LMFDB - Embedded newform 2808.2.q.g.937.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2808,2,Mod(937,2808)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2808.937"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2808, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2808 = 2^{3} \cdot 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2808.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [22,0,0,0,3,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$22.4219928876$
Analytic rank:	$0$
Dimension:	$22$
Relative dimension:	$11$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 936)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			937.4
Character	$\chi$	$=$	2808.937
Dual form			2808.2.q.g.1873.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.618351 - 1.07102i) q^{5} +(2.31092 - 4.00263i) q^{7} +(0.517400 - 0.896163i) q^{11} +(-0.500000 - 0.866025i) q^{13} -7.43820 q^{17} +4.41618 q^{19} +(-2.42822 - 4.20580i) q^{23} +(1.73528 - 3.00560i) q^{25} +(3.04612 - 5.27603i) q^{29} +(3.58751 + 6.21374i) q^{31} -5.71584 q^{35} -8.61130 q^{37} +(0.532259 + 0.921899i) q^{41} +(1.34187 - 2.32419i) q^{43} +(-4.00605 + 6.93867i) q^{47} +(-7.18071 - 12.4374i) q^{49} +10.4399 q^{53} -1.27974 q^{55} +(-4.04267 - 7.00211i) q^{59} +(-5.92025 + 10.2542i) q^{61} +(-0.618351 + 1.07102i) q^{65} +(-0.406963 - 0.704881i) q^{67} -10.3144 q^{71} +8.15391 q^{73} +(-2.39134 - 4.14192i) q^{77} +(-3.78968 + 6.56392i) q^{79} +(-2.56200 + 4.43752i) q^{83} +(4.59942 + 7.96643i) q^{85} +3.37420 q^{89} -4.62184 q^{91} +(-2.73075 - 4.72980i) q^{95} +(4.96536 - 8.60025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 3 q^{5} - 4 q^{7} - 5 q^{11} - 11 q^{13} - 8 q^{17} + 10 q^{19} - 9 q^{23} - 24 q^{25} + 16 q^{29} - q^{31} + 18 q^{37} + 6 q^{41} - 7 q^{43} - 21 q^{47} - 27 q^{49} - 32 q^{53} + 34 q^{55} - 11 q^{59}+ \cdots - 10 q^{97}+O(q^{100}) $ 22 * q + 3 * q^5 - 4 * q^7 - 5 * q^11 - 11 * q^13 - 8 * q^17 + 10 * q^19 - 9 * q^23 - 24 * q^25 + 16 * q^29 - q^31 + 18 * q^37 + 6 * q^41 - 7 * q^43 - 21 * q^47 - 27 * q^49 - 32 * q^53 + 34 * q^55 - 11 * q^59 - 2 * q^61 + 3 * q^65 - 18 * q^67 + 30 * q^71 + 18 * q^73 + 9 * q^77 - 30 * q^79 + 16 * q^83 - 32 * q^85 - 26 * q^89 + 8 * q^91 - 20 * q^95 - 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2808\mathbb{Z}\right)^\times$.

$n$	$703$	$1081$	$1405$	$2081$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.618351	−	1.07102i	−0.276535	−	0.478973i	0.693986	−	0.719988i	$-0.255853\pi$
$5$	−0.618351	−	1.07102i	−0.276535	−	0.478973i	−0.970521	+	0.241016i	$0.922519\pi$
$6$		0			0
$7$	2.31092	−	4.00263i	0.873446	−	1.51285i	0.0150373	−	0.999887i	$-0.495213\pi$
$7$	2.31092	−	4.00263i	0.873446	−	1.51285i	0.858409	−	0.512966i	$-0.171453\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.517400	−	0.896163i	0.156002	−	0.270203i	−0.777422	−	0.628980i	$-0.783473\pi$
$11$	0.517400	−	0.896163i	0.156002	−	0.270203i	0.933423	+	0.358777i	$0.116806\pi$
$12$		0			0
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−7.43820			−1.80403			−0.902015	−	0.431705i	$-0.857912\pi$
$17$	−7.43820			−1.80403			−0.902015	+	0.431705i	$0.857912\pi$
$18$		0			0
$19$	4.41618			1.01314			0.506571	−	0.862198i	$-0.330913\pi$
$19$	4.41618			1.01314			0.506571	+	0.862198i	$0.330913\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.42822	−	4.20580i	−0.506318	−	0.876969i	−0.999973	−	0.00731127i	$-0.997673\pi$
$23$	−2.42822	−	4.20580i	−0.506318	−	0.876969i	0.493655	−	0.869658i	$-0.335661\pi$
$24$		0			0
$25$	1.73528	−	3.00560i	0.347057	−	0.601120i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.04612	−	5.27603i	0.565650	−	0.979734i	−0.431339	−	0.902190i	$-0.641959\pi$
$29$	3.04612	−	5.27603i	0.565650	−	0.979734i	0.996989	−	0.0775442i	$-0.0247079\pi$
$30$		0			0
$31$	3.58751	+	6.21374i	0.644335	+	1.11602i	0.984455	+	0.175639i	$0.0561992\pi$
$31$	3.58751	+	6.21374i	0.644335	+	1.11602i	−0.340119	+	0.940382i	$0.610467\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.71584			−0.966154
$36$		0			0
$37$	−8.61130			−1.41569			−0.707844	−	0.706368i	$-0.750332\pi$
$37$	−8.61130			−1.41569			−0.707844	+	0.706368i	$0.750332\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.532259	+	0.921899i	0.0831249	+	0.143976i	0.904591	−	0.426281i	$-0.140177\pi$
$41$	0.532259	+	0.921899i	0.0831249	+	0.143976i	−0.821466	+	0.570258i	$0.806843\pi$
$42$		0			0
$43$	1.34187	−	2.32419i	0.204634	−	0.354436i	−0.745382	−	0.666637i	$-0.767733\pi$
$43$	1.34187	−	2.32419i	0.204634	−	0.354436i	0.950016	+	0.312201i	$0.101066\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.00605	+	6.93867i	−0.584342	+	1.01211i	0.410615	+	0.911809i	$0.365314\pi$
$47$	−4.00605	+	6.93867i	−0.584342	+	1.01211i	−0.994957	+	0.100301i	$0.968019\pi$
$48$		0			0
$49$	−7.18071	−	12.4374i	−1.02582	−	1.77677i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	10.4399			1.43403			0.717017	−	0.697056i	$-0.245507\pi$
$53$	10.4399			1.43403			0.717017	+	0.697056i	$0.245507\pi$
$54$		0			0
$55$	−1.27974			−0.172560
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.04267	−	7.00211i	−0.526311	−	0.911597i	−0.999530	−	0.0306521i	$-0.990242\pi$
$59$	−4.04267	−	7.00211i	−0.526311	−	0.911597i	0.473220	−	0.880945i	$-0.343092\pi$
$60$		0			0
$61$	−5.92025	+	10.2542i	−0.758011	+	1.31291i	0.185853	+	0.982578i	$0.440495\pi$
$61$	−5.92025	+	10.2542i	−0.758011	+	1.31291i	−0.943864	+	0.330335i	$0.892838\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.618351	+	1.07102i	−0.0766970	+	0.132843i
$66$		0			0
$67$	−0.406963	−	0.704881i	−0.0497185	−	0.0861149i	0.840095	−	0.542439i	$-0.182499\pi$
$67$	−0.406963	−	0.704881i	−0.0497185	−	0.0861149i	−0.889814	+	0.456324i	$0.849166\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−10.3144			−1.22409			−0.612044	−	0.790824i	$-0.709653\pi$
