LMFDB - Embedded newform 1792.2.e.g.895.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(895,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1792.895");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$14.3091920422$
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:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.2
Root			$-0.923880 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	1792.895
Dual form			1792.2.e.g.895.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61313i q^{3} +1.08239 q^{5} +(1.08239 + 2.41421i) q^{7} -3.82843 q^{9} +O(q^{10})$	$q-2.61313i q^{3} +1.08239 q^{5} +(1.08239 + 2.41421i) q^{7} -3.82843 q^{9} +2.00000 q^{11} +4.14386 q^{13} -2.82843i q^{15} +7.39104i q^{17} +4.77791i q^{19} +(6.30864 - 2.82843i) q^{21} -3.65685i q^{23} -3.82843 q^{25} +2.16478i q^{27} +7.65685i q^{29} +7.39104 q^{31} -5.22625i q^{33} +(1.17157 + 2.61313i) q^{35} +3.65685i q^{37} -10.8284i q^{39} +8.28772i q^{41} +7.65685 q^{43} -4.14386 q^{45} -3.06147 q^{47} +(-4.65685 + 5.22625i) q^{49} +19.3137 q^{51} -2.00000i q^{53} +2.16478 q^{55} +12.4853 q^{57} -5.67459i q^{59} +1.08239 q^{61} +(-4.14386 - 9.24264i) q^{63} +4.48528 q^{65} -4.34315 q^{67} -9.55582 q^{69} -3.17157i q^{71} -0.896683i q^{73} +10.0042i q^{75} +(2.16478 + 4.82843i) q^{77} -7.17157i q^{79} -5.82843 q^{81} +1.71644i q^{83} +8.00000i q^{85} +20.0083 q^{87} -5.22625i q^{89} +(4.48528 + 10.0042i) q^{91} -19.3137i q^{93} +5.17157i q^{95} -11.7206i q^{97} -7.65685 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} + 16 q^{11} - 8 q^{25} + 32 q^{35} + 16 q^{43} + 8 q^{49} + 64 q^{51} + 32 q^{57} - 32 q^{65} - 80 q^{67} - 24 q^{81} - 32 q^{91} - 16 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 + 16 * q^11 - 8 * q^25 + 32 * q^35 + 16 * q^43 + 8 * q^49 + 64 * q^51 + 32 * q^57 - 32 * q^65 - 80 * q^67 - 24 * q^81 - 32 * q^91 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$		−	2.61313i		−	1.50869i	−0.656479	−	0.754344i	$-0.727955\pi$
$3$		−	2.61313i		−	1.50869i	0.656479	−	0.754344i	$-0.272045\pi$
$4$		0			0
$5$	1.08239			0.484061			0.242030	−	0.970269i	$-0.422187\pi$
$5$	1.08239			0.484061			0.242030	+	0.970269i	$0.422187\pi$
$6$		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
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$	4.14386			1.14930			0.574650	−	0.818399i	$-0.305138\pi$
$13$	4.14386			1.14930			0.574650	+	0.818399i	$0.305138\pi$
$14$		0			0
$15$		−	2.82843i		−	0.730297i
$16$		0			0
$17$			7.39104i			1.79259i	0.443459	+	0.896295i	$0.353751\pi$
$17$			7.39104i			1.79259i	−0.443459	+	0.896295i	$0.646249\pi$
$18$		0			0
$19$			4.77791i			1.09613i	0.836436	+	0.548064i	$0.184635\pi$
$19$			4.77791i			1.09613i	−0.836436	+	0.548064i	$0.815365\pi$
$20$		0			0
$21$	6.30864	−	2.82843i	1.37666	−	0.617213i
$22$		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
$27$			2.16478i			0.416613i
$28$		0			0
$29$			7.65685i			1.42184i	0.703272	+	0.710921i	$0.251722\pi$
$29$			7.65685i			1.42184i	−0.703272	+	0.710921i	$0.748278\pi$
$30$		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
$35$	1.17157	+	2.61313i	0.198032	+	0.441699i
$36$		0			0
$37$			3.65685i			0.601183i	0.953753	+	0.300592i	$0.0971841\pi$
$37$			3.65685i			0.601183i	−0.953753	+	0.300592i	$0.902816\pi$
$38$		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
$43$	7.65685			1.16766			0.583830	−	0.811876i	$-0.301554\pi$
$43$	7.65685			1.16766			0.583830	+	0.811876i	$0.301554\pi$
$44$		0			0
$45$	−4.14386			−0.617730
$46$		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
$51$	19.3137			2.70446
$52$		0			0
$53$		−	2.00000i		−	0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$		−	2.00000i		−	0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$		0			0
$55$	2.16478			0.291899
$56$		0			0
$57$	12.4853			1.65372
$58$		0			0
$59$		−	5.67459i		−	0.738769i	−0.929277	−	0.369385i	$-0.879569\pi$
$59$		−	5.67459i		−	0.738769i	0.929277	−	0.369385i	$-0.120431\pi$
$60$		0			0
$61$	1.08239			0.138586			0.0692931	−	0.997596i	$-0.477926\pi$
$61$	1.08239			0.138586			0.0692931	+	0.997596i	$0.477926\pi$
$62$		0			0
$63$	−4.14386	−	9.24264i	−0.522077	−	1.16446i
$64$		0			0
$65$	4.48528			0.556331
$66$		0			0
$67$	−4.34315			−0.530600			−0.265300	−	0.964166i	$-0.585471\pi$
$67$	−4.34315			−0.530600			−0.265300	+	0.964166i	$0.585471\pi$
$68$		0			0
$69$	−9.55582			−1.15039
$70$		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
$75$			10.0042i			1.15518i
$76$		0			0
$77$	2.16478	+	4.82843i	0.246700	+	0.550250i
$78$		0			0
$79$		−	7.17157i		−	0.806865i	−0.915009	−	0.403432i	$-0.867817\pi$
$79$		−	7.17157i		−	0.806865i	0.915009	−	0.403432i	$-0.132183\pi$
$80$		0			0
$81$	−5.82843			−0.647603
$82$		0			0
$83$			1.71644i			0.188404i	0.995553	+	0.0942020i	$0.0300300\pi$
$83$			1.71644i			0.188404i	−0.995553	+	0.0942020i	$0.969970\pi$
$84$		0			0
$85$			8.00000i			0.867722i
