LMFDB - Embedded newform 6975.2.a.bv.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] 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("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,3,0,3,0,0,-8,9,0,0,0,0,-6,-16,0,3,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.126032.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} + 6x - 2 $ x^5 - 6x^3 + 6x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.815403$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.697747 q^{2} -1.51315 q^{4} -0.302253 q^{7} +2.45129 q^{8} -2.71947 q^{11} -3.96444 q^{13} +0.210896 q^{14} +1.31592 q^{16} +6.05458 q^{17} +0.452774 q^{19} +1.89750 q^{22} -4.37050 q^{23} +2.76617 q^{26} +0.457355 q^{28} +3.48564 q^{29} -1.00000 q^{31} -5.82076 q^{32} -4.22456 q^{34} +1.56089 q^{37} -0.315922 q^{38} +9.00441 q^{41} -8.93693 q^{43} +4.11496 q^{44} +3.04950 q^{46} +6.64299 q^{47} -6.90864 q^{49} +5.99879 q^{52} +8.49423 q^{53} -0.740910 q^{56} -2.43209 q^{58} +3.25033 q^{59} -5.35535 q^{61} +0.697747 q^{62} +1.42957 q^{64} +4.26548 q^{67} -9.16149 q^{68} -8.83190 q^{71} -10.4968 q^{73} -1.08911 q^{74} -0.685115 q^{76} +0.821968 q^{77} +14.6348 q^{79} -6.28279 q^{82} +10.2063 q^{83} +6.23571 q^{86} -6.66619 q^{88} -15.6279 q^{89} +1.19827 q^{91} +6.61323 q^{92} -4.63512 q^{94} +8.85267 q^{97} +4.82048 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} + 3 q^{4} - 8 q^{7} + 9 q^{8} - 6 q^{13} - 16 q^{14} + 3 q^{16} + 8 q^{17} - 4 q^{19} - 6 q^{22} + 4 q^{23} + 6 q^{26} - 18 q^{28} - 4 q^{29} - 5 q^{31} + q^{32} + 6 q^{34} + 4 q^{37} + 2 q^{38}+ \cdots + 29 q^{98}+O(q^{100}) $ 5 * q + 3 * q^2 + 3 * q^4 - 8 * q^7 + 9 * q^8 - 6 * q^13 - 16 * q^14 + 3 * q^16 + 8 * q^17 - 4 * q^19 - 6 * q^22 + 4 * q^23 + 6 * q^26 - 18 * q^28 - 4 * q^29 - 5 * q^31 + q^32 + 6 * q^34 + 4 * q^37 + 2 * q^38 - 20 * q^43 - 6 * q^44 - 16 * q^46 + 18 * q^47 - 11 * q^49 - 2 * q^52 + 2 * q^53 - 18 * q^56 - 14 * q^58 - 10 * q^59 - 14 * q^61 - 3 * q^62 + 11 * q^64 - 18 * q^67 + 6 * q^68 - 6 * q^71 - 10 * q^73 + 4 * q^74 - 8 * q^76 + 6 * q^77 - 24 * q^79 - 30 * q^82 + 48 * q^83 - 4 * q^86 - 8 * q^88 - 10 * q^89 - 26 * q^92 + 12 * q^94 - 20 * q^97 + 29 * 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 $

$2$	−0.697747	−0.493381	−0.246691	−	0.969094i	$-0.579343\pi$
$2$	−0.697747	−0.493381	−0.246691	+	0.969094i	$0.579343\pi$
$3$	0	0
$3$	0	0
$4$	−1.51315	−0.756575
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.302253	−0.114241	−0.0571205	−	0.998367i	$-0.518192\pi$
$7$	−0.302253	−0.114241	−0.0571205	+	0.998367i	$0.518192\pi$
$8$	2.45129	0.866661
$9$	0	0
$10$	0	0
$11$	−2.71947	−0.819950	−0.409975	−	0.912097i	$-0.634463\pi$
$11$	−2.71947	−0.819950	−0.409975	+	0.912097i	$0.634463\pi$
$12$	0	0
$13$	−3.96444	−1.09954	−0.549769	−	0.835317i	$-0.685284\pi$
$13$	−3.96444	−1.09954	−0.549769	+	0.835317i	$0.685284\pi$
$14$	0.210896	0.0563644
$15$	0	0
$16$	1.31592	0.328980
$17$	6.05458	1.46845	0.734226	−	0.678905i	$-0.237545\pi$
$17$	6.05458	1.46845	0.734226	+	0.678905i	$0.237545\pi$
$18$	0	0
$19$	0.452774	0.103874	0.0519368	−	0.998650i	$-0.483461\pi$
$19$	0.452774	0.103874	0.0519368	+	0.998650i	$0.483461\pi$
$20$	0	0
$21$	0	0
$22$	1.89750	0.404548
$23$	−4.37050	−0.911313	−0.455657	−	0.890156i	$-0.650596\pi$
$23$	−4.37050	−0.911313	−0.455657	+	0.890156i	$0.650596\pi$
$24$	0	0
$25$	0	0
$26$	2.76617	0.542491
$27$	0	0
$28$	0.457355	0.0864319
$29$	3.48564	0.647267	0.323633	−	0.946183i	$-0.395096\pi$
$29$	3.48564	0.647267	0.323633	+	0.946183i	$0.395096\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−5.82076	−1.02897
$33$	0	0
$34$	−4.22456	−0.724507
$35$	0	0
$36$	0	0
$37$	1.56089	0.256609	0.128305	−	0.991735i	$-0.459046\pi$
$37$	1.56089	0.256609	0.128305	+	0.991735i	$0.459046\pi$
$38$	−0.315922	−0.0512493
$39$	0	0
$40$	0	0
$41$	9.00441	1.40625	0.703126	−	0.711065i	$-0.251787\pi$
$41$	9.00441	1.40625	0.703126	+	0.711065i	$0.251787\pi$
$42$	0	0
$43$	−8.93693	−1.36287	−0.681434	−	0.731879i	$-0.738643\pi$
$43$	−8.93693	−1.36287	−0.681434	+	0.731879i	$0.738643\pi$
$44$	4.11496	0.620353
$45$	0	0
$46$	3.04950	0.449625
$47$	6.64299	0.968979	0.484490	−	0.874797i	$-0.339005\pi$
$47$	6.64299	0.968979	0.484490	+	0.874797i	$0.339005\pi$
$48$	0	0
$49$	−6.90864	−0.986949
$50$	0	0
$51$	0	0
$52$	5.99879	0.831882
$53$	8.49423	1.16677	0.583386	−	0.812195i	$-0.301728\pi$
$53$	8.49423	1.16677	0.583386	+	0.812195i	$0.301728\pi$
$54$	0	0
$55$	0	0
$56$	−0.740910	−0.0990083
$57$	0	0
$58$	−2.43209	−0.319349
$59$	3.25033	0.423156	0.211578	−	0.977361i	$-0.432140\pi$
$59$	3.25033	0.423156	0.211578	+	0.977361i	$0.432140\pi$
$60$	0	0
$61$	−5.35535	−0.685682	−0.342841	−	0.939393i	$-0.611389\pi$
$61$	−5.35535	−0.685682	−0.342841	+	0.939393i	$0.611389\pi$
$62$	0.697747	0.0886139
$63$	0	0
$64$	1.42957	0.178696