$71$	−10.3144			−1.22409			−0.612044	+	0.790824i	$0.709653\pi$
$72$		0			0
$73$	8.15391			0.954343			0.477171	−	0.878810i	$-0.341662\pi$
$73$	8.15391			0.954343			0.477171	+	0.878810i	$0.341662\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.39134	−	4.14192i	−0.272518	−	0.472016i
$78$		0			0
$79$	−3.78968	+	6.56392i	−0.426373	+	0.738499i	−0.996548	−	0.0830244i	$-0.973542\pi$
$79$	−3.78968	+	6.56392i	−0.426373	+	0.738499i	0.570175	+	0.821523i	$0.306875\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.56200	+	4.43752i	−0.281216	+	0.487081i	−0.971685	−	0.236282i	$-0.924071\pi$
$83$	−2.56200	+	4.43752i	−0.281216	+	0.487081i	0.690468	+	0.723363i	$0.257405\pi$
$84$		0			0
$85$	4.59942	+	7.96643i	0.498877	+	0.864081i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.37420			0.357665			0.178832	−	0.983880i	$-0.442768\pi$
$89$	3.37420			0.357665			0.178832	+	0.983880i	$0.442768\pi$
$90$		0			0
$91$	−4.62184			−0.484501
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.73075	−	4.72980i	−0.280169	−	0.485267i
$96$		0			0
$97$	4.96536	−	8.60025i	0.504156	−	0.873223i	−0.495833	−	0.868418i	$-0.665137\pi$
$97$	4.96536	−	8.60025i	0.504156	−	0.873223i	0.999988	−	0.00480529i	$-0.00152958\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.98103	+	5.16330i	−0.296624	+	0.513768i	−0.975361	−	0.220613i	$-0.929194\pi$
$101$	−2.98103	+	5.16330i	−0.296624	+	0.513768i	0.678737	+	0.734381i	$0.262527\pi$
$102$		0			0
$103$	−2.65513	−	4.59883i	−0.261618	−	0.453136i	0.705054	−	0.709154i	$-0.250923\pi$
$103$	−2.65513	−	4.59883i	−0.261618	−	0.453136i	−0.966672	+	0.256018i	$0.917589\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.79266			0.269977			0.134988	−	0.990847i	$-0.456900\pi$
$107$	2.79266			0.269977			0.134988	+	0.990847i	$0.456900\pi$
$108$		0			0
$109$	−11.3696			−1.08901			−0.544507	−	0.838756i	$-0.683283\pi$
$109$	−11.3696			−1.08901			−0.544507	+	0.838756i	$0.683283\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.99219	+	15.5749i	0.845914	+	1.46517i	0.884825	+	0.465924i	$0.154278\pi$
$113$	8.99219	+	15.5749i	0.845914	+	1.46517i	−0.0389108	+	0.999243i	$0.512389\pi$
$114$		0			0
$115$	−3.00298	+	5.20132i	−0.280029	+	0.485025i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−17.1891	+	29.7724i	−1.57572	+	2.72923i
$120$		0			0
$121$	4.96460	+	8.59893i	0.451327	+	0.781721i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.4756			−0.936963
$126$		0			0
$127$	−8.20363			−0.727954			−0.363977	−	0.931408i	$-0.618581\pi$
$127$	−8.20363			−0.727954			−0.363977	+	0.931408i	$0.618581\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.73112	−	9.92658i	−0.500730	−	0.867290i	−1.00000		0.000843000i	$-0.999732\pi$
$131$	−5.73112	−	9.92658i	−0.500730	−	0.867290i	0.499270	−	0.866447i	$-0.333602\pi$
$132$		0			0
$133$	10.2055	−	17.6764i	0.884925	−	1.53274i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.5319	−	18.2418i	0.899800	−	1.55850i	0.0720520	−	0.997401i	$-0.477045\pi$
$137$	10.5319	−	18.2418i	0.899800	−	1.55850i	0.827749	−	0.561099i	$-0.189621\pi$
$138$		0			0
$139$	−11.7804	−	20.4042i	−0.999200	−	1.73067i	−0.534227	−	0.845341i	$-0.679397\pi$
$139$	−11.7804	−	20.4042i	−0.999200	−	1.73067i	−0.464973	−	0.885325i	$-0.653936\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.03480			−0.0865343
$144$		0			0
$145$	−7.53428			−0.625688
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.17486	−	2.03492i	−0.0962485	−	0.166707i	0.813880	−	0.581032i	$-0.197351\pi$
$149$	−1.17486	−	2.03492i	−0.0962485	−	0.166707i	−0.910129	+	0.414325i	$0.864018\pi$
$150$		0			0
$151$	0.914469	−	1.58391i	0.0744184	−	0.128896i	−0.826415	−	0.563062i	$-0.809623\pi$
$151$	0.914469	−	1.58391i	0.0744184	−	0.128896i	0.900833	+	0.434165i	$0.142957\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.43668	−	7.68455i	0.356363	−	0.617238i
$156$		0			0
$157$	−9.53466	−	16.5145i	−0.760948	−	1.31800i	−0.942362	−	0.334594i	$-0.891401\pi$
$157$	−9.53466	−	16.5145i	−0.760948	−	1.31800i	0.181414	−	0.983407i	$-0.441933\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−22.4457			−1.76897
$162$		0			0
$163$	5.40033			0.422986			0.211493	−	0.977379i	$-0.432167\pi$
$163$	5.40033			0.422986			0.211493	+	0.977379i	$0.432167\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.80911	−	8.32963i	−0.372140	−	0.644566i	0.617754	−	0.786371i	$-0.288043\pi$
$167$	−4.80911	−	8.32963i	−0.372140	−	0.644566i	−0.989895	+	0.141805i	$0.954709\pi$
$168$		0			0
$169$	−0.500000	+	0.866025i	−0.0384615	+	0.0666173i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.182732	−	0.316501i	0.0138929	−	0.0240632i	−0.858995	−	0.511983i	$-0.828911\pi$
$173$	0.182732	−	0.316501i	0.0138929	−	0.0240632i	0.872888	+	0.487920i	$0.162244\pi$
$174$		0			0
$175$	−8.02021	−	13.8914i	−0.606271	−	1.05009i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3.94330			−0.294736			−0.147368	−	0.989082i	$-0.547080\pi$
$179$	−3.94330			−0.294736			−0.147368	+	0.989082i	$0.547080\pi$
$180$		0			0
$181$	5.11469			0.380172			0.190086	−	0.981767i	$-0.439123\pi$
$181$	5.11469			0.380172			0.190086	+	0.981767i	$0.439123\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.32481	+	9.22283i	0.391487	+	0.678076i
$186$		0			0
$187$	−3.84852	+	6.66584i	−0.281432	+	0.487454i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.217758	−	0.377168i	0.0157564	−	0.0272909i	−0.858040	−	0.513583i	$-0.828318\pi$
$191$	0.217758	−	0.377168i	0.0157564	−	0.0272909i	0.873796	+	0.486292i	$0.161651\pi$
$192$		0			0
$193$	−8.14261	−	14.1034i	−0.586118	−	1.01519i	−0.994735	−	0.102480i	$-0.967322\pi$
$193$	−8.14261	−	14.1034i	−0.586118	−	1.01519i	0.408617	−	0.912706i	$-0.366011\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	15.9126			1.13373			0.566863	−	0.823812i	$-0.308157\pi$
$197$	15.9126			1.13373			0.566863	+	0.823812i	$0.308157\pi$
