LMFDB - Embedded newform 224.3.h.d.209.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,3,Mod(209,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("224.209");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	224.h (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:	$6.10355792167$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.976966189056.51
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{6} + 123x^{4} - 300x^{3} + 86x^{2} + 300x + 526 $ x^8 - 10x^6 + 123x^4 - 300x^3 + 86x^2 + 300*x + 526
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.6
Root			$-3.26020 - 1.99551i$ of defining polynomial
Character	$\chi$	$=$	224.209
Dual form			224.3.h.d.209.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.75265 q^{3} -6.54099 q^{5} +(0.267949 + 6.99487i) q^{7} -5.92820 q^{9} +O(q^{10})$	$q+1.75265 q^{3} -6.54099 q^{5} +(0.267949 + 6.99487i) q^{7} -5.92820 q^{9} +2.13878i q^{11} -12.6124 q^{13} -11.4641 q^{15} +24.2309i q^{17} -13.8954 q^{19} +(0.469622 + 12.2596i) q^{21} +24.7846 q^{23} +17.7846 q^{25} -26.1640 q^{27} -43.6146i q^{29} +34.4721i q^{31} +3.74854i q^{33} +(-1.75265 - 45.7534i) q^{35} +27.6506i q^{37} -22.1051 q^{39} -27.9795i q^{41} +42.6220i q^{43} +38.7763 q^{45} -34.4721i q^{47} +(-48.8564 + 3.74854i) q^{49} +42.4685i q^{51} +35.0595i q^{53} -13.9897i q^{55} -24.3538 q^{57} +55.7074 q^{59} -6.54099 q^{61} +(-1.58846 - 41.4670i) q^{63} +82.4974 q^{65} -42.6220i q^{67} +43.4389 q^{69} -22.9667 q^{71} -59.7075i q^{73} +31.1703 q^{75} +(-14.9605 + 0.573084i) q^{77} +35.1769 q^{79} +7.49742 q^{81} +144.715 q^{83} -158.494i q^{85} -76.4414i q^{87} -108.169i q^{89} +(-3.37947 - 88.2219i) q^{91} +60.4177i q^{93} +90.8897 q^{95} +136.149i q^{97} -12.6791i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q + 16 * q^7 + 8 * q^9	$ 8 q + 16 q^{7} + 8 q^{9} - 64 q^{15} + 32 q^{23} - 24 q^{25} + 128 q^{39} - 280 q^{49} + 304 q^{57} + 112 q^{63} + 272 q^{65} - 544 q^{71} + 32 q^{79} - 328 q^{81} + 256 q^{95}+O(q^{100}) $ 8 * q + 16 * q^7 + 8 * q^9 - 64 * q^15 + 32 * q^23 - 24 * q^25 + 128 * q^39 - 280 * q^49 + 304 * q^57 + 112 * q^63 + 272 * q^65 - 544 * q^71 + 32 * q^79 - 328 * q^81 + 256 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	1.75265			0.584218			0.292109	−	0.956385i	$-0.405643\pi$
$3$	1.75265			0.584218			0.292109	+	0.956385i	$0.405643\pi$
$4$		0			0
$5$	−6.54099			−1.30820			−0.654099	−	0.756409i	$-0.726952\pi$
$5$	−6.54099			−1.30820			−0.654099	+	0.756409i	$0.726952\pi$
$6$		0			0
$7$	0.267949	+	6.99487i	0.0382785	+	0.999267i
$7$	0.267949	+	6.99487i	0.0382785	+	0.999267i
$8$		0			0
$9$	−5.92820			−0.658689
$10$		0			0
$11$			2.13878i			0.194434i	0.995263	+	0.0972172i	$0.0309942\pi$
$11$			2.13878i			0.194434i	−0.995263	+	0.0972172i	$0.969006\pi$
$12$		0			0
$13$	−12.6124			−0.970182			−0.485091	−	0.874464i	$-0.661214\pi$
$13$	−12.6124			−0.970182			−0.485091	+	0.874464i	$0.661214\pi$
$14$		0			0
$15$	−11.4641			−0.764273
$16$		0			0
$17$			24.2309i			1.42535i	0.701495	+	0.712675i	$0.252516\pi$
$17$			24.2309i			1.42535i	−0.701495	+	0.712675i	$0.747484\pi$
$18$		0			0
$19$	−13.8954			−0.731337			−0.365668	−	0.930745i	$-0.619160\pi$
$19$	−13.8954			−0.731337			−0.365668	+	0.930745i	$0.619160\pi$
$20$		0			0
$21$	0.469622	+	12.2596i	0.0223630	+	0.583790i
$22$		0			0
$23$	24.7846			1.07759			0.538796	−	0.842436i	$-0.318879\pi$
$23$	24.7846			1.07759			0.538796	+	0.842436i	$0.318879\pi$
$24$		0			0
$25$	17.7846			0.711384
$26$		0			0
$27$	−26.1640			−0.969036
$28$		0			0
$29$		−	43.6146i		−	1.50395i	−0.659190	−	0.751976i	$-0.729101\pi$
$29$		−	43.6146i		−	1.50395i	0.659190	−	0.751976i	$-0.270899\pi$
$30$		0			0
$31$			34.4721i			1.11200i	0.831181	+	0.556002i	$0.187665\pi$
$31$			34.4721i			1.11200i	−0.831181	+	0.556002i	$0.812335\pi$
$32$		0			0
$33$			3.74854i			0.113592i
$34$		0			0
$35$	−1.75265	−	45.7534i	−0.0500758	−	1.30724i
$36$		0			0
$37$			27.6506i			0.747313i	0.927567	+	0.373656i	$0.121896\pi$
$37$			27.6506i			0.747313i	−0.927567	+	0.373656i	$0.878104\pi$
$38$		0			0
$39$	−22.1051			−0.566798
$40$		0			0
$41$		−	27.9795i		−	0.682426i	−0.939986	−	0.341213i	$-0.889162\pi$
$41$		−	27.9795i		−	0.682426i	0.939986	−	0.341213i	$-0.110838\pi$
$42$		0			0
$43$			42.6220i			0.991210i	0.868548	+	0.495605i	$0.165054\pi$
$43$			42.6220i			0.991210i	−0.868548	+	0.495605i	$0.834946\pi$
$44$		0			0
$45$	38.7763			0.861697
$46$		0			0
$47$		−	34.4721i		−	0.733450i	−0.930329	−	0.366725i	$-0.880479\pi$
$47$		−	34.4721i		−	0.733450i	0.930329	−	0.366725i	$-0.119521\pi$
$48$		0			0
$49$	−48.8564	+	3.74854i	−0.997070	+	0.0765008i
$50$		0			0
$51$			42.4685i			0.832715i
$52$		0			0
$53$			35.0595i			0.661500i	0.943718	+	0.330750i	$0.107302\pi$
$53$			35.0595i			0.661500i	−0.943718	+	0.330750i	$0.892698\pi$
$54$		0			0
$55$		−	13.9897i		−	0.254359i
$56$		0			0
$57$	−24.3538			−0.427260
$58$		0			0
$59$	55.7074			0.944194			0.472097	−	0.881547i	$-0.343497\pi$
$59$	55.7074			0.944194			0.472097	+	0.881547i	$0.343497\pi$
$60$		0			0
$61$	−6.54099			−0.107229			−0.0536147	−	0.998562i	$-0.517074\pi$
$61$	−6.54099			−0.107229			−0.0536147	+	0.998562i	$0.517074\pi$
$62$		0			0
$63$	−1.58846	−	41.4670i	−0.0252136	−	0.658207i
$64$		0			0
$65$	82.4974			1.26919
$66$		0			0
$67$		−	42.6220i		−	0.636149i	−0.948066	−	0.318075i	$-0.896964\pi$
$67$		−	42.6220i		−	0.636149i	0.948066	−	0.318075i	$-0.103036\pi$
$68$		0			0
$69$	43.4389			0.629549
$70$		0			0
$71$	−22.9667			−0.323474			−0.161737	−	0.986834i	$-0.551710\pi$
$71$	−22.9667			−0.323474			−0.161737	+	0.986834i	$0.551710\pi$
