LMFDB - Embedded newform 1848.2.v.b.881.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1848,2,Mod(881,1848)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1848, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1848.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1848.v (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$14.7563542935$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.342102016.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 $ x^8 + x^6 + 4x^4 + 4x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.2
Root			$-1.17915 + 0.780776i$ of defining polynomial
Character	$\chi$	$=$	1848.881
Dual form			1848.2.v.b.881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.51022 + 0.848071i) q^{3} +0.662153 q^{5} +(-2.56155 + 0.662153i) q^{7} +(1.56155 - 2.56155i) q^{9} +O(q^{10})$	\(q+(-1.51022 + 0.848071i) q^{3} +0.662153 q^{5} +(-2.56155 + 0.662153i) q^{7} +(1.56155 - 2.56155i) q^{9} +1.00000i q^{11} +0.662153i q^{13} +(-1.00000 + 0.561553i) q^{15} +4.71659 q^{17} -4.05444i q^{19} +(3.30697 - 3.17238i) q^{21} -5.12311i q^{23} -4.56155 q^{25} +(-0.185917 + 5.19283i) q^{27} -3.56155i q^{29} +8.68951i q^{31} +(-0.848071 - 1.51022i) q^{33} +(-1.69614 + 0.438447i) q^{35} +4.43845 q^{37} +(-0.561553 - 1.00000i) q^{39} +9.43318 q^{41} -8.00000 q^{43} +(1.03399 - 1.69614i) q^{45} +5.75058 q^{47} +(6.12311 - 3.39228i) q^{49} +(-7.12311 + 4.00000i) q^{51} +8.24621i q^{53} +0.662153i q^{55} +(3.43845 + 6.12311i) q^{57} +1.40582 q^{59} +1.69614i q^{61} +(-2.30386 + 7.59554i) q^{63} +0.438447i q^{65} +9.56155 q^{67} +(4.34475 + 7.73704i) q^{69} +5.00691i q^{73} +(6.88897 - 3.86852i) q^{75} +(-0.662153 - 2.56155i) q^{77} -8.24621 q^{79} +(-4.12311 - 8.00000i) q^{81} +15.1022 q^{83} +3.12311 q^{85} +(3.02045 + 5.37874i) q^{87} +12.0818 q^{89} +(-0.438447 - 1.69614i) q^{91} +(-7.36932 - 13.1231i) q^{93} -2.68466i q^{95} -6.04090i q^{97} +(2.56155 + 1.56155i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7} - 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^7 - 4 * q^9	$ 8 q - 4 q^{7} - 4 q^{9} - 8 q^{15} + 12 q^{21} - 20 q^{25} + 52 q^{37} + 12 q^{39} - 64 q^{43} + 16 q^{49} - 24 q^{51} + 44 q^{57} - 32 q^{63} + 60 q^{67} - 8 q^{85} - 20 q^{91} + 40 q^{93} + 4 q^{99}+O(q^{100}) $ 8 * q - 4 * q^7 - 4 * q^9 - 8 * q^15 + 12 * q^21 - 20 * q^25 + 52 * q^37 + 12 * q^39 - 64 * q^43 + 16 * q^49 - 24 * q^51 + 44 * q^57 - 32 * q^63 + 60 * q^67 - 8 * q^85 - 20 * q^91 + 40 * q^93 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1848\mathbb{Z}\right)^\times$.

$n$	$463$	$617$	$673$	$925$	$1585$
$\chi(n)$	$1$	$-1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.51022	+	0.848071i	−0.871928	+	0.489634i
$3$	−1.51022	+	0.848071i	−0.871928	+	0.489634i
$4$		0			0
$5$	0.662153			0.296124			0.148062	−	0.988978i	$-0.452696\pi$
$5$	0.662153			0.296124			0.148062	+	0.988978i	$0.452696\pi$
$6$		0			0
$7$	−2.56155	+	0.662153i	−0.968176	+	0.250270i
$7$	−2.56155	+	0.662153i	−0.968176	+	0.250270i
$8$		0			0
$9$	1.56155	−	2.56155i	0.520518	−	0.853851i
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$			0.662153i			0.183648i	0.995775	+	0.0918242i	$0.0292698\pi$
$13$			0.662153i			0.183648i	−0.995775	+	0.0918242i	$0.970730\pi$
$14$		0			0
$15$	−1.00000	+	0.561553i	−0.258199	+	0.144992i
$16$		0			0
$17$	4.71659			1.14394			0.571970	−	0.820274i	$-0.306179\pi$
$17$	4.71659			1.14394			0.571970	+	0.820274i	$0.306179\pi$
$18$		0			0
$19$		−	4.05444i		−	0.930151i	−0.885271	−	0.465076i	$-0.846027\pi$
$19$		−	4.05444i		−	0.930151i	0.885271	−	0.465076i	$-0.153973\pi$
$20$		0			0
$21$	3.30697	−	3.17238i	0.721639	−	0.692270i
$22$		0			0
$23$		−	5.12311i		−	1.06824i	−0.845408	−	0.534121i	$-0.820643\pi$
$23$		−	5.12311i		−	1.06824i	0.845408	−	0.534121i	$-0.179357\pi$
$24$		0			0
$25$	−4.56155			−0.912311
$26$		0			0
$27$	−0.185917	+	5.19283i	−0.0357798	+	0.999360i
$28$		0			0
$29$		−	3.56155i		−	0.661364i	−0.943742	−	0.330682i	$-0.892721\pi$
$29$		−	3.56155i		−	0.661364i	0.943742	−	0.330682i	$-0.107279\pi$
$30$		0			0
$31$			8.68951i			1.56068i	0.625354	+	0.780341i	$0.284954\pi$
$31$			8.68951i			1.56068i	−0.625354	+	0.780341i	$0.715046\pi$
$32$		0			0
$33$	−0.848071	−	1.51022i	−0.147630	−	0.262896i
$34$		0			0
$35$	−1.69614	+	0.438447i	−0.286700	+	0.0741111i
$36$		0			0
$37$	4.43845			0.729676			0.364838	−	0.931071i	$-0.381124\pi$
$37$	4.43845			0.729676			0.364838	+	0.931071i	$0.381124\pi$
$38$		0			0
$39$	−0.561553	−	1.00000i	−0.0899204	−	0.160128i
$40$		0			0
$41$	9.43318			1.47321			0.736607	−	0.676321i	$-0.236427\pi$
$41$	9.43318			1.47321			0.736607	+	0.676321i	$0.236427\pi$
$42$		0			0
$43$	−8.00000			−1.21999			−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000			−1.21999			−0.609994	+	0.792406i	$0.708828\pi$
$44$		0			0
$45$	1.03399	−	1.69614i	0.154138	−	0.252846i
$46$		0			0
$47$	5.75058			0.838808			0.419404	−	0.907800i	$-0.362239\pi$
$47$	5.75058			0.838808			0.419404	+	0.907800i	$0.362239\pi$
$48$		0			0
$49$	6.12311	−	3.39228i	0.874729	−	0.484612i
$50$		0			0
$51$	−7.12311	+	4.00000i	−0.997434	+	0.560112i
$52$		0			0
$53$			8.24621i			1.13270i	0.824163	+	0.566352i	$0.191646\pi$
$53$			8.24621i			1.13270i	−0.824163	+	0.566352i	$0.808354\pi$
$54$		0			0
$55$			0.662153i			0.0892848i
$56$		0			0
$57$	3.43845	+	6.12311i	0.455433	+	0.811025i
$58$		0			0
$59$	1.40582			0.183022			0.0915112	−	0.995804i	$-0.470830\pi$
$59$	1.40582			0.183022			0.0915112	+	0.995804i	$0.470830\pi$
$60$		0			0
$61$			1.69614i			0.217169i	0.994087	+	0.108584i	$0.0346317\pi$
$61$			1.69614i			0.217169i	−0.994087	+	0.108584i	$0.965368\pi$
$62$		0			0
$63$	−2.30386	+	7.59554i	−0.290259	+	0.956948i
$64$		0			0
$65$			0.438447i			0.0543827i
$66$		0			0
$67$	9.56155			1.16813			0.584065	−	0.811707i	$-0.301461\pi$
