LMFDB - Embedded newform 2800.2.g.r.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(449,2800)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2800.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.g (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$22.3581125660$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 17x^{2} + 64 $ x^4 + 17*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$-2.37228i$ of defining polynomial
Character	$\chi$	$=$	2800.449
Dual form			2800.2.g.r.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.37228i q^{3} -1.00000i q^{7} -2.62772 q^{9} +O(q^{10})$	$q-2.37228i q^{3} -1.00000i q^{7} -2.62772 q^{9} -6.37228 q^{11} +4.37228i q^{13} +0.372281i q^{17} -4.74456 q^{19} -2.37228 q^{21} +4.74456i q^{23} -0.883156i q^{27} +4.37228 q^{29} +8.00000 q^{31} +15.1168i q^{33} +2.00000i q^{37} +10.3723 q^{39} +6.74456 q^{41} +8.74456i q^{43} -7.11684i q^{47} -1.00000 q^{49} +0.883156 q^{51} +10.7446i q^{53} +11.2554i q^{57} +8.00000 q^{59} -2.74456 q^{61} +2.62772i q^{63} -4.00000i q^{67} +11.2554 q^{69} -8.00000 q^{71} -6.00000i q^{73} +6.37228i q^{77} +15.1168 q^{79} -9.97825 q^{81} +9.48913i q^{83} -10.3723i q^{87} -14.7446 q^{89} +4.37228 q^{91} -18.9783i q^{93} +9.86141i q^{97} +16.7446 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 22 q^{9}+O(q^{10}) $ 4 * q - 22 * q^9	$ 4 q - 22 q^{9} - 14 q^{11} + 4 q^{19} + 2 q^{21} + 6 q^{29} + 32 q^{31} + 30 q^{39} + 4 q^{41} - 4 q^{49} + 38 q^{51} + 32 q^{59} + 12 q^{61} + 68 q^{69} - 32 q^{71} + 26 q^{79} + 52 q^{81} - 36 q^{89} + 6 q^{91} + 44 q^{99}+O(q^{100}) $ 4 * q - 22 * q^9 - 14 * q^11 + 4 * q^19 + 2 * q^21 + 6 * q^29 + 32 * q^31 + 30 * q^39 + 4 * q^41 - 4 * q^49 + 38 * q^51 + 32 * q^59 + 12 * q^61 + 68 * q^69 - 32 * q^71 + 26 * q^79 + 52 * q^81 - 36 * q^89 + 6 * q^91 + 44 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$.

$n$	$351$	$801$	$2101$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 2.37228i	− 1.36964i	−0.728714	−	0.684819i	$-0.759881\pi$
$3$	− 2.37228i	− 1.36964i	0.728714	−	0.684819i	$-0.240119\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	−2.62772	−0.875906
$10$	0	0
$11$	−6.37228	−1.92132	−0.960658	−	0.277736i	$-0.910416\pi$
$11$	−6.37228	−1.92132	−0.960658	+	0.277736i	$0.910416\pi$
$12$	0	0
$13$	4.37228i	1.21265i	0.795216	+	0.606326i	$0.207357\pi$
$13$	4.37228i	1.21265i	−0.795216	+	0.606326i	$0.792643\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.372281i	0.0902915i	0.998980	+	0.0451457i	$0.0143752\pi$
$17$	0.372281i	0.0902915i	−0.998980	+	0.0451457i	$0.985625\pi$
$18$	0	0
$19$	−4.74456	−1.08848	−0.544239	−	0.838930i	$-0.683181\pi$
$19$	−4.74456	−1.08848	−0.544239	+	0.838930i	$0.683181\pi$
$20$	0	0
$21$	−2.37228	−0.517674
$22$	0	0
$23$	4.74456i	0.989310i	0.869090	+	0.494655i	$0.164706\pi$
$23$	4.74456i	0.989310i	−0.869090	+	0.494655i	$0.835294\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 0.883156i	− 0.169963i
$28$	0	0
$29$	4.37228	0.811912	0.405956	−	0.913893i	$-0.366939\pi$
$29$	4.37228	0.811912	0.405956	+	0.913893i	$0.366939\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	15.1168i	2.63150i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	10.3723	1.66089
$40$	0	0
$41$	6.74456	1.05332	0.526662	−	0.850075i	$-0.323443\pi$
$41$	6.74456	1.05332	0.526662	+	0.850075i	$0.323443\pi$
$42$	0	0
$43$	8.74456i	1.33353i	0.745267	+	0.666767i	$0.232322\pi$
$43$	8.74456i	1.33353i	−0.745267	+	0.666767i	$0.767678\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 7.11684i	− 1.03810i	−0.854744	−	0.519049i	$-0.826286\pi$
$47$	− 7.11684i	− 1.03810i	0.854744	−	0.519049i	$-0.173714\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0.883156	0.123667
$52$	0	0
$53$	10.7446i	1.47588i	0.674867	+	0.737940i	$0.264201\pi$
$53$	10.7446i	1.47588i	−0.674867	+	0.737940i	$0.735799\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	11.2554i	1.49082i
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−2.74456	−0.351405	−0.175703	−	0.984443i	$-0.556220\pi$
$61$	−2.74456	−0.351405	−0.175703	+	0.984443i	$0.556220\pi$
$62$	0	0
$63$	2.62772i	0.331061i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	11.2554	1.35500
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.37228i	0.726189i
$78$	0	0
$79$	15.1168	1.70078	0.850389	−	0.526155i	$-0.176367\pi$
$79$	15.1168	1.70078	0.850389	+	0.526155i	$0.176367\pi$
$80$	0	0
$81$	−9.97825	−1.10869
$82$	0	0
$83$	9.48913i	1.04157i	0.853689	+	0.520783i	$0.174360\pi$
