LMFDB - Embedded newform 2800.2.e.i.2799.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(2799,2800)]

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([1, 0, 1, 1]))

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

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

chi := DirichletCharacter("2800.2799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.e (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:	$8$
Coefficient field:	8.0.303595776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 $ x^8 + 5x^6 + 16x^4 + 45*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2799.1
Root			$-1.26217 + 1.18614i$ of defining polynomial
Character	$\chi$	$=$	2800.2799
Dual form			2800.2.e.i.2799.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52434i q^{3} +(-0.792287 + 2.52434i) q^{7} -3.37228 q^{9} +O(q^{10})$	$q-2.52434i q^{3} +(-0.792287 + 2.52434i) q^{7} -3.37228 q^{9} +2.52434i q^{11} +0.372281 q^{13} -2.37228 q^{17} -3.46410 q^{19} +(6.37228 + 2.00000i) q^{21} +8.51278 q^{23} +0.939764i q^{27} -0.372281 q^{29} +1.87953 q^{31} +6.37228 q^{33} -10.7446i q^{37} -0.939764i q^{39} -8.74456i q^{41} +3.46410 q^{43} -7.86797i q^{47} +(-5.74456 - 4.00000i) q^{49} +5.98844i q^{51} -11.4891i q^{53} +8.74456i q^{57} -6.63325 q^{59} -10.7446i q^{61} +(2.67181 - 8.51278i) q^{63} +6.63325 q^{67} -21.4891i q^{69} +6.63325i q^{71} +8.74456 q^{73} +(-6.37228 - 2.00000i) q^{77} +15.7908i q^{79} -7.74456 q^{81} -10.3923i q^{83} +0.939764i q^{87} -5.48913i q^{89} +(-0.294954 + 0.939764i) q^{91} -4.74456i q^{93} +15.1168 q^{97} -8.51278i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^9	$ 8 q - 4 q^{9} - 20 q^{13} + 4 q^{17} + 28 q^{21} + 20 q^{29} + 28 q^{33} + 24 q^{73} - 28 q^{77} - 16 q^{81} + 52 q^{97}+O(q^{100}) $ 8 * q - 4 * q^9 - 20 * q^13 + 4 * q^17 + 28 * q^21 + 20 * q^29 + 28 * q^33 + 24 * q^73 - 28 * q^77 - 16 * q^81 + 52 * q^97

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
$3$		−	2.52434i		−	1.45743i	−0.684819	−	0.728714i	$-0.740119\pi$
$3$		−	2.52434i		−	1.45743i	0.684819	−	0.728714i	$-0.259881\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.792287	+	2.52434i	−0.299456	+	0.954110i
$7$	−0.792287	+	2.52434i	−0.299456	+	0.954110i
$8$		0			0
$9$	−3.37228			−1.12409
$10$		0			0
$11$			2.52434i			0.761116i	0.924757	+	0.380558i	$0.124268\pi$
$11$			2.52434i			0.761116i	−0.924757	+	0.380558i	$0.875732\pi$
$12$		0			0
$13$	0.372281			0.103252			0.0516261	−	0.998666i	$-0.483560\pi$
$13$	0.372281			0.103252			0.0516261	+	0.998666i	$0.483560\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.37228			−0.575363			−0.287681	−	0.957726i	$-0.592884\pi$
$17$	−2.37228			−0.575363			−0.287681	+	0.957726i	$0.592884\pi$
$18$		0			0
$19$	−3.46410			−0.794719			−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410			−0.794719			−0.397360	+	0.917663i	$0.630073\pi$
$20$		0			0
$21$	6.37228	+	2.00000i	1.39055	+	0.436436i
$22$		0			0
$23$	8.51278			1.77504			0.887518	−	0.460772i	$-0.152428\pi$
$23$	8.51278			1.77504			0.887518	+	0.460772i	$0.152428\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			0.939764i			0.180858i
$28$		0			0
$29$	−0.372281			−0.0691309			−0.0345655	−	0.999402i	$-0.511005\pi$
$29$	−0.372281			−0.0691309			−0.0345655	+	0.999402i	$0.511005\pi$
$30$		0			0
$31$	1.87953			0.337573			0.168787	−	0.985653i	$-0.446015\pi$
$31$	1.87953			0.337573			0.168787	+	0.985653i	$0.446015\pi$
$32$		0			0
$33$	6.37228			1.10927
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	10.7446i		−	1.76640i	−0.469001	−	0.883198i	$-0.655386\pi$
$37$		−	10.7446i		−	1.76640i	0.469001	−	0.883198i	$-0.344614\pi$
$38$		0			0
$39$		−	0.939764i		−	0.150483i
$40$		0			0
$41$		−	8.74456i		−	1.36567i	−0.730572	−	0.682836i	$-0.760747\pi$
$41$		−	8.74456i		−	1.36567i	0.730572	−	0.682836i	$-0.239253\pi$
$42$		0			0
$43$	3.46410			0.528271			0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410			0.528271			0.264135	+	0.964486i	$0.414913\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	7.86797i		−	1.14766i	−0.818974	−	0.573830i	$-0.805457\pi$
$47$		−	7.86797i		−	1.14766i	0.818974	−	0.573830i	$-0.194543\pi$
$48$		0			0
$49$	−5.74456	−	4.00000i	−0.820652	−	0.571429i
$50$		0			0
$51$			5.98844i			0.838549i
$52$		0			0
$53$		−	11.4891i		−	1.57815i	−0.614295	−	0.789076i	$-0.710560\pi$
$53$		−	11.4891i		−	1.57815i	0.614295	−	0.789076i	$-0.289440\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			8.74456i			1.15825i
$58$		0			0
$59$	−6.63325			−0.863576			−0.431788	−	0.901975i	$-0.642117\pi$
$59$	−6.63325			−0.863576			−0.431788	+	0.901975i	$0.642117\pi$
$60$		0			0
$61$		−	10.7446i		−	1.37570i	−0.725853	−	0.687850i	$-0.758555\pi$
$61$		−	10.7446i		−	1.37570i	0.725853	−	0.687850i	$-0.241445\pi$
$62$		0			0
$63$	2.67181	−	8.51278i	0.336617	−	1.07251i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.63325			0.810380			0.405190	−	0.914232i	$-0.367205\pi$
$67$	6.63325			0.810380			0.405190	+	0.914232i	$0.367205\pi$
$68$		0			0
$69$		−	21.4891i		−	2.58699i
$70$		0			0
$71$			6.63325i			0.787222i	0.919277	+	0.393611i	$0.128774\pi$
$71$			6.63325i			0.787222i	−0.919277	+	0.393611i	$0.871226\pi$
$72$		0			0
$73$	8.74456			1.02347			0.511737	−	0.859142i	$-0.329002\pi$
$73$	8.74456			1.02347			0.511737	+	0.859142i	$0.329002\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.37228	−	2.00000i	−0.726189	−	0.227921i
