LMFDB - Embedded newform 1200.3.bg.a.1057.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(193,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))

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

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

chi := DirichletCharacter("1200.193");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.bg (of order $4$, degree $2$, 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:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 150)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1057.2
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1200.1057
Dual form			1200.3.bg.a.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 - 1.22474i) q^{3} +(-7.22474 - 7.22474i) q^{7} -3.00000i q^{9} +O(q^{10})$	$q+(1.22474 - 1.22474i) q^{3} +(-7.22474 - 7.22474i) q^{7} -3.00000i q^{9} +8.69694 q^{11} +(-15.6742 + 15.6742i) q^{13} +(13.3485 + 13.3485i) q^{17} +4.30306i q^{19} -17.6969 q^{21} +(-28.0454 + 28.0454i) q^{23} +(-3.67423 - 3.67423i) q^{27} -20.6969i q^{29} -39.0908 q^{31} +(10.6515 - 10.6515i) q^{33} +(12.4949 + 12.4949i) q^{37} +38.3939i q^{39} +62.6969 q^{41} +(11.8763 - 11.8763i) q^{43} +(58.0454 + 58.0454i) q^{47} +55.3939i q^{49} +32.6969 q^{51} +(0.606123 - 0.606123i) q^{53} +(5.27015 + 5.27015i) q^{57} +30.0000i q^{59} +69.7878 q^{61} +(-21.6742 + 21.6742i) q^{63} +(5.02270 + 5.02270i) q^{67} +68.6969i q^{69} +38.6969 q^{71} +(-46.2929 + 46.2929i) q^{73} +(-62.8332 - 62.8332i) q^{77} +31.3939i q^{79} -9.00000 q^{81} +(-39.4393 + 39.4393i) q^{83} +(-25.3485 - 25.3485i) q^{87} -41.3939i q^{89} +226.485 q^{91} +(-47.8763 + 47.8763i) q^{93} +(54.1237 + 54.1237i) q^{97} -26.0908i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 24 q^{7}+O(q^{10}) $ 4 * q - 24 * q^7	$ 4 q - 24 q^{7} - 24 q^{11} - 48 q^{13} + 24 q^{17} - 12 q^{21} - 24 q^{23} + 20 q^{31} + 72 q^{33} - 48 q^{37} + 192 q^{41} + 72 q^{43} + 144 q^{47} + 72 q^{51} + 120 q^{53} - 72 q^{57} + 44 q^{61} - 72 q^{63} - 24 q^{67} + 96 q^{71} - 48 q^{73} + 72 q^{77} - 36 q^{81} + 48 q^{83} - 72 q^{87} + 612 q^{91} - 216 q^{93} + 192 q^{97}+O(q^{100}) $ 4 * q - 24 * q^7 - 24 * q^11 - 48 * q^13 + 24 * q^17 - 12 * q^21 - 24 * q^23 + 20 * q^31 + 72 * q^33 - 48 * q^37 + 192 * q^41 + 72 * q^43 + 144 * q^47 + 72 * q^51 + 120 * q^53 - 72 * q^57 + 44 * q^61 - 72 * q^63 - 24 * q^67 + 96 * q^71 - 48 * q^73 + 72 * q^77 - 36 * q^81 + 48 * q^83 - 72 * q^87 + 612 * q^91 - 216 * q^93 + 192 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

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
$3$	1.22474	−	1.22474i	0.408248	−	0.408248i
$3$	1.22474	−	1.22474i	0.408248	−	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−7.22474	−	7.22474i	−1.03211	−	1.03211i	−0.999467	−	0.0326392i	$-0.989609\pi$
$7$	−7.22474	−	7.22474i	−1.03211	−	1.03211i	−0.0326392	−	0.999467i	$-0.510391\pi$
$8$		0			0
$9$		−	3.00000i		−	0.333333i
$10$		0			0
$11$	8.69694			0.790631			0.395315	−	0.918545i	$-0.370635\pi$
$11$	8.69694			0.790631			0.395315	+	0.918545i	$0.370635\pi$
$12$		0			0
$13$	−15.6742	+	15.6742i	−1.20571	+	1.20571i	−0.233307	+	0.972403i	$0.574955\pi$
$13$	−15.6742	+	15.6742i	−1.20571	+	1.20571i	−0.972403	+	0.233307i	$0.925045\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	13.3485	+	13.3485i	0.785204	+	0.785204i	0.980704	−	0.195500i	$-0.0626329\pi$
$17$	13.3485	+	13.3485i	0.785204	+	0.785204i	−0.195500	+	0.980704i	$0.562633\pi$
$18$		0			0
$19$			4.30306i			0.226477i	0.993568	+	0.113238i	$0.0361224\pi$
$19$			4.30306i			0.226477i	−0.993568	+	0.113238i	$0.963878\pi$
$20$		0			0
$21$	−17.6969			−0.842711
$22$		0			0
$23$	−28.0454	+	28.0454i	−1.21937	+	1.21937i	−0.251511	+	0.967854i	$0.580928\pi$
$23$	−28.0454	+	28.0454i	−1.21937	+	1.21937i	−0.967854	+	0.251511i	$0.919072\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	−	3.67423i	−0.136083	−	0.136083i
$28$		0			0
$29$		−	20.6969i		−	0.713688i	−0.934164	−	0.356844i	$-0.883853\pi$
$29$		−	20.6969i		−	0.713688i	0.934164	−	0.356844i	$-0.116147\pi$