$83$	9.48913i	1.04157i	−0.853689	+	0.520783i	$0.825640\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 10.3723i	− 1.11203i
$88$	0	0
$89$	−14.7446	−1.56292	−0.781460	−	0.623955i	$-0.785525\pi$
$89$	−14.7446	−1.56292	−0.781460	+	0.623955i	$0.785525\pi$
$90$	0	0
$91$	4.37228	0.458340
$92$	0	0
$93$	− 18.9783i	− 1.96795i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.86141i	1.00127i	0.865657	+	0.500637i	$0.166901\pi$
$97$	9.86141i	1.00127i	−0.865657	+	0.500637i	$0.833099\pi$
$98$	0	0
$99$	16.7446	1.68289
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	5.62772i	0.554516i	0.960796	+	0.277258i	$0.0894256\pi$
$103$	5.62772i	0.554516i	−0.960796	+	0.277258i	$0.910574\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.74456i	0.845369i	0.906277	+	0.422684i	$0.138912\pi$
$107$	8.74456i	0.845369i	−0.906277	+	0.422684i	$0.861088\pi$
$108$	0	0
$109$	−0.372281	−0.0356581	−0.0178290	−	0.999841i	$-0.505675\pi$
$109$	−0.372281	−0.0356581	−0.0178290	+	0.999841i	$0.505675\pi$
$110$	0	0
$111$	4.74456	0.450334
$112$	0	0
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 11.4891i	− 1.06217i
$118$	0	0
$119$	0.372281	0.0341270
$120$	0	0
$121$	29.6060	2.69145
$122$	0	0
$123$	− 16.0000i	− 1.44267i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	0	0
$129$	20.7446	1.82646
$130$	0	0
$131$	4.74456	0.414534	0.207267	−	0.978284i	$-0.433543\pi$
$131$	4.74456	0.414534	0.207267	+	0.978284i	$0.433543\pi$
$132$	0	0
$133$	4.74456i	0.411406i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 14.7446i	− 1.25971i	−0.776712	−	0.629857i	$-0.783114\pi$
$137$	− 14.7446i	− 1.25971i	0.776712	−	0.629857i	$-0.216886\pi$
$138$	0	0
$139$	−4.74456	−0.402429	−0.201214	−	0.979547i	$-0.564489\pi$
$139$	−4.74456	−0.402429	−0.201214	+	0.979547i	$0.564489\pi$
$140$	0	0
$141$	−16.8832	−1.42182
$142$	0	0
$143$	− 27.8614i	− 2.32989i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.37228i	0.195662i
$148$	0	0
$149$	−15.4891	−1.26892	−0.634459	−	0.772956i	$-0.718777\pi$
$149$	−15.4891	−1.26892	−0.634459	+	0.772956i	$0.718777\pi$
$150$	0	0
$151$	−15.1168	−1.23019	−0.615096	−	0.788452i	$-0.710883\pi$
$151$	−15.1168	−1.23019	−0.615096	+	0.788452i	$0.710883\pi$
$152$	0	0
$153$	− 0.978251i	− 0.0790869i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.4891i	1.23617i	0.786113	+	0.618083i	$0.212091\pi$
$157$	15.4891i	1.23617i	−0.786113	+	0.618083i	$0.787909\pi$
$158$	0	0
$159$	25.4891	2.02142
$160$	0	0
$161$	4.74456	0.373924
$162$	0	0
$163$	16.7446i	1.31154i	0.754963	+	0.655768i	$0.227655\pi$
$163$	16.7446i	1.31154i	−0.754963	+	0.655768i	$0.772345\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 5.62772i	− 0.435486i	−0.976006	−	0.217743i	$-0.930131\pi$
$167$	− 5.62772i	− 0.435486i	0.976006	−	0.217743i	$-0.0698695\pi$
$168$	0	0
$169$	−6.11684	−0.470526
$170$	0	0
$171$	12.4674	0.953404
$172$	0	0
$173$	− 0.372281i	− 0.0283040i	−0.999900	−	0.0141520i	$-0.995495\pi$
$173$	− 0.372281i	− 0.0283040i	0.999900	−	0.0141520i	$-0.00450488\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 18.9783i	− 1.42649i
$178$	0	0
$179$	−22.9783	−1.71748	−0.858738	−	0.512416i	$-0.828751\pi$
$179$	−22.9783	−1.71748	−0.858738	+	0.512416i	$0.828751\pi$
$180$	0	0
$181$	16.2337	1.20664	0.603320	−	0.797499i	$-0.293844\pi$
$181$	16.2337	1.20664	0.603320	+	0.797499i	$0.293844\pi$
$182$	0	0
$183$	6.51087i	0.481298i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.37228i	− 0.173478i
$188$	0	0
$189$	−0.883156	−0.0642401
$190$	0	0
$191$	−3.86141	−0.279402	−0.139701	−	0.990194i	$-0.544614\pi$
$191$	−3.86141	−0.279402	−0.139701	+	0.990194i	$0.544614\pi$
$192$	0	0
$193$	6.74456i	0.485484i	0.970091	+	0.242742i	$0.0780469\pi$
$193$	6.74456i	0.485484i	−0.970091	+	0.242742i	$0.921953\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.23369i	0.586626i	0.956016	+	0.293313i	$0.0947578\pi$
$197$	8.23369i	0.586626i	−0.956016	+	0.293313i	$0.905242\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−9.48913	−0.669311
$202$	0	0
$203$	− 4.37228i	− 0.306874i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 12.4674i	− 0.866543i
$208$	0	0
$209$	30.2337	2.09131
$210$	0	0
$211$	14.3723	0.989429	0.494714	−	0.869056i	$-0.335273\pi$
$211$	14.3723	0.989429	0.494714	+	0.869056i	$0.335273\pi$
$212$	0	0
$213$	18.9783i	1.30037i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 8.00000i	− 0.543075i
$218$	0	0
$219$	−14.2337	−0.961823
$220$	0	0
$221$	−1.62772	−0.109492
$222$	0	0
$223$	5.62772i	0.376860i	0.982087	+	0.188430i	$0.0603398\pi$
