LMFDB - Embedded newform 224.2.f.a.223.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,2,Mod(223,224)]

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([1, 0, 1]))

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

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

chi := DirichletCharacter("224.223");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	224.f (of order $2$, degree $1$, 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:	$1.78864900528$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			223.8
Root			$-0.923880 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	224.223
Dual form			224.2.f.a.223.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61313 q^{3} +1.08239i q^{5} +(-1.08239 + 2.41421i) q^{7} +3.82843 q^{9} +O(q^{10})$	$q+2.61313 q^{3} +1.08239i q^{5} +(-1.08239 + 2.41421i) q^{7} +3.82843 q^{9} +2.00000i q^{11} -4.14386i q^{13} +2.82843i q^{15} -7.39104i q^{17} -4.77791 q^{19} +(-2.82843 + 6.30864i) q^{21} -3.65685i q^{23} +3.82843 q^{25} +2.16478 q^{27} -7.65685 q^{29} +7.39104 q^{31} +5.22625i q^{33} +(-2.61313 - 1.17157i) q^{35} +3.65685 q^{37} -10.8284i q^{39} +8.28772i q^{41} +7.65685i q^{43} +4.14386i q^{45} -3.06147 q^{47} +(-4.65685 - 5.22625i) q^{49} -19.3137i q^{51} -2.00000 q^{53} -2.16478 q^{55} -12.4853 q^{57} -5.67459 q^{59} -1.08239i q^{61} +(-4.14386 + 9.24264i) q^{63} +4.48528 q^{65} +4.34315i q^{67} -9.55582i q^{69} -3.17157i q^{71} -0.896683i q^{73} +10.0042 q^{75} +(-4.82843 - 2.16478i) q^{77} +7.17157i q^{79} -5.82843 q^{81} -1.71644 q^{83} +8.00000 q^{85} -20.0083 q^{87} -5.22625i q^{89} +(10.0042 + 4.48528i) q^{91} +19.3137 q^{93} -5.17157i q^{95} +11.7206i q^{97} +7.65685i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} + 8 q^{25} - 16 q^{29} - 16 q^{37} + 8 q^{49} - 16 q^{53} - 32 q^{57} - 32 q^{65} - 16 q^{77} - 24 q^{81} + 64 q^{85} + 64 q^{93}+O(q^{100}) $ 8 * q + 8 * q^9 + 8 * q^25 - 16 * q^29 - 16 * q^37 + 8 * q^49 - 16 * q^53 - 32 * q^57 - 32 * q^65 - 16 * q^77 - 24 * q^81 + 64 * q^85 + 64 * q^93

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
$3$	2.61313			1.50869			0.754344	−	0.656479i	$-0.227955\pi$
$3$	2.61313			1.50869			0.754344	+	0.656479i	$0.227955\pi$
$4$		0			0
$5$			1.08239i			0.484061i	0.970269	+	0.242030i	$0.0778133\pi$
$5$			1.08239i			0.484061i	−0.970269	+	0.242030i	$0.922187\pi$
$6$		0			0
$7$	−1.08239	+	2.41421i	−0.409106	+	0.912487i
$7$	−1.08239	+	2.41421i	−0.409106	+	0.912487i
$8$		0			0
$9$	3.82843			1.27614
$10$		0			0
$11$			2.00000i			0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$			2.00000i			0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$		0			0
$13$		−	4.14386i		−	1.14930i	−0.818399	−	0.574650i	$-0.805138\pi$
$13$		−	4.14386i		−	1.14930i	0.818399	−	0.574650i	$-0.194862\pi$
$14$		0			0
$15$			2.82843i			0.730297i
$16$		0			0
$17$		−	7.39104i		−	1.79259i	−0.443459	−	0.896295i	$-0.646249\pi$
$17$		−	7.39104i		−	1.79259i	0.443459	−	0.896295i	$-0.353751\pi$
$18$		0			0
$19$	−4.77791			−1.09613			−0.548064	−	0.836436i	$-0.684635\pi$
$19$	−4.77791			−1.09613			−0.548064	+	0.836436i	$0.684635\pi$
$20$		0			0
$21$	−2.82843	+	6.30864i	−0.617213	+	1.37666i
$22$		0			0
$23$		−	3.65685i		−	0.762507i	−0.924471	−	0.381253i	$-0.875493\pi$
$23$		−	3.65685i		−	0.762507i	0.924471	−	0.381253i	$-0.124507\pi$
$24$		0			0
$25$	3.82843			0.765685
$26$		0			0
$27$	2.16478			0.416613
$28$		0			0
$29$	−7.65685			−1.42184			−0.710921	−	0.703272i	$-0.751722\pi$
$29$	−7.65685			−1.42184			−0.710921	+	0.703272i	$0.751722\pi$
$30$		0			0
$31$	7.39104			1.32747			0.663735	−	0.747968i	$-0.268970\pi$
$31$	7.39104			1.32747			0.663735	+	0.747968i	$0.268970\pi$
$32$		0			0
$33$			5.22625i			0.909774i
$34$		0			0
$35$	−2.61313	−	1.17157i	−0.441699	−	0.198032i
$36$		0			0
$37$	3.65685			0.601183			0.300592	−	0.953753i	$-0.402816\pi$
$37$	3.65685			0.601183			0.300592	+	0.953753i	$0.402816\pi$
$38$		0			0
$39$		−	10.8284i		−	1.73394i
$40$		0			0
$41$			8.28772i			1.29432i	0.762352	+	0.647162i	$0.224044\pi$
$41$			8.28772i			1.29432i	−0.762352	+	0.647162i	$0.775956\pi$
$42$		0			0
$43$			7.65685i			1.16766i	0.811876	+	0.583830i	$0.198446\pi$
$43$			7.65685i			1.16766i	−0.811876	+	0.583830i	$0.801554\pi$
$44$		0			0
$45$			4.14386i			0.617730i
$46$		0			0
$47$	−3.06147			−0.446561			−0.223280	−	0.974754i	$-0.571677\pi$
$47$	−3.06147			−0.446561			−0.223280	+	0.974754i	$0.571677\pi$
$48$		0			0
$49$	−4.65685	−	5.22625i	−0.665265	−	0.746607i
$50$		0			0
$51$		−	19.3137i		−	2.70446i
$52$		0			0
$53$	−2.00000			−0.274721			−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000			−0.274721			−0.137361	+	0.990521i	$0.543862\pi$
$54$		0			0
$55$	−2.16478			−0.291899
$56$		0			0
$57$	−12.4853			−1.65372
$58$		0			0
$59$	−5.67459			−0.738769			−0.369385	−	0.929277i	$-0.620431\pi$
$59$	−5.67459			−0.738769			−0.369385	+	0.929277i	$0.620431\pi$
$60$		0			0
$61$		−	1.08239i		−	0.138586i	−0.997596	−	0.0692931i	$-0.977926\pi$
$61$		−	1.08239i		−	0.138586i	0.997596	−	0.0692931i	$-0.0220744\pi$
$62$		0			0
$63$	−4.14386	+	9.24264i	−0.522077	+	1.16446i
$64$		0			0
$65$	4.48528			0.556331
$66$		0			0
$67$			4.34315i			0.530600i	0.964166	+	0.265300i	$0.0854709\pi$
$67$			4.34315i			0.530600i	−0.964166	+	0.265300i	$0.914529\pi$
$68$		0			0
$69$		−	9.55582i		−	1.15039i
$70$		0			0
$71$		−	3.17157i		−	0.376396i	−0.982131	−	0.188198i	$-0.939735\pi$
$71$		−	3.17157i		−	0.376396i	0.982131	−	0.188198i	$-0.0602647\pi$
$72$		0			0
$73$		−	0.896683i		−	0.104949i	−0.998622	−	0.0524744i	$-0.983289\pi$
$73$		−	0.896683i		−	0.104949i	0.998622	−	0.0524744i	$-0.0167108\pi$