$67$	9.56155			1.16813			0.584065	+	0.811707i	$0.301461\pi$
$68$		0			0
$69$	4.34475	+	7.73704i	0.523047	+	0.931430i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$			5.00691i			0.586014i	0.956110	+	0.293007i	$0.0946560\pi$
$73$			5.00691i			0.586014i	−0.956110	+	0.293007i	$0.905344\pi$
$74$		0			0
$75$	6.88897	−	3.86852i	0.795469	−	0.446698i
$76$		0			0
$77$	−0.662153	−	2.56155i	−0.0754594	−	0.291916i
$78$		0			0
$79$	−8.24621			−0.927771			−0.463886	−	0.885895i	$-0.653545\pi$
$79$	−8.24621			−0.927771			−0.463886	+	0.885895i	$0.653545\pi$
$80$		0			0
$81$	−4.12311	−	8.00000i	−0.458123	−	0.888889i
$82$		0			0
$83$	15.1022			1.65769			0.828843	−	0.559481i	$-0.189000\pi$
$83$	15.1022			1.65769			0.828843	+	0.559481i	$0.189000\pi$
$84$		0			0
$85$	3.12311			0.338748
$86$		0			0
$87$	3.02045	+	5.37874i	0.323826	+	0.576662i
$88$		0			0
$89$	12.0818			1.28067			0.640334	−	0.768097i	$-0.278796\pi$
$89$	12.0818			1.28067			0.640334	+	0.768097i	$0.278796\pi$
$90$		0			0
$91$	−0.438447	−	1.69614i	−0.0459618	−	0.177804i
$92$		0			0
$93$	−7.36932	−	13.1231i	−0.764163	−	1.36080i
$94$		0			0
$95$		−	2.68466i		−	0.275440i
$96$		0			0
$97$		−	6.04090i		−	0.613360i	−0.951813	−	0.306680i	$-0.900782\pi$
$97$		−	6.04090i		−	0.613360i	0.951813	−	0.306680i	$-0.0992181\pi$
$98$		0			0
$99$	2.56155	+	1.56155i	0.257446	+	0.156942i
$100$		0			0
$101$	−11.1293			−1.10741			−0.553704	−	0.832713i	$-0.686786\pi$
$101$	−11.1293			−1.10741			−0.553704	+	0.832713i	$0.686786\pi$
$102$		0			0
$103$			20.1907i			1.98945i	0.102601	+	0.994723i	$0.467283\pi$
$103$			20.1907i			1.98945i	−0.102601	+	0.994723i	$0.532717\pi$
$104$		0			0
$105$	2.18972	−	2.10060i	0.213695	−	0.204998i
$106$		0			0
$107$			7.31534i			0.707201i	0.935397	+	0.353600i	$0.115043\pi$
$107$			7.31534i			0.707201i	−0.935397	+	0.353600i	$0.884957\pi$
$108$		0			0
$109$	6.87689			0.658687			0.329344	−	0.944210i	$-0.393173\pi$
$109$	6.87689			0.658687			0.329344	+	0.944210i	$0.393173\pi$
$110$		0			0
$111$	−6.70305	+	3.76412i	−0.636225	+	0.357274i
$112$		0			0
$113$			14.4924i			1.36333i	0.731663	+	0.681666i	$0.238744\pi$
$113$			14.4924i			1.36333i	−0.731663	+	0.681666i	$0.761256\pi$
$114$		0			0
$115$		−	3.39228i		−	0.316332i
$116$		0			0
$117$	1.69614	+	1.03399i	0.156808	+	0.0955922i
$118$		0			0
$119$	−12.0818	+	3.12311i	−1.10754	+	0.286295i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	−14.2462	+	8.00000i	−1.28454	+	0.721336i
$124$		0			0
$125$	−6.33122			−0.566281
$126$		0			0
$127$	19.1231			1.69690			0.848451	−	0.529275i	$-0.177536\pi$
$127$	19.1231			1.69690			0.848451	+	0.529275i	$0.177536\pi$
$128$		0			0
$129$	12.0818	−	6.78456i	1.06374	−	0.597348i
$130$		0			0
$131$	−2.43981			−0.213167			−0.106584	−	0.994304i	$-0.533991\pi$
$131$	−2.43981			−0.213167			−0.106584	+	0.994304i	$0.533991\pi$
$132$		0			0
$133$	2.68466	+	10.3857i	0.232789	+	0.900550i
$134$		0			0
$135$	−0.123106	+	3.43845i	−0.0105952	+	0.295934i
$136$		0			0
$137$			5.36932i			0.458732i	0.973340	+	0.229366i	$0.0736652\pi$
$137$			5.36932i			0.458732i	−0.973340	+	0.229366i	$0.926335\pi$
$138$		0			0
$139$			18.4945i			1.56868i	0.620328	+	0.784342i	$0.286999\pi$
$139$			18.4945i			1.56868i	−0.620328	+	0.784342i	$0.713001\pi$
$140$		0			0
$141$	−8.68466	+	4.87689i	−0.731380	+	0.410709i
$142$		0			0
$143$	−0.662153			−0.0553721
$144$		0			0
$145$		−	2.35829i		−	0.195846i
$146$		0			0
$147$	−6.37037	+	10.3159i	−0.525419	+	0.850844i
$148$		0			0
$149$			3.56155i			0.291774i	0.989301	+	0.145887i	$0.0466036\pi$
$149$			3.56155i			0.291774i	−0.989301	+	0.145887i	$0.953396\pi$
$150$		0			0
$151$	19.1231			1.55622			0.778108	−	0.628130i	$-0.216180\pi$
$151$	19.1231			1.55622			0.778108	+	0.628130i	$0.216180\pi$
$152$		0			0
$153$	7.36520	−	12.0818i	0.595441	−	0.976755i
$154$		0			0
$155$			5.75379i			0.462155i
$156$		0			0
$157$		−	3.02045i		−	0.241058i	−0.992710	−	0.120529i	$-0.961541\pi$
$157$		−	3.02045i		−	0.241058i	0.992710	−	0.120529i	$-0.0384591\pi$
$158$		0			0
$159$	−6.99337	−	12.4536i	−0.554610	−	0.987637i
$160$		0			0
$161$	3.39228	+	13.1231i	0.267349	+	1.03425i
$162$		0			0
$163$	−16.6847			−1.30684			−0.653422	−	0.756994i	$-0.726667\pi$
$163$	−16.6847			−1.30684			−0.653422	+	0.756994i	$0.726667\pi$
$164$		0			0
$165$	−0.561553	−	1.00000i	−0.0437168	−	0.0778499i
$166$		0			0
$167$	18.1227			1.40238			0.701188	−	0.712977i	$-0.252653\pi$
$167$	18.1227			1.40238			0.701188	+	0.712977i	$0.252653\pi$
$168$		0			0
$169$	12.5616			0.966273
$170$		0			0
$171$	−10.3857	−	6.33122i	−0.794211	−	0.484160i
$172$		0			0
$173$	5.66906			0.431011			0.215505	−	0.976503i	$-0.430860\pi$
$173$	5.66906			0.431011			0.215505	+	0.976503i	$0.430860\pi$
$174$		0			0
$175$	11.6847	−	3.02045i	0.883277	−	0.228324i
$176$		0			0
$177$	−2.12311	+	1.19224i	−0.159582	+	0.0896139i
$178$		0			0
$179$		−	23.6155i		−	1.76511i	−0.470212	−	0.882554i	$-0.655822\pi$
$179$		−	23.6155i		−	1.76511i	0.470212	−	0.882554i	$-0.344178\pi$
$180$		0			0
$181$			3.76412i			0.279785i	0.990167	+	0.139892i	$0.0446756\pi$
$181$			3.76412i			0.279785i	−0.990167	+	0.139892i	$0.955324\pi$
$182$		0			0
$183$	−1.43845	−	2.56155i	−0.106333	−	0.189355i
$184$		0			0
$185$	2.93893			0.216075
$186$		0			0
$187$			4.71659i			0.344911i
$188$		0			0
$189$	−2.96221	−	13.4248i	−0.215469	−	0.976511i
$190$		0			0
$191$		−	27.1231i		−	1.96256i	−0.192589	−	0.981280i	$-0.561688\pi$
$191$		−	27.1231i		−	1.96256i	0.192589	−	0.981280i	$-0.438312\pi$
$192$		0			0
$193$	11.1231			0.800659			0.400329	−	0.916371i	$-0.368896\pi$
$193$	11.1231			0.800659			0.400329	+	0.916371i	$0.368896\pi$
$194$		0			0
$195$	−0.371834	−	0.662153i	−0.0266276	−	0.0474178i
$196$		0			0
$197$			16.2462i			1.15749i	0.815507	+	0.578747i	$0.196458\pi$
$197$			16.2462i			1.15749i	−0.815507	+	0.578747i	$0.803542\pi$
$198$		0			0