$223$	5.62772i	0.376860i	−0.982087	+	0.188430i	$0.939660\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.8614i	1.31825i	0.752034	+	0.659124i	$0.229073\pi$
$227$	19.8614i	1.31825i	−0.752034	+	0.659124i	$0.770927\pi$
$228$	0	0
$229$	12.2337	0.808425	0.404212	−	0.914665i	$-0.367546\pi$
$229$	12.2337	0.808425	0.404212	+	0.914665i	$0.367546\pi$
$230$	0	0
$231$	15.1168	0.994615
$232$	0	0
$233$	− 1.25544i	− 0.0822464i	−0.999154	−	0.0411232i	$-0.986906\pi$
$233$	− 1.25544i	− 0.0822464i	0.999154	−	0.0411232i	$-0.0130936\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 35.8614i	− 2.32945i
$238$	0	0
$239$	13.6277	0.881504	0.440752	−	0.897629i	$-0.354712\pi$
$239$	13.6277	0.881504	0.440752	+	0.897629i	$0.354712\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	0	0
$243$	21.0217i	1.34855i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 20.7446i	− 1.31994i
$248$	0	0
$249$	22.5109	1.42657
$250$	0	0
$251$	−4.74456	−0.299474	−0.149737	−	0.988726i	$-0.547843\pi$
$251$	−4.74456	−0.299474	−0.149737	+	0.988726i	$0.547843\pi$
$252$	0	0
$253$	− 30.2337i	− 1.90078i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	23.4891i	1.46521i	0.680653	+	0.732606i	$0.261696\pi$
$257$	23.4891i	1.46521i	−0.680653	+	0.732606i	$0.738304\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	−11.4891	−0.711159
$262$	0	0
$263$	− 22.2337i	− 1.37099i	−0.728078	−	0.685494i	$-0.759586\pi$
$263$	− 22.2337i	− 1.37099i	0.728078	−	0.685494i	$-0.240414\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	34.9783i	2.14063i
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−9.48913	−0.576423	−0.288212	−	0.957567i	$-0.593061\pi$
$271$	−9.48913	−0.576423	−0.288212	+	0.957567i	$0.593061\pi$
$272$	0	0
$273$	− 10.3723i	− 0.627759i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 24.9783i	− 1.50080i	−0.660985	−	0.750399i	$-0.729861\pi$
$277$	− 24.9783i	− 1.50080i	0.660985	−	0.750399i	$-0.270139\pi$
$278$	0	0
$279$	−21.0217	−1.25854
$280$	0	0
$281$	18.6060	1.10994	0.554970	−	0.831871i	$-0.312730\pi$
$281$	18.6060	1.10994	0.554970	+	0.831871i	$0.312730\pi$
$282$	0	0
$283$	8.88316i	0.528049i	0.964516	+	0.264024i	$0.0850500\pi$
$283$	8.88316i	0.528049i	−0.964516	+	0.264024i	$0.914950\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 6.74456i	− 0.398119i
$288$	0	0
$289$	16.8614	0.991847
$290$	0	0
$291$	23.3940	1.37138
$292$	0	0
$293$	25.1168i	1.46734i	0.679505	+	0.733671i	$0.262195\pi$
$293$	25.1168i	1.46734i	−0.679505	+	0.733671i	$0.737805\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.62772i	0.326553i
$298$	0	0
$299$	−20.7446	−1.19969
$300$	0	0
$301$	8.74456	0.504028
$302$	0	0
$303$	14.2337i	0.817704i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	31.1168i	1.77593i	0.459909	+	0.887966i	$0.347882\pi$
$307$	31.1168i	1.77593i	−0.459909	+	0.887966i	$0.652118\pi$
$308$	0	0
$309$	13.3505	0.759485
$310$	0	0
$311$	12.7446	0.722678	0.361339	−	0.932435i	$-0.382320\pi$
$311$	12.7446	0.722678	0.361339	+	0.932435i	$0.382320\pi$
$312$	0	0
$313$	2.88316i	0.162966i	0.996675	+	0.0814828i	$0.0259656\pi$
$313$	2.88316i	0.162966i	−0.996675	+	0.0814828i	$0.974034\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 14.0000i	− 0.786318i	−0.919470	−	0.393159i	$-0.871382\pi$
$317$	− 14.0000i	− 0.786318i	0.919470	−	0.393159i	$-0.128618\pi$
$318$	0	0
$319$	−27.8614	−1.55994
$320$	0	0
$321$	20.7446	1.15785
$322$	0	0
$323$	− 1.76631i	− 0.0982802i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.883156i	0.0488386i
$328$	0	0
$329$	−7.11684	−0.392364
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	0	0
$333$	− 5.25544i	− 0.287996i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7.48913i	0.407959i	0.978975	+	0.203979i	$0.0653875\pi$
$337$	7.48913i	0.407959i	−0.978975	+	0.203979i	$0.934612\pi$
$338$	0	0
$339$	4.74456	0.257689
$340$	0	0
$341$	−50.9783	−2.76063
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 24.7446i	− 1.32836i	−0.747574	−	0.664179i	$-0.768781\pi$
$347$	− 24.7446i	− 1.32836i	0.747574	−	0.664179i	$-0.231219\pi$
$348$	0	0
$349$	−19.4891	−1.04323	−0.521614	−	0.853181i	$-0.674670\pi$
$349$	−19.4891	−1.04323	−0.521614	+	0.853181i	$0.674670\pi$
$350$	0	0
$351$	3.86141	0.206107
$352$	0	0
$353$	− 1.86141i	− 0.0990727i	−0.998772	−	0.0495363i	$-0.984226\pi$
$353$	− 1.86141i	− 0.0990727i	0.998772	−	0.0495363i	$-0.0157744\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 0.883156i	− 0.0467416i
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	3.51087	0.184783
$362$	0	0
$363$	− 70.2337i	− 3.68631i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	19.8614i	1.03676i	0.855151	+	0.518378i	$0.173464\pi$