$74$		0			0
$75$	10.0042			1.15518
$76$		0			0
$77$	−4.82843	−	2.16478i	−0.550250	−	0.246700i
$78$		0			0
$79$			7.17157i			0.806865i	0.915009	+	0.403432i	$0.132183\pi$
$79$			7.17157i			0.806865i	−0.915009	+	0.403432i	$0.867817\pi$
$80$		0			0
$81$	−5.82843			−0.647603
$82$		0			0
$83$	−1.71644			−0.188404			−0.0942020	−	0.995553i	$-0.530030\pi$
$83$	−1.71644			−0.188404			−0.0942020	+	0.995553i	$0.530030\pi$
$84$		0			0
$85$	8.00000			0.867722
$86$		0			0
$87$	−20.0083			−2.14512
$88$		0			0
$89$		−	5.22625i		−	0.553982i	−0.960873	−	0.276991i	$-0.910663\pi$
$89$		−	5.22625i		−	0.553982i	0.960873	−	0.276991i	$-0.0893372\pi$
$90$		0			0
$91$	10.0042	+	4.48528i	1.04872	+	0.470185i
$92$		0			0
$93$	19.3137			2.00274
$94$		0			0
$95$		−	5.17157i		−	0.530592i
$96$		0			0
$97$			11.7206i			1.19005i	0.803708	+	0.595024i	$0.202857\pi$
$97$			11.7206i			1.19005i	−0.803708	+	0.595024i	$0.797143\pi$
$98$		0			0
$99$			7.65685i			0.769543i
$100$		0			0
$101$			8.47343i			0.843138i	0.906796	+	0.421569i	$0.138520\pi$
$101$			8.47343i			0.843138i	−0.906796	+	0.421569i	$0.861480\pi$
$102$		0			0
$103$	11.3492			1.11827			0.559134	−	0.829077i	$-0.311134\pi$
$103$	11.3492			1.11827			0.559134	+	0.829077i	$0.311134\pi$
$104$		0			0
$105$	−6.82843	−	3.06147i	−0.666386	−	0.298769i
$106$		0			0
$107$		−	11.6569i		−	1.12691i	−0.826147	−	0.563455i	$-0.809472\pi$
$107$		−	11.6569i		−	1.12691i	0.826147	−	0.563455i	$-0.190528\pi$
$108$		0			0
$109$	17.3137			1.65835			0.829176	−	0.558987i	$-0.188810\pi$
$109$	17.3137			1.65835			0.829176	+	0.558987i	$0.188810\pi$
$110$		0			0
$111$	9.55582			0.906998
$112$		0			0
$113$	3.17157			0.298356			0.149178	−	0.988810i	$-0.452337\pi$
$113$	3.17157			0.298356			0.149178	+	0.988810i	$0.452337\pi$
$114$		0			0
$115$	3.95815			0.369099
$116$		0			0
$117$		−	15.8645i		−	1.46667i
$118$		0			0
$119$	17.8435	+	8.00000i	1.63571	+	0.733359i
$120$		0			0
$121$	7.00000			0.636364
$122$		0			0
$123$			21.6569i			1.95273i
$124$		0			0
$125$			9.55582i			0.854699i
$126$		0			0
$127$			7.65685i			0.679436i	0.940527	+	0.339718i	$0.110332\pi$
$127$			7.65685i			0.679436i	−0.940527	+	0.339718i	$0.889668\pi$
$128$		0			0
$129$			20.0083i			1.76163i
$130$		0			0
$131$	−3.50981			−0.306653			−0.153327	−	0.988176i	$-0.548999\pi$
$131$	−3.50981			−0.306653			−0.153327	+	0.988176i	$0.548999\pi$
$132$		0			0
$133$	5.17157	−	11.5349i	0.448432	−	1.00020i
$134$		0			0
$135$			2.34315i			0.201666i
$136$		0			0
$137$	2.00000			0.170872			0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000			0.170872			0.0854358	+	0.996344i	$0.472772\pi$
$138$		0			0
$139$	−10.0042			−0.848542			−0.424271	−	0.905535i	$-0.639470\pi$
$139$	−10.0042			−0.848542			−0.424271	+	0.905535i	$0.639470\pi$
$140$		0			0
$141$	−8.00000			−0.673722
$142$		0			0
$143$	8.28772			0.693054
$144$		0			0
$145$		−	8.28772i		−	0.688258i
$146$		0			0
$147$	−12.1689	−	13.6569i	−1.00368	−	1.12640i
$148$		0			0
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$		−	19.6569i		−	1.59965i	−0.600232	−	0.799826i	$-0.704925\pi$
$151$		−	19.6569i		−	1.59965i	0.600232	−	0.799826i	$-0.295075\pi$
$152$		0			0
$153$		−	28.2960i		−	2.28760i
$154$		0			0
$155$			8.00000i			0.642575i
$156$		0			0
$157$		−	8.47343i		−	0.676253i	−0.941101	−	0.338127i	$-0.890207\pi$
$157$		−	8.47343i		−	0.676253i	0.941101	−	0.338127i	$-0.109793\pi$
$158$		0			0
$159$	−5.22625			−0.414469
$160$		0			0
$161$	8.82843	+	3.95815i	0.695778	+	0.311946i
$162$		0			0
$163$			5.31371i			0.416202i	0.978107	+	0.208101i	$0.0667282\pi$
$163$			5.31371i			0.416202i	−0.978107	+	0.208101i	$0.933272\pi$
$164$		0			0
$165$	−5.65685			−0.440386
$166$		0			0
$167$	−15.6788			−1.21326			−0.606629	−	0.794985i	$-0.707479\pi$
$167$	−15.6788			−1.21326			−0.606629	+	0.794985i	$0.707479\pi$
$168$		0			0
$169$	−4.17157			−0.320890
$170$		0			0
$171$	−18.2919			−1.39882
$172$		0			0
$173$			10.6382i			0.808808i	0.914580	+	0.404404i	$0.132521\pi$
$173$			10.6382i			0.808808i	−0.914580	+	0.404404i	$0.867479\pi$
$174$		0			0
$175$	−4.14386	+	9.24264i	−0.313246	+	0.698678i
$176$		0			0
$177$	−14.8284			−1.11457
$178$		0			0
$179$			2.00000i			0.149487i	0.997203	+	0.0747435i	$0.0238138\pi$
$179$			2.00000i			0.149487i	−0.997203	+	0.0747435i	$0.976186\pi$
$180$		0			0
$181$			11.5349i			0.857382i	0.903451	+	0.428691i	$0.141025\pi$
$181$			11.5349i			0.857382i	−0.903451	+	0.428691i	$0.858975\pi$
$182$		0			0
$183$		−	2.82843i		−	0.209083i
$184$		0			0
$185$			3.95815i			0.291009i
$186$		0			0
$187$	14.7821			1.08097
$188$		0			0
$189$	−2.34315	+	5.22625i	−0.170439	+	0.380154i
$190$		0			0
$191$		−	0.828427i		−	0.0599429i	−0.999551	−	0.0299714i	$-0.990458\pi$
$191$		−	0.828427i		−	0.0599429i	0.999551	−	0.0299714i	$-0.00954164\pi$
$192$		0			0
$193$	6.48528			0.466821			0.233410	−	0.972378i	$-0.425011\pi$
$193$	6.48528			0.466821			0.233410	+	0.972378i	$0.425011\pi$
$194$		0			0
$195$	11.7206			0.839330
$196$		0			0
$197$	−7.65685			−0.545528			−0.272764	−	0.962081i	$-0.587938\pi$
$197$	−7.65685			−0.545528			−0.272764	+	0.962081i	$0.587938\pi$
$198$		0			0
$199$	−0.896683			−0.0635642			−0.0317821	−	0.999495i	$-0.510118\pi$
$199$	−0.896683			−0.0635642			−0.0317821	+	0.999495i	$0.510118\pi$
$200$		0			0
$201$			11.3492i			0.800510i
$202$		0			0
$203$	8.28772	−	18.4853i	0.581684	−	1.29741i
$204$		0			0
$205$	−8.97056			−0.626531
$206$		0			0
$207$		−	14.0000i		−	0.973067i
$208$		0			0
$209$		−	9.55582i		−	0.660990i
$210$		0			0
$211$		−	22.9706i		−	1.58136i	−0.612230	−	0.790679i	$-0.709728\pi$