$199$			0.743668i			0.0527172i	0.999653	+	0.0263586i	$0.00839118\pi$
$199$			0.743668i			0.0527172i	−0.999653	+	0.0263586i	$0.991609\pi$
$200$		0			0
$201$	−14.4401	+	8.10887i	−1.01853	+	0.571956i
$202$		0			0
$203$	2.35829	+	9.12311i	0.165520	+	0.640316i
$204$		0			0
$205$	6.24621			0.436254
$206$		0			0
$207$	−13.1231	−	8.00000i	−0.912119	−	0.556038i
$208$		0			0
$209$	4.05444			0.280451
$210$		0			0
$211$	20.4924			1.41076			0.705378	−	0.708831i	$-0.250777\pi$
$211$	20.4924			1.41076			0.705378	+	0.708831i	$0.250777\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−5.29723			−0.361268
$216$		0			0
$217$	−5.75379	−	22.2586i	−0.390593	−	1.51101i
$218$		0			0
$219$	−4.24621	−	7.56155i	−0.286932	−	0.510962i
$220$		0			0
$221$			3.12311i			0.210083i
$222$		0			0
$223$			14.7304i			0.986421i	0.869910	+	0.493210i	$0.164177\pi$
$223$			14.7304i			0.986421i	−0.869910	+	0.493210i	$0.835823\pi$
$224$		0			0
$225$	−7.12311	+	11.6847i	−0.474874	+	0.778977i
$226$		0			0
$227$	−20.5625			−1.36478			−0.682390	−	0.730988i	$-0.739060\pi$
$227$	−20.5625			−1.36478			−0.682390	+	0.730988i	$0.739060\pi$
$228$		0			0
$229$			6.24970i			0.412992i	0.978447	+	0.206496i	$0.0662060\pi$
$229$			6.24970i			0.412992i	−0.978447	+	0.206496i	$0.933794\pi$
$230$		0			0
$231$	3.17238	+	3.30697i	0.208727	+	0.217582i
$232$		0			0
$233$		−	21.3693i		−	1.39995i	−0.714167	−	0.699975i	$-0.753194\pi$
$233$		−	21.3693i		−	1.39995i	0.714167	−	0.699975i	$-0.246806\pi$
$234$		0			0
$235$	3.80776			0.248391
$236$		0			0
$237$	12.4536	−	6.99337i	0.808950	−	0.454268i
$238$		0			0
$239$			1.80776i			0.116935i	0.998289	+	0.0584673i	$0.0186213\pi$
$239$			1.80776i			0.116935i	−0.998289	+	0.0584673i	$0.981379\pi$
$240$		0			0
$241$		−	21.9683i		−	1.41510i	−0.706661	−	0.707552i	$-0.749800\pi$
$241$		−	21.9683i		−	1.41510i	0.706661	−	0.707552i	$-0.250200\pi$
$242$		0			0
$243$	13.0114	+	8.58511i	0.834680	+	0.550735i
$244$		0			0
$245$	4.05444	−	2.24621i	0.259028	−	0.143505i
$246$		0			0
$247$	2.68466			0.170821
$248$		0			0
$249$	−22.8078	+	12.8078i	−1.44538	+	0.811659i
$250$		0			0
$251$	−4.05444			−0.255914			−0.127957	−	0.991780i	$-0.540842\pi$
$251$	−4.05444			−0.255914			−0.127957	+	0.991780i	$0.540842\pi$
$252$		0			0
$253$	5.12311			0.322087
$254$		0			0
$255$	−4.71659	+	2.64861i	−0.295364	+	0.165863i
$256$		0			0
$257$	−0.453349			−0.0282791			−0.0141396	−	0.999900i	$-0.504501\pi$
$257$	−0.453349			−0.0282791			−0.0141396	+	0.999900i	$0.504501\pi$
$258$		0			0
$259$	−11.3693	+	2.93893i	−0.706455	+	0.182616i
$260$		0			0
$261$	−9.12311	−	5.56155i	−0.564706	−	0.344251i
$262$		0			0
$263$			5.56155i			0.342940i	0.985189	+	0.171470i	$0.0548517\pi$
$263$			5.56155i			0.342940i	−0.985189	+	0.171470i	$0.945148\pi$
$264$		0			0
$265$			5.46026i			0.335421i
$266$		0			0
$267$	−18.2462	+	10.2462i	−1.11665	+	0.627058i
$268$		0			0
$269$	−5.08842			−0.310247			−0.155123	−	0.987895i	$-0.549577\pi$
$269$	−5.08842			−0.310247			−0.155123	+	0.987895i	$0.549577\pi$
$270$		0			0
$271$		−	5.00691i		−	0.304148i	−0.988369	−	0.152074i	$-0.951405\pi$
$271$		−	5.00691i		−	0.304148i	0.988369	−	0.152074i	$-0.0485952\pi$
$272$		0			0
$273$	2.10060	+	2.18972i	0.127134	+	0.132528i
$274$		0			0
$275$		−	4.56155i		−	0.275072i
$276$		0			0
$277$	−14.0000			−0.841178			−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000			−0.841178			−0.420589	+	0.907251i	$0.638177\pi$
$278$		0			0
$279$	22.2586	+	13.5691i	1.33259	+	0.812362i
$280$		0			0
$281$		−	19.3153i		−	1.15226i	−0.817359	−	0.576128i	$-0.804563\pi$
$281$		−	19.3153i		−	1.15226i	0.817359	−	0.576128i	$-0.195437\pi$
$282$		0			0
$283$		−	8.77102i		−	0.521383i	−0.965422	−	0.260692i	$-0.916049\pi$
$283$		−	8.77102i		−	0.521383i	0.965422	−	0.260692i	$-0.0839506\pi$
$284$		0			0
$285$	2.27678	+	4.05444i	0.134865	+	0.240164i
$286$		0			0
$287$	−24.1636	+	6.24621i	−1.42633	+	0.368702i
$288$		0			0
$289$	5.24621			0.308601
$290$		0			0
$291$	5.12311	+	9.12311i	0.300322	+	0.534806i
$292$		0			0
$293$	11.1293			0.650182			0.325091	−	0.945683i	$-0.394605\pi$
$293$	11.1293			0.650182			0.325091	+	0.945683i	$0.394605\pi$
$294$		0			0
$295$	0.930870			0.0541973
$296$		0			0
$297$	−5.19283	−	0.185917i	−0.301318	−	0.0107880i
$298$		0			0
$299$	3.39228			0.196181
$300$		0			0
$301$	20.4924	−	5.29723i	1.18116	−	0.305327i
$302$		0			0
$303$	16.8078	−	9.43845i	0.965581	−	0.542225i
$304$		0			0
$305$			1.12311i			0.0643088i
$306$		0			0
$307$			13.1973i			0.753209i	0.926374	+	0.376605i	$0.122908\pi$
$307$			13.1973i			0.753209i	−0.926374	+	0.376605i	$0.877092\pi$
$308$		0			0
$309$	−17.1231	−	30.4924i	−0.974099	−	1.73465i
$310$		0			0
$311$	−18.7033			−1.06057			−0.530284	−	0.847820i	$-0.677915\pi$
$311$	−18.7033			−1.06057			−0.530284	+	0.847820i	$0.677915\pi$
$312$		0			0
$313$		−	10.7575i		−	0.608049i	−0.952664	−	0.304024i	$-0.901670\pi$
$313$		−	10.7575i		−	0.608049i	0.952664	−	0.304024i	$-0.0983305\pi$
$314$		0			0
$315$	−1.52551	+	5.02941i	−0.0859526	+	0.283375i
$316$		0			0
$317$			10.0000i			0.561656i	0.959758	+	0.280828i	$0.0906090\pi$
$317$			10.0000i			0.561656i	−0.959758	+	0.280828i	$0.909391\pi$
$318$		0			0
$319$	3.56155			0.199409
$320$		0			0
$321$	−6.20393	−	11.0478i	−0.346269	−	0.616628i
$322$		0			0
$323$		−	19.1231i		−	1.06404i
$324$		0			0
$325$		−	3.02045i		−	0.167544i
$326$		0			0
$327$	−10.3857	+	5.83209i	−0.574328	+	0.322515i
$328$		0			0
$329$	−14.7304	+	3.80776i	−0.812114	+	0.209929i
$330$		0			0
$331$	−10.2462			−0.563183			−0.281591	−	0.959534i	$-0.590862\pi$
$331$	−10.2462			−0.563183			−0.281591	+	0.959534i	$0.590862\pi$
$332$		0			0
$333$	6.93087	−	11.3693i	0.379809	−	0.623035i
$334$		0			0
$335$	6.33122			0.345911
$336$		0			0
$337$	−10.2462			−0.558147			−0.279073	−	0.960270i	$-0.590027\pi$
$337$	−10.2462			−0.558147			−0.279073	+	0.960270i	$0.590027\pi$
