LMFDB - Embedded newform 1200.3.c.k.449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(449,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1200.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$-1.14412 - 1.14412i$ of defining polynomial
Character	$\chi$	$=$	1200.449
Dual form			1200.3.c.k.449.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.52896 - 2.58114i) q^{3} -7.48683i q^{7} +(-4.32456 + 7.89292i) q^{9} +O(q^{10})$	$q+(-1.52896 - 2.58114i) q^{3} -7.48683i q^{7} +(-4.32456 + 7.89292i) q^{9} +8.48528i q^{11} -10.0000i q^{13} +30.3870 q^{17} +26.9737 q^{19} +(-19.3246 + 11.4471i) q^{21} -9.17377 q^{23} +(26.9848 - 0.905694i) q^{27} -26.8328i q^{29} -8.00000 q^{31} +(21.9017 - 12.9737i) q^{33} -15.9473i q^{37} +(-25.8114 + 15.2896i) q^{39} -47.3575i q^{41} +14.4605i q^{43} +45.8688 q^{47} -7.05267 q^{49} +(-46.4605 - 78.4330i) q^{51} -30.3870 q^{53} +(-41.2417 - 69.6228i) q^{57} +24.0789i q^{59} -53.9473 q^{61} +(59.0930 + 32.3772i) q^{63} -110.460i q^{67} +(14.0263 + 23.6788i) q^{69} +15.5936i q^{71} +87.9473i q^{73} +63.5279 q^{77} -46.9737 q^{79} +(-43.5964 - 68.2668i) q^{81} -26.1443 q^{83} +(-69.2592 + 41.0263i) q^{87} +60.7739i q^{89} -74.8683 q^{91} +(12.2317 + 20.6491i) q^{93} -36.0527i q^{97} +(-66.9737 - 36.6951i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{9}+O(q^{10}) $ 8 * q + 16 * q^9	$ 8 q + 16 q^{9} + 64 q^{19} - 104 q^{21} - 64 q^{31} - 80 q^{39} - 360 q^{49} - 144 q^{51} - 128 q^{61} + 264 q^{69} - 224 q^{79} + 56 q^{81} + 160 q^{91} - 384 q^{99}+O(q^{100}) $ 8 * q + 16 * q^9 + 64 * q^19 - 104 * q^21 - 64 * q^31 - 80 * q^39 - 360 * q^49 - 144 * q^51 - 128 * q^61 + 264 * q^69 - 224 * q^79 + 56 * q^81 + 160 * q^91 - 384 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−1.52896	−	2.58114i	−0.509654	−	0.860380i
$3$	−1.52896	−	2.58114i	−0.509654	−	0.860380i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	7.48683i		−	1.06955i	−0.844995	−	0.534774i	$-0.820397\pi$
$7$		−	7.48683i		−	1.06955i	0.844995	−	0.534774i	$-0.179603\pi$
$8$		0			0
$9$	−4.32456	+	7.89292i	−0.480506	+	0.876991i
$10$		0			0
$11$			8.48528i			0.771389i	0.922627	+	0.385695i	$0.126038\pi$
$11$			8.48528i			0.771389i	−0.922627	+	0.385695i	$0.873962\pi$
$12$		0			0
$13$		−	10.0000i		−	0.769231i	−0.923077	−	0.384615i	$-0.874334\pi$
$13$		−	10.0000i		−	0.769231i	0.923077	−	0.384615i	$-0.125666\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	30.3870			1.78747			0.893734	−	0.448596i	$-0.148076\pi$
$17$	30.3870			1.78747			0.893734	+	0.448596i	$0.148076\pi$
$18$		0			0
$19$	26.9737			1.41967			0.709833	−	0.704370i	$-0.248770\pi$
$19$	26.9737			1.41967			0.709833	+	0.704370i	$0.248770\pi$
$20$		0			0
$21$	−19.3246	+	11.4471i	−0.920217	+	0.545099i
$22$		0			0
$23$	−9.17377			−0.398859			−0.199430	−	0.979912i	$-0.563909\pi$
$23$	−9.17377			−0.398859			−0.199430	+	0.979912i	$0.563909\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	26.9848	−	0.905694i	0.999437	−	0.0335442i
$28$		0			0
$29$		−	26.8328i		−	0.925270i	−0.886549	−	0.462635i	$-0.846904\pi$
$29$		−	26.8328i		−	0.925270i	0.886549	−	0.462635i	$-0.153096\pi$
$30$		0			0
$31$	−8.00000			−0.258065			−0.129032	−	0.991640i	$-0.541187\pi$
$31$	−8.00000			−0.258065			−0.129032	+	0.991640i	$0.541187\pi$
$32$		0			0
$33$	21.9017	−	12.9737i	0.663688	−	0.393141i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	15.9473i		−	0.431009i	−0.976503	−	0.215504i	$-0.930860\pi$
$37$		−	15.9473i		−	0.431009i	0.976503	−	0.215504i	$-0.0691396\pi$
$38$		0			0
$39$	−25.8114	+	15.2896i	−0.661830	+	0.392041i
$40$		0			0
$41$		−	47.3575i		−	1.15506i	−0.816369	−	0.577531i	$-0.804016\pi$
$41$		−	47.3575i		−	1.15506i	0.816369	−	0.577531i	$-0.195984\pi$
$42$		0			0
$43$			14.4605i			0.336291i	0.985762	+	0.168145i	$0.0537778\pi$
$43$			14.4605i			0.336291i	−0.985762	+	0.168145i	$0.946222\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	45.8688			0.975933			0.487966	−	0.872862i	$-0.337739\pi$
$47$	45.8688			0.975933			0.487966	+	0.872862i	$0.337739\pi$
$48$		0			0
$49$	−7.05267			−0.143932
$50$		0			0
$51$	−46.4605	−	78.4330i	−0.910990	−	1.53790i
$52$		0			0
$53$	−30.3870			−0.573339			−0.286670	−	0.958030i	$-0.592548\pi$
$53$	−30.3870			−0.573339			−0.286670	+	0.958030i	$0.592548\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−41.2417	−	69.6228i	−0.723538	−	1.22145i
$58$		0			0
$59$			24.0789i			0.408116i	0.978959	+	0.204058i	$0.0654132\pi$
$59$			24.0789i			0.408116i	−0.978959	+	0.204058i	$0.934587\pi$
$60$		0			0
$61$	−53.9473			−0.884382			−0.442191	−	0.896921i	$-0.645799\pi$
$61$	−53.9473			−0.884382			−0.442191	+	0.896921i	$0.645799\pi$
$62$		0			0
$63$	59.0930	+	32.3772i	0.937984	+	0.513924i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	110.460i		−	1.64866i	−0.566107	−	0.824332i	$-0.691551\pi$
$67$		−	110.460i		−	1.64866i	0.566107	−	0.824332i	$-0.308449\pi$
$68$		0			0
$69$	14.0263	+	23.6788i	0.203280	+	0.343171i
$70$		0			0
$71$			15.5936i			0.219628i	0.993952	+	0.109814i	$0.0350255\pi$
$71$			15.5936i			0.219628i	−0.993952	+	0.109814i	$0.964974\pi$
$72$		0			0
$73$			87.9473i			1.20476i	0.798210	+	0.602379i	$0.205780\pi$
$73$			87.9473i			1.20476i	−0.798210	+	0.602379i	$0.794220\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	63.5279			0.825037
$78$		0			0
$79$	−46.9737			−0.594603			−0.297302	−	0.954784i	$-0.596087\pi$
$79$	−46.9737			−0.594603			−0.297302	+	0.954784i	$0.596087\pi$
$80$		0			0
$81$	−43.5964	−	68.2668i	−0.538228	−	0.842799i
$82$		0			0