$211$		−	22.9706i		−	1.58136i	0.612230	−	0.790679i	$-0.290272\pi$
$212$		0			0
$213$		−	8.28772i		−	0.567865i
$214$		0			0
$215$	−8.28772			−0.565218
$216$		0			0
$217$	−8.00000	+	17.8435i	−0.543075	+	1.21130i
$218$		0			0
$219$		−	2.34315i		−	0.158335i
$220$		0			0
$221$	−30.6274			−2.06022
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	14.6569			0.977124
$226$		0			0
$227$	29.1158			1.93248			0.966242	−	0.257637i	$-0.0829437\pi$
$227$	29.1158			1.93248			0.966242	+	0.257637i	$0.0829437\pi$
$228$		0			0
$229$			29.3784i			1.94138i	0.240332	+	0.970691i	$0.422744\pi$
$229$			29.3784i			1.94138i	−0.240332	+	0.970691i	$0.577256\pi$
$230$		0			0
$231$	−12.6173	−	5.65685i	−0.830157	−	0.372194i
$232$		0			0
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$		−	3.31371i		−	0.216163i
$236$		0			0
$237$			18.7402i			1.21731i
$238$		0			0
$239$		−	19.6569i		−	1.27150i	−0.771897	−	0.635748i	$-0.780692\pi$
$239$		−	19.6569i		−	1.27150i	0.771897	−	0.635748i	$-0.219308\pi$
$240$		0			0
$241$		−	16.0502i		−	1.03388i	−0.856021	−	0.516941i	$-0.827071\pi$
$241$		−	16.0502i		−	1.03388i	0.856021	−	0.516941i	$-0.172929\pi$
$242$		0			0
$243$	−21.7248			−1.39364
$244$		0			0
$245$	5.65685	−	5.04054i	0.361403	−	0.322028i
$246$		0			0
$247$			19.7990i			1.25978i
$248$		0			0
$249$	−4.48528			−0.284243
$250$		0			0
$251$	−2.61313			−0.164939			−0.0824695	−	0.996594i	$-0.526281\pi$
$251$	−2.61313			−0.164939			−0.0824695	+	0.996594i	$0.526281\pi$
$252$		0			0
$253$	7.31371			0.459809
$254$		0			0
$255$	20.9050			1.30912
$256$		0			0
$257$		−	14.7821i		−	0.922080i	−0.887379	−	0.461040i	$-0.847476\pi$
$257$		−	14.7821i		−	0.922080i	0.887379	−	0.461040i	$-0.152524\pi$
$258$		0			0
$259$	−3.95815	+	8.82843i	−0.245948	+	0.548572i
$260$		0			0
$261$	−29.3137			−1.81447
$262$		0			0
$263$			24.1421i			1.48867i	0.667808	+	0.744334i	$0.267233\pi$
$263$			24.1421i			1.48867i	−0.667808	+	0.744334i	$0.732767\pi$
$264$		0			0
$265$		−	2.16478i		−	0.132982i
$266$		0			0
$267$		−	13.6569i		−	0.835786i
$268$		0			0
$269$		−	25.0489i		−	1.52726i	−0.645656	−	0.763628i	$-0.723416\pi$
$269$		−	25.0489i		−	1.52726i	0.645656	−	0.763628i	$-0.276584\pi$
$270$		0			0
$271$	−25.2346			−1.53289			−0.766446	−	0.642309i	$-0.777977\pi$
$271$	−25.2346			−1.53289			−0.766446	+	0.642309i	$0.777977\pi$
$272$		0			0
$273$	26.1421	+	11.7206i	1.58219	+	0.709363i
$274$		0			0
$275$			7.65685i			0.461726i
$276$		0			0
$277$	6.97056			0.418821			0.209410	−	0.977828i	$-0.432846\pi$
$277$	6.97056			0.418821			0.209410	+	0.977828i	$0.432846\pi$
$278$		0			0
$279$	28.2960			1.69404
$280$		0			0
$281$	16.6274			0.991909			0.495954	−	0.868349i	$-0.334818\pi$
$281$	16.6274			0.991909			0.495954	+	0.868349i	$0.334818\pi$
$282$		0			0
$283$	−2.61313			−0.155334			−0.0776671	−	0.996979i	$-0.524747\pi$
$283$	−2.61313			−0.155334			−0.0776671	+	0.996979i	$0.524747\pi$
$284$		0			0
$285$		−	13.5140i		−	0.800499i
$286$		0			0
$287$	−20.0083	−	8.97056i	−1.18105	−	0.529516i
$288$		0			0
$289$	−37.6274			−2.21338
$290$		0			0
$291$			30.6274i			1.79541i
$292$		0			0
$293$			8.47343i			0.495023i	0.968885	+	0.247511i	$0.0796128\pi$
$293$			8.47343i			0.495023i	−0.968885	+	0.247511i	$0.920387\pi$
$294$		0			0
$295$		−	6.14214i		−	0.357609i
$296$		0			0
$297$			4.32957i			0.251227i
$298$		0			0
$299$	−15.1535			−0.876349
$300$		0			0
$301$	−18.4853	−	8.28772i	−1.06547	−	0.477696i
$302$		0			0
$303$			22.1421i			1.27203i
$304$		0			0
$305$	1.17157			0.0670841
$306$		0			0
$307$	−1.71644			−0.0979626			−0.0489813	−	0.998800i	$-0.515597\pi$
$307$	−1.71644			−0.0979626			−0.0489813	+	0.998800i	$0.515597\pi$
$308$		0			0
$309$	29.6569			1.68712
$310$		0			0
$311$	−3.95815			−0.224446			−0.112223	−	0.993683i	$-0.535797\pi$
$311$	−3.95815			−0.224446			−0.112223	+	0.993683i	$0.535797\pi$
$312$		0			0
$313$			6.49435i			0.367083i	0.983012	+	0.183541i	$0.0587561\pi$
$313$			6.49435i			0.367083i	−0.983012	+	0.183541i	$0.941244\pi$
$314$		0			0
$315$	−10.0042	−	4.48528i	−0.563671	−	0.252717i
$316$		0			0
$317$	16.3431			0.917923			0.458961	−	0.888456i	$-0.348222\pi$
$317$	16.3431			0.917923			0.458961	+	0.888456i	$0.348222\pi$
$318$		0			0
$319$		−	15.3137i		−	0.857403i
$320$		0			0
$321$		−	30.4608i		−	1.70016i
$322$		0			0
$323$			35.3137i			1.96491i
$324$		0			0
$325$		−	15.8645i		−	0.880002i
$326$		0			0
$327$	45.2429			2.50194
$328$		0			0
$329$	3.31371	−	7.39104i	0.182691	−	0.407481i
$330$		0			0
$331$			5.31371i			0.292068i	0.989280	+	0.146034i	$0.0466509\pi$
$331$			5.31371i			0.292068i	−0.989280	+	0.146034i	$0.953349\pi$
$332$		0			0
$333$	14.0000			0.767195
$334$		0			0
$335$	−4.70099			−0.256842
$336$		0			0
$337$	3.17157			0.172767			0.0863833	−	0.996262i	$-0.472469\pi$
$337$	3.17157			0.172767			0.0863833	+	0.996262i	$0.472469\pi$
$338$		0			0
$339$	8.28772			0.450127
$340$		0			0
$341$			14.7821i			0.800494i
$342$		0			0
$343$	17.6578	−	5.58579i	0.953433	−	0.301604i
$344$		0			0
$345$	10.3431			0.556856
$346$		0			0
$347$		−	8.34315i		−	0.447884i	−0.974603	−	0.223942i	$-0.928107\pi$
$347$		−	8.34315i		−	0.447884i	0.974603	−	0.223942i	$-0.0718925\pi$
$348$		0			0
$349$		−	15.8645i		−	0.849205i	−0.905380	−	0.424603i	$-0.860414\pi$
$349$		−	15.8645i		−	0.849205i	0.905380	−	0.424603i	$-0.139586\pi$
$350$		0			0
$351$		−	8.97056i		−	0.478813i
$352$		0			0
$353$		−	12.2459i		−	0.651782i	−0.945407	−	0.325891i	$-0.894336\pi$
$353$		−	12.2459i		−	0.651782i	0.945407	−	0.325891i	$-0.105664\pi$
$354$		0			0
$355$	3.43289			0.182199
$356$		0			0
$357$	46.6274	+	20.9050i	2.46778	+	1.10641i
$358$		0			0
