71535	0.857674	−	0.514194i	$-0.171909\pi$
$709$	45.6747	1.71535	0.857674	+	0.514194i	$0.171909\pi$
$710$	0	0
$711$	13.9756	0.524125
$712$	45.1754	1.69302
$713$	3.84955	0.144167
$714$	−6.66433	−0.249406
$715$	0	0
$716$	−2.07522	−0.0775547
$717$	−12.3733	−0.462089
$718$	−12.7757	−0.476787
$719$	16.2520	0.606098	0.303049	−	0.952975i	$-0.401995\pi$
$719$	16.2520	0.606098	0.303049	+	0.952975i	$0.401995\pi$
$720$	0	0
$721$	20.2228	0.753138
$722$	−14.2569	−0.530588
$723$	−24.5501	−0.913027
$724$	−1.51247	−0.0562104
$725$	0	0
$726$	−1.48119	−0.0549723
$727$	15.2243	0.564636	0.282318	−	0.959321i	$-0.408897\pi$
$727$	15.2243	0.564636	0.282318	+	0.959321i	$0.408897\pi$
$728$	2.57452	0.0954179
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.6497	0.615813
$732$	−0.0145884	−0.000539205 0
$733$	−43.5066	−1.60695	−0.803476	−	0.595337i	$-0.797019\pi$
$733$	−43.5066	−1.60695	−0.803476	+	0.595337i	$0.797019\pi$
$734$	−34.0870	−1.25817
$735$	0	0
$736$	4.37328	0.161201
$737$	−2.70052	−0.0994751
$738$	−13.4617	−0.495531
$739$	−7.02302	−0.258346	−0.129173	−	0.991622i	$-0.541232\pi$
$739$	−7.02302	−0.258346	−0.129173	+	0.991622i	$0.541232\pi$
$740$	0	0
$741$	−4.31265	−0.158429
$742$	−2.52090	−0.0925452
$743$	−2.94192	−0.107929	−0.0539643	−	0.998543i	$-0.517186\pi$
$743$	−2.94192	−0.107929	−0.0539643	+	0.998543i	$0.517186\pi$
$744$	2.57452	0.0943863
$745$	0	0
$746$	−41.7586	−1.52889
$747$	−9.89446	−0.362019
$748$	0.730841	0.0267222
$749$	9.88858	0.361321
$750$	0	0
$751$	24.1016	0.879479	0.439739	−	0.898125i	$-0.355071\pi$
$751$	24.1016	0.879479	0.439739	+	0.898125i	$0.355071\pi$
$752$	−53.5633	−1.95325
$753$	13.9003	0.506557
$754$	5.14903	0.187517
$755$	0	0
$756$	0.231548	0.00842132
$757$	−16.3127	−0.592893	−0.296447	−	0.955049i	$-0.595802\pi$
$757$	−16.3127	−0.592893	−0.296447	+	0.955049i	$0.595802\pi$
$758$	−5.55149	−0.201639
$759$	4.00000	0.145191
$760$	0	0
$761$	−5.08840	−0.184454	−0.0922271	−	0.995738i	$-0.529399\pi$
$761$	−5.08840	−0.184454	−0.0922271	+	0.995738i	$0.529399\pi$
$762$	20.4690	0.741513
$763$	−11.9394	−0.432234
$764$	0.252016	0.00911762
$765$	0	0
$766$	−2.61071	−0.0943289
$767$	10.7005	0.386374
$768$	4.61213	0.166426
$769$	2.10157	0.0757846	0.0378923	−	0.999282i	$-0.487936\pi$
$769$	2.10157	0.0757846	0.0378923	+	0.999282i	$0.487936\pi$
$770$	0	0
$771$	−18.8872	−0.680205
$772$	1.66688	0.0599924
$773$	46.0625	1.65675	0.828377	−	0.560171i	$-0.189264\pi$
$773$	46.0625	1.65675	0.828377	+	0.560171i	$0.189264\pi$
$774$	−6.54420	−0.235226
$775$	0	0
$776$	30.7005	1.10208
$777$	1.92478	0.0690510
$778$	−9.66291	−0.346432
$779$	−48.6253	−1.74218
$780$	0	0
$781$	−14.0508	−0.502777
$782$	−22.3272	−0.798420
$783$	−4.31265	−0.154122
$784$	24.2506	0.866093
$785$	0	0
$786$	8.77575	0.313021