$30$		0			0
$31$	−39.0908			−1.26099			−0.630497	−	0.776192i	$-0.717149\pi$
$31$	−39.0908			−1.26099			−0.630497	+	0.776192i	$0.717149\pi$
$32$		0			0
$33$	10.6515	−	10.6515i	0.322774	−	0.322774i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	12.4949	+	12.4949i	0.337700	+	0.337700i	0.855501	−	0.517801i	$-0.173249\pi$
$37$	12.4949	+	12.4949i	0.337700	+	0.337700i	−0.517801	+	0.855501i	$0.673249\pi$
$38$		0			0
$39$			38.3939i			0.984458i
$40$		0			0
$41$	62.6969			1.52919			0.764597	−	0.644509i	$-0.222938\pi$
$41$	62.6969			1.52919			0.764597	+	0.644509i	$0.222938\pi$
$42$		0			0
$43$	11.8763	−	11.8763i	0.276192	−	0.276192i	−0.555395	−	0.831587i	$-0.687433\pi$
$43$	11.8763	−	11.8763i	0.276192	−	0.276192i	0.831587	+	0.555395i	$0.187433\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	58.0454	+	58.0454i	1.23501	+	1.23501i	0.962016	+	0.272993i	$0.0880134\pi$
$47$	58.0454	+	58.0454i	1.23501	+	1.23501i	0.272993	+	0.962016i	$0.411987\pi$
$48$		0			0
$49$			55.3939i			1.13049i
$50$		0			0
$51$	32.6969			0.641116
$52$		0			0
$53$	0.606123	−	0.606123i	0.0114363	−	0.0114363i	−0.701366	−	0.712802i	$-0.747426\pi$
$53$	0.606123	−	0.606123i	0.0114363	−	0.0114363i	0.712802	+	0.701366i	$0.247426\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	5.27015	+	5.27015i	0.0924588	+	0.0924588i
$58$		0			0
$59$			30.0000i			0.508475i	0.967142	+	0.254237i	$0.0818244\pi$
$59$			30.0000i			0.508475i	−0.967142	+	0.254237i	$0.918176\pi$
$60$		0			0
$61$	69.7878			1.14406			0.572031	−	0.820232i	$-0.306156\pi$
$61$	69.7878			1.14406			0.572031	+	0.820232i	$0.306156\pi$
$62$		0			0
$63$	−21.6742	+	21.6742i	−0.344035	+	0.344035i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.02270	+	5.02270i	0.0749657	+	0.0749657i	0.743595	−	0.668630i	$-0.233119\pi$
$67$	5.02270	+	5.02270i	0.0749657	+	0.0749657i	−0.668630	+	0.743595i	$0.733119\pi$
$68$		0			0
$69$			68.6969i			0.995608i
$70$		0			0
$71$	38.6969			0.545027			0.272514	−	0.962152i	$-0.412145\pi$
$71$	38.6969			0.545027			0.272514	+	0.962152i	$0.412145\pi$
$72$		0			0
$73$	−46.2929	+	46.2929i	−0.634149	+	0.634149i	−0.949106	−	0.314957i	$-0.898010\pi$
$73$	−46.2929	+	46.2929i	−0.634149	+	0.634149i	0.314957	+	0.949106i	$0.398010\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−62.8332	−	62.8332i	−0.816015	−	0.816015i
$78$		0			0
$79$			31.3939i			0.397391i	0.980061	+	0.198695i	$0.0636705\pi$
$79$			31.3939i			0.397391i	−0.980061	+	0.198695i	$0.936330\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	−39.4393	+	39.4393i	−0.475172	+	0.475172i	−0.903584	−	0.428412i	$-0.859073\pi$
$83$	−39.4393	+	39.4393i	−0.475172	+	0.475172i	0.428412	+	0.903584i	$0.359073\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−25.3485	−	25.3485i	−0.291362	−	0.291362i
$88$		0			0
$89$		−	41.3939i		−	0.465100i	−0.972584	−	0.232550i	$-0.925293\pi$
$89$		−	41.3939i		−	0.465100i	0.972584	−	0.232550i	$-0.0747069\pi$
$90$		0			0
$91$	226.485			2.48884
$92$		0			0
$93$	−47.8763	+	47.8763i	−0.514799	+	0.514799i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	54.1237	+	54.1237i	0.557977	+	0.557977i	0.928731	−	0.370754i	$-0.120901\pi$
$97$	54.1237	+	54.1237i	0.557977	+	0.557977i	−0.370754	+	0.928731i	$0.620901\pi$
$98$		0			0
$99$		−	26.0908i		−	0.263544i
$100$		0			0
$101$	−42.8786			−0.424540			−0.212270	−	0.977211i	$-0.568086\pi$
$101$	−42.8786			−0.424540			−0.212270	+	0.977211i	$0.568086\pi$
$102$		0			0
$103$	−60.9898	+	60.9898i	−0.592134	+	0.592134i	−0.938207	−	0.346073i	$-0.887515\pi$
$103$	−60.9898	+	60.9898i	−0.592134	+	0.592134i	0.346073	+	0.938207i	$0.387515\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−45.9092	−	45.9092i	−0.429058	−	0.429058i	0.459250	−	0.888307i	$-0.348118\pi$
$107$	−45.9092	−	45.9092i	−0.429058	−	0.429058i	−0.888307	+	0.459250i	$0.848118\pi$
$108$		0			0
$109$			145.000i			1.33028i	0.746721	+	0.665138i	$0.231627\pi$
$109$			145.000i			1.33028i	−0.746721	+	0.665138i	$0.768373\pi$
$110$		0			0
$111$	30.6061			0.275731
$112$		0			0