$359$			18.9706i			1.00123i	0.865671	+	0.500614i	$0.166892\pi$
$359$			18.9706i			1.00123i	−0.865671	+	0.500614i	$0.833108\pi$
$360$		0			0
$361$	3.82843			0.201496
$362$		0			0
$363$	18.2919			0.960075
$364$		0			0
$365$	0.970563			0.0508016
$366$		0			0
$367$	4.32957			0.226002			0.113001	−	0.993595i	$-0.463954\pi$
$367$	4.32957			0.226002			0.113001	+	0.993595i	$0.463954\pi$
$368$		0			0
$369$			31.7289i			1.65174i
$370$		0			0
$371$	2.16478	−	4.82843i	0.112390	−	0.250679i
$372$		0			0
$373$	−26.9706			−1.39648			−0.698241	−	0.715862i	$-0.746034\pi$
$373$	−26.9706			−1.39648			−0.698241	+	0.715862i	$0.746034\pi$
$374$		0			0
$375$			24.9706i			1.28947i
$376$		0			0
$377$			31.7289i			1.63412i
$378$		0			0
$379$		−	17.3137i		−	0.889345i	−0.895693	−	0.444673i	$-0.853320\pi$
$379$		−	17.3137i		−	0.889345i	0.895693	−	0.444673i	$-0.146680\pi$
$380$		0			0
$381$			20.0083i			1.02506i
$382$		0			0
$383$	13.5140			0.690532			0.345266	−	0.938505i	$-0.387789\pi$
$383$	13.5140			0.690532			0.345266	+	0.938505i	$0.387789\pi$
$384$		0			0
$385$	2.34315	−	5.22625i	0.119418	−	0.266354i
$386$		0			0
$387$			29.3137i			1.49010i
$388$		0			0
$389$	8.34315			0.423014			0.211507	−	0.977376i	$-0.432163\pi$
$389$	8.34315			0.423014			0.211507	+	0.977376i	$0.432163\pi$
$390$		0			0
$391$	−27.0279			−1.36686
$392$		0			0
$393$	−9.17157			−0.462645
$394$		0			0
$395$	−7.76245			−0.390571
$396$		0			0
$397$		−	14.0711i		−	0.706208i	−0.935584	−	0.353104i	$-0.885126\pi$
$397$		−	14.0711i		−	0.706208i	0.935584	−	0.353104i	$-0.114874\pi$
$398$		0			0
$399$	13.5140	−	30.1421i	0.676545	−	1.50899i
$400$		0			0
$401$	−20.8284			−1.04012			−0.520061	−	0.854129i	$-0.674091\pi$
$401$	−20.8284			−1.04012			−0.520061	+	0.854129i	$0.674091\pi$
$402$		0			0
$403$		−	30.6274i		−	1.52566i
$404$		0			0
$405$		−	6.30864i		−	0.313479i
$406$		0			0
$407$			7.31371i			0.362527i
$408$		0			0
$409$			12.6173i			0.623885i	0.950101	+	0.311942i	$0.100980\pi$
$409$			12.6173i			0.623885i	−0.950101	+	0.311942i	$0.899020\pi$
$410$		0			0
$411$	5.22625			0.257792
$412$		0			0
$413$	6.14214	−	13.6997i	0.302235	−	0.674117i
$414$		0			0
$415$		−	1.85786i		−	0.0911990i
$416$		0			0
$417$	−26.1421			−1.28019
$418$		0			0
$419$	5.14933			0.251561			0.125781	−	0.992058i	$-0.459856\pi$
$419$	5.14933			0.251561			0.125781	+	0.992058i	$0.459856\pi$
$420$		0			0
$421$	1.31371			0.0640262			0.0320131	−	0.999487i	$-0.489808\pi$
$421$	1.31371			0.0640262			0.0320131	+	0.999487i	$0.489808\pi$
$422$		0			0
$423$	−11.7206			−0.569875
$424$		0			0
$425$		−	28.2960i		−	1.37256i
$426$		0			0
$427$	2.61313	+	1.17157i	0.126458	+	0.0566964i
$428$		0			0
$429$	21.6569			1.04560
$430$		0			0
$431$			23.6569i			1.13951i	0.821814	+	0.569755i	$0.192962\pi$
$431$			23.6569i			1.13951i	−0.821814	+	0.569755i	$0.807038\pi$
$432$		0			0
$433$			28.2960i			1.35982i	0.733295	+	0.679911i	$0.237981\pi$
$433$			28.2960i			1.35982i	−0.733295	+	0.679911i	$0.762019\pi$
$434$		0			0
$435$		−	21.6569i		−	1.03837i
$436$		0			0
$437$			17.4721i			0.835805i
$438$		0			0
$439$	−39.6452			−1.89216			−0.946082	−	0.323928i	$-0.894997\pi$
$439$	−39.6452			−1.89216			−0.946082	+	0.323928i	$0.894997\pi$
$440$		0			0
$441$	−17.8284	−	20.0083i	−0.848973	−	0.952777i
$442$		0			0
$443$			10.0000i			0.475114i	0.971374	+	0.237557i	$0.0763467\pi$
$443$			10.0000i			0.475114i	−0.971374	+	0.237557i	$0.923653\pi$
$444$		0			0
$445$	5.65685			0.268161
$446$		0			0
$447$	−26.1313			−1.23597
$448$		0			0
$449$	−1.31371			−0.0619977			−0.0309989	−	0.999519i	$-0.509869\pi$
$449$	−1.31371			−0.0619977			−0.0309989	+	0.999519i	$0.509869\pi$
$450$		0			0
$451$	−16.5754			−0.780507
$452$		0			0
$453$		−	51.3658i		−	2.41338i
$454$		0			0
$455$	−4.85483	+	10.8284i	−0.227598	+	0.507644i
$456$		0			0
$457$	−25.1127			−1.17472			−0.587361	−	0.809325i	$-0.699833\pi$
$457$	−25.1127			−1.17472			−0.587361	+	0.809325i	$0.699833\pi$
$458$		0			0
$459$		−	16.0000i		−	0.746816i
$460$		0			0
$461$		−	32.4399i		−	1.51088i	−0.655220	−	0.755438i	$-0.727424\pi$
$461$		−	32.4399i		−	1.51088i	0.655220	−	0.755438i	$-0.272576\pi$
$462$		0			0
$463$		−	12.1421i		−	0.564293i	−0.959371	−	0.282146i	$-0.908954\pi$
$463$		−	12.1421i		−	0.564293i	0.959371	−	0.282146i	$-0.0910464\pi$
$464$		0			0
$465$			20.9050i			0.969447i
$466$		0			0
$467$	24.7862			1.14697			0.573485	−	0.819216i	$-0.305591\pi$
$467$	24.7862			1.14697			0.573485	+	0.819216i	$0.305591\pi$
$468$		0			0
$469$	−10.4853	−	4.70099i	−0.484165	−	0.217071i
$470$		0			0
$471$		−	22.1421i		−	1.02026i
$472$		0			0
$473$	−15.3137			−0.704125
$474$		0			0
$475$	−18.2919			−0.839289
$476$		0			0
$477$	−7.65685			−0.350583
$478$		0			0
$479$	−30.8322			−1.40876			−0.704381	−	0.709822i	$-0.748775\pi$
$479$	−30.8322			−1.40876			−0.704381	+	0.709822i	$0.748775\pi$
$480$		0			0
$481$		−	15.1535i		−	0.690940i
$482$		0			0
$483$	23.0698	+	10.3431i	1.04971	+	0.470629i
$484$		0			0
$485$	−12.6863			−0.576055
$486$		0			0
$487$		−	26.2843i		−	1.19105i	−0.803335	−	0.595527i	$-0.796943\pi$
$487$		−	26.2843i		−	1.19105i	0.803335	−	0.595527i	$-0.203057\pi$
$488$		0			0
$489$			13.8854i			0.627919i
$490$		0			0
$491$		−	26.2843i		−	1.18619i	−0.805132	−	0.593096i	$-0.797905\pi$
$491$		−	26.2843i		−	1.18619i	0.805132	−	0.593096i	$-0.202095\pi$
$492$		0			0
$493$			56.5921i			2.54878i
$494$		0			0
$495$	−8.28772			−0.372505
$496$		0			0
$497$	7.65685	+	3.43289i	0.343457	+	0.153986i
$498$		0			0
$499$			5.31371i			0.237874i	0.992902	+	0.118937i	$0.0379487\pi$
$499$			5.31371i			0.237874i	−0.992902	+	0.118937i	$0.962051\pi$