$65$	0	0
$66$	0	0
$67$	4.26548	0.521111	0.260556	−	0.965459i	$-0.416094\pi$
$67$	4.26548	0.521111	0.260556	+	0.965459i	$0.416094\pi$
$68$	−9.16149	−1.11099
$69$	0	0
$70$	0	0
$71$	−8.83190	−1.04815	−0.524077	−	0.851671i	$-0.675590\pi$
$71$	−8.83190	−1.04815	−0.524077	+	0.851671i	$0.675590\pi$
$72$	0	0
$73$	−10.4968	−1.22855	−0.614276	−	0.789091i	$-0.710552\pi$
$73$	−10.4968	−1.22855	−0.614276	+	0.789091i	$0.710552\pi$
$74$	−1.08911	−0.126606
$75$	0	0
$76$	−0.685115	−0.0785881
$77$	0.821968	0.0936719
$78$	0	0
$79$	14.6348	1.64654	0.823270	−	0.567650i	$-0.192147\pi$
$79$	14.6348	1.64654	0.823270	+	0.567650i	$0.192147\pi$
$80$	0	0
$81$	0	0
$82$	−6.28279	−0.693818
$83$	10.2063	1.12029	0.560144	−	0.828395i	$-0.310746\pi$
$83$	10.2063	1.12029	0.560144	+	0.828395i	$0.310746\pi$
$84$	0	0
$85$	0	0
$86$	6.23571	0.672414
$87$	0	0
$88$	−6.66619	−0.710619
$89$	−15.6279	−1.65656	−0.828279	−	0.560316i	$-0.810680\pi$
$89$	−15.6279	−1.65656	−0.828279	+	0.560316i	$0.810680\pi$
$90$	0	0
$91$	1.19827	0.125612
$92$	6.61323	0.689477
$93$	0	0
$94$	−4.63512	−0.478076
$95$	0	0
$96$	0	0
$97$	8.85267	0.898853	0.449426	−	0.893317i	$-0.351628\pi$
$97$	8.85267	0.898853	0.449426	+	0.893317i	$0.351628\pi$
$98$	4.82048	0.486942
$99$	0	0
$100$	0	0
$101$	18.9217	1.88278	0.941390	−	0.337321i	$-0.109521\pi$
$101$	18.9217	1.88278	0.941390	+	0.337321i	$0.109521\pi$
$102$	0	0
$103$	1.14577	0.112896	0.0564478	−	0.998406i	$-0.482023\pi$
$103$	1.14577	0.112896	0.0564478	+	0.998406i	$0.482023\pi$
$104$	−9.71798	−0.952926
$105$	0	0
$106$	−5.92682	−0.575663
$107$	8.34317	0.806565	0.403282	−	0.915076i	$-0.367869\pi$
$107$	8.34317	0.806565	0.403282	+	0.915076i	$0.367869\pi$
$108$	0	0
$109$	−15.8135	−1.51466	−0.757328	−	0.653034i	$-0.773496\pi$
$109$	−15.8135	−1.51466	−0.757328	+	0.653034i	$0.773496\pi$
$110$	0	0
$111$	0	0
$112$	−0.397742	−0.0375831
$113$	−3.55793	−0.334702	−0.167351	−	0.985897i	$-0.553521\pi$
$113$	−3.55793	−0.334702	−0.167351	+	0.985897i	$0.553521\pi$
$114$	0	0
$115$	0	0
$116$	−5.27429	−0.489706
$117$	0	0
$118$	−2.26790	−0.208777
$119$	−1.83002	−0.167758
$120$	0	0
$121$	−3.60451	−0.327682
$122$	3.73668	0.338303
$123$	0	0
$124$	1.51315	0.135885
$125$	0	0
$126$	0	0
$127$	5.58891	0.495936	0.247968	−	0.968768i	$-0.420237\pi$
$127$	5.58891	0.495936	0.247968	+	0.968768i	$0.420237\pi$
$128$	10.6440	0.940809
$129$	0	0
$130$	0	0
$131$	−14.5221	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$131$	−14.5221	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$132$	0	0
$133$	−0.136853	−0.0118666
$134$	−2.97622	−0.257107
$135$	0	0
$136$	14.8415	1.27265
$137$	−2.20929	−0.188752	−0.0943761	−	0.995537i	$-0.530086\pi$
$137$	−2.20929	−0.188752	−0.0943761	+	0.995537i	$0.530086\pi$
$138$	0	0
$139$	−1.96283	−0.166485	−0.0832425	−	0.996529i	$-0.526528\pi$
$139$	−1.96283	−0.166485	−0.0832425	+	0.996529i	$0.526528\pi$
$140$	0	0
$141$	0	0
$142$	6.16243	0.517140
$143$	10.7812	0.901565
$144$	0	0
$145$	0	0
$146$	7.32407	0.606145
$147$	0	0
$148$	−2.36187	−0.194144
$149$	−15.5134	−1.27091	−0.635455	−	0.772138i	$-0.719187\pi$
$149$	−15.5134	−1.27091	−0.635455	+	0.772138i	$0.719187\pi$
$150$	0	0
$151$	3.50914	0.285570	0.142785	−	0.989754i	$-0.454394\pi$
$151$	3.50914	0.285570	0.142785	+	0.989754i	$0.454394\pi$
$152$	1.10988	0.0900232
$153$	0	0
$154$	−0.573525	−0.0462160
$155$	0	0
$156$	0	0
$157$	−23.0764	−1.84170	−0.920849	−	0.389920i	$-0.872503\pi$
$157$	−23.0764	−1.84170	−0.920849	+	0.389920i	$0.872503\pi$
$158$	−10.2114	−0.812372
$159$	0	0
$160$	0	0
$161$	1.32100	0.104109
$162$	0	0
$163$	16.6343	1.30290	0.651450	−	0.758691i	$-0.274161\pi$
$163$	16.6343	1.30290	0.651450	+	0.758691i	$0.274161\pi$
$164$	−13.6250	−1.06393
$165$	0	0
$166$	−7.12142	−0.552730
$167$	5.65485	0.437585	0.218793	−	0.975771i	$-0.429788\pi$
$167$	5.65485	0.437585	0.218793	+	0.975771i	$0.429788\pi$
$168$	0	0
$169$	2.71677	0.208982
$170$	0	0
$171$	0	0
$172$	13.5229	1.03111
$173$	14.1686	1.07722	0.538610	−	0.842555i	$-0.318949\pi$
$173$	14.1686	1.07722	0.538610	+	0.842555i	$0.318949\pi$
$174$	0	0
$175$	0	0
$176$	−3.57860	−0.269747
$177$	0	0
$178$	10.9043	0.817315
$179$	5.04745	0.377264	0.188632	−	0.982048i	$-0.439595\pi$
$179$	5.04745	0.377264	0.188632	+	0.982048i	$0.439595\pi$
$180$	0	0
$181$	−2.43964	−0.181337	−0.0906687	−	0.995881i	$-0.528900\pi$
$181$	−2.43964	−0.181337	−0.0906687	+	0.995881i	$0.528900\pi$
$182$	−0.836085	−0.0619748
$183$	0	0
$184$	−10.7134	−0.789800
$185$	0	0
$186$	0	0
$187$	−16.4652	−1.20406
$188$	−10.0518	−0.733106
$189$	0	0
$190$	0	0
$191$	−4.10435	−0.296981	−0.148490	−	0.988914i	$-0.547441\pi$
$191$	−4.10435	−0.296981	−0.148490	+	0.988914i	$0.547441\pi$
$192$	0	0
$193$	−16.5430	−1.19079	−0.595396	−	0.803432i	$-0.703005\pi$
$193$	−16.5430	−1.19079	−0.595396	+	0.803432i	$0.703005\pi$