$78$		0			0
$79$			15.7908i			1.77661i	0.459256	+	0.888304i	$0.348116\pi$
$79$			15.7908i			1.77661i	−0.459256	+	0.888304i	$0.651884\pi$
$80$		0			0
$81$	−7.74456			−0.860507
$82$		0			0
$83$		−	10.3923i		−	1.14070i	−0.821401	−	0.570352i	$-0.806807\pi$
$83$		−	10.3923i		−	1.14070i	0.821401	−	0.570352i	$-0.193193\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			0.939764i			0.100753i
$88$		0			0
$89$		−	5.48913i		−	0.581846i	−0.956746	−	0.290923i	$-0.906038\pi$
$89$		−	5.48913i		−	0.581846i	0.956746	−	0.290923i	$-0.0939624\pi$
$90$		0			0
$91$	−0.294954	+	0.939764i	−0.0309195	+	0.0985140i
$92$		0			0
$93$		−	4.74456i		−	0.491988i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	15.1168			1.53488			0.767441	−	0.641119i	$-0.221530\pi$
$97$	15.1168			1.53488			0.767441	+	0.641119i	$0.221530\pi$
$98$		0			0
$99$		−	8.51278i		−	0.855566i
$100$		0			0
$101$			7.48913i			0.745196i	0.927993	+	0.372598i	$0.121533\pi$
$101$			7.48913i			0.745196i	−0.927993	+	0.372598i	$0.878467\pi$
$102$		0			0
$103$			2.81929i			0.277793i	0.990307	+	0.138897i	$0.0443555\pi$
$103$			2.81929i			0.277793i	−0.990307	+	0.138897i	$0.955644\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	16.7306			1.61741			0.808704	−	0.588216i	$-0.200169\pi$
$107$	16.7306			1.61741			0.808704	+	0.588216i	$0.200169\pi$
$108$		0			0
$109$	7.62772			0.730603			0.365301	−	0.930889i	$-0.380966\pi$
$109$	7.62772			0.730603			0.365301	+	0.930889i	$0.380966\pi$
$110$		0			0
$111$	−27.1229			−2.57439
$112$		0			0
$113$			1.25544i			0.118102i	0.998255	+	0.0590508i	$0.0188074\pi$
$113$			1.25544i			0.118102i	−0.998255	+	0.0590508i	$0.981193\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.25544			−0.116065
$118$		0			0
$119$	1.87953	−	5.98844i	0.172296	−	0.548959i
$120$		0			0
$121$	4.62772			0.420702
$122$		0			0
$123$	−22.0742			−1.99037
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−6.63325			−0.588606			−0.294303	−	0.955712i	$-0.595087\pi$
$127$	−6.63325			−0.588606			−0.294303	+	0.955712i	$0.595087\pi$
$128$		0			0
$129$		−	8.74456i		−	0.769916i
$130$		0			0
$131$	15.4410			1.34908			0.674542	−	0.738236i	$-0.264341\pi$
$131$	15.4410			1.34908			0.674542	+	0.738236i	$0.264341\pi$
$132$		0			0
$133$	2.74456	−	8.74456i	0.237984	−	0.758250i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			10.7446i			0.917970i	0.888444	+	0.458985i	$0.151787\pi$
$137$			10.7446i			0.917970i	−0.888444	+	0.458985i	$0.848213\pi$
$138$		0			0
$139$	−4.75372			−0.403205			−0.201603	−	0.979467i	$-0.564615\pi$
$139$	−4.75372			−0.403205			−0.201603	+	0.979467i	$0.564615\pi$
$140$		0			0
$141$	−19.8614			−1.67263
$142$		0			0
$143$			0.939764i			0.0785870i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−10.0974	+	14.5012i	−0.832815	+	1.19604i
$148$		0			0
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$		−	18.2603i		−	1.48600i	−0.669291	−	0.743000i	$-0.733402\pi$
$151$		−	18.2603i		−	1.48600i	0.669291	−	0.743000i	$-0.266598\pi$
$152$		0			0
$153$	8.00000			0.646762
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−23.4891			−1.87464			−0.937318	−	0.348475i	$-0.886700\pi$
$157$	−23.4891			−1.87464			−0.937318	+	0.348475i	$0.886700\pi$
$158$		0			0
$159$	−29.0024			−2.30004
$160$		0			0
$161$	−6.74456	+	21.4891i	−0.531546	+	1.69358i
$162$		0			0
$163$	−4.75372			−0.372340			−0.186170	−	0.982518i	$-0.559608\pi$
$163$	−4.75372			−0.372340			−0.186170	+	0.982518i	$0.559608\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	7.27806i		−	0.563193i	−0.959533	−	0.281597i	$-0.909136\pi$
$167$		−	7.27806i		−	0.563193i	0.959533	−	0.281597i	$-0.0908640\pi$
$168$		0			0
$169$	−12.8614			−0.989339
$170$		0			0
$171$	11.6819			0.893339
$172$		0			0
$173$	16.3723			1.24476			0.622381	−	0.782715i	$-0.286166\pi$
$173$	16.3723			1.24476			0.622381	+	0.782715i	$0.286166\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			16.7446i			1.25860i
$178$		0			0
$179$			13.5615i			1.01363i	0.862055	+	0.506815i	$0.169177\pi$
$179$			13.5615i			1.01363i	−0.862055	+	0.506815i	$0.830823\pi$
$180$		0			0
$181$		−	20.9783i		−	1.55930i	−0.626215	−	0.779651i	$-0.715397\pi$
$181$		−	20.9783i		−	1.55930i	0.626215	−	0.779651i	$-0.284603\pi$
$182$		0			0
$183$	−27.1229			−2.00498
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	5.98844i		−	0.437918i
$188$		0			0
$189$	−2.37228	−	0.744563i	−0.172558	−	0.0541590i
$190$		0			0
$191$		−	7.57301i		−	0.547964i	−0.961735	−	0.273982i	$-0.911659\pi$
$191$		−	7.57301i		−	0.547964i	0.961735	−	0.273982i	$-0.0883409\pi$
$192$		0			0
$193$			26.7446i			1.92512i	0.271076	+	0.962558i	$0.412620\pi$
$193$			26.7446i			1.92512i	−0.271076	+	0.962558i	$0.587380\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	6.00000i		−	0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$		−	6.00000i		−	0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$		0			0
$199$	1.28962			0.0914188			0.0457094	−	0.998955i	$-0.485445\pi$
$199$	1.28962			0.0914188			0.0457094	+	0.998955i	$0.485445\pi$
$200$		0			0
$201$		−	16.7446i		−	1.18107i
$202$		0			0
$203$	0.294954	−	0.939764i	0.0207017	−	0.0659585i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−28.7075			−1.99531
$208$		0			0
$209$		−	8.74456i		−	0.604874i
$210$		0			0