$500$		0			0
$501$	−40.9706			−1.83043
$502$		0			0
$503$	31.7289			1.41472			0.707362	−	0.706852i	$-0.249885\pi$
$503$	31.7289			1.41472			0.707362	+	0.706852i	$0.249885\pi$
$504$		0			0
$505$	−9.17157			−0.408130
$506$		0			0
$507$	−10.9008			−0.484124
$508$		0			0
$509$			25.9456i			1.15002i	0.818148	+	0.575008i	$0.195001\pi$
$509$			25.9456i			1.15002i	−0.818148	+	0.575008i	$0.804999\pi$
$510$		0			0
$511$	2.16478	+	0.970563i	0.0957644	+	0.0429352i
$512$		0			0
$513$	−10.3431			−0.456661
$514$		0			0
$515$			12.2843i			0.541310i
$516$		0			0
$517$		−	6.12293i		−	0.269286i
$518$		0			0
$519$			27.7990i			1.22024i
$520$		0			0
$521$		−	12.6173i		−	0.552773i	−0.961047	−	0.276387i	$-0.910863\pi$
$521$		−	12.6173i		−	0.552773i	0.961047	−	0.276387i	$-0.0891371\pi$
$522$		0			0
$523$	−1.34502			−0.0588138			−0.0294069	−	0.999568i	$-0.509362\pi$
$523$	−1.34502			−0.0588138			−0.0294069	+	0.999568i	$0.509362\pi$
$524$		0			0
$525$	−10.8284	+	24.1522i	−0.472591	+	1.05409i
$526$		0			0
$527$		−	54.6274i		−	2.37961i
$528$		0			0
$529$	9.62742			0.418583
$530$		0			0
$531$	−21.7248			−0.942775
$532$		0			0
$533$	34.3431			1.48757
$534$		0			0
$535$	12.6173			0.545493
$536$		0			0
$537$			5.22625i			0.225529i
$538$		0			0
$539$	10.4525	−	9.31371i	0.450221	−	0.401170i
$540$		0			0
$541$	2.68629			0.115493			0.0577463	−	0.998331i	$-0.481609\pi$
$541$	2.68629			0.115493			0.0577463	+	0.998331i	$0.481609\pi$
$542$		0			0
$543$			30.1421i			1.29352i
$544$		0			0
$545$			18.7402i			0.802743i
$546$		0			0
$547$		−	10.6863i		−	0.456913i	−0.973554	−	0.228456i	$-0.926632\pi$
$547$		−	10.6863i		−	0.456913i	0.973554	−	0.228456i	$-0.0733678\pi$
$548$		0			0
$549$		−	4.14386i		−	0.176856i
$550$		0			0
$551$	36.5838			1.55852
$552$		0			0
$553$	−17.3137	−	7.76245i	−0.736254	−	0.330093i
$554$		0			0
$555$			10.3431i			0.439042i
$556$		0			0
$557$	17.3137			0.733605			0.366803	−	0.930299i	$-0.380452\pi$
$557$	17.3137			0.733605			0.366803	+	0.930299i	$0.380452\pi$
$558$		0			0
$559$	31.7289			1.34199
$560$		0			0
$561$	38.6274			1.63085
$562$		0			0
$563$	−25.6829			−1.08241			−0.541203	−	0.840892i	$-0.682031\pi$
$563$	−25.6829			−1.08241			−0.541203	+	0.840892i	$0.682031\pi$
$564$		0			0
$565$			3.43289i			0.144423i
$566$		0			0
$567$	6.30864	−	14.0711i	0.264938	−	0.590929i
$568$		0			0
$569$	36.1421			1.51516			0.757579	−	0.652744i	$-0.226382\pi$
$569$	36.1421			1.51516			0.757579	+	0.652744i	$0.226382\pi$
$570$		0			0
$571$		−	19.6569i		−	0.822614i	−0.911497	−	0.411307i	$-0.865072\pi$
$571$		−	19.6569i		−	0.822614i	0.911497	−	0.411307i	$-0.134928\pi$
$572$		0			0
$573$		−	2.16478i		−	0.0904352i
$574$		0			0
$575$		−	14.0000i		−	0.583840i
$576$		0			0
$577$			2.53620i			0.105583i	0.998606	+	0.0527917i	$0.0168120\pi$
$577$			2.53620i			0.105583i	−0.998606	+	0.0527917i	$0.983188\pi$
$578$		0			0
$579$	16.9469			0.704287
$580$		0			0
$581$	1.85786	−	4.14386i	0.0770772	−	0.171916i
$582$		0			0
$583$		−	4.00000i		−	0.165663i
$584$		0			0
$585$	17.1716			0.709957
$586$		0			0
$587$	−33.4454			−1.38044			−0.690219	−	0.723600i	$-0.742486\pi$
$587$	−33.4454			−1.38044			−0.690219	+	0.723600i	$0.742486\pi$
$588$		0			0
$589$	−35.3137			−1.45508
$590$		0			0
$591$	−20.0083			−0.823032
$592$		0			0
$593$			25.2346i			1.03626i	0.855302	+	0.518130i	$0.173372\pi$
$593$			25.2346i			1.03626i	−0.855302	+	0.518130i	$0.826628\pi$
$594$		0			0
$595$	−8.65914	+	19.3137i	−0.354990	+	0.791785i
$596$		0			0
$597$	−2.34315			−0.0958986
$598$		0			0
$599$			36.8284i			1.50477i	0.658724	+	0.752384i	$0.271096\pi$
$599$			36.8284i			1.50477i	−0.658724	+	0.752384i	$0.728904\pi$
$600$		0			0
$601$			13.1426i			0.536096i	0.963406	+	0.268048i	$0.0863786\pi$
$601$			13.1426i			0.536096i	−0.963406	+	0.268048i	$0.913621\pi$
$602$		0			0
$603$			16.6274i			0.677121i
$604$		0			0
$605$			7.57675i			0.308039i
$606$		0			0
$607$	41.8100			1.69702			0.848508	−	0.529182i	$-0.177501\pi$
$607$	41.8100			1.69702			0.848508	+	0.529182i	$0.177501\pi$
$608$		0			0
$609$	21.6569	−	48.3044i	0.877580	−	1.95739i
$610$		0			0
$611$			12.6863i			0.513232i
$612$		0			0
$613$	17.3137			0.699294			0.349647	−	0.936881i	$-0.386302\pi$
$613$	17.3137			0.699294			0.349647	+	0.936881i	$0.386302\pi$
$614$		0			0
$615$	−23.4412			−0.945241
$616$		0			0
$617$	−49.1127			−1.97720			−0.988601	−	0.150557i	$-0.951893\pi$
$617$	−49.1127			−1.97720			−0.988601	+	0.150557i	$0.951893\pi$
$618$		0			0
$619$	22.6215			0.909233			0.454616	−	0.890687i	$-0.349776\pi$
$619$	22.6215			0.909233			0.454616	+	0.890687i	$0.349776\pi$
$620$		0			0
$621$		−	7.91630i		−	0.317670i
$622$		0			0
$623$	12.6173	+	5.65685i	0.505501	+	0.226637i
$624$		0			0
$625$	8.79899			0.351960
$626$		0			0
$627$		−	24.9706i		−	0.997228i
$628$		0			0
$629$		−	27.0279i		−	1.07767i
$630$		0			0
$631$			9.51472i			0.378775i	0.981902	+	0.189387i	$0.0606502\pi$
$631$			9.51472i			0.378775i	−0.981902	+	0.189387i	$0.939350\pi$
$632$		0			0
$633$		−	60.0250i		−	2.38578i
$634$		0			0
$635$	−8.28772			−0.328888
$636$		0			0
$637$	−21.6569	+	19.2974i	−0.858076	+	0.764589i
$638$		0			0
$639$		−	12.1421i		−	0.480335i
$640$		0			0
$641$	−0.142136			−0.00561402			−0.00280701	−	0.999996i	$-0.500894\pi$
$641$	−0.142136			−0.00561402			−0.00280701	+	0.999996i	$0.500894\pi$
$642$		0			0
$643$	26.5796			1.04820			0.524099	−	0.851658i	$-0.324402\pi$
$643$	26.5796			1.04820			0.524099	+	0.851658i	$0.324402\pi$
$644$		0			0
$645$	−21.6569			−0.852738
$646$		0			0
$647$	40.9133			1.60847			0.804235	−	0.594312i	$-0.202576\pi$
$647$	40.9133			1.60847			0.804235	+	0.594312i	$0.202576\pi$