$367$	19.8614i	1.03676i	−0.855151	+	0.518378i	$0.826536\pi$
$368$	0	0
$369$	−17.7228	−0.922613
$370$	0	0
$371$	10.7446	0.557830
$372$	0	0
$373$	− 30.7446i	− 1.59189i	−0.605367	−	0.795947i	$-0.706974\pi$
$373$	− 30.7446i	− 1.59189i	0.605367	−	0.795947i	$-0.293026\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	19.1168i	0.984568i
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	0	0
$381$	18.9783	0.972285
$382$	0	0
$383$	17.4891i	0.893653i	0.894621	+	0.446826i	$0.147446\pi$
$383$	17.4891i	0.893653i	−0.894621	+	0.446826i	$0.852554\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 22.9783i	− 1.16805i
$388$	0	0
$389$	−17.8614	−0.905609	−0.452805	−	0.891610i	$-0.649576\pi$
$389$	−17.8614	−0.905609	−0.452805	+	0.891610i	$0.649576\pi$
$390$	0	0
$391$	−1.76631	−0.0893262
$392$	0	0
$393$	− 11.2554i	− 0.567762i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 31.6277i	− 1.58735i	−0.608342	−	0.793675i	$-0.708165\pi$
$397$	− 31.6277i	− 1.58735i	0.608342	−	0.793675i	$-0.291835\pi$
$398$	0	0
$399$	11.2554	0.563477
$400$	0	0
$401$	−38.6060	−1.92789	−0.963945	−	0.266101i	$-0.914264\pi$
$401$	−38.6060	−1.92789	−0.963945	+	0.266101i	$0.914264\pi$
$402$	0	0
$403$	34.9783i	1.74239i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 12.7446i	− 0.631725i
$408$	0	0
$409$	−11.4891	−0.568101	−0.284050	−	0.958809i	$-0.591678\pi$
$409$	−11.4891	−0.568101	−0.284050	+	0.958809i	$0.591678\pi$
$410$	0	0
$411$	−34.9783	−1.72535
$412$	0	0
$413$	− 8.00000i	− 0.393654i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11.2554i	0.551181i
$418$	0	0
$419$	−14.5109	−0.708903	−0.354451	−	0.935074i	$-0.615332\pi$
$419$	−14.5109	−0.708903	−0.354451	+	0.935074i	$0.615332\pi$
$420$	0	0
$421$	−18.6060	−0.906799	−0.453400	−	0.891307i	$-0.649789\pi$
$421$	−18.6060	−0.906799	−0.453400	+	0.891307i	$0.649789\pi$
$422$	0	0
$423$	18.7011i	0.909277i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.74456i	0.132819i
$428$	0	0
$429$	−66.0951	−3.19110
$430$	0	0
$431$	−18.3723	−0.884962	−0.442481	−	0.896778i	$-0.645902\pi$
$431$	−18.3723	−0.884962	−0.442481	+	0.896778i	$0.645902\pi$
$432$	0	0
$433$	28.9783i	1.39261i	0.717748	+	0.696303i	$0.245173\pi$
$433$	28.9783i	1.39261i	−0.717748	+	0.696303i	$0.754827\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 22.5109i	− 1.07684i
$438$	0	0
$439$	−6.23369	−0.297518	−0.148759	−	0.988874i	$-0.547528\pi$
$439$	−6.23369	−0.297518	−0.148759	+	0.988874i	$0.547528\pi$
$440$	0	0
$441$	2.62772	0.125129
$442$	0	0
$443$	16.7446i	0.795558i	0.917481	+	0.397779i	$0.130219\pi$
$443$	16.7446i	0.795558i	−0.917481	+	0.397779i	$0.869781\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	36.7446i	1.73796i
$448$	0	0
$449$	−17.1168	−0.807794	−0.403897	−	0.914805i	$-0.632345\pi$
$449$	−17.1168	−0.807794	−0.403897	+	0.914805i	$0.632345\pi$
$450$	0	0
$451$	−42.9783	−2.02377
$452$	0	0
$453$	35.8614i	1.68492i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 5.25544i	− 0.245839i	−0.992417	−	0.122919i	$-0.960774\pi$
$457$	− 5.25544i	− 0.245839i	0.992417	−	0.122919i	$-0.0392257\pi$
$458$	0	0
$459$	0.328782	0.0153463
$460$	0	0
$461$	−1.25544	−0.0584715	−0.0292358	−	0.999573i	$-0.509307\pi$
$461$	−1.25544	−0.0584715	−0.0292358	+	0.999573i	$0.509307\pi$
$462$	0	0
$463$	6.51087i	0.302586i	0.988489	+	0.151293i	$0.0483437\pi$
$463$	6.51087i	0.302586i	−0.988489	+	0.151293i	$0.951656\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.60597i	0.398237i	0.979975	+	0.199118i	$0.0638078\pi$
$467$	8.60597i	0.398237i	−0.979975	+	0.199118i	$0.936192\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	36.7446	1.69310
$472$	0	0
$473$	− 55.7228i	− 2.56214i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 28.2337i	− 1.29273i
$478$	0	0
$479$	22.2337	1.01588	0.507942	−	0.861392i	$-0.330407\pi$
$479$	22.2337	1.01588	0.507942	+	0.861392i	$0.330407\pi$
$480$	0	0
$481$	−8.74456	−0.398718
$482$	0	0
$483$	− 11.2554i	− 0.512140i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	30.2337i	1.37002i	0.728534	+	0.685010i	$0.240202\pi$
$487$	30.2337i	1.37002i	−0.728534	+	0.685010i	$0.759798\pi$
$488$	0	0
$489$	39.7228	1.79633
$490$	0	0
$491$	−35.1168	−1.58480	−0.792400	−	0.610001i	$-0.791169\pi$
$491$	−35.1168	−1.58480	−0.792400	+	0.610001i	$0.791169\pi$
$492$	0	0
$493$	1.62772i	0.0733088i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.00000i	0.358849i
$498$	0	0
$499$	31.8614	1.42631	0.713156	−	0.701005i	$-0.247265\pi$
$499$	31.8614	1.42631	0.713156	+	0.701005i	$0.247265\pi$
$500$	0	0
