LMFDB - Embedded newform 2800.2.a.bo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(1,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, 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("2800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.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:	$22.3581125660$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1400)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	2800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{3} +1.00000 q^{7} +3.56155 q^{9} +O(q^{10})$	$q+2.56155 q^{3} +1.00000 q^{7} +3.56155 q^{9} -2.12311 q^{11} -2.00000 q^{13} +2.56155 q^{17} +0.561553 q^{19} +2.56155 q^{21} +5.56155 q^{23} +1.43845 q^{27} +7.56155 q^{29} +0.876894 q^{31} -5.43845 q^{33} +11.8078 q^{37} -5.12311 q^{39} -6.56155 q^{41} +2.43845 q^{43} +8.24621 q^{47} +1.00000 q^{49} +6.56155 q^{51} -7.12311 q^{53} +1.43845 q^{57} +13.3693 q^{59} +2.87689 q^{61} +3.56155 q^{63} +16.1231 q^{67} +14.2462 q^{69} +10.6847 q^{71} -10.5616 q^{73} -2.12311 q^{77} -6.68466 q^{79} -7.00000 q^{81} -13.6847 q^{83} +19.3693 q^{87} -10.8078 q^{89} -2.00000 q^{91} +2.24621 q^{93} -6.00000 q^{97} -7.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q + q^3 + 2 * q^7 + 3 * q^9	$ 2 q + q^{3} + 2 q^{7} + 3 q^{9} + 4 q^{11} - 4 q^{13} + q^{17} - 3 q^{19} + q^{21} + 7 q^{23} + 7 q^{27} + 11 q^{29} + 10 q^{31} - 15 q^{33} + 3 q^{37} - 2 q^{39} - 9 q^{41} + 9 q^{43} + 2 q^{49} + 9 q^{51} - 6 q^{53} + 7 q^{57} + 2 q^{59} + 14 q^{61} + 3 q^{63} + 24 q^{67} + 12 q^{69} + 9 q^{71} - 17 q^{73} + 4 q^{77} - q^{79} - 14 q^{81} - 15 q^{83} + 14 q^{87} - q^{89} - 4 q^{91} - 12 q^{93} - 12 q^{97} - 11 q^{99}+O(q^{100}) $ 2 * q + q^3 + 2 * q^7 + 3 * q^9 + 4 * q^11 - 4 * q^13 + q^17 - 3 * q^19 + q^21 + 7 * q^23 + 7 * q^27 + 11 * q^29 + 10 * q^31 - 15 * q^33 + 3 * q^37 - 2 * q^39 - 9 * q^41 + 9 * q^43 + 2 * q^49 + 9 * q^51 - 6 * q^53 + 7 * q^57 + 2 * q^59 + 14 * q^61 + 3 * q^63 + 24 * q^67 + 12 * q^69 + 9 * q^71 - 17 * q^73 + 4 * q^77 - q^79 - 14 * q^81 - 15 * q^83 + 14 * q^87 - q^89 - 4 * q^91 - 12 * q^93 - 12 * q^97 - 11 * 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
$3$	2.56155	1.47891	0.739457	−	0.673204i	$-0.235083\pi$
$3$	2.56155	1.47891	0.739457	+	0.673204i	$0.235083\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	−2.12311	−0.640140	−0.320070	−	0.947394i	$-0.603707\pi$
$11$	−2.12311	−0.640140	−0.320070	+	0.947394i	$0.603707\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.56155	0.621268	0.310634	−	0.950530i	$-0.399459\pi$
$17$	2.56155	0.621268	0.310634	+	0.950530i	$0.399459\pi$
$18$	0	0
$19$	0.561553	0.128829	0.0644145	−	0.997923i	$-0.479482\pi$
$19$	0.561553	0.128829	0.0644145	+	0.997923i	$0.479482\pi$
$20$	0	0
$21$	2.56155	0.558977
$22$	0	0
$23$	5.56155	1.15966	0.579832	−	0.814736i	$-0.303118\pi$
$23$	5.56155	1.15966	0.579832	+	0.814736i	$0.303118\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.43845	0.276829
$28$	0	0
$29$	7.56155	1.40415	0.702073	−	0.712105i	$-0.252258\pi$
$29$	7.56155	1.40415	0.702073	+	0.712105i	$0.252258\pi$
$30$	0	0
$31$	0.876894	0.157495	0.0787474	−	0.996895i	$-0.474908\pi$
$31$	0.876894	0.157495	0.0787474	+	0.996895i	$0.474908\pi$
$32$	0	0
$33$	−5.43845	−0.946712
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.8078	1.94118	0.970592	−	0.240730i	$-0.0773867\pi$
$37$	11.8078	1.94118	0.970592	+	0.240730i	$0.0773867\pi$
$38$	0	0
$39$	−5.12311	−0.820353
$40$	0	0
$41$	−6.56155	−1.02474	−0.512371	−	0.858764i	$-0.671233\pi$
$41$	−6.56155	−1.02474	−0.512371	+	0.858764i	$0.671233\pi$
$42$	0	0
$43$	2.43845	0.371860	0.185930	−	0.982563i	$-0.440470\pi$
$43$	2.43845	0.371860	0.185930	+	0.982563i	$0.440470\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.24621	1.20283	0.601417	−	0.798935i	$-0.294603\pi$
$47$	8.24621	1.20283	0.601417	+	0.798935i	$0.294603\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.56155	0.918801
$52$	0	0
$53$	−7.12311	−0.978434	−0.489217	−	0.872162i	$-0.662717\pi$
$53$	−7.12311	−0.978434	−0.489217	+	0.872162i	$0.662717\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.43845	0.190527
$58$	0	0
$59$	13.3693	1.74054	0.870268	−	0.492578i	$-0.163945\pi$
$59$	13.3693	1.74054	0.870268	+	0.492578i	$0.163945\pi$
$60$	0	0
$61$	2.87689	0.368349	0.184174	−	0.982894i	$-0.441039\pi$
$61$	2.87689	0.368349	0.184174	+	0.982894i	$0.441039\pi$
$62$	0	0
$63$	3.56155	0.448713
$64$	0	0
$65$	0	0
$66$	0	0
$67$	16.1231	1.96975	0.984875	−	0.173264i	$-0.0554314\pi$
$67$	16.1231	1.96975	0.984875	+	0.173264i	$0.0554314\pi$