$338$		0			0
$339$	−12.2906	−	21.8868i	−0.667534	−	1.18873i
$340$		0			0
$341$	−8.68951			−0.470563
$342$		0			0
$343$	−13.4384	+	12.7439i	−0.725608	+	0.688108i
$344$		0			0
$345$	2.87689	+	5.12311i	0.154887	+	0.275819i
$346$		0			0
$347$		−	20.0000i		−	1.07366i	−0.843692	−	0.536828i	$-0.819622\pi$
$347$		−	20.0000i		−	1.07366i	0.843692	−	0.536828i	$-0.180378\pi$
$348$		0			0
$349$		−	5.37874i		−	0.287917i	−0.989584	−	0.143959i	$-0.954017\pi$
$349$		−	5.37874i		−	0.287917i	0.989584	−	0.143959i	$-0.0459833\pi$
$350$		0			0
$351$	−3.43845	−	0.123106i	−0.183531	−	0.00657089i
$352$		0			0
$353$	−18.4130			−0.980025			−0.490013	−	0.871715i	$-0.663008\pi$
$353$	−18.4130			−0.980025			−0.490013	+	0.871715i	$0.663008\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	15.5976	−	14.9628i	0.825512	−	0.791915i
$358$		0			0
$359$		−	17.6155i		−	0.929712i	−0.885386	−	0.464856i	$-0.846106\pi$
$359$		−	17.6155i		−	0.929712i	0.885386	−	0.464856i	$-0.153894\pi$
$360$		0			0
$361$	2.56155			0.134819
$362$		0			0
$363$	1.51022	−	0.848071i	0.0792662	−	0.0445122i
$364$		0			0
$365$			3.31534i			0.173533i
$366$		0			0
$367$		−	20.9343i		−	1.09276i	−0.837536	−	0.546382i	$-0.816005\pi$
$367$		−	20.9343i		−	1.09276i	0.837536	−	0.546382i	$-0.183995\pi$
$368$		0			0
$369$	14.7304	−	24.1636i	0.766834	−	1.25791i
$370$		0			0
$371$	−5.46026	−	21.1231i	−0.283482	−	1.09666i
$372$		0			0
$373$	−20.2462			−1.04831			−0.524155	−	0.851623i	$-0.675619\pi$
$373$	−20.2462			−1.04831			−0.524155	+	0.851623i	$0.675619\pi$
$374$		0			0
$375$	9.56155	−	5.36932i	0.493756	−	0.277270i
$376$		0			0
$377$	2.35829			0.121458
$378$		0			0
$379$	14.0540			0.721904			0.360952	−	0.932584i	$-0.382452\pi$
$379$	14.0540			0.721904			0.360952	+	0.932584i	$0.382452\pi$
$380$		0			0
$381$	−28.8802	+	16.2177i	−1.47958	+	0.830860i
$382$		0			0
$383$	10.0138			0.511682			0.255841	−	0.966719i	$-0.417648\pi$
$383$	10.0138			0.511682			0.255841	+	0.966719i	$0.417648\pi$
$384$		0			0
$385$	−0.438447	−	1.69614i	−0.0223453	−	0.0864434i
$386$		0			0
$387$	−12.4924	+	20.4924i	−0.635026	+	1.04169i
$388$		0			0
$389$			6.87689i			0.348672i	0.984686	+	0.174336i	$0.0557779\pi$
$389$			6.87689i			0.348672i	−0.984686	+	0.174336i	$0.944222\pi$
$390$		0			0
$391$		−	24.1636i		−	1.22200i
$392$		0			0
$393$	3.68466	−	2.06913i	0.185866	−	0.104374i
$394$		0			0
$395$	−5.46026			−0.274735
$396$		0			0
$397$			35.8735i			1.80044i	0.435434	+	0.900221i	$0.356595\pi$
$397$			35.8735i			1.80044i	−0.435434	+	0.900221i	$0.643405\pi$
$398$		0			0
$399$	−12.8622	−	13.4079i	−0.643915	−	0.671234i
$400$		0			0
$401$		−	2.87689i		−	0.143665i	−0.997417	−	0.0718326i	$-0.977115\pi$
$401$		−	2.87689i		−	0.143665i	0.997417	−	0.0718326i	$-0.0228847\pi$
$402$		0			0
$403$	−5.75379			−0.286617
$404$		0			0
$405$	−2.73013	−	5.29723i	−0.135661	−	0.263221i
$406$		0			0
$407$			4.43845i			0.220006i
$408$		0			0
$409$		−	24.1636i		−	1.19481i	−0.801939	−	0.597406i	$-0.796198\pi$
$409$		−	24.1636i		−	1.19481i	0.801939	−	0.597406i	$-0.203802\pi$
$410$		0			0
$411$	−4.55356	−	8.10887i	−0.224611	−	0.399981i
$412$		0			0
$413$	−3.60109	+	0.930870i	−0.177198	+	0.0458051i
$414$		0			0
$415$	10.0000			0.490881
$416$		0			0
$417$	−15.6847	−	27.9309i	−0.768081	−	1.36778i
$418$		0			0
$419$	−35.5832			−1.73835			−0.869177	−	0.494501i	$-0.835351\pi$
$419$	−35.5832			−1.73835			−0.869177	+	0.494501i	$0.835351\pi$
$420$		0			0
$421$	−10.6847			−0.520738			−0.260369	−	0.965509i	$-0.583844\pi$
$421$	−10.6847			−0.520738			−0.260369	+	0.965509i	$0.583844\pi$
$422$		0			0
$423$	8.97983	−	14.7304i	0.436614	−	0.716217i
$424$		0			0
$425$	−21.5150			−1.04363
$426$		0			0
$427$	−1.12311	−	4.34475i	−0.0543509	−	0.210257i
$428$		0			0
$429$	1.00000	−	0.561553i	0.0482805	−	0.0271120i
$430$		0			0
$431$			6.19224i			0.298270i	0.988817	+	0.149135i	$0.0476488\pi$
$431$			6.19224i			0.298270i	−0.988817	+	0.149135i	$0.952351\pi$
$432$		0			0
$433$			5.46026i			0.262403i	0.991356	+	0.131202i	$0.0418835\pi$
$433$			5.46026i			0.262403i	−0.991356	+	0.131202i	$0.958116\pi$
$434$		0			0
$435$	2.00000	+	3.56155i	0.0958927	+	0.170763i
$436$		0			0
$437$	−20.7713			−0.993626
$438$		0			0
$439$		−	33.8871i		−	1.61734i	−0.588261	−	0.808671i	$-0.700187\pi$
$439$		−	33.8871i		−	1.61734i	0.588261	−	0.808671i	$-0.299813\pi$
$440$		0			0
$441$	0.872043	−	20.9819i	0.0415259	−	0.999137i
$442$		0			0
$443$			7.12311i			0.338429i	0.985579	+	0.169214i	$0.0541231\pi$
$443$			7.12311i			0.338429i	−0.985579	+	0.169214i	$0.945877\pi$
$444$		0			0
$445$	8.00000			0.379236
$446$		0			0
$447$	−3.02045	−	5.37874i	−0.142862	−	0.254406i
$448$		0			0
$449$			15.1231i			0.713703i	0.934161	+	0.356852i	$0.116150\pi$
$449$			15.1231i			0.713703i	−0.934161	+	0.356852i	$0.883850\pi$
$450$		0			0
$451$			9.43318i			0.444191i
$452$		0			0
$453$	−28.8802	+	16.2177i	−1.35691	+	0.761976i
$454$		0			0
$455$	−0.290319	−	1.12311i	−0.0136104	−	0.0526520i
$456$		0			0
$457$	38.2462			1.78908			0.894541	−	0.446986i	$-0.147502\pi$
$457$	38.2462			1.78908			0.894541	+	0.446986i	$0.147502\pi$
$458$		0			0
$459$	−0.876894	+	24.4924i	−0.0409299	+	1.14321i
$460$		0			0
$461$	18.4945			0.861376			0.430688	−	0.902501i	$-0.358271\pi$
$461$	18.4945			0.861376			0.430688	+	0.902501i	$0.358271\pi$
$462$		0			0
$463$	15.1771			0.705339			0.352669	−	0.935748i	$-0.385274\pi$
$463$	15.1771			0.705339			0.352669	+	0.935748i	$0.385274\pi$
$464$		0			0
$465$	−4.87962	−	8.68951i	−0.226287	−	0.402966i
$466$		0			0
$467$	34.8396			1.61218			0.806091	−	0.591791i	$-0.201579\pi$
$467$	34.8396			1.61218			0.806091	+	0.591791i	$0.201579\pi$
$468$		0			0
$469$	−24.4924	+	6.33122i	−1.13095	+	0.292348i
$470$		0			0
$471$	2.56155	+	4.56155i	0.118030	+	0.210185i
$472$		0			0
$473$		−	8.00000i		−	0.367840i
$474$		0			0
