LMFDB - Embedded newform 4400.2.a.bn.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4400,2,Mod(1,4400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4400.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-4,0,10,0,-2,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	4400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{3} -2.00000 q^{7} +5.00000 q^{9} -1.00000 q^{11} +6.82843 q^{13} -1.17157 q^{17} -5.65685 q^{21} +2.82843 q^{23} +5.65685 q^{27} +7.65685 q^{29} -2.82843 q^{33} -3.65685 q^{37} +19.3137 q^{39} +6.00000 q^{41} -6.00000 q^{43} -2.82843 q^{47} -3.00000 q^{49} -3.31371 q^{51} -0.343146 q^{53} +9.65685 q^{59} +13.3137 q^{61} -10.0000 q^{63} -4.48528 q^{67} +8.00000 q^{69} +11.3137 q^{71} +6.82843 q^{73} +2.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -6.00000 q^{83} +21.6569 q^{87} +9.31371 q^{89} -13.6569 q^{91} +7.65685 q^{97} -5.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{13} - 8 q^{17} + 4 q^{29} + 4 q^{37} + 16 q^{39} + 12 q^{41} - 12 q^{43} - 6 q^{49} + 16 q^{51} - 12 q^{53} + 8 q^{59} + 4 q^{61} - 20 q^{63} + 8 q^{67} + 16 q^{69}+ \cdots - 10 q^{99}+O(q^{100}) $ 2 * q - 4 * q^7 + 10 * q^9 - 2 * q^11 + 8 * q^13 - 8 * q^17 + 4 * q^29 + 4 * q^37 + 16 * q^39 + 12 * q^41 - 12 * q^43 - 6 * q^49 + 16 * q^51 - 12 * q^53 + 8 * q^59 + 4 * q^61 - 20 * q^63 + 8 * q^67 + 16 * q^69 + 8 * q^73 + 4 * q^77 - 8 * q^79 + 2 * q^81 - 12 * q^83 + 32 * q^87 - 4 * q^89 - 16 * q^91 + 4 * q^97 - 10 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.82843	1.63299	0.816497	−	0.577350i	$-0.195913\pi$
$3$	2.82843	1.63299	0.816497	+	0.577350i	$0.195913\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	5.00000	1.66667
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.82843	1.89386	0.946932	−	0.321433i	$-0.104164\pi$
$13$	6.82843	1.89386	0.946932	+	0.321433i	$0.104164\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.17157	−0.284148	−0.142074	−	0.989856i	$-0.545377\pi$
$17$	−1.17157	−0.284148	−0.142074	+	0.989856i	$0.545377\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−5.65685	−1.23443
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	7.65685	1.42184	0.710921	−	0.703272i	$-0.248278\pi$
$29$	7.65685	1.42184	0.710921	+	0.703272i	$0.248278\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−2.82843	−0.492366
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.65685	−0.601183	−0.300592	−	0.953753i	$-0.597184\pi$
$37$	−3.65685	−0.601183	−0.300592	+	0.953753i	$0.597184\pi$
$38$	0	0
$39$	19.3137	3.09267
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−3.31371	−0.464012
$52$	0	0
$53$	−0.343146	−0.0471347	−0.0235673	−	0.999722i	$-0.507502\pi$
$53$	−0.343146	−0.0471347	−0.0235673	+	0.999722i	$0.507502\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.65685	1.25722	0.628608	−	0.777723i	$-0.283625\pi$
$59$	9.65685	1.25722	0.628608	+	0.777723i	$0.283625\pi$
$60$	0	0
$61$	13.3137	1.70465	0.852323	−	0.523016i	$-0.175193\pi$
$61$	13.3137	1.70465	0.852323	+	0.523016i	$0.175193\pi$
$62$	0	0
$63$	−10.0000	−1.25988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.48528	−0.547964	−0.273982	−	0.961735i	$-0.588341\pi$
$67$	−4.48528	−0.547964	−0.273982	+	0.961735i	$0.588341\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	11.3137	1.34269	0.671345	−	0.741145i	$-0.265717\pi$
$71$	11.3137	1.34269	0.671345	+	0.741145i	$0.265717\pi$
$72$	0	0
$73$	6.82843	0.799207	0.399603	−	0.916688i	$-0.369148\pi$
$73$	6.82843	0.799207	0.399603	+	0.916688i	$0.369148\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	21.6569	2.32186
$88$	0	0
$89$	9.31371	0.987251	0.493626	−	0.869675i	$-0.335671\pi$
$89$	9.31371	0.987251	0.493626	+	0.869675i	$0.335671\pi$
$90$	0	0
$91$	−13.6569	−1.43163
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.65685	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$97$	7.65685	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	0	0
$101$	−13.3137	−1.32476	−0.662382	−	0.749166i	$-0.730454\pi$
$101$	−13.3137	−1.32476	−0.662382	+	0.749166i	$0.730454\pi$
$102$	0	0
$103$	1.17157	0.115439	0.0577193	−	0.998333i	$-0.481617\pi$
$103$	1.17157	0.115439	0.0577193	+	0.998333i	$0.481617\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.65685	−0.353521	−0.176761	−	0.984254i	$-0.556562\pi$
