LMFDB - Embedded newform 5600.2.a.bw.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5600,2,Mod(1,5600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5600 = 2^{5} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5600.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:	$44.7162251319$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.504568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 2 $ x^5 - x^4 - 6x^3 + 3x^2 + 7*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1120)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.52064$ of defining polynomial
Character	$\chi$	$=$	5600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.83297 q^{3} -1.00000 q^{7} +0.359777 q^{9} +O(q^{10})$	$q+1.83297 q^{3} -1.00000 q^{7} +0.359777 q^{9} -4.40105 q^{11} +3.20830 q^{13} -1.14821 q^{17} +1.72603 q^{19} -1.83297 q^{21} -3.25284 q^{23} -4.83945 q^{27} +4.18948 q^{29} -1.36952 q^{31} -8.06699 q^{33} -4.19923 q^{37} +5.88072 q^{39} +11.7485 q^{41} +2.64997 q^{43} -0.106937 q^{47} +1.00000 q^{49} -2.10463 q^{51} -7.86516 q^{53} +3.16376 q^{57} -13.3029 q^{59} -10.0224 q^{61} -0.359777 q^{63} -1.28045 q^{67} -5.96236 q^{69} -14.1616 q^{71} -11.8652 q^{73} +4.40105 q^{77} -5.48227 q^{79} -9.94989 q^{81} +17.1586 q^{83} +7.67919 q^{87} +4.31591 q^{89} -3.20830 q^{91} -2.51029 q^{93} -7.65389 q^{97} -1.58340 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{3} - 5 q^{7} + 7 q^{9}+O(q^{10}) $ 5 * q + 2 * q^3 - 5 * q^7 + 7 * q^9	$ 5 q + 2 q^{3} - 5 q^{7} + 7 q^{9} - 4 q^{11} + 4 q^{17} - 12 q^{19} - 2 q^{21} - 8 q^{23} + 14 q^{27} - 12 q^{29} - 12 q^{31} - 8 q^{33} + 12 q^{37} - 32 q^{39} - 2 q^{41} + 8 q^{43} - 14 q^{47} + 5 q^{49} - 12 q^{51} + 8 q^{53} - 8 q^{57} - 16 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{67} + 4 q^{69} - 4 q^{71} - 12 q^{73} + 4 q^{77} - 32 q^{79} + q^{81} + 24 q^{83} - 2 q^{87} + 2 q^{89} - 20 q^{93} - 12 q^{97} - 40 q^{99}+O(q^{100}) $ 5 * q + 2 * q^3 - 5 * q^7 + 7 * q^9 - 4 * q^11 + 4 * q^17 - 12 * q^19 - 2 * q^21 - 8 * q^23 + 14 * q^27 - 12 * q^29 - 12 * q^31 - 8 * q^33 + 12 * q^37 - 32 * q^39 - 2 * q^41 + 8 * q^43 - 14 * q^47 + 5 * q^49 - 12 * q^51 + 8 * q^53 - 8 * q^57 - 16 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^67 + 4 * q^69 - 4 * q^71 - 12 * q^73 + 4 * q^77 - 32 * q^79 + q^81 + 24 * q^83 - 2 * q^87 + 2 * q^89 - 20 * q^93 - 12 * q^97 - 40 * q^99

Display 100 coefficients 10 coefficients

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	0
$2$	0	0
$3$	1.83297	1.05827	0.529133	−	0.848539i	$-0.322517\pi$
$3$	1.83297	1.05827	0.529133	+	0.848539i	$0.322517\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0.359777	0.119926
$10$	0	0
$11$	−4.40105	−1.32697	−0.663483	−	0.748191i	$-0.730923\pi$
$11$	−4.40105	−1.32697	−0.663483	+	0.748191i	$0.730923\pi$
$12$	0	0
$13$	3.20830	0.889823	0.444911	−	0.895575i	$-0.353235\pi$
$13$	3.20830	0.889823	0.444911	+	0.895575i	$0.353235\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.14821	−0.278482	−0.139241	−	0.990259i	$-0.544466\pi$
$17$	−1.14821	−0.278482	−0.139241	+	0.990259i	$0.544466\pi$
$18$	0	0
$19$	1.72603	0.395979	0.197990	−	0.980204i	$-0.436559\pi$
$19$	1.72603	0.395979	0.197990	+	0.980204i	$0.436559\pi$
$20$	0	0
$21$	−1.83297	−0.399987
$22$	0	0
$23$	−3.25284	−0.678264	−0.339132	−	0.940739i	$-0.610133\pi$
$23$	−3.25284	−0.678264	−0.339132	+	0.940739i	$0.610133\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−4.83945	−0.931352
$28$	0	0
$29$	4.18948	0.777967	0.388983	−	0.921245i	$-0.372826\pi$
$29$	4.18948	0.777967	0.388983	+	0.921245i	$0.372826\pi$
$30$	0	0
$31$	−1.36952	−0.245973	−0.122987	−	0.992408i	$-0.539247\pi$
$31$	−1.36952	−0.245973	−0.122987	+	0.992408i	$0.539247\pi$
$32$	0	0
$33$	−8.06699	−1.40428
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.19923	−0.690348	−0.345174	−	0.938539i	$-0.612180\pi$
$37$	−4.19923	−0.690348	−0.345174	+	0.938539i	$0.612180\pi$
$38$	0	0
$39$	5.88072	0.941669
$40$	0	0
$41$	11.7485	1.83480	0.917402	−	0.397961i	$-0.130282\pi$
$41$	11.7485	1.83480	0.917402	+	0.397961i	$0.130282\pi$
$42$	0	0
$43$	2.64997	0.404116	0.202058	−	0.979374i	$-0.435237\pi$
$43$	2.64997	0.404116	0.202058	+	0.979374i	$0.435237\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.106937	−0.0155983	−0.00779917	−	0.999970i	$-0.502483\pi$
$47$	−0.106937	−0.0155983	−0.00779917	+	0.999970i	$0.502483\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.10463	−0.294707
$52$	0	0
$53$	−7.86516	−1.08036	−0.540182	−	0.841548i	$-0.681644\pi$
$53$	−7.86516	−1.08036	−0.540182	+	0.841548i	$0.681644\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.16376	0.419051
$58$	0	0
$59$	−13.3029	−1.73189	−0.865945	−	0.500140i	$-0.833282\pi$
$59$	−13.3029	−1.73189	−0.865945	+	0.500140i	$0.833282\pi$
$60$	0	0