$86$		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
$91$	4.48528	+	10.0042i	0.470185	+	1.04872i
$92$		0			0
$93$		−	19.3137i		−	2.00274i
$94$		0			0
$95$			5.17157i			0.530592i
$96$		0			0
$97$		−	11.7206i		−	1.19005i	−0.803708	−	0.595024i	$-0.797143\pi$
$97$		−	11.7206i		−	1.19005i	0.803708	−	0.595024i	$-0.202857\pi$
$98$		0			0
$99$	−7.65685			−0.769543
$100$		0			0
$101$	8.47343			0.843138			0.421569	−	0.906796i	$-0.361480\pi$
$101$	8.47343			0.843138			0.421569	+	0.906796i	$0.361480\pi$
$102$		0			0
$103$	−11.3492			−1.11827			−0.559134	−	0.829077i	$-0.688866\pi$
$103$	−11.3492			−1.11827			−0.559134	+	0.829077i	$0.688866\pi$
$104$		0			0
$105$	6.82843	−	3.06147i	0.666386	−	0.298769i
$106$		0			0
$107$	−11.6569			−1.12691			−0.563455	−	0.826147i	$-0.690528\pi$
$107$	−11.6569			−1.12691			−0.563455	+	0.826147i	$0.690528\pi$
$108$		0			0
$109$		−	17.3137i		−	1.65835i	−0.558987	−	0.829176i	$-0.688810\pi$
$109$		−	17.3137i		−	1.65835i	0.558987	−	0.829176i	$-0.311190\pi$
$110$		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
$115$		−	3.95815i		−	0.369099i
$116$		0			0
$117$	−15.8645			−1.46667
$118$		0			0
$119$	−17.8435	+	8.00000i	−1.63571	+	0.733359i
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$	21.6569			1.95273
$124$		0			0
$125$	−9.55582			−0.854699
$126$		0			0
$127$		−	7.65685i		−	0.679436i	−0.940527	−	0.339718i	$-0.889668\pi$
$127$		−	7.65685i		−	0.679436i	0.940527	−	0.339718i	$-0.110332\pi$
$128$		0			0
$129$		−	20.0083i		−	1.76163i
$130$		0			0
$131$			3.50981i			0.306653i	0.988176	+	0.153327i	$0.0489987\pi$
$131$			3.50981i			0.306653i	−0.988176	+	0.153327i	$0.951001\pi$
$132$		0			0
$133$	−11.5349	+	5.17157i	−1.00020	+	0.448432i
$134$		0			0
$135$			2.34315i			0.201666i
$136$		0			0
$137$	−2.00000			−0.170872			−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000			−0.170872			−0.0854358	+	0.996344i	$0.527228\pi$
$138$		0			0
$139$		−	10.0042i		−	0.848542i	−0.905535	−	0.424271i	$-0.860530\pi$
$139$		−	10.0042i		−	0.848542i	0.905535	−	0.424271i	$-0.139470\pi$
$140$		0			0
$141$			8.00000i			0.673722i
$142$		0			0
$143$	8.28772			0.693054
$144$		0			0
$145$			8.28772i			0.688258i
$146$		0			0
$147$	13.6569	+	12.1689i	1.12640	+	1.00368i
$148$		0			0
$149$		−	10.0000i		−	0.819232i	−0.912258	−	0.409616i	$-0.865663\pi$
$149$		−	10.0000i		−	0.819232i	0.912258	−	0.409616i	$-0.134337\pi$
$150$		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
$155$	8.00000			0.642575
$156$		0			0
$157$	8.47343			0.676253			0.338127	−	0.941101i	$-0.390207\pi$
$157$	8.47343			0.676253			0.338127	+	0.941101i	$0.390207\pi$
$158$		0			0
$159$	−5.22625			−0.414469
$160$		0			0
$161$	8.82843	−	3.95815i	0.695778	−	0.311946i
$162$		0			0
$163$	−5.31371			−0.416202			−0.208101	−	0.978107i	$-0.566728\pi$
$163$	−5.31371			−0.416202			−0.208101	+	0.978107i	$0.566728\pi$
$164$		0			0
$165$		−	5.65685i		−	0.440386i
$166$		0			0
$167$	15.6788			1.21326			0.606629	−	0.794985i	$-0.292521\pi$
$167$	15.6788			1.21326			0.606629	+	0.794985i	$0.292521\pi$
$168$		0			0
$169$	4.17157			0.320890
$170$		0			0
$171$		−	18.2919i		−	1.39882i
$172$		0			0
$173$	−10.6382			−0.808808			−0.404404	−	0.914580i	$-0.632521\pi$
$173$	−10.6382			−0.808808			−0.404404	+	0.914580i	$0.632521\pi$
$174$		0			0
$175$	−4.14386	−	9.24264i	−0.313246	−	0.698678i
$176$		0			0
$177$	−14.8284			−1.11457
$178$		0			0
$179$	−2.00000			−0.149487			−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000			−0.149487			−0.0747435	+	0.997203i	$0.523814\pi$
$180$		0			0
$181$	11.5349			0.857382			0.428691	−	0.903451i	$-0.358975\pi$
$181$	11.5349			0.857382			0.428691	+	0.903451i	$0.358975\pi$
$182$		0			0
$183$		−	2.82843i		−	0.209083i
$184$		0			0
$185$			3.95815i			0.291009i
$186$		0			0
$187$			14.7821i			1.08097i
$188$		0			0
$189$	−5.22625	+	2.34315i	−0.380154	+	0.170439i
$190$		0			0
$191$			0.828427i			0.0599429i	0.999551	+	0.0299714i	$0.00954164\pi$
$191$			0.828427i			0.0599429i	−0.999551	+	0.0299714i	$0.990458\pi$
$192$		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
$195$		−	11.7206i		−	0.839330i
$196$		0			0
$197$		−	7.65685i		−	0.545528i	−0.962081	−	0.272764i	$-0.912062\pi$
$197$		−	7.65685i		−	0.545528i	0.962081	−	0.272764i	$-0.0879379\pi$
$198$		0			0
$199$	0.896683			0.0635642			0.0317821	−	0.999495i	$-0.489882\pi$
$199$	0.896683			0.0635642			0.0317821	+	0.999495i	$0.489882\pi$
$200$		0			0
$201$			11.3492i			0.800510i
$202$		0			0
$203$	−18.4853	+	8.28772i	−1.29741	+	0.581684i
$204$		0			0
$205$			8.97056i			0.626531i
$206$		0			0
$207$			14.0000i			0.973067i
$208$		0			0
$209$			9.55582i			0.660990i
$210$		0			0
$211$	22.9706			1.58136			0.790679	−	0.612230i	$-0.209728\pi$
$211$	22.9706			1.58136			0.790679	+	0.612230i	$0.209728\pi$
$212$		0			0
$213$	−8.28772			−0.567865
$214$		0			0
$215$	8.28772			0.565218
$216$		0			0
$217$	8.00000	+	17.8435i	0.543075	+	1.21130i