$194$	−6.17692	−0.443477
$195$	0	0
$196$	10.4538	0.746701
$197$	−15.5640	−1.10889	−0.554443	−	0.832221i	$-0.687069\pi$
$197$	−15.5640	−1.10889	−0.554443	+	0.832221i	$0.687069\pi$
$198$	0	0
$199$	−2.17021	−0.153842	−0.0769212	−	0.997037i	$-0.524509\pi$
$199$	−2.17021	−0.153842	−0.0769212	+	0.997037i	$0.524509\pi$
$200$	0	0
$201$	0	0
$202$	−13.2025	−0.928928
$203$	−1.05355	−0.0739445
$204$	0	0
$205$	0	0
$206$	−0.799454	−0.0557006
$207$	0	0
$208$	−5.21689	−0.361726
$209$	−1.23130	−0.0851711
$210$	0	0
$211$	19.8604	1.36724	0.683622	−	0.729836i	$-0.260404\pi$
$211$	19.8604	1.36724	0.683622	+	0.729836i	$0.260404\pi$
$212$	−12.8530	−0.882750
$213$	0	0
$214$	−5.82142	−0.397944
$215$	0	0
$216$	0	0
$217$	0.302253	0.0205183
$218$	11.0338	0.747303
$219$	0	0
$220$	0	0
$221$	−24.0030	−1.61462
$222$	0	0
$223$	2.56154	0.171534	0.0857668	−	0.996315i	$-0.472666\pi$
$223$	2.56154	0.171534	0.0857668	+	0.996315i	$0.472666\pi$
$224$	1.75934	0.117551
$225$	0	0
$226$	2.48253	0.165136
$227$	4.46459	0.296325	0.148163	−	0.988963i	$-0.452664\pi$
$227$	4.46459	0.296325	0.148163	+	0.988963i	$0.452664\pi$
$228$	0	0
$229$	−5.62097	−0.371444	−0.185722	−	0.982602i	$-0.559462\pi$
$229$	−5.62097	−0.371444	−0.185722	+	0.982602i	$0.559462\pi$
$230$	0	0
$231$	0	0
$232$	8.54430	0.560961
$233$	4.15427	0.272155	0.136077	−	0.990698i	$-0.456550\pi$
$233$	4.15427	0.272155	0.136077	+	0.990698i	$0.456550\pi$
$234$	0	0
$235$	0	0
$236$	−4.91823	−0.320149
$237$	0	0
$238$	1.27689	0.0827684
$239$	−19.9161	−1.28827	−0.644134	−	0.764913i	$-0.722782\pi$
$239$	−19.9161	−1.28827	−0.644134	+	0.764913i	$0.722782\pi$
$240$	0	0
$241$	−23.9190	−1.54076	−0.770380	−	0.637584i	$-0.779934\pi$
$241$	−23.9190	−1.54076	−0.770380	+	0.637584i	$0.779934\pi$
$242$	2.51503	0.161672
$243$	0	0
$244$	8.10345	0.518770
$245$	0	0
$246$	0	0
$247$	−1.79500	−0.114213
$248$	−2.45129	−0.155657
$249$	0	0
$250$	0	0
$251$	−25.3168	−1.59798	−0.798992	−	0.601341i	$-0.794633\pi$
$251$	−25.3168	−1.59798	−0.798992	+	0.601341i	$0.794633\pi$
$252$	0	0
$253$	11.8854	0.747231
$254$	−3.89964	−0.244685
$255$	0	0
$256$	−10.2860	−0.642874
$257$	3.55326	0.221646	0.110823	−	0.993840i	$-0.464651\pi$
$257$	3.55326	0.221646	0.110823	+	0.993840i	$0.464651\pi$
$258$	0	0
$259$	−0.471786	−0.0293153
$260$	0	0
$261$	0	0
$262$	10.1327	0.626003
$263$	10.2740	0.633524	0.316762	−	0.948505i	$-0.397404\pi$
$263$	10.2740	0.633524	0.316762	+	0.948505i	$0.397404\pi$
$264$	0	0
$265$	0	0
$266$	0.0954884	0.00585477
$267$	0	0
$268$	−6.45431	−0.394260
$269$	−8.38317	−0.511131	−0.255565	−	0.966792i	$-0.582262\pi$
$269$	−8.38317	−0.511131	−0.255565	+	0.966792i	$0.582262\pi$
$270$	0	0
$271$	8.34595	0.506980	0.253490	−	0.967338i	$-0.418421\pi$
$271$	8.34595	0.506980	0.253490	+	0.967338i	$0.418421\pi$
$272$	7.96736	0.483092
$273$	0	0
$274$	1.54152	0.0931268
$275$	0	0
$276$	0	0
$277$	6.19221	0.372054	0.186027	−	0.982545i	$-0.440439\pi$
$277$	6.19221	0.372054	0.186027	+	0.982545i	$0.440439\pi$
$278$	1.36956	0.0821406
$279$	0	0
$280$	0	0
$281$	−19.6063	−1.16962	−0.584808	−	0.811172i	$-0.698830\pi$
$281$	−19.6063	−1.16962	−0.584808	+	0.811172i	$0.698830\pi$
$282$	0	0
$283$	32.4535	1.92916	0.964581	−	0.263789i	$-0.0849721\pi$
$283$	32.4535	1.92916	0.964581	+	0.263789i	$0.0849721\pi$
$284$	13.3640	0.793007
$285$	0	0
$286$	−7.52251	−0.444815
$287$	−2.72161	−0.160652
$288$	0	0
$289$	19.6580	1.15635
$290$	0	0
$291$	0	0
$292$	15.8832	0.929491
$293$	−14.7700	−0.862873	−0.431437	−	0.902143i	$-0.641993\pi$
$293$	−14.7700	−0.862873	−0.431437	+	0.902143i	$0.641993\pi$
$294$	0	0
$295$	0	0
$296$	3.82620	0.222393
$297$	0	0
$298$	10.8244	0.627043
$299$	17.3266	1.00202
$300$	0	0
$301$	2.70122	0.155696
$302$	−2.44849	−0.140895
$303$	0	0
$304$	0.595815	0.0341724
$305$	0	0
$306$	0	0
$307$	−23.5695	−1.34518	−0.672590	−	0.740015i	$-0.734818\pi$
$307$	−23.5695	−1.34518	−0.672590	+	0.740015i	$0.734818\pi$
$308$	−1.24376	−0.0708698
$309$	0	0
$310$	0	0
$311$	9.96325	0.564964	0.282482	−	0.959273i	$-0.408842\pi$
$311$	9.96325	0.564964	0.282482	+	0.959273i	$0.408842\pi$
$312$	0	0
$313$	10.1209	0.572067	0.286033	−	0.958220i	$-0.407663\pi$
$313$	10.1209	0.572067	0.286033	+	0.958220i	$0.407663\pi$
$314$	16.1015	0.908659
$315$	0	0
$316$	−22.1446	−1.24573
$317$	2.12717	0.119474	0.0597370	−	0.998214i	$-0.480974\pi$
$317$	2.12717	0.119474	0.0597370	+	0.998214i	$0.480974\pi$
$318$	0	0
$319$	−9.47907	−0.530726
$320$	0	0
$321$	0	0
$322$	−0.921723	−0.0513656
$323$	2.74136	0.152533
$324$	0	0
$325$	0	0
$326$	−11.6065	−0.642827
$327$	0	0
$328$	22.0724	1.21874
$329$	−2.00787	−0.110697
$330$	0	0
$331$	5.67365	0.311852	0.155926	−	0.987769i	$-0.450164\pi$
$331$	5.67365	0.311852	0.155926	+	0.987769i	$0.450164\pi$
$332$	−15.4437	−0.847582
$333$	0	0
$334$	−3.94565	−0.215896
$335$	0	0
$336$	0	0
$337$	−8.30083	−0.452175	−0.226087	−	0.974107i	$-0.572593\pi$