$787$	−26.0303	−0.927881	−0.463940	−	0.885866i	$-0.653565\pi$
$787$	−26.0303	−0.927881	−0.463940	+	0.885866i	$0.653565\pi$
$788$	−4.01573	−0.143054
$789$	20.8061	0.740715
$790$	0	0
$791$	2.70052	0.0960196
$792$	2.67513	0.0950566
$793$	−0.0606343	−0.00215319
$794$	35.1030	1.24576
$795$	0	0
$796$	1.07664	0.0381604
$797$	−29.9902	−1.06231	−0.531153	−	0.847276i	$-0.678241\pi$
$797$	−29.9902	−1.06231	−0.531153	+	0.847276i	$0.678241\pi$
$798$	9.46168	0.334940
$799$	46.3996	1.64150
$800$	0	0
$801$	16.8872	0.596679
$802$	−13.1636	−0.464824
$803$	−10.7308	−0.378683
$804$	−0.523730	−0.0184705
$805$	0	0
$806$	−1.14903	−0.0404729
$807$	−32.3996	−1.14052
$808$	28.7875	1.01274
$809$	39.7597	1.39788	0.698939	−	0.715181i	$-0.253656\pi$
$809$	39.7597	1.39788	0.698939	+	0.715181i	$0.253656\pi$
$810$	0	0
$811$	−10.2765	−0.360855	−0.180428	−	0.983588i	$-0.557748\pi$
$811$	−10.2765	−0.360855	−0.180428	+	0.983588i	$0.557748\pi$
$812$	−0.998585	−0.0350435
$813$	−16.8265	−0.590132
$814$	−2.38787	−0.0836949
$815$	0	0
$816$	−16.3938	−0.573896
$817$	−23.6385	−0.827006
$818$	7.18523	0.251226
$819$	0.962389	0.0336286
$820$	0	0
$821$	50.0870	1.74805	0.874024	−	0.485883i	$-0.161502\pi$
$821$	50.0870	1.74805	0.874024	+	0.485883i	$0.161502\pi$
$822$	−4.96239	−0.173083
$823$	29.7137	1.03575	0.517877	−	0.855455i	$-0.326722\pi$
$823$	29.7137	1.03575	0.517877	+	0.855455i	$0.326722\pi$
$824$	45.3112	1.57849
$825$	0	0
$826$	−23.4763	−0.816844
$827$	−2.41819	−0.0840887	−0.0420444	−	0.999116i	$-0.513387\pi$
$827$	−2.41819	−0.0840887	−0.0420444	+	0.999116i	$0.513387\pi$
$828$	0.775746	0.0269590
$829$	−23.2750	−0.808376	−0.404188	−	0.914676i	$-0.632446\pi$
$829$	−23.2750	−0.808376	−0.404188	+	0.914676i	$0.632446\pi$
$830$	0	0
$831$	16.9076	0.586519
$832$	5.70782	0.197883
$833$	−21.0073	−0.727860
$834$	31.4010	1.08733
$835$	0	0
$836$	−1.03761	−0.0358865
$837$	0.962389	0.0332650
$838$	15.8496	0.547514
$839$	−18.8265	−0.649964	−0.324982	−	0.945720i	$-0.605358\pi$
$839$	−18.8265	−0.649964	−0.324982	+	0.945720i	$0.605358\pi$
$840$	0	0
$841$	−10.4010	−0.358657
$842$	45.3258	1.56203
$843$	5.61213	0.193292
$844$	−3.58769	−0.123493
$845$	0	0
$846$	−18.2374	−0.627016
$847$	1.19394	0.0410241
$848$	−6.20123	−0.212951
$849$	5.81924	0.199716
$850$	0	0
$851$	6.44851	0.221052
$852$	−2.72496	−0.0933556
$853$	−24.9076	−0.852821	−0.426411	−	0.904530i	$-0.640222\pi$
$853$	−24.9076	−0.852821	−0.426411	+	0.904530i	$0.640222\pi$
$854$	0.133028	0.00455212
$855$	0	0
$856$	22.1563	0.757288
$857$	−7.60625	−0.259824	−0.129912	−	0.991525i	$-0.541470\pi$
$857$	−7.60625	−0.259824	−0.129912	+	0.991525i	$0.541470\pi$
$858$	−1.19394	−0.0407603
$859$	−2.14060	−0.0730362	−0.0365181	−	0.999333i	$-0.511627\pi$
$859$	−2.14060	−0.0730362	−0.0365181	+	0.999333i	$0.511627\pi$
$860$	0	0