$501$	−13.3505	−0.596458
$502$	0	0
$503$	18.3723i	0.819180i	0.912270	+	0.409590i	$0.134328\pi$
$503$	18.3723i	0.819180i	−0.912270	+	0.409590i	$0.865672\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	14.5109i	0.644451i
$508$	0	0
$509$	40.9783	1.81633	0.908165	−	0.418613i	$-0.137484\pi$
$509$	40.9783	1.81633	0.908165	+	0.418613i	$0.137484\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	4.19019i	0.185001i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	45.3505i	1.99451i
$518$	0	0
$519$	−0.883156	−0.0387662
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	36.4674i	1.59461i	0.603579	+	0.797304i	$0.293741\pi$
$523$	36.4674i	1.59461i	−0.603579	+	0.797304i	$0.706259\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.97825i	0.129735i
$528$	0	0
$529$	0.489125	0.0212663
$530$	0	0
$531$	−21.0217	−0.912266
$532$	0	0
$533$	29.4891i	1.27732i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	54.5109i	2.35232i
$538$	0	0
$539$	6.37228	0.274474
$540$	0	0
$541$	−31.3505	−1.34786	−0.673932	−	0.738793i	$-0.735396\pi$
$541$	−31.3505	−1.34786	−0.673932	+	0.738793i	$0.735396\pi$
$542$	0	0
$543$	− 38.5109i	− 1.65266i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 30.9783i	− 1.32453i	−0.749268	−	0.662267i	$-0.769594\pi$
$547$	− 30.9783i	− 1.32453i	0.749268	−	0.662267i	$-0.230406\pi$
$548$	0	0
$549$	7.21194	0.307798
$550$	0	0
$551$	−20.7446	−0.883748
$552$	0	0
$553$	− 15.1168i	− 0.642834i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.76631i	0.159584i	0.996812	+	0.0797919i	$0.0254256\pi$
$557$	3.76631i	0.159584i	−0.996812	+	0.0797919i	$0.974574\pi$
$558$	0	0
$559$	−38.2337	−1.61711
$560$	0	0
$561$	−5.62772	−0.237602
$562$	0	0
$563$	− 17.4891i	− 0.737079i	−0.929612	−	0.368539i	$-0.879858\pi$
$563$	− 17.4891i	− 0.737079i	0.929612	−	0.368539i	$-0.120142\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	9.97825i	0.419047i
$568$	0	0
$569$	24.9783	1.04714	0.523571	−	0.851982i	$-0.324599\pi$
$569$	24.9783	1.04714	0.523571	+	0.851982i	$0.324599\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	9.16034i	0.382679i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	22.6060i	0.941099i	0.882374	+	0.470549i	$0.155944\pi$
$577$	22.6060i	0.941099i	−0.882374	+	0.470549i	$0.844056\pi$
$578$	0	0
$579$	16.0000	0.664937
$580$	0	0
$581$	9.48913	0.393675
$582$	0	0
$583$	− 68.4674i	− 2.83563i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 34.9783i	− 1.44371i	−0.692046	−	0.721853i	$-0.743290\pi$
$587$	− 34.9783i	− 1.44371i	0.692046	−	0.721853i	$-0.256710\pi$
$588$	0	0
$589$	−37.9565	−1.56397
$590$	0	0
$591$	19.5326	0.803465
$592$	0	0
$593$	− 19.6277i	− 0.806014i	−0.915197	−	0.403007i	$-0.867965\pi$
$593$	− 19.6277i	− 0.806014i	0.915197	−	0.403007i	$-0.132035\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	37.9565i	1.55346i
$598$	0	0
$599$	−32.6060	−1.33224	−0.666122	−	0.745843i	$-0.732047\pi$
$599$	−32.6060	−1.33224	−0.666122	+	0.745843i	$0.732047\pi$
$600$	0	0
$601$	16.5109	0.673493	0.336746	−	0.941595i	$-0.390674\pi$
$601$	16.5109	0.673493	0.336746	+	0.941595i	$0.390674\pi$
$602$	0	0
$603$	10.5109i	0.428036i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	24.6060i	0.998725i	0.866393	+	0.499363i	$0.166432\pi$
$607$	24.6060i	0.998725i	−0.866393	+	0.499363i	$0.833568\pi$
$608$	0	0
$609$	−10.3723	−0.420306
$610$	0	0
$611$	31.1168	1.25885
$612$	0	0
$613$	− 8.51087i	− 0.343751i	−0.985119	−	0.171875i	$-0.945017\pi$
$613$	− 8.51087i	− 0.343751i	0.985119	−	0.171875i	$-0.0549827\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	0	0
$619$	−12.7446	−0.512247	−0.256124	−	0.966644i	$-0.582445\pi$
$619$	−12.7446	−0.512247	−0.256124	+	0.966644i	$0.582445\pi$
$620$	0	0
$621$	4.19019	0.168146
$622$	0	0
$623$	14.7446i	0.590728i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 71.7228i	− 2.86433i
$628$	0	0
$629$	−0.744563	−0.0296877
$630$	0	0
$631$	−2.37228	−0.0944390	−0.0472195	−	0.998885i	$-0.515036\pi$
$631$	−2.37228	−0.0944390	−0.0472195	+	0.998885i	$0.515036\pi$
$632$	0	0
$633$	− 34.0951i	− 1.35516i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 4.37228i	− 0.173236i
$638$	0	0
$639$	21.0217	0.831608
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	− 18.3723i	− 0.724532i	−0.932075	−	0.362266i	$-0.882003\pi$
$643$	− 18.3723i	− 0.724532i	0.932075	−	0.362266i	$-0.117997\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 16.0000i	− 0.629025i	−0.949253	−	0.314512i	$-0.898159\pi$
$647$	− 16.0000i	− 0.629025i	0.949253	−	0.314512i	$-0.101841\pi$
$648$	0	0