$107$	−3.65685	−0.353521	−0.176761	+	0.984254i	$0.556562\pi$
$108$	0	0
$109$	3.65685	0.350263	0.175132	−	0.984545i	$-0.443965\pi$
$109$	3.65685	0.350263	0.175132	+	0.984545i	$0.443965\pi$
$110$	0	0
$111$	−10.3431	−0.981728
$112$	0	0
$113$	−8.34315	−0.784857	−0.392429	−	0.919782i	$-0.628365\pi$
$113$	−8.34315	−0.784857	−0.392429	+	0.919782i	$0.628365\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	34.1421	3.15644
$118$	0	0
$119$	2.34315	0.214796
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	16.9706	1.53018
$124$	0	0
$125$	0	0
$126$	0	0
$127$	15.6569	1.38932	0.694661	−	0.719338i	$-0.255555\pi$
$127$	15.6569	1.38932	0.694661	+	0.719338i	$0.255555\pi$
$128$	0	0
$129$	−16.9706	−1.49417
$130$	0	0
$131$	−11.3137	−0.988483	−0.494242	−	0.869325i	$-0.664554\pi$
$131$	−11.3137	−0.988483	−0.494242	+	0.869325i	$0.664554\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−22.9706	−1.96251	−0.981254	−	0.192720i	$-0.938269\pi$
$137$	−22.9706	−1.96251	−0.981254	+	0.192720i	$0.938269\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	−6.82843	−0.571022
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−8.48528	−0.699854
$148$	0	0
$149$	11.6569	0.954967	0.477483	−	0.878641i	$-0.341549\pi$
$149$	11.6569	0.954967	0.477483	+	0.878641i	$0.341549\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	−5.85786	−0.473580
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	−0.970563	−0.0769706
$160$	0	0
$161$	−5.65685	−0.445823
$162$	0	0
$163$	−0.485281	−0.0380102	−0.0190051	−	0.999819i	$-0.506050\pi$
$163$	−0.485281	−0.0380102	−0.0190051	+	0.999819i	$0.506050\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.9706	0.848928	0.424464	−	0.905445i	$-0.360463\pi$
$167$	10.9706	0.848928	0.424464	+	0.905445i	$0.360463\pi$
$168$	0	0
$169$	33.6274	2.58672
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.14214	−0.466978	−0.233489	−	0.972359i	$-0.575014\pi$
$173$	−6.14214	−0.466978	−0.233489	+	0.972359i	$0.575014\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	27.3137	2.05302
$178$	0	0
$179$	1.65685	0.123839	0.0619196	−	0.998081i	$-0.480278\pi$
$179$	1.65685	0.123839	0.0619196	+	0.998081i	$0.480278\pi$
$180$	0	0
$181$	−1.31371	−0.0976472	−0.0488236	−	0.998807i	$-0.515547\pi$
$181$	−1.31371	−0.0976472	−0.0488236	+	0.998807i	$0.515547\pi$
$182$	0	0
$183$	37.6569	2.78367
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.17157	0.0856739
$188$	0	0
$189$	−11.3137	−0.822951
$190$	0	0
$191$	19.3137	1.39749	0.698745	−	0.715370i	$-0.253742\pi$
$191$	19.3137	1.39749	0.698745	+	0.715370i	$0.253742\pi$
$192$	0	0
$193$	6.82843	0.491521	0.245760	−	0.969331i	$-0.420962\pi$
$193$	6.82843	0.491521	0.245760	+	0.969331i	$0.420962\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.17157	0.368459	0.184230	−	0.982883i	$-0.441021\pi$
$197$	5.17157	0.368459	0.184230	+	0.982883i	$0.441021\pi$
$198$	0	0
$199$	−21.6569	−1.53521	−0.767607	−	0.640921i	$-0.778553\pi$
$199$	−21.6569	−1.53521	−0.767607	+	0.640921i	$0.778553\pi$
$200$	0	0
$201$	−12.6863	−0.894822
$202$	0	0
$203$	−15.3137	−1.07481
$204$	0	0
$205$	0	0
$206$	0	0
$207$	14.1421	0.982946
$208$	0	0
$209$	0	0
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	0	0
$213$	32.0000	2.19260
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	19.3137	1.30510
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−5.17157	−0.346314	−0.173157	−	0.984894i	$-0.555397\pi$
$223$	−5.17157	−0.346314	−0.173157	+	0.984894i	$0.555397\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.68629	0.178295	0.0891477	−	0.996018i	$-0.471586\pi$
$227$	2.68629	0.178295	0.0891477	+	0.996018i	$0.471586\pi$
$228$	0	0
$229$	−21.3137	−1.40845	−0.704225	−	0.709977i	$-0.748705\pi$
$229$	−21.3137	−1.40845	−0.704225	+	0.709977i	$0.748705\pi$
$230$	0	0
$231$	5.65685	0.372194
$232$	0	0
$233$	−22.1421	−1.45058	−0.725290	−	0.688444i	$-0.758294\pi$
$233$	−22.1421	−1.45058	−0.725290	+	0.688444i	$0.758294\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−11.3137	−0.734904
$238$	0	0
$239$	0.686292	0.0443925	0.0221963	−	0.999754i	$-0.492934\pi$
$239$	0.686292	0.0443925	0.0221963	+	0.999754i	$0.492934\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	0	0
$243$	−14.1421	−0.907218
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−16.9706	−1.07547