$72$		0			0
$73$		−	59.7075i		−	0.817911i	−0.912554	−	0.408955i	$-0.865893\pi$
$73$		−	59.7075i		−	0.817911i	0.912554	−	0.408955i	$-0.134107\pi$
$74$		0			0
$75$	31.1703			0.415604
$76$		0			0
$77$	−14.9605	+	0.573084i	−0.194292	+	0.00744265i
$78$		0			0
$79$	35.1769			0.445277			0.222639	−	0.974901i	$-0.428533\pi$
$79$	35.1769			0.445277			0.222639	+	0.974901i	$0.428533\pi$
$80$		0			0
$81$	7.49742			0.0925608
$82$		0			0
$83$	144.715			1.74356			0.871779	−	0.489900i	$-0.162967\pi$
$83$	144.715			1.74356			0.871779	+	0.489900i	$0.162967\pi$
$84$		0			0
$85$		−	158.494i		−	1.86464i
$86$		0			0
$87$		−	76.4414i		−	0.878636i
$88$		0			0
$89$		−	108.169i		−	1.21539i	−0.794172	−	0.607693i	$-0.792095\pi$
$89$		−	108.169i		−	1.21539i	0.794172	−	0.607693i	$-0.207905\pi$
$90$		0			0
$91$	−3.37947	−	88.2219i	−0.0371371	−	0.969471i
$92$		0			0
$93$			60.4177i			0.649653i
$94$		0			0
$95$	90.8897			0.956734
$96$		0			0
$97$			136.149i			1.40360i	0.712376	+	0.701798i	$0.247619\pi$
$97$			136.149i			1.40360i	−0.712376	+	0.701798i	$0.752381\pi$
$98$		0			0
$99$		−	12.6791i		−	0.128072i
$100$		0			0
$101$	−66.8188			−0.661572			−0.330786	−	0.943706i	$-0.607314\pi$
$101$	−66.8188			−0.661572			−0.330786	+	0.943706i	$0.607314\pi$
$102$		0			0
$103$			76.4414i			0.742149i	0.928603	+	0.371075i	$0.121011\pi$
$103$			76.4414i			0.742149i	−0.928603	+	0.371075i	$0.878989\pi$
$104$		0			0
$105$	−3.07180	−	80.1899i	−0.0292552	−	0.763713i
$106$		0			0
$107$			187.138i			1.74895i	0.485071	+	0.874475i	$0.338794\pi$
$107$			187.138i			1.74895i	−0.485071	+	0.874475i	$0.661206\pi$
$108$		0			0
$109$			156.509i			1.43586i	0.696113	+	0.717932i	$0.254911\pi$
$109$			156.509i			1.43586i	−0.696113	+	0.717932i	$0.745089\pi$
$110$		0			0
$111$			48.4619i			0.436594i
$112$		0			0
$113$	−114.354			−1.01198			−0.505990	−	0.862539i	$-0.668873\pi$
$113$	−114.354			−1.01198			−0.505990	+	0.862539i	$0.668873\pi$
$114$		0			0
$115$	−162.116			−1.40970
$116$		0			0
$117$	74.7687			0.639048
$118$		0			0
$119$	−169.492	+	6.49266i	−1.42430	+	0.0545602i
$120$		0			0
$121$	116.426			0.962195
$122$		0			0
$123$		−	49.0384i		−	0.398686i
$124$		0			0
$125$	47.1958			0.377567
$126$		0			0
$127$	−151.636			−1.19398			−0.596992	−	0.802248i	$-0.703637\pi$
$127$	−151.636			−1.19398			−0.596992	+	0.802248i	$0.703637\pi$
$128$		0			0
$129$			74.7017i			0.579083i
$130$		0			0
$131$	−15.5222			−0.118490			−0.0592451	−	0.998243i	$-0.518869\pi$
$131$	−15.5222			−0.118490			−0.0592451	+	0.998243i	$0.518869\pi$
$132$		0			0
$133$	−3.72326	−	97.1965i	−0.0279944	−	0.730801i
$134$		0			0
$135$	171.138			1.26769
$136$		0			0
$137$	−72.8513			−0.531761			−0.265880	−	0.964006i	$-0.585663\pi$
$137$	−72.8513			−0.531761			−0.265880	+	0.964006i	$0.585663\pi$
$138$		0			0
$139$	−240.482			−1.73009			−0.865044	−	0.501697i	$-0.832709\pi$
$139$	−240.482			−1.73009			−0.865044	+	0.501697i	$0.832709\pi$
$140$		0			0
$141$		−	60.4177i		−	0.428495i
$142$		0			0
$143$		−	26.9751i		−	0.188637i
$144$		0			0
$145$			285.283i			1.96747i
$146$		0			0
$147$	−85.6284	+	6.56989i	−0.582506	+	0.0446932i
$148$		0			0
$149$			80.6594i			0.541338i	0.962672	+	0.270669i	$0.0872449\pi$
$149$			80.6594i			0.541338i	−0.962672	+	0.270669i	$0.912755\pi$
$150$		0			0
$151$	114.354			0.757310			0.378655	−	0.925538i	$-0.376387\pi$
$151$	114.354			0.757310			0.378655	+	0.925538i	$0.376387\pi$
$152$		0			0
$153$		−	143.646i		−	0.938862i
$154$		0			0
$155$		−	225.482i		−	1.45472i
$156$		0			0
$157$	−55.5479			−0.353808			−0.176904	−	0.984228i	$-0.556608\pi$
$157$	−55.5479			−0.353808			−0.176904	+	0.984228i	$0.556608\pi$
$158$		0			0
$159$			61.4472i			0.386460i
$160$		0			0
$161$	6.64102	+	173.365i	0.0412485	+	1.07680i
$162$		0			0
$163$			117.019i			0.717906i	0.933356	+	0.358953i	$0.116866\pi$
$163$			117.019i			0.717906i	−0.933356	+	0.358953i	$0.883134\pi$
$164$		0			0
$165$		−	24.5192i		−	0.148601i
$166$		0			0
$167$			63.4560i			0.379976i	0.981786	+	0.189988i	$0.0608450\pi$
$167$			63.4560i			0.379976i	−0.981786	+	0.189988i	$0.939155\pi$
$168$		0			0
$169$	−9.92820			−0.0587468
$170$		0			0
$171$	82.3747			0.481724
$172$		0			0
$173$	−138.803			−0.802332			−0.401166	−	0.916005i	$-0.631395\pi$
$173$	−138.803			−0.802332			−0.401166	+	0.916005i	$0.631395\pi$
$174$		0			0
$175$	4.76537	+	124.401i	0.0272307	+	0.710863i
$176$		0			0
$177$	97.6359			0.551615
$178$		0			0
$179$			129.851i			0.725426i	0.931901	+	0.362713i	$0.118149\pi$
$179$			129.851i			0.725426i	−0.931901	+	0.362713i	$0.881851\pi$
$180$		0			0
$181$	−248.524			−1.37306			−0.686531	−	0.727101i	$-0.740867\pi$
$181$	−248.524			−1.37306			−0.686531	+	0.727101i	$0.740867\pi$
$182$		0			0
$183$	−11.4641			−0.0626454
$184$		0			0
$185$		−	180.862i		−	0.977633i
$186$		0			0
$187$	−51.8246			−0.277137
$188$		0			0
$189$	−7.01062	−	183.014i	−0.0370932	−	0.968326i
$190$		0			0
$191$	−16.0385			−0.0839711			−0.0419855	−	0.999118i	$-0.513368\pi$
$191$	−16.0385			−0.0839711			−0.0419855	+	0.999118i	$0.513368\pi$
$192$		0			0
$193$	52.7846			0.273495			0.136748	−	0.990606i	$-0.456335\pi$
$193$	52.7846			0.273495			0.136748	+	0.990606i	$0.456335\pi$
$194$		0			0
$195$	144.589			0.741484
$196$		0			0
$197$			189.584i			0.962353i	0.876624	+	0.481176i	$0.159790\pi$
$197$			189.584i			0.962353i	−0.876624	+	0.481176i	$0.840210\pi$
$198$		0			0
$199$			216.339i			1.08713i	0.839367	+	0.543565i	$0.182926\pi$
$199$			216.339i			1.08713i	−0.839367	+	0.543565i	$0.817074\pi$
$200$		0			0
$201$		−	74.7017i		−	0.371650i
$202$		0			0
$203$	305.079	−	11.6865i	1.50285	−	0.0575690i