$68$	0	0
$69$	14.2462	1.71504
$70$	0	0
$71$	10.6847	1.26804	0.634018	−	0.773318i	$-0.281405\pi$
$71$	10.6847	1.26804	0.634018	+	0.773318i	$0.281405\pi$
$72$	0	0
$73$	−10.5616	−1.23614	−0.618068	−	0.786125i	$-0.712084\pi$
$73$	−10.5616	−1.23614	−0.618068	+	0.786125i	$0.712084\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.12311	−0.241950
$78$	0	0
$79$	−6.68466	−0.752083	−0.376041	−	0.926603i	$-0.622715\pi$
$79$	−6.68466	−0.752083	−0.376041	+	0.926603i	$0.622715\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−13.6847	−1.50209	−0.751043	−	0.660253i	$-0.770449\pi$
$83$	−13.6847	−1.50209	−0.751043	+	0.660253i	$0.770449\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	19.3693	2.07661
$88$	0	0
$89$	−10.8078	−1.14562	−0.572810	−	0.819688i	$-0.694147\pi$
$89$	−10.8078	−1.14562	−0.572810	+	0.819688i	$0.694147\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	2.24621	0.232921
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	−7.56155	−0.759965
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	7.12311	0.701860	0.350930	−	0.936402i	$-0.385865\pi$
$103$	7.12311	0.701860	0.350930	+	0.936402i	$0.385865\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.8078	−1.23817	−0.619087	−	0.785323i	$-0.712497\pi$
$107$	−12.8078	−1.23817	−0.619087	+	0.785323i	$0.712497\pi$
$108$	0	0
$109$	−8.93087	−0.855422	−0.427711	−	0.903915i	$-0.640680\pi$
$109$	−8.93087	−0.855422	−0.427711	+	0.903915i	$0.640680\pi$
$110$	0	0
$111$	30.2462	2.87084
$112$	0	0
$113$	−0.753789	−0.0709105	−0.0354552	−	0.999371i	$-0.511288\pi$
$113$	−0.753789	−0.0709105	−0.0354552	+	0.999371i	$0.511288\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−7.12311	−0.658531
$118$	0	0
$119$	2.56155	0.234817
$120$	0	0
$121$	−6.49242	−0.590220
$122$	0	0
$123$	−16.8078	−1.51551
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−9.80776	−0.870298	−0.435149	−	0.900358i	$-0.643304\pi$
$127$	−9.80776	−0.870298	−0.435149	+	0.900358i	$0.643304\pi$
$128$	0	0
$129$	6.24621	0.549948
$130$	0	0
$131$	1.75379	0.153229	0.0766146	−	0.997061i	$-0.475589\pi$
$131$	1.75379	0.153229	0.0766146	+	0.997061i	$0.475589\pi$
$132$	0	0
$133$	0.561553	0.0486928
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.31534	0.197813	0.0989065	−	0.995097i	$-0.468466\pi$
$137$	2.31534	0.197813	0.0989065	+	0.995097i	$0.468466\pi$
$138$	0	0
$139$	19.6847	1.66963	0.834815	−	0.550530i	$-0.185574\pi$
$139$	19.6847	1.66963	0.834815	+	0.550530i	$0.185574\pi$
$140$	0	0
$141$	21.1231	1.77889
$142$	0	0
$143$	4.24621	0.355086
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.56155	0.211273
$148$	0	0
$149$	17.5616	1.43870	0.719349	−	0.694649i	$-0.244440\pi$
$149$	17.5616	1.43870	0.719349	+	0.694649i	$0.244440\pi$
$150$	0	0
$151$	5.56155	0.452593	0.226296	−	0.974058i	$-0.427338\pi$
$151$	5.56155	0.452593	0.226296	+	0.974058i	$0.427338\pi$
$152$	0	0
$153$	9.12311	0.737559
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−24.4924	−1.95471	−0.977354	−	0.211611i	$-0.932129\pi$
$157$	−24.4924	−1.95471	−0.977354	+	0.211611i	$0.932129\pi$
$158$	0	0
$159$	−18.2462	−1.44702
$160$	0	0
$161$	5.56155	0.438312
$162$	0	0
$163$	−13.9309	−1.09115	−0.545575	−	0.838062i	$-0.683689\pi$
$163$	−13.9309	−1.09115	−0.545575	+	0.838062i	$0.683689\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.6155	−1.51790	−0.758948	−	0.651152i	$-0.774286\pi$
$167$	−19.6155	−1.51790	−0.758948	+	0.651152i	$0.774286\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	−14.4924	−1.10184	−0.550919	−	0.834559i	$-0.685723\pi$
$173$	−14.4924	−1.10184	−0.550919	+	0.834559i	$0.685723\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	34.2462	2.57410
$178$	0	0
$179$	8.31534	0.621518	0.310759	−	0.950489i	$-0.399417\pi$
$179$	8.31534	0.621518	0.310759	+	0.950489i	$0.399417\pi$
$180$	0	0
$181$	−2.87689	−0.213838	−0.106919	−	0.994268i	$-0.534099\pi$
$181$	−2.87689	−0.213838	−0.106919	+	0.994268i	$0.534099\pi$
$182$	0	0
$183$	7.36932	0.544756
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5.43845	−0.397699
$188$	0	0
$189$	1.43845	0.104632
$190$	0	0
$191$	−0.630683	−0.0456346	−0.0228173	−	0.999740i	$-0.507264\pi$
$191$	−0.630683	−0.0456346	−0.0228173	+	0.999740i	$0.507264\pi$
$192$	0	0
$193$	−20.6155	−1.48394	−0.741969	−	0.670434i	$-0.766108\pi$
$193$	−20.6155	−1.48394	−0.741969	+	0.670434i	$0.766108\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.56155	0.253750	0.126875	−	0.991919i	$-0.459505\pi$
$197$	3.56155	0.253750	0.126875	+	0.991919i	$0.459505\pi$
$198$	0	0
$199$	13.1231	0.930272	0.465136	−	0.885239i	$-0.346005\pi$