$211$		−	17.6704i		−	1.21648i	−0.793754	−	0.608239i	$-0.791876\pi$
$211$		−	17.6704i		−	1.21648i	0.793754	−	0.608239i	$-0.208124\pi$
$212$		0			0
$213$	16.7446			1.14732
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.48913	+	4.74456i	−0.101088	+	0.322082i
$218$		0			0
$219$		−	22.0742i		−	1.49164i
$220$		0			0
$221$	−0.883156			−0.0594075
$222$		0			0
$223$		−	4.10891i		−	0.275153i	−0.990491	−	0.137577i	$-0.956069\pi$
$223$		−	4.10891i		−	0.275153i	0.990491	−	0.137577i	$-0.0439313\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			20.8395i			1.38317i	0.722297	+	0.691584i	$0.243087\pi$
$227$			20.8395i			1.38317i	−0.722297	+	0.691584i	$0.756913\pi$
$228$		0			0
$229$		−	14.0000i		−	0.925146i	−0.886581	−	0.462573i	$-0.846926\pi$
$229$		−	14.0000i		−	0.925146i	0.886581	−	0.462573i	$-0.153074\pi$
$230$		0			0
$231$	−5.04868	+	16.0858i	−0.332178	+	1.05837i
$232$		0			0
$233$		−	10.7446i		−	0.703900i	−0.936019	−	0.351950i	$-0.885519\pi$
$233$		−	10.7446i		−	0.703900i	0.936019	−	0.351950i	$-0.114481\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	39.8614			2.58928
$238$		0			0
$239$			12.6217i			0.816429i	0.912886	+	0.408215i	$0.133848\pi$
$239$			12.6217i			0.816429i	−0.912886	+	0.408215i	$0.866152\pi$
$240$		0			0
$241$			16.0000i			1.03065i	0.856995	+	0.515325i	$0.172329\pi$
$241$			16.0000i			1.03065i	−0.856995	+	0.515325i	$0.827671\pi$
$242$		0			0
$243$			22.3692i			1.43498i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−1.28962			−0.0820566
$248$		0			0
$249$	−26.2337			−1.66249
$250$		0			0
$251$	−15.4410			−0.974626			−0.487313	−	0.873227i	$-0.662023\pi$
$251$	−15.4410			−0.974626			−0.487313	+	0.873227i	$0.662023\pi$
$252$		0			0
$253$			21.4891i			1.35101i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−12.7446			−0.794984			−0.397492	−	0.917606i	$-0.630119\pi$
$257$	−12.7446			−0.794984			−0.397492	+	0.917606i	$0.630119\pi$
$258$		0			0
$259$	27.1229	+	8.51278i	1.68534	+	0.528958i
$260$		0			0
$261$	1.25544			0.0777096
$262$		0			0
$263$	−14.1514			−0.872610			−0.436305	−	0.899799i	$-0.643713\pi$
$263$	−14.1514			−0.872610			−0.436305	+	0.899799i	$0.643713\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−13.8564			−0.847998
$268$		0			0
$269$			10.7446i			0.655108i	0.944833	+	0.327554i	$0.106224\pi$
$269$			10.7446i			0.655108i	−0.944833	+	0.327554i	$0.893776\pi$
$270$		0			0
$271$	−26.5330			−1.61176			−0.805882	−	0.592076i	$-0.798309\pi$
$271$	−26.5330			−1.61176			−0.805882	+	0.592076i	$0.798309\pi$
$272$		0			0
$273$	2.37228	+	0.744563i	0.143577	+	0.0450630i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	20.2337i		−	1.21572i	−0.794043	−	0.607862i	$-0.792027\pi$
$277$		−	20.2337i		−	1.21572i	0.794043	−	0.607862i	$-0.207973\pi$
$278$		0			0
$279$	−6.33830			−0.379464
$280$		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
$283$		−	5.69349i		−	0.338443i	−0.985578	−	0.169221i	$-0.945875\pi$
$283$		−	5.69349i		−	0.338443i	0.985578	−	0.169221i	$-0.0541253\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	22.0742	+	6.92820i	1.30300	+	0.408959i
$288$		0			0
$289$	−11.3723			−0.668958
$290$		0			0
$291$		−	38.1600i		−	2.23698i
$292$		0			0
$293$	−19.6277			−1.14666			−0.573332	−	0.819323i	$-0.694349\pi$
$293$	−19.6277			−1.14666			−0.573332	+	0.819323i	$0.694349\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−2.37228			−0.137654
$298$		0			0
$299$	3.16915			0.183277
$300$		0			0
$301$	−2.74456	+	8.74456i	−0.158194	+	0.504028i
$302$		0			0
$303$	18.9051			1.08607
$304$		0			0
$305$		0			0
$306$		0			0
$307$			0.644810i			0.0368013i	0.999831	+	0.0184006i	$0.00585743\pi$
$307$			0.644810i			0.0368013i	−0.999831	+	0.0184006i	$0.994143\pi$
$308$		0			0
$309$	7.11684			0.404863
$310$		0			0
$311$	11.3870			0.645696			0.322848	−	0.946451i	$-0.395360\pi$
$311$	11.3870			0.645696			0.322848	+	0.946451i	$0.395360\pi$
$312$		0			0
$313$	3.11684			0.176174			0.0880872	−	0.996113i	$-0.471925\pi$
$313$	3.11684			0.176174			0.0880872	+	0.996113i	$0.471925\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	4.51087i		−	0.253356i	−0.991944	−	0.126678i	$-0.959569\pi$
$317$		−	4.51087i		−	0.253356i	0.991944	−	0.126678i	$-0.0404315\pi$
$318$		0			0
$319$		−	0.939764i		−	0.0526167i
$320$		0			0
$321$		−	42.2337i		−	2.35725i
$322$		0			0
$323$	8.21782			0.457252
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	19.2549i		−	1.06480i
$328$		0			0
$329$	19.8614	+	6.23369i	1.09499	+	0.343674i
$330$		0			0
$331$			24.2487i			1.33283i	0.745581	+	0.666415i	$0.232172\pi$
$331$			24.2487i			1.33283i	−0.745581	+	0.666415i	$0.767828\pi$
$332$		0			0
$333$			36.2337i			1.98559i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	20.2337i		−	1.10220i	−0.834439	−	0.551100i	$-0.814208\pi$
$337$		−	20.2337i		−	1.10220i	0.834439	−	0.551100i	$-0.185792\pi$
$338$		0			0
$339$	3.16915			0.172124
$340$		0			0
$341$			4.74456i			0.256932i
$342$		0			0
$343$	14.6487	−	11.3321i	0.790955	−	0.611874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.994667			0.0533965			0.0266983	−	0.999644i	$-0.491501\pi$
$347$	0.994667			0.0533965			0.0266983	+	0.999644i	$0.491501\pi$
$348$		0			0
$349$		−	10.0000i		−	0.535288i	−0.963518	−	0.267644i	$-0.913755\pi$
$349$		−	10.0000i		−	0.535288i	0.963518	−	0.267644i	$-0.0862451\pi$
$350$		0			0
$351$			0.349857i			0.0186740i
$352$		0			0