$218$		0			0
$219$	−2.34315			−0.158335
$220$		0			0
$221$			30.6274i			2.06022i
$222$		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
$227$		−	29.1158i		−	1.93248i	−0.257637	−	0.966242i	$-0.582944\pi$
$227$		−	29.1158i		−	1.93248i	0.257637	−	0.966242i	$-0.417056\pi$
$228$		0			0
$229$	29.3784			1.94138			0.970691	−	0.240332i	$-0.0772562\pi$
$229$	29.3784			1.94138			0.970691	+	0.240332i	$0.0772562\pi$
$230$		0			0
$231$	12.6173	−	5.65685i	0.830157	−	0.372194i
$232$		0			0
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$		0			0
$235$	−3.31371			−0.216163
$236$		0			0
$237$	−18.7402			−1.21731
$238$		0			0
$239$			19.6569i			1.27150i	0.771897	+	0.635748i	$0.219308\pi$
$239$			19.6569i			1.27150i	−0.771897	+	0.635748i	$0.780692\pi$
$240$		0			0
$241$			16.0502i			1.03388i	0.856021	+	0.516941i	$0.172929\pi$
$241$			16.0502i			1.03388i	−0.856021	+	0.516941i	$0.827071\pi$
$242$		0			0
$243$			21.7248i			1.39364i
$244$		0			0
$245$	−5.04054	+	5.65685i	−0.322028	+	0.361403i
$246$		0			0
$247$			19.7990i			1.25978i
$248$		0			0
$249$	4.48528			0.284243
$250$		0			0
$251$		−	2.61313i		−	0.164939i	−0.996594	−	0.0824695i	$-0.973719\pi$
$251$		−	2.61313i		−	0.164939i	0.996594	−	0.0824695i	$-0.0262807\pi$
$252$		0			0
$253$		−	7.31371i		−	0.459809i
$254$		0			0
$255$	20.9050			1.30912
$256$		0			0
$257$			14.7821i			0.922080i	0.887379	+	0.461040i	$0.152524\pi$
$257$			14.7821i			0.922080i	−0.887379	+	0.461040i	$0.847476\pi$
$258$		0			0
$259$	−8.82843	+	3.95815i	−0.548572	+	0.245948i
$260$		0			0
$261$		−	29.3137i		−	1.81447i
$262$		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
$267$	−13.6569			−0.835786
$268$		0			0
$269$	25.0489			1.52726			0.763628	−	0.645656i	$-0.223416\pi$
$269$	25.0489			1.52726			0.763628	+	0.645656i	$0.223416\pi$
$270$		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
$275$	−7.65685			−0.461726
$276$		0			0
$277$			6.97056i			0.418821i	0.977828	+	0.209410i	$0.0671544\pi$
$277$			6.97056i			0.418821i	−0.977828	+	0.209410i	$0.932846\pi$
$278$		0			0
$279$	−28.2960			−1.69404
$280$		0			0
$281$	−16.6274			−0.991909			−0.495954	−	0.868349i	$-0.665182\pi$
$281$	−16.6274			−0.991909			−0.495954	+	0.868349i	$0.665182\pi$
$282$		0			0
$283$		−	2.61313i		−	0.155334i	−0.996979	−	0.0776671i	$-0.975253\pi$
$283$		−	2.61313i		−	0.155334i	0.996979	−	0.0776671i	$-0.0247471\pi$
$284$		0			0
$285$	13.5140			0.800499
$286$		0			0
$287$	−20.0083	+	8.97056i	−1.18105	+	0.529516i
$288$		0			0
$289$	−37.6274			−2.21338
$290$		0			0
$291$	−30.6274			−1.79541
$292$		0			0
$293$	8.47343			0.495023			0.247511	−	0.968885i	$-0.420387\pi$
$293$	8.47343			0.495023			0.247511	+	0.968885i	$0.420387\pi$
$294$		0			0
$295$		−	6.14214i		−	0.357609i
$296$		0			0
$297$			4.32957i			0.251227i
$298$		0			0
$299$		−	15.1535i		−	0.876349i
$300$		0			0
$301$	8.28772	+	18.4853i	0.477696	+	1.06547i
$302$		0			0
$303$		−	22.1421i		−	1.27203i
$304$		0			0
$305$	1.17157			0.0670841
$306$		0			0
$307$			1.71644i			0.0979626i	0.998800	+	0.0489813i	$0.0155975\pi$
$307$			1.71644i			0.0979626i	−0.998800	+	0.0489813i	$0.984403\pi$
$308$		0			0
$309$			29.6569i			1.68712i
$310$		0			0
$311$	3.95815			0.224446			0.112223	−	0.993683i	$-0.464203\pi$
$311$	3.95815			0.224446			0.112223	+	0.993683i	$0.464203\pi$
$312$		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
$315$	−4.48528	−	10.0042i	−0.252717	−	0.563671i
$316$		0			0
$317$		−	16.3431i		−	0.917923i	−0.888456	−	0.458961i	$-0.848222\pi$
$317$		−	16.3431i		−	0.917923i	0.888456	−	0.458961i	$-0.151778\pi$
$318$		0			0
$319$			15.3137i			0.857403i
$320$		0			0
$321$			30.4608i			1.70016i
$322$		0			0
$323$	−35.3137			−1.96491
$324$		0			0
$325$	−15.8645			−0.880002
$326$		0			0
$327$	−45.2429			−2.50194
$328$		0			0
$329$	−3.31371	−	7.39104i	−0.182691	−	0.407481i
$330$		0			0
$331$	5.31371			0.292068			0.146034	−	0.989280i	$-0.453349\pi$
$331$	5.31371			0.292068			0.146034	+	0.989280i	$0.453349\pi$
$332$		0			0
$333$		−	14.0000i		−	0.767195i
$334$		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
$339$		−	8.28772i		−	0.450127i
$340$		0			0
$341$	14.7821			0.800494
$342$		0			0
$343$	−17.6578	−	5.58579i	−0.953433	−	0.301604i
$344$		0			0
$345$	−10.3431			−0.556856
$346$		0			0
$347$	−8.34315			−0.447884			−0.223942	−	0.974603i	$-0.571893\pi$
$347$	−8.34315			−0.447884			−0.223942	+	0.974603i	$0.571893\pi$
$348$		0			0
$349$	15.8645			0.849205			0.424603	−	0.905380i	$-0.360414\pi$
$349$	15.8645			0.849205			0.424603	+	0.905380i	$0.360414\pi$
$350$		0			0
$351$			8.97056i			0.478813i
$352$		0			0
$353$			12.2459i			0.651782i	0.945407	+	0.325891i	$0.105664\pi$
$353$			12.2459i			0.651782i	−0.945407	+	0.325891i	$0.894336\pi$
$354$		0			0
$355$		−	3.43289i		−	0.182199i
$356$		0			0
$357$	20.9050	+	46.6274i	1.10641	+	2.46778i
$358$		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
