LMFDB - Embedded newform 4680.2.l.e.2809.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(2809,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("4680.2809");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.l (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:	$37.3699881460$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5161984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 4x^{3} + 25x^{2} - 20x + 8 $ x^6 - 4x^3 + 25x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2809.2
Root			$1.32001 + 1.32001i$ of defining polynomial
Character	$\chi$	$=$	4680.2809
Dual form			4680.2.l.e.2809.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.32001 + 1.80487i) q^{5} +O(q^{10})$	$q+(-1.32001 + 1.80487i) q^{5} +4.64002 q^{11} +1.00000i q^{13} +4.24977i q^{17} +6.24977 q^{19} -2.24977i q^{23} +(-1.51514 - 4.76491i) q^{25} -9.21949 q^{29} +9.28005 q^{31} -7.28005i q^{37} +5.67030 q^{41} -4.24977i q^{43} +2.88979i q^{47} +7.00000 q^{49} +9.21949i q^{53} +(-6.12489 + 8.37466i) q^{55} +5.92007 q^{59} +0.969724 q^{61} +(-1.80487 - 1.32001i) q^{65} +1.93945i q^{67} -5.60975 q^{71} -12.5601i q^{73} -12.2498 q^{79} +3.67030i q^{83} +(-7.67030 - 5.60975i) q^{85} -9.67030 q^{89} +(-8.24977 + 11.2800i) q^{95} +6.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 12 q^{11} + 4 q^{19} - 10 q^{25} - 20 q^{29} + 24 q^{31} + 20 q^{41} + 42 q^{49} - 20 q^{55} - 12 q^{59} + 4 q^{61} - 2 q^{65} - 16 q^{71} - 40 q^{79} - 32 q^{85} - 44 q^{89} - 16 q^{95}+O(q^{100}) $ 6 * q + 12 * q^11 + 4 * q^19 - 10 * q^25 - 20 * q^29 + 24 * q^31 + 20 * q^41 + 42 * q^49 - 20 * q^55 - 12 * q^59 + 4 * q^61 - 2 * q^65 - 16 * q^71 - 40 * q^79 - 32 * q^85 - 44 * q^89 - 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$2081$	$2341$	$3511$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.32001	+	1.80487i	−0.590327	+	0.807164i
$5$	−1.32001	+	1.80487i	−0.590327	+	0.807164i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.64002			1.39902			0.699510	−	0.714623i	$-0.253402\pi$
$11$	4.64002			1.39902			0.699510	+	0.714623i	$0.253402\pi$
$12$		0			0
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.24977i			1.03072i	0.856974	+	0.515360i	$0.172342\pi$
$17$			4.24977i			1.03072i	−0.856974	+	0.515360i	$0.827658\pi$
$18$		0			0
$19$	6.24977			1.43380			0.716898	−	0.697178i	$-0.245561\pi$
$19$	6.24977			1.43380			0.716898	+	0.697178i	$0.245561\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.24977i		−	0.469110i	−0.972103	−	0.234555i	$-0.924637\pi$
$23$		−	2.24977i		−	0.469110i	0.972103	−	0.234555i	$-0.0753632\pi$
$24$		0			0
$25$	−1.51514	−	4.76491i	−0.303028	−	0.952982i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−9.21949			−1.71202			−0.856009	−	0.516962i	$-0.827063\pi$
$29$	−9.21949			−1.71202			−0.856009	+	0.516962i	$0.827063\pi$
$30$		0			0
$31$	9.28005			1.66675			0.833373	−	0.552711i	$-0.186407\pi$
$31$	9.28005			1.66675			0.833373	+	0.552711i	$0.186407\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	7.28005i		−	1.19683i	−0.801185	−	0.598416i	$-0.795797\pi$
$37$		−	7.28005i		−	1.19683i	0.801185	−	0.598416i	$-0.204203\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.67030			0.885552			0.442776	−	0.896632i	$-0.353994\pi$
$41$	5.67030			0.885552			0.442776	+	0.896632i	$0.353994\pi$
$42$		0			0
$43$		−	4.24977i		−	0.648084i	−0.946043	−	0.324042i	$-0.894958\pi$
$43$		−	4.24977i		−	0.648084i	0.946043	−	0.324042i	$-0.105042\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2.88979i			0.421520i	0.977538	+	0.210760i	$0.0675938\pi$
$47$			2.88979i			0.421520i	−0.977538	+	0.210760i	$0.932406\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$			9.21949i			1.26639i	0.773990	+	0.633197i	$0.218258\pi$
$53$			9.21949i			1.26639i	−0.773990	+	0.633197i	$0.781742\pi$
$54$		0			0
$55$	−6.12489	+	8.37466i	−0.825879	+	1.12924i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.92007			0.770728			0.385364	−	0.922765i	$-0.374076\pi$
$59$	5.92007			0.770728			0.385364	+	0.922765i	$0.374076\pi$
$60$		0			0
$61$	0.969724			0.124160			0.0620802	−	0.998071i	$-0.480227\pi$
$61$	0.969724			0.124160			0.0620802	+	0.998071i	$0.480227\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.80487	−	1.32001i	−0.223867	−	0.163727i
$66$		0			0
$67$			1.93945i			0.236941i	0.992958	+	0.118471i	$0.0377991\pi$
$67$			1.93945i			0.236941i	−0.992958	+	0.118471i	$0.962201\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−5.60975			−0.665755			−0.332877	−	0.942970i	$-0.608019\pi$
$71$	−5.60975			−0.665755			−0.332877	+	0.942970i	$0.608019\pi$
$72$		0			0
$73$		−	12.5601i		−	1.47005i	−0.678041	−	0.735024i	$-0.737171\pi$
$73$		−	12.5601i		−	1.47005i	0.678041	−	0.735024i	$-0.262829\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−12.2498			−1.37821			−0.689103	−	0.724663i	$-0.741995\pi$