$337$	−8.30083	−0.452175	−0.226087	+	0.974107i	$0.572593\pi$
$338$	−1.89562	−0.103108
$339$	0	0
$340$	0	0
$341$	2.71947	0.147267
$342$	0	0
$343$	4.20394	0.226991
$344$	−21.9070	−1.18115
$345$	0	0
$346$	−9.88611	−0.531481
$347$	16.8933	0.906879	0.453440	−	0.891287i	$-0.350197\pi$
$347$	16.8933	0.906879	0.453440	+	0.891287i	$0.350197\pi$
$348$	0	0
$349$	−11.1328	−0.595925	−0.297963	−	0.954578i	$-0.596307\pi$
$349$	−11.1328	−0.595925	−0.297963	+	0.954578i	$0.596307\pi$
$350$	0	0
$351$	0	0
$352$	15.8293	0.843707
$353$	2.73101	0.145357	0.0726784	−	0.997355i	$-0.476845\pi$
$353$	2.73101	0.145357	0.0726784	+	0.997355i	$0.476845\pi$
$354$	0	0
$355$	0	0
$356$	23.6474	1.25331
$357$	0	0
$358$	−3.52184	−0.186135
$359$	−7.53229	−0.397539	−0.198770	−	0.980046i	$-0.563695\pi$
$359$	−7.53229	−0.397539	−0.198770	+	0.980046i	$0.563695\pi$
$360$	0	0
$361$	−18.7950	−0.989210
$362$	1.70225	0.0894685
$363$	0	0
$364$	−1.81315	−0.0950351
$365$	0	0
$366$	0	0
$367$	−25.4201	−1.32692	−0.663458	−	0.748213i	$-0.730912\pi$
$367$	−25.4201	−1.32692	−0.663458	+	0.748213i	$0.730912\pi$
$368$	−5.75124	−0.299804
$369$	0	0
$370$	0	0
$371$	−2.56741	−0.133293
$372$	0	0
$373$	12.8612	0.665929	0.332965	−	0.942939i	$-0.391951\pi$
$373$	12.8612	0.665929	0.332965	+	0.942939i	$0.391951\pi$
$374$	11.4886	0.594059
$375$	0	0
$376$	16.2839	0.839777
$377$	−13.8186	−0.711694
$378$	0	0
$379$	27.2868	1.40163	0.700816	−	0.713342i	$-0.252820\pi$
$379$	27.2868	1.40163	0.700816	+	0.713342i	$0.252820\pi$
$380$	0	0
$381$	0	0
$382$	2.86380	0.146525
$383$	−19.4253	−0.992586	−0.496293	−	0.868155i	$-0.665306\pi$
$383$	−19.4253	−0.992586	−0.496293	+	0.868155i	$0.665306\pi$
$384$	0	0
$385$	0	0
$386$	11.5428	0.587515
$387$	0	0
$388$	−13.3954	−0.680049
$389$	−33.8195	−1.71471	−0.857357	−	0.514722i	$-0.827895\pi$
$389$	−33.8195	−1.71471	−0.857357	+	0.514722i	$0.827895\pi$
$390$	0	0
$391$	−26.4616	−1.33822
$392$	−16.9351	−0.855350
$393$	0	0
$394$	10.8597	0.547104
$395$	0	0
$396$	0	0
$397$	5.75310	0.288740	0.144370	−	0.989524i	$-0.453884\pi$
$397$	5.75310	0.288740	0.144370	+	0.989524i	$0.453884\pi$
$398$	1.51426	0.0759030
$399$	0	0
$400$	0	0
$401$	−30.9973	−1.54793	−0.773965	−	0.633228i	$-0.781730\pi$
$401$	−30.9973	−1.54793	−0.773965	+	0.633228i	$0.781730\pi$
$402$	0	0
$403$	3.96444	0.197483
$404$	−28.6314	−1.42446
$405$	0	0
$406$	0.735108	0.0364828
$407$	−4.24480	−0.210407
$408$	0	0
$409$	0.233213	0.0115316	0.00576582	−	0.999983i	$-0.498165\pi$
$409$	0.233213	0.0115316	0.00576582	+	0.999983i	$0.498165\pi$
$410$	0	0
$411$	0	0
$412$	−1.73371	−0.0854140
$413$	−0.982422	−0.0483418
$414$	0	0
$415$	0	0
$416$	23.0760	1.13140
$417$	0	0
$418$	0.859138	0.0420218
$419$	−3.34632	−0.163478	−0.0817391	−	0.996654i	$-0.526047\pi$
$419$	−3.34632	−0.163478	−0.0817391	+	0.996654i	$0.526047\pi$
$420$	0	0
$421$	−9.12325	−0.444640	−0.222320	−	0.974974i	$-0.571363\pi$
$421$	−9.12325	−0.444640	−0.222320	+	0.974974i	$0.571363\pi$
$422$	−13.8575	−0.674573
$423$	0	0
$424$	20.8218	1.01120
$425$	0	0
$426$	0	0
$427$	1.61867	0.0783331
$428$	−12.6245	−0.610226
$429$	0	0
$430$	0	0
$431$	−21.4495	−1.03318	−0.516592	−	0.856231i	$-0.672800\pi$
$431$	−21.4495	−1.03318	−0.516592	+	0.856231i	$0.672800\pi$
$432$	0	0
$433$	−12.9480	−0.622242	−0.311121	−	0.950370i	$-0.600704\pi$
$433$	−12.9480	−0.622242	−0.311121	+	0.950370i	$0.600704\pi$
$434$	−0.210896	−0.0101233
$435$	0	0
$436$	23.9282	1.14595
$437$	−1.97885	−0.0946613
$438$	0	0
$439$	−36.5832	−1.74602	−0.873011	−	0.487701i	$-0.837836\pi$
$439$	−36.5832	−1.74602	−0.873011	+	0.487701i	$0.837836\pi$
$440$	0	0
$441$	0	0
$442$	16.7480	0.796622
$443$	−18.4925	−0.878604	−0.439302	−	0.898339i	$-0.644774\pi$
$443$	−18.4925	−0.878604	−0.439302	+	0.898339i	$0.644774\pi$
$444$	0	0
$445$	0	0
$446$	−1.78731	−0.0846315
$447$	0	0
$448$	−0.432092	−0.0204144
$449$	−14.1531	−0.667924	−0.333962	−	0.942587i	$-0.608386\pi$
$449$	−14.1531	−0.667924	−0.333962	+	0.942587i	$0.608386\pi$
$450$	0	0
$451$	−24.4872	−1.15306
$452$	5.38368	0.253227
$453$	0	0
$454$	−3.11515	−0.146201
$455$	0	0
$456$	0	0
$457$	−14.2373	−0.665994	−0.332997	−	0.942928i	$-0.608060\pi$
$457$	−14.2373	−0.665994	−0.332997	+	0.942928i	$0.608060\pi$
$458$	3.92201	0.183264
$459$	0	0
$460$	0	0
$461$	−5.69980	−0.265466	−0.132733	−	0.991152i	$-0.542375\pi$
$461$	−5.69980	−0.265466	−0.132733	+	0.991152i	$0.542375\pi$
$462$	0	0
$463$	−10.1278	−0.470677	−0.235339	−	0.971913i	$-0.575620\pi$
$463$	−10.1278	−0.470677	−0.235339	+	0.971913i	$0.575620\pi$
$464$	4.58683	0.212938
$465$	0	0
$466$	−2.89862	−0.134276
$467$	−14.9241	−0.690604	−0.345302	−	0.938492i	$-0.612224\pi$
$467$	−14.9241	−0.690604	−0.345302	+	0.938492i	$0.612224\pi$
$468$	0	0
$469$	−1.28926	−0.0595323
$470$	0	0
$471$	0	0
$472$	7.96748	0.366733
$473$	24.3037	1.11748