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−2.82843	−0.177822
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.3137	−0.830486	−0.415243	−	0.909710i	$-0.636304\pi$
$257$	−13.3137	−0.830486	−0.415243	+	0.909710i	$0.636304\pi$
$258$	0	0
$259$	7.31371	0.454452
$260$	0	0
$261$	38.2843	2.36974
$262$	0	0
$263$	22.9706	1.41643	0.708213	−	0.705999i	$-0.249502\pi$
$263$	22.9706	1.41643	0.708213	+	0.705999i	$0.249502\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	26.3431	1.61217
$268$	0	0
$269$	−5.31371	−0.323983	−0.161991	−	0.986792i	$-0.551792\pi$
$269$	−5.31371	−0.323983	−0.161991	+	0.986792i	$0.551792\pi$
$270$	0	0
$271$	15.3137	0.930242	0.465121	−	0.885247i	$-0.346011\pi$
$271$	15.3137	0.930242	0.465121	+	0.885247i	$0.346011\pi$
$272$	0	0
$273$	−38.6274	−2.33784
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.17157	−0.0703930	−0.0351965	−	0.999380i	$-0.511206\pi$
$277$	−1.17157	−0.0703930	−0.0351965	+	0.999380i	$0.511206\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.31371	−0.316989	−0.158495	−	0.987360i	$-0.550664\pi$
$281$	−5.31371	−0.316989	−0.158495	+	0.987360i	$0.550664\pi$
$282$	0	0
$283$	−12.6274	−0.750622	−0.375311	−	0.926899i	$-0.622464\pi$
$283$	−12.6274	−0.750622	−0.375311	+	0.926899i	$0.622464\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−15.6274	−0.919260
$290$	0	0
$291$	21.6569	1.26955
$292$	0	0
$293$	14.8284	0.866286	0.433143	−	0.901325i	$-0.357405\pi$
$293$	14.8284	0.866286	0.433143	+	0.901325i	$0.357405\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.65685	−0.328244
$298$	0	0
$299$	19.3137	1.11694
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	−37.6569	−2.16333
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−27.6569	−1.57846	−0.789230	−	0.614098i	$-0.789520\pi$
$307$	−27.6569	−1.57846	−0.789230	+	0.614098i	$0.789520\pi$
$308$	0	0
$309$	3.31371	0.188510
$310$	0	0
$311$	−27.3137	−1.54882	−0.774409	−	0.632685i	$-0.781953\pi$
$311$	−27.3137	−1.54882	−0.774409	+	0.632685i	$0.781953\pi$
$312$	0	0
$313$	−21.3137	−1.20472	−0.602361	−	0.798224i	$-0.705773\pi$
$313$	−21.3137	−1.20472	−0.602361	+	0.798224i	$0.705773\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.3137	−1.19710	−0.598549	−	0.801087i	$-0.704256\pi$
$317$	−21.3137	−1.19710	−0.598549	+	0.801087i	$0.704256\pi$
$318$	0	0
$319$	−7.65685	−0.428702
$320$	0	0
$321$	−10.3431	−0.577298
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.3431	0.571977
$328$	0	0
$329$	5.65685	0.311872
$330$	0	0
$331$	−15.3137	−0.841718	−0.420859	−	0.907126i	$-0.638271\pi$
$331$	−15.3137	−0.841718	−0.420859	+	0.907126i	$0.638271\pi$
$332$	0	0
$333$	−18.2843	−1.00197
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.51472	0.191459	0.0957295	−	0.995407i	$-0.469482\pi$
$337$	3.51472	0.191459	0.0957295	+	0.995407i	$0.469482\pi$
$338$	0	0
$339$	−23.5980	−1.28167
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.9706	−1.23312	−0.616562	−	0.787306i	$-0.711475\pi$
$347$	−22.9706	−1.23312	−0.616562	+	0.787306i	$0.711475\pi$
$348$	0	0
$349$	−6.97056	−0.373126	−0.186563	−	0.982443i	$-0.559735\pi$
$349$	−6.97056	−0.373126	−0.186563	+	0.982443i	$0.559735\pi$
$350$	0	0
$351$	38.6274	2.06178
$352$	0	0
$353$	1.31371	0.0699216	0.0349608	−	0.999389i	$-0.488869\pi$
$353$	1.31371	0.0699216	0.0349608	+	0.999389i	$0.488869\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.62742	0.350760
$358$	0	0
$359$	−23.3137	−1.23045	−0.615225	−	0.788351i	$-0.710935\pi$
$359$	−23.3137	−1.23045	−0.615225	+	0.788351i	$0.710935\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	2.82843	0.148454
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.48528	0.442928	0.221464	−	0.975169i	$-0.428916\pi$
$367$	8.48528	0.442928	0.221464	+	0.975169i	$0.428916\pi$
$368$	0	0
$369$	30.0000	1.56174
$370$	0	0
$371$	0.686292	0.0356305
$372$	0	0
$373$	3.79899	0.196704	0.0983521	−	0.995152i	$-0.468643\pi$
$373$	3.79899	0.196704	0.0983521	+	0.995152i	$0.468643\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	52.2843	2.69278
$378$	0	0
$379$	−22.3431	−1.14769	−0.573845	−	0.818964i	$-0.694549\pi$
$379$	−22.3431	−1.14769	−0.573845	+	0.818964i	$0.694549\pi$
$380$	0	0
$381$	44.2843	2.26875
$382$	0	0
$383$	−34.1421	−1.74458	−0.872291	−	0.488987i	$-0.837366\pi$