$79$	−12.2498			−1.37821			−0.689103	+	0.724663i	$0.741995\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			3.67030i			0.402868i	0.979502	+	0.201434i	$0.0645601\pi$
$83$			3.67030i			0.402868i	−0.979502	+	0.201434i	$0.935440\pi$
$84$		0			0
$85$	−7.67030	−	5.60975i	−0.831961	−	0.608463i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.67030			−1.02505			−0.512525	−	0.858672i	$-0.671290\pi$
$89$	−9.67030			−1.02505			−0.512525	+	0.858672i	$0.671290\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.24977	+	11.2800i	−0.846409	+	1.15731i
$96$		0			0
$97$			6.00000i			0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$			6.00000i			0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	15.2800			1.52042			0.760211	−	0.649677i	$-0.225096\pi$
$101$	15.2800			1.52042			0.760211	+	0.649677i	$0.225096\pi$
$102$		0			0
$103$			1.21949i			0.120160i	0.998194	+	0.0600802i	$0.0191356\pi$
$103$			1.21949i			0.120160i	−0.998194	+	0.0600802i	$0.980864\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	5.28005i		−	0.510441i	−0.966883	−	0.255221i	$-0.917852\pi$
$107$		−	5.28005i		−	0.510441i	0.966883	−	0.255221i	$-0.0821481\pi$
$108$		0			0
$109$	10.1892			0.975950			0.487975	−	0.872858i	$-0.337736\pi$
$109$	10.1892			0.975950			0.487975	+	0.872858i	$0.337736\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1.03028i		−	0.0969202i	−0.998825	−	0.0484601i	$-0.984569\pi$
$113$		−	1.03028i		−	0.0969202i	0.998825	−	0.0484601i	$-0.0154314\pi$
$114$		0			0
$115$	4.06055	+	2.96972i	0.378648	+	0.276928i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.5298			0.957256
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.6001	+	3.55510i	0.948098	+	0.317978i
$126$		0			0
$127$			11.0303i			0.978779i	0.872065	+	0.489389i	$0.162780\pi$
$127$			11.0303i			0.978779i	−0.872065	+	0.489389i	$0.837220\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.21949			−0.630770			−0.315385	−	0.948964i	$-0.602134\pi$
$131$	−7.21949			−0.630770			−0.315385	+	0.948964i	$0.602134\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			8.10929i			0.692823i	0.938083	+	0.346412i	$0.112600\pi$
$137$			8.10929i			0.692823i	−0.938083	+	0.346412i	$0.887400\pi$
$138$		0			0
$139$	19.0596			1.61662			0.808309	−	0.588759i	$-0.200383\pi$
$139$	19.0596			1.61662			0.808309	+	0.588759i	$0.200383\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			4.64002i			0.388018i
$144$		0			0
$145$	12.1698	−	16.6400i	1.01065	−	1.38188i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.92007			0.321145			0.160572	−	0.987024i	$-0.448666\pi$
$149$	3.92007			0.321145			0.160572	+	0.987024i	$0.448666\pi$
$150$		0			0
$151$	21.7796			1.77240			0.886199	−	0.463305i	$-0.153337\pi$
$151$	21.7796			1.77240			0.886199	+	0.463305i	$0.153337\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−12.2498	+	16.7493i	−0.983925	+	1.34534i
$156$		0			0
$157$			21.5904i			1.72310i	0.507673	+	0.861550i	$0.330506\pi$
$157$			21.5904i			1.72310i	−0.507673	+	0.861550i	$0.669494\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$			1.28005i			0.100261i	0.998743	+	0.0501305i	$0.0159637\pi$
$163$			1.28005i			0.100261i	−0.998743	+	0.0501305i	$0.984036\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	18.8898i		−	1.46174i	−0.682519	−	0.730868i	$-0.739115\pi$
$167$		−	18.8898i		−	1.46174i	0.682519	−	0.730868i	$-0.260885\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$			22.9991i			1.74859i	0.485397	+	0.874294i	$0.338675\pi$
$173$			22.9991i			1.74859i	−0.485397	+	0.874294i	$0.661325\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.49954			−0.336312			−0.168156	−	0.985760i	$-0.553781\pi$
$179$	−4.49954			−0.336312			−0.168156	+	0.985760i	$0.553781\pi$
$180$		0			0
$181$	−21.0596			−1.56535			−0.782675	−	0.622430i	$-0.786145\pi$
$181$	−21.0596			−1.56535			−0.782675	+	0.622430i	$0.786145\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	13.1396	+	9.60975i	0.966040	+	0.706523i
$186$		0			0
$187$			19.7190i			1.44200i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	14.4390			1.04477			0.522384	−	0.852710i	$-0.325043\pi$
$191$	14.4390			1.04477			0.522384	+	0.852710i	$0.325043\pi$
$192$		0			0
$193$			21.7190i			1.56337i	0.623673	+	0.781685i	$0.285640\pi$
$193$			21.7190i			1.56337i	−0.623673	+	0.781685i	$0.714360\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	13.0109i		−	0.926988i	−0.886100	−	0.463494i	$-0.846596\pi$
$197$		−	13.0109i		−	0.926988i	0.886100	−	0.463494i	$-0.153404\pi$
$198$		0			0
$199$	−18.6888			−1.32481			−0.662406	−	0.749145i	$-0.730464\pi$
$199$	−18.6888			−1.32481			−0.662406	+	0.749145i	$0.730464\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−7.48486	+	10.2342i	−0.522765	+	0.714786i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	28.9991			2.00591
$210$		0			0
