LMFDB - Embedded newform 5600.2.a.bq.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:	$4$
Coefficient field:	4.4.48396.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 6x + 6 $ x^4 - 2x^3 - 7x^2 + 6*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.08183$ 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)

$f(q)$	$=$	$q+2.08183 q^{3} -1.00000 q^{7} +1.33402 q^{9} +O(q^{10})$	$q+2.08183 q^{3} -1.00000 q^{7} +1.33402 q^{9} -3.41585 q^{11} +3.28158 q^{13} +6.49768 q^{17} -7.36341 q^{19} -2.08183 q^{21} -4.61560 q^{23} -3.46829 q^{27} -9.66134 q^{29} -1.28158 q^{31} -7.11123 q^{33} +11.6613 q^{37} +6.83170 q^{39} -1.21610 q^{41} -10.3798 q^{43} +2.83170 q^{47} +1.00000 q^{49} +13.5271 q^{51} -6.56317 q^{53} -15.3294 q^{57} -2.71842 q^{59} -5.94962 q^{61} -1.33402 q^{63} -11.3110 q^{67} -9.60891 q^{69} +4.45194 q^{71} -13.5434 q^{73} +3.41585 q^{77} -2.21610 q^{79} -11.2225 q^{81} +4.91353 q^{83} -20.1133 q^{87} -3.21610 q^{89} -3.28158 q^{91} -2.66804 q^{93} +11.6089 q^{97} -4.55682 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 4 q^{7} + 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 4 * q^7 + 6 * q^9	$ 4 q - 2 q^{3} - 4 q^{7} + 6 q^{9} - 4 q^{11} - 2 q^{13} + 6 q^{17} - 4 q^{19} + 2 q^{21} - 4 q^{23} - 20 q^{27} + 2 q^{29} + 10 q^{31} + 8 q^{33} + 6 q^{37} + 8 q^{39} - 16 q^{43} - 8 q^{47} + 4 q^{49} + 8 q^{51} + 4 q^{53} - 22 q^{57} - 26 q^{59} - 10 q^{61} - 6 q^{63} - 4 q^{67} + 18 q^{69} + 24 q^{71} - 8 q^{73} + 4 q^{77} - 4 q^{79} + 28 q^{81} - 10 q^{83} - 46 q^{87} - 8 q^{89} + 2 q^{91} - 12 q^{93} - 10 q^{97} - 56 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 4 * q^7 + 6 * q^9 - 4 * q^11 - 2 * q^13 + 6 * q^17 - 4 * q^19 + 2 * q^21 - 4 * q^23 - 20 * q^27 + 2 * q^29 + 10 * q^31 + 8 * q^33 + 6 * q^37 + 8 * q^39 - 16 * q^43 - 8 * q^47 + 4 * q^49 + 8 * q^51 + 4 * q^53 - 22 * q^57 - 26 * q^59 - 10 * q^61 - 6 * q^63 - 4 * q^67 + 18 * q^69 + 24 * q^71 - 8 * q^73 + 4 * q^77 - 4 * q^79 + 28 * q^81 - 10 * q^83 - 46 * q^87 - 8 * q^89 + 2 * q^91 - 12 * q^93 - 10 * q^97 - 56 * 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
$3$	2.08183	1.20195	0.600973	−	0.799269i	$-0.294780\pi$
$3$	2.08183	1.20195	0.600973	+	0.799269i	$0.294780\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.33402	0.444673
$10$	0	0
$11$	−3.41585	−1.02992	−0.514959	−	0.857215i	$-0.672193\pi$
$11$	−3.41585	−1.02992	−0.514959	+	0.857215i	$0.672193\pi$
$12$	0	0
$13$	3.28158	0.910148	0.455074	−	0.890454i	$-0.349613\pi$
$13$	3.28158	0.910148	0.455074	+	0.890454i	$0.349613\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.49768	1.57592	0.787960	−	0.615727i	$-0.211137\pi$
$17$	6.49768	1.57592	0.787960	+	0.615727i	$0.211137\pi$
$18$	0	0
$19$	−7.36341	−1.68928	−0.844642	−	0.535332i	$-0.820186\pi$
$19$	−7.36341	−1.68928	−0.844642	+	0.535332i	$0.820186\pi$
$20$	0	0
$21$	−2.08183	−0.454293
$22$	0	0
$23$	−4.61560	−0.962420	−0.481210	−	0.876605i	$-0.659803\pi$
$23$	−4.61560	−0.962420	−0.481210	+	0.876605i	$0.659803\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.46829	−0.667472
$28$	0	0
$29$	−9.66134	−1.79407	−0.897033	−	0.441963i	$-0.854282\pi$
$29$	−9.66134	−1.79407	−0.897033	+	0.441963i	$0.854282\pi$
$30$	0	0
$31$	−1.28158	−0.230179	−0.115090	−	0.993355i	$-0.536716\pi$
$31$	−1.28158	−0.230179	−0.115090	+	0.993355i	$0.536716\pi$
$32$	0	0
$33$	−7.11123	−1.23791
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.6613	1.91711	0.958557	−	0.284902i	$-0.0919611\pi$
$37$	11.6613	1.91711	0.958557	+	0.284902i	$0.0919611\pi$
$38$	0	0
$39$	6.83170	1.09395
$40$	0	0
$41$	−1.21610	−0.189923	−0.0949614	−	0.995481i	$-0.530273\pi$
$41$	−1.21610	−0.189923	−0.0949614	+	0.995481i	$0.530273\pi$
$42$	0	0
$43$	−10.3798	−1.58290	−0.791449	−	0.611235i	$-0.790673\pi$
$43$	−10.3798	−1.58290	−0.791449	+	0.611235i	$0.790673\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.83170	0.413046	0.206523	−	0.978442i	$-0.433785\pi$
$47$	2.83170	0.413046	0.206523	+	0.978442i	$0.433785\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	13.5271	1.89417
$52$	0	0
$53$	−6.56317	−0.901520	−0.450760	−	0.892645i	$-0.648847\pi$
$53$	−6.56317	−0.901520	−0.450760	+	0.892645i	$0.648847\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−15.3294	−2.03043
$58$	0	0
$59$	−2.71842	−0.353908	−0.176954	−	0.984219i	$-0.556624\pi$
$59$	−2.71842	−0.353908	−0.176954	+	0.984219i	$0.556624\pi$
$60$	0	0
$61$	−5.94962	−0.761771	−0.380886	−	0.924622i	$-0.624381\pi$
$61$	−5.94962	−0.761771	−0.380886	+	0.924622i	$0.624381\pi$
$62$	0	0
$63$	−1.33402	−0.168071
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.3110	−1.38186	−0.690928	−	0.722924i	$-0.742798\pi$
