LMFDB - Embedded newform 1344.4.p.c.223.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(223,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1344.223");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.p (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:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			223.1
Character	$\chi$	$=$	1344.223
Dual form			1344.4.p.c.223.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -12.5168 q^{5} +(18.2936 + 2.88837i) q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -12.5168 q^{5} +(18.2936 + 2.88837i) q^{7} -9.00000 q^{9} -55.4008 q^{11} -92.1564 q^{13} +37.5504i q^{15} +118.220i q^{17} -155.587i q^{19} +(8.66510 - 54.8809i) q^{21} -125.358i q^{23} +31.6706 q^{25} +27.0000i q^{27} -131.643i q^{29} -66.0511 q^{31} +166.202i q^{33} +(-228.978 - 36.1532i) q^{35} +147.117i q^{37} +276.469i q^{39} +20.3097i q^{41} +355.210 q^{43} +112.651 q^{45} -79.5977 q^{47} +(326.315 + 105.678i) q^{49} +354.659 q^{51} +463.012i q^{53} +693.441 q^{55} -466.762 q^{57} +580.211i q^{59} -587.607 q^{61} +(-164.643 - 25.9953i) q^{63} +1153.50 q^{65} +496.198 q^{67} -376.073 q^{69} +232.774i q^{71} -551.640i q^{73} -95.0119i q^{75} +(-1013.48 - 160.018i) q^{77} -437.936i q^{79} +81.0000 q^{81} -191.266i q^{83} -1479.73i q^{85} -394.929 q^{87} +93.7894i q^{89} +(-1685.88 - 266.182i) q^{91} +198.153i q^{93} +1947.46i q^{95} -758.892i q^{97} +498.607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 288 q^{9}+O(q^{10}) $ 32 * q - 288 * q^9	$ 32 q - 288 q^{9} - 224 q^{13} - 72 q^{21} + 1120 q^{25} - 752 q^{49} - 672 q^{57} - 544 q^{61} + 1536 q^{65} - 144 q^{69} - 1632 q^{77} + 2592 q^{81}+O(q^{100}) $ 32 * q - 288 * q^9 - 224 * q^13 - 72 * q^21 + 1120 * q^25 - 752 * q^49 - 672 * q^57 - 544 * q^61 + 1536 * q^65 - 144 * q^69 - 1632 * q^77 + 2592 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$		−	3.00000i		−	0.577350i
$3$		−	3.00000i		−	0.577350i
$4$		0			0
$5$	−12.5168			−1.11954			−0.559769	−	0.828649i	$-0.689110\pi$
$5$	−12.5168			−1.11954			−0.559769	+	0.828649i	$0.689110\pi$
$6$		0			0
$7$	18.2936	+	2.88837i	0.987764	+	0.155957i
$7$	18.2936	+	2.88837i	0.987764	+	0.155957i
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	−55.4008			−1.51854			−0.759271	−	0.650774i	$-0.774444\pi$
$11$	−55.4008			−1.51854			−0.759271	+	0.650774i	$0.774444\pi$
$12$		0			0
$13$	−92.1564			−1.96612			−0.983061	−	0.183276i	$-0.941330\pi$
$13$	−92.1564			−1.96612			−0.983061	+	0.183276i	$0.941330\pi$
$14$		0			0
$15$			37.5504i			0.646365i
$16$		0			0
$17$			118.220i			1.68662i	0.537430	+	0.843309i	$0.319395\pi$
$17$			118.220i			1.68662i	−0.537430	+	0.843309i	$0.680605\pi$
$18$		0			0
$19$		−	155.587i		−	1.87864i	−0.343041	−	0.939321i	$-0.611457\pi$
$19$		−	155.587i		−	1.87864i	0.343041	−	0.939321i	$-0.388543\pi$
$20$		0			0
$21$	8.66510	−	54.8809i	0.0900419	−	0.570286i
$22$		0			0
$23$		−	125.358i		−	1.13647i	−0.822865	−	0.568236i	$-0.807626\pi$
$23$		−	125.358i		−	1.13647i	0.822865	−	0.568236i	$-0.192374\pi$
$24$		0			0
$25$	31.6706			0.253365
$26$		0			0
$27$			27.0000i			0.192450i
$28$		0			0
$29$		−	131.643i		−	0.842947i	−0.906841	−	0.421474i	$-0.861513\pi$
$29$		−	131.643i		−	0.842947i	0.906841	−	0.421474i	$-0.138487\pi$
$30$		0			0
$31$	−66.0511			−0.382682			−0.191341	−	0.981524i	$-0.561284\pi$
$31$	−66.0511			−0.382682			−0.191341	+	0.981524i	$0.561284\pi$
$32$		0			0
$33$			166.202i			0.876731i
$34$		0			0
$35$	−228.978	−	36.1532i	−1.10584	−	0.174600i
$36$		0			0
$37$			147.117i			0.653671i	0.945081	+	0.326835i	$0.105982\pi$
$37$			147.117i			0.653671i	−0.945081	+	0.326835i	$0.894018\pi$
$38$		0			0
$39$			276.469i			1.13514i
$40$		0			0
$41$			20.3097i			0.0773622i	0.999252	+	0.0386811i	$0.0123156\pi$
$41$			20.3097i			0.0773622i	−0.999252	+	0.0386811i	$0.987684\pi$
$42$		0			0
$43$	355.210			1.25974			0.629872	−	0.776699i	$-0.283107\pi$
$43$	355.210			1.25974			0.629872	+	0.776699i	$0.283107\pi$
$44$		0			0
$45$	112.651			0.373179
$46$		0			0
$47$	−79.5977			−0.247032			−0.123516	−	0.992343i	$-0.539417\pi$
$47$	−79.5977			−0.247032			−0.123516	+	0.992343i	$0.539417\pi$
$48$		0			0
$49$	326.315	+	105.678i	0.951355	+	0.308098i
$50$		0			0
$51$	354.659			0.973769
$52$		0			0
$53$			463.012i			1.19999i	0.800003	+	0.599996i	$0.204831\pi$
$53$			463.012i			1.19999i	−0.800003	+	0.599996i	$0.795169\pi$
$54$		0			0
$55$	693.441			1.70007
$56$		0			0
$57$	−466.762			−1.08463
$58$		0			0
$59$			580.211i			1.28029i	0.768255	+	0.640144i	$0.221125\pi$
$59$			580.211i			1.28029i	−0.768255	+	0.640144i	$0.778875\pi$
$60$		0			0
$61$	−587.607			−1.23337			−0.616683	−	0.787212i	$-0.711524\pi$
$61$	−587.607			−1.23337			−0.616683	+	0.787212i	$0.711524\pi$
$62$		0			0
$63$	−164.643	−	25.9953i	−0.329255	−	0.0519857i
$64$		0			0
$65$	1153.50			2.20115
$66$		0			0
$67$	496.198			0.904779			0.452390	−	0.891820i	$-0.350572\pi$
$67$	496.198			0.904779			0.452390	+	0.891820i	$0.350572\pi$
$68$		0			0
$69$	−376.073			−0.656143
$70$		0			0
$71$			232.774i			0.389087i	0.980894	+	0.194543i	$0.0623225\pi$
$71$			232.774i			0.389087i	−0.980894	+	0.194543i	$0.937677\pi$
$72$		0			0
$73$		−	551.640i		−	0.884447i	−0.896905	−	0.442223i	$-0.854190\pi$
$73$		−	551.640i		−	0.884447i	0.896905	−	0.442223i	$-0.145810\pi$
$74$		0			0
$75$		−	95.0119i		−	0.146280i
$76$		0			0
$77$	−1013.48	−	160.018i	−1.49996	−	0.236828i