$211$	−8.24977			−0.567938			−0.283969	−	0.958834i	$-0.591651\pi$
$211$	−8.24977			−0.567938			−0.283969	+	0.958834i	$0.591651\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	7.67030	+	5.60975i	0.523110	+	0.382582i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.24977			−0.285871
$222$		0			0
$223$			9.77959i			0.654890i	0.944870	+	0.327445i	$0.106188\pi$
$223$			9.77959i			0.654890i	−0.944870	+	0.327445i	$0.893812\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			25.4499i			1.68917i	0.535423	+	0.844584i	$0.320152\pi$
$227$			25.4499i			1.68917i	−0.535423	+	0.844584i	$0.679848\pi$
$228$		0			0
$229$	8.24977			0.545160			0.272580	−	0.962133i	$-0.412123\pi$
$229$	8.24977			0.545160			0.272580	+	0.962133i	$0.412123\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	11.5904i		−	0.759310i	−0.925128	−	0.379655i	$-0.876043\pi$
$233$		−	11.5904i		−	0.759310i	0.925128	−	0.379655i	$-0.123957\pi$
$234$		0			0
$235$	−5.21571	−	3.81456i	−0.340236	−	0.248835i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.88979			−0.445664			−0.222832	−	0.974857i	$-0.571530\pi$
$239$	−6.88979			−0.445664			−0.222832	+	0.974857i	$0.571530\pi$
$240$		0			0
$241$	21.2195			1.36687			0.683434	−	0.730012i	$-0.260486\pi$
$241$	21.2195			1.36687			0.683434	+	0.730012i	$0.260486\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.24008	+	12.6341i	−0.590327	+	0.807164i
$246$		0			0
$247$			6.24977i			0.397663i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	24.3397			1.53631			0.768154	−	0.640266i	$-0.221176\pi$
$251$	24.3397			1.53631			0.768154	+	0.640266i	$0.221176\pi$
$252$		0			0
$253$		−	10.4390i		−	0.656294i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	23.4693i		−	1.46397i	−0.681319	−	0.731986i	$-0.738593\pi$
$257$		−	23.4693i		−	1.46397i	0.681319	−	0.731986i	$-0.261407\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$			22.6282i			1.39532i	0.716431	+	0.697658i	$0.245774\pi$
$263$			22.6282i			1.39532i	−0.716431	+	0.697658i	$0.754226\pi$
$264$		0			0
$265$	−16.6400	−	12.1698i	−1.02219	−	0.747587i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.2791			0.748672			0.374336	−	0.927293i	$-0.377871\pi$
$269$	12.2791			0.748672			0.374336	+	0.927293i	$0.377871\pi$
$270$		0			0
$271$	−22.9385			−1.39342			−0.696708	−	0.717355i	$-0.745353\pi$
$271$	−22.9385			−1.39342			−0.696708	+	0.717355i	$0.745353\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7.03028	−	22.1093i	−0.423942	−	1.33324i
$276$		0			0
$277$		−	11.5298i		−	0.692760i	−0.938094	−	0.346380i	$-0.887411\pi$
$277$		−	11.5298i		−	0.692760i	0.938094	−	0.346380i	$-0.112589\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	11.3288			0.675819			0.337909	−	0.941179i	$-0.390280\pi$
$281$	11.3288			0.675819			0.337909	+	0.941179i	$0.390280\pi$
$282$		0			0
$283$			20.9991i			1.24827i	0.781318	+	0.624133i	$0.214548\pi$
$283$			20.9991i			1.24827i	−0.781318	+	0.624133i	$0.785452\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.06055			−0.0623854
$290$		0			0
$291$		0			0
$292$		0			0
$293$			4.26915i			0.249406i	0.992194	+	0.124703i	$0.0397979\pi$
$293$			4.26915i			0.249406i	−0.992194	+	0.124703i	$0.960202\pi$
$294$		0			0
$295$	−7.81456	+	10.6850i	−0.454981	+	0.622104i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.24977			0.130108
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.28005	+	1.75023i	−0.0732953	+	0.100218i
$306$		0			0
$307$			20.0000i			1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$			20.0000i			1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3.71904			−0.210887			−0.105444	−	0.994425i	$-0.533626\pi$
$311$	−3.71904			−0.210887			−0.105444	+	0.994425i	$0.533626\pi$
$312$		0			0
$313$			1.52982i			0.0864704i	0.999065	+	0.0432352i	$0.0137665\pi$
$313$			1.52982i			0.0864704i	−0.999065	+	0.0432352i	$0.986234\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			16.1093i			0.904788i	0.891818	+	0.452394i	$0.149430\pi$
$317$			16.1093i			0.904788i	−0.891818	+	0.452394i	$0.850570\pi$
$318$		0			0
$319$	−42.7787			−2.39515
$320$		0			0
$321$		0			0
$322$		0			0
$323$			26.5601i			1.47784i
$324$		0			0
$325$	4.76491	−	1.51514i	0.264310	−	0.0840447i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	15.5298			0.853596			0.426798	−	0.904347i	$-0.359642\pi$
$331$	15.5298			0.853596			0.426798	+	0.904347i	$0.359642\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.50046	−	2.56009i	−0.191250	−	0.139873i
$336$		0			0
$337$			21.4087i			1.16621i	0.812398	+	0.583103i	$0.198162\pi$
$337$			21.4087i			1.16621i	−0.812398	+	0.583103i	$0.801838\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	43.0596			2.33181
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	4.62065i		−	0.248049i	−0.992279	−	0.124025i	$-0.960420\pi$
$347$		−	4.62065i		−	0.248049i	0.992279	−	0.124025i	$-0.0395802\pi$