$383$	−34.1421	−1.74458	−0.872291	+	0.488987i	$0.837366\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−30.0000	−1.52499
$388$	0	0
$389$	−24.6274	−1.24866	−0.624330	−	0.781161i	$-0.714628\pi$
$389$	−24.6274	−1.24866	−0.624330	+	0.781161i	$0.714628\pi$
$390$	0	0
$391$	−3.31371	−0.167581
$392$	0	0
$393$	−32.0000	−1.61419
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−13.3137	−0.668196	−0.334098	−	0.942538i	$-0.608432\pi$
$397$	−13.3137	−0.668196	−0.334098	+	0.942538i	$0.608432\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.3137	0.864605	0.432303	−	0.901729i	$-0.357701\pi$
$401$	17.3137	0.864605	0.432303	+	0.901729i	$0.357701\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.65685	0.181264
$408$	0	0
$409$	34.9706	1.72918	0.864592	−	0.502475i	$-0.167577\pi$
$409$	34.9706	1.72918	0.864592	+	0.502475i	$0.167577\pi$
$410$	0	0
$411$	−64.9706	−3.20476
$412$	0	0
$413$	−19.3137	−0.950365
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11.3137	0.554035
$418$	0	0
$419$	14.3431	0.700709	0.350354	−	0.936617i	$-0.386061\pi$
$419$	14.3431	0.700709	0.350354	+	0.936617i	$0.386061\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	−14.1421	−0.687614
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−26.6274	−1.28859
$428$	0	0
$429$	−19.3137	−0.932475
$430$	0	0
$431$	−11.3137	−0.544962	−0.272481	−	0.962161i	$-0.587844\pi$
$431$	−11.3137	−0.544962	−0.272481	+	0.962161i	$0.587844\pi$
$432$	0	0
$433$	−3.65685	−0.175737	−0.0878686	−	0.996132i	$-0.528006\pi$
$433$	−3.65685	−0.175737	−0.0878686	+	0.996132i	$0.528006\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	−15.0000	−0.714286
$442$	0	0
$443$	−21.1716	−1.00589	−0.502946	−	0.864318i	$-0.667751\pi$
$443$	−21.1716	−1.00589	−0.502946	+	0.864318i	$0.667751\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	32.9706	1.55945
$448$	0	0
$449$	−16.6274	−0.784696	−0.392348	−	0.919817i	$-0.628337\pi$
$449$	−16.6274	−0.784696	−0.392348	+	0.919817i	$0.628337\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	0	0
$453$	33.9411	1.59469
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16.4853	0.771149	0.385574	−	0.922677i	$-0.374003\pi$
$457$	16.4853	0.771149	0.385574	+	0.922677i	$0.374003\pi$
$458$	0	0
$459$	−6.62742	−0.309341
$460$	0	0
$461$	−32.6274	−1.51961	−0.759805	−	0.650151i	$-0.774706\pi$
$461$	−32.6274	−1.51961	−0.759805	+	0.650151i	$0.774706\pi$
$462$	0	0
$463$	22.1421	1.02903	0.514516	−	0.857481i	$-0.327972\pi$
$463$	22.1421	1.02903	0.514516	+	0.857481i	$0.327972\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.17157	−0.424410	−0.212205	−	0.977225i	$-0.568064\pi$
$467$	−9.17157	−0.424410	−0.212205	+	0.977225i	$0.568064\pi$
$468$	0	0
$469$	8.97056	0.414222
$470$	0	0
$471$	39.5980	1.82458
$472$	0	0
$473$	6.00000	0.275880
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.71573	−0.0785578
$478$	0	0
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	−24.9706	−1.13856
$482$	0	0
$483$	−16.0000	−0.728025
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7.51472	−0.340524	−0.170262	−	0.985399i	$-0.554461\pi$
$487$	−7.51472	−0.340524	−0.170262	+	0.985399i	$0.554461\pi$
$488$	0	0
$489$	−1.37258	−0.0620703
$490$	0	0
$491$	23.3137	1.05213	0.526066	−	0.850443i	$-0.323666\pi$
$491$	23.3137	1.05213	0.526066	+	0.850443i	$0.323666\pi$
$492$	0	0
$493$	−8.97056	−0.404014
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−22.6274	−1.01498
$498$	0	0
$499$	1.65685	0.0741710	0.0370855	−	0.999312i	$-0.488193\pi$
$499$	1.65685	0.0741710	0.0370855	+	0.999312i	$0.488193\pi$
$500$	0	0
$501$	31.0294	1.38629
$502$	0	0
$503$	−28.6274	−1.27643	−0.638217	−	0.769857i	$-0.720328\pi$
$503$	−28.6274	−1.27643	−0.638217	+	0.769857i	$0.720328\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	95.1127	4.22410
$508$	0	0
$509$	9.31371	0.412823	0.206411	−	0.978465i	$-0.433821\pi$
$509$	9.31371	0.412823	0.206411	+	0.978465i	$0.433821\pi$
$510$	0	0
$511$	−13.6569	−0.604144
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.82843	0.124394
$518$	0	0
$519$	−17.3726	−0.762572
$520$	0	0
$521$	2.68629	0.117689	0.0588443	−	0.998267i	$-0.481258\pi$
$521$	2.68629	0.117689	0.0588443	+	0.998267i	$0.481258\pi$
$522$	0	0
$523$	37.5980	1.64404	0.822022	−	0.569455i	$-0.192846\pi$