$113$	−130.788	+	130.788i	−1.15741	+	1.15741i	−0.172384	+	0.985030i	$0.555147\pi$
$113$	−130.788	+	130.788i	−1.15741	+	1.15741i	−0.985030	+	0.172384i	$0.944853\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	47.0227	+	47.0227i	0.401903	+	0.401903i
$118$		0			0
$119$		−	192.879i		−	1.62083i
$120$		0			0
$121$	−45.3633			−0.374903
$122$		0			0
$123$	76.7878	−	76.7878i	0.624291	−	0.624291i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	21.3031	+	21.3031i	0.167741	+	0.167741i	0.785986	−	0.618245i	$-0.212156\pi$
$127$	21.3031	+	21.3031i	0.167741	+	0.167741i	−0.618245	+	0.785986i	$0.712156\pi$
$128$		0			0
$129$		−	29.0908i		−	0.225510i
$130$		0			0
$131$	−108.272			−0.826507			−0.413254	−	0.910616i	$-0.635608\pi$
$131$	−108.272			−0.826507			−0.413254	+	0.910616i	$0.635608\pi$
$132$		0			0
$133$	31.0885	−	31.0885i	0.233748	−	0.233748i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−76.6515	−	76.6515i	−0.559500	−	0.559500i	0.369665	−	0.929165i	$-0.379472\pi$
$137$	−76.6515	−	76.6515i	−0.559500	−	0.559500i	−0.929165	+	0.369665i	$0.879472\pi$
$138$		0			0
$139$			152.788i			1.09919i	0.835430	+	0.549596i	$0.185218\pi$
$139$			152.788i			1.09919i	−0.835430	+	0.549596i	$0.814782\pi$
$140$		0			0
$141$	142.182			1.00838
$142$		0			0
$143$	−136.318	+	136.318i	−0.953272	+	0.953272i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	67.8434	+	67.8434i	0.461520	+	0.461520i
$148$		0			0
$149$			112.788i			0.756965i	0.925608	+	0.378482i	$0.123554\pi$
$149$			112.788i			0.756965i	−0.925608	+	0.378482i	$0.876446\pi$
$150$		0			0
$151$	167.879			1.11178			0.555889	−	0.831256i	$-0.312378\pi$
$151$	167.879			1.11178			0.555889	+	0.831256i	$0.312378\pi$
$152$		0			0
$153$	40.0454	−	40.0454i	0.261735	−	0.261735i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−114.866	−	114.866i	−0.731631	−	0.731631i	0.239312	−	0.970943i	$-0.423078\pi$
$157$	−114.866	−	114.866i	−0.731631	−	0.731631i	−0.970943	+	0.239312i	$0.923078\pi$
$158$		0			0
$159$		−	1.48469i		−	0.00933769i
$160$		0			0
$161$	405.242			2.51703
$162$		0			0
$163$	−10.1441	+	10.1441i	−0.0622340	+	0.0622340i	−0.737539	−	0.675305i	$-0.764012\pi$
$163$	−10.1441	+	10.1441i	−0.0622340	+	0.0622340i	0.675305	+	0.737539i	$0.264012\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−155.666	−	155.666i	−0.932134	−	0.932134i	0.0657054	−	0.997839i	$-0.479070\pi$
$167$	−155.666	−	155.666i	−0.932134	−	0.932134i	−0.997839	+	0.0657054i	$0.979070\pi$
$168$		0			0
$169$		−	322.363i		−	1.90747i
$170$		0			0
$171$	12.9092			0.0754923
$172$		0			0
$173$	−86.8332	+	86.8332i	−0.501926	+	0.501926i	−0.912036	−	0.410110i	$-0.865490\pi$
$173$	−86.8332	+	86.8332i	−0.501926	+	0.501926i	0.410110	+	0.912036i	$0.365490\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	36.7423	+	36.7423i	0.207584	+	0.207584i
$178$		0			0
$179$			241.151i			1.34721i	0.739090	+	0.673606i	$0.235256\pi$
$179$			241.151i			1.34721i	−0.739090	+	0.673606i	$0.764744\pi$
$180$		0			0
$181$	41.1816			0.227523			0.113761	−	0.993508i	$-0.463710\pi$
$181$	41.1816			0.227523			0.113761	+	0.993508i	$0.463710\pi$
$182$		0			0
$183$	85.4722	−	85.4722i	0.467061	−	0.467061i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	116.091	+	116.091i	0.620806	+	0.620806i
$188$		0			0
$189$			53.0908i			0.280904i
$190$		0			0
$191$	110.091			0.576392			0.288196	−	0.957571i	$-0.406945\pi$
$191$	110.091			0.576392			0.288196	+	0.957571i	$0.406945\pi$
$192$		0			0
$193$	266.487	−	266.487i	1.38076	−	1.38076i	0.537494	−	0.843268i	$-0.319371\pi$
$193$	266.487	−	266.487i	1.38076	−	1.38076i	0.843268	−	0.537494i	$-0.180629\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−39.4393	−	39.4393i	−0.200199	−	0.200199i	0.599886	−	0.800085i	$-0.295213\pi$
$197$	−39.4393	−	39.4393i	−0.200199	−	0.200199i	−0.800085	+	0.599886i	$0.795213\pi$
$198$		0			0
$199$			92.9092i			0.466880i	0.972371	+	0.233440i	$0.0749983\pi$
$199$			92.9092i			0.466880i	−0.972371	+	0.233440i	$0.925002\pi$
$200$		0			0
$201$	12.3031			0.0612093
$202$		0			0