$348$		0			0
$349$	−23.9688			−1.28302			−0.641510	−	0.767114i	$-0.721692\pi$
$349$	−23.9688			−1.28302			−0.641510	+	0.767114i	$0.721692\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			0.511357i			0.0272168i	0.999907	+	0.0136084i	$0.00433182\pi$
$353$			0.511357i			0.0272168i	−0.999907	+	0.0136084i	$0.995668\pi$
$354$		0			0
$355$	7.40493	−	10.1249i	0.393013	−	0.537373i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.6694			−0.668664			−0.334332	−	0.942455i	$-0.608511\pi$
$359$	−12.6694			−0.668664			−0.334332	+	0.942455i	$0.608511\pi$
$360$		0			0
$361$	20.0596			1.05577
$362$		0			0
$363$		0			0
$364$		0			0
$365$	22.6694	+	16.5795i	1.18657	+	0.867809i
$366$		0			0
$367$			0.841057i			0.0439028i	0.999759	+	0.0219514i	$0.00698792\pi$
$367$			0.841057i			0.0439028i	−0.999759	+	0.0219514i	$0.993012\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	29.0596i		−	1.50465i	−0.658792	−	0.752325i	$-0.728932\pi$
$373$		−	29.0596i		−	1.50465i	0.658792	−	0.752325i	$-0.271068\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	9.21949i		−	0.474828i
$378$		0			0
$379$	−9.75023			−0.500836			−0.250418	−	0.968138i	$-0.580568\pi$
$379$	−9.75023			−0.500836			−0.250418	+	0.968138i	$0.580568\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			7.38934i			0.377577i	0.982018	+	0.188789i	$0.0604561\pi$
$383$			7.38934i			0.377577i	−0.982018	+	0.188789i	$0.939544\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−14.0000			−0.709828			−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000			−0.709828			−0.354914	+	0.934899i	$0.615490\pi$
$390$		0			0
$391$	9.56101			0.483521
$392$		0			0
$393$		0			0
$394$		0			0
$395$	16.1698	−	22.1093i	0.813593	−	1.11244i
$396$		0			0
$397$		−	26.2186i		−	1.31587i	−0.753074	−	0.657936i	$-0.771430\pi$
$397$		−	26.2186i		−	1.31587i	0.753074	−	0.657936i	$-0.228570\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10.2909			−0.513905			−0.256953	−	0.966424i	$-0.582718\pi$
$401$	−10.2909			−0.513905			−0.256953	+	0.966424i	$0.582718\pi$
$402$		0			0
$403$			9.28005i			0.462272i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	33.7796i		−	1.67439i
$408$		0			0
$409$	11.9007			0.588451			0.294226	−	0.955736i	$-0.404938\pi$
$409$	11.9007			0.588451			0.294226	+	0.955736i	$0.404938\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−6.62443	−	4.84484i	−0.325180	−	0.237824i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	26.2791			1.28382			0.641910	−	0.766780i	$-0.278142\pi$
$419$	26.2791			1.28382			0.641910	+	0.766780i	$0.278142\pi$
$420$		0			0
$421$	22.3103			1.08734			0.543669	−	0.839300i	$-0.317035\pi$
$421$	22.3103			1.08734			0.543669	+	0.839300i	$0.317035\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	20.2498	−	6.43899i	0.982258	−	0.312337i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12.0487			−0.580367			−0.290184	−	0.956971i	$-0.593716\pi$
$431$	−12.0487			−0.580367			−0.290184	+	0.956971i	$0.593716\pi$
$432$		0			0
$433$			27.5904i			1.32591i	0.748660	+	0.662954i	$0.230698\pi$
$433$			27.5904i			1.32591i	−0.748660	+	0.662954i	$0.769302\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	14.0606i		−	0.672607i
$438$		0			0
$439$	−19.8789			−0.948768			−0.474384	−	0.880318i	$-0.657329\pi$
$439$	−19.8789			−0.948768			−0.474384	+	0.880318i	$0.657329\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	17.9007i		−	0.850488i	−0.905079	−	0.425244i	$-0.860188\pi$
$443$		−	17.9007i		−	0.850488i	0.905079	−	0.425244i	$-0.139812\pi$
$444$		0			0
$445$	12.7649	−	17.4537i	0.605115	−	0.827383i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−17.2682			−0.814938			−0.407469	−	0.913219i	$-0.633589\pi$
$449$	−17.2682			−0.814938			−0.407469	+	0.913219i	$0.633589\pi$
$450$		0			0
$451$	26.3103			1.23890
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			30.4995i			1.42671i	0.700804	+	0.713354i	$0.252825\pi$
$457$			30.4995i			1.42671i	−0.700804	+	0.713354i	$0.747175\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.64002			0.122958			0.0614791	−	0.998108i	$-0.480418\pi$
$461$	2.64002			0.122958			0.0614791	+	0.998108i	$0.480418\pi$
$462$		0			0
$463$		−	16.0000i		−	0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$		−	16.0000i		−	0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	4.78051i		−	0.221215i	−0.993864	−	0.110608i	$-0.964720\pi$
$467$		−	4.78051i		−	0.221215i	0.993864	−	0.110608i	$-0.0352797\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	19.7190i		−	0.906682i
$474$		0			0
$475$	−9.46927	−	29.7796i	−0.434480	−	1.36638i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.11021			0.416256			0.208128	−	0.978102i	$-0.433263\pi$
$479$	9.11021			0.416256			0.208128	+	0.978102i	$0.433263\pi$
$480$		0			0
$481$	7.28005			0.331942