$353$	−18.3723			−0.977858			−0.488929	−	0.872324i	$-0.662612\pi$
$353$	−18.3723			−0.977858			−0.488929	+	0.872324i	$0.662612\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−15.1168	−	4.74456i	−0.800068	−	0.251109i
$358$		0			0
$359$			0.294954i			0.0155671i	0.999970	+	0.00778353i	$0.00247760\pi$
$359$			0.294954i			0.0155671i	−0.999970	+	0.00778353i	$0.997522\pi$
$360$		0			0
$361$	−7.00000			−0.368421
$362$		0			0
$363$		−	11.6819i		−	0.613142i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	21.7244i		−	1.13400i	−0.823717	−	0.567002i	$-0.808103\pi$
$367$		−	21.7244i		−	1.13400i	0.823717	−	0.567002i	$-0.191897\pi$
$368$		0			0
$369$			29.4891i			1.53514i
$370$		0			0
$371$	29.0024	+	9.10268i	1.50573	+	0.472588i
$372$		0			0
$373$		−	21.2554i		−	1.10056i	−0.834979	−	0.550282i	$-0.814520\pi$
$373$		−	21.2554i		−	1.10056i	0.834979	−	0.550282i	$-0.185480\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.138593			−0.00713792
$378$		0			0
$379$			23.6588i			1.21527i	0.794216	+	0.607636i	$0.207882\pi$
$379$			23.6588i			1.21527i	−0.794216	+	0.607636i	$0.792118\pi$
$380$		0			0
$381$			16.7446i			0.857850i
$382$		0			0
$383$		−	11.9769i		−	0.611990i	−0.952033	−	0.305995i	$-0.901011\pi$
$383$		−	11.9769i		−	0.611990i	0.952033	−	0.305995i	$-0.0989891\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−11.6819			−0.593826
$388$		0			0
$389$	1.11684			0.0566262			0.0283131	−	0.999599i	$-0.490986\pi$
$389$	1.11684			0.0566262			0.0283131	+	0.999599i	$0.490986\pi$
$390$		0			0
$391$	−20.1947			−1.02129
$392$		0			0
$393$		−	38.9783i		−	1.96619i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	12.3723			0.620947			0.310473	−	0.950582i	$-0.399512\pi$
$397$	12.3723			0.620947			0.310473	+	0.950582i	$0.399512\pi$
$398$		0			0
$399$	−22.0742	−	6.92820i	−1.10509	−	0.346844i
$400$		0			0
$401$	5.11684			0.255523			0.127761	−	0.991805i	$-0.459221\pi$
$401$	5.11684			0.255523			0.127761	+	0.991805i	$0.459221\pi$
$402$		0			0
$403$	0.699713			0.0348552
$404$		0			0
$405$		0			0
$406$		0			0
$407$	27.1229			1.34443
$408$		0			0
$409$			16.7446i			0.827965i	0.910285	+	0.413983i	$0.135863\pi$
$409$			16.7446i			0.827965i	−0.910285	+	0.413983i	$0.864137\pi$
$410$		0			0
$411$	27.1229			1.33787
$412$		0			0
$413$	5.25544	−	16.7446i	0.258603	−	0.823946i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			12.0000i			0.587643i
$418$		0			0
$419$	12.9715			0.633701			0.316851	−	0.948475i	$-0.397375\pi$
$419$	12.9715			0.633701			0.316851	+	0.948475i	$0.397375\pi$
$420$		0			0
$421$	21.1168			1.02917			0.514586	−	0.857439i	$-0.327946\pi$
$421$	21.1168			1.02917			0.514586	+	0.857439i	$0.327946\pi$
$422$		0			0
$423$			26.5330i			1.29008i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	27.1229	+	8.51278i	1.31257	+	0.411962i
$428$		0			0
$429$	2.37228			0.114535
$430$		0			0
$431$			12.0318i			0.579551i	0.957095	+	0.289775i	$0.0935806\pi$
$431$			12.0318i			0.579551i	−0.957095	+	0.289775i	$0.906419\pi$
$432$		0			0
$433$	−6.23369			−0.299572			−0.149786	−	0.988718i	$-0.547858\pi$
$433$	−6.23369			−0.299572			−0.149786	+	0.988718i	$0.547858\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−29.4891			−1.41066
$438$		0			0
$439$	0.589907			0.0281547			0.0140774	−	0.999901i	$-0.495519\pi$
$439$	0.589907			0.0281547			0.0140774	+	0.999901i	$0.495519\pi$
$440$		0			0
$441$	19.3723	+	13.4891i	0.922490	+	0.642339i
$442$		0			0
$443$	21.0796			1.00152			0.500760	−	0.865586i	$-0.333054\pi$
$443$	21.0796			1.00152			0.500760	+	0.865586i	$0.333054\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	25.2434i		−	1.19397i
$448$		0			0
$449$	22.6060			1.06684			0.533421	−	0.845850i	$-0.320906\pi$
$449$	22.6060			1.06684			0.533421	+	0.845850i	$0.320906\pi$
$450$		0			0
$451$	22.0742			1.03943
$452$		0			0
$453$	−46.0951			−2.16574
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	15.4891i		−	0.724551i	−0.932071	−	0.362275i	$-0.882000\pi$
$457$		−	15.4891i		−	0.724551i	0.932071	−	0.362275i	$-0.118000\pi$
$458$		0			0
$459$		−	2.22938i		−	0.104059i
$460$		0			0
$461$			22.7446i			1.05932i	0.848210	+	0.529660i	$0.177680\pi$
$461$			22.7446i			1.05932i	−0.848210	+	0.529660i	$0.822320\pi$
$462$		0			0
$463$	21.7793			1.01217			0.506084	−	0.862484i	$-0.331092\pi$
$463$	21.7793			1.01217			0.506084	+	0.862484i	$0.331092\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			10.7422i			0.497088i	0.968621	+	0.248544i	$0.0799520\pi$
$467$			10.7422i			0.497088i	−0.968621	+	0.248544i	$0.920048\pi$
$468$		0			0
$469$	−5.25544	+	16.7446i	−0.242674	+	0.773192i
$470$		0			0
$471$			59.2945i			2.73215i
$472$		0			0
$473$			8.74456i			0.402075i
$474$		0			0
$475$		0			0
$476$		0			0
$477$			38.7446i			1.77399i
$478$		0			0
$479$	37.8102			1.72759			0.863795	−	0.503843i	$-0.168081\pi$
$479$	37.8102			1.72759			0.863795	+	0.503843i	$0.168081\pi$
$480$		0			0
$481$		−	4.00000i		−	0.182384i
$482$		0			0
$483$	54.2458	+	17.0256i	2.46827	+	0.774690i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−4.75372			−0.215412			−0.107706	−	0.994183i	$-0.534350\pi$
$487$	−4.75372			−0.215412			−0.107706	+	0.994183i	$0.534350\pi$
$488$		0			0
$489$			12.0000i			0.542659i
$490$		0			0
$491$		−	24.0087i		−	1.08350i	−0.840541	−	0.541748i	$-0.817763\pi$
$491$		−	24.0087i		−	1.08350i	0.840541	−	0.541748i	$-0.182237\pi$