$523$	37.5980	1.64404	0.822022	+	0.569455i	$0.192846\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	48.2843	2.09536
$532$	0	0
$533$	40.9706	1.77463
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.68629	0.202228
$538$	0	0
$539$	3.00000	0.129219
$540$	0	0
$541$	6.00000	0.257960	0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000	0.257960	0.128980	+	0.991647i	$0.458830\pi$
$542$	0	0
$543$	−3.71573	−0.159457
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−34.0000	−1.45374	−0.726868	−	0.686778i	$-0.759025\pi$
$547$	−34.0000	−1.45374	−0.726868	+	0.686778i	$0.759025\pi$
$548$	0	0
$549$	66.5685	2.84108
$550$	0	0
$551$	0	0
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−38.1421	−1.61613	−0.808067	−	0.589090i	$-0.799486\pi$
$557$	−38.1421	−1.61613	−0.808067	+	0.589090i	$0.799486\pi$
$558$	0	0
$559$	−40.9706	−1.73287
$560$	0	0
$561$	3.31371	0.139905
$562$	0	0
$563$	11.6569	0.491278	0.245639	−	0.969361i	$-0.421002\pi$
$563$	11.6569	0.491278	0.245639	+	0.969361i	$0.421002\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	20.3431	0.852829	0.426415	−	0.904528i	$-0.359776\pi$
$569$	20.3431	0.852829	0.426415	+	0.904528i	$0.359776\pi$
$570$	0	0
$571$	−45.9411	−1.92258	−0.961288	−	0.275545i	$-0.911142\pi$
$571$	−45.9411	−1.92258	−0.961288	+	0.275545i	$0.911142\pi$
$572$	0	0
$573$	54.6274	2.28209
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−6.97056	−0.290188	−0.145094	−	0.989418i	$-0.546349\pi$
$577$	−6.97056	−0.290188	−0.145094	+	0.989418i	$0.546349\pi$
$578$	0	0
$579$	19.3137	0.802650
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	0.343146	0.0142116
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.1421	1.07900	0.539501	−	0.841985i	$-0.318613\pi$
$587$	26.1421	1.07900	0.539501	+	0.841985i	$0.318613\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	14.6274	0.601692
$592$	0	0
$593$	−20.4853	−0.841230	−0.420615	−	0.907239i	$-0.638186\pi$
$593$	−20.4853	−0.841230	−0.420615	+	0.907239i	$0.638186\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−61.2548	−2.50699
$598$	0	0
$599$	−5.65685	−0.231133	−0.115566	−	0.993300i	$-0.536868\pi$
$599$	−5.65685	−0.231133	−0.115566	+	0.993300i	$0.536868\pi$
$600$	0	0
$601$	−43.9411	−1.79240	−0.896198	−	0.443654i	$-0.853682\pi$
$601$	−43.9411	−1.79240	−0.896198	+	0.443654i	$0.853682\pi$
$602$	0	0
$603$	−22.4264	−0.913274
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−18.2843	−0.742136	−0.371068	−	0.928606i	$-0.621008\pi$
$607$	−18.2843	−0.742136	−0.371068	+	0.928606i	$0.621008\pi$
$608$	0	0
$609$	−43.3137	−1.75516
$610$	0	0
$611$	−19.3137	−0.781349
$612$	0	0
$613$	−25.4558	−1.02815	−0.514076	−	0.857745i	$-0.671865\pi$
$613$	−25.4558	−1.02815	−0.514076	+	0.857745i	$0.671865\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11.6569	−0.469287	−0.234644	−	0.972081i	$-0.575392\pi$
$617$	−11.6569	−0.469287	−0.234644	+	0.972081i	$0.575392\pi$
$618$	0	0
$619$	25.6569	1.03124	0.515618	−	0.856819i	$-0.327562\pi$
$619$	25.6569	1.03124	0.515618	+	0.856819i	$0.327562\pi$
$620$	0	0
$621$	16.0000	0.642058
$622$	0	0
$623$	−18.6274	−0.746292
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.28427	0.170825
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	45.2548	1.79872
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−20.4853	−0.811656
$638$	0	0
$639$	56.5685	2.23782
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	49.4558	1.95035	0.975174	−	0.221440i	$-0.0710756\pi$
$643$	49.4558	1.95035	0.975174	+	0.221440i	$0.0710756\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−35.1127	−1.38042	−0.690211	−	0.723608i	$-0.742482\pi$
$647$	−35.1127	−1.38042	−0.690211	+	0.723608i	$0.742482\pi$
$648$	0	0
$649$	−9.65685	−0.379065
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−0.343146	−0.0134283	−0.00671417	−	0.999977i	$-0.502137\pi$
$653$	−0.343146	−0.0134283	−0.00671417	+	0.999977i	$0.502137\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	34.1421	1.33201
$658$	0	0
$659$	21.9411	0.854705	0.427352	−	0.904085i	$-0.359446\pi$
$659$	21.9411	0.854705	0.427352	+	0.904085i	$0.359446\pi$
$660$	0	0
$661$	−0.627417	−0.0244037	−0.0122018	−	0.999926i	$-0.503884\pi$
$661$	−0.627417	−0.0244037	−0.0122018	+	0.999926i	$0.503884\pi$
$662$	0	0
$663$	−22.6274	−0.878776
$664$	0	0
$665$	0	0
$666$	0	0