$474$	0	0
$475$	0	0
$476$	2.76909	0.126921
$477$	0	0
$478$	13.8964	0.635607
$479$	5.73952	0.262246	0.131123	−	0.991366i	$-0.458142\pi$
$479$	5.73952	0.262246	0.131123	+	0.991366i	$0.458142\pi$
$480$	0	0
$481$	−6.18807	−0.282152
$482$	16.6894	0.760183
$483$	0	0
$484$	5.45416	0.247916
$485$	0	0
$486$	0	0
$487$	29.2234	1.32424	0.662120	−	0.749398i	$-0.269657\pi$
$487$	29.2234	1.32424	0.662120	+	0.749398i	$0.269657\pi$
$488$	−13.1275	−0.594254
$489$	0	0
$490$	0	0
$491$	24.4759	1.10458	0.552292	−	0.833651i	$-0.313753\pi$
$491$	24.4759	1.10458	0.552292	+	0.833651i	$0.313753\pi$
$492$	0	0
$493$	21.1041	0.950480
$494$	1.25245	0.0563505
$495$	0	0
$496$	−1.31592	−0.0590866
$497$	2.66947	0.119742
$498$	0	0
$499$	−43.1187	−1.93026	−0.965128	−	0.261777i	$-0.915691\pi$
$499$	−43.1187	−1.93026	−0.965128	+	0.261777i	$0.915691\pi$
$500$	0	0
$501$	0	0
$502$	17.6647	0.788416
$503$	3.57111	0.159228	0.0796140	−	0.996826i	$-0.474631\pi$
$503$	3.57111	0.159228	0.0796140	+	0.996826i	$0.474631\pi$
$504$	0	0
$505$	0	0
$506$	−8.29302	−0.368670
$507$	0	0
$508$	−8.45686	−0.375213
$509$	−36.1976	−1.60443	−0.802215	−	0.597036i	$-0.796345\pi$
$509$	−36.1976	−1.60443	−0.802215	+	0.597036i	$0.796345\pi$
$510$	0	0
$511$	3.17268	0.140351
$512$	−14.1111	−0.623627
$513$	0	0
$514$	−2.47927	−0.109356
$515$	0	0
$516$	0	0
$517$	−18.0654	−0.794514
$518$	0.329187	0.0144636
$519$	0	0
$520$	0	0
$521$	−20.1464	−0.882630	−0.441315	−	0.897352i	$-0.645488\pi$
$521$	−20.1464	−0.882630	−0.441315	+	0.897352i	$0.645488\pi$
$522$	0	0
$523$	−26.2012	−1.14570	−0.572848	−	0.819661i	$-0.694162\pi$
$523$	−26.2012	−1.14570	−0.572848	+	0.819661i	$0.694162\pi$
$524$	21.9741	0.959943
$525$	0	0
$526$	−7.16867	−0.312569
$527$	−6.05458	−0.263742
$528$	0	0
$529$	−3.89869	−0.169508
$530$	0	0
$531$	0	0
$532$	0.207078	0.00897799
$533$	−35.6974	−1.54623
$534$	0	0
$535$	0	0
$536$	10.4559	0.451627
$537$	0	0
$538$	5.84933	0.252182
$539$	18.7878	0.809249
$540$	0	0
$541$	13.9835	0.601196	0.300598	−	0.953751i	$-0.402814\pi$
$541$	13.9835	0.601196	0.300598	+	0.953751i	$0.402814\pi$
$542$	−5.82335	−0.250134
$543$	0	0
$544$	−35.2423	−1.51100
$545$	0	0
$546$	0	0
$547$	−41.0507	−1.75520	−0.877601	−	0.479392i	$-0.840857\pi$
$547$	−41.0507	−1.75520	−0.877601	+	0.479392i	$0.840857\pi$
$548$	3.34298	0.142805
$549$	0	0
$550$	0	0
$551$	1.57821	0.0672339
$552$	0	0
$553$	−4.42341	−0.188103
$554$	−4.32059	−0.183564
$555$	0	0
$556$	2.97006	0.125958
$557$	−6.60110	−0.279697	−0.139849	−	0.990173i	$-0.544662\pi$
$557$	−6.60110	−0.279697	−0.139849	+	0.990173i	$0.544662\pi$
$558$	0	0
$559$	35.4299	1.49852
$560$	0	0
$561$	0	0
$562$	13.6803	0.577067
$563$	−21.0671	−0.887872	−0.443936	−	0.896059i	$-0.646418\pi$
$563$	−21.0671	−0.887872	−0.443936	+	0.896059i	$0.646418\pi$
$564$	0	0
$565$	0	0
$566$	−22.6443	−0.951812
$567$	0	0
$568$	−21.6495	−0.908394
$569$	10.5055	0.440412	0.220206	−	0.975453i	$-0.429327\pi$
$569$	10.5055	0.440412	0.220206	+	0.975453i	$0.429327\pi$
$570$	0	0
$571$	−16.4974	−0.690393	−0.345196	−	0.938530i	$-0.612188\pi$
$571$	−16.4974	−0.690393	−0.345196	+	0.938530i	$0.612188\pi$
$572$	−16.3135	−0.682102
$573$	0	0
$574$	1.89900	0.0792626
$575$	0	0
$576$	0	0
$577$	−21.4180	−0.891642	−0.445821	−	0.895122i	$-0.647088\pi$
$577$	−21.4180	−0.891642	−0.445821	+	0.895122i	$0.647088\pi$
$578$	−13.7163	−0.570522
$579$	0	0
$580$	0	0
$581$	−3.08489	−0.127983
$582$	0	0
$583$	−23.0998	−0.956694
$584$	−25.7306	−1.06474
$585$	0	0
$586$	10.3057	0.425725
$587$	46.8914	1.93542	0.967708	−	0.252076i	$-0.0811132\pi$
$587$	46.8914	1.93542	0.967708	+	0.252076i	$0.0811132\pi$
$588$	0	0
$589$	−0.452774	−0.0186562
$590$	0	0
$591$	0	0
$592$	2.05401	0.0844195
$593$	−41.4585	−1.70250	−0.851248	−	0.524764i	$-0.824154\pi$
$593$	−41.4585	−1.70250	−0.851248	+	0.524764i	$0.824154\pi$
$594$	0	0
$595$	0	0
$596$	23.4741	0.961538
$597$	0	0
$598$	−12.0896	−0.494379
$599$	7.10882	0.290459	0.145229	−	0.989398i	$-0.453608\pi$
$599$	7.10882	0.290459	0.145229	+	0.989398i	$0.453608\pi$
$600$	0	0
$601$	29.2479	1.19305	0.596523	−	0.802596i	$-0.296548\pi$
$601$	29.2479	1.19305	0.596523	+	0.802596i	$0.296548\pi$
$602$	−1.88476	−0.0768173
$603$	0	0
$604$	−5.30986	−0.216055
$605$	0	0
$606$	0	0
$607$	0.717581	0.0291257	0.0145629	−	0.999894i	$-0.495364\pi$
$607$	0.717581	0.0291257	0.0145629	+	0.999894i	$0.495364\pi$
$608$	−2.63549	−0.106883
$609$	0	0
$610$	0	0
$611$	−26.3357	−1.06543
$612$	0	0
$613$	38.5954	1.55885	0.779427	−	0.626493i	$-0.215510\pi$
$613$	38.5954	1.55885	0.779427	+	0.626493i	$0.215510\pi$
$614$	16.4455	0.663687
$615$	0	0
$616$	2.01488	0.0811818
$617$	40.7474	1.64043	0.820214	−	0.572056i	$-0.193854\pi$
$617$	40.7474	1.64043	0.820214	+	0.572056i	$0.193854\pi$
$618$	0	0
$619$	−34.8629	−1.40126	−0.700628	−	0.713526i	$-0.747097\pi$