$78$		0			0
$79$		−	437.936i		−	0.623692i	−0.950133	−	0.311846i	$-0.899053\pi$
$79$		−	437.936i		−	0.623692i	0.950133	−	0.311846i	$-0.100947\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$		−	191.266i		−	0.252941i	−0.991970	−	0.126471i	$-0.959635\pi$
$83$		−	191.266i		−	0.252941i	0.991970	−	0.126471i	$-0.0403650\pi$
$84$		0			0
$85$		−	1479.73i		−	1.88823i
$86$		0			0
$87$	−394.929			−0.486676
$88$		0			0
$89$			93.7894i			0.111704i	0.998439	+	0.0558520i	$0.0177875\pi$
$89$			93.7894i			0.111704i	−0.998439	+	0.0558520i	$0.982213\pi$
$90$		0			0
$91$	−1685.88	−	266.182i	−1.94207	−	0.306631i
$92$		0			0
$93$			198.153i			0.220941i
$94$		0			0
$95$			1947.46i			2.10321i
$96$		0			0
$97$		−	758.892i		−	0.794369i	−0.917739	−	0.397184i	$-0.869987\pi$
$97$		−	758.892i		−	0.794369i	0.917739	−	0.397184i	$-0.130013\pi$
$98$		0			0
$99$	498.607			0.506181
$100$		0			0
$101$	803.819			0.791910			0.395955	−	0.918270i	$-0.370414\pi$
$101$	803.819			0.791910			0.395955	+	0.918270i	$0.370414\pi$
$102$		0			0
$103$	−946.857			−0.905792			−0.452896	−	0.891563i	$-0.649609\pi$
$103$	−946.857			−0.905792			−0.452896	+	0.891563i	$0.649609\pi$
$104$		0			0
$105$	−108.459	+	686.934i	−0.100805	+	0.638456i
$106$		0			0
$107$	1693.10			1.52970			0.764850	−	0.644208i	$-0.222813\pi$
$107$	1693.10			1.52970			0.764850	+	0.644208i	$0.222813\pi$
$108$		0			0
$109$		−	103.927i		−	0.0913251i	−0.998957	−	0.0456626i	$-0.985460\pi$
$109$		−	103.927i		−	0.0913251i	0.998957	−	0.0456626i	$-0.0145399\pi$
$110$		0			0
$111$	441.350			0.377397
$112$		0			0
$113$	−17.0288			−0.0141764			−0.00708819	−	0.999975i	$-0.502256\pi$
$113$	−17.0288			−0.0141764			−0.00708819	+	0.999975i	$0.502256\pi$
$114$		0			0
$115$			1569.08i			1.27232i
$116$		0			0
$117$	829.408			0.655374
$118$		0			0
$119$	−341.462	+	2162.67i	−0.263040	+	1.66598i
$120$		0			0
$121$	1738.25			1.30597
$122$		0			0
$123$	60.9292			0.0446651
$124$		0			0
$125$	1168.19			0.835886
$126$		0			0
$127$			481.156i			0.336187i	0.985771	+	0.168093i	$0.0537610\pi$
$127$			481.156i			0.336187i	−0.985771	+	0.168093i	$0.946239\pi$
$128$		0			0
$129$		−	1065.63i		−	0.727314i
$130$		0			0
$131$			1599.97i			1.06710i	0.845769	+	0.533550i	$0.179142\pi$
$131$			1599.97i			1.06710i	−0.845769	+	0.533550i	$0.820858\pi$
$132$		0			0
$133$	449.393	−	2846.26i	0.292988	−	1.85565i
$134$		0			0
$135$		−	337.954i		−	0.215455i
$136$		0			0
$137$	−867.738			−0.541138			−0.270569	−	0.962701i	$-0.587212\pi$
$137$	−867.738			−0.541138			−0.270569	+	0.962701i	$0.587212\pi$
$138$		0			0
$139$			773.391i			0.471929i	0.971762	+	0.235965i	$0.0758249\pi$
$139$			773.391i			0.471929i	−0.971762	+	0.235965i	$0.924175\pi$
$140$		0			0
$141$			238.793i			0.142624i
$142$		0			0
$143$	5105.54			2.98564
$144$		0			0
$145$			1647.75i			0.943711i
$146$		0			0
$147$	317.033	−	978.944i	0.177880	−	0.549265i
$148$		0			0
$149$			1566.08i			0.861063i	0.902576	+	0.430531i	$0.141674\pi$
$149$			1566.08i			0.861063i	−0.902576	+	0.430531i	$0.858326\pi$
$150$		0			0
$151$		−	168.569i		−	0.0908474i	−0.998968	−	0.0454237i	$-0.985536\pi$
$151$		−	168.569i		−	0.0908474i	0.998968	−	0.0454237i	$-0.0144638\pi$
$152$		0			0
$153$		−	1063.98i		−	0.562206i
$154$		0			0
$155$	826.750			0.428427
$156$		0			0
$157$	1378.04			0.700506			0.350253	−	0.936655i	$-0.386096\pi$
$157$	1378.04			0.700506			0.350253	+	0.936655i	$0.386096\pi$
$158$		0			0
$159$	1389.03			0.692815
$160$		0			0
$161$	362.079	−	2293.25i	0.177241	−	1.12257i
$162$		0			0
$163$	2078.67			0.998858			0.499429	−	0.866355i	$-0.333543\pi$
$163$	2078.67			0.998858			0.499429	+	0.866355i	$0.333543\pi$
$164$		0			0
$165$		−	2080.32i		−	0.981534i
$166$		0			0
$167$	−388.656			−0.180090			−0.0900452	−	0.995938i	$-0.528701\pi$
$167$	−388.656			−0.180090			−0.0900452	+	0.995938i	$0.528701\pi$
$168$		0			0
$169$	6295.81			2.86564
$170$		0			0
$171$			1400.29i			0.626214i
$172$		0			0
$173$	−2510.22			−1.10317			−0.551586	−	0.834118i	$-0.685977\pi$
$173$	−2510.22			−1.10317			−0.551586	+	0.834118i	$0.685977\pi$
$174$		0			0
$175$	579.371	+	91.4764i	0.250265	+	0.0395141i
$176$		0			0
$177$	1740.63			0.739175
$178$		0			0
$179$	1287.12			0.537450			0.268725	−	0.963217i	$-0.413398\pi$
$179$	1287.12			0.537450			0.268725	+	0.963217i	$0.413398\pi$
$180$		0			0
$181$	2880.43			1.18288			0.591438	−	0.806351i	$-0.298561\pi$
$181$	2880.43			1.18288			0.591438	+	0.806351i	$0.298561\pi$
$182$		0			0
$183$			1762.82i			0.712084i
$184$		0			0
$185$		−	1841.43i		−	0.731809i
$186$		0			0
$187$		−	6549.47i		−	2.56120i
$188$		0			0
$189$	−77.9859	+	493.928i	−0.0300140	+	0.190095i
$190$		0			0
$191$			1264.81i			0.479155i	0.970877	+	0.239577i	$0.0770089\pi$
$191$			1264.81i			0.479155i	−0.970877	+	0.239577i	$0.922991\pi$
$192$		0			0
$193$	3174.10			1.18382			0.591908	−	0.806005i	$-0.298375\pi$
$193$	3174.10			1.18382			0.591908	+	0.806005i	$0.298375\pi$
$194$		0			0
$195$		−	3460.51i		−	1.27083i
$196$		0			0
$197$			2153.82i			0.778951i	0.921037	+	0.389475i	$0.127344\pi$
$197$			2153.82i			0.778951i	−0.921037	+	0.389475i	$0.872656\pi$
$198$		0			0
$199$	−5051.78			−1.79955			−0.899776	−	0.436352i	$-0.856270\pi$
$199$	−5051.78			−1.79955			−0.899776	+	0.436352i	$0.856270\pi$
$200$		0			0
$201$		−	1488.59i		−	0.522375i
$202$		0			0
$203$	380.233	−	2408.23i	0.131464	−	0.832633i
$204$		0			0
$205$		−	254.213i		−	0.0866099i