$83$	−26.1443			−0.314992			−0.157496	−	0.987520i	$-0.550342\pi$
$83$	−26.1443			−0.314992			−0.157496	+	0.987520i	$0.550342\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−69.2592	+	41.0263i	−0.796083	+	0.471567i
$88$		0			0
$89$			60.7739i			0.682853i	0.939908	+	0.341427i	$0.110910\pi$
$89$			60.7739i			0.682853i	−0.939908	+	0.341427i	$0.889090\pi$
$90$		0			0
$91$	−74.8683			−0.822729
$92$		0			0
$93$	12.2317	+	20.6491i	0.131524	+	0.222033i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	36.0527i		−	0.371677i	−0.982580	−	0.185838i	$-0.940500\pi$
$97$		−	36.0527i		−	0.371677i	0.982580	−	0.185838i	$-0.0595001\pi$
$98$		0			0
$99$	−66.9737	−	36.6951i	−0.676502	−	0.370657i
$100$		0			0
$101$			48.1577i			0.476809i	0.971166	+	0.238405i	$0.0766245\pi$
$101$			48.1577i			0.476809i	−0.971166	+	0.238405i	$0.923376\pi$
$102$		0			0
$103$		−	140.408i		−	1.36318i	−0.731733	−	0.681591i	$-0.761288\pi$
$103$		−	140.408i		−	1.36318i	0.731733	−	0.681591i	$-0.238712\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−43.1149			−0.402943			−0.201471	−	0.979494i	$-0.564572\pi$
$107$	−43.1149			−0.402943			−0.201471	+	0.979494i	$0.564572\pi$
$108$		0			0
$109$	−133.842			−1.22791			−0.613954	−	0.789342i	$-0.710422\pi$
$109$	−133.842			−1.22791			−0.613954	+	0.789342i	$0.710422\pi$
$110$		0			0
$111$	−41.1623	+	24.3829i	−0.370831	+	0.219665i
$112$		0			0
$113$	7.90852			0.0699869			0.0349935	−	0.999388i	$-0.488859\pi$
$113$	7.90852			0.0699869			0.0349935	+	0.999388i	$0.488859\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	78.9292	+	43.2456i	0.674609	+	0.369620i
$118$		0			0
$119$		−	227.502i		−	1.91178i
$120$		0			0
$121$	49.0000			0.404959
$122$		0			0
$123$	−122.236	+	72.4078i	−0.993792	+	0.588682i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	134.460i		−	1.05874i	−0.848390	−	0.529372i	$-0.822428\pi$
$127$		−	134.460i		−	1.05874i	0.848390	−	0.529372i	$-0.177572\pi$
$128$		0			0
$129$	37.3246	−	22.1095i	0.289338	−	0.171392i
$130$		0			0
$131$		−	220.394i		−	1.68240i	−0.540727	−	0.841198i	$-0.681851\pi$
$131$		−	220.394i		−	1.68240i	0.540727	−	0.841198i	$-0.318149\pi$
$132$		0			0
$133$		−	201.947i		−	1.51840i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−95.5153			−0.697192			−0.348596	−	0.937273i	$-0.613341\pi$
$137$	−95.5153			−0.697192			−0.348596	+	0.937273i	$0.613341\pi$
$138$		0			0
$139$	−76.8157			−0.552631			−0.276315	−	0.961067i	$-0.589113\pi$
$139$	−76.8157			−0.552631			−0.276315	+	0.961067i	$0.589113\pi$
$140$		0			0
$141$	−70.1317	−	118.394i	−0.497388	−	0.839673i
$142$		0			0
$143$	84.8528			0.593376
$144$		0			0
$145$		0			0
$146$		0			0
$147$	10.7833	+	18.2039i	0.0733555	+	0.123836i
$148$		0			0
$149$		−	276.237i		−	1.85394i	−0.375139	−	0.926969i	$-0.622405\pi$
$149$		−	276.237i		−	1.85394i	0.375139	−	0.926969i	$-0.377595\pi$
$150$		0			0
$151$	−18.0527			−0.119554			−0.0597770	−	0.998212i	$-0.519039\pi$
$151$	−18.0527			−0.119554			−0.0597770	+	0.998212i	$0.519039\pi$
$152$		0			0
$153$	−131.410	+	239.842i	−0.858890	+	1.56759i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	103.842i		−	0.661414i	−0.943733	−	0.330707i	$-0.892713\pi$
$157$		−	103.842i		−	0.661414i	0.943733	−	0.330707i	$-0.107287\pi$
$158$		0			0
$159$	46.4605	+	78.4330i	0.292204	+	0.493289i
$160$		0			0
$161$			68.6825i			0.426599i
$162$		0			0
$163$			11.3815i			0.0698251i	0.999390	+	0.0349126i	$0.0111153\pi$
$163$			11.3815i			0.0698251i	−0.999390	+	0.0349126i	$0.988885\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	252.270			1.51060			0.755298	−	0.655382i	$-0.227492\pi$
$167$	252.270			1.51060			0.755298	+	0.655382i	$0.227492\pi$
$168$		0			0
$169$	69.0000			0.408284
$170$		0			0
$171$	−116.649	+	212.901i	−0.682159	+	1.24504i
$172$		0			0
$173$	11.8160			0.0683005			0.0341502	−	0.999417i	$-0.489128\pi$
$173$	11.8160			0.0683005			0.0341502	+	0.999417i	$0.489128\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	62.1509	−	36.8157i	0.351135	−	0.207998i
$178$		0			0
$179$			69.0358i			0.385675i	0.981231	+	0.192837i	$0.0617690\pi$
$179$			69.0358i			0.385675i	−0.981231	+	0.192837i	$0.938231\pi$
$180$		0			0
$181$	−189.684			−1.04798			−0.523989	−	0.851725i	$-0.675557\pi$
$181$	−189.684			−1.04798			−0.523989	+	0.851725i	$0.675557\pi$
$182$		0			0
$183$	82.4834	+	139.246i	0.450729	+	0.760905i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			257.842i			1.37883i
$188$		0			0
$189$	−6.78078	−	202.031i	−0.0358771	−	1.06895i
$190$		0			0
$191$			108.708i			0.569153i	0.958653	+	0.284577i	$0.0918530\pi$
$191$			108.708i			0.569153i	−0.958653	+	0.284577i	$0.908147\pi$
$192$		0			0
$193$		−	167.947i		−	0.870193i	−0.900384	−	0.435097i	$-0.856714\pi$
$193$		−	167.947i		−	0.870193i	0.900384	−	0.435097i	$-0.143286\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−171.659			−0.871367			−0.435684	−	0.900100i	$-0.643493\pi$
$197$	−171.659			−0.871367			−0.435684	+	0.900100i	$0.643493\pi$
$198$		0			0
$199$	35.0790			0.176276			0.0881382	−	0.996108i	$-0.471908\pi$
$199$	35.0790			0.176276			0.0881382	+	0.996108i	$0.471908\pi$
$200$		0			0
$201$	−285.114	+	168.890i	−1.41848	+	0.840248i
$202$		0			0
$203$	−200.893			−0.989620
$204$		0			0
$205$		0			0
$206$		0			0
$207$	39.6725	−	72.4078i	0.191654	−	0.349796i
$208$		0			0
$209$			228.879i			1.09512i
$210$		0			0
$211$	58.1580			0.275630			0.137815	−	0.990458i	$-0.455992\pi$
$211$	58.1580			0.275630			0.137815	+	0.990458i	$0.455992\pi$
$212$		0			0
$213$	40.2492	−	23.8420i	0.188963	−	0.111934i
$214$		0			0
$215$		0			0
$216$		0			0