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10.8292	−	7.92007i	−0.491731	−	0.359632i
$486$		0			0
$487$			26.2791i			1.19082i	0.803422	+	0.595411i	$0.203011\pi$
$487$			26.2791i			1.19082i	−0.803422	+	0.595411i	$0.796989\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.40115			−0.243751			−0.121875	−	0.992545i	$-0.538891\pi$
$491$	−5.40115			−0.243751			−0.121875	+	0.992545i	$0.538891\pi$
$492$		0			0
$493$		−	39.1807i		−	1.76461i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−22.9092			−1.02556			−0.512778	−	0.858521i	$-0.671383\pi$
$499$	−22.9092			−1.02556			−0.512778	+	0.858521i	$0.671383\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	37.3094i		−	1.66354i	−0.555117	−	0.831772i	$-0.687327\pi$
$503$		−	37.3094i		−	1.66354i	0.555117	−	0.831772i	$-0.312673\pi$
$504$		0			0
$505$	−20.1698	+	27.5786i	−0.897546	+	1.22723i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.700576			−0.0310525			−0.0155262	−	0.999879i	$-0.504942\pi$
$509$	−0.700576			−0.0310525			−0.0155262	+	0.999879i	$0.504942\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.20103	−	1.60975i	−0.0969891	−	0.0709339i
$516$		0			0
$517$			13.4087i			0.589715i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−13.8401			−0.606348			−0.303174	−	0.952935i	$-0.598046\pi$
$521$	−13.8401			−0.606348			−0.303174	+	0.952935i	$0.598046\pi$
$522$		0			0
$523$		−	43.6878i		−	1.91034i	−0.296064	−	0.955168i	$-0.595674\pi$
$523$		−	43.6878i		−	1.91034i	0.296064	−	0.955168i	$-0.404326\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			39.4381i			1.71795i
$528$		0			0
$529$	17.9385			0.779936
$530$		0			0
$531$		0			0
$532$		0			0
$533$			5.67030i			0.245608i
$534$		0			0
$535$	9.52982	+	6.96972i	0.412010	+	0.301327i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	32.4802			1.39902
$540$		0			0
$541$	14.1892			0.610042			0.305021	−	0.952346i	$-0.401336\pi$
$541$	14.1892			0.610042			0.305021	+	0.952346i	$0.401336\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−13.4499	+	18.3903i	−0.576130	+	0.787752i
$546$		0			0
$547$		−	44.3085i		−	1.89449i	−0.320504	−	0.947247i	$-0.603852\pi$
$547$		−	44.3085i		−	1.89449i	0.320504	−	0.947247i	$-0.396148\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−57.6197			−2.45468
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	39.2077i		−	1.66128i	−0.556808	−	0.830641i	$-0.687974\pi$
$557$		−	39.2077i		−	1.66128i	0.556808	−	0.830641i	$-0.312026\pi$
$558$		0			0
$559$	4.24977			0.179746
$560$		0			0
$561$		0			0
$562$		0			0
$563$			18.0606i			0.761162i	0.924748	+	0.380581i	$0.124276\pi$
$563$			18.0606i			0.761162i	−0.924748	+	0.380581i	$0.875724\pi$
$564$		0			0
$565$	1.85952	+	1.35998i	0.0782305	+	0.0572146i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−42.3397			−1.77497			−0.887486	−	0.460835i	$-0.847550\pi$
$569$	−42.3397			−1.77497			−0.887486	+	0.460835i	$0.847550\pi$
$570$		0			0
$571$	−39.9301			−1.67102			−0.835510	−	0.549475i	$-0.814828\pi$
$571$	−39.9301			−1.67102			−0.835510	+	0.549475i	$0.814828\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−10.7200	+	3.40871i	−0.447053	+	0.142153i
$576$		0			0
$577$		−	21.3406i		−	0.888421i	−0.895923	−	0.444210i	$-0.853484\pi$
$577$		−	21.3406i		−	0.888421i	0.895923	−	0.444210i	$-0.146516\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$			42.7787i			1.77171i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			32.7905i			1.35341i	0.736255	+	0.676704i	$0.236592\pi$
$587$			32.7905i			1.35341i	−0.736255	+	0.676704i	$0.763408\pi$
$588$		0			0
$589$	57.9982			2.38977
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	6.66938i		−	0.273879i	−0.990579	−	0.136939i	$-0.956273\pi$
$593$		−	6.66938i		−	0.273879i	0.990579	−	0.136939i	$-0.0437265\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.40115			0.0572495			0.0286247	−	0.999590i	$-0.490887\pi$
$599$	1.40115			0.0572495			0.0286247	+	0.999590i	$0.490887\pi$
$600$		0			0
$601$	0.349078			0.0142392			0.00711959	−	0.999975i	$-0.497734\pi$
$601$	0.349078			0.0142392			0.00711959	+	0.999975i	$0.497734\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−13.8995	+	19.0050i	−0.565094	+	0.772663i
$606$		0			0
$607$			27.2800i			1.10726i	0.832762	+	0.553631i	$0.186758\pi$
$607$			27.2800i			1.10726i	−0.832762	+	0.553631i	$0.813242\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.88979			−0.116909
$612$		0			0
$613$			9.34060i			0.377263i	0.982048	+	0.188632i	$0.0604052\pi$
$613$			9.34060i			0.377263i	−0.982048	+	0.188632i	$0.939595\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	34.6694i		−	1.39574i	−0.716226	−	0.697868i	$-0.754132\pi$
$617$		−	34.6694i		−	1.39574i	0.716226	−	0.697868i	$-0.245868\pi$
$618$		0			0
$619$	−19.6509			−0.789837			−0.394919	−	0.918716i	$-0.629227\pi$