$61$	−10.0224	−1.28324	−0.641622	−	0.767021i	$-0.721738\pi$
$61$	−10.0224	−1.28324	−0.641622	+	0.767021i	$0.721738\pi$
$62$	0	0
$63$	−0.359777	−0.0453276
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.28045	−0.156431	−0.0782157	−	0.996936i	$-0.524922\pi$
$67$	−1.28045	−0.156431	−0.0782157	+	0.996936i	$0.524922\pi$
$68$	0	0
$69$	−5.96236	−0.717783
$70$	0	0
$71$	−14.1616	−1.68067	−0.840335	−	0.542067i	$-0.817642\pi$
$71$	−14.1616	−1.68067	−0.840335	+	0.542067i	$0.817642\pi$
$72$	0	0
$73$	−11.8652	−1.38871	−0.694356	−	0.719631i	$-0.744311\pi$
$73$	−11.8652	−1.38871	−0.694356	+	0.719631i	$0.744311\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.40105	0.501546
$78$	0	0
$79$	−5.48227	−0.616804	−0.308402	−	0.951256i	$-0.599794\pi$
$79$	−5.48227	−0.616804	−0.308402	+	0.951256i	$0.599794\pi$
$80$	0	0
$81$	−9.94989	−1.10554
$82$	0	0
$83$	17.1586	1.88340	0.941701	−	0.336451i	$-0.109227\pi$
$83$	17.1586	1.88340	0.941701	+	0.336451i	$0.109227\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.67919	0.823296
$88$	0	0
$89$	4.31591	0.457485	0.228743	−	0.973487i	$-0.426539\pi$
$89$	4.31591	0.457485	0.228743	+	0.973487i	$0.426539\pi$
$90$	0	0
$91$	−3.20830	−0.336321
$92$	0	0
$93$	−2.51029	−0.260305
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.65389	−0.777135	−0.388567	−	0.921420i	$-0.627030\pi$
$97$	−7.65389	−0.777135	−0.388567	+	0.921420i	$0.627030\pi$
$98$	0	0
$99$	−1.58340	−0.159137
$100$	0	0
$101$	−2.25581	−0.224462	−0.112231	−	0.993682i	$-0.535800\pi$
$101$	−2.25581	−0.224462	−0.112231	+	0.993682i	$0.535800\pi$
$102$	0	0
$103$	−6.61262	−0.651560	−0.325780	−	0.945446i	$-0.605627\pi$
$103$	−6.61262	−0.651560	−0.325780	+	0.945446i	$0.605627\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.83756	−0.564338	−0.282169	−	0.959365i	$-0.591054\pi$
$107$	−5.83756	−0.564338	−0.282169	+	0.959365i	$0.591054\pi$
$108$	0	0
$109$	−10.6952	−1.02441	−0.512205	−	0.858863i	$-0.671171\pi$
$109$	−10.6952	−1.02441	−0.512205	+	0.858863i	$0.671171\pi$
$110$	0	0
$111$	−7.69705	−0.730572
$112$	0	0
$113$	0.867348	0.0815932	0.0407966	−	0.999167i	$-0.487010\pi$
$113$	0.867348	0.0815932	0.0407966	+	0.999167i	$0.487010\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.15427	0.106713
$118$	0	0
$119$	1.14821	0.105256
$120$	0	0
$121$	8.36923	0.760839
$122$	0	0
$123$	21.5346	1.94171
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.6158	−1.29695	−0.648473	−	0.761238i	$-0.724592\pi$
$127$	−14.6158	−1.29695	−0.648473	+	0.761238i	$0.724592\pi$
$128$	0	0
$129$	4.85731	0.427662
$130$	0	0
$131$	17.8600	1.56044	0.780218	−	0.625508i	$-0.215108\pi$
$131$	17.8600	1.56044	0.780218	+	0.625508i	$0.215108\pi$
$132$	0	0
$133$	−1.72603	−0.149666
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.22523	0.446422	0.223211	−	0.974770i	$-0.428346\pi$
$137$	5.22523	0.446422	0.223211	+	0.974770i	$0.428346\pi$
$138$	0	0
$139$	−5.25800	−0.445977	−0.222989	−	0.974821i	$-0.571581\pi$
$139$	−5.25800	−0.445977	−0.222989	+	0.974821i	$0.571581\pi$
$140$	0	0
$141$	−0.196012	−0.0165072
$142$	0	0
$143$	−14.1199	−1.18076
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.83297	0.151181
$148$	0	0
$149$	−11.8928	−0.974294	−0.487147	−	0.873320i	$-0.661962\pi$
$149$	−11.8928	−0.974294	−0.487147	+	0.873320i	$0.661962\pi$
$150$	0	0
$151$	−8.54534	−0.695410	−0.347705	−	0.937604i	$-0.613039\pi$
$151$	−8.54534	−0.695410	−0.347705	+	0.937604i	$0.613039\pi$
$152$	0	0
$153$	−0.413099	−0.0333971
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3.36436	0.268506	0.134253	−	0.990947i	$-0.457137\pi$
$157$	3.36436	0.268506	0.134253	+	0.990947i	$0.457137\pi$
$158$	0	0
$159$	−14.4166	−1.14331
$160$	0	0
$161$	3.25284	0.256360
$162$	0	0
$163$	−15.8310	−1.23998	−0.619991	−	0.784609i	$-0.712864\pi$
$163$	−15.8310	−1.23998	−0.619991	+	0.784609i	$0.712864\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.8843	1.61608	0.808040	−	0.589128i	$-0.200529\pi$
$167$	20.8843	1.61608	0.808040	+	0.589128i	$0.200529\pi$
$168$	0	0
$169$	−2.70680	−0.208215
$170$	0	0
$171$	0.620986	0.0474880
$172$	0	0
$173$	−5.94735	−0.452168	−0.226084	−	0.974108i	$-0.572592\pi$
$173$	−5.94735	−0.452168	−0.226084	+	0.974108i	$0.572592\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−24.3838	−1.83280
$178$	0	0
$179$	−22.9041	−1.71194	−0.855968	−	0.517030i	$-0.827038\pi$
$179$	−22.9041	−1.71194	−0.855968	+	0.517030i	$0.827038\pi$
$180$	0	0
$181$	−1.77746	−0.132118	−0.0660589	−	0.997816i	$-0.521043\pi$
$181$	−1.77746	−0.132118	−0.0660589	+	0.997816i	$0.521043\pi$
$182$	0	0
$183$	−18.3708	−1.35801
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.05332	0.369536
$188$	0	0
$189$	4.83945	0.352018
$190$	0	0
$191$	19.7183	1.42676	0.713382	−	0.700775i	$-0.247162\pi$
$191$	19.7183	1.42676	0.713382	+	0.700775i	$0.247162\pi$
$192$	0	0