$198$		0			0
$199$	13.8785			0.983822			0.491911	−	0.870646i	$-0.336299\pi$
$199$	13.8785			0.983822			0.491911	+	0.870646i	$0.336299\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−14.0787	−	24.3850i	−0.988129	−	1.71149i
$204$		0			0
$205$	0.658245	−	1.14011i	0.0459739	−	0.0796291i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.28493	−	3.95762i	0.158052	−	0.273754i
$210$		0			0
$211$	−4.72422	−	8.18259i	−0.325229	−	0.563313i	0.656330	−	0.754474i	$-0.272108\pi$
$211$	−4.72422	−	8.18259i	−0.325229	−	0.563313i	−0.981559	+	0.191161i	$0.938775\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.31899			−0.226353
$216$		0			0
$217$	33.1618			2.25117
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.71910	+	6.44167i	0.250174	+	0.433314i
$222$		0			0
$223$	−4.44932	+	7.70644i	−0.297948	+	0.516062i	−0.975666	−	0.219260i	$-0.929636\pi$
$223$	−4.44932	+	7.70644i	−0.297948	+	0.516062i	0.677718	+	0.735322i	$0.262969\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.10801	+	15.7755i	−0.604520	+	1.04706i	0.387607	+	0.921825i	$0.373302\pi$
$227$	−9.10801	+	15.7755i	−0.604520	+	1.04706i	−0.992127	+	0.125235i	$0.960032\pi$
$228$		0			0
$229$	−11.8921	−	20.5977i	−0.785852	−	1.36113i	−0.928489	−	0.371360i	$-0.878892\pi$
$229$	−11.8921	−	20.5977i	−0.785852	−	1.36113i	0.142638	−	0.989775i	$-0.454442\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	8.70410			0.570225			0.285112	−	0.958494i	$-0.407969\pi$
$233$	8.70410			0.570225			0.285112	+	0.958494i	$0.407969\pi$
$234$		0			0
$235$	9.90857			0.646364
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.28917	+	14.3573i	0.536182	+	0.928694i	0.999105	+	0.0422959i	$0.0134672\pi$
$239$	8.28917	+	14.3573i	0.536182	+	0.928694i	−0.462923	+	0.886398i	$0.653199\pi$
$240$		0			0
$241$	−5.97432	+	10.3478i	−0.384839	+	0.666562i	−0.991747	−	0.128211i	$-0.959077\pi$
$241$	−5.97432	+	10.3478i	−0.384839	+	0.666562i	0.606907	+	0.794773i	$0.292410\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−8.88040	+	15.3813i	−0.567348	+	0.982676i
$246$		0			0
$247$	−2.20809	−	3.82453i	−0.140498	−	0.243349i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.09870			0.0693495			0.0346748	−	0.999399i	$-0.488960\pi$
$251$	1.09870			0.0693495			0.0346748	+	0.999399i	$0.488960\pi$
$252$		0			0
$253$	−5.02544			−0.315946
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.63626	+	9.76229i	0.351580	+	0.608955i	0.986526	−	0.163602i	$-0.0523112\pi$
$257$	5.63626	+	9.76229i	0.351580	+	0.608955i	−0.634946	+	0.772556i	$0.718978\pi$
$258$		0			0
$259$	−19.9000	+	34.4679i	−1.23653	+	2.14173i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.05973	−	15.6919i	0.558647	−	0.967605i	−0.438963	−	0.898505i	$-0.644654\pi$
$263$	9.05973	−	15.6919i	0.558647	−	0.967605i	0.997610	−	0.0690994i	$-0.0220126\pi$
$264$		0			0
$265$	−6.45554	−	11.1813i	−0.396561	−	0.686863i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.69587			−0.530197			−0.265098	−	0.964221i	$-0.585404\pi$
$269$	−8.69587			−0.530197			−0.265098	+	0.964221i	$0.585404\pi$
$270$		0			0
$271$	−2.47019			−0.150053			−0.0750266	−	0.997182i	$-0.523904\pi$
$271$	−2.47019			−0.150053			−0.0750266	+	0.997182i	$0.523904\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.79567	−	3.11019i	−0.108283	−	0.187552i
$276$		0			0
$277$	16.0775	−	27.8470i	0.966002	−	1.67316i	0.259103	−	0.965850i	$-0.416573\pi$
$277$	16.0775	−	27.8470i	0.966002	−	1.67316i	0.706899	−	0.707315i	$-0.250094\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.02312	−	10.4323i	0.359309	−	0.622342i	−0.628536	−	0.777780i	$-0.716346\pi$
$281$	6.02312	−	10.4323i	0.359309	−	0.622342i	0.987846	+	0.155438i	$0.0496790\pi$
$282$		0			0
$283$	−8.39750	−	14.5449i	−0.499180	−	0.864605i	0.500820	−	0.865552i	$-0.333032\pi$
$283$	−8.39750	−	14.5449i	−0.499180	−	0.864605i	−1.00000		0.000946989i	$0.999699\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.92003			0.290420
$288$		0			0
$289$	38.3269			2.25452
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5.95928	−	10.3218i	−0.348145	−	0.603004i	0.637775	−	0.770223i	$-0.279855\pi$
$293$	−5.95928	−	10.3218i	−0.348145	−	0.603004i	−0.985920	+	0.167218i	$0.946522\pi$
$294$		0			0
$295$	−4.99958	+	8.65952i	−0.291087	+	0.504177i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.42822	+	4.20580i	−0.140427	+	0.243227i
$300$		0			0
$301$	−6.20192	−	10.7420i	−0.357473	−	0.619161i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	14.6432			0.838466
$306$		0			0
$307$	2.39507			0.136694			0.0683469	−	0.997662i	$-0.478228\pi$
$307$	2.39507			0.136694			0.0683469	+	0.997662i	$0.478228\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.2622	+	21.2388i	0.695327	+	1.20434i	0.970070	+	0.242824i	$0.0780737\pi$
$311$	12.2622	+	21.2388i	0.695327	+	1.20434i	−0.274743	+	0.961518i	$0.588593\pi$
$312$		0			0
$313$	4.17269	−	7.22731i	0.235854	−	0.408512i	−0.723666	−	0.690150i	$-0.757545\pi$
$313$	4.17269	−	7.22731i	0.235854	−	0.408512i	0.959521	+	0.281638i	$0.0908779\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.35033	−	4.07089i	0.132008	−	0.228644i	−0.792443	−	0.609946i	$-0.791191\pi$
$317$	2.35033	−	4.07089i	0.132008	−	0.228644i	0.924450	+	0.381302i	$0.124524\pi$
$318$		0			0
$319$	−3.15212	−	5.45963i	−0.176485	−	0.305681i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−32.8485			−1.82774
$324$		0			0
$325$	−3.47057			−0.192512
$326$		0			0
$327$		0			0
$328$		0			0
$329$	18.5153	+	32.0695i	1.02078	+	1.76805i
$330$		0			0
$331$	−7.41963	+	12.8512i	−0.407820	+	0.706364i	−0.994645	−	0.103349i	$-0.967044\pi$
$331$	−7.41963	+	12.8512i	−0.407820	+	0.706364i	0.586826	+	0.809713i	$0.300377\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.503292	+	0.871727i	−0.0274978	+	0.0476276i
$336$		0			0