$619$	−19.6509			−0.789837			−0.394919	+	0.918716i	$0.629227\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−20.4087	+	14.4390i	−0.816349	+	0.577560i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	30.9385			1.23360
$630$		0			0
$631$	−43.3993			−1.72770			−0.863850	−	0.503749i	$-0.831953\pi$
$631$	−43.3993			−1.72770			−0.863850	+	0.503749i	$0.831953\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−19.9083	−	14.5601i	−0.790035	−	0.577800i
$636$		0			0
$637$			7.00000i			0.277350i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4.56009			0.180113			0.0900564	−	0.995937i	$-0.471295\pi$
$641$	4.56009			0.180113			0.0900564	+	0.995937i	$0.471295\pi$
$642$		0			0
$643$		−	8.62065i		−	0.339965i	−0.985447	−	0.169983i	$-0.945629\pi$
$643$		−	8.62065i		−	0.339965i	0.985447	−	0.169983i	$-0.0543711\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	7.40871i		−	0.291267i	−0.989339	−	0.145633i	$-0.953478\pi$
$647$		−	7.40871i		−	0.291267i	0.989339	−	0.145633i	$-0.0465220\pi$
$648$		0			0
$649$	27.4693			1.07826
$650$		0			0
$651$		0			0
$652$		0			0
$653$			2.00000i			0.0782660i	0.999234	+	0.0391330i	$0.0124596\pi$
$653$			2.00000i			0.0782660i	−0.999234	+	0.0391330i	$0.987540\pi$
$654$		0			0
$655$	9.52982	−	13.0303i	0.372361	−	0.509135i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	40.3784			1.57292			0.786460	−	0.617641i	$-0.211911\pi$
$659$	40.3784			1.57292			0.786460	+	0.617641i	$0.211911\pi$
$660$		0			0
$661$	39.0284			1.51803			0.759015	−	0.651073i	$-0.225681\pi$
$661$	39.0284			1.51803			0.759015	+	0.651073i	$0.225681\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			20.7418i			0.803124i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.49954			0.173703
$672$		0			0
$673$		−	11.1807i		−	0.430986i	−0.976505	−	0.215493i	$-0.930864\pi$
$673$		−	11.1807i		−	0.430986i	0.976505	−	0.215493i	$-0.0691358\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	33.3406i		−	1.28138i	−0.767798	−	0.640692i	$-0.778648\pi$
$677$		−	33.3406i		−	1.28138i	0.767798	−	0.640692i	$-0.221352\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			31.8889i			1.22019i	0.792327	+	0.610097i	$0.208870\pi$
$683$			31.8889i			1.22019i	−0.792327	+	0.610097i	$0.791130\pi$
$684$		0			0
$685$	−14.6362	−	10.7044i	−0.559222	−	0.408992i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−9.21949			−0.351235
$690$		0			0
$691$	39.0303			1.48478			0.742391	−	0.669967i	$-0.233692\pi$
$691$	39.0303			1.48478			0.742391	+	0.669967i	$0.233692\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−25.1589	+	34.4002i	−0.954333	+	1.30488i
$696$		0			0
$697$			24.0975i			0.912757i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	44.5601			1.68301			0.841506	−	0.540248i	$-0.181670\pi$
$701$	44.5601			1.68301			0.841506	+	0.540248i	$0.181670\pi$
$702$		0			0
$703$		−	45.4986i		−	1.71601i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−23.4305			−0.879951			−0.439976	−	0.898010i	$-0.645013\pi$
$709$	−23.4305			−0.879951			−0.439976	+	0.898010i	$0.645013\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	20.8780i		−	0.781886i
$714$		0			0
$715$	−8.37466	−	6.12489i	−0.313194	−	0.229058i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−15.0984			−0.563075			−0.281537	−	0.959550i	$-0.590844\pi$
$719$	−15.0984			−0.563075			−0.281537	+	0.959550i	$0.590844\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	13.9688	+	43.9301i	0.518788	+	1.63152i
$726$		0			0
$727$		−	3.40871i		−	0.126422i	−0.998000	−	0.0632111i	$-0.979866\pi$
$727$		−	3.40871i		−	0.126422i	0.998000	−	0.0632111i	$-0.0201341\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	18.0606			0.667994
$732$		0			0
$733$			12.4390i			0.459445i	0.973256	+	0.229722i	$0.0737818\pi$
$733$			12.4390i			0.459445i	−0.973256	+	0.229722i	$0.926218\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			8.99908i			0.331485i
$738$		0			0
$739$	13.4693			0.495475			0.247737	−	0.968827i	$-0.420313\pi$
$739$	13.4693			0.495475			0.247737	+	0.968827i	$0.420313\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			45.4111i			1.66597i	0.553293	+	0.832986i	$0.313371\pi$
$743$			45.4111i			1.66597i	−0.553293	+	0.832986i	$0.686629\pi$
$744$		0			0
$745$	−5.17454	+	7.07523i	−0.189580	+	0.259216i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.56101			0.0569621			0.0284810	−	0.999594i	$-0.490933\pi$
$751$	1.56101			0.0569621			0.0284810	+	0.999594i	$0.490933\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−28.7493	+	39.3094i	−1.04629	+	1.43062i
$756$		0			0
$757$		−	38.8780i		−	1.41304i	−0.707691	−	0.706522i	$-0.750263\pi$
$757$		−	38.8780i		−	1.41304i	0.707691	−	0.706522i	$-0.249737\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−5.42809			−0.196768			−0.0983841	−	0.995149i	$-0.531367\pi$
$761$	−5.42809			−0.196768			−0.0983841	+	0.995149i	$0.531367\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$			5.92007i			0.213761i