$67$	−11.3110	−1.38186	−0.690928	+	0.722924i	$0.742798\pi$
$68$	0	0
$69$	−9.60891	−1.15678
$70$	0	0
$71$	4.45194	0.528348	0.264174	−	0.964475i	$-0.414901\pi$
$71$	4.45194	0.528348	0.264174	+	0.964475i	$0.414901\pi$
$72$	0	0
$73$	−13.5434	−1.58514	−0.792569	−	0.609782i	$-0.791257\pi$
$73$	−13.5434	−1.58514	−0.792569	+	0.609782i	$0.791257\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.41585	0.389272
$78$	0	0
$79$	−2.21610	−0.249331	−0.124665	−	0.992199i	$-0.539786\pi$
$79$	−2.21610	−0.249331	−0.124665	+	0.992199i	$0.539786\pi$
$80$	0	0
$81$	−11.2225	−1.24694
$82$	0	0
$83$	4.91353	0.539330	0.269665	−	0.962954i	$-0.413087\pi$
$83$	4.91353	0.539330	0.269665	+	0.962954i	$0.413087\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−20.1133	−2.15637
$88$	0	0
$89$	−3.21610	−0.340906	−0.170453	−	0.985366i	$-0.554523\pi$
$89$	−3.21610	−0.340906	−0.170453	+	0.985366i	$0.554523\pi$
$90$	0	0
$91$	−3.28158	−0.344003
$92$	0	0
$93$	−2.66804	−0.276663
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.6089	1.17871	0.589353	−	0.807876i	$-0.299383\pi$
$97$	11.6089	1.17871	0.589353	+	0.807876i	$0.299383\pi$
$98$	0	0
$99$	−4.55682	−0.457977
$100$	0	0
$101$	14.9450	1.48708	0.743541	−	0.668690i	$-0.233145\pi$
$101$	14.9450	1.48708	0.743541	+	0.668690i	$0.233145\pi$
$102$	0	0
$103$	17.5585	1.73009	0.865047	−	0.501691i	$-0.167289\pi$
$103$	17.5585	1.73009	0.865047	+	0.501691i	$0.167289\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.3588	1.19477	0.597384	−	0.801955i	$-0.296207\pi$
$107$	12.3588	1.19477	0.597384	+	0.801955i	$0.296207\pi$
$108$	0	0
$109$	3.56986	0.341931	0.170965	−	0.985277i	$-0.445311\pi$
$109$	3.56986	0.341931	0.170965	+	0.985277i	$0.445311\pi$
$110$	0	0
$111$	24.2769	2.30427
$112$	0	0
$113$	−0.764157	−0.0718858	−0.0359429	−	0.999354i	$-0.511443\pi$
$113$	−0.764157	−0.0718858	−0.0359429	+	0.999354i	$0.511443\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.37770	0.404718
$118$	0	0
$119$	−6.49768	−0.595642
$120$	0	0
$121$	0.668041	0.0607310
$122$	0	0
$123$	−2.53171	−0.228277
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.5481	−1.02472	−0.512362	−	0.858769i	$-0.671229\pi$
$127$	−11.5481	−1.02472	−0.512362	+	0.858769i	$0.671229\pi$
$128$	0	0
$129$	−21.6089	−1.90256
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	7.36341	0.638489
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.22915	−0.703063	−0.351532	−	0.936176i	$-0.614339\pi$
$137$	−8.22915	−0.703063	−0.351532	+	0.936176i	$0.614339\pi$
$138$	0	0
$139$	3.58157	0.303785	0.151893	−	0.988397i	$-0.451463\pi$
$139$	3.58157	0.303785	0.151893	+	0.988397i	$0.451463\pi$
$140$	0	0
$141$	5.89513	0.496459
$142$	0	0
$143$	−11.2094	−0.937377
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.08183	0.171707
$148$	0	0
$149$	−15.6613	−1.28303	−0.641514	−	0.767112i	$-0.721693\pi$
$149$	−15.6613	−1.28303	−0.641514	+	0.767112i	$0.721693\pi$
$150$	0	0
$151$	2.05244	0.167025	0.0835125	−	0.996507i	$-0.473386\pi$
$151$	2.05244	0.167025	0.0835125	+	0.996507i	$0.473386\pi$
$152$	0	0
$153$	8.66804	0.700770
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−8.38182	−0.668942	−0.334471	−	0.942406i	$-0.608558\pi$
$157$	−8.38182	−0.668942	−0.334471	+	0.942406i	$0.608558\pi$
$158$	0	0
$159$	−13.6634	−1.08358
$160$	0	0
$161$	4.61560	0.363761
$162$	0	0
$163$	−17.3634	−1.36001	−0.680004	−	0.733209i	$-0.738022\pi$
$163$	−17.3634	−1.36001	−0.680004	+	0.733209i	$0.738022\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.2816	−1.33729	−0.668645	−	0.743582i	$-0.733125\pi$
$167$	−17.2816	−1.33729	−0.668645	+	0.743582i	$0.733125\pi$
$168$	0	0
$169$	−2.23121	−0.171631
$170$	0	0
$171$	−9.82295	−0.751179
$172$	0	0
$173$	15.6634	1.19087	0.595433	−	0.803405i	$-0.296980\pi$
$173$	15.6634	1.19087	0.595433	+	0.803405i	$0.296980\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.65928	−0.425378
$178$	0	0
$179$	−21.3634	−1.59678	−0.798388	−	0.602143i	$-0.794314\pi$
$179$	−21.3634	−1.59678	−0.798388	+	0.602143i	$0.794314\pi$
$180$	0	0
$181$	−9.94962	−0.739550	−0.369775	−	0.929121i	$-0.620565\pi$
$181$	−9.94962	−0.739550	−0.369775	+	0.929121i	$0.620565\pi$
$182$	0	0
$183$	−12.3861	−0.915608
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−22.1951	−1.62307
$188$	0	0
$189$	3.46829	0.252281
$190$	0	0
$191$	−5.82707	−0.421632	−0.210816	−	0.977526i	$-0.567612\pi$
$191$	−5.82707	−0.421632	−0.210816	+	0.977526i	$0.567612\pi$
$192$	0	0
$193$	−13.2266	−0.952070	−0.476035	−	0.879426i	$-0.657926\pi$
$193$	−13.2266	−0.952070	−0.476035	+	0.879426i	$0.657926\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.2245	0.870960	0.435480	−	0.900198i	$-0.356579\pi$