$193$	18.4956	1.33135	0.665673	−	0.746244i	$-0.268145\pi$
$193$	18.4956	1.33135	0.665673	+	0.746244i	$0.268145\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−19.4179	−1.38347	−0.691735	−	0.722151i	$-0.743153\pi$
$197$	−19.4179	−1.38347	−0.691735	+	0.722151i	$0.743153\pi$
$198$	0	0
$199$	−21.7485	−1.54171	−0.770855	−	0.637011i	$-0.780170\pi$
$199$	−21.7485	−1.54171	−0.770855	+	0.637011i	$0.780170\pi$
$200$	0	0
$201$	−2.34702	−0.165546
$202$	0	0
$203$	−4.18948	−0.294044
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.17030	−0.0813412
$208$	0	0
$209$	−7.59635	−0.525451
$210$	0	0
$211$	−23.2091	−1.59778	−0.798890	−	0.601477i	$-0.794579\pi$
$211$	−23.2091	−1.59778	−0.798890	+	0.601477i	$0.794579\pi$
$212$	0	0
$213$	−25.9577	−1.77860
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.36952	0.0929692
$218$	0	0
$219$	−21.7485	−1.46963
$220$	0	0
$221$	−3.68380	−0.247799
$222$	0	0
$223$	0.784235	0.0525163	0.0262581	−	0.999655i	$-0.491641\pi$
$223$	0.784235	0.0525163	0.0262581	+	0.999655i	$0.491641\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.7257	0.778265	0.389132	−	0.921182i	$-0.372775\pi$
$227$	11.7257	0.778265	0.389132	+	0.921182i	$0.372775\pi$
$228$	0	0
$229$	−1.09555	−0.0723962	−0.0361981	−	0.999345i	$-0.511525\pi$
$229$	−1.09555	−0.0723962	−0.0361981	+	0.999345i	$0.511525\pi$
$230$	0	0
$231$	8.06699	0.530769
$232$	0	0
$233$	25.3345	1.65972	0.829860	−	0.557972i	$-0.188420\pi$
$233$	25.3345	1.65972	0.829860	+	0.557972i	$0.188420\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−10.0488	−0.652742
$238$	0	0
$239$	−28.8886	−1.86865	−0.934323	−	0.356427i	$-0.883995\pi$
$239$	−28.8886	−1.86865	−0.934323	+	0.356427i	$0.883995\pi$
$240$	0	0
$241$	10.2542	0.660529	0.330264	−	0.943888i	$-0.392862\pi$
$241$	10.2542	0.660529	0.330264	+	0.943888i	$0.392862\pi$
$242$	0	0
$243$	−3.71950	−0.238606
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.53763	0.352351
$248$	0	0
$249$	31.4512	1.99314
$250$	0	0
$251$	27.5389	1.73824	0.869120	−	0.494601i	$-0.164686\pi$
$251$	27.5389	1.73824	0.869120	+	0.494601i	$0.164686\pi$
$252$	0	0
$253$	14.3159	0.900033
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.2636	0.764983	0.382492	−	0.923959i	$-0.375066\pi$
$257$	12.2636	0.764983	0.382492	+	0.923959i	$0.375066\pi$
$258$	0	0
$259$	4.19923	0.260927
$260$	0	0
$261$	1.50728	0.0932982
$262$	0	0
$263$	−17.8928	−1.10332	−0.551658	−	0.834071i	$-0.686005\pi$
$263$	−17.8928	−1.10332	−0.551658	+	0.834071i	$0.686005\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	7.91092	0.484141
$268$	0	0
$269$	25.2295	1.53827	0.769136	−	0.639085i	$-0.220687\pi$
$269$	25.2295	1.53827	0.769136	+	0.639085i	$0.220687\pi$
$270$	0	0
$271$	−9.65999	−0.586803	−0.293401	−	0.955989i	$-0.594787\pi$
$271$	−9.65999	−0.586803	−0.293401	+	0.955989i	$0.594787\pi$
$272$	0	0
$273$	−5.88072	−0.355917
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.1970	1.39378	0.696888	−	0.717180i	$-0.254568\pi$
$277$	23.1970	1.39378	0.696888	+	0.717180i	$0.254568\pi$
$278$	0	0
$279$	−0.492722	−0.0294985
$280$	0	0
$281$	26.9734	1.60910	0.804550	−	0.593885i	$-0.202407\pi$
$281$	26.9734	1.60910	0.804550	+	0.593885i	$0.202407\pi$
$282$	0	0
$283$	22.2119	1.32036	0.660181	−	0.751106i	$-0.270480\pi$
$283$	22.2119	1.32036	0.660181	+	0.751106i	$0.270480\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.7485	−0.693491
$288$	0	0
$289$	−15.6816	−0.922448
$290$	0	0
$291$	−14.0293	−0.822415
$292$	0	0
$293$	20.8756	1.21956	0.609782	−	0.792569i	$-0.291257\pi$
$293$	20.8756	1.21956	0.609782	+	0.792569i	$0.291257\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	21.2986	1.23587
$298$	0	0
$299$	−10.4361	−0.603535
$300$	0	0
$301$	−2.64997	−0.152742
$302$	0	0
$303$	−4.13484	−0.237540
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2.97374	−0.169720	−0.0848601	−	0.996393i	$-0.527044\pi$
$307$	−2.97374	−0.169720	−0.0848601	+	0.996393i	$0.527044\pi$
$308$	0	0
$309$	−12.1207	−0.689524
$310$	0	0
$311$	−1.04007	−0.0589770	−0.0294885	−	0.999565i	$-0.509388\pi$
$311$	−1.04007	−0.0589770	−0.0294885	+	0.999565i	$0.509388\pi$
$312$	0	0
$313$	−9.12411	−0.515725	−0.257863	−	0.966182i	$-0.583018\pi$
$313$	−9.12411	−0.515725	−0.257863	+	0.966182i	$0.583018\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.74255	−0.266368	−0.133184	−	0.991091i	$-0.542520\pi$
$317$	−4.74255	−0.266368	−0.133184	+	0.991091i	$0.542520\pi$
$318$	0	0
$319$	−18.4381	−1.03234
$320$	0	0
$321$	−10.7001	−0.597219
$322$	0	0
$323$	−1.98185	−0.110273
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−19.6039	−1.08410
$328$	0	0
$329$	0.106937	0.00589562
$330$	0	0
$331$	−9.38114	−0.515634	−0.257817	−	0.966194i	$-0.583003\pi$
$331$	−9.38114	−0.515634	−0.257817	+	0.966194i	$0.583003\pi$
$332$	0	0
$333$	−1.51078	−0.0827904
$334$	0	0