$475$			18.4945i			0.848587i
$476$		0			0
$477$	21.1231	+	12.8769i	0.967161	+	0.589592i
$478$		0			0
$479$	21.3519			0.975595			0.487798	−	0.872957i	$-0.337800\pi$
$479$	21.3519			0.975595			0.487798	+	0.872957i	$0.337800\pi$
$480$		0			0
$481$			2.93893i			0.134004i
$482$		0			0
$483$	−16.2524	−	16.9419i	−0.739511	−	0.770885i
$484$		0			0
$485$		−	4.00000i		−	0.181631i
$486$		0			0
$487$	24.0000			1.08754			0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000			1.08754			0.543772	+	0.839233i	$0.316996\pi$
$488$		0			0
$489$	25.1976	−	14.1498i	1.13947	−	0.639875i
$490$		0			0
$491$		−	19.4233i		−	0.876561i	−0.898838	−	0.438280i	$-0.855588\pi$
$491$		−	19.4233i		−	0.876561i	0.898838	−	0.438280i	$-0.144412\pi$
$492$		0			0
$493$		−	16.7984i		−	0.756561i
$494$		0			0
$495$	1.69614	+	1.03399i	0.0762359	+	0.0464743i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	16.3002			0.729697			0.364848	−	0.931067i	$-0.381121\pi$
$499$	16.3002			0.729697			0.364848	+	0.931067i	$0.381121\pi$
$500$		0			0
$501$	−27.3693	+	15.3693i	−1.22277	+	0.686650i
$502$		0			0
$503$	3.22925			0.143985			0.0719926	−	0.997405i	$-0.477064\pi$
$503$	3.22925			0.143985			0.0719926	+	0.997405i	$0.477064\pi$
$504$		0			0
$505$	−7.36932			−0.327930
$506$		0			0
$507$	−18.9708	+	10.6531i	−0.842521	+	0.473120i
$508$		0			0
$509$	−34.5492			−1.53137			−0.765684	−	0.643217i	$-0.777599\pi$
$509$	−34.5492			−1.53137			−0.765684	+	0.643217i	$0.777599\pi$
$510$		0			0
$511$	−3.31534	−	12.8255i	−0.146662	−	0.567365i
$512$		0			0
$513$	21.0540	+	0.753789i	0.929556	+	0.0332806i
$514$		0			0
$515$			13.3693i			0.589122i
$516$		0			0
$517$			5.75058i			0.252910i
$518$		0			0
$519$	−8.56155	+	4.80776i	−0.375810	+	0.211037i
$520$		0			0
$521$	1.19702			0.0524423			0.0262211	−	0.999656i	$-0.491653\pi$
$521$	1.19702			0.0524423			0.0262211	+	0.999656i	$0.491653\pi$
$522$		0			0
$523$		−	24.8257i		−	1.08555i	−0.839877	−	0.542777i	$-0.817373\pi$
$523$		−	24.8257i		−	1.08555i	0.839877	−	0.542777i	$-0.182627\pi$
$524$		0			0
$525$	−15.0849	+	14.4710i	−0.658359	+	0.631565i
$526$		0			0
$527$			40.9848i			1.78533i
$528$		0			0
$529$	−3.24621			−0.141140
$530$		0			0
$531$	2.19526	−	3.60109i	0.0952664	−	0.156274i
$532$		0			0
$533$			6.24621i			0.270553i
$534$		0			0
$535$			4.84388i			0.209419i
$536$		0			0
$537$	20.0276	+	35.6647i	0.864256	+	1.53905i
$538$		0			0
$539$	3.39228	+	6.12311i	0.146116	+	0.263741i
$540$		0			0
$541$	−8.63068			−0.371062			−0.185531	−	0.982638i	$-0.559401\pi$
$541$	−8.63068			−0.371062			−0.185531	+	0.982638i	$0.559401\pi$
$542$		0			0
$543$	−3.19224	−	5.68466i	−0.136992	−	0.243952i
$544$		0			0
$545$	4.55356			0.195053
$546$		0			0
$547$	−13.8617			−0.592685			−0.296343	−	0.955082i	$-0.595767\pi$
$547$	−13.8617			−0.592685			−0.296343	+	0.955082i	$0.595767\pi$
$548$		0			0
$549$	4.34475	+	2.64861i	0.185430	+	0.113040i
$550$		0			0
$551$	−14.4401			−0.615168
$552$		0			0
$553$	21.1231	−	5.46026i	0.898246	−	0.232194i
$554$		0			0
$555$	−4.43845	+	2.49242i	−0.188402	+	0.105797i
$556$		0			0
$557$			3.06913i			0.130043i	0.997884	+	0.0650216i	$0.0207116\pi$
$557$			3.06913i			0.130043i	−0.997884	+	0.0650216i	$0.979288\pi$
$558$		0			0
$559$		−	5.29723i		−	0.224049i
$560$		0			0
$561$	−4.00000	−	7.12311i	−0.168880	−	0.300738i
$562$		0			0
$563$	22.6305			0.953761			0.476880	−	0.878968i	$-0.341768\pi$
$563$	22.6305			0.953761			0.476880	+	0.878968i	$0.341768\pi$
$564$		0			0
$565$			9.59621i			0.403715i
$566$		0			0
$567$	15.8588	+	17.7623i	0.666006	+	0.745946i
$568$		0			0
$569$		−	10.0000i		−	0.419222i	−0.977785	−	0.209611i	$-0.932780\pi$
$569$		−	10.0000i		−	0.419222i	0.977785	−	0.209611i	$-0.0672197\pi$
$570$		0			0
$571$	21.8617			0.914885			0.457443	−	0.889239i	$-0.348766\pi$
$571$	21.8617			0.914885			0.457443	+	0.889239i	$0.348766\pi$
$572$		0			0
$573$	23.0023	+	40.9620i	0.960935	+	1.71121i
$574$		0			0
$575$			23.3693i			0.974568i
$576$		0			0
$577$			6.78456i			0.282445i	0.989978	+	0.141223i	$0.0451033\pi$
$577$			6.78456i			0.282445i	−0.989978	+	0.141223i	$0.954897\pi$
$578$		0			0
$579$	−16.7984	+	9.43318i	−0.698117	+	0.392029i
$580$		0			0
$581$	−38.6852	+	10.0000i	−1.60493	+	0.414870i
$582$		0			0
$583$	−8.24621			−0.341523
$584$		0			0
$585$	1.12311	+	0.684658i	0.0464347	+	0.0283071i
$586$		0			0
$587$	−5.54177			−0.228733			−0.114367	−	0.993439i	$-0.536484\pi$
$587$	−5.54177			−0.228733			−0.114367	+	0.993439i	$0.536484\pi$
$588$		0			0
$589$	35.2311			1.45167
$590$		0			0
$591$	−13.7779	−	24.5354i	−0.566748	−	1.00925i
$592$		0			0
$593$	34.9211			1.43404			0.717018	−	0.697054i	$-0.245506\pi$
$593$	34.9211			1.43404			0.717018	+	0.697054i	$0.245506\pi$
$594$		0			0
$595$	−8.00000	+	2.06798i	−0.327968	+	0.0847787i
$596$		0			0
$597$	−0.630683	−	1.12311i	−0.0258121	−	0.0459657i
$598$		0			0
$599$		−	37.8617i		−	1.54699i	−0.633803	−	0.773494i	$-0.718507\pi$
$599$		−	37.8617i		−	1.54699i	0.633803	−	0.773494i	$-0.281493\pi$
$600$		0			0
$601$			15.9274i			0.649693i	0.945767	+	0.324847i	$0.105313\pi$
$601$			15.9274i			0.649693i	−0.945767	+	0.324847i	$0.894687\pi$
$602$		0			0
$603$	14.9309	−	24.4924i	0.608032	−	0.997409i
$604$		0			0
$605$	−0.662153			−0.0269204
$606$		0			0
$607$			1.77766i			0.0721528i	0.999349	+	0.0360764i	$0.0114860\pi$
$607$			1.77766i			0.0721528i	−0.999349	+	0.0360764i	$0.988514\pi$
$608$		0			0
$609$	−11.2986	−	11.7779i	−0.457842	−	0.477266i
$610$		0			0
$611$			3.80776i			0.154046i
$612$		0			0
$613$	−7.36932			−0.297644			−0.148822	−	0.988864i	$-0.547548\pi$
$613$	−7.36932			−0.297644			−0.148822	+	0.988864i	$0.547548\pi$
$614$		0			0
$615$	−9.43318	+	5.29723i	−0.380382	+	0.213605i
$616$		0			0
$617$			14.0000i			0.563619i	0.959470	+	0.281809i	$0.0909346\pi$
$617$			14.0000i			0.563619i	−0.959470	+	0.281809i	$0.909065\pi$