$337$	14.4755	+	25.0722i	0.788529	+	1.36577i	0.926868	+	0.375387i	$0.122490\pi$
$337$	14.4755	+	25.0722i	0.788529	+	1.36577i	−0.138339	+	0.990385i	$0.544176\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	7.42470			0.402070
$342$		0			0
$343$	−34.0234			−1.83709
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.47367	+	4.28452i	0.132793	+	0.230005i	0.924752	−	0.380569i	$-0.124272\pi$
$347$	2.47367	+	4.28452i	0.132793	+	0.230005i	−0.791959	+	0.610574i	$0.790939\pi$
$348$		0			0
$349$	−13.0836	+	22.6615i	−0.700351	+	1.21304i	0.267993	+	0.963421i	$0.413640\pi$
$349$	−13.0836	+	22.6615i	−0.700351	+	1.21304i	−0.968343	+	0.249622i	$0.919694\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	17.0625	−	29.5531i	0.908143	−	1.57295i	0.0915021	−	0.995805i	$-0.470833\pi$
$353$	17.0625	−	29.5531i	0.908143	−	1.57295i	0.816641	−	0.577146i	$-0.195833\pi$
$354$		0			0
$355$	6.37789	+	11.0468i	0.338503	+	0.586305i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	31.0572			1.63914			0.819569	−	0.572980i	$-0.194213\pi$
$359$	31.0572			1.63914			0.819569	+	0.572980i	$0.194213\pi$
$360$		0			0
$361$	0.502686			0.0264572
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.04198	−	8.73296i	−0.263909	−	0.457104i
$366$		0			0
$367$	−13.3961	+	23.2027i	−0.699271	+	1.21117i	0.269449	+	0.963015i	$0.413158\pi$
$367$	−13.3961	+	23.2027i	−0.699271	+	1.21117i	−0.968720	+	0.248158i	$0.920175\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	24.1258	−	41.7872i	1.25255	−	2.16948i
$372$		0			0
$373$	−9.69162	−	16.7864i	−0.501813	−	0.869166i	−0.999998	−	0.00209477i	$-0.999333\pi$
$373$	−9.69162	−	16.7864i	−0.501813	−	0.869166i	0.498185	−	0.867071i	$-0.334000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.09223			−0.313766
$378$		0			0
$379$	9.40917			0.483317			0.241658	−	0.970361i	$-0.422309\pi$
$379$	9.40917			0.483317			0.241658	+	0.970361i	$0.422309\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6.66805	+	11.5494i	0.340721	+	0.590147i	0.984567	−	0.175009i	$-0.0559954\pi$
$383$	6.66805	+	11.5494i	0.340721	+	0.590147i	−0.643845	+	0.765156i	$0.722662\pi$
$384$		0			0
$385$	−2.95737	+	5.12232i	−0.150722	+	0.261058i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.51890	−	2.63081i	0.0770112	−	0.133387i	−0.824948	−	0.565209i	$-0.808796\pi$
$389$	1.51890	−	2.63081i	0.0770112	−	0.133387i	0.901959	+	0.431821i	$0.142129\pi$
$390$		0			0
$391$	18.0616	+	31.2836i	0.913413	+	1.58208i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9.37341			0.471628
$396$		0			0
$397$	26.4484			1.32741			0.663703	−	0.747997i	$-0.268984\pi$
$397$	26.4484			1.32741			0.663703	+	0.747997i	$0.268984\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.85782	−	10.1460i	−0.292525	−	0.506669i	0.681881	−	0.731463i	$-0.261162\pi$
$401$	−5.85782	−	10.1460i	−0.292525	−	0.506669i	−0.974406	+	0.224794i	$0.927829\pi$
$402$		0			0
$403$	3.58751	−	6.21374i	0.178706	−	0.309529i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.45548	+	7.71712i	−0.220850	+	0.382524i
$408$		0			0
$409$	−1.53325	−	2.65567i	−0.0758144	−	0.131314i	0.825626	−	0.564218i	$-0.190822\pi$
$409$	−1.53325	−	2.65567i	−0.0758144	−	0.131314i	−0.901440	+	0.432904i	$0.857489\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−37.3692			−1.83882
$414$		0			0
$415$	6.33687			0.311065
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.61427	+	4.52805i	0.127715	+	0.221209i	0.922791	−	0.385301i	$-0.125902\pi$
$419$	2.61427	+	4.52805i	0.127715	+	0.221209i	−0.795076	+	0.606510i	$0.792569\pi$
$420$		0			0
$421$	−18.4928	+	32.0304i	−0.901283	+	1.56107i	−0.0754524	+	0.997149i	$0.524040\pi$
$421$	−18.4928	+	32.0304i	−0.901283	+	1.56107i	−0.825831	+	0.563918i	$0.809293\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−12.9074	+	22.3563i	−0.626101	+	1.08444i
$426$		0			0
$427$	27.3625	+	47.3932i	1.32416	+	2.29352i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.273718			−0.0131845			−0.00659227	−	0.999978i	$-0.502098\pi$
$431$	−0.273718			−0.0131845			−0.00659227	+	0.999978i	$0.502098\pi$
$432$		0			0
$433$	−14.2228			−0.683504			−0.341752	−	0.939790i	$-0.611020\pi$
$433$	−14.2228			−0.683504			−0.341752	+	0.939790i	$0.611020\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10.7235	−	18.5736i	−0.512973	−	0.888494i
$438$		0			0
$439$	1.86696	−	3.23366i	0.0891049	−	0.154334i	−0.818028	−	0.575178i	$-0.804933\pi$
$439$	1.86696	−	3.23366i	0.0891049	−	0.154334i	0.907133	+	0.420844i	$0.138266\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−19.2608	+	33.3606i	−0.915107	+	1.58501i	−0.108363	+	0.994111i	$0.534561\pi$
$443$	−19.2608	+	33.3606i	−0.915107	+	1.58501i	−0.806744	+	0.590901i	$0.798772\pi$
$444$		0			0
$445$	−2.08644	−	3.61382i	−0.0989068	−	0.171312i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	11.4384			0.539813			0.269907	−	0.962887i	$-0.413007\pi$
$449$	11.4384			0.539813			0.269907	+	0.962887i	$0.413007\pi$
$450$		0			0
$451$	1.10156			0.0518705
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.85792	+	4.95006i	0.133981	+	0.232063i
$456$		0			0
$457$	11.8697	−	20.5589i	0.555240	−	0.961704i	−0.442644	−	0.896697i	$-0.645960\pi$
$457$	11.8697	−	20.5589i	0.555240	−	0.961704i	0.997885	−	0.0650072i	$-0.0207070\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7.04097	−	12.1953i	0.327931	−	0.567993i	−0.654170	−	0.756347i	$-0.726982\pi$
$461$	7.04097	−	12.1953i	0.327931	−	0.567993i	0.982101	+	0.188355i	$0.0603154\pi$
$462$		0			0
$463$	−5.80139	−	10.0483i	−0.269613	−	0.466984i	0.699149	−	0.714976i	$-0.253563\pi$
$463$	−5.80139	−	10.0483i	−0.269613	−	0.466984i	−0.968762	+	0.247992i	$0.920229\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	15.2384			0.705150			0.352575	−	0.935784i	$-0.385306\pi$
$467$	15.2384			0.705150			0.352575	+	0.935784i	$0.385306\pi$
$468$		0			0
$469$	−3.76184			−0.173706