$363$			18.2919i			0.960075i
$364$		0			0
$365$		−	0.970563i		−	0.0508016i
$366$		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
$371$	4.82843	−	2.16478i	0.250679	−	0.112390i
$372$		0			0
$373$		−	26.9706i		−	1.39648i	−0.715862	−	0.698241i	$-0.753966\pi$
$373$		−	26.9706i		−	1.39648i	0.715862	−	0.698241i	$-0.246034\pi$
$374$		0			0
$375$			24.9706i			1.28947i
$376$		0			0
$377$			31.7289i			1.63412i
$378$		0			0
$379$	−17.3137			−0.889345			−0.444673	−	0.895693i	$-0.646680\pi$
$379$	−17.3137			−0.889345			−0.444673	+	0.895693i	$0.646680\pi$
$380$		0			0
$381$	−20.0083			−1.02506
$382$		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
$387$	−29.3137			−1.49010
$388$		0			0
$389$			8.34315i			0.423014i	0.977376	+	0.211507i	$0.0678372\pi$
$389$			8.34315i			0.423014i	−0.977376	+	0.211507i	$0.932163\pi$
$390$		0			0
$391$	27.0279			1.36686
$392$		0			0
$393$	9.17157			0.462645
$394$		0			0
$395$		−	7.76245i		−	0.390571i
$396$		0			0
$397$	14.0711			0.706208			0.353104	−	0.935584i	$-0.385126\pi$
$397$	14.0711			0.706208			0.353104	+	0.935584i	$0.385126\pi$
$398$		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
$403$	30.6274			1.52566
$404$		0			0
$405$	−6.30864			−0.313479
$406$		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
$411$			5.22625i			0.257792i
$412$		0			0
$413$	13.6997	−	6.14214i	0.674117	−	0.302235i
$414$		0			0
$415$			1.85786i			0.0911990i
$416$		0			0
$417$	−26.1421			−1.28019
$418$		0			0
$419$		−	5.14933i		−	0.251561i	−0.992058	−	0.125781i	$-0.959856\pi$
$419$		−	5.14933i		−	0.251561i	0.992058	−	0.125781i	$-0.0401435\pi$
$420$		0			0
$421$			1.31371i			0.0640262i	0.999487	+	0.0320131i	$0.0101918\pi$
$421$			1.31371i			0.0640262i	−0.999487	+	0.0320131i	$0.989808\pi$
$422$		0			0
$423$	11.7206			0.569875
$424$		0			0
$425$		−	28.2960i		−	1.37256i
$426$		0			0
$427$	1.17157	+	2.61313i	0.0566964	+	0.126458i
$428$		0			0
$429$		−	21.6569i		−	1.04560i
$430$		0			0
$431$		−	23.6569i		−	1.13951i	−0.821814	−	0.569755i	$-0.807038\pi$
$431$		−	23.6569i		−	1.13951i	0.821814	−	0.569755i	$-0.192962\pi$
$432$		0			0
$433$		−	28.2960i		−	1.35982i	−0.733295	−	0.679911i	$-0.762019\pi$
$433$		−	28.2960i		−	1.35982i	0.733295	−	0.679911i	$-0.237981\pi$
$434$		0			0
$435$	21.6569			1.03837
$436$		0			0
$437$	17.4721			0.835805
$438$		0			0
$439$	39.6452			1.89216			0.946082	−	0.323928i	$-0.105003\pi$
$439$	39.6452			1.89216			0.946082	+	0.323928i	$0.105003\pi$
$440$		0			0
$441$	17.8284	−	20.0083i	0.848973	−	0.952777i
$442$		0			0
$443$	10.0000			0.475114			0.237557	−	0.971374i	$-0.423653\pi$
$443$	10.0000			0.475114			0.237557	+	0.971374i	$0.423653\pi$
$444$		0			0
$445$		−	5.65685i		−	0.268161i
$446$		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
$451$			16.5754i			0.780507i
$452$		0			0
$453$	−51.3658			−2.41338
$454$		0			0
$455$	4.85483	+	10.8284i	0.227598	+	0.507644i
$456$		0			0
$457$	25.1127			1.17472			0.587361	−	0.809325i	$-0.300167\pi$
$457$	25.1127			1.17472			0.587361	+	0.809325i	$0.300167\pi$
$458$		0			0
$459$	−16.0000			−0.746816
$460$		0			0
$461$	32.4399			1.51088			0.755438	−	0.655220i	$-0.227424\pi$
$461$	32.4399			1.51088			0.755438	+	0.655220i	$0.227424\pi$
$462$		0			0
$463$			12.1421i			0.564293i	0.959371	+	0.282146i	$0.0910464\pi$
$463$			12.1421i			0.564293i	−0.959371	+	0.282146i	$0.908954\pi$
$464$		0			0
$465$		−	20.9050i		−	0.969447i
$466$		0			0
$467$		−	24.7862i		−	1.14697i	−0.819216	−	0.573485i	$-0.805591\pi$
$467$		−	24.7862i		−	1.14697i	0.819216	−	0.573485i	$-0.194409\pi$
$468$		0			0
$469$	−4.70099	−	10.4853i	−0.217071	−	0.484165i
$470$		0			0
$471$		−	22.1421i		−	1.02026i
$472$		0			0
$473$	15.3137			0.704125
$474$		0			0
$475$		−	18.2919i		−	0.839289i
$476$		0			0
$477$			7.65685i			0.350583i
$478$		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
$483$	−10.3431	−	23.0698i	−0.470629	−	1.04971i
$484$		0			0
$485$		−	12.6863i		−	0.576055i
$486$		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
$491$	−26.2843			−1.18619			−0.593096	−	0.805132i	$-0.702095\pi$
$491$	−26.2843			−1.18619			−0.593096	+	0.805132i	$0.702095\pi$
$492$		0			0
$493$	−56.5921			−2.54878
$494$		0			0
$495$	−8.28772			−0.372505
$496$		0			0
$497$	7.65685	−	3.43289i	0.343457	−	0.153986i
$498$		0			0
$499$	−5.31371			−0.237874			−0.118937	−	0.992902i	$-0.537949\pi$
$499$	−5.31371			−0.237874			−0.118937	+	0.992902i	$0.537949\pi$
$500$		0			0
$501$		−	40.9706i		−	1.83043i
$502$		0			0
$503$	−31.7289			−1.41472			−0.707362	−	0.706852i	$-0.750115\pi$
$503$	−31.7289			−1.41472			−0.707362	+	0.706852i	$0.750115\pi$
$504$		0			0
$505$	9.17157			0.408130
$506$		0			0
$507$		−	10.9008i		−	0.484124i
$508$		0			0
$509$	−25.9456			−1.15002			−0.575008	−	0.818148i	$-0.695001\pi$
$509$	−25.9456			−1.15002			−0.575008	+	0.818148i	$0.695001\pi$
$510$		0			0
$511$	2.16478	−	0.970563i	0.0957644	−	0.0429352i
$512$		0			0