$204$		0			0
$205$			183.014i			0.892749i
$206$		0			0
$207$	−146.928			−0.709798
$208$		0			0
$209$		−	29.7192i		−	0.142197i
$210$		0			0
$211$		−	101.894i		−	0.482908i	−0.970412	−	0.241454i	$-0.922376\pi$
$211$		−	101.894i		−	0.482908i	0.970412	−	0.241454i	$-0.0776243\pi$
$212$		0			0
$213$	−40.2526			−0.188980
$214$		0			0
$215$		−	278.790i		−	1.29670i
$216$		0			0
$217$	−241.128	+	9.23678i	−1.11119	+	0.0425658i
$218$		0			0
$219$		−	104.647i		−	0.477838i
$220$		0			0
$221$		−	305.609i		−	1.38285i
$222$		0			0
$223$		−	221.827i		−	0.994740i	−0.867539	−	0.497370i	$-0.834299\pi$
$223$		−	221.827i		−	0.994740i	0.867539	−	0.497370i	$-0.165701\pi$
$224$		0			0
$225$	−105.431			−0.468581
$226$		0			0
$227$	−52.7055			−0.232183			−0.116091	−	0.993239i	$-0.537037\pi$
$227$	−52.7055			−0.232183			−0.116091	+	0.993239i	$0.537037\pi$
$228$		0			0
$229$	95.8005			0.418343			0.209171	−	0.977879i	$-0.432923\pi$
$229$	95.8005			0.418343			0.209171	+	0.977879i	$0.432923\pi$
$230$		0			0
$231$	−26.2205	+	1.00442i	−0.113509	+	0.00434813i
$232$		0			0
$233$	161.713			0.694046			0.347023	−	0.937857i	$-0.387193\pi$
$233$	161.713			0.694046			0.347023	+	0.937857i	$0.387193\pi$
$234$		0			0
$235$			225.482i			0.959498i
$236$		0			0
$237$	61.6530			0.260139
$238$		0			0
$239$	164.077			0.686514			0.343257	−	0.939241i	$-0.388470\pi$
$239$	164.077			0.686514			0.343257	+	0.939241i	$0.388470\pi$
$240$		0			0
$241$			233.073i			0.967106i	0.875315	+	0.483553i	$0.160654\pi$
$241$			233.073i			0.967106i	−0.875315	+	0.483553i	$0.839346\pi$
$242$		0			0
$243$	248.616			1.02311
$244$		0			0
$245$	319.569	−	24.5192i	1.30437	−	0.100078i
$246$		0			0
$247$	175.254			0.709530
$248$		0			0
$249$	253.636			1.01862
$250$		0			0
$251$	−187.031			−0.745142			−0.372571	−	0.928004i	$-0.621524\pi$
$251$	−187.031			−0.745142			−0.372571	+	0.928004i	$0.621524\pi$
$252$		0			0
$253$			53.0088i			0.209521i
$254$		0			0
$255$		−	277.786i		−	1.08936i
$256$		0			0
$257$		−	234.812i		−	0.913667i	−0.889552	−	0.456833i	$-0.848984\pi$
$257$		−	234.812i		−	0.913667i	0.889552	−	0.456833i	$-0.151016\pi$
$258$		0			0
$259$	−193.412	+	7.40895i	−0.746765	+	0.0286060i
$260$		0			0
$261$			258.556i			0.990637i
$262$		0			0
$263$	−167.962			−0.638637			−0.319318	−	0.947647i	$-0.603454\pi$
$263$	−167.962			−0.638637			−0.319318	+	0.947647i	$0.603454\pi$
$264$		0			0
$265$		−	229.324i		−	0.865374i
$266$		0			0
$267$		−	189.584i		−	0.710051i
$268$		0			0
$269$	439.689			1.63453			0.817266	−	0.576260i	$-0.195489\pi$
$269$	439.689			1.63453			0.817266	+	0.576260i	$0.195489\pi$
$270$		0			0
$271$			167.877i			0.619472i	0.950823	+	0.309736i	$0.100241\pi$
$271$			167.877i			0.619472i	−0.950823	+	0.309736i	$0.899759\pi$
$272$		0			0
$273$	−5.92305	−	154.622i	−0.0216961	−	0.566382i
$274$		0			0
$275$			38.0373i			0.138318i
$276$		0			0
$277$		−	69.2800i		−	0.250108i	−0.992150	−	0.125054i	$-0.960090\pi$
$277$		−	69.2800i		−	0.250108i	0.992150	−	0.125054i	$-0.0399104\pi$
$278$		0			0
$279$		−	204.358i		−	0.732465i
$280$		0			0
$281$	−130.862			−0.465700			−0.232850	−	0.972513i	$-0.574805\pi$
$281$	−130.862			−0.465700			−0.232850	+	0.972513i	$0.574805\pi$
$282$		0			0
$283$	329.238			1.16339			0.581693	−	0.813408i	$-0.302390\pi$
$283$	329.238			1.16339			0.581693	+	0.813408i	$0.302390\pi$
$284$		0			0
$285$	159.298			0.558941
$286$		0			0
$287$	195.713	−	7.49708i	0.681926	−	0.0261222i
$288$		0			0
$289$	−298.138			−1.03162
$290$		0			0
$291$			238.622i			0.820006i
$292$		0			0
$293$	235.878			0.805044			0.402522	−	0.915410i	$-0.368134\pi$
$293$	235.878			0.805044			0.402522	+	0.915410i	$0.368134\pi$
$294$		0			0
$295$	−364.382			−1.23519
$296$		0			0
$297$		−	55.9590i		−	0.188414i
$298$		0			0
$299$	−312.593			−1.04546
$300$		0			0
$301$	−298.135	+	11.4205i	−0.990483	+	0.0379420i
$302$		0			0
$303$	−117.110			−0.386503
$304$		0			0
$305$	42.7846			0.140277
$306$		0			0
$307$	201.304			0.655712			0.327856	−	0.944728i	$-0.393674\pi$
$307$	201.304			0.655712			0.327856	+	0.944728i	$0.393674\pi$
$308$		0			0
$309$			133.975i			0.433577i
$310$		0			0
$311$		−	58.9722i		−	0.189621i	−0.995495	−	0.0948106i	$-0.969775\pi$
$311$		−	58.9722i		−	0.189621i	0.995495	−	0.0948106i	$-0.0302246\pi$
$312$		0			0
$313$		−	117.406i		−	0.375100i	−0.982255	−	0.187550i	$-0.939945\pi$
$313$		−	117.406i		−	0.375100i	0.982255	−	0.187550i	$-0.0600546\pi$
$314$		0			0
$315$	10.3901	+	271.235i	0.0329844	+	0.861065i
$316$		0			0
$317$		−	216.927i		−	0.684312i	−0.939643	−	0.342156i	$-0.888843\pi$
$317$		−	216.927i		−	0.684312i	0.939643	−	0.342156i	$-0.111157\pi$
$318$		0			0
$319$	93.2820			0.292420
$320$		0			0
$321$			327.988i			1.02177i
$322$		0			0
$323$		−	336.699i		−	1.04241i
$324$		0			0
$325$	−224.306			−0.690172
$326$		0			0
$327$			274.307i			0.838858i
$328$		0			0
$329$	241.128	−	9.23678i	0.732912	−	0.0280753i
$330$		0			0
$331$		−	376.107i		−	1.13627i	−0.822934	−	0.568137i	$-0.807664\pi$
$331$		−	376.107i		−	1.13627i	0.822934	−	0.568137i	$-0.192336\pi$
$332$		0			0
$333$		−	163.918i		−	0.492247i
$334$		0			0
$335$			278.790i			0.832210i
$336$		0			0
$337$	513.626			1.52411			0.762056	−	0.647511i	$-0.224190\pi$
$337$	513.626			1.52411			0.762056	+	0.647511i	$0.224190\pi$
$338$		0			0
$339$	−200.423			−0.591217
$340$		0			0
$341$	−73.7283			−0.216212
$342$		0			0
$343$	−39.3116	−	340.740i	−0.114611	−	0.993410i
$344$		0			0
$345$	−284.133			−0.823575
$346$		0			0
$347$			47.2067i			0.136042i	0.997684	+	0.0680212i	$0.0216685\pi$
$347$			47.2067i			0.136042i	−0.997684	+	0.0680212i	$0.978331\pi$