$197$	12.2245	0.870960	0.435480	+	0.900198i	$0.356579\pi$
$198$	0	0
$199$	7.31428	0.518495	0.259248	−	0.965811i	$-0.416525\pi$
$199$	7.31428	0.518495	0.259248	+	0.965811i	$0.416525\pi$
$200$	0	0
$201$	−23.5475	−1.66092
$202$	0	0
$203$	9.66134	0.678093
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−6.15731	−0.427963
$208$	0	0
$209$	25.1523	1.73982
$210$	0	0
$211$	−19.1905	−1.32113	−0.660564	−	0.750770i	$-0.729683\pi$
$211$	−19.1905	−1.32113	−0.660564	+	0.750770i	$0.729683\pi$
$212$	0	0
$213$	9.26819	0.635046
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.28158	0.0869996
$218$	0	0
$219$	−28.1951	−1.90525
$220$	0	0
$221$	21.3227	1.43432
$222$	0	0
$223$	18.2048	1.21908	0.609541	−	0.792755i	$-0.291354\pi$
$223$	18.2048	1.21908	0.609541	+	0.792755i	$0.291354\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.09612	−0.205496	−0.102748	−	0.994707i	$-0.532764\pi$
$227$	−3.09612	−0.205496	−0.102748	+	0.994707i	$0.532764\pi$
$228$	0	0
$229$	6.32732	0.418121	0.209061	−	0.977903i	$-0.432959\pi$
$229$	6.32732	0.418121	0.209061	+	0.977903i	$0.432959\pi$
$230$	0	0
$231$	7.11123	0.467884
$232$	0	0
$233$	−21.5565	−1.41221	−0.706106	−	0.708106i	$-0.749550\pi$
$233$	−21.5565	−1.41221	−0.706106	+	0.708106i	$0.749550\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.61354	−0.299682
$238$	0	0
$239$	5.76416	0.372852	0.186426	−	0.982469i	$-0.440309\pi$
$239$	5.76416	0.372852	0.186426	+	0.982469i	$0.440309\pi$
$240$	0	0
$241$	−21.4930	−1.38449	−0.692244	−	0.721663i	$-0.743378\pi$
$241$	−21.4930	−1.38449	−0.692244	+	0.721663i	$0.743378\pi$
$242$	0	0
$243$	−12.9584	−0.831281
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−24.1637	−1.53750
$248$	0	0
$249$	10.2291	0.648246
$250$	0	0
$251$	20.1229	1.27015	0.635074	−	0.772451i	$-0.280969\pi$
$251$	20.1229	1.27015	0.635074	+	0.772451i	$0.280969\pi$
$252$	0	0
$253$	15.7662	0.991214
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.6089	0.973657	0.486828	−	0.873498i	$-0.338154\pi$
$257$	15.6089	0.973657	0.486828	+	0.873498i	$0.338154\pi$
$258$	0	0
$259$	−11.6613	−0.724601
$260$	0	0
$261$	−12.8884	−0.797774
$262$	0	0
$263$	−9.21146	−0.568003	−0.284002	−	0.958824i	$-0.591662\pi$
$263$	−9.21146	−0.568003	−0.284002	+	0.958824i	$0.591662\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.69537	−0.409750
$268$	0	0
$269$	−19.2816	−1.17562	−0.587809	−	0.808999i	$-0.700010\pi$
$269$	−19.2816	−1.17562	−0.587809	+	0.808999i	$0.700010\pi$
$270$	0	0
$271$	27.5367	1.67274	0.836369	−	0.548168i	$-0.184674\pi$
$271$	27.5367	1.67274	0.836369	+	0.548168i	$0.184674\pi$
$272$	0	0
$273$	−6.83170	−0.413473
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4.66392	−0.280228	−0.140114	−	0.990135i	$-0.544747\pi$
$277$	−4.66392	−0.280228	−0.140114	+	0.990135i	$0.544747\pi$
$278$	0	0
$279$	−1.70966	−0.102355
$280$	0	0
$281$	−3.46499	−0.206704	−0.103352	−	0.994645i	$-0.532957\pi$
$281$	−3.46499	−0.206704	−0.103352	+	0.994645i	$0.532957\pi$
$282$	0	0
$283$	−27.6907	−1.64604	−0.823022	−	0.568010i	$-0.807714\pi$
$283$	−27.6907	−1.64604	−0.823022	+	0.568010i	$0.807714\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.21610	0.0717840
$288$	0	0
$289$	25.2199	1.48352
$290$	0	0
$291$	24.1678	1.41674
$292$	0	0
$293$	−4.71842	−0.275653	−0.137826	−	0.990456i	$-0.544012\pi$
$293$	−4.71842	−0.275653	−0.137826	+	0.990456i	$0.544012\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	11.8472	0.687442
$298$	0	0
$299$	−15.1465	−0.875944
$300$	0	0
$301$	10.3798	0.598279
$302$	0	0
$303$	31.1129	1.78739
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20.1229	1.14848	0.574238	−	0.818688i	$-0.305298\pi$
$307$	20.1229	1.14848	0.574238	+	0.818688i	$0.305298\pi$
$308$	0	0
$309$	36.5539	2.07948
$310$	0	0
$311$	17.3907	0.986139	0.493069	−	0.869990i	$-0.335875\pi$
$311$	17.3907	0.986139	0.493069	+	0.869990i	$0.335875\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.2245	−0.574266	−0.287133	−	0.957891i	$-0.592702\pi$
$317$	−10.2245	−0.574266	−0.287133	+	0.957891i	$0.592702\pi$
$318$	0	0
$319$	33.0017	1.84774
$320$	0	0
$321$	25.7289	1.43605
$322$	0	0
$323$	−47.8451	−2.66217
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.43185	0.410982
$328$	0	0
$329$	−2.83170	−0.156117
$330$	0	0
$331$	18.9022	1.03896	0.519480	−	0.854483i	$-0.326126\pi$
$331$	18.9022	1.03896	0.519480	+	0.854483i	$0.326126\pi$
$332$	0	0
$333$	15.5565	0.852489
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−11.1330	−0.606455	−0.303227	−	0.952918i	$-0.598064\pi$
$337$	−11.1330	−0.606455	−0.303227	+	0.952918i	$0.598064\pi$
$338$	0	0
$339$	−1.59084	−0.0864028
$340$	0	0