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.38857	−	2.40507i	−0.0638465	−	0.110585i
$474$		0			0
$475$	7.66333	−	13.2733i	0.351618	−	0.609020i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.21975	+	5.57676i	−0.147114	+	0.254809i	−0.930160	−	0.367155i	$-0.880332\pi$
$479$	−3.21975	+	5.57676i	−0.147114	+	0.254809i	0.783046	+	0.621964i	$0.213665\pi$
$480$		0			0
$481$	4.30565	+	7.45760i	0.196321	+	0.340037i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12.2813			−0.557667
$486$		0			0
$487$	11.6913			0.529783			0.264891	−	0.964278i	$-0.414664\pi$
$487$	11.6913			0.529783			0.264891	+	0.964278i	$0.414664\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6.94879	+	12.0357i	0.313595	+	0.543162i	0.979138	−	0.203198i	$-0.0651334\pi$
$491$	6.94879	+	12.0357i	0.313595	+	0.543162i	−0.665543	+	0.746359i	$0.731800\pi$
$492$		0			0
$493$	−22.6576	+	39.2442i	−1.02045	+	1.76747i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−23.8357	+	41.2846i	−1.06918	+	1.85187i
$498$		0			0
$499$	−2.21828	−	3.84217i	−0.0993036	−	0.171999i	0.812093	−	0.583528i	$-0.198328\pi$
$499$	−2.21828	−	3.84217i	−0.0993036	−	0.171999i	−0.911397	+	0.411529i	$0.864995\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	18.6182			0.830144			0.415072	−	0.909788i	$-0.363756\pi$
$503$	18.6182			0.830144			0.415072	+	0.909788i	$0.363756\pi$
$504$		0			0
$505$	7.37330			0.328107
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6.60765	−	11.4448i	−0.292879	−	0.507281i	0.681610	−	0.731715i	$-0.261280\pi$
$509$	−6.60765	−	11.4448i	−0.292879	−	0.507281i	−0.974489	+	0.224434i	$0.927947\pi$
$510$		0			0
$511$	18.8430	−	32.6371i	0.833567	−	1.44378i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3.28361	+	5.68738i	−0.144693	+	0.250616i
$516$		0			0
$517$	4.14545	+	7.18014i	0.182317	+	0.315782i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	24.3997			1.06897			0.534486	−	0.845177i	$-0.320505\pi$
$521$	24.3997			1.06897			0.534486	+	0.845177i	$0.320505\pi$
$522$		0			0
$523$	29.2553			1.27924			0.639622	−	0.768690i	$-0.279091\pi$
$523$	29.2553			1.27924			0.639622	+	0.768690i	$0.279091\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−26.6846	−	46.2191i	−1.16240	−	2.01334i
$528$		0			0
$529$	−0.292482	+	0.506593i	−0.0127166	+	0.0220258i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	0.532259	−	0.921899i	0.0230547	−	0.0399319i
$534$		0			0
$535$	−1.72685	−	2.99099i	−0.0746581	−	0.129312i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−14.8612			−0.640117
$540$		0			0
$541$	29.9138			1.28609			0.643047	−	0.765827i	$-0.277670\pi$
$541$	29.9138			1.28609			0.643047	+	0.765827i	$0.277670\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.03043	+	12.1771i	0.301150	+	0.521608i
$546$		0			0
$547$	−10.4275	+	18.0610i	−0.445848	+	0.772231i	−0.998111	−	0.0614391i	$-0.980431\pi$
$547$	−10.4275	+	18.0610i	−0.445848	+	0.772231i	0.552263	+	0.833670i	$0.313764\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	13.4522	−	23.2999i	0.573084	−	0.992610i
$552$		0			0
$553$	17.5153	+	30.3374i	0.744827	+	1.29008i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	21.9289			0.929158			0.464579	−	0.885532i	$-0.346206\pi$
$557$	21.9289			0.929158			0.464579	+	0.885532i	$0.346206\pi$
$558$		0			0
$559$	−2.68374			−0.113510
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−9.14597	−	15.8413i	−0.385457	−	0.667631i	0.606376	−	0.795178i	$-0.292623\pi$
$563$	−9.14597	−	15.8413i	−0.385457	−	0.667631i	−0.991832	+	0.127548i	$0.959289\pi$
$564$		0			0
$565$	11.1207	−	19.2615i	0.467850	−	0.810339i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	16.9632	−	29.3811i	0.711135	−	1.23172i	−0.253297	−	0.967389i	$-0.581515\pi$
$569$	16.9632	−	29.3811i	0.711135	−	1.23172i	0.964432	−	0.264333i	$-0.0851517\pi$
$570$		0			0
$571$	−2.01975	−	3.49830i	−0.0845237	−	0.146399i	0.820664	−	0.571410i	$-0.193604\pi$
$571$	−2.01975	−	3.49830i	−0.0845237	−	0.146399i	−0.905188	+	0.425011i	$0.860270\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−16.8546			−0.702885
$576$		0			0
$577$	25.7179			1.07065			0.535325	−	0.844646i	$-0.320189\pi$
$577$	25.7179			1.07065			0.535325	+	0.844646i	$0.320189\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	11.8412	+	20.5095i	0.491255	+	0.850878i
$582$		0			0
$583$	5.40161	−	9.35587i	0.223712	−	0.387481i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4.65906	−	8.06972i	0.192300	−	0.333073i	−0.753712	−	0.657205i	$-0.771739\pi$
$587$	4.65906	−	8.06972i	0.192300	−	0.333073i	0.946012	+	0.324132i	$0.105072\pi$
$588$		0			0
$589$	15.8431	+	27.4410i	0.652803	+	1.13069i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−21.3998			−0.878786			−0.439393	−	0.898295i	$-0.644806\pi$
$593$	−21.3998			−0.878786			−0.439393	+	0.898295i	$0.644806\pi$
$594$		0			0
$595$	42.5156			1.74297
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10.5949	−	18.3510i	−0.432898	−	0.749801i	0.564224	−	0.825622i	$-0.309175\pi$
$599$	−10.5949	−	18.3510i	−0.432898	−	0.749801i	−0.997121	+	0.0758212i	$0.975842\pi$
$600$		0			0
$601$	−11.7940	+	20.4278i	−0.481088	+	0.833268i	−0.999764	−	0.0217021i	$-0.993091\pi$
$601$	−11.7940	+	20.4278i	−0.481088	+	0.833268i	0.518677	+	0.854970i	$0.326425\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.13972	−	10.6343i	0.249615	−	0.432346i
$606$		0			0
$607$	7.90602	+	13.6936i	0.320895	+	0.555807i	0.980673	−	0.195654i	$-0.0626829\pi$
$607$	7.90602	+	13.6936i	0.320895	+	0.555807i	−0.659778	+	0.751461i	$0.729350\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.01209			0.324134
$612$		0			0
$613$	−1.30853			−0.0528510			−0.0264255	−	0.999651i	$-0.508412\pi$
$613$	−1.30853			−0.0528510			−0.0264255	+	0.999651i	$0.508412\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.543935	+	0.942123i	0.0218980	+	0.0379285i	0.876767	−	0.480916i	$-0.159696\pi$