$199$	13.1231	0.930272	0.465136	+	0.885239i	$0.346005\pi$
$200$	0	0
$201$	41.3002	2.91309
$202$	0	0
$203$	7.56155	0.530717
$204$	0	0
$205$	0	0
$206$	0	0
$207$	19.8078	1.37673
$208$	0	0
$209$	−1.19224	−0.0824687
$210$	0	0
$211$	−10.5616	−0.727087	−0.363544	−	0.931577i	$-0.618433\pi$
$211$	−10.5616	−0.727087	−0.363544	+	0.931577i	$0.618433\pi$
$212$	0	0
$213$	27.3693	1.87531
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.876894	0.0595275
$218$	0	0
$219$	−27.0540	−1.82814
$220$	0	0
$221$	−5.12311	−0.344617
$222$	0	0
$223$	−0.630683	−0.0422337	−0.0211168	−	0.999777i	$-0.506722\pi$
$223$	−0.630683	−0.0422337	−0.0211168	+	0.999777i	$0.506722\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.3693	−0.887353	−0.443676	−	0.896187i	$-0.646326\pi$
$227$	−13.3693	−0.887353	−0.443676	+	0.896187i	$0.646326\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	−5.43845	−0.357824
$232$	0	0
$233$	−3.56155	−0.233325	−0.116663	−	0.993172i	$-0.537220\pi$
$233$	−3.56155	−0.233325	−0.116663	+	0.993172i	$0.537220\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−17.1231	−1.11227
$238$	0	0
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	25.0540	1.61387	0.806934	−	0.590641i	$-0.201125\pi$
$241$	25.0540	1.61387	0.806934	+	0.590641i	$0.201125\pi$
$242$	0	0
$243$	−22.2462	−1.42710
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.12311	−0.0714615
$248$	0	0
$249$	−35.0540	−2.22146
$250$	0	0
$251$	12.3153	0.777337	0.388669	−	0.921378i	$-0.372935\pi$
$251$	12.3153	0.777337	0.388669	+	0.921378i	$0.372935\pi$
$252$	0	0
$253$	−11.8078	−0.742348
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.24621	0.264871	0.132436	−	0.991192i	$-0.457720\pi$
$257$	4.24621	0.264871	0.132436	+	0.991192i	$0.457720\pi$
$258$	0	0
$259$	11.8078	0.733699
$260$	0	0
$261$	26.9309	1.66698
$262$	0	0
$263$	−10.6847	−0.658844	−0.329422	−	0.944183i	$-0.606854\pi$
$263$	−10.6847	−0.658844	−0.329422	+	0.944183i	$0.606854\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−27.6847	−1.69427
$268$	0	0
$269$	11.7538	0.716641	0.358321	−	0.933599i	$-0.383349\pi$
$269$	11.7538	0.716641	0.358321	+	0.933599i	$0.383349\pi$
$270$	0	0
$271$	−16.2462	−0.986887	−0.493444	−	0.869778i	$-0.664262\pi$
$271$	−16.2462	−0.986887	−0.493444	+	0.869778i	$0.664262\pi$
$272$	0	0
$273$	−5.12311	−0.310064
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.4924	0.630429	0.315214	−	0.949021i	$-0.397924\pi$
$277$	10.4924	0.630429	0.315214	+	0.949021i	$0.397924\pi$
$278$	0	0
$279$	3.12311	0.186975
$280$	0	0
$281$	11.5616	0.689704	0.344852	−	0.938657i	$-0.387929\pi$
$281$	11.5616	0.689704	0.344852	+	0.938657i	$0.387929\pi$
$282$	0	0
$283$	−13.6847	−0.813469	−0.406734	−	0.913547i	$-0.633333\pi$
$283$	−13.6847	−0.813469	−0.406734	+	0.913547i	$0.633333\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.56155	−0.387316
$288$	0	0
$289$	−10.4384	−0.614026
$290$	0	0
$291$	−15.3693	−0.900965
$292$	0	0
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.05398	−0.177210
$298$	0	0
$299$	−11.1231	−0.643266
$300$	0	0
$301$	2.43845	0.140550
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.31534	0.246290	0.123145	−	0.992389i	$-0.460702\pi$
$307$	4.31534	0.246290	0.123145	+	0.992389i	$0.460702\pi$
$308$	0	0
$309$	18.2462	1.03799
$310$	0	0
$311$	−2.87689	−0.163134	−0.0815669	−	0.996668i	$-0.525992\pi$
$311$	−2.87689	−0.163134	−0.0815669	+	0.996668i	$0.525992\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.6847	1.16177	0.580883	−	0.813987i	$-0.302707\pi$
$317$	20.6847	1.16177	0.580883	+	0.813987i	$0.302707\pi$
$318$	0	0
$319$	−16.0540	−0.898850
$320$	0	0
$321$	−32.8078	−1.83115
$322$	0	0
$323$	1.43845	0.0800373
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−22.8769	−1.26510
$328$	0	0
$329$	8.24621	0.454628
$330$	0	0
$331$	−27.0000	−1.48405	−0.742027	−	0.670370i	$-0.766135\pi$
$331$	−27.0000	−1.48405	−0.742027	+	0.670370i	$0.766135\pi$
$332$	0	0
$333$	42.0540	2.30454
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5.68466	0.309663	0.154832	−	0.987941i	$-0.450517\pi$
$337$	5.68466	0.309663	0.154832	+	0.987941i	$0.450517\pi$
$338$	0	0
$339$	−1.93087	−0.104870
$340$	0	0
$341$	−1.86174	−0.100819
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.6155	1.21407	0.607033	−	0.794677i	$-0.292360\pi$
$347$	22.6155	1.21407	0.607033	+	0.794677i	$0.292360\pi$
$348$	0	0
$349$	−12.8769	−0.689284	−0.344642	−	0.938734i	$-0.612000\pi$
$349$	−12.8769	−0.689284	−0.344642	+	0.938734i	$0.612000\pi$
$350$	0	0
$351$	−2.87689	−0.153557
$352$	0	0