$513$	−10.3431			−0.456661
$514$		0			0
$515$	−12.2843			−0.541310
$516$		0			0
$517$	−6.12293			−0.269286
$518$		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
$523$		−	1.34502i		−	0.0588138i	−0.999568	−	0.0294069i	$-0.990638\pi$
$523$		−	1.34502i		−	0.0588138i	0.999568	−	0.0294069i	$-0.00936186\pi$
$524$		0			0
$525$	−24.1522	+	10.8284i	−1.05409	+	0.472591i
$526$		0			0
$527$			54.6274i			2.37961i
$528$		0			0
$529$	9.62742			0.418583
$530$		0			0
$531$			21.7248i			0.942775i
$532$		0			0
$533$			34.3431i			1.48757i
$534$		0			0
$535$	−12.6173			−0.545493
$536$		0			0
$537$			5.22625i			0.225529i
$538$		0			0
$539$	−9.31371	+	10.4525i	−0.401170	+	0.450221i
$540$		0			0
$541$		−	2.68629i		−	0.115493i	−0.998331	−	0.0577463i	$-0.981609\pi$
$541$		−	2.68629i		−	0.115493i	0.998331	−	0.0577463i	$-0.0183915\pi$
$542$		0			0
$543$		−	30.1421i		−	1.29352i
$544$		0			0
$545$		−	18.7402i		−	0.802743i
$546$		0			0
$547$	10.6863			0.456913			0.228456	−	0.973554i	$-0.426632\pi$
$547$	10.6863			0.456913			0.228456	+	0.973554i	$0.426632\pi$
$548$		0			0
$549$	−4.14386			−0.176856
$550$		0			0
$551$	−36.5838			−1.55852
$552$		0			0
$553$	17.3137	−	7.76245i	0.736254	−	0.330093i
$554$		0			0
$555$	10.3431			0.439042
$556$		0			0
$557$		−	17.3137i		−	0.733605i	−0.930299	−	0.366803i	$-0.880452\pi$
$557$		−	17.3137i		−	0.733605i	0.930299	−	0.366803i	$-0.119548\pi$
$558$		0			0
$559$	31.7289			1.34199
$560$		0			0
$561$	38.6274			1.63085
$562$		0			0
$563$			25.6829i			1.08241i	0.840892	+	0.541203i	$0.182031\pi$
$563$			25.6829i			1.08241i	−0.840892	+	0.541203i	$0.817969\pi$
$564$		0			0
$565$	3.43289			0.144423
$566$		0			0
$567$	−6.30864	−	14.0711i	−0.264938	−	0.590929i
$568$		0			0
$569$	−36.1421			−1.51516			−0.757579	−	0.652744i	$-0.773618\pi$
$569$	−36.1421			−1.51516			−0.757579	+	0.652744i	$0.773618\pi$
$570$		0			0
$571$	−19.6569			−0.822614			−0.411307	−	0.911497i	$-0.634928\pi$
$571$	−19.6569			−0.822614			−0.411307	+	0.911497i	$0.634928\pi$
$572$		0			0
$573$	2.16478			0.0904352
$574$		0			0
$575$			14.0000i			0.583840i
$576$		0			0
$577$		−	2.53620i		−	0.105583i	−0.998606	−	0.0527917i	$-0.983188\pi$
$577$		−	2.53620i		−	0.105583i	0.998606	−	0.0527917i	$-0.0168120\pi$
$578$		0			0
$579$		−	16.9469i		−	0.704287i
$580$		0			0
$581$	−4.14386	+	1.85786i	−0.171916	+	0.0770772i
$582$		0			0
$583$		−	4.00000i		−	0.165663i
$584$		0			0
$585$	−17.1716			−0.709957
$586$		0			0
$587$		−	33.4454i		−	1.38044i	−0.723600	−	0.690219i	$-0.757514\pi$
$587$		−	33.4454i		−	1.38044i	0.723600	−	0.690219i	$-0.242486\pi$
$588$		0			0
$589$			35.3137i			1.45508i
$590$		0			0
$591$	−20.0083			−0.823032
$592$		0			0
$593$		−	25.2346i		−	1.03626i	−0.855302	−	0.518130i	$-0.826628\pi$
$593$		−	25.2346i		−	1.03626i	0.855302	−	0.518130i	$-0.173372\pi$
$594$		0			0
$595$	−19.3137	+	8.65914i	−0.791785	+	0.354990i
$596$		0			0
$597$		−	2.34315i		−	0.0958986i
$598$		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
$603$	16.6274			0.677121
$604$		0			0
$605$	−7.57675			−0.308039
$606$		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
$611$	−12.6863			−0.513232
$612$		0			0
$613$			17.3137i			0.699294i	0.936881	+	0.349647i	$0.113698\pi$
$613$			17.3137i			0.699294i	−0.936881	+	0.349647i	$0.886302\pi$
$614$		0			0
$615$	23.4412			0.945241
$616$		0			0
$617$	49.1127			1.97720			0.988601	−	0.150557i	$-0.0481066\pi$
$617$	49.1127			1.97720			0.988601	+	0.150557i	$0.0481066\pi$
$618$		0			0
$619$			22.6215i			0.909233i	0.890687	+	0.454616i	$0.150224\pi$
$619$			22.6215i			0.909233i	−0.890687	+	0.454616i	$0.849776\pi$
$620$		0			0
$621$	7.91630			0.317670
$622$		0			0
$623$	12.6173	−	5.65685i	0.505501	−	0.226637i
$624$		0			0
$625$	8.79899			0.351960
$626$		0			0
$627$	24.9706			0.997228
$628$		0			0
$629$	−27.0279			−1.07767
$630$		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
$635$		−	8.28772i		−	0.328888i
$636$		0			0
$637$	−19.2974	+	21.6569i	−0.764589	+	0.858076i
$638$		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
$643$		−	26.5796i		−	1.04820i	−0.851658	−	0.524099i	$-0.824402\pi$
$643$		−	26.5796i		−	1.04820i	0.851658	−	0.524099i	$-0.175598\pi$
$644$		0			0
$645$		−	21.6569i		−	0.852738i
$646$		0			0
$647$	−40.9133			−1.60847			−0.804235	−	0.594312i	$-0.797424\pi$
$647$	−40.9133			−1.60847			−0.804235	+	0.594312i	$0.797424\pi$
$648$		0			0
$649$		−	11.3492i		−	0.445495i
$650$		0			0
$651$	46.6274	−	20.9050i	1.82747	−	0.819332i
$652$		0			0
$653$		−	25.3137i		−	0.990602i	−0.868721	−	0.495301i	$-0.835058\pi$
$653$		−	25.3137i		−	0.990602i	0.868721	−	0.495301i	$-0.164942\pi$
$654$		0			0
$655$			3.79899i			0.148439i
$656$		0			0
$657$			3.43289i			0.133930i
$658$		0			0
$659$	−50.9706			−1.98553			−0.992766	−	0.120068i	$-0.961689\pi$
$659$	−50.9706			−1.98553			−0.992766	+	0.120068i	$0.961689\pi$
$660$		0			0
$661$	6.68006			0.259824			0.129912	−	0.991526i	$-0.458530\pi$