$348$		0			0
$349$	11.2372			0.0321983			0.0160992	−	0.999870i	$-0.494875\pi$
$349$	11.2372			0.0321983			0.0160992	+	0.999870i	$0.494875\pi$
$350$		0			0
$351$	329.990			0.940142
$352$		0			0
$353$		−	149.403i		−	0.423239i	−0.977352	−	0.211619i	$-0.932126\pi$
$353$		−	149.403i		−	0.423239i	0.977352	−	0.211619i	$-0.0678737\pi$
$354$		0			0
$355$	150.225			0.423169
$356$		0			0
$357$	−297.061	+	11.3794i	−0.832105	+	0.0318750i
$358$		0			0
$359$	252.918			0.704507			0.352253	−	0.935905i	$-0.385416\pi$
$359$	252.918			0.704507			0.352253	+	0.935905i	$0.385416\pi$
$360$		0			0
$361$	−167.918			−0.465147
$362$		0			0
$363$	204.054			0.562132
$364$		0			0
$365$			390.546i			1.06999i
$366$		0			0
$367$			359.716i			0.980151i	0.871680	+	0.490076i	$0.163031\pi$
$367$			359.716i			0.980151i	−0.871680	+	0.490076i	$0.836969\pi$
$368$		0			0
$369$			165.868i			0.449507i
$370$		0			0
$371$	−245.237	+	9.39417i	−0.661015	+	0.0253212i
$372$		0			0
$373$		−	413.612i		−	1.10888i	−0.832224	−	0.554440i	$-0.812932\pi$
$373$		−	413.612i		−	1.10888i	0.832224	−	0.554440i	$-0.187068\pi$
$374$		0			0
$375$	82.7180			0.220581
$376$		0			0
$377$			550.084i			1.45911i
$378$		0			0
$379$			636.424i			1.67922i	0.543191	+	0.839609i	$0.317216\pi$
$379$			636.424i			1.67922i	−0.543191	+	0.839609i	$0.682784\pi$
$380$		0			0
$381$	−265.765			−0.697547
$382$		0			0
$383$		−	340.238i		−	0.888349i	−0.895940	−	0.444174i	$-0.853497\pi$
$383$		−	340.238i		−	0.888349i	0.895940	−	0.444174i	$-0.146503\pi$
$384$		0			0
$385$	97.8564	−	3.74854i	0.254172	−	0.00973647i
$386$		0			0
$387$		−	252.672i		−	0.652899i
$388$		0			0
$389$		−	58.9645i		−	0.151580i	−0.997124	−	0.0757898i	$-0.975852\pi$
$389$		−	58.9645i		−	0.151580i	0.997124	−	0.0757898i	$-0.0241478\pi$
$390$		0			0
$391$			600.554i			1.53594i
$392$		0			0
$393$	−27.2051			−0.0692241
$394$		0			0
$395$	−230.092			−0.582511
$396$		0			0
$397$	−561.620			−1.41466			−0.707330	−	0.706884i	$-0.750100\pi$
$397$	−561.620			−1.41466			−0.707330	+	0.706884i	$0.750100\pi$
$398$		0			0
$399$	−6.52559	−	170.352i	−0.0163549	−	0.426947i
$400$		0			0
$401$	−111.769			−0.278726			−0.139363	−	0.990241i	$-0.544505\pi$
$401$	−111.769			−0.278726			−0.139363	+	0.990241i	$0.544505\pi$
$402$		0			0
$403$		−	434.775i		−	1.07885i
$404$		0			0
$405$	−49.0406			−0.121088
$406$		0			0
$407$	−59.1384			−0.145303
$408$		0			0
$409$			311.254i			0.761011i	0.924779	+	0.380506i	$0.124250\pi$
$409$			311.254i			0.761011i	−0.924779	+	0.380506i	$0.875750\pi$
$410$		0			0
$411$	−127.683			−0.310664
$412$		0			0
$413$	14.9268	+	389.666i	0.0361423	+	0.943502i
$414$		0			0
$415$	−946.582			−2.28092
$416$		0			0
$417$	−421.482			−1.01075
$418$		0			0
$419$	−15.5222			−0.0370459			−0.0185229	−	0.999828i	$-0.505896\pi$
$419$	−15.5222			−0.0370459			−0.0185229	+	0.999828i	$0.505896\pi$
$420$		0			0
$421$		−	238.622i		−	0.566798i	−0.959002	−	0.283399i	$-0.908538\pi$
$421$		−	238.622i		−	0.566798i	0.959002	−	0.283399i	$-0.0914620\pi$
$422$		0			0
$423$			204.358i			0.483115i
$424$		0			0
$425$			430.938i			1.01397i
$426$		0			0
$427$	−1.75265	−	45.7534i	−0.00410458	−	0.107151i
$428$		0			0
$429$		−	47.2780i		−	0.110205i
$430$		0			0
$431$	−357.492			−0.829448			−0.414724	−	0.909947i	$-0.636122\pi$
$431$	−357.492			−0.829448			−0.414724	+	0.909947i	$0.636122\pi$
$432$		0			0
$433$		−	434.417i		−	1.00327i	−0.865079	−	0.501637i	$-0.832732\pi$
$433$		−	434.417i		−	1.00327i	0.865079	−	0.501637i	$-0.167268\pi$
$434$		0			0
$435$			500.003i			1.14943i
$436$		0			0
$437$	−344.392			−0.788082
$438$		0			0
$439$		−	211.855i		−	0.482585i	−0.970452	−	0.241293i	$-0.922429\pi$
$439$		−	211.855i		−	0.482585i	0.970452	−	0.241293i	$-0.0775714\pi$
$440$		0			0
$441$	289.631	−	22.2221i	0.656759	−	0.0503903i
$442$		0			0
$443$			556.071i			1.25524i	0.778520	+	0.627620i	$0.215971\pi$
$443$			556.071i			1.25524i	−0.778520	+	0.627620i	$0.784029\pi$
$444$		0			0
$445$			707.535i			1.58997i
$446$		0			0
$447$			141.368i			0.316259i
$448$		0			0
$449$	−52.5641			−0.117069			−0.0585346	−	0.998285i	$-0.518643\pi$
$449$	−52.5641			−0.117069			−0.0585346	+	0.998285i	$0.518643\pi$
$450$		0			0
$451$	59.8419			0.132687
$452$		0			0
$453$	200.423			0.442434
$454$		0			0
$455$	22.1051	+	577.059i	0.0485827	+	1.26826i
$456$		0			0
$457$	−21.8001			−0.0477026			−0.0238513	−	0.999716i	$-0.507593\pi$
$457$	−21.8001			−0.0477026			−0.0238513	+	0.999716i	$0.507593\pi$
$458$		0			0
$459$		−	633.978i		−	1.38122i
$460$		0			0
$461$	−98.9193			−0.214575			−0.107288	−	0.994228i	$-0.534217\pi$
$461$	−98.9193			−0.214575			−0.107288	+	0.994228i	$0.534217\pi$
$462$		0			0
$463$	889.069			1.92024			0.960118	−	0.279596i	$-0.0902005\pi$
$463$	889.069			1.92024			0.960118	+	0.279596i	$0.0902005\pi$
$464$		0			0
$465$		−	395.192i		−	0.849876i
$466$		0			0
$467$	−292.307			−0.625924			−0.312962	−	0.949766i	$-0.601321\pi$
$467$	−292.307			−0.625924			−0.312962	+	0.949766i	$0.601321\pi$
$468$		0			0
$469$	298.135	−	11.4205i	0.635683	−	0.0243508i
$470$		0			0
$471$	−97.3562			−0.206701
$472$		0			0
$473$	−91.1591			−0.192725
$474$		0			0
$475$	−247.124			−0.520262
$476$		0			0
$477$		−	207.840i		−	0.435723i
$478$		0			0
$479$			88.4223i			0.184598i	0.995731	+	0.0922988i	$0.0294215\pi$
$479$			88.4223i			0.184598i	−0.995731	+	0.0922988i	$0.970579\pi$
$480$		0			0
$481$		−	348.739i		−	0.725029i
$482$		0			0
$483$	11.6394	+	303.849i	0.0240981	+	0.629087i
$484$		0			0
$485$		−	890.549i		−	1.83618i
$486$		0			0
$487$	819.041			1.68181			0.840904	−	0.541184i	$-0.182024\pi$