$618$		0			0
$619$			19.0752i			0.766695i	0.923604	+	0.383348i	$0.125229\pi$
$619$			19.0752i			0.766695i	−0.923604	+	0.383348i	$0.874771\pi$
$620$		0			0
$621$	26.6034	+	0.952473i	1.06756	+	0.0382214i
$622$		0			0
$623$	−30.9481	+	8.00000i	−1.23991	+	0.320513i
$624$		0			0
$625$	18.6155			0.744621
$626$		0			0
$627$	−6.12311	+	3.43845i	−0.244533	+	0.137318i
$628$		0			0
$629$	20.9343			0.834706
$630$		0			0
$631$	−38.1080			−1.51705			−0.758527	−	0.651642i	$-0.774081\pi$
$631$	−38.1080			−1.51705			−0.758527	+	0.651642i	$0.774081\pi$
$632$		0			0
$633$	−30.9481	+	17.3790i	−1.23008	+	0.690754i
$634$		0			0
$635$	12.6624			0.502493
$636$		0			0
$637$	2.24621	+	4.05444i	0.0889981	+	0.160643i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			25.1231i			0.992303i	0.868236	+	0.496152i	$0.165254\pi$
$641$			25.1231i			0.992303i	−0.868236	+	0.496152i	$0.834746\pi$
$642$		0			0
$643$			9.22437i			0.363774i	0.983319	+	0.181887i	$0.0582205\pi$
$643$			9.22437i			0.363774i	−0.983319	+	0.181887i	$0.941780\pi$
$644$		0			0
$645$	8.00000	−	4.49242i	0.315000	−	0.176889i
$646$		0			0
$647$	10.4672			0.411507			0.205753	−	0.978604i	$-0.434036\pi$
$647$	10.4672			0.411507			0.205753	+	0.978604i	$0.434036\pi$
$648$		0			0
$649$			1.40582i			0.0551833i
$650$		0			0
$651$	27.5664	+	28.7359i	1.08041	+	1.12625i
$652$		0			0
$653$			17.1231i			0.670079i	0.942204	+	0.335039i	$0.108750\pi$
$653$			17.1231i			0.670079i	−0.942204	+	0.335039i	$0.891250\pi$
$654$		0			0
$655$	−1.61553			−0.0631239
$656$		0			0
$657$	12.8255	+	7.81855i	0.500369	+	0.305031i
$658$		0			0
$659$			9.17708i			0.357488i	0.983896	+	0.178744i	$0.0572034\pi$
$659$			9.17708i			0.357488i	−0.983896	+	0.178744i	$0.942797\pi$
$660$		0			0
$661$			41.1708i			1.60136i	0.599094	+	0.800679i	$0.295528\pi$
$661$			41.1708i			1.60136i	−0.599094	+	0.800679i	$0.704472\pi$
$662$		0			0
$663$	−2.64861	−	4.71659i	−0.102864	−	0.183177i
$664$		0			0
$665$	1.77766	+	6.87689i	0.0689345	+	0.266675i
$666$		0			0
$667$	−18.2462			−0.706496
$668$		0			0
$669$	−12.4924	−	22.2462i	−0.482985	−	0.860088i
$670$		0			0
$671$	−1.69614			−0.0654788
$672$		0			0
$673$	−28.2462			−1.08881			−0.544406	−	0.838822i	$-0.683245\pi$
$673$	−28.2462			−1.08881			−0.544406	+	0.838822i	$0.683245\pi$
$674$		0			0
$675$	0.848071	−	23.6873i	0.0326422	−	0.911726i
$676$		0			0
$677$	−12.4536			−0.478632			−0.239316	−	0.970942i	$-0.576923\pi$
$677$	−12.4536			−0.478632			−0.239316	+	0.970942i	$0.576923\pi$
$678$		0			0
$679$	4.00000	+	15.4741i	0.153506	+	0.593840i
$680$		0			0
$681$	31.0540	−	17.4384i	1.18999	−	0.668243i
$682$		0			0
$683$		−	12.4924i		−	0.478009i	−0.971018	−	0.239005i	$-0.923179\pi$
$683$		−	12.4924i		−	0.478009i	0.971018	−	0.239005i	$-0.0768211\pi$
$684$		0			0
$685$			3.55531i			0.135841i
$686$		0			0
$687$	−5.30019	−	9.43845i	−0.202215	−	0.360099i
$688$		0			0
$689$	−5.46026			−0.208019
$690$		0			0
$691$		−	35.8735i		−	1.36469i	−0.731029	−	0.682347i	$-0.760959\pi$
$691$		−	35.8735i		−	1.36469i	0.731029	−	0.682347i	$-0.239041\pi$
$692$		0			0
$693$	−7.59554	−	2.30386i	−0.288531	−	0.0875164i
$694$		0			0
$695$			12.2462i			0.464525i
$696$		0			0
$697$	44.4924			1.68527
$698$		0			0
$699$	18.1227	+	32.2725i	0.685463	+	1.22066i
$700$		0			0
$701$		−	36.2462i		−	1.36900i	−0.729013	−	0.684500i	$-0.760020\pi$
$701$		−	36.2462i		−	1.36900i	0.729013	−	0.684500i	$-0.239980\pi$
$702$		0			0
$703$		−	17.9954i		−	0.678709i
$704$		0			0
$705$	−5.75058	+	3.22925i	−0.216579	+	0.121621i
$706$		0			0
$707$	28.5083	−	7.36932i	1.07217	−	0.277152i
$708$		0			0
$709$	−40.0540			−1.50426			−0.752129	−	0.659016i	$-0.770973\pi$
$709$	−40.0540			−1.50426			−0.752129	+	0.659016i	$0.770973\pi$
$710$		0			0
$711$	−12.8769	+	21.1231i	−0.482921	+	0.792178i
$712$		0			0
$713$	44.5173			1.66719
$714$		0			0
$715$	−0.438447			−0.0163970
$716$		0			0
$717$	−1.53311	−	2.73013i	−0.0572551	−	0.101959i
$718$		0			0
$719$	36.6987			1.36863			0.684316	−	0.729186i	$-0.260101\pi$
$719$	36.6987			1.36863			0.684316	+	0.729186i	$0.260101\pi$
$720$		0			0
$721$	−13.3693	−	51.7194i	−0.497899	−	1.92613i
$722$		0			0
$723$	18.6307	+	33.1771i	0.692883	+	1.23387i
$724$		0			0
$725$			16.2462i			0.603369i
$726$		0			0
$727$			45.2609i			1.67864i	0.543641	+	0.839318i	$0.317045\pi$
$727$			45.2609i			1.67864i	−0.543641	+	0.839318i	$0.682955\pi$
$728$		0			0
$729$	−26.9309	−	1.93087i	−0.997440	−	0.0715137i
$730$		0			0
$731$	−37.7327			−1.39559
$732$		0			0
$733$			10.3857i			0.383603i	0.981434	+	0.191801i	$0.0614329\pi$
$733$			10.3857i			0.383603i	−0.981434	+	0.191801i	$0.938567\pi$
$734$		0			0
$735$	−4.21816	+	6.83073i	−0.155589	+	0.251955i
$736$		0			0
$737$			9.56155i			0.352204i
$738$		0			0
$739$	7.50758			0.276171			0.138085	−	0.990420i	$-0.455905\pi$
$739$	7.50758			0.276171			0.138085	+	0.990420i	$0.455905\pi$
$740$		0			0
$741$	−4.05444	+	2.27678i	−0.148943	+	0.0836396i
$742$		0			0
$743$		−	3.17708i		−	0.116556i	−0.998300	−	0.0582779i	$-0.981439\pi$
$743$		−	3.17708i		−	0.116556i	0.998300	−	0.0582779i	$-0.0185609\pi$
$744$		0			0
$745$			2.35829i			0.0864012i
$746$		0			0
$747$	23.5829	−	38.6852i	0.862855	−	1.41542i
$748$		0			0
$749$	−4.84388	−	18.7386i	−0.176991	−	0.684695i
$750$		0			0
$751$	−23.3153			−0.850789			−0.425394	−	0.905008i	$-0.639865\pi$
$751$	−23.3153			−0.850789			−0.425394	+	0.905008i	$0.639865\pi$
$752$		0			0
$753$	6.12311	−	3.43845i	0.223138	−	0.125304i
$754$		0			0
$755$	12.6624			0.460833
$756$		0			0
$757$	29.3153			1.06548			0.532742	−	0.846278i	$-0.321162\pi$
$757$	29.3153			1.06548			0.532742	+	0.846278i	$0.321162\pi$
$758$		0			0
$759$	−7.73704	+	4.34475i	−0.280837	+	0.157705i
$760$		0			0
$761$	−42.4493			−1.53879			−0.769393	−	0.638776i	$-0.779441\pi$
$761$	−42.4493			−1.53879			−0.769393	+	0.638776i	$0.779441\pi$
$762$		0			0