$649$	−50.9783	−2.00107
$650$	0	0
$651$	−18.9783	−0.743816
$652$	0	0
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	15.7663i	0.615102i
$658$	0	0
$659$	−11.1168	−0.433051	−0.216525	−	0.976277i	$-0.569472\pi$
$659$	−11.1168	−0.433051	−0.216525	+	0.976277i	$0.569472\pi$
$660$	0	0
$661$	14.7446	0.573497	0.286749	−	0.958006i	$-0.407426\pi$
$661$	14.7446	0.573497	0.286749	+	0.958006i	$0.407426\pi$
$662$	0	0
$663$	3.86141i	0.149965i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20.7446i	0.803233i
$668$	0	0
$669$	13.3505	0.516161
$670$	0	0
$671$	17.4891	0.675160
$672$	0	0
$673$	25.7228i	0.991542i	0.868453	+	0.495771i	$0.165114\pi$
$673$	25.7228i	0.991542i	−0.868453	+	0.495771i	$0.834886\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 15.3505i	− 0.589969i	−0.955502	−	0.294984i	$-0.904686\pi$
$677$	− 15.3505i	− 0.589969i	0.955502	−	0.294984i	$-0.0953145\pi$
$678$	0	0
$679$	9.86141	0.378446
$680$	0	0
$681$	47.1168	1.80552
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 29.0217i	− 1.10725i
$688$	0	0
$689$	−46.9783	−1.78973
$690$	0	0
$691$	9.48913	0.360983	0.180492	−	0.983577i	$-0.442231\pi$
$691$	9.48913	0.360983	0.180492	+	0.983577i	$0.442231\pi$
$692$	0	0
$693$	− 16.7446i	− 0.636073i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.51087i	0.0951062i
$698$	0	0
$699$	−2.97825	−0.112648
$700$	0	0
$701$	2.13859	0.0807736	0.0403868	−	0.999184i	$-0.487141\pi$
$701$	2.13859	0.0807736	0.0403868	+	0.999184i	$0.487141\pi$
$702$	0	0
$703$	− 9.48913i	− 0.357889i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000i	0.225653i
$708$	0	0
$709$	−6.60597	−0.248092	−0.124046	−	0.992276i	$-0.539587\pi$
$709$	−6.60597	−0.248092	−0.124046	+	0.992276i	$0.539587\pi$
$710$	0	0
$711$	−39.7228	−1.48972
$712$	0	0
$713$	37.9565i	1.42148i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 32.3288i	− 1.20734i
$718$	0	0
$719$	3.25544	0.121407	0.0607037	−	0.998156i	$-0.480666\pi$
$719$	3.25544	0.121407	0.0607037	+	0.998156i	$0.480666\pi$
$720$	0	0
$721$	5.62772	0.209587
$722$	0	0
$723$	− 61.6793i	− 2.29388i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	32.0000i	1.18681i	0.804902	+	0.593407i	$0.202218\pi$
$727$	32.0000i	1.18681i	−0.804902	+	0.593407i	$0.797782\pi$
$728$	0	0
$729$	19.9348	0.738324
$730$	0	0
$731$	−3.25544	−0.120407
$732$	0	0
$733$	− 10.1386i	− 0.374477i	−0.982314	−	0.187239i	$-0.940046\pi$
$733$	− 10.1386i	− 0.374477i	0.982314	−	0.187239i	$-0.0599538\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.4891i	0.938904i
$738$	0	0
$739$	−20.6060	−0.758003	−0.379001	−	0.925396i	$-0.623732\pi$
$739$	−20.6060	−0.758003	−0.379001	+	0.925396i	$0.623732\pi$
$740$	0	0
$741$	−49.2119	−1.80785
$742$	0	0
$743$	6.51087i	0.238861i	0.992843	+	0.119430i	$0.0381068\pi$
$743$	6.51087i	0.238861i	−0.992843	+	0.119430i	$0.961893\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 24.9348i	− 0.912315i
$748$	0	0
$749$	8.74456	0.319519
$750$	0	0
$751$	−20.1386	−0.734868	−0.367434	−	0.930050i	$-0.619764\pi$
$751$	−20.1386	−0.734868	−0.367434	+	0.930050i	$0.619764\pi$
$752$	0	0
$753$	11.2554i	0.410171i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.76631i	0.136889i	0.997655	+	0.0684445i	$0.0218036\pi$
$757$	3.76631i	0.136889i	−0.997655	+	0.0684445i	$0.978196\pi$
$758$	0	0
$759$	−71.7228	−2.60337
$760$	0	0
$761$	4.97825	0.180461	0.0902307	−	0.995921i	$-0.471240\pi$
$761$	4.97825	0.180461	0.0902307	+	0.995921i	$0.471240\pi$
$762$	0	0
$763$	0.372281i	0.0134775i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	34.9783i	1.26299i
$768$	0	0
$769$	−3.48913	−0.125821	−0.0629105	−	0.998019i	$-0.520038\pi$
$769$	−3.48913	−0.125821	−0.0629105	+	0.998019i	$0.520038\pi$
$770$	0	0
$771$	55.7228	2.00681
$772$	0	0
$773$	4.37228i	0.157260i	0.996904	+	0.0786300i	$0.0250546\pi$
$773$	4.37228i	0.157260i	−0.996904	+	0.0786300i	$0.974945\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 4.74456i	− 0.170210i
$778$	0	0
$779$	−32.0000	−1.14652
$780$	0	0
$781$	50.9783	1.82415
$782$	0	0
$783$	− 3.86141i	− 0.137995i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 31.1168i	− 1.10920i	−0.832119	−	0.554598i	$-0.812872\pi$
$787$	− 31.1168i	− 1.10920i	0.832119	−	0.554598i	$-0.187128\pi$
$788$	0	0
$789$	−52.7446	−1.87776
$790$	0	0
$791$	2.00000	0.0711118
$792$	0	0
$793$	− 12.0000i	− 0.426132i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 15.6277i	− 0.553562i	−0.960933	−	0.276781i	$-0.910732\pi$
$797$	− 15.6277i	− 0.553562i	0.960933	−	0.276781i	$-0.0892677\pi$
$798$	0	0
$799$	2.64947	0.0937314
$800$	0	0