$217$			59.8947i			0.276012i
$218$		0			0
$219$	227.004	−	134.468i	1.03655	−	0.614009i
$220$		0			0
$221$		−	303.870i		−	1.37498i
$222$		0			0
$223$		−	99.3815i		−	0.445657i	−0.974858	−	0.222828i	$-0.928471\pi$
$223$		−	99.3815i		−	0.445657i	0.974858	−	0.222828i	$-0.0715290\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−216.951			−0.955733			−0.477867	−	0.878432i	$-0.658590\pi$
$227$	−216.951			−0.955733			−0.477867	+	0.878432i	$0.658590\pi$
$228$		0			0
$229$	−325.684			−1.42220			−0.711100	−	0.703090i	$-0.751803\pi$
$229$	−325.684			−1.42220			−0.711100	+	0.703090i	$0.751803\pi$
$230$		0			0
$231$	−97.1317	−	163.974i	−0.420483	−	0.709845i
$232$		0			0
$233$	−51.7119			−0.221939			−0.110970	−	0.993824i	$-0.535396\pi$
$233$	−51.7119			−0.221939			−0.110970	+	0.993824i	$0.535396\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	71.8209	+	121.246i	0.303042	+	0.511585i
$238$		0			0
$239$			410.047i			1.71568i	0.513917	+	0.857840i	$0.328194\pi$
$239$			410.047i			1.71568i	−0.513917	+	0.857840i	$0.671806\pi$
$240$		0			0
$241$	445.526			1.84866			0.924328	−	0.381599i	$-0.124627\pi$
$241$	445.526			1.84866			0.924328	+	0.381599i	$0.124627\pi$
$242$		0			0
$243$	−109.549	+	216.906i	−0.450818	+	0.892616i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	269.737i		−	1.09205i
$248$		0			0
$249$	39.9737	+	67.4821i	0.160537	+	0.271013i
$250$		0			0
$251$		−	237.364i		−	0.945675i	−0.881150	−	0.472838i	$-0.843230\pi$
$251$		−	237.364i		−	0.945675i	0.881150	−	0.472838i	$-0.156770\pi$
$252$		0			0
$253$		−	77.8420i		−	0.307676i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	318.887			1.24080			0.620402	−	0.784284i	$-0.286970\pi$
$257$	318.887			1.24080			0.620402	+	0.784284i	$0.286970\pi$
$258$		0			0
$259$	−119.395			−0.460985
$260$		0			0
$261$	211.789	+	116.040i	0.811453	+	0.444598i
$262$		0			0
$263$	36.2300			0.137757			0.0688784	−	0.997625i	$-0.478058\pi$
$263$	36.2300			0.137757			0.0688784	+	0.997625i	$0.478058\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	156.866	−	92.9210i	0.587513	−	0.348019i
$268$		0			0
$269$			528.041i			1.96298i	0.191518	+	0.981489i	$0.438659\pi$
$269$			528.041i			1.96298i	−0.191518	+	0.981489i	$0.561341\pi$
$270$		0			0
$271$	475.895			1.75607			0.878034	−	0.478597i	$-0.158855\pi$
$271$	475.895			1.75607			0.878034	+	0.478597i	$0.158855\pi$
$272$		0			0
$273$	114.471	+	193.246i	0.419307	+	0.707859i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	188.158i		−	0.679271i	−0.940557	−	0.339635i	$-0.889696\pi$
$277$		−	188.158i		−	0.679271i	0.940557	−	0.339635i	$-0.110304\pi$
$278$		0			0
$279$	34.5964	−	63.1434i	0.124002	−	0.226320i
$280$		0			0
$281$			24.4322i			0.0869473i	0.999055	+	0.0434736i	$0.0138424\pi$
$281$			24.4322i			0.0869473i	−0.999055	+	0.0434736i	$0.986158\pi$
$282$		0			0
$283$		−	198.460i		−	0.701274i	−0.936511	−	0.350637i	$-0.885965\pi$
$283$		−	198.460i		−	0.701274i	0.936511	−	0.350637i	$-0.114035\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−354.558			−1.23539
$288$		0			0
$289$	634.368			2.19504
$290$		0			0
$291$	−93.0569	+	55.1231i	−0.319783	+	0.189427i
$292$		0			0
$293$	513.825			1.75367			0.876834	−	0.480794i	$-0.159651\pi$
$293$	513.825			1.75367			0.876834	+	0.480794i	$0.159651\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	7.68507	+	228.974i	0.0258757	+	0.770955i
$298$		0			0
$299$			91.7377i			0.306815i
$300$		0			0
$301$	108.263			0.359679
$302$		0			0
$303$	124.302	−	73.6313i	0.410237	−	0.243008i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	11.3815i		−	0.0370733i	−0.999828	−	0.0185366i	$-0.994099\pi$
$307$		−	11.3815i		−	0.0370733i	0.999828	−	0.0185366i	$-0.00590073\pi$
$308$		0			0
$309$	−362.412	+	214.678i	−1.17285	+	0.694751i
$310$		0			0
$311$			518.756i			1.66802i	0.551746	+	0.834012i	$0.313962\pi$
$311$			518.756i			1.66802i	−0.551746	+	0.834012i	$0.686038\pi$
$312$		0			0
$313$		−	46.3160i		−	0.147974i	−0.997259	−	0.0739872i	$-0.976428\pi$
$313$		−	46.3160i		−	0.147974i	0.997259	−	0.0739872i	$-0.0235724\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	39.0957			0.123330			0.0616651	−	0.998097i	$-0.480359\pi$
$317$	39.0957			0.123330			0.0616651	+	0.998097i	$0.480359\pi$
$318$		0			0
$319$	227.684			0.713743
$320$		0			0
$321$	65.9210	+	111.286i	0.205361	+	0.346684i
$322$		0			0
$323$	819.648			2.53761
$324$		0			0
$325$		0			0
$326$		0			0
$327$	204.639	+	345.465i	0.625808	+	1.05647i
$328$		0			0
$329$		−	343.412i		−	1.04381i
$330$		0			0
$331$	445.421			1.34568			0.672841	−	0.739787i	$-0.265074\pi$
$331$	445.421			1.34568			0.672841	+	0.739787i	$0.265074\pi$
$332$		0			0
$333$	125.871	+	68.9651i	0.377991	+	0.207102i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	325.684i		−	0.966421i	−0.875504	−	0.483211i	$-0.839471\pi$
$337$		−	325.684i		−	0.966421i	0.875504	−	0.483211i	$-0.160529\pi$
$338$		0			0
$339$	−12.0918	−	20.4130i	−0.0356691	−	0.0602153i
$340$		0			0
$341$		−	67.8823i		−	0.199068i
$342$		0			0
$343$		−	314.053i		−	0.915605i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	51.8236			0.149348			0.0746738	−	0.997208i	$-0.476208\pi$
$347$	51.8236			0.149348			0.0746738	+	0.997208i	$0.476208\pi$
$348$		0			0
$349$	97.5787			0.279595			0.139798	−	0.990180i	$-0.455355\pi$
$349$	97.5787			0.279595			0.139798	+	0.990180i	$0.455355\pi$
$350$		0			0
$351$	−9.05694	−	269.848i	−0.0258033	−	0.768798i
$352$		0			0
$353$	−569.797			−1.61416			−0.807078	−	0.590445i	$-0.798952\pi$