$203$	−149.530	+	149.530i	−0.736601	+	0.736601i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	84.1362	+	84.1362i	0.406455	+	0.406455i
$208$		0			0
$209$			37.4235i			0.179060i
$210$		0			0
$211$	−293.272			−1.38992			−0.694958	−	0.719050i	$-0.744577\pi$
$211$	−293.272			−1.38992			−0.694958	+	0.719050i	$0.744577\pi$
$212$		0			0
$213$	47.3939	−	47.3939i	0.222506	−	0.222506i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	282.421	+	282.421i	1.30148	+	1.30148i
$218$		0			0
$219$			113.394i			0.517780i
$220$		0			0
$221$	−418.454			−1.89346
$222$		0			0
$223$	74.8207	−	74.8207i	0.335519	−	0.335519i	−0.519159	−	0.854678i	$-0.673755\pi$
$223$	74.8207	−	74.8207i	0.335519	−	0.335519i	0.854678	+	0.519159i	$0.173755\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	47.1214	+	47.1214i	0.207583	+	0.207583i	0.803240	−	0.595656i	$-0.203108\pi$
$227$	47.1214	+	47.1214i	0.207583	+	0.207583i	−0.595656	+	0.803240i	$0.703108\pi$
$228$		0			0
$229$			275.000i			1.20087i	0.799672	+	0.600437i	$0.205007\pi$
$229$			275.000i			1.20087i	−0.799672	+	0.600437i	$0.794993\pi$
$230$		0			0
$231$	−153.909			−0.666274
$232$		0			0
$233$	−187.757	+	187.757i	−0.805825	+	0.805825i	−0.983999	−	0.178174i	$-0.942981\pi$
$233$	−187.757	+	187.757i	−0.805825	+	0.805825i	0.178174	+	0.983999i	$0.442981\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	38.4495	+	38.4495i	0.162234	+	0.162234i
$238$		0			0
$239$			17.6663i			0.0739177i	0.999317	+	0.0369588i	$0.0117670\pi$
$239$			17.6663i			0.0739177i	−0.999317	+	0.0369588i	$0.988233\pi$
$240$		0			0
$241$	167.000			0.692946			0.346473	−	0.938060i	$-0.387379\pi$
$241$	167.000			0.692946			0.346473	+	0.938060i	$0.387379\pi$
$242$		0			0
$243$	−11.0227	+	11.0227i	−0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−67.4472	−	67.4472i	−0.273066	−	0.273066i
$248$		0			0
$249$			96.6061i			0.387976i
$250$		0			0
$251$	−26.4245			−0.105277			−0.0526384	−	0.998614i	$-0.516763\pi$
$251$	−26.4245			−0.105277			−0.0526384	+	0.998614i	$0.516763\pi$
$252$		0			0
$253$	−243.909	+	243.909i	−0.964068	+	0.964068i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	241.485	+	241.485i	0.939629	+	0.939629i	0.998279	−	0.0586495i	$-0.0186794\pi$
$257$	241.485	+	241.485i	0.939629	+	0.939629i	−0.0586495	+	0.998279i	$0.518679\pi$
$258$		0			0
$259$		−	180.545i		−	0.697085i
$260$		0			0
$261$	−62.0908			−0.237896
$262$		0			0
$263$	142.182	−	142.182i	0.540615	−	0.540615i	−0.383095	−	0.923709i	$-0.625142\pi$
$263$	142.182	−	142.182i	0.540615	−	0.540615i	0.923709	+	0.383095i	$0.125142\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−50.6969	−	50.6969i	−0.189876	−	0.189876i
$268$		0			0
$269$		−	521.605i		−	1.93905i	−0.244988	−	0.969526i	$-0.578784\pi$
$269$		−	521.605i		−	1.93905i	0.244988	−	0.969526i	$-0.421216\pi$
$270$		0			0
$271$	−39.2122			−0.144695			−0.0723473	−	0.997379i	$-0.523049\pi$
$271$	−39.2122			−0.144695			−0.0723473	+	0.997379i	$0.523049\pi$
$272$		0			0
$273$	277.386	−	277.386i	1.01607	−	1.01607i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−69.5880	−	69.5880i	−0.251220	−	0.251220i	0.570251	−	0.821471i	$-0.306846\pi$
$277$	−69.5880	−	69.5880i	−0.251220	−	0.251220i	−0.821471	+	0.570251i	$0.806846\pi$
$278$		0			0
$279$			117.272i			0.420331i
$280$		0			0
$281$	−519.848			−1.84999			−0.924996	−	0.379976i	$-0.875932\pi$
$281$	−519.848			−1.84999			−0.924996	+	0.379976i	$0.875932\pi$
$282$		0			0
$283$	153.427	−	153.427i	0.542144	−	0.542144i	−0.382013	−	0.924157i	$-0.624769\pi$
$283$	153.427	−	153.427i	0.542144	−	0.542144i	0.924157	+	0.382013i	$0.124769\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−452.969	−	452.969i	−1.57829	−	1.57829i
$288$		0			0
$289$			67.3633i			0.233091i
$290$		0			0
$291$	132.576			0.455586
$292$		0			0
$293$	−19.6209	+	19.6209i	−0.0669656	+	0.0669656i	−0.739796	−	0.672831i	$-0.765078\pi$
$293$	−19.6209	+	19.6209i	−0.0669656	+	0.0669656i	0.672831	+	0.739796i	$0.265078\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−31.9546	−	31.9546i	−0.107591	−	0.107591i
$298$		0			0