$648$		0			0
$649$		−	11.3492i		−	0.445495i
$650$		0			0
$651$	−20.9050	+	46.6274i	−0.819332	+	1.82747i
$652$		0			0
$653$	25.3137			0.990602			0.495301	−	0.868721i	$-0.335058\pi$
$653$	25.3137			0.990602			0.495301	+	0.868721i	$0.335058\pi$
$654$		0			0
$655$		−	3.79899i		−	0.148439i
$656$		0			0
$657$		−	3.43289i		−	0.133930i
$658$		0			0
$659$			50.9706i			1.98553i	0.120068	+	0.992766i	$0.461689\pi$
$659$			50.9706i			1.98553i	−0.120068	+	0.992766i	$0.538311\pi$
$660$		0			0
$661$			6.68006i			0.259824i	0.991526	+	0.129912i	$0.0414695\pi$
$661$			6.68006i			0.259824i	−0.991526	+	0.129912i	$0.958530\pi$
$662$		0			0
$663$	−80.0333			−3.10824
$664$		0			0
$665$	12.4853	+	5.59767i	0.484158	+	0.217068i
$666$		0			0
$667$			28.0000i			1.08416i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2.16478			0.0835706
$672$		0			0
$673$	29.3137			1.12996			0.564980	−	0.825104i	$-0.308884\pi$
$673$	29.3137			1.12996			0.564980	+	0.825104i	$0.308884\pi$
$674$		0			0
$675$	8.28772			0.318994
$676$		0			0
$677$		−	22.3588i		−	0.859319i	−0.902991	−	0.429660i	$-0.858634\pi$
$677$		−	22.3588i		−	0.859319i	0.902991	−	0.429660i	$-0.141366\pi$
$678$		0			0
$679$	−28.2960	−	12.6863i	−1.08590	−	0.486855i
$680$		0			0
$681$	76.0833			2.91552
$682$		0			0
$683$		−	32.3431i		−	1.23758i	−0.785558	−	0.618788i	$-0.787624\pi$
$683$		−	32.3431i		−	1.23758i	0.785558	−	0.618788i	$-0.212376\pi$
$684$		0			0
$685$			2.16478i			0.0827122i
$686$		0			0
$687$			76.7696i			2.92894i
$688$		0			0
$689$			8.28772i			0.315737i
$690$		0			0
$691$	−10.3756			−0.394706			−0.197353	−	0.980333i	$-0.563234\pi$
$691$	−10.3756			−0.394706			−0.197353	+	0.980333i	$0.563234\pi$
$692$		0			0
$693$	−18.4853	−	8.28772i	−0.702198	−	0.314824i
$694$		0			0
$695$		−	10.8284i		−	0.410746i
$696$		0			0
$697$	61.2548			2.32019
$698$		0			0
$699$	47.0363			1.77907
$700$		0			0
$701$	−13.3137			−0.502852			−0.251426	−	0.967877i	$-0.580899\pi$
$701$	−13.3137			−0.502852			−0.251426	+	0.967877i	$0.580899\pi$
$702$		0			0
$703$	−17.4721			−0.658974
$704$		0			0
$705$		−	8.65914i		−	0.326122i
$706$		0			0
$707$	−20.4567	−	9.17157i	−0.769352	−	0.344932i
$708$		0			0
$709$	−18.0000			−0.676004			−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000			−0.676004			−0.338002	+	0.941145i	$0.609751\pi$
$710$		0			0
$711$			27.4558i			1.02967i
$712$		0			0
$713$		−	27.0279i		−	1.01220i
$714$		0			0
$715$			8.97056i			0.335480i
$716$		0			0
$717$		−	51.3658i		−	1.91829i
$718$		0			0
$719$	−20.3797			−0.760036			−0.380018	−	0.924979i	$-0.624082\pi$
$719$	−20.3797			−0.760036			−0.380018	+	0.924979i	$0.624082\pi$
$720$		0			0
$721$	−12.2843	+	27.3994i	−0.457490	+	1.02041i
$722$		0			0
$723$		−	41.9411i		−	1.55981i
$724$		0			0
$725$	−29.3137			−1.08868
$726$		0			0
$727$	13.1426			0.487430			0.243715	−	0.969847i	$-0.421634\pi$
$727$	13.1426			0.487430			0.243715	+	0.969847i	$0.421634\pi$
$728$		0			0
$729$	−39.2843			−1.45497
$730$		0			0
$731$	56.5921			2.09313
$732$		0			0
$733$			22.3588i			0.825842i	0.910767	+	0.412921i	$0.135491\pi$
$733$			22.3588i			0.825842i	−0.910767	+	0.412921i	$0.864509\pi$
$734$		0			0
$735$	14.7821	−	13.1716i	0.545245	−	0.485841i
$736$		0			0
$737$	−8.68629			−0.319964
$738$		0			0
$739$		−	28.6274i		−	1.05308i	−0.850151	−	0.526538i	$-0.823490\pi$
$739$		−	28.6274i		−	1.05308i	0.850151	−	0.526538i	$-0.176510\pi$
$740$		0			0
$741$			51.7373i			1.90062i
$742$		0			0
$743$		−	35.6569i		−	1.30812i	−0.756441	−	0.654062i	$-0.773064\pi$
$743$		−	35.6569i		−	1.30812i	0.756441	−	0.654062i	$-0.226936\pi$
$744$		0			0
$745$		−	10.8239i		−	0.396558i
$746$		0			0
$747$	−6.57128			−0.240430
$748$		0			0
$749$	28.1421	+	12.6173i	1.02829	+	0.461026i
$750$		0			0
$751$			18.9706i			0.692246i	0.938189	+	0.346123i	$0.112502\pi$
$751$			18.9706i			0.692246i	−0.938189	+	0.346123i	$0.887498\pi$
$752$		0			0
$753$	−6.82843			−0.248842
$754$		0			0
$755$	21.2764			0.774328
$756$		0			0
$757$	26.6863			0.969930			0.484965	−	0.874534i	$-0.338832\pi$
$757$	26.6863			0.969930			0.484965	+	0.874534i	$0.338832\pi$
$758$		0			0
$759$	19.1116			0.693709
$760$		0			0
$761$			33.5223i			1.21518i	0.794250	+	0.607591i	$0.207864\pi$
$761$			33.5223i			1.21518i	−0.794250	+	0.607591i	$0.792136\pi$
$762$		0			0
$763$	−18.7402	+	41.7990i	−0.678442	+	1.51323i
$764$		0			0
$765$	30.6274			1.10734
$766$		0			0
$767$			23.5147i			0.849067i
$768$		0			0
$769$		−	11.7206i		−	0.422656i	−0.977415	−	0.211328i	$-0.932221\pi$
$769$		−	11.7206i		−	0.422656i	0.977415	−	0.211328i	$-0.0677788\pi$
$770$		0			0
$771$		−	38.6274i		−	1.39113i
$772$		0			0
$773$			13.3283i			0.479384i	0.970849	+	0.239692i	$0.0770465\pi$
$773$			13.3283i			0.479384i	−0.970849	+	0.239692i	$0.922953\pi$
$774$		0			0
$775$	28.2960			1.01642
$776$		0			0
$777$	−10.3431	+	23.0698i	−0.371058	+	0.827624i
$778$		0			0
$779$		−	39.5980i		−	1.41874i
$780$		0			0
$781$	6.34315			0.226976
$782$		0			0
$783$	−16.5754			−0.592358
$784$		0			0
$785$	9.17157			0.327347
$786$		0			0
$787$	−25.6829			−0.915497			−0.457749	−	0.889082i	$-0.651344\pi$
$787$	−25.6829			−0.915497			−0.457749	+	0.889082i	$0.651344\pi$
$788$		0			0
$789$			63.0864i			2.24594i
$790$		0			0
$791$	−3.43289	+	7.65685i	−0.122059	+	0.272246i
$792$		0			0
$793$	−4.48528			−0.159277
$794$		0			0
$795$		−	5.65685i		−	0.200628i
$796$		0			0
$797$		−	11.0096i		−	0.389981i	−0.980805	−	0.194991i	$-0.937532\pi$
$797$		−	11.0096i		−	0.389981i	0.980805	−	0.194991i	$-0.0624676\pi$
$798$		0			0
$799$			22.6274i			0.800500i
$800$		0			0
$801$		−	20.0083i		−	0.706959i