$619$	−34.8629	−1.40126	−0.700628	+	0.713526i	$0.747097\pi$
$620$	0	0
$621$	0	0
$622$	−6.95182	−0.278743
$623$	4.72360	0.189247
$624$	0	0
$625$	0	0
$626$	−7.06182	−0.282247
$627$	0	0
$628$	34.9181	1.39338
$629$	9.45056	0.376819
$630$	0	0
$631$	22.2668	0.886426	0.443213	−	0.896416i	$-0.353839\pi$
$631$	22.2668	0.886426	0.443213	+	0.896416i	$0.353839\pi$
$632$	35.8740	1.42699
$633$	0	0
$634$	−1.48423	−0.0589462
$635$	0	0
$636$	0	0
$637$	27.3889	1.08519
$638$	6.61399	0.261850
$639$	0	0
$640$	0	0
$641$	27.6637	1.09265	0.546326	−	0.837572i	$-0.316026\pi$
$641$	27.6637	1.09265	0.546326	+	0.837572i	$0.316026\pi$
$642$	0	0
$643$	−33.2383	−1.31079	−0.655396	−	0.755285i	$-0.727498\pi$
$643$	−33.2383	−1.31079	−0.655396	+	0.755285i	$0.727498\pi$
$644$	−1.99887	−0.0787666
$645$	0	0
$646$	−1.91277	−0.0752571
$647$	−40.6171	−1.59682	−0.798411	−	0.602113i	$-0.794326\pi$
$647$	−40.6171	−1.59682	−0.798411	+	0.602113i	$0.794326\pi$
$648$	0	0
$649$	−8.83915	−0.346967
$650$	0	0
$651$	0	0
$652$	−25.1702	−0.985742
$653$	−17.9837	−0.703755	−0.351878	−	0.936046i	$-0.614457\pi$
$653$	−17.9837	−0.703755	−0.351878	+	0.936046i	$0.614457\pi$
$654$	0	0
$655$	0	0
$656$	11.8491	0.462629
$657$	0	0
$658$	1.40098	0.0546160
$659$	−11.2464	−0.438096	−0.219048	−	0.975714i	$-0.570295\pi$
$659$	−11.2464	−0.438096	−0.219048	+	0.975714i	$0.570295\pi$
$660$	0	0
$661$	−38.5168	−1.49813	−0.749065	−	0.662496i	$-0.769497\pi$
$661$	−38.5168	−1.49813	−0.749065	+	0.662496i	$0.769497\pi$
$662$	−3.95877	−0.153862
$663$	0	0
$664$	25.0186	0.970911
$665$	0	0
$666$	0	0
$667$	−15.2340	−0.589863
$668$	−8.55663	−0.331066
$669$	0	0
$670$	0	0
$671$	14.5637	0.562225
$672$	0	0
$673$	−40.8275	−1.57378	−0.786892	−	0.617091i	$-0.788311\pi$
$673$	−40.8275	−1.57378	−0.786892	+	0.617091i	$0.788311\pi$
$674$	5.79187	0.223095
$675$	0	0
$676$	−4.11088	−0.158111
$677$	−15.4388	−0.593362	−0.296681	−	0.954977i	$-0.595880\pi$
$677$	−15.4388	−0.593362	−0.296681	+	0.954977i	$0.595880\pi$
$678$	0	0
$679$	−2.67575	−0.102686
$680$	0	0
$681$	0	0
$682$	−1.89750	−0.0726589
$683$	35.3435	1.35238	0.676191	−	0.736726i	$-0.263629\pi$
$683$	35.3435	1.35238	0.676191	+	0.736726i	$0.263629\pi$
$684$	0	0
$685$	0	0
$686$	−2.93328	−0.111993
$687$	0	0
$688$	−11.7603	−0.448357
$689$	−33.6748	−1.28291
$690$	0	0
$691$	43.0687	1.63841	0.819206	−	0.573500i	$-0.194415\pi$
$691$	43.0687	1.63841	0.819206	+	0.573500i	$0.194415\pi$
$692$	−21.4393	−0.814998
$693$	0	0
$694$	−11.7872	−0.447437
$695$	0	0
$696$	0	0
$697$	54.5179	2.06501
$698$	7.76787	0.294018
$699$	0	0
$700$	0	0
$701$	32.6431	1.23291	0.616457	−	0.787388i	$-0.288567\pi$
$701$	32.6431	1.23291	0.616457	+	0.787388i	$0.288567\pi$
$702$	0	0
$703$	0.706733	0.0266549
$704$	−3.88766	−0.146522
$705$	0	0
$706$	−1.90555	−0.0717163
$707$	−5.71915	−0.215091
$708$	0	0
$709$	7.48673	0.281170	0.140585	−	0.990069i	$-0.455102\pi$
$709$	7.48673	0.281170	0.140585	+	0.990069i	$0.455102\pi$
$710$	0	0
$711$	0	0
$712$	−38.3086	−1.43567
$713$	4.37050	0.163677
$714$	0	0
$715$	0	0
$716$	−7.63754	−0.285428
$717$	0	0
$718$	5.25563	0.196138
$719$	22.0914	0.823870	0.411935	−	0.911213i	$-0.364853\pi$
$719$	22.0914	0.823870	0.411935	+	0.911213i	$0.364853\pi$
$720$	0	0
$721$	−0.346312	−0.0128973
$722$	13.1141	0.488058
$723$	0	0
$724$	3.69155	0.137195
$725$	0	0
$726$	0	0
$727$	10.9437	0.405881	0.202940	−	0.979191i	$-0.434950\pi$
$727$	10.9437	0.405881	0.202940	+	0.979191i	$0.434950\pi$
$728$	2.93729	0.108863
$729$	0	0
$730$	0	0
$731$	−54.1094	−2.00131
$732$	0	0
$733$	−16.8981	−0.624145	−0.312072	−	0.950058i	$-0.601023\pi$
$733$	−16.8981	−0.624145	−0.312072	+	0.950058i	$0.601023\pi$
$734$	17.7368	0.654676
$735$	0	0
$736$	25.4396	0.937718
$737$	−11.5998	−0.427285
$738$	0	0
$739$	−43.3677	−1.59531	−0.797654	−	0.603115i	$-0.793926\pi$
$739$	−43.3677	−1.59531	−0.797654	+	0.603115i	$0.793926\pi$
$740$	0	0
$741$	0	0
$742$	1.79140	0.0657644
$743$	−21.6647	−0.794799	−0.397400	−	0.917646i	$-0.630087\pi$
$743$	−21.6647	−0.794799	−0.397400	+	0.917646i	$0.630087\pi$
$744$	0	0
$745$	0	0
$746$	−8.97388	−0.328557
$747$	0	0
$748$	24.9144	0.910959
$749$	−2.52175	−0.0921428
$750$	0	0
$751$	−44.2265	−1.61385	−0.806923	−	0.590656i	$-0.798869\pi$
$751$	−44.2265	−1.61385	−0.806923	+	0.590656i	$0.798869\pi$
$752$	8.74165	0.318775
$753$	0	0
$754$	9.64188	0.351136
$755$	0	0
$756$	0	0
$757$	−13.3596	−0.485565	−0.242782	−	0.970081i	$-0.578060\pi$
$757$	−13.3596	−0.485565	−0.242782	+	0.970081i	$0.578060\pi$
$758$	−19.0393	−0.691539
$759$	0	0
$760$	0	0
$761$	33.7383	1.22301	0.611507	−	0.791239i	$-0.290564\pi$
$761$	33.7383	1.22301	0.611507	+	0.791239i	$0.290564\pi$
$762$	0	0
$763$	4.77968	0.173036
$764$	6.21050	0.224688
$765$	0	0
$766$	13.5539	0.489723
$767$	−12.8857	−0.465276
$768$	0	0
$769$	−46.9399	−1.69269	−0.846347	−	0.532631i	$-0.821203\pi$