$492$		0			0
$493$	0.883156			0.0397753
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−16.7446	−	5.25544i	−0.751096	−	0.235739i
$498$		0			0
$499$		−	17.0805i		−	0.764626i	−0.924033	−	0.382313i	$-0.875128\pi$
$499$		−	17.0805i		−	0.764626i	0.924033	−	0.382313i	$-0.124872\pi$
$500$		0			0
$501$	−18.3723			−0.820813
$502$		0			0
$503$			35.6906i			1.59136i	0.605714	+	0.795682i	$0.292887\pi$
$503$			35.6906i			1.59136i	−0.605714	+	0.795682i	$0.707113\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			32.4665i			1.44189i
$508$		0			0
$509$			10.7446i			0.476244i	0.971235	+	0.238122i	$0.0765319\pi$
$509$			10.7446i			0.476244i	−0.971235	+	0.238122i	$0.923468\pi$
$510$		0			0
$511$	−6.92820	+	22.0742i	−0.306486	+	0.976506i
$512$		0			0
$513$		−	3.25544i		−	0.143731i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	19.8614			0.873504
$518$		0			0
$519$		−	41.3292i		−	1.81415i
$520$		0			0
$521$			35.7228i			1.56504i	0.622622	+	0.782522i	$0.286067\pi$
$521$			35.7228i			1.56504i	−0.622622	+	0.782522i	$0.713933\pi$
$522$		0			0
$523$		−	10.9822i		−	0.480219i	−0.970746	−	0.240109i	$-0.922817\pi$
$523$		−	10.9822i		−	0.480219i	0.970746	−	0.240109i	$-0.0771833\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.45877			−0.194227
$528$		0			0
$529$	49.4674			2.15076
$530$		0			0
$531$	22.3692			0.970740
$532$		0			0
$533$		−	3.25544i		−	0.141009i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	34.2337			1.47729
$538$		0			0
$539$	10.0974	−	14.5012i	0.434924	−	0.624612i
$540$		0			0
$541$	3.62772			0.155968			0.0779839	−	0.996955i	$-0.475152\pi$
$541$	3.62772			0.155968			0.0779839	+	0.996955i	$0.475152\pi$
$542$		0			0
$543$	−52.9562			−2.27257
$544$		0			0
$545$		0			0
$546$		0			0
$547$	24.9484			1.06672			0.533359	−	0.845889i	$-0.320930\pi$
$547$	24.9484			1.06672			0.533359	+	0.845889i	$0.320930\pi$
$548$		0			0
$549$			36.2337i			1.54642i
$550$		0			0
$551$	1.28962			0.0549397
$552$		0			0
$553$	−39.8614	−	12.5109i	−1.69508	−	0.532017i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			33.7228i			1.42888i	0.699696	+	0.714441i	$0.253319\pi$
$557$			33.7228i			1.42888i	−0.699696	+	0.714441i	$0.746681\pi$
$558$		0			0
$559$	1.28962			0.0545451
$560$		0			0
$561$	−15.1168			−0.638234
$562$		0			0
$563$		−	36.9253i		−	1.55622i	−0.628131	−	0.778108i	$-0.716180\pi$
$563$		−	36.9253i		−	1.55622i	0.628131	−	0.778108i	$-0.283820\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	6.13592	−	19.5499i	0.257684	−	0.821018i
$568$		0			0
$569$	−12.5109			−0.524483			−0.262242	−	0.965002i	$-0.584462\pi$
$569$	−12.5109			−0.524483			−0.262242	+	0.965002i	$0.584462\pi$
$570$		0			0
$571$			2.87419i			0.120281i	0.998190	+	0.0601406i	$0.0191549\pi$
$571$			2.87419i			0.120281i	−0.998190	+	0.0601406i	$0.980845\pi$
$572$		0			0
$573$	−19.1168			−0.798618
$574$		0			0
$575$		0			0
$576$		0			0
$577$	23.1168			0.962367			0.481183	−	0.876620i	$-0.340207\pi$
$577$	23.1168			0.962367			0.481183	+	0.876620i	$0.340207\pi$
$578$		0			0
$579$	67.5123			2.80572
$580$		0			0
$581$	26.2337	+	8.23369i	1.08836	+	0.341591i
$582$		0			0
$583$	29.0024			1.20116
$584$		0			0
$585$		0			0
$586$		0			0
$587$			3.46410i			0.142979i	0.997441	+	0.0714894i	$0.0227752\pi$
$587$			3.46410i			0.142979i	−0.997441	+	0.0714894i	$0.977225\pi$
$588$		0			0
$589$	−6.51087			−0.268276
$590$		0			0
$591$	−15.1460			−0.623024
$592$		0			0
$593$	5.62772			0.231103			0.115551	−	0.993302i	$-0.463137\pi$
$593$	5.62772			0.231103			0.115551	+	0.993302i	$0.463137\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	3.25544i		−	0.133236i
$598$		0			0
$599$		−	17.0805i		−	0.697889i	−0.937143	−	0.348944i	$-0.886540\pi$
$599$		−	17.0805i		−	0.697889i	0.937143	−	0.348944i	$-0.113460\pi$
$600$		0			0
$601$		−	24.7446i		−	1.00935i	−0.863309	−	0.504676i	$-0.831612\pi$
$601$		−	24.7446i		−	1.00935i	0.863309	−	0.504676i	$-0.168388\pi$
$602$		0			0
$603$	−22.3692			−0.910944
$604$		0			0
$605$		0			0
$606$		0			0
$607$			3.40920i			0.138375i	0.997604	+	0.0691876i	$0.0220407\pi$
$607$			3.40920i			0.138375i	−0.997604	+	0.0691876i	$0.977959\pi$
$608$		0			0
$609$	−2.37228	−	0.744563i	−0.0961297	−	0.0301712i
$610$		0			0
$611$		−	2.92910i		−	0.118499i
$612$		0			0
$613$		−	20.9783i		−	0.847304i	−0.905825	−	0.423652i	$-0.860748\pi$
$613$		−	20.9783i		−	0.847304i	0.905825	−	0.423652i	$-0.139252\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	42.7446i		−	1.72083i	−0.509593	−	0.860416i	$-0.670204\pi$
$617$		−	42.7446i		−	1.72083i	0.509593	−	0.860416i	$-0.329796\pi$
$618$		0			0
$619$	−42.5639			−1.71079			−0.855394	−	0.517979i	$-0.826685\pi$
$619$	−42.5639			−1.71079			−0.855394	+	0.517979i	$0.826685\pi$
$620$		0			0
$621$			8.00000i			0.321029i
$622$		0			0
$623$	13.8564	+	4.34896i	0.555145	+	0.174238i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−22.0742			−0.881560
$628$		0			0
$629$			25.4891i			1.01632i
$630$		0			0
$631$			8.86263i			0.352816i	0.984317	+	0.176408i	$0.0564478\pi$
$631$			8.86263i			0.352816i	−0.984317	+	0.176408i	$0.943552\pi$
$632$		0			0
$633$	−44.6060			−1.77293
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−2.13859	−	1.48913i	−0.0847342	−	0.0590013i
$638$		0			0
$639$		−	22.3692i		−	0.884911i
$640$		0			0