$802$		0			0
$803$	1.79337			0.0632865
$804$		0			0
$805$	−4.28427	+	9.55582i	−0.151001	+	0.336798i
$806$		0			0
$807$		−	65.4558i		−	2.30415i
$808$		0			0
$809$	−34.4853			−1.21244			−0.606219	−	0.795298i	$-0.707314\pi$
$809$	−34.4853			−1.21244			−0.606219	+	0.795298i	$0.707314\pi$
$810$		0			0
$811$	25.1577			0.883405			0.441702	−	0.897162i	$-0.354375\pi$
$811$	25.1577			0.883405			0.441702	+	0.897162i	$0.354375\pi$
$812$		0			0
$813$	−65.9411			−2.31266
$814$		0			0
$815$	−5.75152			−0.201467
$816$		0			0
$817$		−	36.5838i		−	1.27990i
$818$		0			0
$819$	38.3002	+	17.1716i	1.33832	+	0.600023i
$820$		0			0
$821$	29.5980			1.03298			0.516488	−	0.856294i	$-0.327239\pi$
$821$	29.5980			1.03298			0.516488	+	0.856294i	$0.327239\pi$
$822$		0			0
$823$		−	37.1127i		−	1.29367i	−0.762631	−	0.646834i	$-0.776093\pi$
$823$		−	37.1127i		−	1.29367i	0.762631	−	0.646834i	$-0.223907\pi$
$824$		0			0
$825$			20.0083i			0.696600i
$826$		0			0
$827$			47.2548i			1.64321i	0.570056	+	0.821606i	$0.306921\pi$
$827$			47.2548i			1.64321i	−0.570056	+	0.821606i	$0.693079\pi$
$828$		0			0
$829$		−	51.5515i		−	1.79046i	−0.445605	−	0.895230i	$-0.647011\pi$
$829$		−	51.5515i		−	1.79046i	0.445605	−	0.895230i	$-0.352989\pi$
$830$		0			0
$831$	18.2150			0.631870
$832$		0			0
$833$	−38.6274	+	34.4190i	−1.33836	+	1.19255i
$834$		0			0
$835$		−	16.9706i		−	0.587291i
$836$		0			0
$837$	16.0000			0.553041
$838$		0			0
$839$	−13.1426			−0.453731			−0.226866	−	0.973926i	$-0.572848\pi$
$839$	−13.1426			−0.453731			−0.226866	+	0.973926i	$0.572848\pi$
$840$		0			0
$841$	29.6274			1.02164
$842$		0			0
$843$	43.4495			1.49648
$844$		0			0
$845$		−	4.51528i		−	0.155330i
$846$		0			0
$847$	−7.57675	+	16.8995i	−0.260340	+	0.580674i
$848$		0			0
$849$	−6.82843			−0.234351
$850$		0			0
$851$		−	13.3726i		−	0.458406i
$852$		0			0
$853$		−	19.2974i		−	0.660729i	−0.943853	−	0.330364i	$-0.892828\pi$
$853$		−	19.2974i		−	0.660729i	0.943853	−	0.330364i	$-0.107172\pi$
$854$		0			0
$855$		−	19.7990i		−	0.677111i
$856$		0			0
$857$			27.3994i			0.935944i	0.883743	+	0.467972i	$0.155015\pi$
$857$			27.3994i			0.935944i	−0.883743	+	0.467972i	$0.844985\pi$
$858$		0			0
$859$	−53.0823			−1.81114			−0.905572	−	0.424193i	$-0.860558\pi$
$859$	−53.0823			−1.81114			−0.905572	+	0.424193i	$0.860558\pi$
$860$		0			0
$861$	−52.2843	−	23.4412i	−1.78184	−	0.798874i
$862$		0			0
$863$		−	23.4558i		−	0.798446i	−0.916854	−	0.399223i	$-0.869280\pi$
$863$		−	23.4558i		−	0.798446i	0.916854	−	0.399223i	$-0.130720\pi$
$864$		0			0
$865$	−11.5147			−0.391512
$866$		0			0
$867$	−98.3252			−3.33930
$868$		0			0
$869$	−14.3431			−0.486558
$870$		0			0
$871$	17.9974			0.609818
$872$		0			0
$873$			44.8715i			1.51867i
$874$		0			0
$875$	−23.0698	−	10.3431i	−0.779901	−	0.349662i
$876$		0			0
$877$	11.6569			0.393624			0.196812	−	0.980441i	$-0.436941\pi$
$877$	11.6569			0.393624			0.196812	+	0.980441i	$0.436941\pi$
$878$		0			0
$879$			22.1421i			0.746836i
$880$		0			0
$881$		−	37.4804i		−	1.26275i	−0.775478	−	0.631374i	$-0.782491\pi$
$881$		−	37.4804i		−	1.26275i	0.775478	−	0.631374i	$-0.217509\pi$
$882$		0			0
$883$		−	35.6569i		−	1.19995i	−0.800019	−	0.599974i	$-0.795177\pi$
$883$		−	35.6569i		−	1.19995i	0.800019	−	0.599974i	$-0.204823\pi$
$884$		0			0
$885$		−	16.0502i		−	0.539521i
$886$		0			0
$887$	−44.5001			−1.49417			−0.747083	−	0.664731i	$-0.768546\pi$
$887$	−44.5001			−1.49417			−0.747083	+	0.664731i	$0.768546\pi$
$888$		0			0
$889$	−18.4853	−	8.28772i	−0.619976	−	0.277961i
$890$		0			0
$891$		−	11.6569i		−	0.390519i
$892$		0			0
$893$	14.6274			0.489488
$894$		0			0
$895$	−2.16478			−0.0723608
$896$		0			0
$897$	−39.5980			−1.32214
$898$		0			0
$899$	−56.5921			−1.88745
$900$		0			0
$901$			14.7821i			0.492462i
$902$		0			0
$903$	−48.3044	−	21.6569i	−1.60747	−	0.720695i
$904$		0			0
$905$	−12.4853			−0.415025
$906$		0			0
$907$			26.0000i			0.863316i	0.902037	+	0.431658i	$0.142071\pi$
$907$			26.0000i			0.863316i	−0.902037	+	0.431658i	$0.857929\pi$
$908$		0			0
$909$			32.4399i			1.07596i
$910$		0			0
$911$		−	35.6569i		−	1.18136i	−0.806904	−	0.590682i	$-0.798859\pi$
$911$		−	35.6569i		−	1.18136i	0.806904	−	0.590682i	$-0.201141\pi$
$912$		0			0
$913$		−	3.43289i		−	0.113612i
$914$		0			0
$915$	3.06147			0.101209
$916$		0			0
$917$	3.79899	−	8.47343i	0.125454	−	0.279817i
$918$		0			0
$919$		−	9.79899i		−	0.323239i	−0.986853	−	0.161619i	$-0.948328\pi$
$919$		−	9.79899i		−	0.323239i	0.986853	−	0.161619i	$-0.0516717\pi$
$920$		0			0
$921$	−4.48528			−0.147795
$922$		0			0
$923$	−13.1426			−0.432592
$924$		0			0
$925$	14.0000			0.460317
$926$		0			0
$927$	43.4495			1.42707
$928$		0			0
$929$		−	6.64820i		−	0.218120i	−0.994035	−	0.109060i	$-0.965216\pi$
$929$		−	6.64820i		−	0.218120i	0.994035	−	0.109060i	$-0.0347841\pi$
$930$		0			0
$931$	22.2500	+	24.9706i	0.729215	+	0.818377i
$932$		0			0
$933$	−10.3431			−0.338619
$934$		0			0
$935$			16.0000i			0.523256i
$936$		0			0
$937$		−	48.8296i		−	1.59519i	−0.603190	−	0.797597i	$-0.706104\pi$
$937$		−	48.8296i		−	1.59519i	0.603190	−	0.797597i	$-0.293896\pi$
$938$		0			0
$939$			16.9706i			0.553813i
$940$		0			0
$941$			8.10201i			0.264118i	0.991242	+	0.132059i	$0.0421588\pi$
$941$			8.10201i			0.264118i	−0.991242	+	0.132059i	$0.957841\pi$
$942$		0			0
$943$	30.3070			0.986931
$944$		0			0
$945$	−5.65685	−	2.53620i	−0.184017	−	0.0825027i
$946$		0			0
$947$			39.6569i			1.28867i	0.764741	+	0.644337i	$0.222867\pi$
$947$			39.6569i			1.28867i	−0.764741	+	0.644337i	$0.777133\pi$
$948$		0			0
$949$	−3.71573			−0.120618
$950$		0			0
$951$	42.7067			1.38486