$861$	10.8510	0.369800
$862$	8.77575	0.298903
$863$	29.3112	0.997766	0.498883	−	0.866669i	$-0.333744\pi$
$863$	29.3112	0.997766	0.498883	+	0.866669i	$0.333744\pi$
$864$	1.09332	0.0371955
$865$	0	0
$866$	23.6991	0.805329
$867$	−2.79877	−0.0950512
$868$	0.222839	0.00756365
$869$	13.9756	0.474088
$870$	0	0
$871$	−2.17679	−0.0737578
$872$	−26.7513	−0.905914
$873$	11.4763	0.388413
$874$	31.6991	1.07224
$875$	0	0
$876$	−2.08110	−0.0703139
$877$	31.5183	1.06430	0.532149	−	0.846650i	$-0.321384\pi$
$877$	31.5183	1.06430	0.532149	+	0.846650i	$0.321384\pi$
$878$	−7.92478	−0.267448
$879$	−8.29218	−0.279688
$880$	0	0
$881$	10.6253	0.357975	0.178988	−	0.983851i	$-0.442718\pi$
$881$	10.6253	0.357975	0.178988	+	0.983851i	$0.442718\pi$
$882$	8.25694	0.278026
$883$	−6.29806	−0.211947	−0.105973	−	0.994369i	$-0.533796\pi$
$883$	−6.29806	−0.211947	−0.105973	+	0.994369i	$0.533796\pi$
$884$	0.589104	0.0198137
$885$	0	0
$886$	−29.1608	−0.979676
$887$	29.1187	0.977711	0.488855	−	0.872365i	$-0.337415\pi$
$887$	29.1187	0.977711	0.488855	+	0.872365i	$0.337415\pi$
$888$	4.31265	0.144723
$889$	−16.4993	−0.553369
$890$	0	0
$891$	1.00000	0.0335013
$892$	3.41564	0.114364
$893$	−65.8759	−2.20445
$894$	−9.46168	−0.316446
$895$	0	0
$896$	−15.1333	−0.505568
$897$	3.22425	0.107655
$898$	46.4142	1.54886
$899$	−4.15045	−0.138425
$900$	0	0
$901$	5.37187	0.178963
$902$	−13.4617	−0.448225
$903$	5.27504	0.175542
$904$	6.05079	0.201246
$905$	0	0
$906$	3.92478	0.130392
$907$	−57.3112	−1.90299	−0.951494	−	0.307667i	$-0.900452\pi$
$907$	−57.3112	−1.90299	−0.951494	+	0.307667i	$0.900452\pi$
$908$	−3.38058	−0.112188
$909$	10.7612	0.356925
$910$	0	0
$911$	−5.52232	−0.182962	−0.0914812	−	0.995807i	$-0.529160\pi$
$911$	−5.52232	−0.182962	−0.0914812	+	0.995807i	$0.529160\pi$
$912$	23.2750	0.770714
$913$	−9.89446	−0.327459
$914$	55.6834	1.84184
$915$	0	0
$916$	2.53549	0.0837749
$917$	−7.07381	−0.233598
$918$	−5.58181	−0.184227
$919$	−5.75272	−0.189765	−0.0948824	−	0.995488i	$-0.530248\pi$
$919$	−5.75272	−0.189765	−0.0948824	+	0.995488i	$0.530248\pi$
$920$	0	0
$921$	25.6688	0.845815
$922$	−26.6399	−0.877338
$923$	−11.3258	−0.372794
$924$	0.231548	0.00761737
$925$	0	0
$926$	−57.7861	−1.89897
$927$	16.9380	0.556315
$928$	−4.71511	−0.154781
$929$	41.4274	1.35919	0.679594	−	0.733588i	$-0.262156\pi$
$929$	41.4274	1.35919	0.679594	+	0.733588i	$0.262156\pi$
$930$	0	0
$931$	29.8251	0.977479
$932$	−2.68479	−0.0879434
$933$	−15.7235	−0.514766
$934$	−21.5515	−0.705186
$935$	0	0
$936$	2.15633	0.0704817
$937$	−14.9222	−0.487488	−0.243744	−	0.969840i	$-0.578376\pi$
$937$	−14.9222	−0.487488	−0.243744	+	0.969840i	$0.578376\pi$
$938$	4.77575	0.155934
$939$	−26.8627	−0.876632
$940$	0	0
$941$	57.0395	1.85944	0.929718	−	0.368273i	$-0.120051\pi$
$941$	57.0395	1.85944	0.929718	+	0.368273i	$0.120051\pi$