$353$	11.7538	0.625591	0.312796	−	0.949820i	$-0.398735\pi$
$353$	11.7538	0.625591	0.312796	+	0.949820i	$0.398735\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.56155	0.347274
$358$	0	0
$359$	14.6847	0.775027	0.387513	−	0.921864i	$-0.373334\pi$
$359$	14.6847	0.775027	0.387513	+	0.921864i	$0.373334\pi$
$360$	0	0
$361$	−18.6847	−0.983403
$362$	0	0
$363$	−16.6307	−0.872884
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.7538	0.926740	0.463370	−	0.886165i	$-0.346640\pi$
$367$	17.7538	0.926740	0.463370	+	0.886165i	$0.346640\pi$
$368$	0	0
$369$	−23.3693	−1.21656
$370$	0	0
$371$	−7.12311	−0.369813
$372$	0	0
$373$	−14.0540	−0.727687	−0.363844	−	0.931460i	$-0.618536\pi$
$373$	−14.0540	−0.727687	−0.363844	+	0.931460i	$0.618536\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−15.1231	−0.778880
$378$	0	0
$379$	24.8617	1.27706	0.638531	−	0.769596i	$-0.279542\pi$
$379$	24.8617	1.27706	0.638531	+	0.769596i	$0.279542\pi$
$380$	0	0
$381$	−25.1231	−1.28710
$382$	0	0
$383$	−20.2462	−1.03453	−0.517267	−	0.855824i	$-0.673050\pi$
$383$	−20.2462	−1.03453	−0.517267	+	0.855824i	$0.673050\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.68466	0.441466
$388$	0	0
$389$	2.43845	0.123634	0.0618171	−	0.998087i	$-0.480310\pi$
$389$	2.43845	0.123634	0.0618171	+	0.998087i	$0.480310\pi$
$390$	0	0
$391$	14.2462	0.720462
$392$	0	0
$393$	4.49242	0.226613
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−38.4924	−1.93188	−0.965940	−	0.258767i	$-0.916684\pi$
$397$	−38.4924	−1.93188	−0.965940	+	0.258767i	$0.916684\pi$
$398$	0	0
$399$	1.43845	0.0720124
$400$	0	0
$401$	−31.4924	−1.57266	−0.786328	−	0.617809i	$-0.788021\pi$
$401$	−31.4924	−1.57266	−0.786328	+	0.617809i	$0.788021\pi$
$402$	0	0
$403$	−1.75379	−0.0873624
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−25.0691	−1.24263
$408$	0	0
$409$	−5.19224	−0.256740	−0.128370	−	0.991726i	$-0.540974\pi$
$409$	−5.19224	−0.256740	−0.128370	+	0.991726i	$0.540974\pi$
$410$	0	0
$411$	5.93087	0.292548
$412$	0	0
$413$	13.3693	0.657861
$414$	0	0
$415$	0	0
$416$	0	0
$417$	50.4233	2.46924
$418$	0	0
$419$	18.8078	0.918819	0.459410	−	0.888224i	$-0.348061\pi$
$419$	18.8078	0.918819	0.459410	+	0.888224i	$0.348061\pi$
$420$	0	0
$421$	5.06913	0.247054	0.123527	−	0.992341i	$-0.460579\pi$
$421$	5.06913	0.247054	0.123527	+	0.992341i	$0.460579\pi$
$422$	0	0
$423$	29.3693	1.42799
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.87689	0.139223
$428$	0	0
$429$	10.8769	0.525141
$430$	0	0
$431$	13.7538	0.662497	0.331248	−	0.943544i	$-0.392530\pi$
$431$	13.7538	0.662497	0.331248	+	0.943544i	$0.392530\pi$
$432$	0	0
$433$	35.5464	1.70825	0.854125	−	0.520067i	$-0.174093\pi$
$433$	35.5464	1.70825	0.854125	+	0.520067i	$0.174093\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.12311	0.149398
$438$	0	0
$439$	7.12311	0.339967	0.169984	−	0.985447i	$-0.445629\pi$
$439$	7.12311	0.339967	0.169984	+	0.985447i	$0.445629\pi$
$440$	0	0
$441$	3.56155	0.169598
$442$	0	0
$443$	22.5616	1.07193	0.535966	−	0.844240i	$-0.319948\pi$
$443$	22.5616	1.07193	0.535966	+	0.844240i	$0.319948\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	44.9848	2.12771
$448$	0	0
$449$	10.3693	0.489358	0.244679	−	0.969604i	$-0.421317\pi$
$449$	10.3693	0.489358	0.244679	+	0.969604i	$0.421317\pi$
$450$	0	0
$451$	13.9309	0.655979
$452$	0	0
$453$	14.2462	0.669345
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−37.1080	−1.73584	−0.867918	−	0.496707i	$-0.834542\pi$
$457$	−37.1080	−1.73584	−0.867918	+	0.496707i	$0.834542\pi$
$458$	0	0
$459$	3.68466	0.171985
$460$	0	0
$461$	6.49242	0.302382	0.151191	−	0.988505i	$-0.451689\pi$
$461$	6.49242	0.302382	0.151191	+	0.988505i	$0.451689\pi$
$462$	0	0
$463$	−20.4924	−0.952364	−0.476182	−	0.879347i	$-0.657980\pi$
$463$	−20.4924	−0.952364	−0.476182	+	0.879347i	$0.657980\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−24.8769	−1.15117	−0.575583	−	0.817744i	$-0.695225\pi$
$467$	−24.8769	−1.15117	−0.575583	+	0.817744i	$0.695225\pi$
$468$	0	0
$469$	16.1231	0.744496
$470$	0	0
$471$	−62.7386	−2.89084
$472$	0	0
$473$	−5.17708	−0.238042
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−25.3693	−1.16158
$478$	0	0
$479$	40.7386	1.86140	0.930698	−	0.365789i	$-0.119201\pi$
$479$	40.7386	1.86140	0.930698	+	0.365789i	$0.119201\pi$
$480$	0	0
$481$	−23.6155	−1.07678
$482$	0	0
$483$	14.2462	0.648225
$484$	0	0
$485$	0	0
$486$	0	0
$487$	35.5616	1.61145	0.805724	−	0.592291i	$-0.201777\pi$
$487$	35.5616	1.61145	0.805724	+	0.592291i	$0.201777\pi$
$488$	0	0
$489$	−35.6847	−1.61372
$490$	0	0