$952$		0			0
$953$	−44.6274			−1.44562			−0.722812	−	0.691045i	$-0.757151\pi$
$953$	−44.6274			−1.44562			−0.722812	+	0.691045i	$0.757151\pi$
$954$		0			0
$955$	0.896683			0.0290160
$956$		0			0
$957$		−	40.0166i		−	1.29355i
$958$		0			0
$959$	−2.16478	+	4.82843i	−0.0699045	+	0.155918i
$960$		0			0
$961$	23.6274			0.762175
$962$		0			0
$963$		−	44.6274i		−	1.43810i
$964$		0			0
$965$			7.01962i			0.225969i
$966$		0			0
$967$			50.9706i			1.63910i	0.573006	+	0.819551i	$0.305777\pi$
$967$			50.9706i			1.63910i	−0.573006	+	0.819551i	$0.694223\pi$
$968$		0			0
$969$			92.2792i			2.96443i
$970$		0			0
$971$	4.77791			0.153330			0.0766652	−	0.997057i	$-0.475573\pi$
$971$	4.77791			0.153330			0.0766652	+	0.997057i	$0.475573\pi$
$972$		0			0
$973$	10.8284	−	24.1522i	0.347143	−	0.774283i
$974$		0			0
$975$		−	41.4558i		−	1.32765i
$976$		0			0
$977$	−28.6274			−0.915872			−0.457936	−	0.888985i	$-0.651411\pi$
$977$	−28.6274			−0.915872			−0.457936	+	0.888985i	$0.651411\pi$
$978$		0			0
$979$	10.4525			0.334063
$980$		0			0
$981$	66.2843			2.11629
$982$		0			0
$983$	−7.76245			−0.247584			−0.123792	−	0.992308i	$-0.539506\pi$
$983$	−7.76245			−0.247584			−0.123792	+	0.992308i	$0.539506\pi$
$984$		0			0
$985$		−	8.28772i		−	0.264069i
$986$		0			0
$987$	8.65914	−	19.3137i	0.275623	−	0.614762i
$988$		0			0
$989$	28.0000			0.890348
$990$		0			0
$991$			2.48528i			0.0789476i	0.999221	+	0.0394738i	$0.0125682\pi$
$991$			2.48528i			0.0789476i	−0.999221	+	0.0394738i	$0.987432\pi$
$992$		0			0
$993$			13.8854i			0.440640i
$994$		0			0
$995$		−	0.970563i		−	0.0307689i
$996$		0			0
$997$		−	5.04054i		−	0.159636i	−0.996809	−	0.0798178i	$-0.974566\pi$
$997$		−	5.04054i		−	0.159636i	0.996809	−	0.0798178i	$-0.0254338\pi$
$998$		0			0
$999$	7.91630			0.250461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.2.f.a.223.8	yes	8
3.2	odd	2		2016.2.b.b.1567.3		8
4.3	odd	2	inner	224.2.f.a.223.2	yes	8
7.2	even	3		1568.2.p.a.31.2		16
7.3	odd	6		1568.2.p.a.607.8		16
7.4	even	3		1568.2.p.a.607.1		16
7.5	odd	6		1568.2.p.a.31.7		16
7.6	odd	2	inner	224.2.f.a.223.1	✓	8
8.3	odd	2		448.2.f.d.447.7		8
8.5	even	2		448.2.f.d.447.1		8
12.11	even	2		2016.2.b.b.1567.4		8
16.3	odd	4		1792.2.e.f.895.2		8
16.5	even	4		1792.2.e.f.895.1		8
16.11	odd	4		1792.2.e.g.895.7		8
16.13	even	4		1792.2.e.g.895.8		8
21.20	even	2		2016.2.b.b.1567.6		8
24.5	odd	2		4032.2.b.p.3583.5		8
24.11	even	2		4032.2.b.p.3583.6		8
28.3	even	6		1568.2.p.a.607.2		16
28.11	odd	6		1568.2.p.a.607.7		16
28.19	even	6		1568.2.p.a.31.1		16
28.23	odd	6		1568.2.p.a.31.8		16
28.27	even	2	inner	224.2.f.a.223.7	yes	8
56.13	odd	2		448.2.f.d.447.8		8
56.27	even	2		448.2.f.d.447.2		8
84.83	odd	2		2016.2.b.b.1567.5		8
112.13	odd	4		1792.2.e.g.895.1		8
112.27	even	4		1792.2.e.g.895.2		8
112.69	odd	4		1792.2.e.f.895.8		8
112.83	even	4		1792.2.e.f.895.7		8
168.83	odd	2		4032.2.b.p.3583.3		8
168.125	even	2		4032.2.b.p.3583.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.f.a.223.1	✓	8	7.6	odd	2	inner
224.2.f.a.223.2	yes	8	4.3	odd	2	inner
224.2.f.a.223.7	yes	8	28.27	even	2	inner
224.2.f.a.223.8	yes	8	1.1	even	1	trivial
448.2.f.d.447.1		8	8.5	even	2
448.2.f.d.447.2		8	56.27	even	2
448.2.f.d.447.7		8	8.3	odd	2
448.2.f.d.447.8		8	56.13	odd	2
1568.2.p.a.31.1		16	28.19	even	6
1568.2.p.a.31.2		16	7.2	even	3
1568.2.p.a.31.7		16	7.5	odd	6
1568.2.p.a.31.8		16	28.23	odd	6
1568.2.p.a.607.1		16	7.4	even	3
1568.2.p.a.607.2		16	28.3	even	6
1568.2.p.a.607.7		16	28.11	odd	6
1568.2.p.a.607.8		16	7.3	odd	6
1792.2.e.f.895.1		8	16.5	even	4
1792.2.e.f.895.2		8	16.3	odd	4
1792.2.e.f.895.7		8	112.83	even	4
1792.2.e.f.895.8		8	112.69	odd	4
1792.2.e.g.895.1		8	112.13	odd	4
1792.2.e.g.895.2		8	112.27	even	4
1792.2.e.g.895.7		8	16.11	odd	4
1792.2.e.g.895.8		8	16.13	even	4
2016.2.b.b.1567.3		8	3.2	odd	2
2016.2.b.b.1567.4		8	12.11	even	2
2016.2.b.b.1567.5		8	84.83	odd	2
2016.2.b.b.1567.6		8	21.20	even	2
4032.2.b.p.3583.3		8	168.83	odd	2
4032.2.b.p.3583.4		8	168.125	even	2
4032.2.b.p.3583.5		8	24.5	odd	2
4032.2.b.p.3583.6		8	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	224.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.61313 q^{3} +1.08239i q^{5} +(-1.08239 + 2.41421i) q^{7} +3.82843 q^{9} +O(q^{10})\)	\(q+2.61313 q^{3} +1.08239i q^{5} +(-1.08239 + 2.41421i) q^{7} +3.82843 q^{9} +2.00000i q^{11} -4.14386i q^{13} +2.82843i q^{15} -7.39104i q^{17} -4.77791 q^{19} +(-2.82843 + 6.30864i) q^{21} -3.65685i q^{23} +3.82843 q^{25} +2.16478 q^{27} -7.65685 q^{29} +7.39104 q^{31} +5.22625i q^{33} +(-2.61313 - 1.17157i) q^{35} +3.65685 q^{37} -10.8284i q^{39} +8.28772i q^{41} +7.65685i q^{43} +4.14386i q^{45} -3.06147 q^{47} +(-4.65685 - 5.22625i) q^{49} -19.3137i q^{51} -2.00000 q^{53} -2.16478 q^{55} -12.4853 q^{57} -5.67459 q^{59} -1.08239i q^{61} +(-4.14386 + 9.24264i) q^{63} +4.48528 q^{65} +4.34315i q^{67} -9.55582i q^{69} -3.17157i q^{71} -0.896683i q^{73} +10.0042 q^{75} +(-4.82843 - 2.16478i) q^{77} +7.17157i q^{79} -5.82843 q^{81} -1.71644 q^{83} +8.00000 q^{85} -20.0083 q^{87} -5.22625i q^{89} +(10.0042 + 4.48528i) q^{91} +19.3137 q^{93} -5.17157i q^{95} +11.7206i q^{97} +7.65685i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} + 8 q^{25} - 16 q^{29} - 16 q^{37} + 8 q^{49} - 16 q^{53} - 32 q^{57} - 32 q^{65} - 16 q^{77} - 24 q^{81} + 64 q^{85} + 64 q^{93}+O(q^{100}) \) 8 * q + 8 * q^9 + 8 * q^25 - 16 * q^29 - 16 * q^37 + 8 * q^49 - 16 * q^53 - 32 * q^57 - 32 * q^65 - 16 * q^77 - 24 * q^81 + 64 * q^85 + 64 * q^93

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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