$335$	0	0
$336$	0	0
$337$	17.4797	0.952178	0.476089	−	0.879397i	$-0.342054\pi$
$337$	17.4797	0.952178	0.476089	+	0.879397i	$0.342054\pi$
$338$	0	0
$339$	1.58982	0.0863472
$340$	0	0
$341$	6.02733	0.326398
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	19.2623	1.03405	0.517027	−	0.855969i	$-0.327039\pi$
$347$	19.2623	1.03405	0.517027	+	0.855969i	$0.327039\pi$
$348$	0	0
$349$	28.5730	1.52948	0.764740	−	0.644339i	$-0.222868\pi$
$349$	28.5730	1.52948	0.764740	+	0.644339i	$0.222868\pi$
$350$	0	0
$351$	−15.5264	−0.828739
$352$	0	0
$353$	−23.8293	−1.26831	−0.634153	−	0.773208i	$-0.718651\pi$
$353$	−23.8293	−1.26831	−0.634153	+	0.773208i	$0.718651\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.10463	0.111389
$358$	0	0
$359$	2.03194	0.107242	0.0536209	−	0.998561i	$-0.482924\pi$
$359$	2.03194	0.107242	0.0536209	+	0.998561i	$0.482924\pi$
$360$	0	0
$361$	−16.0208	−0.843201
$362$	0	0
$363$	15.3405	0.805170
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−22.6965	−1.18475	−0.592374	−	0.805663i	$-0.701809\pi$
$367$	−22.6965	−1.18475	−0.592374	+	0.805663i	$0.701809\pi$
$368$	0	0
$369$	4.22683	0.220040
$370$	0	0
$371$	7.86516	0.408339
$372$	0	0
$373$	30.2909	1.56841	0.784203	−	0.620505i	$-0.213072\pi$
$373$	30.2909	1.56841	0.784203	+	0.620505i	$0.213072\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13.4411	0.692253
$378$	0	0
$379$	−14.9783	−0.769385	−0.384692	−	0.923045i	$-0.625693\pi$
$379$	−14.9783	−0.769385	−0.384692	+	0.923045i	$0.625693\pi$
$380$	0	0
$381$	−26.7904	−1.37251
$382$	0	0
$383$	−5.37413	−0.274605	−0.137303	−	0.990529i	$-0.543843\pi$
$383$	−5.37413	−0.274605	−0.137303	+	0.990529i	$0.543843\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.953397	0.0484639
$388$	0	0
$389$	−2.87331	−0.145683	−0.0728413	−	0.997344i	$-0.523207\pi$
$389$	−2.87331	−0.145683	−0.0728413	+	0.997344i	$0.523207\pi$
$390$	0	0
$391$	3.73494	0.188884
$392$	0	0
$393$	32.7368	1.65136
$394$	0	0
$395$	0	0
$396$	0	0
$397$	24.2887	1.21901	0.609507	−	0.792781i	$-0.291368\pi$
$397$	24.2887	1.21901	0.609507	+	0.792781i	$0.291368\pi$
$398$	0	0
$399$	−3.16376	−0.158386
$400$	0	0
$401$	12.7274	0.635575	0.317787	−	0.948162i	$-0.397060\pi$
$401$	12.7274	0.635575	0.317787	+	0.948162i	$0.397060\pi$
$402$	0	0
$403$	−4.39384	−0.218873
$404$	0	0
$405$	0	0
$406$	0	0
$407$	18.4810	0.916069
$408$	0	0
$409$	−10.0455	−0.496717	−0.248359	−	0.968668i	$-0.579891\pi$
$409$	−10.0455	−0.496717	−0.248359	+	0.968668i	$0.579891\pi$
$410$	0	0
$411$	9.57769	0.472433
$412$	0	0
$413$	13.3029	0.654593
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−9.63775	−0.471962
$418$	0	0
$419$	−16.2433	−0.793539	−0.396769	−	0.917918i	$-0.629869\pi$
$419$	−16.2433	−0.793539	−0.396769	+	0.917918i	$0.629869\pi$
$420$	0	0
$421$	−31.8066	−1.55016	−0.775080	−	0.631863i	$-0.782291\pi$
$421$	−31.8066	−1.55016	−0.775080	+	0.631863i	$0.782291\pi$
$422$	0	0
$423$	−0.0384734	−0.00187064
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10.0224	0.485020
$428$	0	0
$429$	−25.8813	−1.24956
$430$	0	0
$431$	18.8177	0.906414	0.453207	−	0.891405i	$-0.350280\pi$
$431$	18.8177	0.906414	0.453207	+	0.891405i	$0.350280\pi$
$432$	0	0
$433$	21.5825	1.03719	0.518595	−	0.855020i	$-0.326455\pi$
$433$	21.5825	1.03719	0.518595	+	0.855020i	$0.326455\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.61451	−0.268578
$438$	0	0
$439$	2.69487	0.128619	0.0643095	−	0.997930i	$-0.479516\pi$
$439$	2.69487	0.128619	0.0643095	+	0.997930i	$0.479516\pi$
$440$	0	0
$441$	0.359777	0.0171322
$442$	0	0
$443$	18.5368	0.880710	0.440355	−	0.897824i	$-0.354853\pi$
$443$	18.5368	0.880710	0.440355	+	0.897824i	$0.354853\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−21.7991	−1.03106
$448$	0	0
$449$	−12.6816	−0.598483	−0.299241	−	0.954177i	$-0.596734\pi$
$449$	−12.6816	−0.598483	−0.299241	+	0.954177i	$0.596734\pi$
$450$	0	0
$451$	−51.7056	−2.43472
$452$	0	0
$453$	−15.6633	−0.735928
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.5381	−0.913954	−0.456977	−	0.889478i	$-0.651068\pi$
$457$	−19.5381	−0.913954	−0.456977	+	0.889478i	$0.651068\pi$
$458$	0	0
$459$	5.55670	0.259364
$460$	0	0
$461$	14.9929	0.698291	0.349145	−	0.937069i	$-0.386472\pi$
$461$	14.9929	0.698291	0.349145	+	0.937069i	$0.386472\pi$
$462$	0	0
$463$	24.9309	1.15864	0.579318	−	0.815102i	$-0.303319\pi$
$463$	24.9309	1.15864	0.579318	+	0.815102i	$0.303319\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.4350	1.40836	0.704181	−	0.710020i	$-0.251314\pi$
$467$	30.4350	1.40836	0.704181	+	0.710020i	$0.251314\pi$
$468$	0	0
$469$	1.28045	0.0591255
$470$	0	0
$471$	6.16678	0.284150
$472$	0	0
$473$	−11.6626	−0.536249
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.82970	−0.129563
$478$	0	0
$479$	17.9694	0.821041	0.410521	−	0.911851i	$-0.365347\pi$