$341$	4.37770	0.237066
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.6798	−1.21751	−0.608756	−	0.793357i	$-0.708331\pi$
$347$	−22.6798	−1.21751	−0.608756	+	0.793357i	$0.708331\pi$
$348$	0	0
$349$	9.33196	0.499528	0.249764	−	0.968307i	$-0.419647\pi$
$349$	9.33196	0.499528	0.249764	+	0.968307i	$0.419647\pi$
$350$	0	0
$351$	−11.3815	−0.607498
$352$	0	0
$353$	25.5996	1.36253	0.681266	−	0.732036i	$-0.261430\pi$
$353$	25.5996	1.36253	0.681266	+	0.732036i	$0.261430\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−13.5271	−0.715929
$358$	0	0
$359$	−13.1066	−0.691739	−0.345870	−	0.938283i	$-0.612416\pi$
$359$	−13.1066	−0.691739	−0.345870	+	0.938283i	$0.612416\pi$
$360$	0	0
$361$	35.2199	1.85368
$362$	0	0
$363$	1.39075	0.0729953
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−2.28622	−0.119340	−0.0596698	−	0.998218i	$-0.519005\pi$
$367$	−2.28622	−0.119340	−0.0596698	+	0.998218i	$0.519005\pi$
$368$	0	0
$369$	−1.62230	−0.0844536
$370$	0	0
$371$	6.56317	0.340743
$372$	0	0
$373$	7.66134	0.396689	0.198345	−	0.980132i	$-0.436443\pi$
$373$	7.66134	0.396689	0.198345	+	0.980132i	$0.436443\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−31.7045	−1.63287
$378$	0	0
$379$	6.84341	0.351523	0.175761	−	0.984433i	$-0.443761\pi$
$379$	6.84341	0.351523	0.175761	+	0.984433i	$0.443761\pi$
$380$	0	0
$381$	−24.0411	−1.23166
$382$	0	0
$383$	−1.91693	−0.0979507	−0.0489753	−	0.998800i	$-0.515596\pi$
$383$	−1.91693	−0.0979507	−0.0489753	+	0.998800i	$0.515596\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−13.8468	−0.703873
$388$	0	0
$389$	29.8577	1.51385	0.756923	−	0.653504i	$-0.226702\pi$
$389$	29.8577	1.51385	0.756923	+	0.653504i	$0.226702\pi$
$390$	0	0
$391$	−29.9907	−1.51670
$392$	0	0
$393$	8.32732	0.420058
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−28.3181	−1.42124	−0.710621	−	0.703575i	$-0.751586\pi$
$397$	−28.3181	−1.42124	−0.710621	+	0.703575i	$0.751586\pi$
$398$	0	0
$399$	15.3294	0.767429
$400$	0	0
$401$	−3.10487	−0.155050	−0.0775250	−	0.996990i	$-0.524702\pi$
$401$	−3.10487	−0.155050	−0.0775250	+	0.996990i	$0.524702\pi$
$402$	0	0
$403$	−4.20562	−0.209497
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−39.8334	−1.97447
$408$	0	0
$409$	10.4884	0.518619	0.259309	−	0.965794i	$-0.416505\pi$
$409$	10.4884	0.518619	0.259309	+	0.965794i	$0.416505\pi$
$410$	0	0
$411$	−17.1317	−0.845044
$412$	0	0
$413$	2.71842	0.133765
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.45623	0.365133
$418$	0	0
$419$	−25.0772	−1.22510	−0.612551	−	0.790431i	$-0.709856\pi$
$419$	−25.0772	−1.22510	−0.612551	+	0.790431i	$0.709856\pi$
$420$	0	0
$421$	31.4516	1.53286	0.766429	−	0.642330i	$-0.222032\pi$
$421$	31.4516	1.53286	0.766429	+	0.642330i	$0.222032\pi$
$422$	0	0
$423$	3.77755	0.183671
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.94962	0.287923
$428$	0	0
$429$	−23.3361	−1.12668
$430$	0	0
$431$	4.89976	0.236013	0.118007	−	0.993013i	$-0.462350\pi$
$431$	4.89976	0.236013	0.118007	+	0.993013i	$0.462350\pi$
$432$	0	0
$433$	12.8343	0.616776	0.308388	−	0.951261i	$-0.400210\pi$
$433$	12.8343	0.616776	0.308388	+	0.951261i	$0.400210\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	33.9866	1.62580
$438$	0	0
$439$	3.19010	0.152255	0.0761277	−	0.997098i	$-0.475744\pi$
$439$	3.19010	0.152255	0.0761277	+	0.997098i	$0.475744\pi$
$440$	0	0
$441$	1.33402	0.0635248
$442$	0	0
$443$	37.7536	1.79373	0.896865	−	0.442304i	$-0.145839\pi$
$443$	37.7536	1.79373	0.896865	+	0.442304i	$0.145839\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−32.6043	−1.54213
$448$	0	0
$449$	41.5498	1.96086	0.980428	−	0.196880i	$-0.0630810\pi$
$449$	41.5498	1.96086	0.980428	+	0.196880i	$0.0630810\pi$
$450$	0	0
$451$	4.15401	0.195605
$452$	0	0
$453$	4.27283	0.200755
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.23584	0.244922	0.122461	−	0.992473i	$-0.460921\pi$
$457$	5.23584	0.244922	0.122461	+	0.992473i	$0.460921\pi$
$458$	0	0
$459$	−22.5358	−1.05188
$460$	0	0
$461$	13.3731	0.622846	0.311423	−	0.950271i	$-0.399194\pi$
$461$	13.3731	0.622846	0.311423	+	0.950271i	$0.399194\pi$
$462$	0	0
$463$	−37.4142	−1.73878	−0.869392	−	0.494123i	$-0.835489\pi$
$463$	−37.4142	−1.73878	−0.869392	+	0.494123i	$0.835489\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	29.5585	1.36781	0.683903	−	0.729573i	$-0.260281\pi$
$467$	29.5585	1.36781	0.683903	+	0.729573i	$0.260281\pi$
$468$	0	0
$469$	11.3110	0.522292
$470$	0	0
$471$	−17.4495	−0.804032
$472$	0	0
$473$	35.4557	1.63026
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−8.75540	−0.400882
$478$	0	0
$479$	26.5716	1.21409	0.607043	−	0.794669i	$-0.292355\pi$