$491$	−20.6847	−0.933486	−0.466743	−	0.884393i	$-0.654573\pi$
$491$	−20.6847	−0.933486	−0.466743	+	0.884393i	$0.654573\pi$
$492$	0	0
$493$	19.3693	0.872350
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.6847	0.479272
$498$	0	0
$499$	16.4924	0.738302	0.369151	−	0.929369i	$-0.379648\pi$
$499$	16.4924	0.738302	0.369151	+	0.929369i	$0.379648\pi$
$500$	0	0
$501$	−50.2462	−2.24484
$502$	0	0
$503$	19.7538	0.880778	0.440389	−	0.897807i	$-0.354841\pi$
$503$	19.7538	0.880778	0.440389	+	0.897807i	$0.354841\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−23.0540	−1.02386
$508$	0	0
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	0	0
$511$	−10.5616	−0.467216
$512$	0	0
$513$	0.807764	0.0356637
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−17.5076	−0.769982
$518$	0	0
$519$	−37.1231	−1.62952
$520$	0	0
$521$	−27.4384	−1.20210	−0.601050	−	0.799211i	$-0.705251\pi$
$521$	−27.4384	−1.20210	−0.601050	+	0.799211i	$0.705251\pi$
$522$	0	0
$523$	28.3153	1.23814	0.619072	−	0.785334i	$-0.287509\pi$
$523$	28.3153	1.23814	0.619072	+	0.785334i	$0.287509\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.24621	0.0978465
$528$	0	0
$529$	7.93087	0.344820
$530$	0	0
$531$	47.6155	2.06634
$532$	0	0
$533$	13.1231	0.568425
$534$	0	0
$535$	0	0
$536$	0	0
$537$	21.3002	0.919171
$538$	0	0
$539$	−2.12311	−0.0914486
$540$	0	0
$541$	41.4233	1.78093	0.890463	−	0.455055i	$-0.150380\pi$
$541$	41.4233	1.78093	0.890463	+	0.455055i	$0.150380\pi$
$542$	0	0
$543$	−7.36932	−0.316248
$544$	0	0
$545$	0	0
$546$	0	0
$547$	33.7386	1.44256	0.721280	−	0.692644i	$-0.243554\pi$
$547$	33.7386	1.44256	0.721280	+	0.692644i	$0.243554\pi$
$548$	0	0
$549$	10.2462	0.437298
$550$	0	0
$551$	4.24621	0.180895
$552$	0	0
$553$	−6.68466	−0.284261
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.56155	−0.405136	−0.202568	−	0.979268i	$-0.564929\pi$
$557$	−9.56155	−0.405136	−0.202568	+	0.979268i	$0.564929\pi$
$558$	0	0
$559$	−4.87689	−0.206271
$560$	0	0
$561$	−13.9309	−0.588162
$562$	0	0
$563$	−7.12311	−0.300203	−0.150102	−	0.988671i	$-0.547960\pi$
$563$	−7.12311	−0.300203	−0.150102	+	0.988671i	$0.547960\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−7.00000	−0.293972
$568$	0	0
$569$	−19.9848	−0.837808	−0.418904	−	0.908030i	$-0.637586\pi$
$569$	−19.9848	−0.837808	−0.418904	+	0.908030i	$0.637586\pi$
$570$	0	0
$571$	−27.3153	−1.14311	−0.571556	−	0.820563i	$-0.693660\pi$
$571$	−27.3153	−1.14311	−0.571556	+	0.820563i	$0.693660\pi$
$572$	0	0
$573$	−1.61553	−0.0674897
$574$	0	0
$575$	0	0
$576$	0	0
$577$	45.5464	1.89612	0.948061	−	0.318090i	$-0.103041\pi$
$577$	45.5464	1.89612	0.948061	+	0.318090i	$0.103041\pi$
$578$	0	0
$579$	−52.8078	−2.19462
$580$	0	0
$581$	−13.6847	−0.567735
$582$	0	0
$583$	15.1231	0.626335
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.17708	0.337504	0.168752	−	0.985659i	$-0.446026\pi$
$587$	8.17708	0.337504	0.168752	+	0.985659i	$0.446026\pi$
$588$	0	0
$589$	0.492423	0.0202899
$590$	0	0
$591$	9.12311	0.375274
$592$	0	0
$593$	−12.3153	−0.505730	−0.252865	−	0.967502i	$-0.581373\pi$
$593$	−12.3153	−0.505730	−0.252865	+	0.967502i	$0.581373\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	33.6155	1.37579
$598$	0	0
$599$	−24.3002	−0.992879	−0.496439	−	0.868071i	$-0.665359\pi$
$599$	−24.3002	−0.992879	−0.496439	+	0.868071i	$0.665359\pi$
$600$	0	0
$601$	−4.94602	−0.201753	−0.100876	−	0.994899i	$-0.532165\pi$
$601$	−4.94602	−0.201753	−0.100876	+	0.994899i	$0.532165\pi$
$602$	0	0
$603$	57.4233	2.33846
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−37.3693	−1.51677	−0.758387	−	0.651805i	$-0.774012\pi$
$607$	−37.3693	−1.51677	−0.758387	+	0.651805i	$0.774012\pi$
$608$	0	0
$609$	19.3693	0.784884
$610$	0	0
$611$	−16.4924	−0.667212
$612$	0	0
$613$	6.68466	0.269991	0.134995	−	0.990846i	$-0.456898\pi$
$613$	6.68466	0.269991	0.134995	+	0.990846i	$0.456898\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−17.8078	−0.716914	−0.358457	−	0.933546i	$-0.616697\pi$
$617$	−17.8078	−0.716914	−0.358457	+	0.933546i	$0.616697\pi$
$618$	0	0
$619$	−29.3693	−1.18045	−0.590226	−	0.807238i	$-0.700961\pi$
$619$	−29.3693	−1.18045	−0.590226	+	0.807238i	$0.700961\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	−10.8078	−0.433004
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.05398	−0.121964
$628$	0	0
$629$	30.2462	1.20600
$630$	0	0
$631$	−34.0540	−1.35567	−0.677834	−	0.735215i	$-0.737081\pi$
$631$	−34.0540	−1.35567	−0.677834	+	0.735215i	$0.737081\pi$
$632$	0	0
$633$	−27.0540	−1.07530
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	38.0540	1.50539
$640$	0	0