$353$	−569.797			−1.61416			−0.807078	+	0.590445i	$0.798952\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−587.215	+	347.842i	−1.64486	+	0.974347i
$358$		0			0
$359$		−	274.283i		−	0.764019i	−0.924158	−	0.382010i	$-0.875232\pi$
$359$		−	274.283i		−	0.764019i	0.924158	−	0.382010i	$-0.124768\pi$
$360$		0			0
$361$	366.579			1.01545
$362$		0			0
$363$	−74.9191	−	126.476i	−0.206389	−	0.348418i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			461.828i			1.25839i	0.777248	+	0.629194i	$0.216615\pi$
$367$			461.828i			1.25839i	−0.777248	+	0.629194i	$0.783385\pi$
$368$		0			0
$369$	373.789	+	204.800i	1.01298	+	0.555014i
$370$		0			0
$371$			227.502i			0.613213i
$372$		0			0
$373$		−	491.947i		−	1.31889i	−0.751751	−	0.659447i	$-0.770791\pi$
$373$		−	491.947i		−	1.31889i	0.751751	−	0.659447i	$-0.229209\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−268.328			−0.711746
$378$		0			0
$379$	−258.763			−0.682752			−0.341376	−	0.939927i	$-0.610893\pi$
$379$	−258.763			−0.682752			−0.341376	+	0.939927i	$0.610893\pi$
$380$		0			0
$381$	−347.061	+	205.585i	−0.910922	+	0.539593i
$382$		0			0
$383$	−522.422			−1.36402			−0.682012	−	0.731341i	$-0.738895\pi$
$383$	−522.422			−1.36402			−0.682012	+	0.731341i	$0.738895\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−114.136	−	62.5352i	−0.294924	−	0.161590i
$388$		0			0
$389$		−	610.847i		−	1.57030i	−0.619306	−	0.785150i	$-0.712586\pi$
$389$		−	610.847i		−	1.57030i	0.619306	−	0.785150i	$-0.287414\pi$
$390$		0			0
$391$	−278.763			−0.712949
$392$		0			0
$393$	−568.867	+	336.974i	−1.44750	+	0.857439i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			214.000i			0.539043i	0.962994	+	0.269521i	$0.0868655\pi$
$397$			214.000i			0.539043i	−0.962994	+	0.269521i	$0.913135\pi$
$398$		0			0
$399$	−521.254	+	308.770i	−1.30640	+	0.773859i
$400$		0			0
$401$			454.557i			1.13356i	0.823869	+	0.566780i	$0.191811\pi$
$401$			454.557i			1.13356i	−0.823869	+	0.566780i	$0.808189\pi$
$402$		0			0
$403$			80.0000i			0.198511i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	135.318			0.332476
$408$		0			0
$409$	573.842			1.40304			0.701518	−	0.712651i	$-0.252506\pi$
$409$	573.842			1.40304			0.701518	+	0.712651i	$0.252506\pi$
$410$		0			0
$411$	146.039	+	246.538i	0.355326	+	0.599850i
$412$		0			0
$413$	180.274			0.436500
$414$		0			0
$415$		0			0
$416$		0			0
$417$	117.448	+	198.272i	0.281650	+	0.475472i
$418$		0			0
$419$			97.9159i			0.233690i	0.993150	+	0.116845i	$0.0372780\pi$
$419$			97.9159i			0.233690i	−0.993150	+	0.116845i	$0.962722\pi$
$420$		0			0
$421$	717.315			1.70384			0.851918	−	0.523675i	$-0.175439\pi$
$421$	717.315			1.70384			0.851918	+	0.523675i	$0.175439\pi$
$422$		0			0
$423$	−198.362	+	362.039i	−0.468942	+	0.855885i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			403.895i			0.945889i
$428$		0			0
$429$	−129.737	−	219.017i	−0.302416	−	0.510529i
$430$		0			0
$431$		−	293.077i		−	0.679994i	−0.940427	−	0.339997i	$-0.889574\pi$
$431$		−	293.077i		−	0.679994i	0.940427	−	0.339997i	$-0.110426\pi$
$432$		0			0
$433$			487.526i			1.12593i	0.826482	+	0.562963i	$0.190339\pi$
$433$			487.526i			1.12593i	−0.826482	+	0.562963i	$0.809661\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−247.450			−0.566247
$438$		0			0
$439$	257.237			0.585961			0.292981	−	0.956118i	$-0.405353\pi$
$439$	257.237			0.585961			0.292981	+	0.956118i	$0.405353\pi$
$440$		0			0
$441$	30.4997	−	55.6662i	0.0691602	−	0.126227i
$442$		0			0
$443$	−293.096			−0.661615			−0.330808	−	0.943698i	$-0.607321\pi$
$443$	−293.096			−0.661615			−0.330808	+	0.943698i	$0.607321\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−713.005	+	422.355i	−1.59509	+	0.944866i
$448$		0			0
$449$			585.614i			1.30426i	0.758106	+	0.652132i	$0.226125\pi$
$449$			585.614i			1.30426i	−0.758106	+	0.652132i	$0.773875\pi$
$450$		0			0
$451$	401.842			0.891002
$452$		0			0
$453$	27.6018	+	46.5964i	0.0609312	+	0.102862i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			813.052i			1.77911i	0.456831	+	0.889554i	$0.348984\pi$
$457$			813.052i			1.77911i	−0.456831	+	0.889554i	$0.651016\pi$
$458$		0			0
$459$	819.986	−	27.5213i	1.78646	−	0.0599593i
$460$		0			0
$461$			554.074i			1.20190i	0.799288	+	0.600948i	$0.205210\pi$
$461$			554.074i			1.20190i	−0.799288	+	0.600948i	$0.794790\pi$
$462$		0			0
$463$			449.723i			0.971324i	0.874147	+	0.485662i	$0.161421\pi$
$463$			449.723i			0.971324i	−0.874147	+	0.485662i	$0.838579\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	30.7221			0.0657862			0.0328931	−	0.999459i	$-0.489528\pi$
$467$	30.7221			0.0657862			0.0328931	+	0.999459i	$0.489528\pi$
$468$		0			0
$469$	−826.999			−1.76332
$470$		0			0
$471$	−268.031	+	158.770i	−0.569067	+	0.337092i
$472$		0			0
$473$	−122.701			−0.259411
$474$		0			0
$475$		0			0
$476$		0			0
$477$	131.410	−	239.842i	0.275493	−	0.502813i
$478$		0			0
$479$		−	735.242i		−	1.53495i	−0.641078	−	0.767476i	$-0.721512\pi$
$479$		−	735.242i		−	1.53495i	0.641078	−	0.767476i	$-0.278488\pi$
$480$		0			0
$481$	−159.473			−0.331545
$482$		0			0
$483$	177.279	−	105.013i	0.367037	−	0.217418i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	92.6185i		−	0.190182i	−0.995469	−	0.0950909i	$-0.969686\pi$
$487$		−	92.6185i		−	0.190182i	0.995469	−	0.0950909i	$-0.0303142\pi$
$488$		0			0
$489$	29.3772	−	17.4019i	0.0600761	−	0.0355866i
$490$		0			0
$491$			898.323i			1.82958i	0.403933	+	0.914789i	$0.367643\pi$
$491$			898.323i			1.82958i	−0.403933	+	0.914789i	$0.632357\pi$
$492$		0			0
$493$		−	815.368i		−	1.65389i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	116.747			0.234903