$617$	0.543935	+	0.942123i	0.0218980	+	0.0379285i	−0.854869	+	0.518844i	$0.826362\pi$
$618$		0			0
$619$	−7.24611	+	12.5506i	−0.291246	+	0.504453i	−0.974105	−	0.226098i	$-0.927403\pi$
$619$	−7.24611	+	12.5506i	−0.291246	+	0.504453i	0.682859	+	0.730550i	$0.260736\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	7.79751	−	13.5057i	0.312401	−	0.541094i
$624$		0			0
$625$	−2.19884	−	3.80851i	−0.0879537	−	0.152340i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	64.0526			2.55394
$630$		0			0
$631$	−41.4115			−1.64856			−0.824282	−	0.566179i	$-0.808421\pi$
$631$	−41.4115			−1.64856			−0.824282	+	0.566179i	$0.808421\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5.07272	+	8.78621i	0.201305	+	0.348670i
$636$		0			0
$637$	−7.18071	+	12.4374i	−0.284510	+	0.492786i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.0907	+	20.9416i	−0.477553	+	0.827145i	−0.999669	−	0.0257289i	$-0.991809\pi$
$641$	−12.0907	+	20.9416i	−0.477553	+	0.827145i	0.522116	+	0.852874i	$0.325143\pi$
$642$		0			0
$643$	18.6005	+	32.2169i	0.733530	+	1.27051i	0.955365	+	0.295428i	$0.0954621\pi$
$643$	18.6005	+	32.2169i	0.733530	+	1.27051i	−0.221835	+	0.975084i	$0.571205\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9.31675			−0.366279			−0.183140	−	0.983087i	$-0.558626\pi$
$647$	−9.31675			−0.366279			−0.183140	+	0.983087i	$0.558626\pi$
$648$		0			0
$649$	−8.36670			−0.328422
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.51790	−	16.4855i	−0.372464	−	0.645127i	0.617480	−	0.786587i	$-0.288154\pi$
$653$	−9.51790	−	16.4855i	−0.372464	−	0.645127i	−0.989944	+	0.141460i	$0.954820\pi$
$654$		0			0
$655$	−7.08768	+	12.2762i	−0.276939	+	0.479672i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	20.1576	−	34.9140i	0.785228	−	1.36006i	−0.143634	−	0.989631i	$-0.545879\pi$
$659$	20.1576	−	34.9140i	0.785228	−	1.36006i	0.928863	−	0.370424i	$-0.120788\pi$
$660$		0			0
$661$	−2.06589	−	3.57822i	−0.0803537	−	0.139177i	0.823048	−	0.567972i	$-0.192272\pi$
$661$	−2.06589	−	3.57822i	−0.0803537	−	0.139177i	−0.903402	+	0.428795i	$0.858938\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−25.2422			−0.978851
$666$		0			0
$667$	−29.5865			−1.14560
$668$		0			0
$669$		0			0
$670$		0			0
$671$	6.12627	+	10.6110i	0.236502	+	0.409634i
$672$		0			0
$673$	6.07150	−	10.5162i	0.234039	−	0.405368i	−0.724954	−	0.688798i	$-0.758139\pi$
$673$	6.07150	−	10.5162i	0.234039	−	0.405368i	0.958993	+	0.283430i	$0.0914722\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	20.5796	−	35.6449i	0.790937	−	1.36994i	−0.134450	−	0.990920i	$-0.542927\pi$
$677$	20.5796	−	35.6449i	0.790937	−	1.36994i	0.925387	−	0.379023i	$-0.123740\pi$
$678$		0			0
$679$	−22.9491	−	39.7490i	−0.880706	−	1.52543i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.71255			−0.0655290			−0.0327645	−	0.999463i	$-0.510431\pi$
$683$	−1.71255			−0.0655290			−0.0327645	+	0.999463i	$0.510431\pi$
$684$		0			0
$685$	−26.0496			−0.995305
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.21996	−	9.04124i	−0.198865	−	0.344444i
$690$		0			0
$691$	15.6971	−	27.1881i	0.597145	−	1.03429i	−0.396095	−	0.918209i	$-0.629635\pi$
$691$	15.6971	−	27.1881i	0.597145	−	1.03429i	0.993240	−	0.116076i	$-0.0370317\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−14.5688	+	25.2340i	−0.552628	+	0.957179i
$696$		0			0
$697$	−3.95905	−	6.85727i	−0.149960	−	0.259738i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−5.17150			−0.195325			−0.0976625	−	0.995220i	$-0.531137\pi$
$701$	−5.17150			−0.195325			−0.0976625	+	0.995220i	$0.531137\pi$
$702$		0			0
$703$	−38.0291			−1.43429
$704$		0			0
$705$		0			0
$706$		0			0
$707$	13.7779	+	23.8640i	0.518170	+	0.897497i
$708$		0			0
$709$	12.3700	−	21.4255i	0.464565	−	0.804650i	−0.534617	−	0.845095i	$-0.679544\pi$
$709$	12.3700	−	21.4255i	0.464565	−	0.804650i	0.999182	+	0.0404443i	$0.0128773\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	17.4225	−	30.1766i	0.652478	−	1.13012i
$714$		0			0
$715$	0.639869	+	1.10829i	0.0239298	+	0.0414475i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	48.5746			1.81153			0.905764	−	0.423783i	$-0.139298\pi$
$719$	48.5746			1.81153			0.905764	+	0.423783i	$0.139298\pi$
$720$		0			0
$721$	−24.5432			−0.914037
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−10.5718	−	18.3108i	−0.392625	−	0.680047i
$726$		0			0
$727$	−15.1119	+	26.1745i	−0.560469	+	0.970760i	0.436987	+	0.899468i	$0.356046\pi$
$727$	−15.1119	+	26.1745i	−0.560469	+	0.970760i	−0.997455	+	0.0712923i	$0.977288\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9.98112	+	17.2878i	−0.369165	+	0.639413i
$732$		0			0
$733$	6.25615	+	10.8360i	0.231076	+	0.400236i	0.958125	−	0.286350i	$-0.0924420\pi$
$733$	6.25615	+	10.8360i	0.231076	+	0.400236i	−0.727049	+	0.686586i	$0.759109\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.842250			−0.0310247
$738$		0			0
$739$	1.67848			0.0617439			0.0308720	−	0.999523i	$-0.490172\pi$
$739$	1.67848			0.0617439			0.0308720	+	0.999523i	$0.490172\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−5.53968	−	9.59501i	−0.203231	−	0.352007i	0.746337	−	0.665569i	$-0.231811\pi$
$743$	−5.53968	−	9.59501i	−0.203231	−	0.352007i	−0.949568	+	0.313562i	$0.898478\pi$
$744$		0			0
$745$	−1.45296	+	2.51659i	−0.0532322	+	0.0922008i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6.45363	−	11.1780i	0.235810	−	0.408435i
$750$		0			0
$751$	−4.76341	−	8.25046i	−0.173819	−	0.301064i	0.765933	−	0.642921i	$-0.222277\pi$
$751$	−4.76341	−	8.25046i	−0.173819	−	0.301064i	−0.939752	+	0.341857i	$0.888944\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.26185			−0.0823172
$756$		0			0
$757$	20.1215			0.731329			0.365664	−	0.930747i	$-0.380842\pi$