$641$	−16.4384	−0.649280	−0.324640	−	0.945838i	$-0.605243\pi$
$641$	−16.4384	−0.649280	−0.324640	+	0.945838i	$0.605243\pi$
$642$	0	0
$643$	2.73863	0.108001	0.0540006	−	0.998541i	$-0.482803\pi$
$643$	2.73863	0.108001	0.0540006	+	0.998541i	$0.482803\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25.3693	−0.997371	−0.498685	−	0.866783i	$-0.666184\pi$
$647$	−25.3693	−0.997371	−0.498685	+	0.866783i	$0.666184\pi$
$648$	0	0
$649$	−28.3845	−1.11419
$650$	0	0
$651$	2.24621	0.0880360
$652$	0	0
$653$	−33.8617	−1.32511	−0.662556	−	0.749012i	$-0.730528\pi$
$653$	−33.8617	−1.32511	−0.662556	+	0.749012i	$0.730528\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−37.6155	−1.46752
$658$	0	0
$659$	−18.5616	−0.723055	−0.361528	−	0.932361i	$-0.617745\pi$
$659$	−18.5616	−0.723055	−0.361528	+	0.932361i	$0.617745\pi$
$660$	0	0
$661$	−22.4924	−0.874854	−0.437427	−	0.899254i	$-0.644110\pi$
$661$	−22.4924	−0.874854	−0.437427	+	0.899254i	$0.644110\pi$
$662$	0	0
$663$	−13.1231	−0.509659
$664$	0	0
$665$	0	0
$666$	0	0
$667$	42.0540	1.62834
$668$	0	0
$669$	−1.61553	−0.0624599
$670$	0	0
$671$	−6.10795	−0.235795
$672$	0	0
$673$	−46.4924	−1.79215	−0.896076	−	0.443901i	$-0.853594\pi$
$673$	−46.4924	−1.79215	−0.896076	+	0.443901i	$0.853594\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14.6307	−0.562303	−0.281151	−	0.959663i	$-0.590716\pi$
$677$	−14.6307	−0.562303	−0.281151	+	0.959663i	$0.590716\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	−34.2462	−1.31232
$682$	0	0
$683$	34.1231	1.30568	0.652842	−	0.757494i	$-0.273576\pi$
$683$	34.1231	1.30568	0.652842	+	0.757494i	$0.273576\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−35.8617	−1.36821
$688$	0	0
$689$	14.2462	0.542737
$690$	0	0
$691$	17.4384	0.663390	0.331695	−	0.943387i	$-0.392380\pi$
$691$	17.4384	0.663390	0.331695	+	0.943387i	$0.392380\pi$
$692$	0	0
$693$	−7.56155	−0.287240
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−16.8078	−0.636639
$698$	0	0
$699$	−9.12311	−0.345068
$700$	0	0
$701$	44.1080	1.66593	0.832967	−	0.553322i	$-0.186640\pi$
$701$	44.1080	1.66593	0.832967	+	0.553322i	$0.186640\pi$
$702$	0	0
$703$	6.63068	0.250081
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	37.3693	1.40343	0.701717	−	0.712456i	$-0.252417\pi$
$709$	37.3693	1.40343	0.701717	+	0.712456i	$0.252417\pi$
$710$	0	0
$711$	−23.8078	−0.892861
$712$	0	0
$713$	4.87689	0.182641
$714$	0	0
$715$	0	0
$716$	0	0
$717$	10.2462	0.382652
$718$	0	0
$719$	−38.2462	−1.42634	−0.713171	−	0.700990i	$-0.752742\pi$
$719$	−38.2462	−1.42634	−0.713171	+	0.700990i	$0.752742\pi$
$720$	0	0
$721$	7.12311	0.265278
$722$	0	0
$723$	64.1771	2.38677
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.61553	−0.0599166	−0.0299583	−	0.999551i	$-0.509537\pi$
$727$	−1.61553	−0.0599166	−0.0299583	+	0.999551i	$0.509537\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	6.24621	0.231024
$732$	0	0
$733$	−22.2462	−0.821683	−0.410841	−	0.911707i	$-0.634765\pi$
$733$	−22.2462	−0.821683	−0.410841	+	0.911707i	$0.634765\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−34.2311	−1.26092
$738$	0	0
$739$	−17.1771	−0.631869	−0.315935	−	0.948781i	$-0.602318\pi$
$739$	−17.1771	−0.631869	−0.315935	+	0.948781i	$0.602318\pi$
$740$	0	0
$741$	−2.87689	−0.105685
$742$	0	0
$743$	−18.7386	−0.687454	−0.343727	−	0.939070i	$-0.611689\pi$
$743$	−18.7386	−0.687454	−0.343727	+	0.939070i	$0.611689\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−48.7386	−1.78325
$748$	0	0
$749$	−12.8078	−0.467986
$750$	0	0
$751$	−27.3693	−0.998721	−0.499360	−	0.866394i	$-0.666432\pi$
$751$	−27.3693	−0.998721	−0.499360	+	0.866394i	$0.666432\pi$
$752$	0	0
$753$	31.5464	1.14961
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−13.3153	−0.483954	−0.241977	−	0.970282i	$-0.577796\pi$
$757$	−13.3153	−0.483954	−0.241977	+	0.970282i	$0.577796\pi$
$758$	0	0
$759$	−30.2462	−1.09787
$760$	0	0
$761$	34.4233	1.24784	0.623922	−	0.781487i	$-0.285538\pi$
$761$	34.4233	1.24784	0.623922	+	0.781487i	$0.285538\pi$
$762$	0	0
$763$	−8.93087	−0.323319
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−26.7386	−0.965476
$768$	0	0
$769$	−33.3002	−1.20084	−0.600418	−	0.799687i	$-0.704999\pi$
$769$	−33.3002	−1.20084	−0.600418	+	0.799687i	$0.704999\pi$
$770$	0	0
$771$	10.8769	0.391722
$772$	0	0
$773$	11.6155	0.417782	0.208891	−	0.977939i	$-0.433015\pi$
$773$	11.6155	0.417782	0.208891	+	0.977939i	$0.433015\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	30.2462	1.08508
$778$	0	0
$779$	−3.68466	−0.132017
$780$	0	0
$781$	−22.6847	−0.811721
$782$	0	0