$299$		−	879.181i		−	2.94040i
$300$		0			0
$301$	−171.606			−0.570120
$302$		0			0
$303$	−52.5153	+	52.5153i	−0.173318	+	0.173318i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	10.9115	+	10.9115i	0.0355423	+	0.0355423i	0.724655	−	0.689112i	$-0.241999\pi$
$307$	10.9115	+	10.9115i	0.0355423	+	0.0355423i	−0.689112	+	0.724655i	$0.741999\pi$
$308$		0			0
$309$			149.394i			0.483475i
$310$		0			0
$311$	−64.7878			−0.208321			−0.104160	−	0.994561i	$-0.533216\pi$
$311$	−64.7878			−0.208321			−0.104160	+	0.994561i	$0.533216\pi$
$312$		0			0
$313$	−25.4472	+	25.4472i	−0.0813009	+	0.0813009i	−0.746588	−	0.665287i	$-0.768309\pi$
$313$	−25.4472	+	25.4472i	−0.0813009	+	0.0813009i	0.665287	+	0.746588i	$0.268309\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−431.060	−	431.060i	−1.35981	−	1.35981i	−0.874143	−	0.485668i	$-0.838576\pi$
$317$	−431.060	−	431.060i	−1.35981	−	1.35981i	−0.485668	−	0.874143i	$-0.661424\pi$
$318$		0			0
$319$		−	180.000i		−	0.564263i
$320$		0			0
$321$	−112.454			−0.350324
$322$		0			0
$323$	−57.4393	+	57.4393i	−0.177831	+	0.177831i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	177.588	+	177.588i	0.543083	+	0.543083i
$328$		0			0
$329$		−	838.727i		−	2.54932i
$330$		0			0
$331$	−295.939			−0.894075			−0.447037	−	0.894515i	$-0.647521\pi$
$331$	−295.939			−0.894075			−0.447037	+	0.894515i	$0.647521\pi$
$332$		0			0
$333$	37.4847	−	37.4847i	0.112567	−	0.112567i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−141.538	−	141.538i	−0.419994	−	0.419994i	0.465208	−	0.885202i	$-0.345980\pi$
$337$	−141.538	−	141.538i	−0.419994	−	0.419994i	−0.885202	+	0.465208i	$0.845980\pi$
$338$		0			0
$339$			320.363i			0.945024i
$340$		0			0
$341$	−339.970			−0.996981
$342$		0			0
$343$	46.1941	−	46.1941i	0.134677	−	0.134677i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	362.227	+	362.227i	1.04388	+	1.04388i	0.998992	+	0.0448900i	$0.0142937\pi$
$347$	362.227	+	362.227i	1.04388	+	1.04388i	0.0448900	+	0.998992i	$0.485706\pi$
$348$		0			0
$349$			403.939i			1.15742i	0.815534	+	0.578709i	$0.196443\pi$
$349$			403.939i			1.15742i	−0.815534	+	0.578709i	$0.803557\pi$
$350$		0			0
$351$	115.182			0.328153
$352$		0			0
$353$	18.7423	−	18.7423i	0.0530945	−	0.0530945i	−0.680061	−	0.733156i	$-0.738047\pi$
$353$	18.7423	−	18.7423i	0.0530945	−	0.0530945i	0.733156	+	0.680061i	$0.238047\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−236.227	−	236.227i	−0.661700	−	0.661700i
$358$		0			0
$359$		−	383.728i		−	1.06888i	−0.845207	−	0.534439i	$-0.820523\pi$
$359$		−	383.728i		−	1.06888i	0.845207	−	0.534439i	$-0.179477\pi$
$360$		0			0
$361$	342.484			0.948708
$362$		0			0
$363$	−55.5584	+	55.5584i	−0.153054	+	0.153054i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	387.341	+	387.341i	1.05542	+	1.05542i	0.998371	+	0.0570527i	$0.0181703\pi$
$367$	387.341	+	387.341i	1.05542	+	1.05542i	0.0570527	+	0.998371i	$0.481830\pi$
$368$		0			0
$369$		−	188.091i		−	0.509731i
$370$		0			0
$371$	−8.75817			−0.0236069
$372$		0			0
$373$	419.815	−	419.815i	1.12551	−	1.12551i	0.134611	−	0.990899i	$-0.457022\pi$
$373$	419.815	−	419.815i	1.12551	−	1.12551i	0.990899	−	0.134611i	$-0.0429785\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	324.409	+	324.409i	0.860500	+	0.860500i
$378$		0			0
$379$			472.181i			1.24586i	0.782278	+	0.622930i	$0.214058\pi$
$379$			472.181i			1.24586i	−0.782278	+	0.622930i	$0.785942\pi$
$380$		0			0
$381$	52.1816			0.136960
$382$		0			0
$383$	−67.8184	+	67.8184i	−0.177071	+	0.177071i	−0.790078	−	0.613006i	$-0.789960\pi$
$383$	−67.8184	+	67.8184i	−0.177071	+	0.177071i	0.613006	+	0.790078i	$0.289960\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−35.6288	−	35.6288i	−0.0920642	−	0.0920642i
$388$		0			0
$389$			603.242i			1.55075i	0.631501	+	0.775375i	$0.282439\pi$
$389$			603.242i			1.55075i	−0.631501	+	0.775375i	$0.717561\pi$
$390$		0			0
$391$	−748.727			−1.91490
$392$		0			0
$393$	−132.606	+	132.606i	−0.337420	+	0.337420i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−188.648	−	188.648i	−0.475184	−	0.475184i	0.428403	−	0.903588i	$-0.359076\pi$