$641$	−42.0000			−1.65890			−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000			−1.65890			−0.829450	+	0.558581i	$0.811346\pi$
$642$		0			0
$643$			31.5268i			1.24329i	0.783297	+	0.621647i	$0.213536\pi$
$643$			31.5268i			1.24329i	−0.783297	+	0.621647i	$0.786464\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			2.46943i			0.0970835i	0.998821	+	0.0485418i	$0.0154574\pi$
$647$			2.46943i			0.0970835i	−0.998821	+	0.0485418i	$0.984543\pi$
$648$		0			0
$649$		−	16.7446i		−	0.657282i
$650$		0			0
$651$	11.9769	+	3.75906i	0.469411	+	0.147329i
$652$		0			0
$653$			18.4674i			0.722684i	0.932433	+	0.361342i	$0.117681\pi$
$653$			18.4674i			0.722684i	−0.932433	+	0.361342i	$0.882319\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−29.4891			−1.15048
$658$		0			0
$659$			16.9707i			0.661083i	0.943792	+	0.330541i	$0.107231\pi$
$659$			16.9707i			0.661083i	−0.943792	+	0.330541i	$0.892769\pi$
$660$		0			0
$661$		−	37.2554i		−	1.44907i	−0.689239	−	0.724534i	$-0.742055\pi$
$661$		−	37.2554i		−	1.44907i	0.689239	−	0.724534i	$-0.257945\pi$
$662$		0			0
$663$			2.22938i			0.0865821i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−3.16915			−0.122710
$668$		0			0
$669$	−10.3723			−0.401016
$670$		0			0
$671$	27.1229			1.04707
$672$		0			0
$673$			16.2337i			0.625763i	0.949792	+	0.312881i	$0.101294\pi$
$673$			16.2337i			0.625763i	−0.949792	+	0.312881i	$0.898706\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.88316			0.264541			0.132271	−	0.991214i	$-0.457773\pi$
$677$	6.88316			0.264541			0.132271	+	0.991214i	$0.457773\pi$
$678$		0			0
$679$	−11.9769	+	38.1600i	−0.459630	+	1.46445i
$680$		0			0
$681$	52.6060			2.01587
$682$		0			0
$683$	−36.2256			−1.38613			−0.693067	−	0.720873i	$-0.743741\pi$
$683$	−36.2256			−1.38613			−0.693067	+	0.720873i	$0.743741\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−35.3407			−1.34833
$688$		0			0
$689$		−	4.27719i		−	0.162948i
$690$		0			0
$691$	−29.8873			−1.13697			−0.568483	−	0.822695i	$-0.692470\pi$
$691$	−29.8873			−1.13697			−0.568483	+	0.822695i	$0.692470\pi$
$692$		0			0
$693$	21.4891	+	6.74456i	0.816304	+	0.256205i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			20.7446i			0.785756i
$698$		0			0
$699$	−27.1229			−1.02588
$700$		0			0
$701$	−18.8832			−0.713207			−0.356603	−	0.934256i	$-0.616065\pi$
$701$	−18.8832			−0.713207			−0.356603	+	0.934256i	$0.616065\pi$
$702$		0			0
$703$			37.2203i			1.40379i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−18.9051	−	5.93354i	−0.710999	−	0.223154i
$708$		0			0
$709$	−0.372281			−0.0139813			−0.00699066	−	0.999976i	$-0.502225\pi$
$709$	−0.372281			−0.0139813			−0.00699066	+	0.999976i	$0.502225\pi$
$710$		0			0
$711$		−	53.2511i		−	1.99707i
$712$		0			0
$713$	16.0000			0.599205
$714$		0			0
$715$		0			0
$716$		0			0
$717$	31.8614			1.18989
$718$		0			0
$719$	−20.1947			−0.753135			−0.376568	−	0.926389i	$-0.622896\pi$
$719$	−20.1947			−0.753135			−0.376568	+	0.926389i	$0.622896\pi$
$720$		0			0
$721$	−7.11684	−	2.23369i	−0.265045	−	0.0831869i
$722$		0			0
$723$	40.3894			1.50210
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	42.8588i		−	1.58955i	−0.606907	−	0.794773i	$-0.707590\pi$
$727$		−	42.8588i		−	1.58955i	0.606907	−	0.794773i	$-0.292410\pi$
$728$		0			0
$729$	33.2337			1.23088
$730$		0			0
$731$	−8.21782			−0.303947
$732$		0			0
$733$	−6.13859			−0.226734			−0.113367	−	0.993553i	$-0.536164\pi$
$733$	−6.13859			−0.226734			−0.113367	+	0.993553i	$0.536164\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			16.7446i			0.616794i
$738$		0			0
$739$		−	17.0805i		−	0.628315i	−0.949371	−	0.314157i	$-0.898278\pi$
$739$		−	17.0805i		−	0.628315i	0.949371	−	0.314157i	$-0.101722\pi$
$740$		0			0
$741$			3.25544i			0.119591i
$742$		0			0
$743$	−7.22316			−0.264992			−0.132496	−	0.991184i	$-0.542299\pi$
$743$	−7.22316			−0.264992			−0.132496	+	0.991184i	$0.542299\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			35.0458i			1.28226i
$748$		0			0
$749$	−13.2554	+	42.2337i	−0.484343	+	1.54319i
$750$		0			0
$751$		−	24.5986i		−	0.897614i	−0.893629	−	0.448807i	$-0.851849\pi$
$751$		−	24.5986i		−	0.897614i	0.893629	−	0.448807i	$-0.148151\pi$
$752$		0			0
$753$			38.9783i			1.42045i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			22.7446i			0.826665i	0.910580	+	0.413333i	$0.135635\pi$
$757$			22.7446i			0.826665i	−0.910580	+	0.413333i	$0.864365\pi$
$758$		0			0
$759$	54.2458			1.96900
$760$		0			0
$761$			36.0000i			1.30500i	0.757789	+	0.652499i	$0.226280\pi$
$761$			36.0000i			1.30500i	−0.757789	+	0.652499i	$0.773720\pi$
$762$		0			0
$763$	−6.04334	+	19.2549i	−0.218784	+	0.697076i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.46943			−0.0891661
$768$		0			0
$769$			38.2337i			1.37874i	0.724408	+	0.689371i	$0.242113\pi$
$769$			38.2337i			1.37874i	−0.724408	+	0.689371i	$0.757887\pi$
$770$		0			0
$771$			32.1716i			1.15863i
$772$		0			0
$773$	−5.11684			−0.184040			−0.0920200	−	0.995757i	$-0.529332\pi$
$773$	−5.11684			−0.184040			−0.0920200	+	0.995757i	$0.529332\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	21.4891	−	68.4674i	0.770918	−	2.45625i
$778$		0			0
$779$			30.2921i			1.08533i
$780$		0			0
$781$	−16.7446			−0.599168
$782$		0			0
$783$		−	0.349857i		−	0.0125029i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	29.6472i		−	1.05681i	−0.848992	−	0.528405i	$-0.822790\pi$