$479$	26.5716	1.21409	0.607043	+	0.794669i	$0.292355\pi$
$480$	0	0
$481$	38.2677	1.74486
$482$	0	0
$483$	9.60891	0.437220
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−8.44782	−0.382807	−0.191404	−	0.981511i	$-0.561304\pi$
$487$	−8.44782	−0.382807	−0.191404	+	0.981511i	$0.561304\pi$
$488$	0	0
$489$	−36.1477	−1.63466
$490$	0	0
$491$	−21.3710	−0.964460	−0.482230	−	0.876045i	$-0.660173\pi$
$491$	−21.3710	−0.964460	−0.482230	+	0.876045i	$0.660173\pi$
$492$	0	0
$493$	−62.7763	−2.82730
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.45194	−0.199697
$498$	0	0
$499$	−4.10487	−0.183759	−0.0918797	−	0.995770i	$-0.529288\pi$
$499$	−4.10487	−0.183759	−0.0918797	+	0.995770i	$0.529288\pi$
$500$	0	0
$501$	−35.9773	−1.60735
$502$	0	0
$503$	35.4864	1.58226	0.791129	−	0.611649i	$-0.209494\pi$
$503$	35.4864	1.58226	0.791129	+	0.611649i	$0.209494\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−4.64500	−0.206292
$508$	0	0
$509$	−41.2723	−1.82936	−0.914682	−	0.404175i	$-0.867559\pi$
$509$	−41.2723	−1.82936	−0.914682	+	0.404175i	$0.867559\pi$
$510$	0	0
$511$	13.5434	0.599126
$512$	0	0
$513$	25.5384	1.12755
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−9.67268	−0.425404
$518$	0	0
$519$	32.6086	1.43136
$520$	0	0
$521$	0.0259956	0.00113889	0.000569444	−	1.00000i	$-0.499819\pi$
$521$	0.0259956	0.00113889	0.000569444		1.00000i	$0.499819\pi$
$522$	0	0
$523$	12.4503	0.544412	0.272206	−	0.962239i	$-0.412247\pi$
$523$	12.4503	0.544412	0.272206	+	0.962239i	$0.412247\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.32732	−0.362744
$528$	0	0
$529$	−1.69620	−0.0737478
$530$	0	0
$531$	−3.62642	−0.157373
$532$	0	0
$533$	−3.99073	−0.172858
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−44.4750	−1.91924
$538$	0	0
$539$	−3.41585	−0.147131
$540$	0	0
$541$	34.3160	1.47536	0.737680	−	0.675151i	$-0.235921\pi$
$541$	34.3160	1.47536	0.737680	+	0.675151i	$0.235921\pi$
$542$	0	0
$543$	−20.7134	−0.888899
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−12.4439	−0.532063	−0.266032	−	0.963964i	$-0.585713\pi$
$547$	−12.4439	−0.532063	−0.266032	+	0.963964i	$0.585713\pi$
$548$	0	0
$549$	−7.93692	−0.338740
$550$	0	0
$551$	71.1405	3.03069
$552$	0	0
$553$	2.21610	0.0942381
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10.3253	−0.437495	−0.218748	−	0.975781i	$-0.570197\pi$
$557$	−10.3253	−0.437495	−0.218748	+	0.975781i	$0.570197\pi$
$558$	0	0
$559$	−34.0621	−1.44067
$560$	0	0
$561$	−46.2065	−1.95084
$562$	0	0
$563$	18.8946	0.796313	0.398157	−	0.917317i	$-0.369650\pi$
$563$	18.8946	0.796313	0.398157	+	0.917317i	$0.369650\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	11.2225	0.471299
$568$	0	0
$569$	−17.4583	−0.731890	−0.365945	−	0.930637i	$-0.619254\pi$
$569$	−17.4583	−0.731890	−0.365945	+	0.930637i	$0.619254\pi$
$570$	0	0
$571$	−30.8120	−1.28944	−0.644720	−	0.764419i	$-0.723026\pi$
$571$	−30.8120	−1.28944	−0.644720	+	0.764419i	$0.723026\pi$
$572$	0	0
$573$	−12.1310	−0.506779
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11.6987	0.487022	0.243511	−	0.969898i	$-0.421701\pi$
$577$	11.6987	0.487022	0.243511	+	0.969898i	$0.421701\pi$
$578$	0	0
$579$	−27.5355	−1.14434
$580$	0	0
$581$	−4.91353	−0.203848
$582$	0	0
$583$	22.4188	0.928492
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−0.317674	−0.0131118	−0.00655591	−	0.999979i	$-0.502087\pi$
$587$	−0.317674	−0.0131118	−0.00655591	+	0.999979i	$0.502087\pi$
$588$	0	0
$589$	9.43683	0.388838
$590$	0	0
$591$	25.4494	1.04685
$592$	0	0
$593$	−13.1616	−0.540482	−0.270241	−	0.962793i	$-0.587103\pi$
$593$	−13.1616	−0.540482	−0.270241	+	0.962793i	$0.587103\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	15.2271	0.623203
$598$	0	0
$599$	−17.7419	−0.724916	−0.362458	−	0.932000i	$-0.618062\pi$
$599$	−17.7419	−0.724916	−0.362458	+	0.932000i	$0.618062\pi$
$600$	0	0
$601$	9.69867	0.395617	0.197809	−	0.980241i	$-0.436618\pi$
$601$	9.69867	0.395617	0.197809	+	0.980241i	$0.436618\pi$
$602$	0	0
$603$	−15.0891	−0.614475
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−9.28158	−0.376728	−0.188364	−	0.982099i	$-0.560318\pi$
$607$	−9.28158	−0.376728	−0.188364	+	0.982099i	$0.560318\pi$
$608$	0	0
$609$	20.1133	0.815032
$610$	0	0
$611$	9.29247	0.375933
$612$	0	0
$613$	25.5606	1.03238	0.516191	−	0.856473i	$-0.327349\pi$
$613$	25.5606	1.03238	0.516191	+	0.856473i	$0.327349\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.2152	0.894352	0.447176	−	0.894446i	$-0.352430\pi$
$617$	22.2152	0.894352	0.447176	+	0.894446i	$0.352430\pi$
$618$	0	0
$619$	−37.8355	−1.52074	−0.760368	−	0.649492i	$-0.774981\pi$
$619$	−37.8355	−1.52074	−0.760368	+	0.649492i	$0.774981\pi$