$397$	−188.648	−	188.648i	−0.475184	−	0.475184i	−0.903588	+	0.428403i	$0.859076\pi$
$398$		0			0
$399$		−	76.1510i		−	0.190855i
$400$		0			0
$401$	302.697			0.754855			0.377428	−	0.926039i	$-0.376809\pi$
$401$	302.697			0.754855			0.377428	+	0.926039i	$0.376809\pi$
$402$		0			0
$403$	612.719	−	612.719i	1.52039	−	1.52039i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	108.667	+	108.667i	0.266996	+	0.266996i
$408$		0			0
$409$		−	20.8184i		−	0.0509007i	−0.999676	−	0.0254503i	$-0.991898\pi$
$409$		−	20.8184i		−	0.0509007i	0.999676	−	0.0254503i	$-0.00810197\pi$
$410$		0			0
$411$	−187.757			−0.456830
$412$		0			0
$413$	216.742	−	216.742i	0.524800	−	0.524800i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	187.126	+	187.126i	0.448743	+	0.448743i
$418$		0			0
$419$		−	466.515i		−	1.11340i	−0.830713	−	0.556701i	$-0.812067\pi$
$419$		−	466.515i		−	1.11340i	0.830713	−	0.556701i	$-0.187933\pi$
$420$		0			0
$421$	−636.120			−1.51097			−0.755487	−	0.655163i	$-0.772600\pi$
$421$	−636.120			−1.51097			−0.755487	+	0.655163i	$0.772600\pi$
$422$		0			0
$423$	174.136	−	174.136i	0.411670	−	0.411670i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−504.199	−	504.199i	−1.18079	−	1.18079i
$428$		0			0
$429$			333.909i			0.778343i
$430$		0			0
$431$	−165.242			−0.383392			−0.191696	−	0.981454i	$-0.561399\pi$
$431$	−165.242			−0.383392			−0.191696	+	0.981454i	$0.561399\pi$
$432$		0			0
$433$	282.619	−	282.619i	0.652699	−	0.652699i	−0.300943	−	0.953642i	$-0.597301\pi$
$433$	282.619	−	282.619i	0.652699	−	0.652699i	0.953642	+	0.300943i	$0.0973015\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−120.681	−	120.681i	−0.276158	−	0.276158i
$438$		0			0
$439$			272.909i			0.621661i	0.950465	+	0.310831i	$0.100607\pi$
$439$			272.909i			0.621661i	−0.950465	+	0.310831i	$0.899393\pi$
$440$		0			0
$441$	166.182			0.376829
$442$		0			0
$443$	−176.636	+	176.636i	−0.398726	+	0.398726i	−0.877784	−	0.479057i	$-0.840979\pi$
$443$	−176.636	+	176.636i	−0.398726	+	0.398726i	0.479057	+	0.877784i	$0.340979\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	138.136	+	138.136i	0.309030	+	0.309030i
$448$		0			0
$449$		−	751.151i		−	1.67294i	−0.548011	−	0.836471i	$-0.684615\pi$
$449$		−	751.151i		−	1.67294i	0.548011	−	0.836471i	$-0.315385\pi$
$450$		0			0
$451$	545.271			1.20903
$452$		0			0
$453$	205.608	−	205.608i	0.453882	−	0.453882i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−26.4245	−	26.4245i	−0.0578216	−	0.0578216i	0.677605	−	0.735426i	$-0.263018\pi$
$457$	−26.4245	−	26.4245i	−0.0578216	−	0.0578216i	−0.735426	+	0.677605i	$0.763018\pi$
$458$		0			0
$459$		−	98.0908i		−	0.213705i
$460$		0			0
$461$	−38.6969			−0.0839413			−0.0419706	−	0.999119i	$-0.513364\pi$
$461$	−38.6969			−0.0839413			−0.0419706	+	0.999119i	$0.513364\pi$
$462$		0			0
$463$	203.505	−	203.505i	0.439536	−	0.439536i	−0.452320	−	0.891856i	$-0.649403\pi$
$463$	203.505	−	203.505i	0.439536	−	0.439536i	0.891856	+	0.452320i	$0.149403\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−557.530	−	557.530i	−1.19385	−	1.19385i	−0.975976	−	0.217879i	$-0.930086\pi$
$467$	−557.530	−	557.530i	−1.19385	−	1.19385i	−0.217879	−	0.975976i	$-0.569914\pi$
$468$		0			0
$469$		−	72.5755i		−	0.154745i
$470$		0			0
$471$	−281.363			−0.597374
$472$		0			0
$473$	103.287	−	103.287i	0.218366	−	0.218366i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.81837	−	1.81837i	−0.00381209	−	0.00381209i
$478$		0			0
$479$			265.818i			0.554944i	0.960734	+	0.277472i	$0.0894966\pi$
$479$			265.818i			0.554944i	−0.960734	+	0.277472i	$0.910503\pi$
$480$		0			0
$481$	−391.696			−0.814337
$482$		0			0
$483$	496.318	−	496.318i	1.02757	−	1.02757i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	304.083	+	304.083i	0.624400	+	0.624400i	0.946653	−	0.322253i	$-0.104440\pi$
$487$	304.083	+	304.083i	0.624400	+	0.624400i	−0.322253	+	0.946653i	$0.604440\pi$
$488$		0			0
$489$			24.8480i			0.0508138i
$490$		0			0