$661$	6.68006			0.259824			0.129912	+	0.991526i	$0.458530\pi$
$662$		0			0
$663$	80.0333			3.10824
$664$		0			0
$665$	−12.4853	+	5.59767i	−0.484158	+	0.217068i
$666$		0			0
$667$	28.0000			1.08416
$668$		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
$675$		−	8.28772i		−	0.318994i
$676$		0			0
$677$	−22.3588			−0.859319			−0.429660	−	0.902991i	$-0.641366\pi$
$677$	−22.3588			−0.859319			−0.429660	+	0.902991i	$0.641366\pi$
$678$		0			0
$679$	28.2960	−	12.6863i	1.08590	−	0.486855i
$680$		0			0
$681$	−76.0833			−2.91552
$682$		0			0
$683$	−32.3431			−1.23758			−0.618788	−	0.785558i	$-0.712376\pi$
$683$	−32.3431			−1.23758			−0.618788	+	0.785558i	$0.712376\pi$
$684$		0			0
$685$	−2.16478			−0.0827122
$686$		0			0
$687$		−	76.7696i		−	2.92894i
$688$		0			0
$689$		−	8.28772i		−	0.315737i
$690$		0			0
$691$			10.3756i			0.394706i	0.980333	+	0.197353i	$0.0632345\pi$
$691$			10.3756i			0.394706i	−0.980333	+	0.197353i	$0.936766\pi$
$692$		0			0
$693$	−8.28772	−	18.4853i	−0.314824	−	0.702198i
$694$		0			0
$695$		−	10.8284i		−	0.410746i
$696$		0			0
$697$	−61.2548			−2.32019
$698$		0			0
$699$			47.0363i			1.77907i
$700$		0			0
$701$			13.3137i			0.502852i	0.967877	+	0.251426i	$0.0808995\pi$
$701$			13.3137i			0.502852i	−0.967877	+	0.251426i	$0.919101\pi$
$702$		0			0
$703$	−17.4721			−0.658974
$704$		0			0
$705$			8.65914i			0.326122i
$706$		0			0
$707$	9.17157	+	20.4567i	0.344932	+	0.769352i
$708$		0			0
$709$		−	18.0000i		−	0.676004i	−0.941145	−	0.338002i	$-0.890249\pi$
$709$		−	18.0000i		−	0.676004i	0.941145	−	0.338002i	$-0.109751\pi$
$710$		0			0
$711$			27.4558i			1.02967i
$712$		0			0
$713$		−	27.0279i		−	1.01220i
$714$		0			0
$715$	8.97056			0.335480
$716$		0			0
$717$	51.3658			1.91829
$718$		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
$723$	41.9411			1.55981
$724$		0			0
$725$		−	29.3137i		−	1.08868i
$726$		0			0
$727$	−13.1426			−0.487430			−0.243715	−	0.969847i	$-0.578366\pi$
$727$	−13.1426			−0.487430			−0.243715	+	0.969847i	$0.578366\pi$
$728$		0			0
$729$	39.2843			1.45497
$730$		0			0
$731$			56.5921i			2.09313i
$732$		0			0
$733$	−22.3588			−0.825842			−0.412921	−	0.910767i	$-0.635491\pi$
$733$	−22.3588			−0.825842			−0.412921	+	0.910767i	$0.635491\pi$
$734$		0			0
$735$	14.7821	+	13.1716i	0.545245	+	0.485841i
$736$		0			0
$737$	−8.68629			−0.319964
$738$		0			0
$739$	28.6274			1.05308			0.526538	−	0.850151i	$-0.323490\pi$
$739$	28.6274			1.05308			0.526538	+	0.850151i	$0.323490\pi$
$740$		0			0
$741$	51.7373			1.90062
$742$		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
$747$		−	6.57128i		−	0.240430i
$748$		0			0
$749$	−12.6173	−	28.1421i	−0.461026	−	1.02829i
$750$		0			0
$751$		−	18.9706i		−	0.692246i	−0.938189	−	0.346123i	$-0.887498\pi$
$751$		−	18.9706i		−	0.692246i	0.938189	−	0.346123i	$-0.112502\pi$
$752$		0			0
$753$	−6.82843			−0.248842
$754$		0			0
$755$		−	21.2764i		−	0.774328i
$756$		0			0
$757$			26.6863i			0.969930i	0.874534	+	0.484965i	$0.161168\pi$
$757$			26.6863i			0.969930i	−0.874534	+	0.484965i	$0.838832\pi$
$758$		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
$763$	41.7990	−	18.7402i	1.51323	−	0.678442i
$764$		0			0
$765$		−	30.6274i		−	1.10734i
$766$		0			0
$767$		−	23.5147i		−	0.849067i
$768$		0			0
$769$			11.7206i			0.422656i	0.977415	+	0.211328i	$0.0677788\pi$
$769$			11.7206i			0.422656i	−0.977415	+	0.211328i	$0.932221\pi$
$770$		0			0
$771$	38.6274			1.39113
$772$		0			0
$773$	13.3283			0.479384			0.239692	−	0.970849i	$-0.422953\pi$
$773$	13.3283			0.479384			0.239692	+	0.970849i	$0.422953\pi$
$774$		0			0
$775$	−28.2960			−1.01642
$776$		0			0
$777$	10.3431	+	23.0698i	0.371058	+	0.827624i
$778$		0			0
$779$	−39.5980			−1.41874
$780$		0			0
$781$		−	6.34315i		−	0.226976i
$782$		0			0
$783$	−16.5754			−0.592358
$784$		0			0
$785$	9.17157			0.327347
$786$		0			0
$787$			25.6829i			0.915497i	0.889082	+	0.457749i	$0.151344\pi$
$787$			25.6829i			0.915497i	−0.889082	+	0.457749i	$0.848656\pi$
$788$		0			0
$789$	63.0864			2.24594
$790$		0			0
$791$	3.43289	+	7.65685i	0.122059	+	0.272246i
$792$		0			0
$793$	4.48528			0.159277
$794$		0			0
$795$	−5.65685			−0.200628
$796$		0			0
$797$	11.0096			0.389981			0.194991	−	0.980805i	$-0.437532\pi$
$797$	11.0096			0.389981			0.194991	+	0.980805i	$0.437532\pi$
$798$		0			0
$799$		−	22.6274i		−	0.800500i
$800$		0			0
$801$			20.0083i			0.706959i
$802$		0			0
$803$		−	1.79337i		−	0.0632865i
$804$		0			0
$805$	9.55582	−	4.28427i	0.336798	−	0.151001i
$806$		0			0
$807$		−	65.4558i		−	2.30415i
$808$		0			0
$809$	34.4853			1.21244			0.606219	−	0.795298i	$-0.292686\pi$
$809$	34.4853			1.21244			0.606219	+	0.795298i	$0.292686\pi$
$810$		0			0
$811$			25.1577i			0.883405i	0.897162	+	0.441702i	$0.145625\pi$
$811$			25.1577i			0.883405i	−0.897162	+	0.441702i	$0.854375\pi$
$812$		0			0
$813$			65.9411i			2.31266i
$814$		0			0
$815$	−5.75152			−0.201467