$206$		0			0
$207$			1128.22i			0.378824i
$208$		0			0
$209$			8619.66i			2.85280i
$210$		0			0
$211$	1946.49			0.635080			0.317540	−	0.948245i	$-0.397143\pi$
$211$	1946.49			0.635080			0.317540	+	0.948245i	$0.397143\pi$
$212$		0			0
$213$	698.321			0.224639
$214$		0			0
$215$	−4446.10			−1.41033
$216$		0			0
$217$	−1208.32	−	190.780i	−0.377999	−	0.0596820i
$218$		0			0
$219$	−1654.92			−0.510636
$220$		0			0
$221$		−	10894.7i		−	3.31610i
$222$		0			0
$223$	901.888			0.270829			0.135415	−	0.990789i	$-0.456763\pi$
$223$	901.888			0.270829			0.135415	+	0.990789i	$0.456763\pi$
$224$		0			0
$225$	−285.036			−0.0844550
$226$		0			0
$227$		−	4975.14i		−	1.45468i	−0.686278	−	0.727339i	$-0.740757\pi$
$227$		−	4975.14i		−	1.45468i	0.686278	−	0.727339i	$-0.259243\pi$
$228$		0			0
$229$	−3689.85			−1.06477			−0.532384	−	0.846503i	$-0.678704\pi$
$229$	−3689.85			−1.06477			−0.532384	+	0.846503i	$0.678704\pi$
$230$		0			0
$231$	−480.054	+	3040.45i	−0.136733	+	0.866003i
$232$		0			0
$233$	3285.97			0.923909			0.461954	−	0.886904i	$-0.347148\pi$
$233$	3285.97			0.923909			0.461954	+	0.886904i	$0.347148\pi$
$234$		0			0
$235$	996.309			0.276562
$236$		0			0
$237$	−1313.81			−0.360089
$238$		0			0
$239$			2662.25i			0.720529i	0.932850	+	0.360264i	$0.117314\pi$
$239$			2662.25i			0.720529i	−0.932850	+	0.360264i	$0.882686\pi$
$240$		0			0
$241$			494.863i			0.132269i	0.997811	+	0.0661347i	$0.0210667\pi$
$241$			494.863i			0.132269i	−0.997811	+	0.0661347i	$0.978933\pi$
$242$		0			0
$243$		−	243.000i		−	0.0641500i
$244$		0			0
$245$	−4084.42	−	1322.75i	−1.06508	−	0.344927i
$246$		0			0
$247$			14338.4i			3.69364i
$248$		0			0
$249$	−573.797			−0.146036
$250$		0			0
$251$			4465.24i			1.12288i	0.827516	+	0.561442i	$0.189753\pi$
$251$			4465.24i			1.12288i	−0.827516	+	0.561442i	$0.810247\pi$
$252$		0			0
$253$			6944.91i			1.72578i
$254$		0			0
$255$	−4439.20			−1.09017
$256$		0			0
$257$		−	3874.34i		−	0.940369i	−0.882568	−	0.470185i	$-0.844187\pi$
$257$		−	3874.34i		−	0.940369i	0.882568	−	0.470185i	$-0.155813\pi$
$258$		0			0
$259$	−424.927	+	2691.30i	−0.101945	+	0.645672i
$260$		0			0
$261$			1184.79i			0.280982i
$262$		0			0
$263$		−	4291.91i		−	1.00628i	−0.864206	−	0.503138i	$-0.832179\pi$
$263$		−	4291.91i		−	1.00628i	0.864206	−	0.503138i	$-0.167821\pi$
$264$		0			0
$265$		−	5795.43i		−	1.34344i
$266$		0			0
$267$	281.368			0.0644923
$268$		0			0
$269$	−4934.08			−1.11835			−0.559175	−	0.829050i	$-0.688882\pi$
$269$	−4934.08			−1.11835			−0.559175	+	0.829050i	$0.688882\pi$
$270$		0			0
$271$	3796.02			0.850893			0.425446	−	0.904984i	$-0.360117\pi$
$271$	3796.02			0.850893			0.425446	+	0.904984i	$0.360117\pi$
$272$		0			0
$273$	−798.545	+	5057.63i	−0.177033	+	1.12125i
$274$		0			0
$275$	−1754.58			−0.384745
$276$		0			0
$277$		−	6822.56i		−	1.47988i	−0.672671	−	0.739942i	$-0.734853\pi$
$277$		−	6822.56i		−	1.47988i	0.672671	−	0.739942i	$-0.265147\pi$
$278$		0			0
$279$	594.460			0.127561
$280$		0			0
$281$	7537.89			1.60026			0.800129	−	0.599827i	$-0.204764\pi$
$281$	7537.89			1.60026			0.800129	+	0.599827i	$0.204764\pi$
$282$		0			0
$283$		−	1945.91i		−	0.408736i	−0.978894	−	0.204368i	$-0.934486\pi$
$283$		−	1945.91i		−	0.408736i	0.978894	−	0.204368i	$-0.0655139\pi$
$284$		0			0
$285$	5842.37			1.21429
$286$		0			0
$287$	−58.6620	+	371.539i	−0.0120652	+	0.0764155i
$288$		0			0
$289$	−9062.90			−1.84468
$290$		0			0
$291$	−2276.67			−0.458629
$292$		0			0
$293$	−4658.84			−0.928916			−0.464458	−	0.885595i	$-0.653751\pi$
$293$	−4658.84			−0.928916			−0.464458	+	0.885595i	$0.653751\pi$
$294$		0			0
$295$		−	7262.39i		−	1.43333i
$296$		0			0
$297$		−	1495.82i		−	0.292244i
$298$		0			0
$299$			11552.5i			2.23444i
$300$		0			0
$301$	6498.09	+	1025.98i	1.24433	+	0.196466i
$302$		0			0
$303$		−	2411.46i		−	0.457210i
$304$		0			0
$305$	7354.96			1.38080
$306$		0			0
$307$		−	3147.04i		−	0.585053i	−0.956257	−	0.292526i	$-0.905504\pi$
$307$		−	3147.04i		−	0.585053i	0.956257	−	0.292526i	$-0.0944959\pi$
$308$		0			0
$309$			2840.57i			0.522959i
$310$		0			0
$311$	9786.74			1.78442			0.892211	−	0.451619i	$-0.149153\pi$
$311$	9786.74			1.78442			0.892211	+	0.451619i	$0.149153\pi$
$312$		0			0
$313$		−	7053.24i		−	1.27372i	−0.770981	−	0.636858i	$-0.780234\pi$
$313$		−	7053.24i		−	1.27372i	0.770981	−	0.636858i	$-0.219766\pi$
$314$		0			0
$315$	2060.80	+	325.378i	0.368613	+	0.0582000i
$316$		0			0
$317$		−	7111.48i		−	1.26000i	−0.776595	−	0.630001i	$-0.783055\pi$
$317$		−	7111.48i		−	1.26000i	0.776595	−	0.630001i	$-0.216945\pi$
$318$		0			0
$319$			7293.12i			1.28005i
$320$		0			0
$321$		−	5079.29i		−	0.883173i
$322$		0			0
$323$	18393.5			3.16855
$324$		0			0
$325$	−2918.65			−0.498147
$326$		0			0
$327$	−311.782			−0.0527266
$328$		0			0
$329$	−1456.13	−	229.907i	−0.244009	−	0.0385265i
$330$		0			0
$331$	−4383.51			−0.727914			−0.363957	−	0.931416i	$-0.618575\pi$
$331$	−4383.51			−0.727914			−0.363957	+	0.931416i	$0.618575\pi$
$332$		0			0
$333$		−	1324.05i		−	0.217890i
$334$		0			0
$335$	−6210.82			−1.01293
$336$		0			0
$337$	3696.74			0.597549			0.298775	−	0.954324i	$-0.403422\pi$
$337$	3696.74			0.597549			0.298775	+	0.954324i	$0.403422\pi$
$338$		0			0
$339$			51.0863i			0.00818474i
$340$		0			0
$341$	3659.29			0.581119
$342$		0			0
$343$	5664.25	+	2875.74i	0.891664	+	0.452698i
$344$		0			0
$345$	4707.23			0.734577
$346$		0			0