$487$	819.041			1.68181			0.840904	+	0.541184i	$0.182024\pi$
$488$		0			0
$489$			205.093i			0.419413i
$490$		0			0
$491$			110.449i			0.224946i	0.993655	+	0.112473i	$0.0358773\pi$
$491$			110.449i			0.224946i	−0.993655	+	0.112473i	$0.964123\pi$
$492$		0			0
$493$	1056.82			2.14366
$494$		0			0
$495$			82.9340i			0.167543i
$496$		0			0
$497$	−6.15390	−	160.649i	−0.0123821	−	0.323237i
$498$		0			0
$499$		−	802.020i		−	1.60725i	−0.595133	−	0.803627i	$-0.702901\pi$
$499$		−	802.020i		−	1.60725i	0.595133	−	0.803627i	$-0.297099\pi$
$500$		0			0
$501$			111.216i			0.221989i
$502$		0			0
$503$		−	875.327i		−	1.74021i	−0.492864	−	0.870106i	$-0.664050\pi$
$503$		−	875.327i		−	1.74021i	0.492864	−	0.870106i	$-0.335950\pi$
$504$		0			0
$505$	437.061			0.865468
$506$		0			0
$507$	−17.4007			−0.0343209
$508$		0			0
$509$	224.171			0.440415			0.220207	−	0.975453i	$-0.429327\pi$
$509$	224.171			0.440415			0.220207	+	0.975453i	$0.429327\pi$
$510$		0			0
$511$	417.646	−	15.9986i	0.817311	−	0.0313084i
$512$		0			0
$513$	363.559			0.708692
$514$		0			0
$515$		−	500.003i		−	0.970879i
$516$		0			0
$517$	73.7283			0.142608
$518$		0			0
$519$	−243.274			−0.468737
$520$		0			0
$521$			755.446i			1.44999i	0.688753	+	0.724996i	$0.258158\pi$
$521$			755.446i			1.44999i	−0.688753	+	0.724996i	$0.741842\pi$
$522$		0			0
$523$	350.387			0.669956			0.334978	−	0.942226i	$-0.391271\pi$
$523$	350.387			0.669956			0.334978	+	0.942226i	$0.391271\pi$
$524$		0			0
$525$	8.35205	+	218.032i	0.0159087	+	0.415299i
$526$		0			0
$527$	−835.292			−1.58499
$528$		0			0
$529$	85.2769			0.161204
$530$		0			0
$531$	−330.245			−0.621930
$532$		0			0
$533$			352.887i			0.662078i
$534$		0			0
$535$		−	1224.07i		−	2.28797i
$536$		0			0
$537$			227.584i			0.423807i
$538$		0			0
$539$	−8.01730	−	104.493i	−0.0148744	−	0.193865i
$540$		0			0
$541$		−	407.881i		−	0.753940i	−0.926225	−	0.376970i	$-0.876966\pi$
$541$		−	407.881i		−	0.753940i	0.926225	−	0.376970i	$-0.123034\pi$
$542$		0			0
$543$	−435.577			−0.802167
$544$		0			0
$545$		−	1023.73i		−	1.87840i
$546$		0			0
$547$			1014.36i			1.85441i	0.374554	+	0.927205i	$0.377796\pi$
$547$			1014.36i			1.85441i	−0.374554	+	0.927205i	$0.622204\pi$
$548$		0			0
$549$	38.7763			0.0706309
$550$		0			0
$551$			606.043i			1.09990i
$552$		0			0
$553$	9.42563	+	246.058i	0.0170445	+	0.444951i
$554$		0			0
$555$		−	316.989i		−	0.571151i
$556$		0			0
$557$		−	300.718i		−	0.539888i	−0.962876	−	0.269944i	$-0.912995\pi$
$557$		−	300.718i		−	0.539888i	0.962876	−	0.269944i	$-0.0870052\pi$
$558$		0			0
$559$		−	537.564i		−	0.961654i
$560$		0			0
$561$	−90.8306			−0.161908
$562$		0			0
$563$	669.118			1.18849			0.594244	−	0.804285i	$-0.297451\pi$
$563$	669.118			1.18849			0.594244	+	0.804285i	$0.297451\pi$
$564$		0			0
$565$	747.988			1.32387
$566$		0			0
$567$	2.00893	+	52.4435i	0.00354308	+	0.0924929i
$568$		0			0
$569$	−831.174			−1.46076			−0.730382	−	0.683039i	$-0.760658\pi$
$569$	−831.174			−1.46076			−0.730382	+	0.683039i	$0.760658\pi$
$570$		0			0
$571$			309.958i			0.542834i	0.962462	+	0.271417i	$0.0874923\pi$
$571$			309.958i			0.542834i	−0.962462	+	0.271417i	$0.912508\pi$
$572$		0			0
$573$	−28.1099			−0.0490574
$574$		0			0
$575$	440.785			0.766582
$576$		0			0
$577$		−	458.648i		−	0.794884i	−0.917627	−	0.397442i	$-0.869898\pi$
$577$		−	458.648i		−	0.794884i	0.917627	−	0.397442i	$-0.130102\pi$
$578$		0			0
$579$	92.5132			0.159781
$580$		0			0
$581$	38.7763	+	1012.26i	0.0667407	+	1.74228i
$582$		0			0
$583$	−74.9845			−0.128618
$584$		0			0
$585$	−489.061			−0.836003
$586$		0			0
$587$	316.107			0.538513			0.269256	−	0.963069i	$-0.413222\pi$
$587$	316.107			0.538513			0.269256	+	0.963069i	$0.413222\pi$
$588$		0			0
$589$		−	479.004i		−	0.813250i
$590$		0			0
$591$			332.274i			0.562224i
$592$		0			0
$593$		−	764.952i		−	1.28997i	−0.764195	−	0.644985i	$-0.776864\pi$
$593$		−	764.952i		−	1.28997i	0.764195	−	0.644985i	$-0.223136\pi$
$594$		0			0
$595$	1108.65	−	42.4685i	1.86327	−	0.0713756i
$596$		0			0
$597$			379.167i			0.635121i
$598$		0			0
$599$	−354.295			−0.591477			−0.295739	−	0.955269i	$-0.595566\pi$
$599$	−354.295			−0.591477			−0.295739	+	0.955269i	$0.595566\pi$
$600$		0			0
$601$		−	193.578i		−	0.322094i	−0.986947	−	0.161047i	$-0.948513\pi$
$601$		−	193.578i		−	0.322094i	0.986947	−	0.161047i	$-0.0514870\pi$
$602$		0			0
$603$			252.672i			0.419025i
$604$		0			0
$605$	−761.539			−1.25874
$606$		0			0
$607$		−	527.592i		−	0.869180i	−0.900628	−	0.434590i	$-0.856893\pi$
$607$		−	527.592i		−	0.869180i	0.900628	−	0.434590i	$-0.143107\pi$
$608$		0			0
$609$	534.697	−	20.4824i	0.877992	−	0.0336328i
$610$		0			0
$611$			434.775i			0.711580i
$612$		0			0
$613$			703.175i			1.14711i	0.819169	+	0.573553i	$0.194435\pi$
$613$			703.175i			1.14711i	−0.819169	+	0.573553i	$0.805565\pi$
$614$		0			0
$615$			320.760i			0.521560i
$616$		0			0
$617$	547.902			0.888010			0.444005	−	0.896024i	$-0.353557\pi$
$617$	547.902			0.888010			0.444005	+	0.896024i	$0.353557\pi$
$618$		0			0
$619$	3.99961			0.00646141			0.00323071	−	0.999995i	$-0.498972\pi$
$619$	3.99961			0.00646141			0.00323071	+	0.999995i	$0.498972\pi$
$620$		0			0
$621$	−648.464			−1.04423
$622$		0			0
$623$	756.631	−	28.9839i	1.21450	−	0.0465231i
$624$		0			0
$625$	−753.323			−1.20532
$626$		0			0
$627$		−	52.0874i		−	0.0830741i
$628$		0			0
$629$	−669.999			−1.06518
$630$		0			0
$631$	−618.028			−0.979442			−0.489721	−	0.871879i	$-0.662901\pi$
$631$	−618.028			−0.979442			−0.489721	+	0.871879i	$0.662901\pi$
$632$		0			0
$633$		−	178.584i		−	0.282124i
$634$		0			0
$635$	991.849			1.56197
$636$		0			0