$757$	20.1215			0.731329			0.365664	+	0.930747i	$0.380842\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	25.5920	+	44.3267i	0.927711	+	1.60684i	0.787142	+	0.616772i	$0.211560\pi$
$761$	25.5920	+	44.3267i	0.927711	+	1.60684i	0.140569	+	0.990071i	$0.455107\pi$
$762$		0			0
$763$	−26.2743	+	45.5085i	−0.951195	+	1.64752i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.04267	+	7.00211i	−0.145972	+	0.252831i
$768$		0			0
$769$	4.27904	+	7.41152i	0.154306	+	0.267266i	0.932806	−	0.360378i	$-0.117352\pi$
$769$	4.27904	+	7.41152i	0.154306	+	0.267266i	−0.778500	+	0.627645i	$0.784019\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−19.6620			−0.707194			−0.353597	−	0.935398i	$-0.615041\pi$
$773$	−19.6620			−0.707194			−0.353597	+	0.935398i	$0.615041\pi$
$774$		0			0
$775$	24.9014			0.894484
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.35055	+	4.07128i	0.0842173	+	0.145869i
$780$		0			0
$781$	−5.33664	+	9.24333i	−0.190960	+	0.330753i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−11.7915	+	20.4235i	−0.420858	+	0.728947i
$786$		0			0
$787$	6.31421	+	10.9365i	0.225077	+	0.389845i	0.956343	−	0.292248i	$-0.0944032\pi$
$787$	6.31421	+	10.9365i	0.225077	+	0.389845i	−0.731265	+	0.682093i	$0.761070\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	83.1210			2.95544
$792$		0			0
$793$	11.8405			0.420469
$794$		0			0
$795$		0			0
$796$		0			0
$797$	21.8643	+	37.8700i	0.774472	+	1.34142i	0.935091	+	0.354408i	$0.115318\pi$
$797$	21.8643	+	37.8700i	0.774472	+	1.34142i	−0.160619	+	0.987016i	$0.551349\pi$
$798$		0			0
$799$	29.7978	−	51.6113i	1.05417	−	1.82588i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.21883	−	7.30723i	0.148879	−	0.257866i
$804$		0			0
$805$	13.8793	+	24.0397i	0.489181	+	0.847287i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	30.7778			1.08209			0.541044	−	0.840994i	$-0.318029\pi$
$809$	30.7778			1.08209			0.541044	+	0.840994i	$0.318029\pi$
$810$		0			0
$811$	0.698137			0.0245149			0.0122575	−	0.999925i	$-0.496098\pi$
$811$	0.698137			0.0245149			0.0122575	+	0.999925i	$0.496098\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.33930	−	5.78384i	−0.116971	−	0.202599i
$816$		0			0
$817$	5.92596	−	10.2641i	0.207323	−	0.359094i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.75886	+	16.9028i	−0.340586	+	0.589913i	−0.984542	−	0.175150i	$-0.943959\pi$
$821$	−9.75886	+	16.9028i	−0.340586	+	0.589913i	0.643955	+	0.765063i	$0.277292\pi$
$822$		0			0
$823$	−21.2528	−	36.8109i	−0.740825	−	1.28315i	−0.952120	−	0.305725i	$-0.901101\pi$
$823$	−21.2528	−	36.8109i	−0.740825	−	1.28315i	0.211295	−	0.977422i	$-0.432232\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−32.1362			−1.11749			−0.558743	−	0.829341i	$-0.688716\pi$
$827$	−32.1362			−1.11749			−0.558743	+	0.829341i	$0.688716\pi$
$828$		0			0
$829$	35.4543			1.23138			0.615690	−	0.787989i	$-0.288878\pi$
$829$	35.4543			1.23138			0.615690	+	0.787989i	$0.288878\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	53.4116	+	92.5116i	1.85060	+	3.20534i
$834$		0			0
$835$	−5.94744	+	10.3013i	−0.205820	+	0.356490i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.46587	−	4.27100i	0.0851311	−	0.147451i	−0.820316	−	0.571911i	$-0.806202\pi$
$839$	2.46587	−	4.27100i	0.0851311	−	0.147451i	0.905447	+	0.424459i	$0.139536\pi$
$840$		0			0
$841$	−4.05765	−	7.02806i	−0.139919	−	0.242347i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.23670			0.0425438
$846$		0			0
$847$	45.8912			1.57684
$848$		0			0
$849$		0			0
$850$		0			0
$851$	20.9101	+	36.2174i	0.716789	+	1.24152i
$852$		0			0
$853$	7.44717	−	12.8989i	0.254986	−	0.441649i	−0.709906	−	0.704297i	$-0.751262\pi$
$853$	7.44717	−	12.8989i	0.254986	−	0.441649i	0.964892	+	0.262648i	$0.0845957\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	10.0396	−	17.3891i	0.342946	−	0.593999i	−0.642033	−	0.766677i	$-0.721909\pi$
$857$	10.0396	−	17.3891i	0.342946	−	0.593999i	0.984978	+	0.172678i	$0.0552420\pi$
$858$		0			0
$859$	6.62665	+	11.4777i	0.226098	+	0.391614i	0.956648	−	0.291245i	$-0.0940696\pi$
$859$	6.62665	+	11.4777i	0.226098	+	0.391614i	−0.730550	+	0.682859i	$0.760736\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−10.8734			−0.370135			−0.185068	−	0.982726i	$-0.559250\pi$
$863$	−10.8734			−0.370135			−0.185068	+	0.982726i	$0.559250\pi$
$864$		0			0
$865$	−0.451971			−0.0153675
$866$		0			0
$867$		0			0
$868$		0			0
$869$	3.92156	+	6.79234i	0.133030	+	0.230414i
$870$		0			0
$871$	−0.406963	+	0.704881i	−0.0137894	+	0.0238840i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−24.2082	+	41.9299i	−0.818387	+	1.41749i
$876$		0			0
$877$	−6.24563	−	10.8178i	−0.210900	−	0.365289i	0.741096	−	0.671399i	$-0.234306\pi$
$877$	−6.24563	−	10.8178i	−0.210900	−	0.365289i	−0.951996	+	0.306109i	$0.900973\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.28015			0.0431293			0.0215646	−	0.999767i	$-0.493135\pi$
$881$	1.28015			0.0431293			0.0215646	+	0.999767i	$0.493135\pi$
$882$		0			0
$883$	−7.14883			−0.240577			−0.120289	−	0.992739i	$-0.538382\pi$
$883$	−7.14883			−0.240577			−0.120289	+	0.992739i	$0.538382\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	6.25551	+	10.8349i	0.210039	+	0.363799i	0.951727	−	0.306947i	$-0.0993074\pi$
$887$	6.25551	+	10.8349i	0.210039	+	0.363799i	−0.741687	+	0.670746i	$0.765974\pi$
$888$		0			0
$889$	−18.9579	+	32.8361i	−0.635829	+	1.10129i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−17.6914	+	30.6425i	−0.592021	+	1.02541i
$894$		0			0
$895$	2.43834	+	4.22333i	0.0815047	+	0.141170i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	43.7119			1.45787
$900$		0			0
$901$	−77.6543			−2.58704
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3.16268	−	5.47792i	−0.105131	−	0.182092i
$906$		0			0
$907$	26.0188	−	45.0659i	0.863940	−	1.49639i	−0.00415607	−	0.999991i	$-0.501323\pi$