$498$		0			0
$499$	136.921			0.274391			0.137195	−	0.990544i	$-0.456191\pi$
$499$	136.921			0.274391			0.137195	+	0.990544i	$0.456191\pi$
$500$		0			0
$501$	−385.710	−	651.143i	−0.769881	−	1.29969i
$502$		0			0
$503$	−443.077			−0.880868			−0.440434	−	0.897785i	$-0.645175\pi$
$503$	−443.077			−0.880868			−0.440434	+	0.897785i	$0.645175\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−105.498	−	178.099i	−0.208083	−	0.351279i
$508$		0			0
$509$		−	213.062i		−	0.418590i	−0.977853	−	0.209295i	$-0.932883\pi$
$509$		−	213.062i		−	0.418590i	0.977853	−	0.209295i	$-0.0671168\pi$
$510$		0			0
$511$	658.447			1.28855
$512$		0			0
$513$	727.879	−	24.4299i	1.41887	−	0.0476216i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			389.210i			0.752824i
$518$		0			0
$519$	−18.0662	−	30.4987i	−0.0348096	−	0.0587643i
$520$		0			0
$521$			3.20085i			0.00614366i	0.999995	+	0.00307183i	$0.000977795\pi$
$521$			3.20085i			0.00614366i	−0.999995	+	0.00307183i	$0.999022\pi$
$522$		0			0
$523$		−	966.644i		−	1.84827i	−0.382069	−	0.924134i	$-0.624788\pi$
$523$		−	966.644i		−	1.84827i	0.382069	−	0.924134i	$-0.375212\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−243.096			−0.461282
$528$		0			0
$529$	−444.842			−0.840911
$530$		0			0
$531$	−190.053	−	104.130i	−0.357915	−	0.196102i
$532$		0			0
$533$	−473.575			−0.888509
$534$		0			0
$535$		0			0
$536$		0			0
$537$	178.191	−	105.553i	0.331827	−	0.196561i
$538$		0			0
$539$		−	59.8439i		−	0.111028i
$540$		0			0
$541$	−186.105			−0.344002			−0.172001	−	0.985097i	$-0.555023\pi$
$541$	−186.105			−0.344002			−0.172001	+	0.985097i	$0.555023\pi$
$542$		0			0
$543$	290.019	+	489.601i	0.534106	+	0.901659i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	309.434i		−	0.565693i	−0.959165	−	0.282847i	$-0.908721\pi$
$547$		−	309.434i		−	0.565693i	0.959165	−	0.282847i	$-0.0912787\pi$
$548$		0			0
$549$	233.298	−	425.802i	0.424951	−	0.775596i
$550$		0			0
$551$		−	723.779i		−	1.31357i
$552$		0			0
$553$			351.684i			0.635957i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.00106			0.00718323			0.00359161	−	0.999994i	$-0.498857\pi$
$557$	4.00106			0.00718323			0.00359161	+	0.999994i	$0.498857\pi$
$558$		0			0
$559$	144.605			0.258685
$560$		0			0
$561$	665.526	−	394.230i	1.18632	−	0.702728i
$562$		0			0
$563$	−166.970			−0.296572			−0.148286	−	0.988945i	$-0.547376\pi$
$563$	−166.970			−0.296572			−0.148286	+	0.988945i	$0.547376\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−511.102	+	326.399i	−0.901414	+	0.575660i
$568$		0			0
$569$			156.289i			0.274673i	0.990524	+	0.137337i	$0.0438542\pi$
$569$			156.289i			0.274673i	−0.990524	+	0.137337i	$0.956146\pi$
$570$		0			0
$571$	144.105			0.252374			0.126187	−	0.992006i	$-0.459726\pi$
$571$	144.105			0.252374			0.126187	+	0.992006i	$0.459726\pi$
$572$		0			0
$573$	280.591	−	166.211i	0.489688	−	0.290071i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	532.947i		−	0.923651i	−0.886971	−	0.461826i	$-0.847195\pi$
$577$		−	532.947i		−	0.923651i	0.886971	−	0.461826i	$-0.152805\pi$
$578$		0			0
$579$	−433.495	+	256.785i	−0.748697	+	0.443497i
$580$		0			0
$581$			195.738i			0.336899i
$582$		0			0
$583$		−	257.842i		−	0.442268i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	190.342			0.324262			0.162131	−	0.986769i	$-0.448163\pi$
$587$	190.342			0.324262			0.162131	+	0.986769i	$0.448163\pi$
$588$		0			0
$589$	−215.789			−0.366366
$590$		0			0
$591$	262.460	+	443.077i	0.444096	+	0.749707i
$592$		0			0
$593$	345.719			0.583001			0.291500	−	0.956571i	$-0.405846\pi$
$593$	345.719			0.583001			0.291500	+	0.956571i	$0.405846\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−53.6344	−	90.5438i	−0.0898399	−	0.151665i
$598$		0			0
$599$			704.055i			1.17538i	0.809085	+	0.587692i	$0.199963\pi$
$599$			704.055i			1.17538i	−0.809085	+	0.587692i	$0.800037\pi$
$600$		0			0
$601$	338.474			0.563185			0.281592	−	0.959534i	$-0.409137\pi$
$601$	338.474			0.563185			0.281592	+	0.959534i	$0.409137\pi$
$602$		0			0
$603$	871.856	+	477.693i	1.44586	+	0.792193i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	816.513i		−	1.34516i	−0.740024	−	0.672581i	$-0.765186\pi$
$607$		−	816.513i		−	1.34516i	0.740024	−	0.672581i	$-0.234814\pi$
$608$		0			0
$609$	307.157	+	518.532i	0.504363	+	0.851449i
$610$		0			0
$611$		−	458.688i		−	0.750717i
$612$		0			0
$613$		−	229.263i		−	0.374001i	−0.982360	−	0.187001i	$-0.940123\pi$
$613$		−	229.263i		−	0.374001i	0.982360	−	0.187001i	$-0.0598766\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1072.25			−1.73785			−0.868924	−	0.494945i	$-0.835188\pi$
$617$	−1072.25			−1.73785			−0.868924	+	0.494945i	$0.835188\pi$
$618$		0			0
$619$	−80.7103			−0.130388			−0.0651941	−	0.997873i	$-0.520767\pi$
$619$	−80.7103			−0.130388			−0.0651941	+	0.997873i	$0.520767\pi$
$620$		0			0
$621$	−247.552	+	8.30863i	−0.398635	+	0.0133794i
$622$		0			0
$623$	455.004			0.730344
$624$		0			0
$625$		0			0
$626$		0			0
$627$	590.769	−	349.947i	0.942215	−	0.558130i
$628$		0			0
$629$		−	484.591i		−	0.770415i
$630$		0			0
$631$	−492.894			−0.781131			−0.390566	−	0.920575i	$-0.627721\pi$
$631$	−492.894			−0.781131			−0.390566	+	0.920575i	$0.627721\pi$
$632$		0			0
$633$	−88.9213	−	150.114i	−0.140476	−	0.237147i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			70.5267i			0.110717i
$638$		0			0
$639$	−123.079	−	67.4353i	−0.192612	−	0.105533i
$640$		0			0
$641$			65.4816i			0.102155i	0.998695	+	0.0510777i	$0.0162656\pi$
$641$			65.4816i			0.102155i	−0.998695	+	0.0510777i	$0.983734\pi$