$620$	0	0
$621$	16.0082	0.642389
$622$	0	0
$623$	3.21610	0.128850
$624$	0	0
$625$	0	0
$626$	0	0
$627$	52.3629	2.09117
$628$	0	0
$629$	75.7717	3.02122
$630$	0	0
$631$	−36.0390	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$631$	−36.0390	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$632$	0	0
$633$	−39.9513	−1.58792
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.28158	0.130021
$638$	0	0
$639$	5.93898	0.234942
$640$	0	0
$641$	−3.10230	−0.122533	−0.0612667	−	0.998121i	$-0.519514\pi$
$641$	−3.10230	−0.122533	−0.0612667	+	0.998121i	$0.519514\pi$
$642$	0	0
$643$	−33.1263	−1.30638	−0.653188	−	0.757196i	$-0.726569\pi$
$643$	−33.1263	−1.30638	−0.653188	+	0.757196i	$0.726569\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22.3575	0.878966	0.439483	−	0.898251i	$-0.355162\pi$
$647$	22.3575	0.878966	0.439483	+	0.898251i	$0.355162\pi$
$648$	0	0
$649$	9.28571	0.364496
$650$	0	0
$651$	2.66804	0.104569
$652$	0	0
$653$	−26.8551	−1.05092	−0.525461	−	0.850818i	$-0.676107\pi$
$653$	−26.8551	−1.05092	−0.525461	+	0.850818i	$0.676107\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−18.0672	−0.704869
$658$	0	0
$659$	47.0042	1.83102	0.915511	−	0.402293i	$-0.131787\pi$
$659$	47.0042	1.83102	0.915511	+	0.402293i	$0.131787\pi$
$660$	0	0
$661$	−8.33659	−0.324256	−0.162128	−	0.986770i	$-0.551836\pi$
$661$	−8.33659	−0.324256	−0.162128	+	0.986770i	$0.551836\pi$
$662$	0	0
$663$	44.3902	1.72397
$664$	0	0
$665$	0	0
$666$	0	0
$667$	44.5929	1.72665
$668$	0	0
$669$	37.8992	1.46527
$670$	0	0
$671$	20.3230	0.784562
$672$	0	0
$673$	35.4142	1.36512	0.682558	−	0.730831i	$-0.260867\pi$
$673$	35.4142	1.36512	0.682558	+	0.730831i	$0.260867\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.66804	0.256274	0.128137	−	0.991757i	$-0.459100\pi$
$677$	6.66804	0.256274	0.128137	+	0.991757i	$0.459100\pi$
$678$	0	0
$679$	−11.6089	−0.445509
$680$	0	0
$681$	−6.44559	−0.246996
$682$	0	0
$683$	−15.2690	−0.584253	−0.292126	−	0.956380i	$-0.594363\pi$
$683$	−15.2690	−0.584253	−0.292126	+	0.956380i	$0.594363\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	13.1724	0.502559
$688$	0	0
$689$	−21.5376	−0.820517
$690$	0	0
$691$	−6.08183	−0.231364	−0.115682	−	0.993286i	$-0.536905\pi$
$691$	−6.08183	−0.231364	−0.115682	+	0.993286i	$0.536905\pi$
$692$	0	0
$693$	4.55682	0.173099
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−7.90182	−0.299303
$698$	0	0
$699$	−44.8769	−1.69740
$700$	0	0
$701$	−28.4490	−1.07450	−0.537252	−	0.843422i	$-0.680538\pi$
$701$	−28.4490	−1.07450	−0.537252	+	0.843422i	$0.680538\pi$
$702$	0	0
$703$	−85.8673	−3.23855
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−14.9450	−0.562064
$708$	0	0
$709$	−33.1263	−1.24409	−0.622043	−	0.782983i	$-0.713697\pi$
$709$	−33.1263	−1.24409	−0.622043	+	0.782983i	$0.713697\pi$
$710$	0	0
$711$	−2.95632	−0.110871
$712$	0	0
$713$	5.91528	0.221529
$714$	0	0
$715$	0	0
$716$	0	0
$717$	12.0000	0.448148
$718$	0	0
$719$	38.5782	1.43872	0.719362	−	0.694636i	$-0.244434\pi$
$719$	38.5782	1.43872	0.719362	+	0.694636i	$0.244434\pi$
$720$	0	0
$721$	−17.5585	−0.653914
$722$	0	0
$723$	−44.7449	−1.66408
$724$	0	0
$725$	0	0
$726$	0	0
$727$	50.1712	1.86075	0.930374	−	0.366612i	$-0.119483\pi$
$727$	50.1712	1.86075	0.930374	+	0.366612i	$0.119483\pi$
$728$	0	0
$729$	6.69019	0.247785
$730$	0	0
$731$	−67.4444	−2.49452
$732$	0	0
$733$	−42.1737	−1.55772	−0.778860	−	0.627197i	$-0.784202\pi$
$733$	−42.1737	−1.55772	−0.778860	+	0.627197i	$0.784202\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	38.6366	1.42320
$738$	0	0
$739$	47.0780	1.73179	0.865896	−	0.500223i	$-0.166749\pi$
$739$	47.0780	1.73179	0.865896	+	0.500223i	$0.166749\pi$
$740$	0	0
$741$	−50.3047	−1.84799
$742$	0	0
$743$	39.6895	1.45607	0.728033	−	0.685542i	$-0.240435\pi$
$743$	39.6895	1.45607	0.728033	+	0.685542i	$0.240435\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.55475	0.239826
$748$	0	0
$749$	−12.3588	−0.451580
$750$	0	0
$751$	−5.56101	−0.202924	−0.101462	−	0.994839i	$-0.532352\pi$
$751$	−5.56101	−0.202924	−0.101462	+	0.994839i	$0.532352\pi$
$752$	0	0
$753$	41.8926	1.52665
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−11.3601	−0.412890	−0.206445	−	0.978458i	$-0.566190\pi$
$757$	−11.3601	−0.412890	−0.206445	+	0.978458i	$0.566190\pi$
$758$	0	0
$759$	32.8226	1.19138
$760$	0	0
$761$	8.29206	0.300587	0.150293	−	0.988641i	$-0.451978\pi$
$761$	8.29206	0.300587	0.150293	+	0.988641i	$0.451978\pi$
$762$	0	0
$763$	−3.56986	−0.129238
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.92071	−0.322108
$768$	0	0
$769$	−26.4078	−0.952290	−0.476145	−	0.879367i	$-0.657966\pi$
$769$	−26.4078	−0.952290	−0.476145	+	0.879367i	$0.657966\pi$