$816$		0			0
$817$			36.5838i			1.27990i
$818$		0			0
$819$	−17.1716	−	38.3002i	−0.600023	−	1.33832i
$820$		0			0
$821$			29.5980i			1.03298i	0.856294	+	0.516488i	$0.172761\pi$
$821$			29.5980i			1.03298i	−0.856294	+	0.516488i	$0.827239\pi$
$822$		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
$827$	47.2548			1.64321			0.821606	−	0.570056i	$-0.193079\pi$
$827$	47.2548			1.64321			0.821606	+	0.570056i	$0.193079\pi$
$828$		0			0
$829$	51.5515			1.79046			0.895230	−	0.445605i	$-0.147011\pi$
$829$	51.5515			1.79046			0.895230	+	0.445605i	$0.147011\pi$
$830$		0			0
$831$	18.2150			0.631870
$832$		0			0
$833$	−38.6274	−	34.4190i	−1.33836	−	1.19255i
$834$		0			0
$835$	16.9706			0.587291
$836$		0			0
$837$			16.0000i			0.553041i
$838$		0			0
$839$	13.1426			0.453731			0.226866	−	0.973926i	$-0.427152\pi$
$839$	13.1426			0.453731			0.226866	+	0.973926i	$0.427152\pi$
$840$		0			0
$841$	−29.6274			−1.02164
$842$		0			0
$843$			43.4495i			1.49648i
$844$		0			0
$845$	4.51528			0.155330
$846$		0			0
$847$	−7.57675	−	16.8995i	−0.260340	−	0.580674i
$848$		0			0
$849$	−6.82843			−0.234351
$850$		0			0
$851$	13.3726			0.458406
$852$		0			0
$853$	−19.2974			−0.660729			−0.330364	−	0.943853i	$-0.607172\pi$
$853$	−19.2974			−0.660729			−0.330364	+	0.943853i	$0.607172\pi$
$854$		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
$859$		−	53.0823i		−	1.81114i	−0.424193	−	0.905572i	$-0.639442\pi$
$859$		−	53.0823i		−	1.81114i	0.424193	−	0.905572i	$-0.360558\pi$
$860$		0			0
$861$	23.4412	+	52.2843i	0.798874	+	1.78184i
$862$		0			0
$863$			23.4558i			0.798446i	0.916854	+	0.399223i	$0.130720\pi$
$863$			23.4558i			0.798446i	−0.916854	+	0.399223i	$0.869280\pi$
$864$		0			0
$865$	−11.5147			−0.391512
$866$		0			0
$867$			98.3252i			3.33930i
$868$		0			0
$869$		−	14.3431i		−	0.486558i
$870$		0			0
$871$	−17.9974			−0.609818
$872$		0			0
$873$			44.8715i			1.51867i
$874$		0			0
$875$	−10.3431	−	23.0698i	−0.349662	−	0.779901i
$876$		0			0
$877$		−	11.6569i		−	0.393624i	−0.980441	−	0.196812i	$-0.936941\pi$
$877$		−	11.6569i		−	0.393624i	0.980441	−	0.196812i	$-0.0630589\pi$
$878$		0			0
$879$		−	22.1421i		−	0.746836i
$880$		0			0
$881$			37.4804i			1.26275i	0.775478	+	0.631374i	$0.217509\pi$
$881$			37.4804i			1.26275i	−0.775478	+	0.631374i	$0.782491\pi$
$882$		0			0
$883$	35.6569			1.19995			0.599974	−	0.800019i	$-0.295177\pi$
$883$	35.6569			1.19995			0.599974	+	0.800019i	$0.295177\pi$
$884$		0			0
$885$	−16.0502			−0.539521
$886$		0			0
$887$	44.5001			1.49417			0.747083	−	0.664731i	$-0.231454\pi$
$887$	44.5001			1.49417			0.747083	+	0.664731i	$0.231454\pi$
$888$		0			0
$889$	18.4853	−	8.28772i	0.619976	−	0.277961i
$890$		0			0
$891$	−11.6569			−0.390519
$892$		0			0
$893$		−	14.6274i		−	0.489488i
$894$		0			0
$895$	−2.16478			−0.0723608
$896$		0			0
$897$	−39.5980			−1.32214
$898$		0			0
$899$			56.5921i			1.88745i
$900$		0			0
$901$	14.7821			0.492462
$902$		0			0
$903$	48.3044	−	21.6569i	1.60747	−	0.720695i
$904$		0			0
$905$	12.4853			0.415025
$906$		0			0
$907$	26.0000			0.863316			0.431658	−	0.902037i	$-0.357929\pi$
$907$	26.0000			0.863316			0.431658	+	0.902037i	$0.357929\pi$
$908$		0			0
$909$	−32.4399			−1.07596
$910$		0			0
$911$			35.6569i			1.18136i	0.806904	+	0.590682i	$0.201141\pi$
$911$			35.6569i			1.18136i	−0.806904	+	0.590682i	$0.798859\pi$
$912$		0			0
$913$			3.43289i			0.113612i
$914$		0			0
$915$		−	3.06147i		−	0.101209i
$916$		0			0
$917$	−8.47343	+	3.79899i	−0.279817	+	0.125454i
$918$		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
$923$		−	13.1426i		−	0.432592i
$924$		0			0
$925$		−	14.0000i		−	0.460317i
$926$		0			0
$927$	43.4495			1.42707
$928$		0			0
$929$			6.64820i			0.218120i	0.994035	+	0.109060i	$0.0347841\pi$
$929$			6.64820i			0.218120i	−0.994035	+	0.109060i	$0.965216\pi$
$930$		0			0
$931$	−24.9706	−	22.2500i	−0.818377	−	0.729215i
$932$		0			0
$933$		−	10.3431i		−	0.338619i
$934$		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
$939$	16.9706			0.553813
$940$		0			0
$941$	−8.10201			−0.264118			−0.132059	−	0.991242i	$-0.542159\pi$
$941$	−8.10201			−0.264118			−0.132059	+	0.991242i	$0.542159\pi$
$942$		0			0
$943$	30.3070			0.986931
$944$		0			0
$945$	−5.65685	+	2.53620i	−0.184017	+	0.0825027i
$946$		0			0
$947$	−39.6569			−1.28867			−0.644337	−	0.764741i	$-0.722867\pi$
$947$	−39.6569			−1.28867			−0.644337	+	0.764741i	$0.722867\pi$
$948$		0			0
$949$		−	3.71573i		−	0.120618i
$950$		0			0
$951$	−42.7067			−1.38486
$952$		0			0
$953$	44.6274			1.44562			0.722812	−	0.691045i	$-0.242849\pi$
$953$	44.6274			1.44562			0.722812	+	0.691045i	$0.242849\pi$
$954$		0			0
$955$			0.896683i			0.0290160i
$956$		0			0
$957$	40.0166			1.29355
$958$		0			0
$959$	−2.16478	−	4.82843i	−0.0699045	−	0.155918i
$960$		0			0
$961$	23.6274			0.762175
$962$		0			0
$963$	44.6274			1.43810
$964$		0			0
$965$	7.01962			0.225969