$769$	−46.9399	−1.69269	−0.846347	+	0.532631i	$0.821203\pi$
$770$	0	0
$771$	0	0
$772$	25.0321	0.900924
$773$	22.0050	0.791463	0.395732	−	0.918366i	$-0.370491\pi$
$773$	22.0050	0.791463	0.395732	+	0.918366i	$0.370491\pi$
$774$	0	0
$775$	0	0
$776$	21.7005	0.779001
$777$	0	0
$778$	23.5974	0.846008
$779$	4.07696	0.146072
$780$	0	0
$781$	24.0181	0.859433
$782$	18.4635	0.660253
$783$	0	0
$784$	−9.09123	−0.324687
$785$	0	0
$786$	0	0
$787$	−9.52883	−0.339666	−0.169833	−	0.985473i	$-0.554323\pi$
$787$	−9.52883	−0.339666	−0.169833	+	0.985473i	$0.554323\pi$
$788$	23.5506	0.838956
$789$	0	0
$790$	0	0
$791$	1.07540	0.0382367
$792$	0	0
$793$	21.2310	0.753933
$794$	−4.01421	−0.142459
$795$	0	0
$796$	3.28386	0.116393
$797$	46.7651	1.65650	0.828252	−	0.560356i	$-0.189336\pi$
$797$	46.7651	1.65650	0.828252	+	0.560356i	$0.189336\pi$
$798$	0	0
$799$	40.2205	1.42290
$800$	0	0
$801$	0	0
$802$	21.6282	0.763720
$803$	28.5456	1.00735
$804$	0	0
$805$	0	0
$806$	−2.76617	−0.0974343
$807$	0	0
$808$	46.3825	1.63173
$809$	41.5538	1.46095	0.730477	−	0.682938i	$-0.239298\pi$
$809$	41.5538	1.46095	0.730477	+	0.682938i	$0.239298\pi$
$810$	0	0
$811$	30.9980	1.08849	0.544244	−	0.838927i	$-0.316817\pi$
$811$	30.9980	1.08849	0.544244	+	0.838927i	$0.316817\pi$
$812$	1.59417	0.0559445
$813$	0	0
$814$	2.96179	0.103811
$815$	0	0
$816$	0	0
$817$	−4.04641	−0.141566
$818$	−0.162723	−0.00568949
$819$	0	0
$820$	0	0
$821$	−0.962809	−0.0336023	−0.0168011	−	0.999859i	$-0.505348\pi$
$821$	−0.962809	−0.0336023	−0.0168011	+	0.999859i	$0.505348\pi$
$822$	0	0
$823$	42.5061	1.48167	0.740835	−	0.671687i	$-0.234430\pi$
$823$	42.5061	1.48167	0.740835	+	0.671687i	$0.234430\pi$
$824$	2.80860	0.0978423
$825$	0	0
$826$	0.685482	0.0238510
$827$	−12.1193	−0.421429	−0.210714	−	0.977548i	$-0.567579\pi$
$827$	−12.1193	−0.421429	−0.210714	+	0.977548i	$0.567579\pi$
$828$	0	0
$829$	−15.8215	−0.549503	−0.274752	−	0.961515i	$-0.588596\pi$
$829$	−15.8215	−0.549503	−0.274752	+	0.961515i	$0.588596\pi$
$830$	0	0
$831$	0	0
$832$	−5.66744	−0.196483
$833$	−41.8290	−1.44929
$834$	0	0
$835$	0	0
$836$	1.86315	0.0644383
$837$	0	0
$838$	2.33488	0.0806571
$839$	−36.7412	−1.26845	−0.634224	−	0.773149i	$-0.718680\pi$
$839$	−36.7412	−1.26845	−0.634224	+	0.773149i	$0.718680\pi$
$840$	0	0
$841$	−16.8503	−0.581046
$842$	6.36572	0.219377
$843$	0	0
$844$	−30.0517	−1.03442
$845$	0	0
$846$	0	0
$847$	1.08947	0.0374348
$848$	11.1777	0.383845
$849$	0	0
$850$	0	0
$851$	−6.82189	−0.233852
$852$	0	0
$853$	1.32430	0.0453430	0.0226715	−	0.999743i	$-0.492783\pi$
$853$	1.32430	0.0453430	0.0226715	+	0.999743i	$0.492783\pi$
$854$	−1.12942	−0.0386481
$855$	0	0
$856$	20.4515	0.699018
$857$	−25.8324	−0.882417	−0.441208	−	0.897405i	$-0.645450\pi$
$857$	−25.8324	−0.882417	−0.441208	+	0.897405i	$0.645450\pi$
$858$	0	0
$859$	55.6807	1.89980	0.949901	−	0.312550i	$-0.101183\pi$
$859$	55.6807	1.89980	0.949901	+	0.312550i	$0.101183\pi$
$860$	0	0
$861$	0	0
$862$	14.9663	0.509754
$863$	28.8255	0.981230	0.490615	−	0.871376i	$-0.336772\pi$
$863$	28.8255	0.981230	0.490615	+	0.871376i	$0.336772\pi$
$864$	0	0
$865$	0	0
$866$	9.03444	0.307003
$867$	0	0
$868$	−0.457355	−0.0155236
$869$	−39.7988	−1.35008
$870$	0	0
$871$	−16.9102	−0.572981
$872$	−38.7634	−1.31269
$873$	0	0
$874$	1.38074	0.0467041
$875$	0	0
$876$	0	0
$877$	30.4326	1.02764	0.513818	−	0.857899i	$-0.328231\pi$
$877$	30.4326	1.02764	0.513818	+	0.857899i	$0.328231\pi$
$878$	25.5258	0.861454
$879$	0	0
$880$	0	0
$881$	−22.0975	−0.744483	−0.372242	−	0.928136i	$-0.621411\pi$
$881$	−22.0975	−0.744483	−0.372242	+	0.928136i	$0.621411\pi$
$882$	0	0
$883$	−57.3452	−1.92982	−0.964910	−	0.262579i	$-0.915427\pi$
$883$	−57.3452	−1.92982	−0.964910	+	0.262579i	$0.915427\pi$
$884$	36.3202	1.22158
$885$	0	0
$886$	12.9031	0.433487
$887$	−5.33250	−0.179048	−0.0895240	−	0.995985i	$-0.528535\pi$
$887$	−5.33250	−0.179048	−0.0895240	+	0.995985i	$0.528535\pi$
$888$	0	0
$889$	−1.68927	−0.0566562
$890$	0	0
$891$	0	0
$892$	−3.87600	−0.129778
$893$	3.00777	0.100651
$894$	0	0
$895$	0	0
$896$	−3.21720	−0.107479
$897$	0	0
$898$	9.87524	0.329541
$899$	−3.48564	−0.116253
$900$	0	0
$901$	51.4290	1.71335
$902$	17.0858	0.568896
$903$	0	0
$904$	−8.72151	−0.290073
$905$	0	0
$906$	0	0
$907$	−1.99170	−0.0661334	−0.0330667	−	0.999453i	$-0.510527\pi$
$907$	−1.99170	−0.0661334	−0.0330667	+	0.999453i	$0.510527\pi$
$908$	−6.75560	−0.224192
$909$	0	0
$910$	0	0
$911$	−21.8337	−0.723383	−0.361691	−	0.932298i	$-0.617801\pi$
$911$	−21.8337	−0.723383	−0.361691	+	0.932298i	$0.617801\pi$
$912$	0	0
$913$	−27.7557	−0.918580
$914$	9.93404	0.328589
$915$	0	0
$916$	8.50537	0.281025
$917$	4.38935	0.144949
$918$	0	0
$919$	32.1622	1.06093	0.530466	−	0.847706i	$-0.322017\pi$
$919$	32.1622	1.06093	0.530466	+	0.847706i	$0.322017\pi$
$920$	0	0
$921$	0	0
$922$	3.97702	0.130976