$763$	−17.6155	+	4.55356i	−0.637725	+	0.164850i
$764$		0			0
$765$	4.87689	−	8.00000i	0.176324	−	0.289241i
$766$		0			0
$767$			0.930870i			0.0336118i
$768$		0			0
$769$		−	15.1838i		−	0.547540i	−0.961795	−	0.273770i	$-0.911729\pi$
$769$		−	15.1838i		−	0.547540i	0.961795	−	0.273770i	$-0.0882707\pi$
$770$		0			0
$771$	0.684658	−	0.384472i	0.0246574	−	0.0138464i
$772$		0			0
$773$	−50.8943			−1.83054			−0.915270	−	0.402842i	$-0.868023\pi$
$773$	−50.8943			−1.83054			−0.915270	+	0.402842i	$0.868023\pi$
$774$		0			0
$775$		−	39.6377i		−	1.42383i
$776$		0			0
$777$	14.6778	−	14.0804i	0.526563	−	0.505133i
$778$		0			0
$779$		−	38.2462i		−	1.37031i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	18.4945	+	0.662153i	0.660940	+	0.0236634i
$784$		0			0
$785$		−	2.00000i		−	0.0713831i
$786$		0			0
$787$		−	49.8960i		−	1.77860i	−0.457323	−	0.889300i	$-0.651192\pi$
$787$		−	49.8960i		−	1.77860i	0.457323	−	0.889300i	$-0.348808\pi$
$788$		0			0
$789$	−4.71659	−	8.39919i	−0.167915	−	0.299019i
$790$		0			0
$791$	−9.59621	−	37.1231i	−0.341202	−	1.31995i
$792$		0			0
$793$	−1.12311			−0.0398827
$794$		0			0
$795$	−4.63068	−	8.24621i	−0.164233	−	0.292463i
$796$		0			0
$797$	0.825183			0.0292295			0.0146147	−	0.999893i	$-0.495348\pi$
$797$	0.825183			0.0292295			0.0146147	+	0.999893i	$0.495348\pi$
$798$		0			0
$799$	27.1231			0.959546
$800$		0			0
$801$	18.8664	−	30.9481i	0.666610	−	1.09350i
$802$		0			0
$803$	−5.00691			−0.176690
$804$		0			0
$805$	2.24621	+	8.68951i	0.0791685	+	0.306265i
$806$		0			0
$807$	7.68466	−	4.31534i	0.270513	−	0.151907i
$808$		0			0
$809$			41.4233i			1.45637i	0.685383	+	0.728183i	$0.259635\pi$
$809$			41.4233i			1.45637i	−0.685383	+	0.728183i	$0.740365\pi$
$810$		0			0
$811$			12.1633i			0.427111i	0.976931	+	0.213556i	$0.0685045\pi$
$811$			12.1633i			0.427111i	−0.976931	+	0.213556i	$0.931496\pi$
$812$		0			0
$813$	4.24621	+	7.56155i	0.148921	+	0.265195i
$814$		0			0
$815$	−11.0478			−0.386988
$816$		0			0
$817$			32.4355i			1.13477i
$818$		0			0
$819$	−5.02941	−	1.52551i	−0.175742	−	0.0533056i
$820$		0			0
$821$			11.5616i			0.403501i	0.979437	+	0.201750i	$0.0646630\pi$
$821$			11.5616i			0.403501i	−0.979437	+	0.201750i	$0.935337\pi$
$822$		0			0
$823$	9.31534			0.324712			0.162356	−	0.986732i	$-0.448091\pi$
$823$	9.31534			0.324712			0.162356	+	0.986732i	$0.448091\pi$
$824$		0			0
$825$	3.86852	+	6.88897i	0.134685	+	0.239843i
$826$		0			0
$827$		−	40.7926i		−	1.41850i	−0.704958	−	0.709249i	$-0.749034\pi$
$827$		−	40.7926i		−	1.41850i	0.704958	−	0.709249i	$-0.250966\pi$
$828$		0			0
$829$			53.4156i			1.85520i	0.373575	+	0.927600i	$0.378132\pi$
$829$			53.4156i			1.85520i	−0.373575	+	0.927600i	$0.621868\pi$
$830$		0			0
$831$	21.1431	−	11.8730i	0.733447	−	0.411869i
$832$		0			0
$833$	28.8802	−	16.0000i	1.00064	−	0.554367i
$834$		0			0
$835$	12.0000			0.415277
$836$		0			0
$837$	−45.1231	−	1.61553i	−1.55968	−	0.0558408i
$838$		0			0
$839$	42.9026			1.48116			0.740582	−	0.671966i	$-0.234550\pi$
$839$	42.9026			1.48116			0.740582	+	0.671966i	$0.234550\pi$
$840$		0			0
$841$	16.3153			0.562598
$842$		0			0
$843$	16.3808	+	29.1705i	0.564184	+	1.00469i
$844$		0			0
$845$	8.31768			0.286137
$846$		0			0
$847$	2.56155	−	0.662153i	0.0880160	−	0.0227519i
$848$		0			0
$849$	7.43845	+	13.2462i	0.255287	+	0.454609i
$850$		0			0
$851$		−	22.7386i		−	0.779470i
$852$		0			0
$853$		−	18.6576i		−	0.638822i	−0.947616	−	0.319411i	$-0.896515\pi$
$853$		−	18.6576i		−	0.638822i	0.947616	−	0.319411i	$-0.103485\pi$
$854$		0			0
$855$	−6.87689	−	4.19224i	−0.235185	−	0.143371i
$856$		0			0
$857$	9.43318			0.322231			0.161116	−	0.986936i	$-0.448491\pi$
$857$	9.43318			0.322231			0.161116	+	0.986936i	$0.448491\pi$
$858$		0			0
$859$			42.6581i			1.45548i	0.685855	+	0.727738i	$0.259428\pi$
$859$			42.6581i			1.45548i	−0.685855	+	0.727738i	$0.740572\pi$
$860$		0			0
$861$	31.1952	−	29.9256i	1.06313	−	1.01986i
$862$		0			0
$863$			50.3542i			1.71408i	0.515254	+	0.857038i	$0.327698\pi$
$863$			50.3542i			1.71408i	−0.515254	+	0.857038i	$0.672302\pi$
$864$		0			0
$865$	3.75379			0.127633
$866$		0			0
$867$	−7.92295	+	4.44916i	−0.269078	+	0.151101i
$868$		0			0
$869$		−	8.24621i		−	0.279734i
$870$		0			0
$871$			6.33122i			0.214525i
$872$		0			0
$873$	−15.4741	−	9.43318i	−0.523718	−	0.319265i
$874$		0			0
$875$	16.2177	−	4.19224i	0.548260	−	0.141723i
$876$		0			0
$877$	−18.1080			−0.611462			−0.305731	−	0.952118i	$-0.598901\pi$
$877$	−18.1080			−0.611462			−0.305731	+	0.952118i	$0.598901\pi$
$878$		0			0
$879$	−16.8078	+	9.43845i	−0.566912	+	0.318351i
$880$		0			0
$881$	9.88653			0.333086			0.166543	−	0.986034i	$-0.446740\pi$
$881$	9.88653			0.333086			0.166543	+	0.986034i	$0.446740\pi$
$882$		0			0
$883$	−36.7926			−1.23817			−0.619085	−	0.785324i	$-0.712496\pi$
$883$	−36.7926			−1.23817			−0.619085	+	0.785324i	$0.712496\pi$
$884$		0			0
$885$	−1.40582	+	0.789443i	−0.0472562	+	0.0265368i
$886$		0			0
$887$	42.6123			1.43078			0.715391	−	0.698724i	$-0.246249\pi$
$887$	42.6123			1.43078			0.715391	+	0.698724i	$0.246249\pi$
$888$		0			0
$889$	−48.9848	+	12.6624i	−1.64290	+	0.424684i
$890$		0			0
$891$	8.00000	−	4.12311i	0.268010	−	0.138129i
$892$		0			0
$893$		−	23.3153i		−	0.780218i
$894$		0			0
$895$		−	15.6371i		−	0.522691i
$896$		0			0
$897$	−5.12311	+	2.87689i	−0.171056	+	0.0960567i
$898$		0			0
$899$	30.9481			1.03218
$900$		0			0
$901$			38.8940i			1.29575i
$902$		0			0
$903$	−26.4557	+	25.3790i	−0.880391	+	0.844561i
$904$		0			0
$905$			2.49242i			0.0828509i
$906$		0			0
$907$	22.2462			0.738673			0.369337	−	0.929296i	$-0.379585\pi$
$907$	22.2462			0.738673			0.369337	+	0.929296i	$0.379585\pi$
$908$		0			0
$909$	−17.3790	+	28.5083i	−0.576426	+	0.945562i
$910$		0			0
$911$			6.00000i			0.198789i	0.995048	+	0.0993944i	$0.0316906\pi$