$801$	38.7446	1.36897
$802$	0	0
$803$	38.2337i	1.34924i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 14.2337i	− 0.501050i
$808$	0	0
$809$	−20.3723	−0.716251	−0.358126	−	0.933673i	$-0.616584\pi$
$809$	−20.3723	−0.716251	−0.358126	+	0.933673i	$0.616584\pi$
$810$	0	0
$811$	−12.7446	−0.447522	−0.223761	−	0.974644i	$-0.571834\pi$
$811$	−12.7446	−0.447522	−0.223761	+	0.974644i	$0.571834\pi$
$812$	0	0
$813$	22.5109i	0.789491i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 41.4891i	− 1.45152i
$818$	0	0
$819$	−11.4891	−0.401463
$820$	0	0
$821$	−25.1168	−0.876584	−0.438292	−	0.898833i	$-0.644416\pi$
$821$	−25.1168	−0.876584	−0.438292	+	0.898833i	$0.644416\pi$
$822$	0	0
$823$	8.00000i	0.278862i	0.990232	+	0.139431i	$0.0445274\pi$
$823$	8.00000i	0.278862i	−0.990232	+	0.139431i	$0.955473\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24.7446i	0.860453i	0.902721	+	0.430226i	$0.141566\pi$
$827$	24.7446i	0.860453i	−0.902721	+	0.430226i	$0.858434\pi$
$828$	0	0
$829$	12.2337	0.424894	0.212447	−	0.977173i	$-0.431857\pi$
$829$	12.2337	0.424894	0.212447	+	0.977173i	$0.431857\pi$
$830$	0	0
$831$	−59.2554	−2.05555
$832$	0	0
$833$	− 0.372281i	− 0.0128988i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 7.06525i	− 0.244211i
$838$	0	0
$839$	−11.2554	−0.388581	−0.194290	−	0.980944i	$-0.562240\pi$
$839$	−11.2554	−0.388581	−0.194290	+	0.980944i	$0.562240\pi$
$840$	0	0
$841$	−9.88316	−0.340798
$842$	0	0
$843$	− 44.1386i	− 1.52021i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 29.6060i	− 1.01727i
$848$	0	0
$849$	21.0733	0.723235
$850$	0	0
$851$	−9.48913	−0.325283
$852$	0	0
$853$	− 39.4891i	− 1.35208i	−0.736864	−	0.676041i	$-0.763694\pi$
$853$	− 39.4891i	− 1.35208i	0.736864	−	0.676041i	$-0.236306\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	0	0
$859$	44.4674	1.51721	0.758604	−	0.651552i	$-0.225882\pi$
$859$	44.4674	1.51721	0.758604	+	0.651552i	$0.225882\pi$
$860$	0	0
$861$	−16.0000	−0.545279
$862$	0	0
$863$	− 25.4891i	− 0.867660i	−0.900995	−	0.433830i	$-0.857162\pi$
$863$	− 25.4891i	− 0.867660i	0.900995	−	0.433830i	$-0.142838\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 40.0000i	− 1.35847i
$868$	0	0
$869$	−96.3288	−3.26773
$870$	0	0
$871$	17.4891	0.592596
$872$	0	0
$873$	− 25.9130i	− 0.877022i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.02175i	0.237108i	0.992948	+	0.118554i	$0.0378258\pi$
$877$	7.02175i	0.237108i	−0.992948	+	0.118554i	$0.962174\pi$
$878$	0	0
$879$	59.5842	2.00973
$880$	0	0
$881$	−25.2554	−0.850877	−0.425439	−	0.904987i	$-0.639880\pi$
$881$	−25.2554	−0.850877	−0.425439	+	0.904987i	$0.639880\pi$
$882$	0	0
$883$	10.5109i	0.353719i	0.984236	+	0.176860i	$0.0565938\pi$
$883$	10.5109i	0.353719i	−0.984236	+	0.176860i	$0.943406\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 13.0217i	− 0.437228i	−0.975811	−	0.218614i	$-0.929847\pi$
$887$	− 13.0217i	− 0.437228i	0.975811	−	0.218614i	$-0.0701535\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	63.5842	2.13015
$892$	0	0
$893$	33.7663i	1.12995i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	49.2119i	1.64314i
$898$	0	0
$899$	34.9783	1.16659
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	− 20.7446i	− 0.690336i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	11.7228i	0.389250i	0.980878	+	0.194625i	$0.0623489\pi$
$907$	11.7228i	0.389250i	−0.980878	+	0.194625i	$0.937651\pi$
$908$	0	0
$909$	15.7663	0.522936
$910$	0	0
$911$	45.9565	1.52261	0.761303	−	0.648396i	$-0.224560\pi$
$911$	45.9565	1.52261	0.761303	+	0.648396i	$0.224560\pi$
$912$	0	0
$913$	− 60.4674i	− 2.00118i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4.74456i	− 0.156679i
$918$	0	0
$919$	13.6277	0.449537	0.224768	−	0.974412i	$-0.427837\pi$
$919$	13.6277	0.449537	0.224768	+	0.974412i	$0.427837\pi$
$920$	0	0
$921$	73.8179	2.43238
$922$	0	0
$923$	− 34.9783i	− 1.15132i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 14.7881i	− 0.485704i
$928$	0	0
$929$	7.76631	0.254804	0.127402	−	0.991851i	$-0.459336\pi$
$929$	7.76631	0.254804	0.127402	+	0.991851i	$0.459336\pi$
$930$	0	0
$931$	4.74456	0.155497
$932$	0	0
$933$	− 30.2337i	− 0.989807i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 28.0951i	− 0.917827i	−0.888481	−	0.458913i	$-0.848239\pi$
$937$	− 28.0951i	− 0.917827i	0.888481	−	0.458913i	$-0.151761\pi$
$938$	0	0
$939$	6.83966	0.223204
$940$	0	0
$941$	32.2337	1.05079	0.525394	−	0.850859i	$-0.323918\pi$
$941$	32.2337	1.05079	0.525394	+	0.850859i	$0.323918\pi$
$942$	0	0