$491$	−169.212			−0.344628			−0.172314	−	0.985042i	$-0.555124\pi$
$491$	−169.212			−0.344628			−0.172314	+	0.985042i	$0.555124\pi$
$492$		0			0
$493$	276.272	−	276.272i	0.560390	−	0.560390i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−279.576	−	279.576i	−0.562526	−	0.562526i
$498$		0			0
$499$		−	699.636i		−	1.40208i	−0.713124	−	0.701038i	$-0.752720\pi$
$499$		−	699.636i		−	1.40208i	0.713124	−	0.701038i	$-0.247280\pi$
$500$		0			0
$501$	−381.303			−0.761084
$502$		0			0
$503$	279.848	−	279.848i	0.556358	−	0.556358i	−0.371911	−	0.928268i	$-0.621297\pi$
$503$	279.848	−	279.848i	0.556358	−	0.556358i	0.928268	+	0.371911i	$0.121297\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−394.813	−	394.813i	−0.778723	−	0.778723i
$508$		0			0
$509$			193.696i			0.380542i	0.981732	+	0.190271i	$0.0609367\pi$
$509$			193.696i			0.380542i	−0.981732	+	0.190271i	$0.939063\pi$
$510$		0			0
$511$	668.908			1.30902
$512$		0			0
$513$	15.8105	−	15.8105i	0.0308196	−	0.0308196i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	504.817	+	504.817i	0.976436	+	0.976436i
$518$		0			0
$519$			212.697i			0.409821i
$520$		0			0
$521$	167.121			0.320771			0.160385	−	0.987054i	$-0.448726\pi$
$521$	167.121			0.320771			0.160385	+	0.987054i	$0.448726\pi$
$522$		0			0
$523$	−675.860	+	675.860i	−1.29228	+	1.29228i	−0.358900	+	0.933376i	$0.616848\pi$
$523$	−675.860	+	675.860i	−1.29228	+	1.29228i	−0.933376	+	0.358900i	$0.883152\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−521.803	−	521.803i	−0.990138	−	0.990138i
$528$		0			0
$529$		−	1044.09i		−	1.97370i
$530$		0			0
$531$	90.0000			0.169492
$532$		0			0
$533$	−982.727	+	982.727i	−1.84376	+	1.84376i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	295.348	+	295.348i	0.549997	+	0.549997i
$538$		0			0
$539$			481.757i			0.893798i
$540$		0			0
$541$	−131.120			−0.242367			−0.121183	−	0.992630i	$-0.538669\pi$
$541$	−131.120			−0.242367			−0.121183	+	0.992630i	$0.538669\pi$
$542$		0			0
$543$	50.4370	−	50.4370i	0.0928858	−	0.0928858i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−423.959	−	423.959i	−0.775062	−	0.775062i	0.203924	−	0.978987i	$-0.434630\pi$
$547$	−423.959	−	423.959i	−0.775062	−	0.775062i	−0.978987	+	0.203924i	$0.934630\pi$
$548$		0			0
$549$		−	209.363i		−	0.381354i
$550$		0			0
$551$	89.0602			0.161634
$552$		0			0
$553$	226.813	−	226.813i	0.410150	−	0.410150i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−341.741	−	341.741i	−0.613539	−	0.613539i	0.330327	−	0.943866i	$-0.392841\pi$
$557$	−341.741	−	341.741i	−0.613539	−	0.613539i	−0.943866	+	0.330327i	$0.892841\pi$
$558$		0			0
$559$			372.303i			0.666016i
$560$		0			0
$561$	284.363			0.506886
$562$		0			0
$563$	−13.6209	+	13.6209i	−0.0241935	+	0.0241935i	−0.719100	−	0.694907i	$-0.755446\pi$
$563$	−13.6209	+	13.6209i	−0.0241935	+	0.0241935i	0.694907	+	0.719100i	$0.255446\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	65.0227	+	65.0227i	0.114678	+	0.114678i
$568$		0			0
$569$			22.9990i			0.0404200i	0.999796	+	0.0202100i	$0.00643348\pi$
$569$			22.9990i			0.0404200i	−0.999796	+	0.0202100i	$0.993567\pi$
$570$		0			0
$571$	660.666			1.15703			0.578517	−	0.815670i	$-0.303632\pi$
$571$	660.666			1.15703			0.578517	+	0.815670i	$0.303632\pi$
$572$		0			0
$573$	134.833	−	134.833i	0.235311	−	0.235311i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	421.547	+	421.547i	0.730584	+	0.730584i	0.970736	−	0.240151i	$-0.0771970\pi$
$577$	421.547	+	421.547i	0.730584	+	0.730584i	−0.240151	+	0.970736i	$0.577197\pi$
$578$		0			0
$579$		−	652.757i		−	1.12739i
$580$		0			0
$581$	569.878			0.980856
$582$		0			0
$583$	5.27142	−	5.27142i	0.00904188	−	0.00904188i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	441.514	+	441.514i	0.752154	+	0.752154i	0.974881	−	0.222727i	$-0.0714959\pi$
$587$	441.514	+	441.514i	0.752154	+	0.752154i	−0.222727	+	0.974881i	$0.571496\pi$
$588$		0			0
$589$		−	168.210i		−	0.285586i
$590$		0			0
$591$	−96.6061			−0.163462
$592$		0			0
$593$	171.303	−	171.303i	0.288875	−	0.288875i	−0.547760	−	0.836635i	$-0.684519\pi$