$667$	21.6569	0.838557
$668$	0	0
$669$	−14.6274	−0.565529
$670$	0	0
$671$	−13.3137	−0.513970
$672$	0	0
$673$	−4.48528	−0.172895	−0.0864474	−	0.996256i	$-0.527551\pi$
$673$	−4.48528	−0.172895	−0.0864474	+	0.996256i	$0.527551\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−17.1716	−0.659957	−0.329979	−	0.943988i	$-0.607042\pi$
$677$	−17.1716	−0.659957	−0.329979	+	0.943988i	$0.607042\pi$
$678$	0	0
$679$	−15.3137	−0.587686
$680$	0	0
$681$	7.59798	0.291155
$682$	0	0
$683$	31.7990	1.21675	0.608377	−	0.793648i	$-0.291821\pi$
$683$	31.7990	1.21675	0.608377	+	0.793648i	$0.291821\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−60.2843	−2.29999
$688$	0	0
$689$	−2.34315	−0.0892667
$690$	0	0
$691$	16.6863	0.634776	0.317388	−	0.948296i	$-0.397194\pi$
$691$	16.6863	0.634776	0.317388	+	0.948296i	$0.397194\pi$
$692$	0	0
$693$	10.0000	0.379869
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−7.02944	−0.266259
$698$	0	0
$699$	−62.6274	−2.36879
$700$	0	0
$701$	32.6274	1.23232	0.616160	−	0.787621i	$-0.288687\pi$
$701$	32.6274	1.23232	0.616160	+	0.787621i	$0.288687\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	26.6274	1.00143
$708$	0	0
$709$	−20.6274	−0.774679	−0.387339	−	0.921937i	$-0.626606\pi$
$709$	−20.6274	−0.774679	−0.387339	+	0.921937i	$0.626606\pi$
$710$	0	0
$711$	−20.0000	−0.750059
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.94113	0.0724927
$718$	0	0
$719$	−29.6569	−1.10601	−0.553007	−	0.833177i	$-0.686520\pi$
$719$	−29.6569	−1.10601	−0.553007	+	0.833177i	$0.686520\pi$
$720$	0	0
$721$	−2.34315	−0.0872633
$722$	0	0
$723$	16.9706	0.631142
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−36.4853	−1.35316	−0.676582	−	0.736367i	$-0.736540\pi$
$727$	−36.4853	−1.35316	−0.676582	+	0.736367i	$0.736540\pi$
$728$	0	0
$729$	−43.0000	−1.59259
$730$	0	0
$731$	7.02944	0.259993
$732$	0	0
$733$	−33.4558	−1.23572	−0.617860	−	0.786288i	$-0.712000\pi$
$733$	−33.4558	−1.23572	−0.617860	+	0.786288i	$0.712000\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.48528	0.165217
$738$	0	0
$739$	37.9411	1.39569	0.697843	−	0.716250i	$-0.254143\pi$
$739$	37.9411	1.39569	0.697843	+	0.716250i	$0.254143\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	29.5980	1.08584	0.542922	−	0.839783i	$-0.317318\pi$
$743$	29.5980	1.08584	0.542922	+	0.839783i	$0.317318\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−30.0000	−1.09764
$748$	0	0
$749$	7.31371	0.267237
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	−33.9411	−1.23688
$754$	0	0
$755$	0	0
$756$	0	0
$757$	9.31371	0.338512	0.169256	−	0.985572i	$-0.445863\pi$
$757$	9.31371	0.338512	0.169256	+	0.985572i	$0.445863\pi$
$758$	0	0
$759$	−8.00000	−0.290382
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	−7.31371	−0.264774
$764$	0	0
$765$	0	0
$766$	0	0
$767$	65.9411	2.38100
$768$	0	0
$769$	14.9706	0.539852	0.269926	−	0.962881i	$-0.413001\pi$
$769$	14.9706	0.539852	0.269926	+	0.962881i	$0.413001\pi$
$770$	0	0
$771$	−37.6569	−1.35618
$772$	0	0
$773$	30.2843	1.08925	0.544625	−	0.838680i	$-0.316672\pi$
$773$	30.2843	1.08925	0.544625	+	0.838680i	$0.316672\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	20.6863	0.742117
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−11.3137	−0.404836
$782$	0	0
$783$	43.3137	1.54791
$784$	0	0
$785$	0	0
$786$	0	0
$787$	18.9706	0.676228	0.338114	−	0.941105i	$-0.390211\pi$
$787$	18.9706	0.676228	0.338114	+	0.941105i	$0.390211\pi$
$788$	0	0
$789$	64.9706	2.31301
$790$	0	0
$791$	16.6863	0.593296
$792$	0	0
$793$	90.9117	3.22837
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.6274	0.447286	0.223643	−	0.974671i	$-0.428205\pi$
$797$	12.6274	0.447286	0.223643	+	0.974671i	$0.428205\pi$
$798$	0	0
$799$	3.31371	0.117231
$800$	0	0
$801$	46.5685	1.64542
$802$	0	0
$803$	−6.82843	−0.240970
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−15.0294	−0.529061
$808$	0	0
$809$	22.9706	0.807602	0.403801	−	0.914847i	$-0.367689\pi$
$809$	22.9706	0.807602	0.403801	+	0.914847i	$0.367689\pi$
$810$	0	0
$811$	13.9411	0.489539	0.244770	−	0.969581i	$-0.421288\pi$
$811$	13.9411	0.489539	0.244770	+	0.969581i	$0.421288\pi$
$812$	0	0
$813$	43.3137	1.51908
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−68.2843	−2.38605
$820$	0	0
$821$	−18.6863	−0.652156	−0.326078	−	0.945343i	$-0.605727\pi$