$347$	−6007.60			−0.929408			−0.464704	−	0.885466i	$-0.653839\pi$
$347$	−6007.60			−0.929408			−0.464704	+	0.885466i	$0.653839\pi$
$348$		0			0
$349$	1376.99			0.211199			0.105600	−	0.994409i	$-0.466324\pi$
$349$	1376.99			0.211199			0.105600	+	0.994409i	$0.466324\pi$
$350$		0			0
$351$		−	2488.22i		−	0.378381i
$352$		0			0
$353$			3533.89i			0.532833i	0.963858	+	0.266417i	$0.0858397\pi$
$353$			3533.89i			0.532833i	−0.963858	+	0.266417i	$0.914160\pi$
$354$		0			0
$355$		−	2913.59i		−	0.435597i
$356$		0			0
$357$	6488.01	+	1024.39i	0.961854	+	0.151866i
$358$		0			0
$359$		−	4792.38i		−	0.704547i	−0.935897	−	0.352273i	$-0.885409\pi$
$359$		−	4792.38i		−	0.704547i	0.935897	−	0.352273i	$-0.114591\pi$
$360$		0			0
$361$	−17348.4			−2.52929
$362$		0			0
$363$		−	5214.75i		−	0.754003i
$364$		0			0
$365$			6904.78i			0.990172i
$366$		0			0
$367$	−10145.7			−1.44305			−0.721526	−	0.692387i	$-0.756559\pi$
$367$	−10145.7			−1.44305			−0.721526	+	0.692387i	$0.756559\pi$
$368$		0			0
$369$		−	182.788i		−	0.0257874i
$370$		0			0
$371$	−1337.35	+	8470.17i	−0.187147	+	1.18531i
$372$		0			0
$373$			8427.27i			1.16983i	0.811094	+	0.584916i	$0.198873\pi$
$373$			8427.27i			1.16983i	−0.811094	+	0.584916i	$0.801127\pi$
$374$		0			0
$375$		−	3504.56i		−	0.482599i
$376$		0			0
$377$			12131.7i			1.65734i
$378$		0			0
$379$	6309.37			0.855121			0.427560	−	0.903987i	$-0.359373\pi$
$379$	6309.37			0.855121			0.427560	+	0.903987i	$0.359373\pi$
$380$		0			0
$381$	1443.47			0.194098
$382$		0			0
$383$	6889.51			0.919158			0.459579	−	0.888137i	$-0.348000\pi$
$383$	6889.51			0.919158			0.459579	+	0.888137i	$0.348000\pi$
$384$		0			0
$385$	12685.6	+	2002.91i	1.67926	+	0.265138i
$386$		0			0
$387$	−3196.89			−0.419915
$388$		0			0
$389$			3025.39i			0.394327i	0.980371	+	0.197164i	$0.0631730\pi$
$389$			3025.39i			0.394327i	−0.980371	+	0.197164i	$0.936827\pi$
$390$		0			0
$391$	14819.7			1.91679
$392$		0			0
$393$	4799.91			0.616090
$394$		0			0
$395$			5481.56i			0.698247i
$396$		0			0
$397$	−1565.33			−0.197889			−0.0989444	−	0.995093i	$-0.531547\pi$
$397$	−1565.33			−0.197889			−0.0989444	+	0.995093i	$0.531547\pi$
$398$		0			0
$399$	−8538.78	−	1348.18i	−1.07136	−	0.169156i
$400$		0			0
$401$	2687.03			0.334623			0.167312	−	0.985904i	$-0.446491\pi$
$401$	2687.03			0.334623			0.167312	+	0.985904i	$0.446491\pi$
$402$		0			0
$403$	6087.04			0.752399
$404$		0			0
$405$	−1013.86			−0.124393
$406$		0			0
$407$		−	8150.38i		−	0.992627i
$408$		0			0
$409$			11421.7i			1.38084i	0.723407	+	0.690422i	$0.242575\pi$
$409$			11421.7i			1.38084i	−0.723407	+	0.690422i	$0.757425\pi$
$410$		0			0
$411$			2603.21i			0.312426i
$412$		0			0
$413$	−1675.86	+	10614.2i	−0.199670	+	1.26462i
$414$		0			0
$415$			2394.04i			0.283178i
$416$		0			0
$417$	2320.17			0.272468
$418$		0			0
$419$			5192.62i			0.605432i	0.953081	+	0.302716i	$0.0978934\pi$
$419$			5192.62i			0.605432i	−0.953081	+	0.302716i	$0.902107\pi$
$420$		0			0
$421$		−	11047.1i		−	1.27887i	−0.768846	−	0.639434i	$-0.779169\pi$
$421$		−	11047.1i		−	1.27887i	0.768846	−	0.639434i	$-0.220831\pi$
$422$		0			0
$423$	716.379			0.0823441
$424$		0			0
$425$			3744.09i			0.427330i
$426$		0			0
$427$	−10749.5	−	1697.22i	−1.21827	−	0.192352i
$428$		0			0
$429$		−	15316.6i		−	1.72376i
$430$		0			0
$431$			7402.32i			0.827279i	0.910441	+	0.413640i	$0.135743\pi$
$431$			7402.32i			0.827279i	−0.910441	+	0.413640i	$0.864257\pi$
$432$		0			0
$433$			304.730i			0.0338208i	0.999857	+	0.0169104i	$0.00538300\pi$
$433$			304.730i			0.0338208i	−0.999857	+	0.0169104i	$0.994617\pi$
$434$		0			0
$435$	4943.25			0.544852
$436$		0			0
$437$	−19504.1			−2.13502
$438$		0			0
$439$	−7752.82			−0.842875			−0.421437	−	0.906857i	$-0.638474\pi$
$439$	−7752.82			−0.842875			−0.421437	+	0.906857i	$0.638474\pi$
$440$		0			0
$441$	−2936.83	−	951.098i	−0.317118	−	0.102699i
$442$		0			0
$443$	−14980.0			−1.60660			−0.803298	−	0.595577i	$-0.796923\pi$
$443$	−14980.0			−1.60660			−0.803298	+	0.595577i	$0.796923\pi$
$444$		0			0
$445$		−	1173.94i		−	0.125057i
$446$		0			0
$447$	4698.24			0.497135
$448$		0			0
$449$	−17523.6			−1.84185			−0.920925	−	0.389740i	$-0.872565\pi$
$449$	−17523.6			−1.84185			−0.920925	+	0.389740i	$0.872565\pi$
$450$		0			0
$451$		−	1125.18i		−	0.117478i
$452$		0			0
$453$	−505.707			−0.0524508
$454$		0			0
$455$	21101.8	+	3331.75i	2.17422	+	0.343285i
$456$		0			0
$457$	−3264.38			−0.334139			−0.167069	−	0.985945i	$-0.553430\pi$
$457$	−3264.38			−0.334139			−0.167069	+	0.985945i	$0.553430\pi$
$458$		0			0
$459$	−3191.93			−0.324590
$460$		0			0
$461$	4096.53			0.413870			0.206935	−	0.978355i	$-0.433651\pi$
$461$	4096.53			0.413870			0.206935	+	0.978355i	$0.433651\pi$
$462$		0			0
$463$		−	12281.3i		−	1.23275i	−0.787455	−	0.616373i	$-0.788602\pi$
$463$		−	12281.3i		−	1.23275i	0.787455	−	0.616373i	$-0.211398\pi$
$464$		0			0
$465$		−	2480.25i		−	0.247352i
$466$		0			0
$467$			14919.7i			1.47838i	0.673499	+	0.739188i	$0.264791\pi$
$467$			14919.7i			1.47838i	−0.673499	+	0.739188i	$0.735209\pi$
$468$		0			0
$469$	9077.26	+	1433.20i	0.893708	+	0.141107i
$470$		0			0
$471$		−	4134.12i		−	0.404438i
$472$		0			0
$473$	−19678.9			−1.91298
$474$		0			0
$475$		−	4927.55i		−	0.475982i
$476$		0			0
$477$		−	4167.10i		−	0.399997i
$478$		0			0
$479$	3253.39			0.310337			0.155168	−	0.987888i	$-0.450408\pi$
$479$	3253.39			0.310337			0.155168	+	0.987888i	$0.450408\pi$