$911$			6.00000i			0.198789i	−0.995048	+	0.0993944i	$0.968309\pi$
$912$		0			0
$913$			15.1022i			0.499811i
$914$		0			0
$915$	−0.952473	−	1.69614i	−0.0314878	−	0.0560727i
$916$		0			0
$917$	6.24970	−	1.61553i	0.206383	−	0.0533494i
$918$		0			0
$919$	−38.0000			−1.25350			−0.626752	−	0.779219i	$-0.715616\pi$
$919$	−38.0000			−1.25350			−0.626752	+	0.779219i	$0.715616\pi$
$920$		0			0
$921$	−11.1922	−	19.9309i	−0.368797	−	0.656744i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−20.2462			−0.665691
$926$		0			0
$927$	51.7194	+	31.5288i	1.69869	+	1.03554i
$928$		0			0
$929$	39.7649			1.30465			0.652323	−	0.757941i	$-0.273795\pi$
$929$	39.7649			1.30465			0.652323	+	0.757941i	$0.273795\pi$
$930$		0			0
$931$	−13.7538	−	24.8257i	−0.450762	−	0.813631i
$932$		0			0
$933$	28.2462	−	15.8617i	0.924739	−	0.519290i
$934$		0			0
$935$			3.12311i			0.102136i
$936$		0			0
$937$		−	22.2586i		−	0.727158i	−0.931563	−	0.363579i	$-0.881555\pi$
$937$		−	22.2586i		−	0.727158i	0.931563	−	0.363579i	$-0.118445\pi$
$938$		0			0
$939$	9.12311	+	16.2462i	0.297721	+	0.530175i
$940$		0			0
$941$	11.7100			0.381734			0.190867	−	0.981616i	$-0.438870\pi$
$941$	11.7100			0.381734			0.190867	+	0.981616i	$0.438870\pi$
$942$		0			0
$943$		−	48.3272i		−	1.57375i
$944$		0			0
$945$	−1.96144	−	8.88928i	−0.0638056	−	0.289168i
$946$		0			0
$947$			14.6307i			0.475433i	0.971335	+	0.237717i	$0.0763990\pi$
$947$			14.6307i			0.475433i	−0.971335	+	0.237717i	$0.923601\pi$
$948$		0			0
$949$	−3.31534			−0.107621
$950$		0			0
$951$	−8.48071	−	15.1022i	−0.275006	−	0.489724i
$952$		0			0
$953$		−	35.4233i		−	1.14747i	−0.819040	−	0.573736i	$-0.805493\pi$
$953$		−	35.4233i		−	1.14747i	0.819040	−	0.573736i	$-0.194507\pi$
$954$		0			0
$955$		−	17.9597i		−	0.581161i
$956$		0			0
$957$	−5.37874	+	3.02045i	−0.173870	+	0.0976372i
$958$		0			0
$959$	−3.55531	−	13.7538i	−0.114807	−	0.444133i
$960$		0			0
$961$	−44.5076			−1.43573
$962$		0			0
$963$	18.7386	+	11.4233i	0.603844	+	0.368110i
$964$		0			0
$965$	7.36520			0.237094
$966$		0			0
$967$	−1.36932			−0.0440343			−0.0220171	−	0.999758i	$-0.507009\pi$
$967$	−1.36932			−0.0440343			−0.0220171	+	0.999758i	$0.507009\pi$
$968$		0			0
$969$	16.2177	+	28.8802i	0.520989	+	0.927765i
$970$		0			0
$971$	5.54177			0.177844			0.0889220	−	0.996039i	$-0.471658\pi$
$971$	5.54177			0.177844			0.0889220	+	0.996039i	$0.471658\pi$
$972$		0			0
$973$	−12.2462	−	47.3747i	−0.392596	−	1.51876i
$974$		0			0
$975$	2.56155	+	4.56155i	0.0820353	+	0.146087i
$976$		0			0
$977$			46.7386i			1.49530i	0.664092	+	0.747651i	$0.268818\pi$
$977$			46.7386i			1.49530i	−0.664092	+	0.747651i	$0.731182\pi$
$978$		0			0
$979$			12.0818i			0.386136i
$980$		0			0
$981$	10.7386	−	17.6155i	0.342858	−	0.562421i
$982$		0			0
$983$	0.580639			0.0185195			0.00925975	−	0.999957i	$-0.497052\pi$
$983$	0.580639			0.0185195			0.00925975	+	0.999957i	$0.497052\pi$
$984$		0			0
$985$			10.7575i			0.342762i
$986$		0			0
$987$	19.0170	−	18.2430i	0.605316	−	0.580681i
$988$		0			0
$989$			40.9848i			1.30324i
$990$		0			0
$991$	−48.9309			−1.55434			−0.777170	−	0.629291i	$-0.783346\pi$
$991$	−48.9309			−1.55434			−0.777170	+	0.629291i	$0.783346\pi$
$992$		0			0
$993$	15.4741	−	8.68951i	0.491055	−	0.275753i
$994$		0			0
$995$			0.492423i			0.0156108i
$996$		0			0
$997$			48.1184i			1.52392i	0.647622	+	0.761962i	$0.275764\pi$
$997$			48.1184i			1.52392i	−0.647622	+	0.761962i	$0.724236\pi$
$998$		0			0
$999$	−0.825183	+	23.0481i	−0.0261076	+	0.729209i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1848.2.v.b.881.2	yes	8
3.2	odd	2	inner	1848.2.v.b.881.8	yes	8
7.6	odd	2	inner	1848.2.v.b.881.7	yes	8
21.20	even	2	inner	1848.2.v.b.881.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1848.2.v.b.881.1	✓	8	21.20	even	2	inner
1848.2.v.b.881.2	yes	8	1.1	even	1	trivial
1848.2.v.b.881.7	yes	8	7.6	odd	2	inner
1848.2.v.b.881.8	yes	8	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1848.v (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.51022 + 0.848071i) q^{3} +0.662153 q^{5} +(-2.56155 + 0.662153i) q^{7} +(1.56155 - 2.56155i) q^{9} +O(q^{10})\)	\(q+(-1.51022 + 0.848071i) q^{3} +0.662153 q^{5} +(-2.56155 + 0.662153i) q^{7} +(1.56155 - 2.56155i) q^{9} +1.00000i q^{11} +0.662153i q^{13} +(-1.00000 + 0.561553i) q^{15} +4.71659 q^{17} -4.05444i q^{19} +(3.30697 - 3.17238i) q^{21} -5.12311i q^{23} -4.56155 q^{25} +(-0.185917 + 5.19283i) q^{27} -3.56155i q^{29} +8.68951i q^{31} +(-0.848071 - 1.51022i) q^{33} +(-1.69614 + 0.438447i) q^{35} +4.43845 q^{37} +(-0.561553 - 1.00000i) q^{39} +9.43318 q^{41} -8.00000 q^{43} +(1.03399 - 1.69614i) q^{45} +5.75058 q^{47} +(6.12311 - 3.39228i) q^{49} +(-7.12311 + 4.00000i) q^{51} +8.24621i q^{53} +0.662153i q^{55} +(3.43845 + 6.12311i) q^{57} +1.40582 q^{59} +1.69614i q^{61} +(-2.30386 + 7.59554i) q^{63} +0.438447i q^{65} +9.56155 q^{67} +(4.34475 + 7.73704i) q^{69} +5.00691i q^{73} +(6.88897 - 3.86852i) q^{75} +(-0.662153 - 2.56155i) q^{77} -8.24621 q^{79} +(-4.12311 - 8.00000i) q^{81} +15.1022 q^{83} +3.12311 q^{85} +(3.02045 + 5.37874i) q^{87} +12.0818 q^{89} +(-0.438447 - 1.69614i) q^{91} +(-7.36932 - 13.1231i) q^{93} -2.68466i q^{95} -6.04090i q^{97} +(2.56155 + 1.56155i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7} - 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^7 - 4 * q^9	\( 8 q - 4 q^{7} - 4 q^{9} - 8 q^{15} + 12 q^{21} - 20 q^{25} + 52 q^{37} + 12 q^{39} - 64 q^{43} + 16 q^{49} - 24 q^{51} + 44 q^{57} - 32 q^{63} + 60 q^{67} - 8 q^{85} - 20 q^{91} + 40 q^{93} + 4 q^{99}+O(q^{100}) \) 8 * q - 4 * q^7 - 4 * q^9 - 8 * q^15 + 12 * q^21 - 20 * q^25 + 52 * q^37 + 12 * q^39 - 64 * q^43 + 16 * q^49 - 24 * q^51 + 44 * q^57 - 32 * q^63 + 60 * q^67 - 8 * q^85 - 20 * q^91 + 40 * q^93 + 4 * q^99

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