$907$	26.0188	−	45.0659i	0.863940	−	1.49639i	0.868096	−	0.496396i	$-0.165344\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−25.0479	+	43.3842i	−0.829873	+	1.43738i	0.0682636	+	0.997667i	$0.478254\pi$
$911$	−25.0479	+	43.3842i	−0.829873	+	1.43738i	−0.898137	+	0.439716i	$0.855079\pi$
$912$		0			0
$913$	2.65116	+	4.59194i	0.0877406	+	0.151971i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−52.9766			−1.74944
$918$		0			0
$919$	31.9154			1.05279			0.526396	−	0.850240i	$-0.323543\pi$
$919$	31.9154			1.05279			0.526396	+	0.850240i	$0.323543\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5.15718	+	8.93249i	0.169750	+	0.294016i
$924$		0			0
$925$	−14.9430	+	25.8821i	−0.491324	+	0.850999i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.968674	−	1.67779i	0.0317812	−	0.0550466i	−0.849697	−	0.527271i	$-0.823215\pi$
$929$	0.968674	−	1.67779i	0.0317812	−	0.0550466i	0.881479	+	0.472224i	$0.156549\pi$
$930$		0			0
$931$	−31.7114	−	54.9257i	−1.03930	−	1.80012i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	9.51896			0.311303
$936$		0			0
$937$	12.4448			0.406553			0.203277	−	0.979121i	$-0.434841\pi$
$937$	12.4448			0.406553			0.203277	+	0.979121i	$0.434841\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−14.0002	−	24.2490i	−0.456393	−	0.790496i	0.542374	−	0.840137i	$-0.317526\pi$
$941$	−14.0002	−	24.2490i	−0.456393	−	0.790496i	−0.998767	+	0.0496411i	$0.984192\pi$
$942$		0			0
$943$	2.58488	−	4.47714i	0.0841753	−	0.145796i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	24.2126	−	41.9375i	0.786804	−	1.36278i	−0.141111	−	0.989994i	$-0.545068\pi$
$947$	24.2126	−	41.9375i	0.786804	−	1.36278i	0.927915	−	0.372791i	$-0.121599\pi$
$948$		0			0
$949$	−4.07695	−	7.06149i	−0.132344	−	0.229226i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	19.9083			0.644894			0.322447	−	0.946588i	$-0.395495\pi$
$953$	19.9083			0.644894			0.322447	+	0.946588i	$0.395495\pi$
$954$		0			0
$955$	−0.538604			−0.0174288
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−48.6768	−	84.3106i	−1.57185	−	2.72253i
$960$		0			0
$961$	−10.2404	+	17.7369i	−0.330336	+	0.572159i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−10.0700	+	17.4417i	−0.324164	+	0.561469i
$966$		0			0
$967$	24.5219	+	42.4732i	0.788572	+	1.36585i	0.926842	+	0.375452i	$0.122512\pi$
$967$	24.5219	+	42.4732i	0.788572	+	1.36585i	−0.138270	+	0.990395i	$0.544154\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	8.47587			0.272004			0.136002	−	0.990709i	$-0.456575\pi$
$971$	8.47587			0.272004			0.136002	+	0.990709i	$0.456575\pi$
$972$		0			0
$973$	−108.894			−3.49099
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−8.15362	−	14.1225i	−0.260857	−	0.451818i	0.705613	−	0.708598i	$-0.250672\pi$
$977$	−8.15362	−	14.1225i	−0.260857	−	0.451818i	−0.966470	+	0.256780i	$0.917339\pi$
$978$		0			0
$979$	1.74581	−	3.02383i	0.0557964	−	0.0966421i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.8494	−	29.1840i	0.537411	−	0.930824i	−0.461631	−	0.887072i	$-0.652736\pi$
$983$	16.8494	−	29.1840i	0.537411	−	0.930824i	0.999042	−	0.0437519i	$-0.0139311\pi$
$984$		0			0
$985$	−9.83957	−	17.0426i	−0.313515	−	0.543023i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−13.0334			−0.414439
$990$		0			0
$991$	−15.0990			−0.479634			−0.239817	−	0.970818i	$-0.577087\pi$
$991$	−15.0990			−0.479634			−0.239817	+	0.970818i	$0.577087\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8.58179	−	14.8641i	−0.272061	−	0.471224i
$996$		0			0
$997$	−7.49064	+	12.9742i	−0.237231	+	0.410896i	−0.959919	−	0.280279i	$-0.909573\pi$
$997$	−7.49064	+	12.9742i	−0.237231	+	0.410896i	0.722688	+	0.691175i	$0.242906\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2808.2.q.g.937.4		22
3.2	odd	2		936.2.q.g.313.7	✓	22
9.2	odd	6		8424.2.a.bf.1.4		11
9.4	even	3	inner	2808.2.q.g.1873.4		22
9.5	odd	6		936.2.q.g.625.7	yes	22
9.7	even	3		8424.2.a.be.1.8		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.q.g.313.7	✓	22	3.2	odd	2
936.2.q.g.625.7	yes	22	9.5	odd	6
2808.2.q.g.937.4		22	1.1	even	1	trivial
2808.2.q.g.1873.4		22	9.4	even	3	inner
8424.2.a.be.1.8		11	9.7	even	3
8424.2.a.bf.1.4		11	9.2	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2808 = 2^{3} \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2808.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.618351 - 1.07102i) q^{5} +(2.31092 - 4.00263i) q^{7} +(0.517400 - 0.896163i) q^{11} +(-0.500000 - 0.866025i) q^{13} -7.43820 q^{17} +4.41618 q^{19} +(-2.42822 - 4.20580i) q^{23} +(1.73528 - 3.00560i) q^{25} +(3.04612 - 5.27603i) q^{29} +(3.58751 + 6.21374i) q^{31} -5.71584 q^{35} -8.61130 q^{37} +(0.532259 + 0.921899i) q^{41} +(1.34187 - 2.32419i) q^{43} +(-4.00605 + 6.93867i) q^{47} +(-7.18071 - 12.4374i) q^{49} +10.4399 q^{53} -1.27974 q^{55} +(-4.04267 - 7.00211i) q^{59} +(-5.92025 + 10.2542i) q^{61} +(-0.618351 + 1.07102i) q^{65} +(-0.406963 - 0.704881i) q^{67} -10.3144 q^{71} +8.15391 q^{73} +(-2.39134 - 4.14192i) q^{77} +(-3.78968 + 6.56392i) q^{79} +(-2.56200 + 4.43752i) q^{83} +(4.59942 + 7.96643i) q^{85} +3.37420 q^{89} -4.62184 q^{91} +(-2.73075 - 4.72980i) q^{95} +(4.96536 - 8.60025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 3 q^{5} - 4 q^{7} - 5 q^{11} - 11 q^{13} - 8 q^{17} + 10 q^{19} - 9 q^{23} - 24 q^{25} + 16 q^{29} - q^{31} + 18 q^{37} + 6 q^{41} - 7 q^{43} - 21 q^{47} - 27 q^{49} - 32 q^{53} + 34 q^{55} - 11 q^{59}+ \cdots - 10 q^{97}+O(q^{100}) \) 22 * q + 3 * q^5 - 4 * q^7 - 5 * q^11 - 11 * q^13 - 8 * q^17 + 10 * q^19 - 9 * q^23 - 24 * q^25 + 16 * q^29 - q^31 + 18 * q^37 + 6 * q^41 - 7 * q^43 - 21 * q^47 - 27 * q^49 - 32 * q^53 + 34 * q^55 - 11 * q^59 - 2 * q^61 + 3 * q^65 - 18 * q^67 + 30 * q^71 + 18 * q^73 + 9 * q^77 - 30 * q^79 + 16 * q^83 - 32 * q^85 - 26 * q^89 + 8 * q^91 - 20 * q^95 - 10 * q^97

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