$637$	616.195	−	47.2780i	0.967339	−	0.0742197i
$638$		0			0
$639$	136.151			0.213069
$640$		0			0
$641$	545.646			0.851242			0.425621	−	0.904902i	$-0.360056\pi$
$641$	545.646			0.851242			0.425621	+	0.904902i	$0.360056\pi$
$642$		0			0
$643$	−133.579			−0.207744			−0.103872	−	0.994591i	$-0.533123\pi$
$643$	−133.579			−0.207744			−0.103872	+	0.994591i	$0.533123\pi$
$644$		0			0
$645$		−	488.623i		−	0.757555i
$646$		0			0
$647$			861.876i			1.33211i	0.745902	+	0.666055i	$0.232019\pi$
$647$			861.876i			1.33211i	−0.745902	+	0.666055i	$0.767981\pi$
$648$		0			0
$649$			119.146i			0.183584i
$650$		0			0
$651$	−422.614	+	16.1889i	−0.649177	+	0.0248677i
$652$		0			0
$653$		−	873.214i		−	1.33723i	−0.743607	−	0.668617i	$-0.766887\pi$
$653$		−	873.214i		−	1.33723i	0.743607	−	0.668617i	$-0.233113\pi$
$654$		0			0
$655$	101.531			0.155009
$656$		0			0
$657$			353.958i			0.538749i
$658$		0			0
$659$		−	71.9507i		−	0.109182i	−0.998509	−	0.0545908i	$-0.982615\pi$
$659$		−	71.9507i		−	0.109182i	0.998509	−	0.0545908i	$-0.0173854\pi$
$660$		0			0
$661$	−198.645			−0.300523			−0.150261	−	0.988646i	$-0.548011\pi$
$661$	−198.645			−0.300523			−0.150261	+	0.988646i	$0.548011\pi$
$662$		0			0
$663$		−	535.628i		−	0.807885i
$664$		0			0
$665$	24.3538	+	635.762i	0.0366223	+	0.956033i
$666$		0			0
$667$		−	1080.97i		−	1.62065i
$668$		0			0
$669$		−	388.786i		−	0.581145i
$670$		0			0
$671$		−	13.9897i		−	0.0208491i
$672$		0			0
$673$	−201.138			−0.298868			−0.149434	−	0.988772i	$-0.547745\pi$
$673$	−201.138			−0.298868			−0.149434	+	0.988772i	$0.547745\pi$
$674$		0			0
$675$	−465.316			−0.689357
$676$		0			0
$677$	−515.664			−0.761690			−0.380845	−	0.924639i	$-0.624367\pi$
$677$	−515.664			−0.761690			−0.380845	+	0.924639i	$0.624367\pi$
$678$		0			0
$679$	−952.344	+	36.4810i	−1.40257	+	0.0537275i
$680$		0			0
$681$	−92.3744			−0.135645
$682$		0			0
$683$		−	233.577i		−	0.341986i	−0.985272	−	0.170993i	$-0.945302\pi$
$683$		−	233.577i		−	0.341986i	0.985272	−	0.170993i	$-0.0546976\pi$
$684$		0			0
$685$	476.520			0.695649
$686$		0			0
$687$	167.905			0.244403
$688$		0			0
$689$		−	442.183i		−	0.641776i
$690$		0			0
$691$	−916.872			−1.32688			−0.663438	−	0.748231i	$-0.730903\pi$
$691$	−916.872			−1.32688			−0.663438	+	0.748231i	$0.730903\pi$
$692$		0			0
$693$	88.6888	−	3.39736i	0.127978	−	0.00490239i
$694$		0			0
$695$	1572.99			2.26330
$696$		0			0
$697$	677.969			0.972696
$698$		0			0
$699$	283.427			0.405474
$700$		0			0
$701$			394.906i			0.563347i	0.959510	+	0.281674i	$0.0908895\pi$
$701$			394.906i			0.563347i	−0.959510	+	0.281674i	$0.909110\pi$
$702$		0			0
$703$		−	384.216i		−	0.546537i
$704$		0			0
$705$			395.192i			0.560556i
$706$		0			0
$707$	−17.9040	−	467.389i	−0.0253240	−	0.661088i
$708$		0			0
$709$			434.243i			0.612473i	0.951955	+	0.306236i	$0.0990698\pi$
$709$			434.243i			0.612473i	−0.951955	+	0.306236i	$0.900930\pi$
$710$		0			0
$711$	−208.536			−0.293299
$712$		0			0
$713$			854.379i			1.19829i
$714$		0			0
$715$			176.444i			0.246774i
$716$		0			0
$717$	287.570			0.401074
$718$		0			0
$719$			1104.65i			1.53637i	0.640227	+	0.768186i	$0.278840\pi$
$719$			1104.65i			1.53637i	−0.640227	+	0.768186i	$0.721160\pi$
$720$		0			0
$721$	−534.697	+	20.4824i	−0.741605	+	0.0284083i
$722$		0			0
$723$			408.496i			0.565001i
$724$		0			0
$725$		−	775.669i		−	1.06989i
$726$		0			0
$727$		−	140.436i		−	0.193171i	−0.995325	−	0.0965857i	$-0.969208\pi$
$727$		−	140.436i		−	0.193171i	0.995325	−	0.0965857i	$-0.0307922\pi$
$728$		0			0
$729$	368.261			0.505160
$730$		0			0
$731$	−1032.77			−1.41282
$732$		0			0
$733$	898.934			1.22638			0.613188	−	0.789937i	$-0.289887\pi$
$733$	898.934			1.22638			0.613188	+	0.789937i	$0.289887\pi$
$734$		0			0
$735$	560.095	−	42.9736i	0.762034	−	0.0584675i
$736$		0			0
$737$	91.1591			0.123689
$738$		0			0
$739$			972.201i			1.31556i	0.753209	+	0.657781i	$0.228505\pi$
$739$			972.201i			1.31556i	−0.753209	+	0.657781i	$0.771495\pi$
$740$		0			0
$741$	307.159			0.414520
$742$		0			0
$743$	−325.072			−0.437513			−0.218756	−	0.975780i	$-0.570200\pi$
$743$	−325.072			−0.437513			−0.218756	+	0.975780i	$0.570200\pi$
$744$		0			0
$745$		−	527.592i		−	0.708178i
$746$		0			0
$747$	−857.902			−1.14846
$748$		0			0
$749$	−1309.00	+	50.1434i	−1.74767	+	0.0669471i
$750$		0			0
$751$	−1192.32			−1.58765			−0.793824	−	0.608148i	$-0.791913\pi$
$751$	−1192.32			−1.58765			−0.793824	+	0.608148i	$0.791913\pi$
$752$		0			0
$753$	−327.800			−0.435325
$754$		0			0
$755$	−747.988			−0.990712
$756$		0			0
$757$			628.022i			0.829620i	0.909908	+	0.414810i	$0.136152\pi$
$757$			628.022i			0.829620i	−0.909908	+	0.414810i	$0.863848\pi$
$758$		0			0
$759$			92.9061i			0.122406i
$760$		0			0
$761$			1176.61i			1.54613i	0.634324	+	0.773067i	$0.281279\pi$
$761$			1176.61i			1.54613i	−0.634324	+	0.773067i	$0.718721\pi$
$762$		0			0
$763$	−1094.76	+	41.9365i	−1.43481	+	0.0549627i
$764$		0			0
$765$			939.587i			1.22822i
$766$		0			0
$767$	−702.603			−0.916040
$768$		0			0
$769$			415.944i			0.540889i	0.962736	+	0.270445i	$0.0871707\pi$
$769$			415.944i			0.540889i	−0.962736	+	0.270445i	$0.912829\pi$
$770$		0			0
$771$		−	411.545i		−	0.533781i
$772$		0			0
$773$	415.840			0.537956			0.268978	−	0.963146i	$-0.413314\pi$
$773$	415.840			0.537956			0.268978	+	0.963146i	$0.413314\pi$
$774$		0			0
$775$			613.074i			0.791063i
$776$		0			0
$777$	−338.985	+	12.9853i	−0.436274	+	0.0167121i
$778$		0			0
$779$			388.786i			0.499083i
$780$		0			0
$781$		−	49.1206i		−	0.0628945i
$782$		0			0
$783$			1141.13i			1.45738i
$784$		0			0