$923$	35.0135	1.15248
$924$	0	0
$925$	0	0
$926$	7.06661	0.232223
$927$	0	0
$928$	−20.2891	−0.666021
$929$	8.94389	0.293439	0.146720	−	0.989178i	$-0.453128\pi$
$929$	8.94389	0.293439	0.146720	+	0.989178i	$0.453128\pi$
$930$	0	0
$931$	−3.12806	−0.102518
$932$	−6.28603	−0.205906
$933$	0	0
$934$	10.4132	0.340731
$935$	0	0
$936$	0	0
$937$	44.9105	1.46716	0.733581	−	0.679602i	$-0.237847\pi$
$937$	44.9105	1.46716	0.733581	+	0.679602i	$0.237847\pi$
$938$	0.899574	0.0293721
$939$	0	0
$940$	0	0
$941$	13.2411	0.431647	0.215824	−	0.976432i	$-0.430756\pi$
$941$	13.2411	0.431647	0.215824	+	0.976432i	$0.430756\pi$
$942$	0	0
$943$	−39.3538	−1.28154
$944$	4.27717	0.139210
$945$	0	0
$946$	−16.9578	−0.551346
$947$	−15.1047	−0.490836	−0.245418	−	0.969417i	$-0.578925\pi$
$947$	−15.1047	−0.490836	−0.245418	+	0.969417i	$0.578925\pi$
$948$	0	0
$949$	41.6137	1.35084
$950$	0	0
$951$	0	0
$952$	−4.48590	−0.145389
$953$	−47.2438	−1.53038	−0.765188	−	0.643807i	$-0.777354\pi$
$953$	−47.2438	−1.53038	−0.765188	+	0.643807i	$0.777354\pi$
$954$	0	0
$955$	0	0
$956$	30.1361	0.974671
$957$	0	0
$958$	−4.00473	−0.129387
$959$	0.667765	0.0215633
$960$	0	0
$961$	1.00000	0.0322581
$962$	4.31770	0.139208
$963$	0	0
$964$	36.1931	1.16570
$965$	0	0
$966$	0	0
$967$	9.85886	0.317040	0.158520	−	0.987356i	$-0.449328\pi$
$967$	9.85886	0.317040	0.158520	+	0.987356i	$0.449328\pi$
$968$	−8.83568	−0.283990
$969$	0	0
$970$	0	0
$971$	−3.20360	−0.102808	−0.0514042	−	0.998678i	$-0.516370\pi$
$971$	−3.20360	−0.102808	−0.0514042	+	0.998678i	$0.516370\pi$
$972$	0	0
$973$	0.593272	0.0190194
$974$	−20.3905	−0.653355
$975$	0	0
$976$	−7.04722	−0.225576
$977$	31.1531	0.996676	0.498338	−	0.866983i	$-0.333944\pi$
$977$	31.1531	0.996676	0.498338	+	0.866983i	$0.333944\pi$
$978$	0	0
$979$	42.4996	1.35829
$980$	0	0
$981$	0	0
$982$	−17.0780	−0.544981
$983$	−33.1726	−1.05804	−0.529021	−	0.848609i	$-0.677441\pi$
$983$	−33.1726	−1.05804	−0.529021	+	0.848609i	$0.677441\pi$
$984$	0	0
$985$	0	0
$986$	−14.7253	−0.468949
$987$	0	0
$988$	2.71610	0.0864106
$989$	39.0589	1.24200
$990$	0	0
$991$	−40.0379	−1.27185	−0.635923	−	0.771752i	$-0.719380\pi$
$991$	−40.0379	−1.27185	−0.635923	+	0.771752i	$0.719380\pi$
$992$	5.82076	0.184809
$993$	0	0
$994$	−1.86262	−0.0590786
$995$	0	0
$996$	0	0
$997$	32.1083	1.01688	0.508441	−	0.861097i	$-0.330222\pi$
$997$	32.1083	1.01688	0.508441	+	0.861097i	$0.330222\pi$
$998$	30.0859	0.952353
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bv.1.2		5
3.2	odd	2		2325.2.a.w.1.4		5
5.2	odd	4		1395.2.c.f.559.4		10
5.3	odd	4		1395.2.c.f.559.7		10
5.4	even	2		6975.2.a.bs.1.4		5
15.2	even	4		465.2.c.a.94.7	yes	10
15.8	even	4		465.2.c.a.94.4	✓	10
15.14	odd	2		2325.2.a.x.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.c.a.94.4	✓	10	15.8	even	4
465.2.c.a.94.7	yes	10	15.2	even	4
1395.2.c.f.559.4		10	5.2	odd	4
1395.2.c.f.559.7		10	5.3	odd	4
2325.2.a.w.1.4		5	3.2	odd	2
2325.2.a.x.1.2		5	15.14	odd	2
6975.2.a.bs.1.4		5	5.4	even	2
6975.2.a.bv.1.2		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q-0.697747 q^{2} -1.51315 q^{4} -0.302253 q^{7} +2.45129 q^{8} -2.71947 q^{11} -3.96444 q^{13} +0.210896 q^{14} +1.31592 q^{16} +6.05458 q^{17} +0.452774 q^{19} +1.89750 q^{22} -4.37050 q^{23} +2.76617 q^{26} +0.457355 q^{28} +3.48564 q^{29} -1.00000 q^{31} -5.82076 q^{32} -4.22456 q^{34} +1.56089 q^{37} -0.315922 q^{38} +9.00441 q^{41} -8.93693 q^{43} +4.11496 q^{44} +3.04950 q^{46} +6.64299 q^{47} -6.90864 q^{49} +5.99879 q^{52} +8.49423 q^{53} -0.740910 q^{56} -2.43209 q^{58} +3.25033 q^{59} -5.35535 q^{61} +0.697747 q^{62} +1.42957 q^{64} +4.26548 q^{67} -9.16149 q^{68} -8.83190 q^{71} -10.4968 q^{73} -1.08911 q^{74} -0.685115 q^{76} +0.821968 q^{77} +14.6348 q^{79} -6.28279 q^{82} +10.2063 q^{83} +6.23571 q^{86} -6.66619 q^{88} -15.6279 q^{89} +1.19827 q^{91} +6.61323 q^{92} -4.63512 q^{94} +8.85267 q^{97} +4.82048 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} + 3 q^{4} - 8 q^{7} + 9 q^{8} - 6 q^{13} - 16 q^{14} + 3 q^{16} + 8 q^{17} - 4 q^{19} - 6 q^{22} + 4 q^{23} + 6 q^{26} - 18 q^{28} - 4 q^{29} - 5 q^{31} + q^{32} + 6 q^{34} + 4 q^{37} + 2 q^{38}+ \cdots + 29 q^{98}+O(q^{100}) \) 5 * q + 3 * q^2 + 3 * q^4 - 8 * q^7 + 9 * q^8 - 6 * q^13 - 16 * q^14 + 3 * q^16 + 8 * q^17 - 4 * q^19 - 6 * q^22 + 4 * q^23 + 6 * q^26 - 18 * q^28 - 4 * q^29 - 5 * q^31 + q^32 + 6 * q^34 + 4 * q^37 + 2 * q^38 - 20 * q^43 - 6 * q^44 - 16 * q^46 + 18 * q^47 - 11 * q^49 - 2 * q^52 + 2 * q^53 - 18 * q^56 - 14 * q^58 - 10 * q^59 - 14 * q^61 - 3 * q^62 + 11 * q^64 - 18 * q^67 + 6 * q^68 - 6 * q^71 - 10 * q^73 + 4 * q^74 - 8 * q^76 + 6 * q^77 - 24 * q^79 - 30 * q^82 + 48 * q^83 - 4 * q^86 - 8 * q^88 - 10 * q^89 - 26 * q^92 + 12 * q^94 - 20 * q^97 + 29 * q^98

Introduction

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