$787$		−	29.6472i		−	1.05681i	0.848992	−	0.528405i	$-0.177210\pi$
$788$		0			0
$789$			35.7228i			1.27177i
$790$		0			0
$791$	−3.16915	−	0.994667i	−0.112682	−	0.0353663i
$792$		0			0
$793$		−	4.00000i		−	0.142044i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−16.0951			−0.570117			−0.285059	−	0.958510i	$-0.592013\pi$
$797$	−16.0951			−0.570117			−0.285059	+	0.958510i	$0.592013\pi$
$798$		0			0
$799$			18.6650i			0.660321i
$800$		0			0
$801$			18.5109i			0.654050i
$802$		0			0
$803$			22.0742i			0.778983i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	27.1229			0.954771
$808$		0			0
$809$	18.6060			0.654151			0.327076	−	0.944998i	$-0.393937\pi$
$809$	18.6060			0.654151			0.327076	+	0.944998i	$0.393937\pi$
$810$		0			0
$811$	22.3692			0.785488			0.392744	−	0.919648i	$-0.371526\pi$
$811$	22.3692			0.785488			0.392744	+	0.919648i	$0.371526\pi$
$812$		0			0
$813$			66.9783i			2.34903i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−12.0000			−0.419827
$818$		0			0
$819$	0.994667	−	3.16915i	0.0347565	−	0.110739i
$820$		0			0
$821$	48.0951			1.67853			0.839265	−	0.543722i	$-0.182986\pi$
$821$	48.0951			1.67853			0.839265	+	0.543722i	$0.182986\pi$
$822$		0			0
$823$	−41.8642			−1.45929			−0.729647	−	0.683824i	$-0.760315\pi$
$823$	−41.8642			−1.45929			−0.729647	+	0.683824i	$0.760315\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.58457			−0.0551010			−0.0275505	−	0.999620i	$-0.508771\pi$
$827$	−1.58457			−0.0551010			−0.0275505	+	0.999620i	$0.508771\pi$
$828$		0			0
$829$		−	3.48913i		−	0.121182i	−0.998163	−	0.0605912i	$-0.980701\pi$
$829$		−	3.48913i		−	0.121182i	0.998163	−	0.0605912i	$-0.0192986\pi$
$830$		0			0
$831$	−51.0767			−1.77183
$832$		0			0
$833$	13.6277	+	9.48913i	0.472172	+	0.328779i
$834$		0			0
$835$		0			0
$836$		0			0
$837$			1.76631i			0.0610527i
$838$		0			0
$839$	29.7021			1.02543			0.512716	−	0.858558i	$-0.328639\pi$
$839$	29.7021			1.02543			0.512716	+	0.858558i	$0.328639\pi$
$840$		0			0
$841$	−28.8614			−0.995221
$842$		0			0
$843$		−	46.9678i		−	1.61766i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−3.66648	+	11.6819i	−0.125982	+	0.401396i
$848$		0			0
$849$	−14.3723			−0.493255
$850$		0			0
$851$		−	91.4661i		−	3.13542i
$852$		0			0
$853$	−15.4891			−0.530338			−0.265169	−	0.964202i	$-0.585428\pi$
$853$	−15.4891			−0.530338			−0.265169	+	0.964202i	$0.585428\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−37.2119			−1.27114			−0.635568	−	0.772045i	$-0.719234\pi$
$857$	−37.2119			−1.27114			−0.635568	+	0.772045i	$0.719234\pi$
$858$		0			0
$859$	43.1538			1.47239			0.736194	−	0.676770i	$-0.236621\pi$
$859$	43.1538			1.47239			0.736194	+	0.676770i	$0.236621\pi$
$860$		0			0
$861$	17.4891	−	55.7228i	0.596028	−	1.89903i
$862$		0			0
$863$	0.884861			0.0301210			0.0150605	−	0.999887i	$-0.495206\pi$
$863$	0.884861			0.0301210			0.0150605	+	0.999887i	$0.495206\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			28.7075i			0.974957i
$868$		0			0
$869$	−39.8614			−1.35221
$870$		0			0
$871$	2.46943			0.0836736
$872$		0			0
$873$	−50.9783			−1.72535
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	32.9783i		−	1.11360i	−0.830648	−	0.556798i	$-0.812030\pi$
$877$		−	32.9783i		−	1.11360i	0.830648	−	0.556798i	$-0.187970\pi$
$878$		0			0
$879$			49.5470i			1.67118i
$880$		0			0
$881$			45.9565i			1.54831i	0.632994	+	0.774157i	$0.281826\pi$
$881$			45.9565i			1.54831i	−0.632994	+	0.774157i	$0.718174\pi$
$882$		0			0
$883$	−46.3229			−1.55889			−0.779446	−	0.626470i	$-0.784499\pi$
$883$	−46.3229			−1.55889			−0.779446	+	0.626470i	$0.784499\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			25.2434i			0.847590i	0.905758	+	0.423795i	$0.139302\pi$
$887$			25.2434i			0.847590i	−0.905758	+	0.423795i	$0.860698\pi$
$888$		0			0
$889$	5.25544	−	16.7446i	0.176262	−	0.561595i
$890$		0			0
$891$		−	19.5499i		−	0.654946i
$892$		0			0
$893$			27.2554i			0.912068i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	8.00000i		−	0.267112i
$898$		0			0
$899$	−0.699713			−0.0233367
$900$		0			0
$901$			27.2554i			0.908010i
$902$		0			0
$903$	22.0742	+	6.92820i	0.734584	+	0.230556i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−9.10268			−0.302250			−0.151125	−	0.988515i	$-0.548290\pi$
$907$	−9.10268			−0.302250			−0.151125	+	0.988515i	$0.548290\pi$
$908$		0			0
$909$		−	25.2554i		−	0.837670i
$910$		0			0
$911$		−	50.7817i		−	1.68247i	−0.540667	−	0.841237i	$-0.681828\pi$
$911$		−	50.7817i		−	1.68247i	0.540667	−	0.841237i	$-0.318172\pi$
$912$		0			0
$913$	26.2337			0.868208
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−12.2337	+	38.9783i	−0.403992	+	1.28718i
$918$		0			0
$919$		−	24.5986i		−	0.811432i	−0.913999	−	0.405716i	$-0.867022\pi$
$919$		−	24.5986i		−	0.811432i	0.913999	−	0.405716i	$-0.132978\pi$
$920$		0			0
$921$	1.62772			0.0536352
$922$		0			0
$923$			2.46943i			0.0812824i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	9.50744i		−	0.312265i
$928$		0			0
$929$		−	19.2554i		−	0.631750i	−0.948801	−	0.315875i	$-0.897702\pi$
$929$		−	19.2554i		−	0.631750i	0.948801	−	0.315875i	$-0.102298\pi$
$930$		0			0
$931$	19.8997	+	13.8564i	0.652188	+	0.454125i
$932$		0			0
$933$		−	28.7446i		−	0.941055i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−20.8832			−0.682223			−0.341111	−	0.940023i	$-0.610803\pi$