$480$		0			0
$481$		−	13557.7i		−	1.28520i
$482$		0			0
$483$	−6879.74	−	1086.24i	−0.648114	−	0.102330i
$484$		0			0
$485$			9498.90i			0.889326i
$486$		0			0
$487$			367.609i			0.0342053i	0.999854	+	0.0171026i	$0.00544420\pi$
$487$			367.609i			0.0342053i	−0.999854	+	0.0171026i	$0.994556\pi$
$488$		0			0
$489$		−	6236.01i		−	0.576691i
$490$		0			0
$491$	−8656.19			−0.795618			−0.397809	−	0.917468i	$-0.630229\pi$
$491$	−8656.19			−0.795618			−0.397809	+	0.917468i	$0.630229\pi$
$492$		0			0
$493$	15562.8			1.42173
$494$		0			0
$495$	−6240.97			−0.566689
$496$		0			0
$497$	−672.336	+	4258.28i	−0.0606809	+	0.384326i
$498$		0			0
$499$	14675.5			1.31656			0.658280	−	0.752773i	$-0.271284\pi$
$499$	14675.5			1.31656			0.658280	+	0.752773i	$0.271284\pi$
$500$		0			0
$501$			1165.97i			0.103975i
$502$		0			0
$503$	4451.92			0.394635			0.197317	−	0.980340i	$-0.436777\pi$
$503$	4451.92			0.394635			0.197317	+	0.980340i	$0.436777\pi$
$504$		0			0
$505$	−10061.2			−0.886573
$506$		0			0
$507$		−	18887.4i		−	1.65448i
$508$		0			0
$509$	−17750.3			−1.54572			−0.772858	−	0.634579i	$-0.781174\pi$
$509$	−17750.3			−1.54572			−0.772858	+	0.634579i	$0.781174\pi$
$510$		0			0
$511$	1593.34	−	10091.5i	0.137936	−	0.873624i
$512$		0			0
$513$	4200.86			0.361545
$514$		0			0
$515$	11851.6			1.01407
$516$		0			0
$517$	4409.78			0.375129
$518$		0			0
$519$			7530.67i			0.636916i
$520$		0			0
$521$		−	8202.49i		−	0.689746i	−0.938649	−	0.344873i	$-0.887922\pi$
$521$		−	8202.49i		−	0.689746i	0.938649	−	0.344873i	$-0.112078\pi$
$522$		0			0
$523$			14583.3i			1.21928i	0.792678	+	0.609641i	$0.208686\pi$
$523$			14583.3i			1.21928i	−0.792678	+	0.609641i	$0.791314\pi$
$524$		0			0
$525$	274.429	−	1738.11i	0.0228135	−	0.144490i
$526$		0			0
$527$		−	7808.55i		−	0.645438i
$528$		0			0
$529$	−3547.53			−0.291570
$530$		0			0
$531$		−	5221.90i		−	0.426763i
$532$		0			0
$533$		−	1871.67i		−	0.152104i
$534$		0			0
$535$	−21192.2			−1.71256
$536$		0			0
$537$		−	3861.35i		−	0.310297i
$538$		0			0
$539$	−18078.1	−	5854.62i	−1.44467	−	0.467860i
$540$		0			0
$541$			18146.0i			1.44207i	0.692900	+	0.721033i	$0.256333\pi$
$541$			18146.0i			1.44207i	−0.692900	+	0.721033i	$0.743667\pi$
$542$		0			0
$543$		−	8641.28i		−	0.682933i
$544$		0			0
$545$			1300.84i			0.102242i
$546$		0			0
$547$	−10223.9			−0.799164			−0.399582	−	0.916698i	$-0.630845\pi$
$547$	−10223.9			−0.799164			−0.399582	+	0.916698i	$0.630845\pi$
$548$		0			0
$549$	5288.46			0.411122
$550$		0			0
$551$	−20482.0			−1.58360
$552$		0			0
$553$	1264.92	−	8011.44i	0.0972692	−	0.616060i
$554$		0			0
$555$	−5524.29			−0.422510
$556$		0			0
$557$			18894.8i			1.43734i	0.695352	+	0.718669i	$0.255248\pi$
$557$			18894.8i			1.43734i	−0.695352	+	0.718669i	$0.744752\pi$
$558$		0			0
$559$	−32734.9			−2.47681
$560$		0			0
$561$	−19648.4			−1.47871
$562$		0			0
$563$			4811.00i			0.360141i	0.983654	+	0.180071i	$0.0576326\pi$
$563$			4811.00i			0.360141i	−0.983654	+	0.180071i	$0.942367\pi$
$564$		0			0
$565$	213.146			0.0158710
$566$		0			0
$567$	1481.78	+	233.958i	0.109752	+	0.0173286i
$568$		0			0
$569$	15811.2			1.16492			0.582460	−	0.812859i	$-0.302090\pi$
$569$	15811.2			1.16492			0.582460	+	0.812859i	$0.302090\pi$
$570$		0			0
$571$	−12377.3			−0.907131			−0.453566	−	0.891223i	$-0.649848\pi$
$571$	−12377.3			−0.907131			−0.453566	+	0.891223i	$0.649848\pi$
$572$		0			0
$573$	3794.43			0.276640
$574$		0			0
$575$		−	3970.15i		−	0.287942i
$576$		0			0
$577$			17225.6i			1.24283i	0.783484	+	0.621413i	$0.213441\pi$
$577$			17225.6i			1.24283i	−0.783484	+	0.621413i	$0.786559\pi$
$578$		0			0
$579$		−	9522.30i		−	0.683477i
$580$		0			0
$581$	552.446	−	3498.95i	0.0394480	−	0.249846i
$582$		0			0
$583$		−	25651.2i		−	1.82224i
$584$		0			0
$585$	−10381.5			−0.733716
$586$		0			0
$587$			17524.8i			1.23224i	0.787651	+	0.616121i	$0.211297\pi$
$587$			17524.8i			1.23224i	−0.787651	+	0.616121i	$0.788703\pi$
$588$		0			0
$589$			10276.7i			0.718922i
$590$		0			0
$591$	6461.46			0.449727
$592$		0			0
$593$			17080.1i			1.18279i	0.806382	+	0.591394i	$0.201422\pi$
$593$			17080.1i			1.18279i	−0.806382	+	0.591394i	$0.798578\pi$
$594$		0			0
$595$	4274.02	−	27069.7i	0.294483	−	1.86513i
$596$		0			0
$597$			15155.3i			1.03897i
$598$		0			0
$599$		−	18593.0i		−	1.26826i	−0.773225	−	0.634132i	$-0.781358\pi$
$599$		−	18593.0i		−	1.26826i	0.773225	−	0.634132i	$-0.218642\pi$
$600$		0			0
$601$			22600.9i			1.53396i	0.641670	+	0.766981i	$0.278242\pi$
$601$			22600.9i			1.53396i	−0.641670	+	0.766981i	$0.721758\pi$
$602$		0			0
$603$	−4465.78			−0.301593
$604$		0			0
$605$	−21757.3			−1.46209
$606$		0			0
$607$	−7870.14			−0.526259			−0.263130	−	0.964761i	$-0.584755\pi$
$607$	−7870.14			−0.526259			−0.263130	+	0.964761i	$0.584755\pi$
$608$		0			0
$609$	−7224.68	−	1140.70i	−0.480721	−	0.0759006i
$610$		0			0
$611$	7335.44			0.485696
$612$		0			0
$613$		−	5829.16i		−	0.384074i	−0.981388	−	0.192037i	$-0.938491\pi$
$613$		−	5829.16i		−	0.384074i	0.981388	−	0.192037i	$-0.0615094\pi$
$614$		0			0
$615$	−762.640			−0.0500042
$616$		0			0
$617$	3132.87			0.204416			0.102208	−	0.994763i	$-0.467409\pi$
$617$	3132.87			0.204416			0.102208	+	0.994763i	$0.467409\pi$
$618$		0			0
$619$			1967.51i			0.127756i	0.997958	+	0.0638779i	$0.0203468\pi$
$619$			1967.51i			0.127756i	−0.997958	+	0.0638779i	$0.979653\pi$
$620$		0			0
$621$	3384.66			0.218714