$785$	363.338			0.462851
$786$		0			0
$787$	97.7711			0.124233			0.0621163	−	0.998069i	$-0.480215\pi$
$787$	97.7711			0.124233			0.0621163	+	0.998069i	$0.480215\pi$
$788$		0			0
$789$	−294.378			−0.373103
$790$		0			0
$791$	−30.6410	−	799.890i	−0.0387371	−	1.01124i
$792$		0			0
$793$	82.4974			0.104032
$794$		0			0
$795$		−	401.926i		−	0.505567i
$796$		0			0
$797$	−344.358			−0.432068			−0.216034	−	0.976386i	$-0.569312\pi$
$797$	−344.358			−0.432068			−0.216034	+	0.976386i	$0.569312\pi$
$798$		0			0
$799$	835.292			1.04542
$800$		0			0
$801$			641.250i			0.800562i
$802$		0			0
$803$	127.701			0.159030
$804$		0			0
$805$	−43.4389	−	1133.98i	−0.0539613	−	1.40867i
$806$		0			0
$807$	770.623			0.954923
$808$		0			0
$809$	−565.184			−0.698621			−0.349311	−	0.937007i	$-0.613584\pi$
$809$	−565.184			−0.698621			−0.349311	+	0.937007i	$0.613584\pi$
$810$		0			0
$811$	1167.49			1.43957			0.719786	−	0.694196i	$-0.244240\pi$
$811$	1167.49			1.43957			0.719786	+	0.694196i	$0.244240\pi$
$812$		0			0
$813$			294.230i			0.361907i
$814$		0			0
$815$		−	765.418i		−	0.939163i
$816$		0			0
$817$		−	592.250i		−	0.724908i
$818$		0			0
$819$	20.0342	+	522.997i	0.0244618	+	0.638580i
$820$		0			0
$821$			137.107i			0.167000i	0.996508	+	0.0834998i	$0.0266098\pi$
$821$			137.107i			0.167000i	−0.996508	+	0.0834998i	$0.973390\pi$
$822$		0			0
$823$	336.767			0.409194			0.204597	−	0.978846i	$-0.434412\pi$
$823$	336.767			0.409194			0.204597	+	0.978846i	$0.434412\pi$
$824$		0			0
$825$			66.6663i			0.0808076i
$826$		0			0
$827$		−	861.599i		−	1.04184i	−0.853607	−	0.520918i	$-0.825590\pi$
$827$		−	861.599i		−	1.04184i	0.853607	−	0.520918i	$-0.174410\pi$
$828$		0			0
$829$	−1290.59			−1.55680			−0.778401	−	0.627767i	$-0.783969\pi$
$829$	−1290.59			−1.55680			−0.778401	+	0.627767i	$0.783969\pi$
$830$		0			0
$831$		−	121.424i		−	0.146118i
$832$		0			0
$833$	−90.8306	−	1183.84i	−0.109040	−	1.42117i
$834$		0			0
$835$		−	415.066i		−	0.497085i
$836$		0			0
$837$		−	901.928i		−	1.07757i
$838$		0			0
$839$			917.835i			1.09396i	0.837145	+	0.546981i	$0.184223\pi$
$839$			917.835i			1.09396i	−0.837145	+	0.546981i	$0.815777\pi$
$840$		0			0
$841$	−1061.24			−1.26187
$842$		0			0
$843$	−229.355			−0.272070
$844$		0			0
$845$	64.9403			0.0768525
$846$		0			0
$847$	31.1962	+	814.382i	0.0368313	+	0.961490i
$848$		0			0
$849$	577.041			0.679671
$850$		0			0
$851$			685.308i			0.805298i
$852$		0			0
$853$	519.053			0.608503			0.304251	−	0.952592i	$-0.401594\pi$
$853$	519.053			0.608503			0.304251	+	0.952592i	$0.401594\pi$
$854$		0			0
$855$	−538.813			−0.630190
$856$		0			0
$857$			1090.66i			1.27265i	0.771421	+	0.636325i	$0.219546\pi$
$857$			1090.66i			1.27265i	−0.771421	+	0.636325i	$0.780454\pi$
$858$		0			0
$859$	1162.50			1.35331			0.676656	−	0.736299i	$-0.263428\pi$
$859$	1162.50			1.35331			0.676656	+	0.736299i	$0.263428\pi$
$860$		0			0
$861$	343.017	−	13.1398i	0.398394	−	0.0152611i
$862$		0			0
$863$	−1307.60			−1.51518			−0.757588	−	0.652733i	$-0.773622\pi$
$863$	−1307.60			−1.51518			−0.757588	+	0.652733i	$0.773622\pi$
$864$		0			0
$865$	907.913			1.04961
$866$		0			0
$867$	−522.534			−0.602692
$868$		0			0
$869$			75.2356i			0.0865773i
$870$		0			0
$871$			537.564i			0.617181i
$872$		0			0
$873$		−	807.118i		−	0.924534i
$874$		0			0
$875$	12.6461	+	330.129i	0.0144527	+	0.377290i
$876$		0			0
$877$			331.500i			0.377993i	0.981978	+	0.188996i	$0.0605234\pi$
$877$			331.500i			0.377993i	−0.981978	+	0.188996i	$0.939477\pi$
$878$		0			0
$879$	413.413			0.470321
$880$		0			0
$881$		−	332.274i		−	0.377156i	−0.982058	−	0.188578i	$-0.939612\pi$
$881$		−	332.274i		−	0.377156i	0.982058	−	0.188578i	$-0.0603878\pi$
$882$		0			0
$883$		−	1188.37i		−	1.34583i	−0.739718	−	0.672917i	$-0.765041\pi$
$883$		−	1188.37i		−	1.34583i	0.739718	−	0.672917i	$-0.234959\pi$
$884$		0			0
$885$	−638.636			−0.721622
$886$		0			0
$887$		−	989.720i		−	1.11581i	−0.829906	−	0.557903i	$-0.811606\pi$
$887$		−	989.720i		−	1.11581i	0.829906	−	0.557903i	$-0.188394\pi$
$888$		0			0
$889$	−40.6307	−	1060.67i	−0.0457038	−	1.19311i
$890$		0			0
$891$			16.0353i			0.0179970i
$892$		0			0
$893$			479.004i			0.536399i
$894$		0			0
$895$		−	849.356i		−	0.949002i
$896$		0			0
$897$	−547.867			−0.610777
$898$		0			0
$899$	1503.49			1.67240
$900$		0			0
$901$	−849.525			−0.942869
$902$		0			0
$903$	−522.528	+	20.0162i	−0.578658	+	0.0221664i
$904$		0			0
$905$	1625.59			1.79624
$906$		0			0
$907$			1265.20i			1.39493i	0.716618	+	0.697466i	$0.245689\pi$
$907$			1265.20i			1.39493i	−0.716618	+	0.697466i	$0.754311\pi$
$908$		0			0
$909$	396.115			0.435771
$910$		0			0
$911$	−344.169			−0.377793			−0.188896	−	0.981997i	$-0.560491\pi$
$911$	−344.169			−0.377793			−0.188896	+	0.981997i	$0.560491\pi$
$912$		0			0
$913$			309.514i			0.339008i
$914$		0			0
$915$	74.9866			0.0819526
$916$		0			0
$917$	−4.15917	−	108.576i	−0.00453562	−	0.118403i
$918$		0			0
$919$	1432.88			1.55917			0.779586	−	0.626295i	$-0.215429\pi$
$919$	1432.88			1.55917			0.779586	+	0.626295i	$0.215429\pi$
$920$		0			0
$921$	352.816			0.383079
$922$		0			0
$923$	289.664			0.313829
$924$		0			0
$925$			491.755i			0.531626i
$926$		0			0
$927$		−	453.160i		−	0.488846i
$928$		0			0
$929$			624.247i			0.671956i	0.941870	+	0.335978i	$0.109067\pi$
$929$			624.247i			0.671956i	−0.941870	+	0.335978i	$0.890933\pi$
$930$		0			0
$931$	678.879	−	52.0874i	0.729194	−	0.0559479i
$932$		0			0
$933$		−	103.358i		−	0.110780i
$934$		0			0
$935$	338.985			0.362550
$936$		0			0
$937$		−	220.626i		−	0.235460i	−0.993046	−	0.117730i	$-0.962438\pi$