$821$	−18.6863	−0.652156	−0.326078	+	0.945343i	$0.605727\pi$
$822$	0	0
$823$	36.4853	1.27180	0.635898	−	0.771773i	$-0.280630\pi$
$823$	36.4853	1.27180	0.635898	+	0.771773i	$0.280630\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−34.2843	−1.19218	−0.596090	−	0.802917i	$-0.703280\pi$
$827$	−34.2843	−1.19218	−0.596090	+	0.802917i	$0.703280\pi$
$828$	0	0
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0	0
$831$	−3.31371	−0.114951
$832$	0	0
$833$	3.51472	0.121778
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−37.6569	−1.30006	−0.650029	−	0.759909i	$-0.725243\pi$
$839$	−37.6569	−1.30006	−0.650029	+	0.759909i	$0.725243\pi$
$840$	0	0
$841$	29.6274	1.02164
$842$	0	0
$843$	−15.0294	−0.517641
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	−35.7157	−1.22576
$850$	0	0
$851$	−10.3431	−0.354558
$852$	0	0
$853$	32.4853	1.11227	0.556137	−	0.831090i	$-0.312283\pi$
$853$	32.4853	1.11227	0.556137	+	0.831090i	$0.312283\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48.7696	1.66594	0.832968	−	0.553321i	$-0.186640\pi$
$857$	48.7696	1.66594	0.832968	+	0.553321i	$0.186640\pi$
$858$	0	0
$859$	32.2843	1.10153	0.550763	−	0.834662i	$-0.314337\pi$
$859$	32.2843	1.10153	0.550763	+	0.834662i	$0.314337\pi$
$860$	0	0
$861$	−33.9411	−1.15671
$862$	0	0
$863$	−14.8284	−0.504766	−0.252383	−	0.967627i	$-0.581214\pi$
$863$	−14.8284	−0.504766	−0.252383	+	0.967627i	$0.581214\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−44.2010	−1.50115
$868$	0	0
$869$	4.00000	0.135691
$870$	0	0
$871$	−30.6274	−1.03777
$872$	0	0
$873$	38.2843	1.29573
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.45584	−0.0491604	−0.0245802	−	0.999698i	$-0.507825\pi$
$877$	−1.45584	−0.0491604	−0.0245802	+	0.999698i	$0.507825\pi$
$878$	0	0
$879$	41.9411	1.41464
$880$	0	0
$881$	−52.6274	−1.77306	−0.886531	−	0.462668i	$-0.846892\pi$
$881$	−52.6274	−1.77306	−0.886531	+	0.462668i	$0.846892\pi$
$882$	0	0
$883$	42.8284	1.44129	0.720646	−	0.693304i	$-0.243845\pi$
$883$	42.8284	1.44129	0.720646	+	0.693304i	$0.243845\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−18.2843	−0.613926	−0.306963	−	0.951721i	$-0.599313\pi$
$887$	−18.2843	−0.613926	−0.306963	+	0.951721i	$0.599313\pi$
$888$	0	0
$889$	−31.3137	−1.05023
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	54.6274	1.82396
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0.402020	0.0133932
$902$	0	0
$903$	33.9411	1.12949
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−44.4853	−1.47711	−0.738555	−	0.674193i	$-0.764491\pi$
$907$	−44.4853	−1.47711	−0.738555	+	0.674193i	$0.764491\pi$
$908$	0	0
$909$	−66.5685	−2.20794
$910$	0	0
$911$	−57.9411	−1.91968	−0.959838	−	0.280556i	$-0.909481\pi$
$911$	−57.9411	−1.91968	−0.959838	+	0.280556i	$0.909481\pi$
$912$	0	0
$913$	6.00000	0.198571
$914$	0	0
$915$	0	0
$916$	0	0
$917$	22.6274	0.747223
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	−78.2254	−2.57761
$922$	0	0
$923$	77.2548	2.54287
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.85786	0.192398
$928$	0	0
$929$	−17.3137	−0.568044	−0.284022	−	0.958818i	$-0.591669\pi$
$929$	−17.3137	−0.568044	−0.284022	+	0.958818i	$0.591669\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−77.2548	−2.52921
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−49.4558	−1.61565	−0.807826	−	0.589421i	$-0.799356\pi$
$937$	−49.4558	−1.61565	−0.807826	+	0.589421i	$0.799356\pi$
$938$	0	0
$939$	−60.2843	−1.96730
$940$	0	0
$941$	−29.3137	−0.955600	−0.477800	−	0.878469i	$-0.658566\pi$
$941$	−29.3137	−0.955600	−0.477800	+	0.878469i	$0.658566\pi$
$942$	0	0
$943$	16.9706	0.552638
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.8284	1.52172	0.760860	−	0.648916i	$-0.224778\pi$
$947$	46.8284	1.52172	0.760860	+	0.648916i	$0.224778\pi$
$948$	0	0
$949$	46.6274	1.51359
$950$	0	0
$951$	−60.2843	−1.95485
$952$	0	0
$953$	−58.8284	−1.90564	−0.952820	−	0.303536i	$-0.901833\pi$
$953$	−58.8284	−1.90564	−0.952820	+	0.303536i	$0.901833\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−21.6569	−0.700067
$958$	0	0
$959$	45.9411	1.48352
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−18.2843	−0.589202
$964$	0	0
$965$	0	0
$966$	0	0
$967$	18.9706	0.610052	0.305026	−	0.952344i	$-0.401335\pi$