$768$		0			0
$769$	−16.4390			−0.592805			−0.296403	−	0.955063i	$-0.595787\pi$
$769$	−16.4390			−0.592805			−0.296403	+	0.955063i	$0.595787\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	42.3884i		−	1.52461i	−0.647221	−	0.762303i	$-0.724069\pi$
$773$		−	42.3884i		−	1.52461i	0.647221	−	0.762303i	$-0.275931\pi$
$774$		0			0
$775$	−14.0606	−	44.2186i	−0.505070	−	1.58838i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	35.4381			1.26970
$780$		0			0
$781$	−26.0294			−0.931404
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−38.9679	−	28.4995i	−1.39082	−	1.01719i
$786$		0			0
$787$			15.5592i			0.554625i	0.960780	+	0.277312i	$0.0894437\pi$
$787$			15.5592i			0.554625i	−0.960780	+	0.277312i	$0.910556\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$			0.969724i			0.0344359i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			36.4002i			1.28936i	0.764452	+	0.644681i	$0.223010\pi$
$797$			36.4002i			1.28936i	−0.764452	+	0.644681i	$0.776990\pi$
$798$		0			0
$799$	−12.2810			−0.434469
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	58.2791i		−	2.05663i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5.59793			0.196813			0.0984064	−	0.995146i	$-0.468625\pi$
$809$	5.59793			0.196813			0.0984064	+	0.995146i	$0.468625\pi$
$810$		0			0
$811$	28.1892			0.989857			0.494929	−	0.868934i	$-0.335194\pi$
$811$	28.1892			0.989857			0.494929	+	0.868934i	$0.335194\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.31032	−	1.68968i	−0.0809271	−	0.0591868i
$816$		0			0
$817$		−	26.5601i		−	0.929220i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−33.8208			−1.18035			−0.590176	−	0.807274i	$-0.700942\pi$
$821$	−33.8208			−1.18035			−0.590176	+	0.807274i	$0.700942\pi$
$822$		0			0
$823$		−	2.53830i		−	0.0884795i	−0.999021	−	0.0442397i	$-0.985913\pi$
$823$		−	2.53830i		−	0.0884795i	0.999021	−	0.0442397i	$-0.0140865\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	12.6694i		−	0.440558i	−0.975437	−	0.220279i	$-0.929303\pi$
$827$		−	12.6694i		−	0.440558i	0.975437	−	0.220279i	$-0.0706967\pi$
$828$		0			0
$829$	37.4693			1.30136			0.650681	−	0.759351i	$-0.274484\pi$
$829$	37.4693			1.30136			0.650681	+	0.759351i	$0.274484\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			29.7484i			1.03072i
$834$		0			0
$835$	34.0937	+	24.9348i	1.17986	+	0.862903i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−19.3893			−0.669394			−0.334697	−	0.942326i	$-0.608634\pi$
$839$	−19.3893			−0.669394			−0.334697	+	0.942326i	$0.608634\pi$
$840$		0			0
$841$	55.9991			1.93100
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.32001	−	1.80487i	0.0454098	−	0.0620895i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−16.3784			−0.561446
$852$		0			0
$853$		−	7.77959i		−	0.266368i	−0.991091	−	0.133184i	$-0.957480\pi$
$853$		−	7.77959i		−	0.266368i	0.991091	−	0.133184i	$-0.0425201\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	26.9697i		−	0.921268i	−0.887590	−	0.460634i	$-0.847622\pi$
$857$		−	26.9697i		−	0.921268i	0.887590	−	0.460634i	$-0.152378\pi$
$858$		0			0
$859$	−5.93189			−0.202393			−0.101197	−	0.994866i	$-0.532267\pi$
$859$	−5.93189			−0.202393			−0.101197	+	0.994866i	$0.532267\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	33.3288i		−	1.13452i	−0.823537	−	0.567262i	$-0.808002\pi$
$863$		−	33.3288i		−	1.13452i	0.823537	−	0.567262i	$-0.191998\pi$
$864$		0			0
$865$	−41.5104	−	30.3591i	−1.41140	−	1.03224i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−56.8392			−1.92814
$870$		0			0
$871$	−1.93945			−0.0657157
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	19.1589i		−	0.646952i	−0.946236	−	0.323476i	$-0.895149\pi$
$877$		−	19.1589i		−	0.646952i	0.946236	−	0.323476i	$-0.104851\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−53.8989			−1.81590			−0.907949	−	0.419080i	$-0.862353\pi$
$881$	−53.8989			−1.81590			−0.907949	+	0.419080i	$0.862353\pi$
$882$		0			0
$883$		−	22.9385i		−	0.771943i	−0.922511	−	0.385972i	$-0.873866\pi$
$883$		−	22.9385i		−	0.771943i	0.922511	−	0.385972i	$-0.126134\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			47.5298i			1.59590i	0.602727	+	0.797948i	$0.294081\pi$
$887$			47.5298i			1.59590i	−0.602727	+	0.797948i	$0.705919\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$			18.0606i			0.604373i
$894$		0			0
$895$	5.93945	−	8.12110i	0.198534	−	0.271459i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−85.5573			−2.85350
$900$		0			0
$901$	−39.1807			−1.30530
$902$		0			0
$903$		0			0
$904$		0			0
$905$	27.7990	−	38.0100i	0.924069	−	1.26349i
$906$		0			0
$907$		−	20.6206i		−	0.684697i	−0.939573	−	0.342349i	$-0.888778\pi$
$907$		−	20.6206i		−	0.684697i	0.939573	−	0.342349i	$-0.111222\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	41.2413			1.36638			0.683192	−	0.730238i	$-0.260591\pi$