$942$	−2.38787	−0.0778011
$943$	36.3536	1.18384
$944$	−57.7499	−1.87960
$945$	0	0
$946$	−6.54420	−0.212770
$947$	30.2374	0.982584	0.491292	−	0.870995i	$-0.336525\pi$
$947$	30.2374	0.982584	0.491292	+	0.870995i	$0.336525\pi$
$948$	2.71037	0.0880288
$949$	−8.64974	−0.280782
$950$	0	0
$951$	0.710373	0.0230354
$952$	12.0362	0.390095
$953$	−46.2697	−1.49882	−0.749411	−	0.662106i	$-0.769663\pi$
$953$	−46.2697	−1.49882	−0.749411	+	0.662106i	$0.769663\pi$
$954$	−2.11142	−0.0683597
$955$	0	0
$956$	−2.39963	−0.0776097
$957$	−4.31265	−0.139408
$958$	−42.3996	−1.36987
$959$	4.00000	0.129167
$960$	0	0
$961$	−30.0738	−0.970123
$962$	−1.92478	−0.0620573
$963$	8.28233	0.266894
$964$	−4.76116	−0.153347
$965$	0	0
$966$	−7.07381	−0.227596
$967$	49.5790	1.59435	0.797176	−	0.603747i	$-0.206326\pi$
$967$	49.5790	1.59435	0.797176	+	0.603747i	$0.206326\pi$
$968$	2.67513	0.0859820
$969$	−20.1622	−0.647703
$970$	0	0
$971$	−3.12268	−0.100212	−0.0501058	−	0.998744i	$-0.515956\pi$
$971$	−3.12268	−0.100212	−0.0501058	+	0.998744i	$0.515956\pi$
$972$	0.193937	0.00622052
$973$	−25.3112	−0.811441
$974$	−2.14762	−0.0688141
$975$	0	0
$976$	0.327239	0.0104747
$977$	5.15377	0.164884	0.0824419	−	0.996596i	$-0.473728\pi$
$977$	5.15377	0.164884	0.0824419	+	0.996596i	$0.473728\pi$
$978$	0.463096	0.0148082
$979$	16.8872	0.539716
$980$	0	0
$981$	−10.0000	−0.319275
$982$	11.8496	0.378134
$983$	14.8627	0.474047	0.237024	−	0.971504i	$-0.423828\pi$
$983$	14.8627	0.474047	0.237024	+	0.971504i	$0.423828\pi$
$984$	24.3127	0.775059
$985$	0	0
$986$	24.0724	0.766621
$987$	14.7005	0.467923
$988$	−0.836381	−0.0266088
$989$	17.6728	0.561961
$990$	0	0
$991$	4.43866	0.140999	0.0704993	−	0.997512i	$-0.477541\pi$
$991$	4.43866	0.140999	0.0704993	+	0.997512i	$0.477541\pi$
$992$	1.05220	0.0334074
$993$	0.962389	0.0305405
$994$	24.8481	0.788135
$995$	0	0
$996$	−1.91890	−0.0608026
$997$	−38.8324	−1.22983	−0.614917	−	0.788592i	$-0.710811\pi$
$997$	−38.8324	−1.22983	−0.614917	+	0.788592i	$0.710811\pi$
$998$	−45.4734	−1.43944
$999$	1.61213	0.0510054

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.a.l.1.1		3
3.2	odd	2		2475.2.a.ba.1.3		3
5.2	odd	4		165.2.c.b.34.2	✓	6
5.3	odd	4		165.2.c.b.34.5	yes	6
5.4	even	2		825.2.a.j.1.3		3
11.10	odd	2		9075.2.a.cg.1.3		3
15.2	even	4		495.2.c.e.199.5		6
15.8	even	4		495.2.c.e.199.2		6
15.14	odd	2		2475.2.a.bc.1.1		3
20.3	even	4		2640.2.d.h.529.1		6
20.7	even	4		2640.2.d.h.529.4		6
55.32	even	4		1815.2.c.e.364.5		6
55.43	even	4		1815.2.c.e.364.2		6
55.54	odd	2		9075.2.a.ch.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.c.b.34.2	✓	6	5.2	odd	4
165.2.c.b.34.5	yes	6	5.3	odd	4
495.2.c.e.199.2		6	15.8	even	4
495.2.c.e.199.5		6	15.2	even	4
825.2.a.j.1.3		3	5.4	even	2
825.2.a.l.1.1		3	1.1	even	1	trivial