$783$	10.8769	0.388708
$784$	0	0
$785$	0	0
$786$	0	0
$787$	26.2462	0.935576	0.467788	−	0.883841i	$-0.345051\pi$
$787$	26.2462	0.935576	0.467788	+	0.883841i	$0.345051\pi$
$788$	0	0
$789$	−27.3693	−0.974373
$790$	0	0
$791$	−0.753789	−0.0268016
$792$	0	0
$793$	−5.75379	−0.204323
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−41.6155	−1.47410	−0.737049	−	0.675840i	$-0.763781\pi$
$797$	−41.6155	−1.47410	−0.737049	+	0.675840i	$0.763781\pi$
$798$	0	0
$799$	21.1231	0.747282
$800$	0	0
$801$	−38.4924	−1.36006
$802$	0	0
$803$	22.4233	0.791301
$804$	0	0
$805$	0	0
$806$	0	0
$807$	30.1080	1.05985
$808$	0	0
$809$	49.4233	1.73763	0.868815	−	0.495136i	$-0.164882\pi$
$809$	49.4233	1.73763	0.868815	+	0.495136i	$0.164882\pi$
$810$	0	0
$811$	−43.6155	−1.53155	−0.765774	−	0.643110i	$-0.777644\pi$
$811$	−43.6155	−1.53155	−0.765774	+	0.643110i	$0.777644\pi$
$812$	0	0
$813$	−41.6155	−1.45952
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.36932	0.0479063
$818$	0	0
$819$	−7.12311	−0.248901
$820$	0	0
$821$	45.8617	1.60059	0.800293	−	0.599609i	$-0.204677\pi$
$821$	45.8617	1.60059	0.800293	+	0.599609i	$0.204677\pi$
$822$	0	0
$823$	12.4384	0.433577	0.216789	−	0.976219i	$-0.430442\pi$
$823$	12.4384	0.433577	0.216789	+	0.976219i	$0.430442\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.24621	−0.321522	−0.160761	−	0.986993i	$-0.551395\pi$
$827$	−9.24621	−0.321522	−0.160761	+	0.986993i	$0.551395\pi$
$828$	0	0
$829$	−2.24621	−0.0780141	−0.0390071	−	0.999239i	$-0.512419\pi$
$829$	−2.24621	−0.0780141	−0.0390071	+	0.999239i	$0.512419\pi$
$830$	0	0
$831$	26.8769	0.932349
$832$	0	0
$833$	2.56155	0.0887525
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.26137	0.0435992
$838$	0	0
$839$	1.12311	0.0387739	0.0193870	−	0.999812i	$-0.493829\pi$
$839$	1.12311	0.0387739	0.0193870	+	0.999812i	$0.493829\pi$
$840$	0	0
$841$	28.1771	0.971623
$842$	0	0
$843$	29.6155	1.02001
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−6.49242	−0.223082
$848$	0	0
$849$	−35.0540	−1.20305
$850$	0	0
$851$	65.6695	2.25112
$852$	0	0
$853$	−10.8769	−0.372418	−0.186209	−	0.982510i	$-0.559620\pi$
$853$	−10.8769	−0.372418	−0.186209	+	0.982510i	$0.559620\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.94602	−0.168953	−0.0844765	−	0.996425i	$-0.526922\pi$
$857$	−4.94602	−0.168953	−0.0844765	+	0.996425i	$0.526922\pi$
$858$	0	0
$859$	−34.1771	−1.16611	−0.583053	−	0.812434i	$-0.698142\pi$
$859$	−34.1771	−1.16611	−0.583053	+	0.812434i	$0.698142\pi$
$860$	0	0
$861$	−16.8078	−0.572807
$862$	0	0
$863$	−23.8078	−0.810426	−0.405213	−	0.914222i	$-0.632803\pi$
$863$	−23.8078	−0.810426	−0.405213	+	0.914222i	$0.632803\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−26.7386	−0.908092
$868$	0	0
$869$	14.1922	0.481439
$870$	0	0
$871$	−32.2462	−1.09262
$872$	0	0
$873$	−21.3693	−0.723242
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−23.6155	−0.797440	−0.398720	−	0.917073i	$-0.630545\pi$
$877$	−23.6155	−0.797440	−0.398720	+	0.917073i	$0.630545\pi$
$878$	0	0
$879$	76.8466	2.59197
$880$	0	0
$881$	−40.7386	−1.37252	−0.686260	−	0.727357i	$-0.740749\pi$
$881$	−40.7386	−1.37252	−0.686260	+	0.727357i	$0.740749\pi$
$882$	0	0
$883$	−1.49242	−0.0502240	−0.0251120	−	0.999685i	$-0.507994\pi$
$883$	−1.49242	−0.0502240	−0.0251120	+	0.999685i	$0.507994\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−49.6155	−1.66593	−0.832963	−	0.553328i	$-0.813357\pi$
$887$	−49.6155	−1.66593	−0.832963	+	0.553328i	$0.813357\pi$
$888$	0	0
$889$	−9.80776	−0.328942
$890$	0	0
$891$	14.8617	0.497887
$892$	0	0
$893$	4.63068	0.154960
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−28.4924	−0.951334
$898$	0	0
$899$	6.63068	0.221146
$900$	0	0
$901$	−18.2462	−0.607869
$902$	0	0
$903$	6.24621	0.207861
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	39.4233	1.30615	0.653076	−	0.757292i	$-0.273478\pi$
$911$	39.4233	1.30615	0.653076	+	0.757292i	$0.273478\pi$
$912$	0	0
$913$	29.0540	0.961546
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.75379	0.0579152
$918$	0	0
$919$	−18.1922	−0.600106	−0.300053	−	0.953922i	$-0.597004\pi$
$919$	−18.1922	−0.600106	−0.300053	+	0.953922i	$0.597004\pi$
$920$	0	0
$921$	11.0540	0.364241
$922$	0	0
$923$	−21.3693	−0.703380
$924$	0	0
$925$	0	0
$926$	0	0
$927$	25.3693	0.833238
$928$	0	0
$929$	56.3542	1.84892	0.924460	−	0.381279i	$-0.124516\pi$
$929$	56.3542	1.84892	0.924460	+	0.381279i	$0.124516\pi$
$930$	0	0
$931$	0.561553	0.0184042
$932$	0	0
$933$	−7.36932	−0.241261
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−21.3002	−0.695847	−0.347923	−	0.937523i	$-0.613113\pi$