$966$		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
$971$			4.77791i			0.153330i	0.997057	+	0.0766652i	$0.0244273\pi$
$971$			4.77791i			0.153330i	−0.997057	+	0.0766652i	$0.975573\pi$
$972$		0			0
$973$	24.1522	−	10.8284i	0.774283	−	0.347143i
$974$		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
$979$		−	10.4525i		−	0.334063i
$980$		0			0
$981$			66.2843i			2.11629i
$982$		0			0
$983$	7.76245			0.247584			0.123792	−	0.992308i	$-0.460494\pi$
$983$	7.76245			0.247584			0.123792	+	0.992308i	$0.460494\pi$
$984$		0			0
$985$		−	8.28772i		−	0.264069i
$986$		0			0
$987$	−19.3137	+	8.65914i	−0.614762	+	0.275623i
$988$		0			0
$989$		−	28.0000i		−	0.890348i
$990$		0			0
$991$		−	2.48528i		−	0.0789476i	−0.999221	−	0.0394738i	$-0.987432\pi$
$991$		−	2.48528i		−	0.0789476i	0.999221	−	0.0394738i	$-0.0125682\pi$
$992$		0			0
$993$		−	13.8854i		−	0.440640i
$994$		0			0
$995$	0.970563			0.0307689
$996$		0			0
$997$	−5.04054			−0.159636			−0.0798178	−	0.996809i	$-0.525434\pi$
$997$	−5.04054			−0.159636			−0.0798178	+	0.996809i	$0.525434\pi$
$998$		0			0
$999$	−7.91630			−0.250461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.e.g.895.2		8
4.3	odd	2		1792.2.e.f.895.8		8
7.6	odd	2	inner	1792.2.e.g.895.7		8
8.3	odd	2	inner	1792.2.e.g.895.1		8
8.5	even	2		1792.2.e.f.895.7		8
16.3	odd	4		224.2.f.a.223.1	✓	8
16.5	even	4		448.2.f.d.447.2		8
16.11	odd	4		448.2.f.d.447.8		8
16.13	even	4		224.2.f.a.223.7	yes	8
28.27	even	2		1792.2.e.f.895.1		8
48.5	odd	4		4032.2.b.p.3583.3		8
48.11	even	4		4032.2.b.p.3583.4		8
48.29	odd	4		2016.2.b.b.1567.5		8
48.35	even	4		2016.2.b.b.1567.6		8
56.13	odd	2		1792.2.e.f.895.2		8
56.27	even	2	inner	1792.2.e.g.895.8		8
112.3	even	12		1568.2.p.a.607.1		16
112.13	odd	4		224.2.f.a.223.2	yes	8
112.19	even	12		1568.2.p.a.31.2		16
112.27	even	4		448.2.f.d.447.1		8
112.45	odd	12		1568.2.p.a.607.7		16
112.51	odd	12		1568.2.p.a.31.7		16
112.61	odd	12		1568.2.p.a.31.8		16
112.67	odd	12		1568.2.p.a.607.8		16
112.69	odd	4		448.2.f.d.447.7		8
112.83	even	4		224.2.f.a.223.8	yes	8
112.93	even	12		1568.2.p.a.31.1		16
112.109	even	12		1568.2.p.a.607.2		16
336.83	odd	4		2016.2.b.b.1567.3		8
336.125	even	4		2016.2.b.b.1567.4		8
336.251	odd	4		4032.2.b.p.3583.5		8
336.293	even	4		4032.2.b.p.3583.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.f.a.223.1	✓	8	16.3	odd	4
224.2.f.a.223.2	yes	8	112.13	odd	4
224.2.f.a.223.7	yes	8	16.13	even	4
224.2.f.a.223.8	yes	8	112.83	even	4
448.2.f.d.447.1		8	112.27	even	4
448.2.f.d.447.2		8	16.5	even	4
448.2.f.d.447.7		8	112.69	odd	4
448.2.f.d.447.8		8	16.11	odd	4
1568.2.p.a.31.1		16	112.93	even	12
1568.2.p.a.31.2		16	112.19	even	12
1568.2.p.a.31.7		16	112.51	odd	12
1568.2.p.a.31.8		16	112.61	odd	12
1568.2.p.a.607.1		16	112.3	even	12
1568.2.p.a.607.2		16	112.109	even	12
1568.2.p.a.607.7		16	112.45	odd	12
1568.2.p.a.607.8		16	112.67	odd	12
1792.2.e.f.895.1		8	28.27	even	2
1792.2.e.f.895.2		8	56.13	odd	2
1792.2.e.f.895.7		8	8.5	even	2
1792.2.e.f.895.8		8	4.3	odd	2
1792.2.e.g.895.1		8	8.3	odd	2	inner
1792.2.e.g.895.2		8	1.1	even	1	trivial
1792.2.e.g.895.7		8	7.6	odd	2	inner
1792.2.e.g.895.8		8	56.27	even	2	inner
2016.2.b.b.1567.3		8	336.83	odd	4
2016.2.b.b.1567.4		8	336.125	even	4
2016.2.b.b.1567.5		8	48.29	odd	4
2016.2.b.b.1567.6		8	48.35	even	4
4032.2.b.p.3583.3		8	48.5	odd	4
4032.2.b.p.3583.4		8	48.11	even	4
4032.2.b.p.3583.5		8	336.251	odd	4
4032.2.b.p.3583.6		8	336.293	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.61313i q^{3} +1.08239 q^{5} +(1.08239 + 2.41421i) q^{7} -3.82843 q^{9} +O(q^{10})\)	\(q-2.61313i q^{3} +1.08239 q^{5} +(1.08239 + 2.41421i) q^{7} -3.82843 q^{9} +2.00000 q^{11} +4.14386 q^{13} -2.82843i q^{15} +7.39104i q^{17} +4.77791i q^{19} +(6.30864 - 2.82843i) q^{21} -3.65685i q^{23} -3.82843 q^{25} +2.16478i q^{27} +7.65685i q^{29} +7.39104 q^{31} -5.22625i q^{33} +(1.17157 + 2.61313i) q^{35} +3.65685i q^{37} -10.8284i q^{39} +8.28772i q^{41} +7.65685 q^{43} -4.14386 q^{45} -3.06147 q^{47} +(-4.65685 + 5.22625i) q^{49} +19.3137 q^{51} -2.00000i q^{53} +2.16478 q^{55} +12.4853 q^{57} -5.67459i q^{59} +1.08239 q^{61} +(-4.14386 - 9.24264i) q^{63} +4.48528 q^{65} -4.34315 q^{67} -9.55582 q^{69} -3.17157i q^{71} -0.896683i q^{73} +10.0042i q^{75} +(2.16478 + 4.82843i) q^{77} -7.17157i q^{79} -5.82843 q^{81} +1.71644i q^{83} +8.00000i q^{85} +20.0083 q^{87} -5.22625i q^{89} +(4.48528 + 10.0042i) q^{91} -19.3137i q^{93} +5.17157i q^{95} -11.7206i q^{97} -7.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} + 16 q^{11} - 8 q^{25} + 32 q^{35} + 16 q^{43} + 8 q^{49} + 64 q^{51} + 32 q^{57} - 32 q^{65} - 80 q^{67} - 24 q^{81} - 32 q^{91} - 16 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 + 16 * q^11 - 8 * q^25 + 32 * q^35 + 16 * q^43 + 8 * q^49 + 64 * q^51 + 32 * q^57 - 32 * q^65 - 80 * q^67 - 24 * q^81 - 32 * q^91 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more