$770$	0	0
$771$	32.4951	1.17028
$772$	0	0
$773$	−25.6048	−0.920940	−0.460470	−	0.887675i	$-0.652319\pi$
$773$	−25.6048	−0.920940	−0.460470	+	0.887675i	$0.652319\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−24.2769	−0.870931
$778$	0	0
$779$	8.95464	0.320833
$780$	0	0
$781$	−15.2072	−0.544155
$782$	0	0
$783$	33.5083	1.19749
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−31.9311	−1.13822	−0.569110	−	0.822261i	$-0.692712\pi$
$787$	−31.9311	−1.13822	−0.569110	+	0.822261i	$0.692712\pi$
$788$	0	0
$789$	−19.1767	−0.682709
$790$	0	0
$791$	0.764157	0.0271703
$792$	0	0
$793$	−19.5242	−0.693324
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15.6593	−0.554680	−0.277340	−	0.960772i	$-0.589453\pi$
$797$	−15.6593	−0.554680	−0.277340	+	0.960772i	$0.589453\pi$
$798$	0	0
$799$	18.3995	0.650928
$800$	0	0
$801$	−4.29034	−0.151592
$802$	0	0
$803$	46.2623	1.63256
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−40.1410	−1.41303
$808$	0	0
$809$	−13.1938	−0.463869	−0.231934	−	0.972731i	$-0.574505\pi$
$809$	−13.1938	−0.463869	−0.231934	+	0.972731i	$0.574505\pi$
$810$	0	0
$811$	19.5493	0.686467	0.343234	−	0.939250i	$-0.388478\pi$
$811$	19.5493	0.686467	0.343234	+	0.939250i	$0.388478\pi$
$812$	0	0
$813$	57.3268	2.01054
$814$	0	0
$815$	0	0
$816$	0	0
$817$	76.4305	2.67396
$818$	0	0
$819$	−4.37770	−0.152969
$820$	0	0
$821$	−16.8644	−0.588571	−0.294286	−	0.955717i	$-0.595082\pi$
$821$	−16.8644	−0.588571	−0.294286	+	0.955717i	$0.595082\pi$
$822$	0	0
$823$	0.447820	0.0156100	0.00780501	−	0.999970i	$-0.497516\pi$
$823$	0.447820	0.0156100	0.00780501	+	0.999970i	$0.497516\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.88290	−0.100248	−0.0501242	−	0.998743i	$-0.515962\pi$
$827$	−2.88290	−0.100248	−0.0501242	+	0.998743i	$0.515962\pi$
$828$	0	0
$829$	−43.7903	−1.52090	−0.760449	−	0.649397i	$-0.775021\pi$
$829$	−43.7903	−1.52090	−0.760449	+	0.649397i	$0.775021\pi$
$830$	0	0
$831$	−9.70949	−0.336818
$832$	0	0
$833$	6.49768	0.225131
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.44490	0.153638
$838$	0	0
$839$	19.2489	0.664546	0.332273	−	0.943183i	$-0.392185\pi$
$839$	19.2489	0.664546	0.332273	+	0.943183i	$0.392185\pi$
$840$	0	0
$841$	64.3416	2.21868
$842$	0	0
$843$	−7.21352	−0.248447
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−0.668041	−0.0229542
$848$	0	0
$849$	−57.6474	−1.97845
$850$	0	0
$851$	−53.8241	−1.84507
$852$	0	0
$853$	−11.6593	−0.399206	−0.199603	−	0.979877i	$-0.563965\pi$
$853$	−11.6593	−0.399206	−0.199603	+	0.979877i	$0.563965\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41.0064	1.40075	0.700375	−	0.713775i	$-0.253016\pi$
$857$	41.0064	1.40075	0.700375	+	0.713775i	$0.253016\pi$
$858$	0	0
$859$	20.4268	0.696955	0.348477	−	0.937317i	$-0.386699\pi$
$859$	20.4268	0.696955	0.348477	+	0.937317i	$0.386699\pi$
$860$	0	0
$861$	2.53171	0.0862805
$862$	0	0
$863$	45.1300	1.53624	0.768122	−	0.640304i	$-0.221192\pi$
$863$	45.1300	1.53624	0.768122	+	0.640304i	$0.221192\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	52.5035	1.78311
$868$	0	0
$869$	7.56986	0.256790
$870$	0	0
$871$	−37.1179	−1.25769
$872$	0	0
$873$	15.4865	0.524139
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3.42756	−0.115741	−0.0578703	−	0.998324i	$-0.518431\pi$
$877$	−3.42756	−0.115741	−0.0578703	+	0.998324i	$0.518431\pi$
$878$	0	0
$879$	−9.82295	−0.331320
$880$	0	0
$881$	8.39109	0.282703	0.141352	−	0.989959i	$-0.454855\pi$
$881$	8.39109	0.282703	0.141352	+	0.989959i	$0.454855\pi$
$882$	0	0
$883$	−36.8066	−1.23864	−0.619320	−	0.785138i	$-0.712592\pi$
$883$	−36.8066	−1.23864	−0.619320	+	0.785138i	$0.712592\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.55012	0.320662	0.160331	−	0.987063i	$-0.448744\pi$
$887$	9.55012	0.320662	0.160331	+	0.987063i	$0.448744\pi$
$888$	0	0
$889$	11.5481	0.387309
$890$	0	0
$891$	38.3342	1.28424
$892$	0	0
$893$	−20.8510	−0.697752
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−31.5324	−1.05284
$898$	0	0
$899$	12.3818	0.412957
$900$	0	0
$901$	−42.6454	−1.42072
$902$	0	0
$903$	21.6089	0.719099
$904$	0	0
$905$	0	0
$906$	0	0
$907$	13.3361	0.442817	0.221409	−	0.975181i	$-0.428935\pi$
$907$	13.3361	0.442817	0.221409	+	0.975181i	$0.428935\pi$
$908$	0	0
$909$	19.9369	0.661266
$910$	0	0
$911$	−1.70245	−0.0564047	−0.0282023	−	0.999602i	$-0.508978\pi$
$911$	−1.70245	−0.0564047	−0.0282023	+	0.999602i	$0.508978\pi$
$912$	0	0
$913$	−16.7839	−0.555466
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.00000	−0.132092
$918$	0	0
$919$	33.5783	1.10765	0.553823	−	0.832635i	$-0.313169\pi$
$919$	33.5783	1.10765	0.553823	+	0.832635i	$0.313169\pi$