$967$	18.9706	0.610052	0.305026	+	0.952344i	$0.401335\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31.3137	1.00490	0.502452	−	0.864605i	$-0.332431\pi$
$971$	31.3137	1.00490	0.502452	+	0.864605i	$0.332431\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−43.6569	−1.39671	−0.698353	−	0.715753i	$-0.746084\pi$
$977$	−43.6569	−1.39671	−0.698353	+	0.715753i	$0.746084\pi$
$978$	0	0
$979$	−9.31371	−0.297667
$980$	0	0
$981$	18.2843	0.583772
$982$	0	0
$983$	−50.1421	−1.59929	−0.799643	−	0.600476i	$-0.794978\pi$
$983$	−50.1421	−1.59929	−0.799643	+	0.600476i	$0.794978\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	16.0000	0.509286
$988$	0	0
$989$	−16.9706	−0.539633
$990$	0	0
$991$	−9.94113	−0.315790	−0.157895	−	0.987456i	$-0.550471\pi$
$991$	−9.94113	−0.315790	−0.157895	+	0.987456i	$0.550471\pi$
$992$	0	0
$993$	−43.3137	−1.37452
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−9.45584	−0.299470	−0.149735	−	0.988726i	$-0.547842\pi$
$997$	−9.45584	−0.299470	−0.149735	+	0.988726i	$0.547842\pi$
$998$	0	0
$999$	−20.6863	−0.654485

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bn.1.2		2
4.3	odd	2		275.2.a.c.1.2		2
5.2	odd	4		4400.2.b.q.4049.1		4
5.3	odd	4		4400.2.b.q.4049.4		4
5.4	even	2		880.2.a.m.1.1		2
12.11	even	2		2475.2.a.x.1.1		2
15.14	odd	2		7920.2.a.ch.1.1		2
20.3	even	4		275.2.b.d.199.2		4
20.7	even	4		275.2.b.d.199.3		4
20.19	odd	2		55.2.a.b.1.1	✓	2
40.19	odd	2		3520.2.a.bn.1.1		2
40.29	even	2		3520.2.a.bo.1.2		2
44.43	even	2		3025.2.a.o.1.1		2
55.54	odd	2		9680.2.a.bn.1.1		2
60.23	odd	4		2475.2.c.l.199.3		4
60.47	odd	4		2475.2.c.l.199.2		4
60.59	even	2		495.2.a.b.1.2		2
140.139	even	2		2695.2.a.f.1.1		2
220.19	even	10		605.2.g.l.251.2		8
220.39	even	10		605.2.g.l.366.1		8
220.59	odd	10		605.2.g.f.511.1		8
220.79	even	10		605.2.g.l.81.1		8
220.119	odd	10		605.2.g.f.81.2		8
220.139	even	10		605.2.g.l.511.2		8
220.159	odd	10		605.2.g.f.366.2		8
220.179	odd	10		605.2.g.f.251.1		8
220.219	even	2		605.2.a.d.1.2		2
260.259	odd	2		9295.2.a.g.1.2		2
660.659	odd	2		5445.2.a.y.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.a.b.1.1	✓	2	20.19	odd	2
275.2.a.c.1.2		2	4.3	odd	2
275.2.b.d.199.2		4	20.3	even	4
275.2.b.d.199.3		4	20.7	even	4
495.2.a.b.1.2		2	60.59	even	2
605.2.a.d.1.2		2	220.219	even	2
605.2.g.f.81.2		8	220.119	odd	10
605.2.g.f.251.1		8	220.179	odd	10
605.2.g.f.366.2		8	220.159	odd	10
605.2.g.f.511.1		8	220.59	odd	10
605.2.g.l.81.1		8	220.79	even	10
605.2.g.l.251.2		8	220.19	even	10
605.2.g.l.366.1		8	220.39	even	10
605.2.g.l.511.2		8	220.139	even	10
880.2.a.m.1.1		2	5.4	even	2
2475.2.a.x.1.1		2	12.11	even	2
2475.2.c.l.199.2		4	60.47	odd	4
2475.2.c.l.199.3		4	60.23	odd	4
2695.2.a.f.1.1		2	140.139	even	2
3025.2.a.o.1.1		2	44.43	even	2
3520.2.a.bn.1.1		2	40.19	odd	2
3520.2.a.bo.1.2		2	40.29	even	2
4400.2.a.bn.1.2		2	1.1	even	1	trivial
4400.2.b.q.4049.1		4	5.2	odd	4
4400.2.b.q.4049.4		4	5.3	odd	4
5445.2.a.y.1.1		2	660.659	odd	2
7920.2.a.ch.1.1		2	15.14	odd	2
9295.2.a.g.1.2		2	260.259	odd	2
9680.2.a.bn.1.1		2	55.54	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

\(f(q)\)	\(=\)	\(q+2.82843 q^{3} -2.00000 q^{7} +5.00000 q^{9} -1.00000 q^{11} +6.82843 q^{13} -1.17157 q^{17} -5.65685 q^{21} +2.82843 q^{23} +5.65685 q^{27} +7.65685 q^{29} -2.82843 q^{33} -3.65685 q^{37} +19.3137 q^{39} +6.00000 q^{41} -6.00000 q^{43} -2.82843 q^{47} -3.00000 q^{49} -3.31371 q^{51} -0.343146 q^{53} +9.65685 q^{59} +13.3137 q^{61} -10.0000 q^{63} -4.48528 q^{67} +8.00000 q^{69} +11.3137 q^{71} +6.82843 q^{73} +2.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -6.00000 q^{83} +21.6569 q^{87} +9.31371 q^{89} -13.6569 q^{91} +7.65685 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{13} - 8 q^{17} + 4 q^{29} + 4 q^{37} + 16 q^{39} + 12 q^{41} - 12 q^{43} - 6 q^{49} + 16 q^{51} - 12 q^{53} + 8 q^{59} + 4 q^{61} - 20 q^{63} + 8 q^{67} + 16 q^{69}+ \cdots - 10 q^{99}+O(q^{100}) \) 2 * q - 4 * q^7 + 10 * q^9 - 2 * q^11 + 8 * q^13 - 8 * q^17 + 4 * q^29 + 4 * q^37 + 16 * q^39 + 12 * q^41 - 12 * q^43 - 6 * q^49 + 16 * q^51 - 12 * q^53 + 8 * q^59 + 4 * q^61 - 20 * q^63 + 8 * q^67 + 16 * q^69 + 8 * q^73 + 4 * q^77 - 8 * q^79 + 2 * q^81 - 12 * q^83 + 32 * q^87 - 4 * q^89 - 16 * q^91 + 4 * q^97 - 10 * q^99

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