$622$		0			0
$623$	−270.898	+	1715.75i	−0.0174210	+	0.110337i
$624$		0			0
$625$	−18580.8			−1.18917
$626$		0			0
$627$	25859.0			1.64706
$628$		0			0
$629$	−17392.1			−1.10249
$630$		0			0
$631$			12821.2i			0.808883i	0.914564	+	0.404441i	$0.132534\pi$
$631$			12821.2i			0.808883i	−0.914564	+	0.404441i	$0.867466\pi$
$632$		0			0
$633$		−	5839.47i		−	0.366664i
$634$		0			0
$635$		−	6022.54i		−	0.376374i
$636$		0			0
$637$	−30072.0	−	9738.86i	−1.87048	−	0.605758i
$638$		0			0
$639$		−	2094.96i		−	0.129696i
$640$		0			0
$641$	7100.25			0.437509			0.218754	−	0.975780i	$-0.429801\pi$
$641$	7100.25			0.437509			0.218754	+	0.975780i	$0.429801\pi$
$642$		0			0
$643$		−	22717.2i		−	1.39328i	−0.717421	−	0.696640i	$-0.754677\pi$
$643$		−	22717.2i		−	1.39328i	0.717421	−	0.696640i	$-0.245323\pi$
$644$		0			0
$645$			13338.3i			0.814256i
$646$		0			0
$647$	333.262			0.0202502			0.0101251	−	0.999949i	$-0.496777\pi$
$647$	333.262			0.0202502			0.0101251	+	0.999949i	$0.496777\pi$
$648$		0			0
$649$		−	32144.1i		−	1.94417i
$650$		0			0
$651$	−572.340	+	3624.95i	−0.0344574	+	0.218238i
$652$		0			0
$653$			27002.5i			1.61821i	0.587666	+	0.809103i	$0.300047\pi$
$653$			27002.5i			1.61821i	−0.587666	+	0.809103i	$0.699953\pi$
$654$		0			0
$655$		−	20026.5i		−	1.19466i
$656$		0			0
$657$			4964.76i			0.294816i
$658$		0			0
$659$	−17790.1			−1.05160			−0.525799	−	0.850609i	$-0.676234\pi$
$659$	−17790.1			−1.05160			−0.525799	+	0.850609i	$0.676234\pi$
$660$		0			0
$661$	28922.6			1.70190			0.850951	−	0.525246i	$-0.176027\pi$
$661$	28922.6			1.70190			0.850951	+	0.525246i	$0.176027\pi$
$662$		0			0
$663$	−32684.1			−1.91455
$664$		0			0
$665$	−5624.97	+	35626.1i	−0.328011	+	2.07747i
$666$		0			0
$667$	−16502.4			−0.957987
$668$		0			0
$669$		−	2705.66i		−	0.156363i
$670$		0			0
$671$	32553.9			1.87292
$672$		0			0
$673$	−9655.76			−0.553049			−0.276525	−	0.961007i	$-0.589183\pi$
$673$	−9655.76			−0.553049			−0.276525	+	0.961007i	$0.589183\pi$
$674$		0			0
$675$			855.107i			0.0487601i
$676$		0			0
$677$	−97.6916			−0.00554593			−0.00277296	−	0.999996i	$-0.500883\pi$
$677$	−97.6916			−0.00554593			−0.00277296	+	0.999996i	$0.500883\pi$
$678$		0			0
$679$	2191.96	−	13882.9i	0.123888	−	0.784649i
$680$		0			0
$681$	−14925.4			−0.839859
$682$		0			0
$683$	9136.04			0.511831			0.255916	−	0.966699i	$-0.417623\pi$
$683$	9136.04			0.511831			0.255916	+	0.966699i	$0.417623\pi$
$684$		0			0
$685$	10861.3			0.605824
$686$		0			0
$687$			11069.5i			0.614744i
$688$		0			0
$689$		−	42669.5i		−	2.35933i
$690$		0			0
$691$		−	20090.0i		−	1.10602i	−0.833174	−	0.553011i	$-0.813479\pi$
$691$		−	20090.0i		−	1.10602i	0.833174	−	0.553011i	$-0.186521\pi$
$692$		0			0
$693$	9121.34	+	1440.16i	0.499987	+	0.0789426i
$694$		0			0
$695$		−	9680.39i		−	0.528342i
$696$		0			0
$697$	−2401.01			−0.130480
$698$		0			0
$699$		−	9857.90i		−	0.533419i
$700$		0			0
$701$		−	28733.6i		−	1.54815i	−0.633095	−	0.774074i	$-0.718216\pi$
$701$		−	28733.6i		−	1.54815i	0.633095	−	0.774074i	$-0.281784\pi$
$702$		0			0
$703$	22889.5			1.22801
$704$		0			0
$705$		−	2988.93i		−	0.159673i
$706$		0			0
$707$	14704.8	+	2321.72i	0.782220	+	0.123504i
$708$		0			0
$709$		−	1873.43i		−	0.0992360i	−0.998768	−	0.0496180i	$-0.984200\pi$
$709$		−	1873.43i		−	0.0992360i	0.998768	−	0.0496180i	$-0.0158004\pi$
$710$		0			0
$711$			3941.42i			0.207897i
$712$		0			0
$713$			8280.01i			0.434907i
$714$		0			0
$715$	−63905.1			−3.34254
$716$		0			0
$717$	7986.74			0.415998
$718$		0			0
$719$	−2073.02			−0.107525			−0.0537625	−	0.998554i	$-0.517121\pi$
$719$	−2073.02			−0.107525			−0.0537625	+	0.998554i	$0.517121\pi$
$720$		0			0
$721$	−17321.5	−	2734.87i	−0.894709	−	0.141265i
$722$		0			0
$723$	1484.59			0.0763658
$724$		0			0
$725$		−	4169.21i		−	0.213573i
$726$		0			0
$727$	26274.3			1.34038			0.670191	−	0.742188i	$-0.266212\pi$
$727$	26274.3			1.34038			0.670191	+	0.742188i	$0.266212\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$			41992.8i			2.12471i
$732$		0			0
$733$	18164.4			0.915304			0.457652	−	0.889131i	$-0.348691\pi$
$733$	18164.4			0.915304			0.457652	+	0.889131i	$0.348691\pi$
$734$		0			0
$735$	−3968.24	+	12253.3i	−0.199144	+	0.614923i
$736$		0			0
$737$	−27489.8			−1.37395
$738$		0			0
$739$	9438.47			0.469824			0.234912	−	0.972017i	$-0.424520\pi$
$739$	9438.47			0.469824			0.234912	+	0.972017i	$0.424520\pi$
$740$		0			0
$741$	43015.1			2.13252
$742$		0			0
$743$			8531.76i			0.421265i	0.977565	+	0.210633i	$0.0675524\pi$
$743$			8531.76i			0.421265i	−0.977565	+	0.210633i	$0.932448\pi$
$744$		0			0
$745$		−	19602.3i		−	0.963992i
$746$		0			0
$747$			1721.39i			0.0843138i
$748$		0			0
$749$	30972.9	+	4890.29i	1.51098	+	0.238568i
$750$		0			0
$751$			20258.1i			0.984325i	0.870503	+	0.492163i	$0.163794\pi$
$751$			20258.1i			0.984325i	−0.870503	+	0.492163i	$0.836206\pi$
$752$		0			0
$753$	13395.7			0.648297
$754$		0			0
$755$			2109.95i			0.101707i
$756$		0			0
$757$		−	30742.9i		−	1.47605i	−0.674772	−	0.738026i	$-0.735758\pi$
$757$		−	30742.9i		−	1.47605i	0.674772	−	0.738026i	$-0.264242\pi$
$758$		0			0
$759$	20834.7			0.996381
$760$		0			0
$761$			22602.8i			1.07667i	0.842730	+	0.538337i	$0.180947\pi$
$761$			22602.8i			1.07667i	−0.842730	+	0.538337i	$0.819053\pi$
$762$		0			0
$763$	300.181	−	1901.21i	0.0142428	−	0.0902077i
$764$		0			0
$765$			13317.6i			0.629411i
$766$		0			0
$767$		−	53470.2i		−	2.51720i