$920$	0	0
$921$	41.8926	1.38041
$922$	0	0
$923$	14.6094	0.480875
$924$	0	0
$925$	0	0
$926$	0	0
$927$	23.4234	0.769327
$928$	0	0
$929$	−15.5040	−0.508671	−0.254335	−	0.967116i	$-0.581857\pi$
$929$	−15.5040	−0.508671	−0.254335	+	0.967116i	$0.581857\pi$
$930$	0	0
$931$	−7.36341	−0.241326
$932$	0	0
$933$	36.2046	1.18529
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42.5556	1.39023	0.695116	−	0.718898i	$-0.255353\pi$
$937$	42.5556	1.39023	0.695116	+	0.718898i	$0.255353\pi$
$938$	0	0
$939$	−12.4910	−0.407628
$940$	0	0
$941$	−20.7486	−0.676386	−0.338193	−	0.941077i	$-0.609816\pi$
$941$	−20.7486	−0.676386	−0.338193	+	0.941077i	$0.609816\pi$
$942$	0	0
$943$	5.61303	0.182785
$944$	0	0
$945$	0	0
$946$	0	0
$947$	59.9732	1.94887	0.974434	−	0.224675i	$-0.0721320\pi$
$947$	59.9732	1.94887	0.974434	+	0.224675i	$0.0721320\pi$
$948$	0	0
$949$	−44.4439	−1.44271
$950$	0	0
$951$	−21.2857	−0.690236
$952$	0	0
$953$	−60.8807	−1.97212	−0.986060	−	0.166391i	$-0.946788\pi$
$953$	−60.8807	−1.97212	−0.986060	+	0.166391i	$0.946788\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	68.7040	2.22088
$958$	0	0
$959$	8.22915	0.265733
$960$	0	0
$961$	−29.3575	−0.947018
$962$	0	0
$963$	16.4869	0.531282
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−33.2546	−1.06940	−0.534698	−	0.845043i	$-0.679575\pi$
$967$	−33.2546	−1.06940	−0.534698	+	0.845043i	$0.679575\pi$
$968$	0	0
$969$	−99.6055	−3.19979
$970$	0	0
$971$	−20.3639	−0.653510	−0.326755	−	0.945109i	$-0.605955\pi$
$971$	−20.3639	−0.653510	−0.326755	+	0.945109i	$0.605955\pi$
$972$	0	0
$973$	−3.58157	−0.114820
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−22.7683	−0.728422	−0.364211	−	0.931317i	$-0.618661\pi$
$977$	−22.7683	−0.728422	−0.364211	+	0.931317i	$0.618661\pi$
$978$	0	0
$979$	10.9857	0.351105
$980$	0	0
$981$	4.76227	0.152048
$982$	0	0
$983$	16.4172	0.523627	0.261814	−	0.965118i	$-0.415679\pi$
$983$	16.4172	0.523627	0.261814	+	0.965118i	$0.415679\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−5.89513	−0.187644
$988$	0	0
$989$	47.9089	1.52341
$990$	0	0
$991$	41.0251	1.30321	0.651603	−	0.758560i	$-0.274097\pi$
$991$	41.0251	1.30321	0.651603	+	0.758560i	$0.274097\pi$
$992$	0	0
$993$	39.3512	1.24877
$994$	0	0
$995$	0	0
$996$	0	0
$997$	25.2010	0.798123	0.399062	−	0.916924i	$-0.369336\pi$
$997$	25.2010	0.798123	0.399062	+	0.916924i	$0.369336\pi$
$998$	0	0
$999$	−40.4449	−1.27962

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5600.2.a.bq.1.4	✓	4
4.3	odd	2		5600.2.a.bt.1.1	yes	4
5.4	even	2		5600.2.a.bs.1.1	yes	4
20.19	odd	2		5600.2.a.br.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5600.2.a.bq.1.4	✓	4	1.1	even	1	trivial
5600.2.a.br.1.4	yes	4	20.19	odd	2
5600.2.a.bs.1.1	yes	4	5.4	even	2
5600.2.a.bt.1.1	yes	4	4.3	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)

\(f(q)\)	\(=\)	\(q+2.08183 q^{3} -1.00000 q^{7} +1.33402 q^{9} +O(q^{10})\)	\(q+2.08183 q^{3} -1.00000 q^{7} +1.33402 q^{9} -3.41585 q^{11} +3.28158 q^{13} +6.49768 q^{17} -7.36341 q^{19} -2.08183 q^{21} -4.61560 q^{23} -3.46829 q^{27} -9.66134 q^{29} -1.28158 q^{31} -7.11123 q^{33} +11.6613 q^{37} +6.83170 q^{39} -1.21610 q^{41} -10.3798 q^{43} +2.83170 q^{47} +1.00000 q^{49} +13.5271 q^{51} -6.56317 q^{53} -15.3294 q^{57} -2.71842 q^{59} -5.94962 q^{61} -1.33402 q^{63} -11.3110 q^{67} -9.60891 q^{69} +4.45194 q^{71} -13.5434 q^{73} +3.41585 q^{77} -2.21610 q^{79} -11.2225 q^{81} +4.91353 q^{83} -20.1133 q^{87} -3.21610 q^{89} -3.28158 q^{91} -2.66804 q^{93} +11.6089 q^{97} -4.55682 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 4 q^{7} + 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 4 * q^7 + 6 * q^9	\( 4 q - 2 q^{3} - 4 q^{7} + 6 q^{9} - 4 q^{11} - 2 q^{13} + 6 q^{17} - 4 q^{19} + 2 q^{21} - 4 q^{23} - 20 q^{27} + 2 q^{29} + 10 q^{31} + 8 q^{33} + 6 q^{37} + 8 q^{39} - 16 q^{43} - 8 q^{47} + 4 q^{49} + 8 q^{51} + 4 q^{53} - 22 q^{57} - 26 q^{59} - 10 q^{61} - 6 q^{63} - 4 q^{67} + 18 q^{69} + 24 q^{71} - 8 q^{73} + 4 q^{77} - 4 q^{79} + 28 q^{81} - 10 q^{83} - 46 q^{87} - 8 q^{89} + 2 q^{91} - 12 q^{93} - 10 q^{97} - 56 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 4 * q^7 + 6 * q^9 - 4 * q^11 - 2 * q^13 + 6 * q^17 - 4 * q^19 + 2 * q^21 - 4 * q^23 - 20 * q^27 + 2 * q^29 + 10 * q^31 + 8 * q^33 + 6 * q^37 + 8 * q^39 - 16 * q^43 - 8 * q^47 + 4 * q^49 + 8 * q^51 + 4 * q^53 - 22 * q^57 - 26 * q^59 - 10 * q^61 - 6 * q^63 - 4 * q^67 + 18 * q^69 + 24 * q^71 - 8 * q^73 + 4 * q^77 - 4 * q^79 + 28 * q^81 - 10 * q^83 - 46 * q^87 - 8 * q^89 + 2 * q^91 - 12 * q^93 - 10 * q^97 - 56 * q^99

Introduction

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