$642$		0			0
$643$			428.619i			0.666592i	0.942822	+	0.333296i	$0.108161\pi$
$643$			428.619i			0.666592i	−0.942822	+	0.333296i	$0.891839\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	462.801			0.715303			0.357652	−	0.933855i	$-0.383578\pi$
$647$	462.801			0.715303			0.357652	+	0.933855i	$0.383578\pi$
$648$		0			0
$649$	−204.316			−0.314817
$650$		0			0
$651$	154.596	−	91.5766i	0.237475	−	0.140671i
$652$		0			0
$653$	425.064			0.650941			0.325470	−	0.945552i	$-0.394477\pi$
$653$	425.064			0.650941			0.325470	+	0.945552i	$0.394477\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−694.161	−	380.333i	−1.05656	−	0.578894i
$658$		0			0
$659$			182.769i			0.277343i	0.990338	+	0.138671i	$0.0442831\pi$
$659$			182.769i			0.277343i	−0.990338	+	0.138671i	$0.955717\pi$
$660$		0			0
$661$	−482.053			−0.729278			−0.364639	−	0.931149i	$-0.618808\pi$
$661$	−482.053			−0.729278			−0.364639	+	0.931149i	$0.618808\pi$
$662$		0			0
$663$	−784.330	+	464.605i	−1.18300	+	0.700762i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			246.158i			0.369052i
$668$		0			0
$669$	−256.517	+	151.950i	−0.383434	+	0.227131i
$670$		0			0
$671$		−	457.758i		−	0.682203i
$672$		0			0
$673$			184.579i			0.274264i	0.990553	+	0.137132i	$0.0437884\pi$
$673$			184.579i			0.274264i	−0.990553	+	0.137132i	$0.956212\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1065.85			1.57437			0.787187	−	0.616715i	$-0.211537\pi$
$677$	1065.85			1.57437			0.787187	+	0.616715i	$0.211537\pi$
$678$		0			0
$679$	−269.920			−0.397526
$680$		0			0
$681$	331.710	+	559.982i	0.487093	+	0.822293i
$682$		0			0
$683$	788.926			1.15509			0.577545	−	0.816359i	$-0.304011\pi$
$683$	788.926			1.15509			0.577545	+	0.816359i	$0.304011\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	497.958	+	840.636i	0.724830	+	1.22363i
$688$		0			0
$689$			303.870i			0.441030i
$690$		0			0
$691$	−932.000			−1.34877			−0.674385	−	0.738380i	$-0.735591\pi$
$691$	−932.000			−1.34877			−0.674385	+	0.738380i	$0.735591\pi$
$692$		0			0
$693$	−274.730	+	501.421i	−0.396436	+	0.723551i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	1439.05i		−	2.06464i
$698$		0			0
$699$	79.0655	+	133.476i	0.113112	+	0.190952i
$700$		0			0
$701$		−	1352.75i		−	1.92974i	−0.262721	−	0.964872i	$-0.584620\pi$
$701$		−	1352.75i		−	1.92974i	0.262721	−	0.964872i	$-0.415380\pi$
$702$		0			0
$703$		−	430.158i		−	0.611889i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	360.549			0.509970
$708$		0			0
$709$	269.473			0.380075			0.190038	−	0.981777i	$-0.439139\pi$
$709$	269.473			0.380075			0.190038	+	0.981777i	$0.439139\pi$
$710$		0			0
$711$	203.140	−	370.759i	0.285711	−	0.521462i
$712$		0			0
$713$	73.3901			0.102931
$714$		0			0
$715$		0			0
$716$		0			0
$717$	1058.39	−	626.947i	1.47614	−	0.874403i
$718$		0			0
$719$			537.103i			0.747014i	0.927627	+	0.373507i	$0.121845\pi$
$719$			537.103i			0.747014i	−0.927627	+	0.373507i	$0.878155\pi$
$720$		0			0
$721$	−1051.21			−1.45799
$722$		0			0
$723$	−681.192	−	1149.96i	−0.942174	−	1.59055i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	1117.83i		−	1.53759i	−0.639495	−	0.768795i	$-0.720856\pi$
$727$		−	1117.83i		−	1.53759i	0.639495	−	0.768795i	$-0.279144\pi$
$728$		0			0
$729$	727.359	−	48.8800i	0.997750	−	0.0670507i
$730$		0			0
$731$			439.411i			0.601109i
$732$		0			0
$733$			7.52599i			0.0102674i	0.999987	+	0.00513369i	$0.00163411\pi$
$733$			7.52599i			0.0102674i	−0.999987	+	0.00513369i	$0.998366\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	937.288			1.27176
$738$		0			0
$739$	−823.079			−1.11377			−0.556887	−	0.830588i	$-0.688004\pi$
$739$	−823.079			−1.11377			−0.556887	+	0.830588i	$0.688004\pi$
$740$		0			0
$741$	−696.228	+	412.417i	−0.939579	+	0.556568i
$742$		0			0
$743$	3.21898			0.00433241			0.00216620	−	0.999998i	$-0.499310\pi$
$743$	3.21898			0.00433241			0.00216620	+	0.999998i	$0.499310\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	113.063	−	206.355i	0.151356	−	0.276245i
$748$		0			0
$749$			322.794i			0.430967i
$750$		0			0
$751$	−1185.63			−1.57874			−0.789368	−	0.613920i	$-0.789592\pi$
$751$	−1185.63			−1.57874			−0.789368	+	0.613920i	$0.789592\pi$
$752$		0			0
$753$	−612.671	+	362.921i	−0.813639	+	0.481967i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			863.315i			1.14044i	0.821491	+	0.570221i	$0.193143\pi$
$757$			863.315i			1.14044i	−0.821491	+	0.570221i	$0.806857\pi$
$758$		0			0
$759$	−200.921	+	119.017i	−0.264718	+	0.156808i
$760$		0			0
$761$		−	570.597i		−	0.749800i	−0.927065	−	0.374900i	$-0.877677\pi$
$761$		−	570.597i		−	0.749800i	0.927065	−	0.374900i	$-0.122323\pi$
$762$		0			0
$763$			1002.05i			1.31331i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	240.789			0.313936
$768$		0			0
$769$	741.684			0.964479			0.482239	−	0.876040i	$-0.339824\pi$
$769$	741.684			0.964479			0.482239	+	0.876040i	$0.339824\pi$
$770$		0			0
$771$	−487.565	−	823.090i	−0.632380	−	1.06756i
$772$		0			0
$773$	−623.203			−0.806214			−0.403107	−	0.915153i	$-0.632070\pi$
$773$	−623.203			−0.806214			−0.403107	+	0.915153i	$0.632070\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	182.550	+	308.175i	0.234943	+	0.396622i
$778$		0			0
$779$		−	1277.41i		−	1.63980i
$780$		0			0
$781$	−132.316			−0.169419
$782$		0			0
$783$	−24.3023	−	724.078i	−0.0310375	−	0.924749i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			335.303i			0.426052i	0.977046	+	0.213026i	$0.0683320\pi$
$787$			335.303i			0.426052i	−0.977046	+	0.213026i	$0.931668\pi$
$788$		0			0