$911$	41.2413			1.36638			0.683192	+	0.730238i	$0.260591\pi$
$912$		0			0
$913$			17.0303i			0.563620i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−37.2489			−1.22873			−0.614363	−	0.789023i	$-0.710587\pi$
$919$	−37.2489			−1.22873			−0.614363	+	0.789023i	$0.710587\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	5.60975i		−	0.184647i
$924$		0			0
$925$	−34.6888	+	11.0303i	−1.14056	+	0.362673i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−59.0091			−1.93602			−0.968012	−	0.250903i	$-0.919273\pi$
$929$	−59.0091			−1.93602			−0.968012	+	0.250903i	$0.919273\pi$
$930$		0			0
$931$	43.7484			1.43380
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−35.5904	−	26.0294i	−1.16393	−	0.851251i
$936$		0			0
$937$		−	56.9679i		−	1.86106i	−0.366216	−	0.930530i	$-0.619347\pi$
$937$		−	56.9679i		−	1.86106i	0.366216	−	0.930530i	$-0.380653\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−2.35906			−0.0769032			−0.0384516	−	0.999260i	$-0.512243\pi$
$941$	−2.35906			−0.0769032			−0.0384516	+	0.999260i	$0.512243\pi$
$942$		0			0
$943$		−	12.7569i		−	0.415421i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	43.4481i		−	1.41187i	−0.708276	−	0.705936i	$-0.750527\pi$
$947$		−	43.4481i		−	1.41187i	0.708276	−	0.705936i	$-0.249473\pi$
$948$		0			0
$949$	12.5601			0.407718
$950$		0			0
$951$		0			0
$952$		0			0
$953$			23.5904i			0.764167i	0.924128	+	0.382084i	$0.124793\pi$
$953$			23.5904i			0.764167i	−0.924128	+	0.382084i	$0.875207\pi$
$954$		0			0
$955$	−19.0596	+	26.0606i	−0.616755	+	0.843300i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	55.1193			1.77804
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−39.2001	−	28.6694i	−1.26190	−	0.922900i
$966$		0			0
$967$			31.5979i			1.01612i	0.861321	+	0.508060i	$0.169637\pi$
$967$			31.5979i			1.01612i	−0.861321	+	0.508060i	$0.830363\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.499542			0.0160311			0.00801553	−	0.999968i	$-0.497449\pi$
$971$	0.499542			0.0160311			0.00801553	+	0.999968i	$0.497449\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	18.7081i		−	0.598526i	−0.954171	−	0.299263i	$-0.903259\pi$
$977$		−	18.7081i		−	0.598526i	0.954171	−	0.299263i	$-0.0967409\pi$
$978$		0			0
$979$	−44.8704			−1.43406
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	3.83016i		−	0.122163i	−0.998133	−	0.0610815i	$-0.980545\pi$
$983$		−	3.83016i		−	0.122163i	0.998133	−	0.0610815i	$-0.0194550\pi$
$984$		0			0
$985$	23.4830	+	17.1745i	0.748232	+	0.547226i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−9.56101			−0.304022
$990$		0			0
$991$	6.43899			0.204541			0.102271	−	0.994757i	$-0.467389\pi$
$991$	6.43899			0.204541			0.102271	+	0.994757i	$0.467389\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	24.6694	−	33.7309i	0.782072	−	1.06934i
$996$		0			0
$997$		−	12.1505i		−	0.384809i	−0.981316	−	0.192405i	$-0.938371\pi$
$997$		−	12.1505i		−	0.384809i	0.981316	−	0.192405i	$-0.0616286\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.l.e.2809.2		6
3.2	odd	2		1560.2.l.c.1249.6	yes	6
5.4	even	2	inner	4680.2.l.e.2809.1		6
12.11	even	2		3120.2.l.m.1249.3		6
15.2	even	4		7800.2.a.bp.1.1		3
15.8	even	4		7800.2.a.bj.1.1		3
15.14	odd	2		1560.2.l.c.1249.3	✓	6
60.59	even	2		3120.2.l.m.1249.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.c.1249.3	✓	6	15.14	odd	2
1560.2.l.c.1249.6	yes	6	3.2	odd	2
3120.2.l.m.1249.3		6	12.11	even	2
3120.2.l.m.1249.6		6	60.59	even	2
4680.2.l.e.2809.1		6	5.4	even	2	inner
4680.2.l.e.2809.2		6	1.1	even	1	trivial
7800.2.a.bj.1.1		3	15.8	even	4
7800.2.a.bp.1.1		3	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.32001 + 1.80487i) q^{5} +O(q^{10})\)	\(q+(-1.32001 + 1.80487i) q^{5} +4.64002 q^{11} +1.00000i q^{13} +4.24977i q^{17} +6.24977 q^{19} -2.24977i q^{23} +(-1.51514 - 4.76491i) q^{25} -9.21949 q^{29} +9.28005 q^{31} -7.28005i q^{37} +5.67030 q^{41} -4.24977i q^{43} +2.88979i q^{47} +7.00000 q^{49} +9.21949i q^{53} +(-6.12489 + 8.37466i) q^{55} +5.92007 q^{59} +0.969724 q^{61} +(-1.80487 - 1.32001i) q^{65} +1.93945i q^{67} -5.60975 q^{71} -12.5601i q^{73} -12.2498 q^{79} +3.67030i q^{83} +(-7.67030 - 5.60975i) q^{85} -9.67030 q^{89} +(-8.24977 + 11.2800i) q^{95} +6.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 12 q^{11} + 4 q^{19} - 10 q^{25} - 20 q^{29} + 24 q^{31} + 20 q^{41} + 42 q^{49} - 20 q^{55} - 12 q^{59} + 4 q^{61} - 2 q^{65} - 16 q^{71} - 40 q^{79} - 32 q^{85} - 44 q^{89} - 16 q^{95}+O(q^{100}) \) 6 * q + 12 * q^11 + 4 * q^19 - 10 * q^25 - 20 * q^29 + 24 * q^31 + 20 * q^41 + 42 * q^49 - 20 * q^55 - 12 * q^59 + 4 * q^61 - 2 * q^65 - 16 * q^71 - 40 * q^79 - 32 * q^85 - 44 * q^89 - 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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