$937$	−20.8832			−0.682223			−0.341111	+	0.940023i	$0.610803\pi$
$938$		0			0
$939$		−	7.86797i		−	0.256761i
$940$		0			0
$941$			11.7663i			0.383571i	0.981437	+	0.191785i	$0.0614278\pi$
$941$			11.7663i			0.383571i	−0.981437	+	0.191785i	$0.938572\pi$
$942$		0			0
$943$		−	74.4405i		−	2.42412i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	26.8280			0.871791			0.435896	−	0.899997i	$-0.356432\pi$
$947$	26.8280			0.871791			0.435896	+	0.899997i	$0.356432\pi$
$948$		0			0
$949$	3.25544			0.105676
$950$		0			0
$951$	−11.3870			−0.369248
$952$		0			0
$953$			28.9783i			0.938698i	0.883013	+	0.469349i	$0.155511\pi$
$953$			28.9783i			0.938698i	−0.883013	+	0.469349i	$0.844489\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−2.37228			−0.0766850
$958$		0			0
$959$	−27.1229	−	8.51278i	−0.875844	−	0.274892i
$960$		0			0
$961$	−27.4674			−0.886044
$962$		0			0
$963$	−56.4203			−1.81812
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−40.6844			−1.30832			−0.654160	−	0.756356i	$-0.726978\pi$
$967$	−40.6844			−1.30832			−0.654160	+	0.756356i	$0.726978\pi$
$968$		0			0
$969$		−	20.7446i		−	0.666411i
$970$		0			0
$971$	33.1662			1.06436			0.532178	−	0.846633i	$-0.321374\pi$
$971$	33.1662			1.06436			0.532178	+	0.846633i	$0.321374\pi$
$972$		0			0
$973$	3.76631	−	12.0000i	0.120742	−	0.384702i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	20.9783i		−	0.671154i	−0.942013	−	0.335577i	$-0.891069\pi$
$977$		−	20.9783i		−	0.671154i	0.942013	−	0.335577i	$-0.108931\pi$
$978$		0			0
$979$	13.8564			0.442853
$980$		0			0
$981$	−25.7228			−0.821266
$982$		0			0
$983$			53.3060i			1.70020i	0.526622	+	0.850099i	$0.323458\pi$
$983$			53.3060i			1.70020i	−0.526622	+	0.850099i	$0.676542\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	15.7359	−	50.1369i	0.500880	−	1.59588i
$988$		0			0
$989$	29.4891			0.937700
$990$		0			0
$991$		−	4.64392i		−	0.147519i	−0.997276	−	0.0737594i	$-0.976500\pi$
$991$		−	4.64392i		−	0.147519i	0.997276	−	0.0737594i	$-0.0234997\pi$
$992$		0			0
$993$	61.2119			1.94250
$994$		0			0
$995$		0			0
$996$		0			0
$997$	19.3505			0.612837			0.306419	−	0.951897i	$-0.400869\pi$
$997$	19.3505			0.612837			0.306419	+	0.951897i	$0.400869\pi$
$998$		0			0
$999$	10.0974			0.319466

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.e.i.2799.1		8
4.3	odd	2	inner	2800.2.e.i.2799.8		8
5.2	odd	4		560.2.k.a.111.2	yes	8
5.3	odd	4		2800.2.k.l.2351.8		8
5.4	even	2		2800.2.e.j.2799.8		8
7.6	odd	2		2800.2.e.j.2799.7		8
15.2	even	4		5040.2.d.e.4591.1		8
20.3	even	4		2800.2.k.l.2351.1		8
20.7	even	4		560.2.k.a.111.8	yes	8
20.19	odd	2		2800.2.e.j.2799.1		8
28.27	even	2		2800.2.e.j.2799.2		8
35.13	even	4		2800.2.k.l.2351.2		8
35.27	even	4		560.2.k.a.111.7	yes	8
35.34	odd	2	inner	2800.2.e.i.2799.2		8
40.27	even	4		2240.2.k.c.1791.1		8
40.37	odd	4		2240.2.k.c.1791.7		8
60.47	odd	4		5040.2.d.e.4591.4		8
105.62	odd	4		5040.2.d.e.4591.8		8
140.27	odd	4		560.2.k.a.111.1	✓	8
140.83	odd	4		2800.2.k.l.2351.7		8
140.139	even	2	inner	2800.2.e.i.2799.7		8
280.27	odd	4		2240.2.k.c.1791.8		8
280.237	even	4		2240.2.k.c.1791.2		8
420.167	even	4		5040.2.d.e.4591.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.k.a.111.1	✓	8	140.27	odd	4
560.2.k.a.111.2	yes	8	5.2	odd	4
560.2.k.a.111.7	yes	8	35.27	even	4
560.2.k.a.111.8	yes	8	20.7	even	4
2240.2.k.c.1791.1		8	40.27	even	4
2240.2.k.c.1791.2		8	280.237	even	4
2240.2.k.c.1791.7		8	40.37	odd	4
2240.2.k.c.1791.8		8	280.27	odd	4
2800.2.e.i.2799.1		8	1.1	even	1	trivial
2800.2.e.i.2799.2		8	35.34	odd	2	inner
2800.2.e.i.2799.7		8	140.139	even	2	inner
2800.2.e.i.2799.8		8	4.3	odd	2	inner
2800.2.e.j.2799.1		8	20.19	odd	2
2800.2.e.j.2799.2		8	28.27	even	2
2800.2.e.j.2799.7		8	7.6	odd	2
2800.2.e.j.2799.8		8	5.4	even	2
2800.2.k.l.2351.1		8	20.3	even	4
2800.2.k.l.2351.2		8	35.13	even	4
2800.2.k.l.2351.7		8	140.83	odd	4
2800.2.k.l.2351.8		8	5.3	odd	4
5040.2.d.e.4591.1		8	15.2	even	4
5040.2.d.e.4591.4		8	60.47	odd	4
5040.2.d.e.4591.5		8	420.167	even	4
5040.2.d.e.4591.8		8	105.62	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.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.52434i q^{3} +(-0.792287 + 2.52434i) q^{7} -3.37228 q^{9} +O(q^{10})\)	\(q-2.52434i q^{3} +(-0.792287 + 2.52434i) q^{7} -3.37228 q^{9} +2.52434i q^{11} +0.372281 q^{13} -2.37228 q^{17} -3.46410 q^{19} +(6.37228 + 2.00000i) q^{21} +8.51278 q^{23} +0.939764i q^{27} -0.372281 q^{29} +1.87953 q^{31} +6.37228 q^{33} -10.7446i q^{37} -0.939764i q^{39} -8.74456i q^{41} +3.46410 q^{43} -7.86797i q^{47} +(-5.74456 - 4.00000i) q^{49} +5.98844i q^{51} -11.4891i q^{53} +8.74456i q^{57} -6.63325 q^{59} -10.7446i q^{61} +(2.67181 - 8.51278i) q^{63} +6.63325 q^{67} -21.4891i q^{69} +6.63325i q^{71} +8.74456 q^{73} +(-6.37228 - 2.00000i) q^{77} +15.7908i q^{79} -7.74456 q^{81} -10.3923i q^{83} +0.939764i q^{87} -5.48913i q^{89} +(-0.294954 + 0.939764i) q^{91} -4.74456i q^{93} +15.1168 q^{97} -8.51278i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^9	\( 8 q - 4 q^{9} - 20 q^{13} + 4 q^{17} + 28 q^{21} + 20 q^{29} + 28 q^{33} + 24 q^{73} - 28 q^{77} - 16 q^{81} + 52 q^{97}+O(q^{100}) \) 8 * q - 4 * q^9 - 20 * q^13 + 4 * q^17 + 28 * q^21 + 20 * q^29 + 28 * q^33 + 24 * q^73 - 28 * q^77 - 16 * q^81 + 52 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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