$479$	17.9694	0.821041	0.410521	+	0.911851i	$0.365347\pi$
$480$	0	0
$481$	−13.4724	−0.614288
$482$	0	0
$483$	5.96236	0.271297
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−9.51295	−0.431073	−0.215536	−	0.976496i	$-0.569150\pi$
$487$	−9.51295	−0.431073	−0.215536	+	0.976496i	$0.569150\pi$
$488$	0	0
$489$	−29.0178	−1.31223
$490$	0	0
$491$	−21.0466	−0.949822	−0.474911	−	0.880034i	$-0.657520\pi$
$491$	−21.0466	−0.949822	−0.474911	+	0.880034i	$0.657520\pi$
$492$	0	0
$493$	−4.81040	−0.216649
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.1616	0.635234
$498$	0	0
$499$	36.4408	1.63131	0.815656	−	0.578537i	$-0.196376\pi$
$499$	36.4408	1.63131	0.815656	+	0.578537i	$0.196376\pi$
$500$	0	0
$501$	38.2804	1.71024
$502$	0	0
$503$	−32.1783	−1.43476	−0.717381	−	0.696681i	$-0.754659\pi$
$503$	−32.1783	−1.43476	−0.717381	+	0.696681i	$0.754659\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−4.96147	−0.220347
$508$	0	0
$509$	2.06604	0.0915756	0.0457878	−	0.998951i	$-0.485420\pi$
$509$	2.06604	0.0915756	0.0457878	+	0.998951i	$0.485420\pi$
$510$	0	0
$511$	11.8652	0.524884
$512$	0	0
$513$	−8.35305	−0.368796
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.470634	0.0206985
$518$	0	0
$519$	−10.9013	−0.478514
$520$	0	0
$521$	−23.0609	−1.01032	−0.505158	−	0.863027i	$-0.668566\pi$
$521$	−23.0609	−1.01032	−0.505158	+	0.863027i	$0.668566\pi$
$522$	0	0
$523$	26.1602	1.14391	0.571953	−	0.820286i	$-0.306186\pi$
$523$	26.1602	1.14391	0.571953	+	0.820286i	$0.306186\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.57250	0.0684990
$528$	0	0
$529$	−12.4190	−0.539958
$530$	0	0
$531$	−4.78607	−0.207698
$532$	0	0
$533$	37.6927	1.63265
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−41.9826	−1.81168
$538$	0	0
$539$	−4.40105	−0.189567
$540$	0	0
$541$	41.2094	1.77173	0.885866	−	0.463941i	$-0.153565\pi$
$541$	41.2094	1.77173	0.885866	+	0.463941i	$0.153565\pi$
$542$	0	0
$543$	−3.25804	−0.139816
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7.67055	−0.327969	−0.163985	−	0.986463i	$-0.552435\pi$
$547$	−7.67055	−0.327969	−0.163985	+	0.986463i	$0.552435\pi$
$548$	0	0
$549$	−3.60585	−0.153894
$550$	0	0
$551$	7.23118	0.308059
$552$	0	0
$553$	5.48227	0.233130
$554$	0	0
$555$	0	0
$556$	0	0
$557$	31.8600	1.34995	0.674975	−	0.737841i	$-0.264154\pi$
$557$	31.8600	1.34995	0.674975	+	0.737841i	$0.264154\pi$
$558$	0	0
$559$	8.50190	0.359592
$560$	0	0
$561$	9.26258	0.391067
$562$	0	0
$563$	−9.20351	−0.387882	−0.193941	−	0.981013i	$-0.562127\pi$
$563$	−9.20351	−0.387882	−0.193941	+	0.981013i	$0.562127\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	9.94989	0.417856
$568$	0	0
$569$	8.47615	0.355339	0.177669	−	0.984090i	$-0.443144\pi$
$569$	8.47615	0.355339	0.177669	+	0.984090i	$0.443144\pi$
$570$	0	0
$571$	−13.1245	−0.549245	−0.274622	−	0.961552i	$-0.588553\pi$
$571$	−13.1245	−0.549245	−0.274622	+	0.961552i	$0.588553\pi$
$572$	0	0
$573$	36.1430	1.50990
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16.5640	0.689567	0.344783	−	0.938682i	$-0.387952\pi$
$577$	16.5640	0.689567	0.344783	+	0.938682i	$0.387952\pi$
$578$	0	0
$579$	33.9019	1.40892
$580$	0	0
$581$	−17.1586	−0.711859
$582$	0	0
$583$	34.6150	1.43361
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0.890659	0.0367614	0.0183807	−	0.999831i	$-0.494149\pi$
$587$	0.890659	0.0367614	0.0183807	+	0.999831i	$0.494149\pi$
$588$	0	0
$589$	−2.36384	−0.0974003
$590$	0	0
$591$	−35.5925	−1.46408
$592$	0	0
$593$	8.12654	0.333717	0.166858	−	0.985981i	$-0.446638\pi$
$593$	8.12654	0.333717	0.166858	+	0.985981i	$0.446638\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−39.8643	−1.63154
$598$	0	0
$599$	−9.38288	−0.383374	−0.191687	−	0.981456i	$-0.561396\pi$
$599$	−9.38288	−0.383374	−0.191687	+	0.981456i	$0.561396\pi$
$600$	0	0
$601$	−5.44289	−0.222020	−0.111010	−	0.993819i	$-0.535409\pi$
$601$	−5.44289	−0.222020	−0.111010	+	0.993819i	$0.535409\pi$
$602$	0	0
$603$	−0.460675	−0.0187601
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−36.7479	−1.49155	−0.745776	−	0.666197i	$-0.767921\pi$
$607$	−36.7479	−1.49155	−0.745776	+	0.666197i	$0.767921\pi$
$608$	0	0
$609$	−7.67919	−0.311176
$610$	0	0
$611$	−0.343086	−0.0138798
$612$	0	0
$613$	16.8238	0.679505	0.339753	−	0.940515i	$-0.389657\pi$
$613$	16.8238	0.679505	0.339753	+	0.940515i	$0.389657\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.67406	0.228429	0.114214	−	0.993456i	$-0.463565\pi$
$617$	5.67406	0.228429	0.114214	+	0.993456i	$0.463565\pi$
$618$	0	0
$619$	4.33702	0.174320	0.0871598	−	0.996194i	$-0.472221\pi$
$619$	4.33702	0.174320	0.0871598	+	0.996194i	$0.472221\pi$
$620$	0	0
$621$	15.7420	0.631703
$622$	0	0
$623$	−4.31591	−0.172913
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−13.9239	−0.556066
$628$	0	0
$629$	4.82159	0.192249
$630$	0	0