$937$	−21.3002	−0.695847	−0.347923	+	0.937523i	$0.613113\pi$
$938$	0	0
$939$	−56.3542	−1.83905
$940$	0	0
$941$	−1.36932	−0.0446385	−0.0223192	−	0.999751i	$-0.507105\pi$
$941$	−1.36932	−0.0446385	−0.0223192	+	0.999751i	$0.507105\pi$
$942$	0	0
$943$	−36.4924	−1.18836
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.4924	0.535932	0.267966	−	0.963428i	$-0.413649\pi$
$947$	16.4924	0.535932	0.267966	+	0.963428i	$0.413649\pi$
$948$	0	0
$949$	21.1231	0.685685
$950$	0	0
$951$	52.9848	1.71815
$952$	0	0
$953$	27.8769	0.903021	0.451511	−	0.892266i	$-0.350885\pi$
$953$	27.8769	0.903021	0.451511	+	0.892266i	$0.350885\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−41.1231	−1.32932
$958$	0	0
$959$	2.31534	0.0747663
$960$	0	0
$961$	−30.2311	−0.975195
$962$	0	0
$963$	−45.6155	−1.46994
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−16.6307	−0.534807	−0.267403	−	0.963585i	$-0.586166\pi$
$967$	−16.6307	−0.534807	−0.267403	+	0.963585i	$0.586166\pi$
$968$	0	0
$969$	3.68466	0.118368
$970$	0	0
$971$	2.31534	0.0743028	0.0371514	−	0.999310i	$-0.488172\pi$
$971$	2.31534	0.0743028	0.0371514	+	0.999310i	$0.488172\pi$
$972$	0	0
$973$	19.6847	0.631061
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33.2462	1.06364	0.531820	−	0.846857i	$-0.321508\pi$
$977$	33.2462	1.06364	0.531820	+	0.846857i	$0.321508\pi$
$978$	0	0
$979$	22.9460	0.733358
$980$	0	0
$981$	−31.8078	−1.01554
$982$	0	0
$983$	−62.3542	−1.98879	−0.994394	−	0.105734i	$-0.966281\pi$
$983$	−62.3542	−1.98879	−0.994394	+	0.105734i	$0.966281\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	21.1231	0.672356
$988$	0	0
$989$	13.5616	0.431232
$990$	0	0
$991$	29.6695	0.942483	0.471241	−	0.882004i	$-0.343806\pi$
$991$	29.6695	0.942483	0.471241	+	0.882004i	$0.343806\pi$
$992$	0	0
$993$	−69.1619	−2.19479
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−12.2462	−0.387841	−0.193921	−	0.981017i	$-0.562120\pi$
$997$	−12.2462	−0.387841	−0.193921	+	0.981017i	$0.562120\pi$
$998$	0	0
$999$	16.9848	0.537377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.a.bo.1.2		2
4.3	odd	2		1400.2.a.o.1.1	✓	2
5.2	odd	4		2800.2.g.v.449.1		4
5.3	odd	4		2800.2.g.v.449.4		4
5.4	even	2		2800.2.a.bj.1.1		2
20.3	even	4		1400.2.g.j.449.1		4
20.7	even	4		1400.2.g.j.449.4		4
20.19	odd	2		1400.2.a.q.1.2	yes	2
28.27	even	2		9800.2.a.bx.1.2		2
140.139	even	2		9800.2.a.bt.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1400.2.a.o.1.1	✓	2	4.3	odd	2
1400.2.a.q.1.2	yes	2	20.19	odd	2
1400.2.g.j.449.1		4	20.3	even	4
1400.2.g.j.449.4		4	20.7	even	4
2800.2.a.bj.1.1		2	5.4	even	2
2800.2.a.bo.1.2		2	1.1	even	1	trivial
2800.2.g.v.449.1		4	5.2	odd	4
2800.2.g.v.449.4		4	5.3	odd	4
9800.2.a.bt.1.1		2	140.139	even	2
9800.2.a.bx.1.2		2	28.27	even	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:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{3} +1.00000 q^{7} +3.56155 q^{9} +O(q^{10})\)	\(q+2.56155 q^{3} +1.00000 q^{7} +3.56155 q^{9} -2.12311 q^{11} -2.00000 q^{13} +2.56155 q^{17} +0.561553 q^{19} +2.56155 q^{21} +5.56155 q^{23} +1.43845 q^{27} +7.56155 q^{29} +0.876894 q^{31} -5.43845 q^{33} +11.8078 q^{37} -5.12311 q^{39} -6.56155 q^{41} +2.43845 q^{43} +8.24621 q^{47} +1.00000 q^{49} +6.56155 q^{51} -7.12311 q^{53} +1.43845 q^{57} +13.3693 q^{59} +2.87689 q^{61} +3.56155 q^{63} +16.1231 q^{67} +14.2462 q^{69} +10.6847 q^{71} -10.5616 q^{73} -2.12311 q^{77} -6.68466 q^{79} -7.00000 q^{81} -13.6847 q^{83} +19.3693 q^{87} -10.8078 q^{89} -2.00000 q^{91} +2.24621 q^{93} -6.00000 q^{97} -7.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q + q^3 + 2 * q^7 + 3 * q^9	\( 2 q + q^{3} + 2 q^{7} + 3 q^{9} + 4 q^{11} - 4 q^{13} + q^{17} - 3 q^{19} + q^{21} + 7 q^{23} + 7 q^{27} + 11 q^{29} + 10 q^{31} - 15 q^{33} + 3 q^{37} - 2 q^{39} - 9 q^{41} + 9 q^{43} + 2 q^{49} + 9 q^{51} - 6 q^{53} + 7 q^{57} + 2 q^{59} + 14 q^{61} + 3 q^{63} + 24 q^{67} + 12 q^{69} + 9 q^{71} - 17 q^{73} + 4 q^{77} - q^{79} - 14 q^{81} - 15 q^{83} + 14 q^{87} - q^{89} - 4 q^{91} - 12 q^{93} - 12 q^{97} - 11 q^{99}+O(q^{100}) \) 2 * q + q^3 + 2 * q^7 + 3 * q^9 + 4 * q^11 - 4 * q^13 + q^17 - 3 * q^19 + q^21 + 7 * q^23 + 7 * q^27 + 11 * q^29 + 10 * q^31 - 15 * q^33 + 3 * q^37 - 2 * q^39 - 9 * q^41 + 9 * q^43 + 2 * q^49 + 9 * q^51 - 6 * q^53 + 7 * q^57 + 2 * q^59 + 14 * q^61 + 3 * q^63 + 24 * q^67 + 12 * q^69 + 9 * q^71 - 17 * q^73 + 4 * q^77 - q^79 - 14 * q^81 - 15 * q^83 + 14 * q^87 - q^89 - 4 * q^91 - 12 * q^93 - 12 * q^97 - 11 * q^99

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