$789$	−55.3943	−	93.5147i	−0.0702083	−	0.118523i
$790$		0			0
$791$		−	59.2098i		−	0.0748543i
$792$		0			0
$793$			539.473i			0.680294i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	550.520			0.690740			0.345370	−	0.938467i	$-0.387753\pi$
$797$	550.520			0.690740			0.345370	+	0.938467i	$0.387753\pi$
$798$		0			0
$799$	1393.81			1.74445
$800$		0			0
$801$	−479.684	−	262.820i	−0.598856	−	0.328115i
$802$		0			0
$803$	−746.258			−0.929337
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1362.95	−	807.354i	1.68891	−	1.00044i
$808$		0			0
$809$			560.288i			0.692569i	0.938130	+	0.346284i	$0.112557\pi$
$809$			560.288i			0.692569i	−0.938130	+	0.346284i	$0.887443\pi$
$810$		0			0
$811$	237.842			0.293270			0.146635	−	0.989191i	$-0.453156\pi$
$811$	237.842			0.293270			0.146635	+	0.989191i	$0.453156\pi$
$812$		0			0
$813$	−727.624	−	1228.35i	−0.894987	−	1.51089i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			390.053i			0.477421i
$818$		0			0
$819$	323.772	−	590.930i	0.395326	−	0.721526i
$820$		0			0
$821$			65.4816i			0.0797584i	0.999205	+	0.0398792i	$0.0126973\pi$
$821$			65.4816i			0.0797584i	−0.999205	+	0.0398792i	$0.987303\pi$
$822$		0			0
$823$		−	521.512i		−	0.633673i	−0.948480	−	0.316836i	$-0.897379\pi$
$823$		−	521.512i		−	0.633673i	0.948480	−	0.316836i	$-0.102621\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−987.512			−1.19409			−0.597045	−	0.802208i	$-0.703658\pi$
$827$	−987.512			−1.19409			−0.597045	+	0.802208i	$0.703658\pi$
$828$		0			0
$829$	−333.631			−0.402450			−0.201225	−	0.979545i	$-0.564492\pi$
$829$	−333.631			−0.402450			−0.201225	+	0.979545i	$0.564492\pi$
$830$		0			0
$831$	−485.662	+	287.686i	−0.584431	+	0.346193i
$832$		0			0
$833$	−214.309			−0.257274
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−215.878	+	7.24555i	−0.257919	+	0.00865657i
$838$		0			0
$839$		−	129.363i		−	0.154187i	−0.997024	−	0.0770934i	$-0.975436\pi$
$839$		−	129.363i		−	0.154187i	0.997024	−	0.0770934i	$-0.0245640\pi$
$840$		0			0
$841$	121.000			0.143876
$842$		0			0
$843$	63.0629	−	37.3559i	0.0748077	−	0.0443130i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	366.855i		−	0.433123i
$848$		0			0
$849$	−512.254	+	303.438i	−0.603362	+	0.357407i
$850$		0			0
$851$			146.297i			0.171912i
$852$		0			0
$853$			1080.42i			1.26661i	0.773902	+	0.633306i	$0.218303\pi$
$853$			1080.42i			1.26661i	−0.773902	+	0.633306i	$0.781697\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	702.548			0.819776			0.409888	−	0.912136i	$-0.365568\pi$
$857$	702.548			0.819776			0.409888	+	0.912136i	$0.365568\pi$
$858$		0			0
$859$	−281.132			−0.327278			−0.163639	−	0.986520i	$-0.552323\pi$
$859$	−281.132			−0.327278			−0.163639	+	0.986520i	$0.552323\pi$
$860$		0			0
$861$	542.105	+	915.163i	0.629623	+	1.06291i
$862$		0			0
$863$	419.221			0.485772			0.242886	−	0.970055i	$-0.421906\pi$
$863$	419.221			0.485772			0.242886	+	0.970055i	$0.421906\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−969.924	−	1637.39i	−1.11871	−	1.88857i
$868$		0			0
$869$		−	398.585i		−	0.458671i
$870$		0			0
$871$	−1104.60			−1.26820
$872$		0			0
$873$	284.561	+	155.912i	0.325958	+	0.178593i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	1079.42i		−	1.23081i	−0.788211	−	0.615405i	$-0.788992\pi$
$877$		−	1079.42i		−	1.23081i	0.788211	−	0.615405i	$-0.211008\pi$
$878$		0			0
$879$	−785.618	−	1326.25i	−0.893763	−	1.50882i
$880$		0			0
$881$		−	748.212i		−	0.849275i	−0.905363	−	0.424638i	$-0.860401\pi$
$881$		−	748.212i		−	0.849275i	0.905363	−	0.424638i	$-0.139599\pi$
$882$		0			0
$883$			875.749i			0.991788i	0.868383	+	0.495894i	$0.165160\pi$
$883$			875.749i			0.991788i	−0.868383	+	0.495894i	$0.834840\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1015.05			−1.14436			−0.572182	−	0.820127i	$-0.693903\pi$
$887$	−1015.05			−1.14436			−0.572182	+	0.820127i	$0.693903\pi$
$888$		0			0
$889$	−1006.68			−1.13238
$890$		0			0
$891$	579.263	−	369.928i	0.650126	−	0.415183i
$892$		0			0
$893$	1237.25			1.38550
$894$		0			0
$895$		0			0
$896$		0			0
$897$	236.788	−	140.263i	0.263977	−	0.156369i
$898$		0			0
$899$			214.663i			0.238779i
$900$		0			0
$901$	−923.368			−1.02483
$902$		0			0
$903$	−165.530	−	279.443i	−0.183312	−	0.309460i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			1504.70i			1.65898i	0.558520	+	0.829491i	$0.311369\pi$
$907$			1504.70i			1.65898i	−0.558520	+	0.829491i	$0.688631\pi$
$908$		0			0
$909$	−380.105	−	208.261i	−0.418158	−	0.229110i
$910$		0			0
$911$			1002.19i			1.10010i	0.835131	+	0.550051i	$0.185392\pi$
$911$			1002.19i			1.10010i	−0.835131	+	0.550051i	$0.814608\pi$
$912$		0			0
$913$		−	221.842i		−	0.242981i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1650.05			−1.79940
$918$		0			0
$919$	−780.289			−0.849063			−0.424532	−	0.905413i	$-0.639561\pi$
$919$	−780.289			−0.849063			−0.424532	+	0.905413i	$0.639561\pi$
$920$		0			0
$921$	−29.3772	+	17.4019i	−0.0318971	+	0.0188945i
$922$		0			0
$923$	155.936			0.168945
$924$		0			0
$925$		0			0
$926$		0			0
$927$	1108.23	+	607.201i	1.19550	+	0.655018i
$928$		0			0
$929$		−	1093.84i		−	1.17744i	−0.808339	−	0.588718i	$-0.799633\pi$
$929$		−	1093.84i		−	1.17744i	0.808339	−	0.588718i	$-0.200367\pi$
$930$		0			0
$931$	−190.236			−0.204335
$932$		0			0
$933$	1338.98	−	793.157i	1.43513	−	0.850115i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			407.947i			0.435376i	0.976018	+	0.217688i	$0.0698515\pi$
$937$			407.947i			0.435376i	−0.976018	+	0.217688i	$0.930148\pi$
$938$		0			0