$631$	−41.9784	−1.67113	−0.835567	−	0.549389i	$-0.814860\pi$
$631$	−41.9784	−1.67113	−0.835567	+	0.549389i	$0.814860\pi$
$632$	0	0
$633$	−42.5416	−1.69088
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.20830	0.127118
$638$	0	0
$639$	−5.09501	−0.201555
$640$	0	0
$641$	−40.4035	−1.59584	−0.797922	−	0.602761i	$-0.794067\pi$
$641$	−40.4035	−1.59584	−0.797922	+	0.602761i	$0.794067\pi$
$642$	0	0
$643$	−28.7826	−1.13507	−0.567537	−	0.823348i	$-0.692104\pi$
$643$	−28.7826	−1.13507	−0.567537	+	0.823348i	$0.692104\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−50.0475	−1.96757	−0.983786	−	0.179346i	$-0.942602\pi$
$647$	−50.0475	−1.96757	−0.983786	+	0.179346i	$0.942602\pi$
$648$	0	0
$649$	58.5467	2.29816
$650$	0	0
$651$	2.51029	0.0983861
$652$	0	0
$653$	−39.5622	−1.54819	−0.774095	−	0.633070i	$-0.781795\pi$
$653$	−39.5622	−1.54819	−0.774095	+	0.633070i	$0.781795\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4.26881	−0.166542
$658$	0	0
$659$	−10.0994	−0.393418	−0.196709	−	0.980462i	$-0.563025\pi$
$659$	−10.0994	−0.393418	−0.196709	+	0.980462i	$0.563025\pi$
$660$	0	0
$661$	−4.51434	−0.175588	−0.0877938	−	0.996139i	$-0.527982\pi$
$661$	−4.51434	−0.175588	−0.0877938	+	0.996139i	$0.527982\pi$
$662$	0	0
$663$	−6.75229	−0.262237
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−13.6277	−0.527667
$668$	0	0
$669$	1.43748	0.0555762
$670$	0	0
$671$	44.1093	1.70282
$672$	0	0
$673$	3.62104	0.139581	0.0697904	−	0.997562i	$-0.477767\pi$
$673$	3.62104	0.139581	0.0697904	+	0.997562i	$0.477767\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.1617	1.35137	0.675687	−	0.737189i	$-0.263847\pi$
$677$	35.1617	1.35137	0.675687	+	0.737189i	$0.263847\pi$
$678$	0	0
$679$	7.65389	0.293729
$680$	0	0
$681$	21.4929	0.823611
$682$	0	0
$683$	−4.39058	−0.168001	−0.0840005	−	0.996466i	$-0.526770\pi$
$683$	−4.39058	−0.168001	−0.0840005	+	0.996466i	$0.526770\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.00812	−0.0766144
$688$	0	0
$689$	−25.2338	−0.961332
$690$	0	0
$691$	6.28314	0.239022	0.119511	−	0.992833i	$-0.461867\pi$
$691$	6.28314	0.239022	0.119511	+	0.992833i	$0.461867\pi$
$692$	0	0
$693$	1.58340	0.0601482
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−13.4897	−0.510959
$698$	0	0
$699$	46.4374	1.75642
$700$	0	0
$701$	5.41413	0.204489	0.102244	−	0.994759i	$-0.467398\pi$
$701$	5.41413	0.204489	0.102244	+	0.994759i	$0.467398\pi$
$702$	0	0
$703$	−7.24800	−0.273363
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.25581	0.0848386
$708$	0	0
$709$	15.5284	0.583180	0.291590	−	0.956543i	$-0.405816\pi$
$709$	15.5284	0.583180	0.291590	+	0.956543i	$0.405816\pi$
$710$	0	0
$711$	−1.97239	−0.0739705
$712$	0	0
$713$	4.45483	0.166835
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−52.9519	−1.97752
$718$	0	0
$719$	19.9304	0.743279	0.371640	−	0.928377i	$-0.378796\pi$
$719$	19.9304	0.743279	0.371640	+	0.928377i	$0.378796\pi$
$720$	0	0
$721$	6.61262	0.246267
$722$	0	0
$723$	18.7956	0.699015
$724$	0	0
$725$	0	0
$726$	0	0
$727$	24.0839	0.893222	0.446611	−	0.894728i	$-0.352631\pi$
$727$	24.0839	0.893222	0.446611	+	0.894728i	$0.352631\pi$
$728$	0	0
$729$	23.0319	0.853035
$730$	0	0
$731$	−3.04272	−0.112539
$732$	0	0
$733$	−35.8771	−1.32515	−0.662576	−	0.748994i	$-0.730537\pi$
$733$	−35.8771	−1.32515	−0.662576	+	0.748994i	$0.730537\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.63531	0.207579
$738$	0	0
$739$	1.29734	0.0477233	0.0238617	−	0.999715i	$-0.492404\pi$
$739$	1.29734	0.0477233	0.0238617	+	0.999715i	$0.492404\pi$
$740$	0	0
$741$	10.1503	0.372881
$742$	0	0
$743$	−8.53790	−0.313225	−0.156613	−	0.987660i	$-0.550057\pi$
$743$	−8.53790	−0.313225	−0.156613	+	0.987660i	$0.550057\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.17327	0.225868
$748$	0	0
$749$	5.83756	0.213300
$750$	0	0
$751$	−17.5930	−0.641978	−0.320989	−	0.947083i	$-0.604015\pi$
$751$	−17.5930	−0.641978	−0.320989	+	0.947083i	$0.604015\pi$
$752$	0	0
$753$	50.4780	1.83952
$754$	0	0
$755$	0	0
$756$	0	0
$757$	33.5195	1.21829	0.609143	−	0.793061i	$-0.291514\pi$
$757$	33.5195	1.21829	0.609143	+	0.793061i	$0.291514\pi$
$758$	0	0
$759$	26.2406	0.952474
$760$	0	0
$761$	−25.8063	−0.935479	−0.467740	−	0.883866i	$-0.654931\pi$
$761$	−25.8063	−0.935479	−0.467740	+	0.883866i	$0.654931\pi$
$762$	0	0
$763$	10.6952	0.387191
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−42.6797	−1.54108
$768$	0	0
$769$	−27.6616	−0.997502	−0.498751	−	0.866745i	$-0.666208\pi$
$769$	−27.6616	−0.997502	−0.498751	+	0.866745i	$0.666208\pi$
$770$	0	0
$771$	22.4788	0.809555
$772$	0	0
$773$	−49.4294	−1.77785	−0.888927	−	0.458050i	$-0.848548\pi$
$773$	−49.4294	−1.77785	−0.888927	+	0.458050i	$0.848548\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.69705	0.276130
$778$	0	0
$779$	20.2783	0.726544