$937$		−	220.626i		−	0.235460i	0.993046	−	0.117730i	$-0.0375617\pi$
$938$		0			0
$939$		−	205.772i		−	0.219140i
$940$		0			0
$941$	1457.94			1.54935			0.774674	−	0.632361i	$-0.217914\pi$
$941$	1457.94			1.54935			0.774674	+	0.632361i	$0.217914\pi$
$942$		0			0
$943$		−	693.460i		−	0.735377i
$944$		0			0
$945$	45.8564	+	1197.09i	0.0485253	+	1.26676i
$946$		0			0
$947$			1241.52i			1.31101i	0.755193	+	0.655503i	$0.227543\pi$
$947$			1241.52i			1.31101i	−0.755193	+	0.655503i	$0.772457\pi$
$948$		0			0
$949$			753.053i			0.793523i
$950$		0			0
$951$		−	380.198i		−	0.399788i
$952$		0			0
$953$	−1484.28			−1.55748			−0.778739	−	0.627348i	$-0.784140\pi$
$953$	−1484.28			−1.55748			−0.778739	+	0.627348i	$0.784140\pi$
$954$		0			0
$955$	104.908			0.109851
$956$		0			0
$957$	163.491			0.170837
$958$		0			0
$959$	−19.5204	−	509.585i	−0.0203550	−	0.531371i
$960$		0			0
$961$	−227.328			−0.236554
$962$		0			0
$963$		−	1109.39i		−	1.15201i
$964$		0			0
$965$	−345.264			−0.357786
$966$		0			0
$967$	11.4613			0.0118525			0.00592624	−	0.999982i	$-0.498114\pi$
$967$	11.4613			0.0118525			0.00592624	+	0.999982i	$0.498114\pi$
$968$		0			0
$969$		−	590.116i		−	0.608995i
$970$		0			0
$971$	−707.946			−0.729090			−0.364545	−	0.931186i	$-0.618775\pi$
$971$	−707.946			−0.729090			−0.364545	+	0.931186i	$0.618775\pi$
$972$		0			0
$973$	−64.4370	−	1682.14i	−0.0662251	−	1.72882i
$974$		0			0
$975$	−393.131			−0.403211
$976$		0			0
$977$	−1643.07			−1.68175			−0.840873	−	0.541232i	$-0.817958\pi$
$977$	−1643.07			−1.68175			−0.840873	+	0.541232i	$0.817958\pi$
$978$		0			0
$979$	231.350			0.236313
$980$		0			0
$981$		−	927.818i		−	0.945788i
$982$		0			0
$983$		−	30.5266i		−	0.0310545i	−0.999879	−	0.0155273i	$-0.995057\pi$
$983$		−	30.5266i		−	0.0310545i	0.999879	−	0.0155273i	$-0.00494268\pi$
$984$		0			0
$985$		−	1240.06i		−	1.25895i
$986$		0			0
$987$	422.614	−	16.1889i	0.428181	−	0.0164021i
$988$		0			0
$989$			1056.37i			1.06812i
$990$		0			0
$991$	911.115			0.919390			0.459695	−	0.888077i	$-0.347959\pi$
$991$	911.115			0.919390			0.459695	+	0.888077i	$0.347959\pi$
$992$		0			0
$993$		−	659.185i		−	0.663832i
$994$		0			0
$995$		−	1415.07i		−	1.42218i
$996$		0			0
$997$	1924.96			1.93076			0.965378	−	0.260853i	$-0.0840039\pi$
$997$	1924.96			1.93076			0.965378	+	0.260853i	$0.0840039\pi$
$998$		0			0
$999$		−	723.449i		−	0.724173i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.h.d.209.6		8
3.2	odd	2		2016.3.l.f.433.8		8
4.3	odd	2		56.3.h.d.13.3	yes	8
7.6	odd	2	inner	224.3.h.d.209.3		8
8.3	odd	2		56.3.h.d.13.2	yes	8
8.5	even	2	inner	224.3.h.d.209.4		8
12.11	even	2		504.3.l.f.181.6		8
21.20	even	2		2016.3.l.f.433.1		8
24.5	odd	2		2016.3.l.f.433.2		8
24.11	even	2		504.3.l.f.181.7		8
28.3	even	6		392.3.j.d.117.3		16
28.11	odd	6		392.3.j.d.117.4		16
28.19	even	6		392.3.j.d.325.7		16
28.23	odd	6		392.3.j.d.325.8		16
28.27	even	2		56.3.h.d.13.4	yes	8
56.3	even	6		392.3.j.d.117.8		16
56.11	odd	6		392.3.j.d.117.7		16
56.13	odd	2	inner	224.3.h.d.209.5		8
56.19	even	6		392.3.j.d.325.4		16
56.27	even	2		56.3.h.d.13.1	✓	8
56.51	odd	6		392.3.j.d.325.3		16
84.83	odd	2		504.3.l.f.181.5		8
168.83	odd	2		504.3.l.f.181.8		8
168.125	even	2		2016.3.l.f.433.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.h.d.13.1	✓	8	56.27	even	2
56.3.h.d.13.2	yes	8	8.3	odd	2
56.3.h.d.13.3	yes	8	4.3	odd	2
56.3.h.d.13.4	yes	8	28.27	even	2
224.3.h.d.209.3		8	7.6	odd	2	inner
224.3.h.d.209.4		8	8.5	even	2	inner
224.3.h.d.209.5		8	56.13	odd	2	inner
224.3.h.d.209.6		8	1.1	even	1	trivial
392.3.j.d.117.3		16	28.3	even	6
392.3.j.d.117.4		16	28.11	odd	6
392.3.j.d.117.7		16	56.11	odd	6
392.3.j.d.117.8		16	56.3	even	6
392.3.j.d.325.3		16	56.51	odd	6
392.3.j.d.325.4		16	56.19	even	6
392.3.j.d.325.7		16	28.19	even	6
392.3.j.d.325.8		16	28.23	odd	6
504.3.l.f.181.5		8	84.83	odd	2
504.3.l.f.181.6		8	12.11	even	2
504.3.l.f.181.7		8	24.11	even	2
504.3.l.f.181.8		8	168.83	odd	2
2016.3.l.f.433.1		8	21.20	even	2
2016.3.l.f.433.2		8	24.5	odd	2
2016.3.l.f.433.7		8	168.125	even	2
2016.3.l.f.433.8		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.75265 q^{3} -6.54099 q^{5} +(0.267949 + 6.99487i) q^{7} -5.92820 q^{9} +O(q^{10})\)	\(q+1.75265 q^{3} -6.54099 q^{5} +(0.267949 + 6.99487i) q^{7} -5.92820 q^{9} +2.13878i q^{11} -12.6124 q^{13} -11.4641 q^{15} +24.2309i q^{17} -13.8954 q^{19} +(0.469622 + 12.2596i) q^{21} +24.7846 q^{23} +17.7846 q^{25} -26.1640 q^{27} -43.6146i q^{29} +34.4721i q^{31} +3.74854i q^{33} +(-1.75265 - 45.7534i) q^{35} +27.6506i q^{37} -22.1051 q^{39} -27.9795i q^{41} +42.6220i q^{43} +38.7763 q^{45} -34.4721i q^{47} +(-48.8564 + 3.74854i) q^{49} +42.4685i q^{51} +35.0595i q^{53} -13.9897i q^{55} -24.3538 q^{57} +55.7074 q^{59} -6.54099 q^{61} +(-1.58846 - 41.4670i) q^{63} +82.4974 q^{65} -42.6220i q^{67} +43.4389 q^{69} -22.9667 q^{71} -59.7075i q^{73} +31.1703 q^{75} +(-14.9605 + 0.573084i) q^{77} +35.1769 q^{79} +7.49742 q^{81} +144.715 q^{83} -158.494i q^{85} -76.4414i q^{87} -108.169i q^{89} +(-3.37947 - 88.2219i) q^{91} +60.4177i q^{93} +90.8897 q^{95} +136.149i q^{97} -12.6791i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q + 16 * q^7 + 8 * q^9	\( 8 q + 16 q^{7} + 8 q^{9} - 64 q^{15} + 32 q^{23} - 24 q^{25} + 128 q^{39} - 280 q^{49} + 304 q^{57} + 112 q^{63} + 272 q^{65} - 544 q^{71} + 32 q^{79} - 328 q^{81} + 256 q^{95}+O(q^{100}) \) 8 * q + 16 * q^7 + 8 * q^9 - 64 * q^15 + 32 * q^23 - 24 * q^25 + 128 * q^39 - 280 * q^49 + 304 * q^57 + 112 * q^63 + 272 * q^65 - 544 * q^71 + 32 * q^79 - 328 * q^81 + 256 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more