1815.2.c.e.364.2		6	55.43	even	4
1815.2.c.e.364.5		6	55.32	even	4
2475.2.a.ba.1.3		3	3.2	odd	2
2475.2.a.bc.1.1		3	15.14	odd	2
2640.2.d.h.529.1		6	20.3	even	4
2640.2.d.h.529.4		6	20.7	even	4
9075.2.a.cg.1.3		3	11.10	odd	2
9075.2.a.ch.1.1		3	55.54	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q-1.48119 q^{2} +1.00000 q^{3} +0.193937 q^{4} -1.48119 q^{6} +1.19394 q^{7} +2.67513 q^{8} +1.00000 q^{9} +1.00000 q^{11} +0.193937 q^{12} +0.806063 q^{13} -1.76845 q^{14} -4.35026 q^{16} +3.76845 q^{17} -1.48119 q^{18} -5.35026 q^{19} +1.19394 q^{21} -1.48119 q^{22} +4.00000 q^{23} +2.67513 q^{24} -1.19394 q^{26} +1.00000 q^{27} +0.231548 q^{28} -4.31265 q^{29} +0.962389 q^{31} +1.09332 q^{32} +1.00000 q^{33} -5.58181 q^{34} +0.193937 q^{36} +1.61213 q^{37} +7.92478 q^{38} +0.806063 q^{39} +9.08840 q^{41} -1.76845 q^{42} +4.41819 q^{43} +0.193937 q^{44} -5.92478 q^{46} +12.3127 q^{47} -4.35026 q^{48} -5.57452 q^{49} +3.76845 q^{51} +0.156325 q^{52} +1.42548 q^{53} -1.48119 q^{54} +3.19394 q^{56} -5.35026 q^{57} +6.38787 q^{58} +13.2750 q^{59} -0.0752228 q^{61} -1.42548 q^{62} +1.19394 q^{63} +7.08110 q^{64} -1.48119 q^{66} -2.70052 q^{67} +0.730841 q^{68} +4.00000 q^{69} -14.0508 q^{71} +2.67513 q^{72} -10.7308 q^{73} -2.38787 q^{74} -1.03761 q^{76} +1.19394 q^{77} -1.19394 q^{78} +13.9756 q^{79} +1.00000 q^{81} -13.4617 q^{82} -9.89446 q^{83} +0.231548 q^{84} -6.54420 q^{86} -4.31265 q^{87} +2.67513 q^{88} +16.8872 q^{89} +0.962389 q^{91} +0.775746 q^{92} +0.962389 q^{93} -18.2374 q^{94} +1.09332 q^{96} +11.4763 q^{97} +8.25694 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{3} + q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 3 q^{9} + 3 q^{11} + q^{12} + 2 q^{13} + 6 q^{14} - 3 q^{16} + q^{18} - 6 q^{19} + 4 q^{21} + q^{22} + 12 q^{23} + 3 q^{24} - 4 q^{26} + 3 q^{27}+ \cdots + 3 q^{99}+O(q^{100}) \) 3 * q + q^2 + 3 * q^3 + q^4 + q^6 + 4 * q^7 + 3 * q^8 + 3 * q^9 + 3 * q^11 + q^12 + 2 * q^13 + 6 * q^14 - 3 * q^16 + q^18 - 6 * q^19 + 4 * q^21 + q^22 + 12 * q^23 + 3 * q^24 - 4 * q^26 + 3 * q^27 + 12 * q^28 + 8 * q^29 - 8 * q^31 - 3 * q^32 + 3 * q^33 - 18 * q^34 + q^36 + 4 * q^37 + 2 * q^38 + 2 * q^39 + 8 * q^41 + 6 * q^42 + 12 * q^43 + q^44 + 4 * q^46 + 16 * q^47 - 3 * q^48 - 5 * q^49 - 10 * q^52 + 16 * q^53 + q^54 + 10 * q^56 - 6 * q^57 + 20 * q^58 + 8 * q^59 - 22 * q^61 - 16 * q^62 + 4 * q^63 - 11 * q^64 + q^66 + 12 * q^67 - 20 * q^68 + 12 * q^69 - 12 * q^71 + 3 * q^72 - 10 * q^73 - 8 * q^74 - 14 * q^76 + 4 * q^77 - 4 * q^78 - 10 * q^79 + 3 * q^81 + 4 * q^82 - 10 * q^83 + 12 * q^84 - 10 * q^86 + 8 * q^87 + 3 * q^88 + 18 * q^89 - 8 * q^91 + 4 * q^92 - 8 * q^93 - 12 * q^94 - 3 * q^96 + 16 * q^97 + 21 * q^98 + 3 * q^99

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