$768$		0			0
$769$			6329.77i			0.296823i	0.988926	+	0.148412i	$0.0474161\pi$
$769$			6329.77i			0.296823i	−0.988926	+	0.148412i	$0.952584\pi$
$770$		0			0
$771$	−11623.0			−0.542922
$772$		0			0
$773$	13945.2			0.648867			0.324433	−	0.945909i	$-0.394826\pi$
$773$	13945.2			0.648867			0.324433	+	0.945909i	$0.394826\pi$
$774$		0			0
$775$	−2091.88			−0.0969581
$776$		0			0
$777$	8073.89	+	1274.78i	0.372779	+	0.0588578i
$778$		0			0
$779$	3159.94			0.145336
$780$		0			0
$781$		−	12895.9i		−	0.590845i
$782$		0			0
$783$	3554.36			0.162225
$784$		0			0
$785$	−17248.7			−0.784243
$786$		0			0
$787$			38929.4i			1.76326i	0.471946	+	0.881628i	$0.343552\pi$
$787$			38929.4i			1.76326i	−0.471946	+	0.881628i	$0.656448\pi$
$788$		0			0
$789$	−12875.7			−0.580973
$790$		0			0
$791$	−311.518	−	49.1853i	−0.0140029	−	0.00221091i
$792$		0			0
$793$	54151.7			2.42495
$794$		0			0
$795$	−17386.3			−0.775633
$796$		0			0
$797$	5612.71			0.249451			0.124726	−	0.992191i	$-0.460195\pi$
$797$	5612.71			0.249451			0.124726	+	0.992191i	$0.460195\pi$
$798$		0			0
$799$		−	9410.02i		−	0.416649i
$800$		0			0
$801$		−	844.104i		−	0.0372346i
$802$		0			0
$803$			30561.3i			1.34307i
$804$		0			0
$805$	−4532.07	+	28704.2i	−0.198428	+	1.25676i
$806$		0			0
$807$			14802.2i			0.645679i
$808$		0			0
$809$	−13074.7			−0.568210			−0.284105	−	0.958793i	$-0.591696\pi$
$809$	−13074.7			−0.568210			−0.284105	+	0.958793i	$0.591696\pi$
$810$		0			0
$811$		−	3773.85i		−	0.163400i	−0.996657	−	0.0817002i	$-0.973965\pi$
$811$		−	3773.85i		−	0.163400i	0.996657	−	0.0817002i	$-0.0260350\pi$
$812$		0			0
$813$		−	11388.1i		−	0.491263i
$814$		0			0
$815$	−26018.3			−1.11826
$816$		0			0
$817$		−	55266.2i		−	2.36661i
$818$		0			0
$819$	15172.9	+	2395.64i	0.647355	+	0.102210i
$820$		0			0
$821$		−	15494.5i		−	0.658661i	−0.944215	−	0.329331i	$-0.893177\pi$
$821$		−	15494.5i		−	0.658661i	0.944215	−	0.329331i	$-0.106823\pi$
$822$		0			0
$823$		−	32765.2i		−	1.38776i	−0.720092	−	0.693879i	$-0.755900\pi$
$823$		−	32765.2i		−	1.38776i	0.720092	−	0.693879i	$-0.244100\pi$
$824$		0			0
$825$			5263.73i			0.222133i
$826$		0			0
$827$	−8067.05			−0.339200			−0.169600	−	0.985513i	$-0.554248\pi$
$827$	−8067.05			−0.339200			−0.169600	+	0.985513i	$0.554248\pi$
$828$		0			0
$829$	19574.3			0.820078			0.410039	−	0.912068i	$-0.365515\pi$
$829$	19574.3			0.820078			0.410039	+	0.912068i	$0.365515\pi$
$830$		0			0
$831$	−20467.7			−0.854411
$832$		0			0
$833$	−12493.2	+	38576.8i	−0.519643	+	1.60457i
$834$		0			0
$835$	4864.74			0.201618
$836$		0			0
$837$		−	1783.38i		−	0.0736471i
$838$		0			0
$839$	41602.1			1.71187			0.855937	−	0.517079i	$-0.172981\pi$
$839$	41602.1			1.71187			0.855937	+	0.517079i	$0.172981\pi$
$840$		0			0
$841$	7059.15			0.289440
$842$		0			0
$843$		−	22613.7i		−	0.923910i
$844$		0			0
$845$	−78803.5			−3.20819
$846$		0			0
$847$	31798.9	+	5020.70i	1.28999	+	0.203676i
$848$		0			0
$849$	−5837.73			−0.235984
$850$		0			0
$851$	18442.2			0.742879
$852$		0			0
$853$	21967.3			0.881765			0.440882	−	0.897565i	$-0.354666\pi$
$853$	21967.3			0.881765			0.440882	+	0.897565i	$0.354666\pi$
$854$		0			0
$855$		−	17527.1i		−	0.701070i
$856$		0			0
$857$			40486.3i			1.61375i	0.590720	+	0.806877i	$0.298844\pi$
$857$			40486.3i			1.61375i	−0.590720	+	0.806877i	$0.701156\pi$
$858$		0			0
$859$			26419.8i			1.04940i	0.851289	+	0.524698i	$0.175822\pi$
$859$			26419.8i			1.04940i	−0.851289	+	0.524698i	$0.824178\pi$
$860$		0			0
$861$	1114.62	+	175.986i	0.0441185	+	0.00696584i
$862$		0			0
$863$			31789.9i			1.25393i	0.779048	+	0.626964i	$0.215703\pi$
$863$			31789.9i			1.25393i	−0.779048	+	0.626964i	$0.784297\pi$
$864$		0			0
$865$	31420.0			1.23504
$866$		0			0
$867$			27188.7i			1.06503i
$868$		0			0
$869$			24262.0i			0.947103i
$870$		0			0
$871$	−45727.8			−1.77891
$872$		0			0
$873$			6830.02i			0.264790i
$874$		0			0
$875$	21370.4	+	3374.15i	0.825658	+	0.130362i
$876$		0			0
$877$			8034.36i			0.309351i	0.987965	+	0.154676i	$0.0494332\pi$
$877$			8034.36i			0.309351i	−0.987965	+	0.154676i	$0.950567\pi$
$878$		0			0
$879$			13976.5i			0.536310i
$880$		0			0
$881$		−	17579.3i		−	0.672262i	−0.941815	−	0.336131i	$-0.890882\pi$
$881$		−	17579.3i		−	0.672262i	0.941815	−	0.336131i	$-0.109118\pi$
$882$		0			0
$883$	31775.9			1.21104			0.605518	−	0.795832i	$-0.292966\pi$
$883$	31775.9			1.21104			0.605518	+	0.795832i	$0.292966\pi$
$884$		0			0
$885$	−21787.2			−0.827534
$886$		0			0
$887$	−6353.28			−0.240498			−0.120249	−	0.992744i	$-0.538369\pi$
$887$	−6353.28			−0.240498			−0.120249	+	0.992744i	$0.538369\pi$
$888$		0			0
$889$	−1389.76	+	8802.10i	−0.0524308	+	0.332073i
$890$		0			0
$891$	−4487.46			−0.168727
$892$		0			0
$893$			12384.4i			0.464085i
$894$		0			0
$895$	−16110.6			−0.601696
$896$		0			0
$897$	34657.5			1.29006
$898$		0			0
$899$			8695.16i			0.322581i
$900$		0			0
$901$	−54737.1			−2.02393
$902$		0			0
$903$	3077.93	−	19494.3i	0.113430	−	0.718414i
$904$		0			0
$905$	−36053.8			−1.32427
$906$		0			0
$907$	21510.5			0.787480			0.393740	−	0.919222i	$-0.371181\pi$
$907$	21510.5			0.787480			0.393740	+	0.919222i	$0.371181\pi$
$908$		0			0
$909$	−7234.37			−0.263970
$910$		0			0
$911$		−	14005.7i		−	0.509363i	−0.967025	−	0.254681i	$-0.918029\pi$
$911$		−	14005.7i		−	0.509363i	0.967025	−	0.254681i	$-0.0819706\pi$
$912$		0			0
$913$			10596.3i			0.384102i
$914$		0			0