$780$	0	0
$781$	62.3258	2.23019
$782$	0	0
$783$	−20.2748	−0.724561
$784$	0	0
$785$	0	0
$786$	0	0
$787$	44.2790	1.57837	0.789187	−	0.614153i	$-0.210502\pi$
$787$	44.2790	1.57837	0.789187	+	0.614153i	$0.210502\pi$
$788$	0	0
$789$	−32.7969	−1.16760
$790$	0	0
$791$	−0.867348	−0.0308393
$792$	0	0
$793$	−32.1550	−1.14186
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17.7589	−0.629052	−0.314526	−	0.949249i	$-0.601846\pi$
$797$	−17.7589	−0.629052	−0.314526	+	0.949249i	$0.601846\pi$
$798$	0	0
$799$	0.122786	0.00434385
$800$	0	0
$801$	1.55276	0.0548642
$802$	0	0
$803$	52.2192	1.84277
$804$	0	0
$805$	0	0
$806$	0	0
$807$	46.2450	1.62790
$808$	0	0
$809$	31.2427	1.09844	0.549218	−	0.835679i	$-0.314926\pi$
$809$	31.2427	1.09844	0.549218	+	0.835679i	$0.314926\pi$
$810$	0	0
$811$	6.66672	0.234100	0.117050	−	0.993126i	$-0.462656\pi$
$811$	6.66672	0.234100	0.117050	+	0.993126i	$0.462656\pi$
$812$	0	0
$813$	−17.7065	−0.620993
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.57393	0.160022
$818$	0	0
$819$	−1.15427	−0.0403336
$820$	0	0
$821$	−43.2138	−1.50817	−0.754086	−	0.656776i	$-0.771920\pi$
$821$	−43.2138	−1.50817	−0.754086	+	0.656776i	$0.771920\pi$
$822$	0	0
$823$	−17.8129	−0.620918	−0.310459	−	0.950587i	$-0.600483\pi$
$823$	−17.8129	−0.620918	−0.310459	+	0.950587i	$0.600483\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18.9139	0.657702	0.328851	−	0.944382i	$-0.393339\pi$
$827$	18.9139	0.657702	0.328851	+	0.944382i	$0.393339\pi$
$828$	0	0
$829$	37.5969	1.30579	0.652897	−	0.757447i	$-0.273554\pi$
$829$	37.5969	1.30579	0.652897	+	0.757447i	$0.273554\pi$
$830$	0	0
$831$	42.5195	1.47498
$832$	0	0
$833$	−1.14821	−0.0397831
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.62773	0.229088
$838$	0	0
$839$	−32.2054	−1.11185	−0.555927	−	0.831231i	$-0.687636\pi$
$839$	−32.2054	−1.11185	−0.555927	+	0.831231i	$0.687636\pi$
$840$	0	0
$841$	−11.4483	−0.394767
$842$	0	0
$843$	49.4415	1.70285
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−8.36923	−0.287570
$848$	0	0
$849$	40.7138	1.39729
$850$	0	0
$851$	13.6594	0.468238
$852$	0	0
$853$	−20.6540	−0.707179	−0.353590	−	0.935401i	$-0.615039\pi$
$853$	−20.6540	−0.707179	−0.353590	+	0.935401i	$0.615039\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−42.6561	−1.45711	−0.728553	−	0.684990i	$-0.759807\pi$
$857$	−42.6561	−1.45711	−0.728553	+	0.684990i	$0.759807\pi$
$858$	0	0
$859$	7.95286	0.271348	0.135674	−	0.990754i	$-0.456680\pi$
$859$	7.95286	0.271348	0.135674	+	0.990754i	$0.456680\pi$
$860$	0	0
$861$	−21.5346	−0.733897
$862$	0	0
$863$	−29.0419	−0.988597	−0.494299	−	0.869292i	$-0.664575\pi$
$863$	−29.0419	−0.988597	−0.494299	+	0.869292i	$0.664575\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−28.7439	−0.976195
$868$	0	0
$869$	24.1277	0.818477
$870$	0	0
$871$	−4.10806	−0.139196
$872$	0	0
$873$	−2.75369	−0.0931983
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.1827	0.512682	0.256341	−	0.966586i	$-0.417483\pi$
$877$	15.1827	0.512682	0.256341	+	0.966586i	$0.417483\pi$
$878$	0	0
$879$	38.2643	1.29062
$880$	0	0
$881$	−17.3877	−0.585806	−0.292903	−	0.956142i	$-0.594621\pi$
$881$	−17.3877	−0.585806	−0.292903	+	0.956142i	$0.594621\pi$
$882$	0	0
$883$	8.46587	0.284899	0.142450	−	0.989802i	$-0.454502\pi$
$883$	8.46587	0.284899	0.142450	+	0.989802i	$0.454502\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.8229	1.10208	0.551042	−	0.834477i	$-0.314230\pi$
$887$	32.8229	1.10208	0.551042	+	0.834477i	$0.314230\pi$
$888$	0	0
$889$	14.6158	0.490199
$890$	0	0
$891$	43.7900	1.46702
$892$	0	0
$893$	−0.184577	−0.00617662
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−19.1290	−0.638700
$898$	0	0
$899$	−5.73758	−0.191359
$900$	0	0
$901$	9.03085	0.300861
$902$	0	0
$903$	−4.85731	−0.161641
$904$	0	0
$905$	0	0
$906$	0	0
$907$	54.6260	1.81383	0.906913	−	0.421317i	$-0.138432\pi$
$907$	54.6260	1.81383	0.906913	+	0.421317i	$0.138432\pi$
$908$	0	0
$909$	−0.811589	−0.0269187
$910$	0	0
$911$	−4.21329	−0.139592	−0.0697962	−	0.997561i	$-0.522235\pi$
$911$	−4.21329	−0.139592	−0.0697962	+	0.997561i	$0.522235\pi$
$912$	0	0
$913$	−75.5159	−2.49921
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−17.8600	−0.589789
$918$	0	0
$919$	25.0033	0.824784	0.412392	−	0.911007i	$-0.364693\pi$
$919$	25.0033	0.824784	0.412392	+	0.911007i	$0.364693\pi$
$920$	0	0
$921$	−5.45077	−0.179609
$922$	0	0
$923$	−45.4346	−1.49550
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2.37907	−0.0781388
$928$	0	0
$929$	−34.5462	−1.13342	−0.566712	−	0.823916i	$-0.691785\pi$
$929$	−34.5462	−1.13342	−0.566712	+	0.823916i	$0.691785\pi$
$930$	0	0
$931$	1.72603	0.0565684
$932$	0	0
$933$	−1.90642	−0.0624134
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19.7689	0.645821	0.322911	−	0.946429i	$-0.395339\pi$