$939$	−119.548	+	70.8154i	−0.127314	+	0.0754157i
$940$		0			0
$941$			671.008i			0.713079i	0.934280	+	0.356540i	$0.116044\pi$
$941$			671.008i			0.713079i	−0.934280	+	0.356540i	$0.883956\pi$
$942$		0			0
$943$			434.447i			0.460707i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1608.13			1.69813			0.849064	−	0.528290i	$-0.177167\pi$
$947$	1608.13			1.69813			0.849064	+	0.528290i	$0.177167\pi$
$948$		0			0
$949$	879.473			0.926737
$950$		0			0
$951$	−59.7758	−	100.911i	−0.0628557	−	0.106111i
$952$		0			0
$953$	695.440			0.729737			0.364869	−	0.931059i	$-0.381114\pi$
$953$	695.440			0.729737			0.364869	+	0.931059i	$0.381114\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−348.120	−	587.684i	−0.363762	−	0.614090i
$958$		0			0
$959$			715.107i			0.745680i
$960$		0			0
$961$	−897.000			−0.933403
$962$		0			0
$963$	186.453	−	340.302i	0.193617	−	0.353377i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1030.07i		−	1.06522i	−0.846361	−	0.532609i	$-0.821212\pi$
$967$		−	1030.07i		−	1.06522i	0.846361	−	0.532609i	$-0.178788\pi$
$968$		0			0
$969$	−1253.21	−	2115.63i	−1.29330	−	2.18331i
$970$		0			0
$971$		−	1165.24i		−	1.20004i	−0.799986	−	0.600019i	$-0.795160\pi$
$971$		−	1165.24i		−	1.20004i	0.799986	−	0.600019i	$-0.204840\pi$
$972$		0			0
$973$			575.106i			0.591065i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	726.440			0.743541			0.371771	−	0.928325i	$-0.378751\pi$
$977$	726.440			0.743541			0.371771	+	0.928325i	$0.378751\pi$
$978$		0			0
$979$	−515.684			−0.526746
$980$		0			0
$981$	578.807	−	1056.40i	0.590017	−	1.07686i
$982$		0			0
$983$	1024.21			1.04192			0.520960	−	0.853581i	$-0.325574\pi$
$983$	1024.21			1.04192			0.520960	+	0.853581i	$0.325574\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−886.395	+	525.064i	−0.898070	+	0.531980i
$988$		0			0
$989$		−	132.657i		−	0.134133i
$990$		0			0
$991$	1797.89			1.81422			0.907111	−	0.420892i	$-0.138283\pi$
$991$	1797.89			1.81422			0.907111	+	0.420892i	$0.138283\pi$
$992$		0			0
$993$	−681.031	−	1149.69i	−0.685832	−	1.15780i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	901.368i		−	0.904080i	−0.891998	−	0.452040i	$-0.850696\pi$
$997$		−	901.368i		−	0.904080i	0.891998	−	0.452040i	$-0.149304\pi$
$998$		0			0
$999$	−14.4434	−	430.336i	−0.0144579	−	0.430766i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.c.k.449.3		8
3.2	odd	2	inner	1200.3.c.k.449.5		8
4.3	odd	2		150.3.b.b.149.8		8
5.2	odd	4		240.3.l.c.161.2		4
5.3	odd	4		1200.3.l.u.401.3		4
5.4	even	2	inner	1200.3.c.k.449.6		8
12.11	even	2		150.3.b.b.149.2		8
15.2	even	4		240.3.l.c.161.1		4
15.8	even	4		1200.3.l.u.401.4		4
15.14	odd	2	inner	1200.3.c.k.449.4		8
20.3	even	4		150.3.d.c.101.1		4
20.7	even	4		30.3.d.a.11.4	yes	4
20.19	odd	2		150.3.b.b.149.1		8
40.27	even	4		960.3.l.e.641.2		4
40.37	odd	4		960.3.l.f.641.3		4
60.23	odd	4		150.3.d.c.101.3		4
60.47	odd	4		30.3.d.a.11.2	✓	4
60.59	even	2		150.3.b.b.149.7		8
120.77	even	4		960.3.l.f.641.4		4
120.107	odd	4		960.3.l.e.641.1		4
180.7	even	12		810.3.h.a.701.3		8
180.47	odd	12		810.3.h.a.701.2		8
180.67	even	12		810.3.h.a.431.2		8
180.167	odd	12		810.3.h.a.431.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.3.d.a.11.2	✓	4	60.47	odd	4
30.3.d.a.11.4	yes	4	20.7	even	4
150.3.b.b.149.1		8	20.19	odd	2
150.3.b.b.149.2		8	12.11	even	2
150.3.b.b.149.7		8	60.59	even	2
150.3.b.b.149.8		8	4.3	odd	2
150.3.d.c.101.1		4	20.3	even	4
150.3.d.c.101.3		4	60.23	odd	4
240.3.l.c.161.1		4	15.2	even	4
240.3.l.c.161.2		4	5.2	odd	4
810.3.h.a.431.2		8	180.67	even	12
810.3.h.a.431.3		8	180.167	odd	12
810.3.h.a.701.2		8	180.47	odd	12
810.3.h.a.701.3		8	180.7	even	12
960.3.l.e.641.1		4	120.107	odd	4
960.3.l.e.641.2		4	40.27	even	4
960.3.l.f.641.3		4	40.37	odd	4
960.3.l.f.641.4		4	120.77	even	4
1200.3.c.k.449.3		8	1.1	even	1	trivial
1200.3.c.k.449.4		8	15.14	odd	2	inner
1200.3.c.k.449.5		8	3.2	odd	2	inner
1200.3.c.k.449.6		8	5.4	even	2	inner
1200.3.l.u.401.3		4	5.3	odd	4
1200.3.l.u.401.4		4	15.8	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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.52896 - 2.58114i) q^{3} -7.48683i q^{7} +(-4.32456 + 7.89292i) q^{9} +O(q^{10})\)	\(q+(-1.52896 - 2.58114i) q^{3} -7.48683i q^{7} +(-4.32456 + 7.89292i) q^{9} +8.48528i q^{11} -10.0000i q^{13} +30.3870 q^{17} +26.9737 q^{19} +(-19.3246 + 11.4471i) q^{21} -9.17377 q^{23} +(26.9848 - 0.905694i) q^{27} -26.8328i q^{29} -8.00000 q^{31} +(21.9017 - 12.9737i) q^{33} -15.9473i q^{37} +(-25.8114 + 15.2896i) q^{39} -47.3575i q^{41} +14.4605i q^{43} +45.8688 q^{47} -7.05267 q^{49} +(-46.4605 - 78.4330i) q^{51} -30.3870 q^{53} +(-41.2417 - 69.6228i) q^{57} +24.0789i q^{59} -53.9473 q^{61} +(59.0930 + 32.3772i) q^{63} -110.460i q^{67} +(14.0263 + 23.6788i) q^{69} +15.5936i q^{71} +87.9473i q^{73} +63.5279 q^{77} -46.9737 q^{79} +(-43.5964 - 68.2668i) q^{81} -26.1443 q^{83} +(-69.2592 + 41.0263i) q^{87} +60.7739i q^{89} -74.8683 q^{91} +(12.2317 + 20.6491i) q^{93} -36.0527i q^{97} +(-66.9737 - 36.6951i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{9}+O(q^{10}) \) 8 * q + 16 * q^9	\( 8 q + 16 q^{9} + 64 q^{19} - 104 q^{21} - 64 q^{31} - 80 q^{39} - 360 q^{49} - 144 q^{51} - 128 q^{61} + 264 q^{69} - 224 q^{79} + 56 q^{81} + 160 q^{91} - 384 q^{99}+O(q^{100}) \) 8 * q + 16 * q^9 + 64 * q^19 - 104 * q^21 - 64 * q^31 - 80 * q^39 - 360 * q^49 - 144 * q^51 - 128 * q^61 + 264 * q^69 - 224 * q^79 + 56 * q^81 + 160 * q^91 - 384 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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