$915$		−	22064.9i		−	0.797205i
$916$		0			0
$917$	−4621.30	+	29269.3i	−0.166422	+	1.05404i
$918$		0			0
$919$		−	17108.7i		−	0.614106i	−0.951692	−	0.307053i	$-0.900657\pi$
$919$		−	17108.7i		−	0.614106i	0.951692	−	0.307053i	$-0.0993428\pi$
$920$		0			0
$921$	−9441.12			−0.337780
$922$		0			0
$923$		−	21451.6i		−	0.764993i
$924$		0			0
$925$			4659.27i			0.165617i
$926$		0			0
$927$	8521.71			0.301931
$928$		0			0
$929$		−	50366.6i		−	1.77877i	−0.457162	−	0.889384i	$-0.651134\pi$
$929$		−	50366.6i		−	1.77877i	0.457162	−	0.889384i	$-0.348866\pi$
$930$		0			0
$931$	16442.1	−	50770.4i	0.578805	−	1.78725i
$932$		0			0
$933$		−	29360.2i		−	1.03024i
$934$		0			0
$935$			81978.5i			2.86736i
$936$		0			0
$937$			34447.3i			1.20101i	0.799621	+	0.600505i	$0.205034\pi$
$937$			34447.3i			1.20101i	−0.799621	+	0.600505i	$0.794966\pi$
$938$		0			0
$939$	−21159.7			−0.735380
$940$		0			0
$941$	−21746.7			−0.753372			−0.376686	−	0.926341i	$-0.622936\pi$
$941$	−21746.7			−0.753372			−0.376686	+	0.926341i	$0.622936\pi$
$942$		0			0
$943$	2545.98			0.0879200
$944$		0			0
$945$	976.135	−	6182.41i	0.0336018	−	0.212819i
$946$		0			0
$947$	20179.9			0.692460			0.346230	−	0.938150i	$-0.387462\pi$
$947$	20179.9			0.692460			0.346230	+	0.938150i	$0.387462\pi$
$948$		0			0
$949$			50837.2i			1.73893i
$950$		0			0
$951$	−21334.4			−0.727462
$952$		0			0
$953$	−49322.8			−1.67652			−0.838259	−	0.545272i	$-0.816426\pi$
$953$	−49322.8			−1.67652			−0.838259	+	0.545272i	$0.816426\pi$
$954$		0			0
$955$		−	15831.4i		−	0.536432i
$956$		0			0
$957$	21879.4			0.739038
$958$		0			0
$959$	−15874.1	−	2506.35i	−0.534516	−	0.0843943i
$960$		0			0
$961$	−25428.2			−0.853555
$962$		0			0
$963$	−15237.9			−0.509900
$964$		0			0
$965$	−39729.6			−1.32533
$966$		0			0
$967$			41458.2i			1.37870i	0.724427	+	0.689352i	$0.242105\pi$
$967$			41458.2i			1.37870i	−0.724427	+	0.689352i	$0.757895\pi$
$968$		0			0
$969$		−	55180.5i		−	1.82936i
$970$		0			0
$971$		−	19762.1i		−	0.653136i	−0.945174	−	0.326568i	$-0.894108\pi$
$971$		−	19762.1i		−	0.653136i	0.945174	−	0.326568i	$-0.105892\pi$
$972$		0			0
$973$	−2233.84	+	14148.1i	−0.0736007	+	0.466154i
$974$		0			0
$975$			8755.95i			0.287605i
$976$		0			0
$977$	59693.7			1.95473			0.977365	−	0.211561i	$-0.0678547\pi$
$977$	59693.7			1.95473			0.977365	+	0.211561i	$0.0678547\pi$
$978$		0			0
$979$		−	5196.01i		−	0.169627i
$980$		0			0
$981$			935.347i			0.0304417i
$982$		0			0
$983$	−6590.24			−0.213831			−0.106916	−	0.994268i	$-0.534097\pi$
$983$	−6590.24			−0.213831			−0.106916	+	0.994268i	$0.534097\pi$
$984$		0			0
$985$		−	26959.0i		−	0.872065i
$986$		0			0
$987$	−689.722	+	4368.39i	−0.0222433	+	0.140879i
$988$		0			0
$989$		−	44528.3i		−	1.43167i
$990$		0			0
$991$		−	8611.31i		−	0.276031i	−0.990430	−	0.138016i	$-0.955928\pi$
$991$		−	8611.31i		−	0.276031i	0.990430	−	0.138016i	$-0.0440724\pi$
$992$		0			0
$993$			13150.5i			0.420261i
$994$		0			0
$995$	63232.1			2.01467
$996$		0			0
$997$	−13666.7			−0.434130			−0.217065	−	0.976157i	$-0.569648\pi$
$997$	−13666.7			−0.434130			−0.217065	+	0.976157i	$0.569648\pi$
$998$		0			0
$999$	−3972.15			−0.125799

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.p.c.223.1	✓	32
4.3	odd	2	inner	1344.4.p.c.223.8	yes	32
7.6	odd	2		1344.4.p.d.223.2	yes	32
8.3	odd	2		1344.4.p.d.223.1	yes	32
8.5	even	2		1344.4.p.d.223.32	yes	32
28.27	even	2		1344.4.p.d.223.31	yes	32
56.13	odd	2	inner	1344.4.p.c.223.7	yes	32
56.27	even	2	inner	1344.4.p.c.223.2	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.p.c.223.1	✓	32	1.1	even	1	trivial
1344.4.p.c.223.2	yes	32	56.27	even	2	inner
1344.4.p.c.223.7	yes	32	56.13	odd	2	inner
1344.4.p.c.223.8	yes	32	4.3	odd	2	inner
1344.4.p.d.223.1	yes	32	8.3	odd	2
1344.4.p.d.223.2	yes	32	7.6	odd	2
1344.4.p.d.223.31	yes	32	28.27	even	2
1344.4.p.d.223.32	yes	32	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -12.5168 q^{5} +(18.2936 + 2.88837i) q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -12.5168 q^{5} +(18.2936 + 2.88837i) q^{7} -9.00000 q^{9} -55.4008 q^{11} -92.1564 q^{13} +37.5504i q^{15} +118.220i q^{17} -155.587i q^{19} +(8.66510 - 54.8809i) q^{21} -125.358i q^{23} +31.6706 q^{25} +27.0000i q^{27} -131.643i q^{29} -66.0511 q^{31} +166.202i q^{33} +(-228.978 - 36.1532i) q^{35} +147.117i q^{37} +276.469i q^{39} +20.3097i q^{41} +355.210 q^{43} +112.651 q^{45} -79.5977 q^{47} +(326.315 + 105.678i) q^{49} +354.659 q^{51} +463.012i q^{53} +693.441 q^{55} -466.762 q^{57} +580.211i q^{59} -587.607 q^{61} +(-164.643 - 25.9953i) q^{63} +1153.50 q^{65} +496.198 q^{67} -376.073 q^{69} +232.774i q^{71} -551.640i q^{73} -95.0119i q^{75} +(-1013.48 - 160.018i) q^{77} -437.936i q^{79} +81.0000 q^{81} -191.266i q^{83} -1479.73i q^{85} -394.929 q^{87} +93.7894i q^{89} +(-1685.88 - 266.182i) q^{91} +198.153i q^{93} +1947.46i q^{95} -758.892i q^{97} +498.607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 288 q^{9}+O(q^{10}) \) 32 * q - 288 * q^9	\( 32 q - 288 q^{9} - 224 q^{13} - 72 q^{21} + 1120 q^{25} - 752 q^{49} - 672 q^{57} - 544 q^{61} + 1536 q^{65} - 144 q^{69} - 1632 q^{77} + 2592 q^{81}+O(q^{100}) \) 32 * q - 288 * q^9 - 224 * q^13 - 72 * q^21 + 1120 * q^25 - 752 * q^49 - 672 * q^57 - 544 * q^61 + 1536 * q^65 - 144 * q^69 - 1632 * q^77 + 2592 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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