$943$	32.0000i	1.04206i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 28.0000i	− 0.909878i	−0.890523	−	0.454939i	$-0.849661\pi$
$947$	− 28.0000i	− 0.909878i	0.890523	−	0.454939i	$-0.150339\pi$
$948$	0	0
$949$	26.2337	0.851582
$950$	0	0
$951$	−33.2119	−1.07697
$952$	0	0
$953$	37.2554i	1.20682i	0.797430	+	0.603411i	$0.206192\pi$
$953$	37.2554i	1.20682i	−0.797430	+	0.603411i	$0.793808\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	66.0951i	2.13655i
$958$	0	0
$959$	−14.7446	−0.476127
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	− 22.9783i	− 0.740464i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.76631i	0.0568008i	0.999597	+	0.0284004i	$0.00904134\pi$
$967$	1.76631i	0.0568008i	−0.999597	+	0.0284004i	$0.990959\pi$
$968$	0	0
$969$	−4.19019	−0.134608
$970$	0	0
$971$	−33.4891	−1.07472	−0.537359	−	0.843354i	$-0.680578\pi$
$971$	−33.4891	−1.07472	−0.537359	+	0.843354i	$0.680578\pi$
$972$	0	0
$973$	4.74456i	0.152104i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 52.9783i	− 1.69492i	−0.530856	−	0.847462i	$-0.678129\pi$
$977$	− 52.9783i	− 1.69492i	0.530856	−	0.847462i	$-0.321871\pi$
$978$	0	0
$979$	93.9565	3.00286
$980$	0	0
$981$	0.978251	0.0312331
$982$	0	0
$983$	10.3723i	0.330824i	0.986225	+	0.165412i	$0.0528954\pi$
$983$	10.3723i	0.330824i	−0.986225	+	0.165412i	$0.947105\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	16.8832i	0.537397i
$988$	0	0
$989$	−41.4891	−1.31928
$990$	0	0
$991$	37.9565	1.20573	0.602864	−	0.797844i	$-0.294026\pi$
$991$	37.9565	1.20573	0.602864	+	0.797844i	$0.294026\pi$
$992$	0	0
$993$	28.4674i	0.903385i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	6.88316i	0.217992i	0.994042	+	0.108996i	$0.0347635\pi$
$997$	6.88316i	0.217992i	−0.994042	+	0.108996i	$0.965236\pi$
$998$	0	0
$999$	1.76631	0.0558836

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.g.r.449.2		4
4.3	odd	2		1400.2.g.i.449.3		4
5.2	odd	4		560.2.a.h.1.1		2
5.3	odd	4		2800.2.a.bk.1.2		2
5.4	even	2	inner	2800.2.g.r.449.3		4
15.2	even	4		5040.2.a.by.1.2		2
20.3	even	4		1400.2.a.r.1.1		2
20.7	even	4		280.2.a.c.1.2	✓	2
20.19	odd	2		1400.2.g.i.449.2		4
35.27	even	4		3920.2.a.bt.1.2		2
40.27	even	4		2240.2.a.bk.1.1		2
40.37	odd	4		2240.2.a.bg.1.2		2
60.47	odd	4		2520.2.a.x.1.1		2
140.27	odd	4		1960.2.a.s.1.1		2
140.47	odd	12		1960.2.q.r.361.2		4
140.67	even	12		1960.2.q.t.961.1		4
140.83	odd	4		9800.2.a.bu.1.2		2
140.87	odd	12		1960.2.q.r.961.2		4
140.107	even	12		1960.2.q.t.361.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.a.c.1.2	✓	2	20.7	even	4
560.2.a.h.1.1		2	5.2	odd	4
1400.2.a.r.1.1		2	20.3	even	4
1400.2.g.i.449.2		4	20.19	odd	2
1400.2.g.i.449.3		4	4.3	odd	2
1960.2.a.s.1.1		2	140.27	odd	4
1960.2.q.r.361.2		4	140.47	odd	12
1960.2.q.r.961.2		4	140.87	odd	12
1960.2.q.t.361.1		4	140.107	even	12
1960.2.q.t.961.1		4	140.67	even	12
2240.2.a.bg.1.2		2	40.37	odd	4
2240.2.a.bk.1.1		2	40.27	even	4
2520.2.a.x.1.1		2	60.47	odd	4
2800.2.a.bk.1.2		2	5.3	odd	4
2800.2.g.r.449.2		4	1.1	even	1	trivial
2800.2.g.r.449.3		4	5.4	even	2	inner
3920.2.a.bt.1.2		2	35.27	even	4
5040.2.a.by.1.2		2	15.2	even	4
9800.2.a.bu.1.2		2	140.83	odd	4

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.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.37228i q^{3} -1.00000i q^{7} -2.62772 q^{9} +O(q^{10})\)	\(q-2.37228i q^{3} -1.00000i q^{7} -2.62772 q^{9} -6.37228 q^{11} +4.37228i q^{13} +0.372281i q^{17} -4.74456 q^{19} -2.37228 q^{21} +4.74456i q^{23} -0.883156i q^{27} +4.37228 q^{29} +8.00000 q^{31} +15.1168i q^{33} +2.00000i q^{37} +10.3723 q^{39} +6.74456 q^{41} +8.74456i q^{43} -7.11684i q^{47} -1.00000 q^{49} +0.883156 q^{51} +10.7446i q^{53} +11.2554i q^{57} +8.00000 q^{59} -2.74456 q^{61} +2.62772i q^{63} -4.00000i q^{67} +11.2554 q^{69} -8.00000 q^{71} -6.00000i q^{73} +6.37228i q^{77} +15.1168 q^{79} -9.97825 q^{81} +9.48913i q^{83} -10.3723i q^{87} -14.7446 q^{89} +4.37228 q^{91} -18.9783i q^{93} +9.86141i q^{97} +16.7446 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 22 q^{9}+O(q^{10}) \) 4 * q - 22 * q^9	\( 4 q - 22 q^{9} - 14 q^{11} + 4 q^{19} + 2 q^{21} + 6 q^{29} + 32 q^{31} + 30 q^{39} + 4 q^{41} - 4 q^{49} + 38 q^{51} + 32 q^{59} + 12 q^{61} + 68 q^{69} - 32 q^{71} + 26 q^{79} + 52 q^{81} - 36 q^{89} + 6 q^{91} + 44 q^{99}+O(q^{100}) \) 4 * q - 22 * q^9 - 14 * q^11 + 4 * q^19 + 2 * q^21 + 6 * q^29 + 32 * q^31 + 30 * q^39 + 4 * q^41 - 4 * q^49 + 38 * q^51 + 32 * q^59 + 12 * q^61 + 68 * q^69 - 32 * q^71 + 26 * q^79 + 52 * q^81 - 36 * q^89 + 6 * q^91 + 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more