$937$	19.7689	0.645821	0.322911	+	0.946429i	$0.395339\pi$
$938$	0	0
$939$	−16.7242	−0.545774
$940$	0	0
$941$	23.0820	0.752452	0.376226	−	0.926528i	$-0.377222\pi$
$941$	23.0820	0.752452	0.376226	+	0.926528i	$0.377222\pi$
$942$	0	0
$943$	−38.2159	−1.24448
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−60.0386	−1.95099	−0.975496	−	0.220019i	$-0.929388\pi$
$947$	−60.0386	−1.95099	−0.975496	+	0.220019i	$0.929388\pi$
$948$	0	0
$949$	−38.0670	−1.23571
$950$	0	0
$951$	−8.69295	−0.281888
$952$	0	0
$953$	11.4188	0.369891	0.184945	−	0.982749i	$-0.440789\pi$
$953$	11.4188	0.369891	0.184945	+	0.982749i	$0.440789\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−33.7965	−1.09249
$958$	0	0
$959$	−5.22523	−0.168732
$960$	0	0
$961$	−29.1244	−0.939497
$962$	0	0
$963$	−2.10022	−0.0676786
$964$	0	0
$965$	0	0
$966$	0	0
$967$	28.9169	0.929903	0.464952	−	0.885336i	$-0.346072\pi$
$967$	28.9169	0.929903	0.464952	+	0.885336i	$0.346072\pi$
$968$	0	0
$969$	−3.63266	−0.116698
$970$	0	0
$971$	43.1410	1.38446	0.692230	−	0.721677i	$-0.256629\pi$
$971$	43.1410	1.38446	0.692230	+	0.721677i	$0.256629\pi$
$972$	0	0
$973$	5.25800	0.168564
$974$	0	0
$975$	0	0
$976$	0	0
$977$	38.0221	1.21644	0.608218	−	0.793770i	$-0.291885\pi$
$977$	38.0221	1.21644	0.608218	+	0.793770i	$0.291885\pi$
$978$	0	0
$979$	−18.9945	−0.607067
$980$	0	0
$981$	−3.84787	−0.122853
$982$	0	0
$983$	−12.7030	−0.405164	−0.202582	−	0.979265i	$-0.564933\pi$
$983$	−12.7030	−0.405164	−0.202582	+	0.979265i	$0.564933\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.196012	0.00623913
$988$	0	0
$989$	−8.61992	−0.274098
$990$	0	0
$991$	−25.4382	−0.808073	−0.404036	−	0.914743i	$-0.632393\pi$
$991$	−25.4382	−0.808073	−0.404036	+	0.914743i	$0.632393\pi$
$992$	0	0
$993$	−17.1954	−0.545678
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−13.5489	−0.429098	−0.214549	−	0.976713i	$-0.568828\pi$
$997$	−13.5489	−0.429098	−0.214549	+	0.976713i	$0.568828\pi$
$998$	0	0
$999$	20.3219	0.642958

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5600.2.a.bw.1.4		5
4.3	odd	2		5600.2.a.bv.1.2		5
5.2	odd	4		1120.2.g.b.449.3	✓	10
5.3	odd	4		1120.2.g.b.449.8	yes	10
5.4	even	2		5600.2.a.bu.1.2		5
20.3	even	4		1120.2.g.c.449.3	yes	10
20.7	even	4		1120.2.g.c.449.8	yes	10
20.19	odd	2		5600.2.a.bx.1.4		5
40.3	even	4		2240.2.g.n.449.8		10
40.13	odd	4		2240.2.g.o.449.3		10
40.27	even	4		2240.2.g.n.449.3		10
40.37	odd	4		2240.2.g.o.449.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.g.b.449.3	✓	10	5.2	odd	4
1120.2.g.b.449.8	yes	10	5.3	odd	4
1120.2.g.c.449.3	yes	10	20.3	even	4
1120.2.g.c.449.8	yes	10	20.7	even	4
2240.2.g.n.449.3		10	40.27	even	4
2240.2.g.n.449.8		10	40.3	even	4
2240.2.g.o.449.3		10	40.13	odd	4
2240.2.g.o.449.8		10	40.37	odd	4
5600.2.a.bu.1.2		5	5.4	even	2
5600.2.a.bv.1.2		5	4.3	odd	2
5600.2.a.bw.1.4		5	1.1	even	1	trivial
5600.2.a.bx.1.4		5	20.19	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:	\( N \)	\(=\)	\( 5600 = 2^{5} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5600.a (trivial)

\(f(q)\)	\(=\)	\(q+1.83297 q^{3} -1.00000 q^{7} +0.359777 q^{9} +O(q^{10})\)	\(q+1.83297 q^{3} -1.00000 q^{7} +0.359777 q^{9} -4.40105 q^{11} +3.20830 q^{13} -1.14821 q^{17} +1.72603 q^{19} -1.83297 q^{21} -3.25284 q^{23} -4.83945 q^{27} +4.18948 q^{29} -1.36952 q^{31} -8.06699 q^{33} -4.19923 q^{37} +5.88072 q^{39} +11.7485 q^{41} +2.64997 q^{43} -0.106937 q^{47} +1.00000 q^{49} -2.10463 q^{51} -7.86516 q^{53} +3.16376 q^{57} -13.3029 q^{59} -10.0224 q^{61} -0.359777 q^{63} -1.28045 q^{67} -5.96236 q^{69} -14.1616 q^{71} -11.8652 q^{73} +4.40105 q^{77} -5.48227 q^{79} -9.94989 q^{81} +17.1586 q^{83} +7.67919 q^{87} +4.31591 q^{89} -3.20830 q^{91} -2.51029 q^{93} -7.65389 q^{97} -1.58340 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) 5 * q + 2 * q^3 - 5 * q^7 + 7 * q^9	\( 5 q + 2 q^{3} - 5 q^{7} + 7 q^{9} - 4 q^{11} + 4 q^{17} - 12 q^{19} - 2 q^{21} - 8 q^{23} + 14 q^{27} - 12 q^{29} - 12 q^{31} - 8 q^{33} + 12 q^{37} - 32 q^{39} - 2 q^{41} + 8 q^{43} - 14 q^{47} + 5 q^{49} - 12 q^{51} + 8 q^{53} - 8 q^{57} - 16 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{67} + 4 q^{69} - 4 q^{71} - 12 q^{73} + 4 q^{77} - 32 q^{79} + q^{81} + 24 q^{83} - 2 q^{87} + 2 q^{89} - 20 q^{93} - 12 q^{97} - 40 q^{99}+O(q^{100}) \) 5 * q + 2 * q^3 - 5 * q^7 + 7 * q^9 - 4 * q^11 + 4 * q^17 - 12 * q^19 - 2 * q^21 - 8 * q^23 + 14 * q^27 - 12 * q^29 - 12 * q^31 - 8 * q^33 + 12 * q^37 - 32 * q^39 - 2 * q^41 + 8 * q^43 - 14 * q^47 + 5 * q^49 - 12 * q^51 + 8 * q^53 - 8 * q^57 - 16 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^67 + 4 * q^69 - 4 * q^71 - 12 * q^73 + 4 * q^77 - 32 * q^79 + q^81 + 24 * q^83 - 2 * q^87 + 2 * q^89 - 20 * q^93 - 12 * q^97 - 40 * q^99

Introduction

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