$593$	171.303	−	171.303i	0.288875	−	0.288875i	0.836635	+	0.547760i	$0.184519\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	113.790	+	113.790i	0.190603	+	0.190603i
$598$		0			0
$599$		−	64.1816i		−	0.107148i	−0.998564	−	0.0535740i	$-0.982939\pi$
$599$		−	64.1816i		−	0.107148i	0.998564	−	0.0535740i	$-0.0170613\pi$
$600$		0			0
$601$	469.545			0.781273			0.390636	−	0.920545i	$-0.372255\pi$
$601$	469.545			0.781273			0.390636	+	0.920545i	$0.372255\pi$
$602$		0			0
$603$	15.0681	−	15.0681i	0.0249886	−	0.0249886i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	121.930	+	121.930i	0.200872	+	0.200872i	0.800374	−	0.599501i	$-0.204634\pi$
$607$	121.930	+	121.930i	0.200872	+	0.200872i	−0.599501	+	0.800374i	$0.704634\pi$
$608$		0			0
$609$			366.272i			0.601433i
$610$		0			0
$611$	−1819.63			−2.97813
$612$		0			0
$613$	−542.252	+	542.252i	−0.884587	+	0.884587i	−0.993997	−	0.109409i	$-0.965104\pi$
$613$	−542.252	+	542.252i	−0.884587	+	0.884587i	0.109409	+	0.993997i	$0.465104\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−117.364	−	117.364i	−0.190218	−	0.190218i	0.605572	−	0.795790i	$-0.292944\pi$
$617$	−117.364	−	117.364i	−0.190218	−	0.190218i	−0.795790	+	0.605572i	$0.792944\pi$
$618$		0			0
$619$			295.697i			0.477701i	0.971056	+	0.238851i	$0.0767706\pi$
$619$			295.697i			0.477701i	−0.971056	+	0.238851i	$0.923229\pi$
$620$		0			0
$621$	206.091			0.331869
$622$		0			0
$623$	−299.060	+	299.060i	−0.480032	+	0.480032i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	45.8342	+	45.8342i	0.0731008	+	0.0731008i
$628$		0			0
$629$			333.576i			0.530327i
$630$		0			0
$631$	−897.454			−1.42227			−0.711136	−	0.703054i	$-0.751819\pi$
$631$	−897.454			−1.42227			−0.711136	+	0.703054i	$0.751819\pi$
$632$		0			0
$633$	−359.184	+	359.184i	−0.567431	+	0.567431i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−868.257	−	868.257i	−1.36304	−	1.36304i
$638$		0			0
$639$		−	116.091i		−	0.181676i
$640$		0			0
$641$	−226.120			−0.352762			−0.176381	−	0.984322i	$-0.556439\pi$
$641$	−226.120			−0.352762			−0.176381	+	0.984322i	$0.556439\pi$
$642$		0			0
$643$	−472.454	+	472.454i	−0.734765	+	0.734765i	−0.971560	−	0.236794i	$-0.923903\pi$
$643$	−472.454	+	472.454i	−0.734765	+	0.734765i	0.236794	+	0.971560i	$0.423903\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	583.151	+	583.151i	0.901315	+	0.901315i	0.995550	−	0.0942347i	$-0.0300404\pi$
$647$	583.151	+	583.151i	0.901315	+	0.901315i	−0.0942347	+	0.995550i	$0.530040\pi$
$648$		0			0
$649$			260.908i			0.402016i
$650$		0			0
$651$	691.788			1.06265
$652$		0			0
$653$	36.1975	−	36.1975i	0.0554325	−	0.0554325i	−0.678847	−	0.734280i	$-0.737520\pi$
$653$	36.1975	−	36.1975i	0.0554325	−	0.0554325i	0.734280	+	0.678847i	$0.237520\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	138.879	+	138.879i	0.211383	+	0.211383i
$658$		0			0
$659$			615.787i			0.934426i	0.884145	+	0.467213i	$0.154742\pi$
$659$			615.787i			0.934426i	−0.884145	+	0.467213i	$0.845258\pi$
$660$		0			0
$661$	−426.849			−0.645762			−0.322881	−	0.946439i	$-0.604651\pi$
$661$	−426.849			−0.645762			−0.322881	+	0.946439i	$0.604651\pi$
$662$		0			0
$663$	−512.499	+	512.499i	−0.773001	+	0.773001i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	580.454	+	580.454i	0.870246	+	0.870246i
$668$		0			0
$669$		−	183.272i		−	0.273950i
$670$		0			0
$671$	606.940			0.904530
$672$		0			0
$673$	−350.474	+	350.474i	−0.520764	+	0.520764i	−0.917802	−	0.397038i	$-0.870038\pi$
$673$	−350.474	+	350.474i	−0.520764	+	0.520764i	0.397038	+	0.917802i	$0.370038\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−208.985	−	208.985i	−0.308693	−	0.308693i	0.535709	−	0.844402i	$-0.320044\pi$
$677$	−208.985	−	208.985i	−0.308693	−	0.308693i	−0.844402	+	0.535709i	$0.820044\pi$
$678$		0			0
$679$		−	782.060i		−	1.15178i
$680$		0			0
$681$	115.423			0.169491
$682$		0			0
$683$	−701.271	+	701.271i	−1.02675	+	1.02675i	−0.0271195	+	0.999632i	$0.508633\pi$
$683$	−701.271	+	701.271i	−1.02675	+	1.02675i	−0.999632	+	0.0271195i	$0.991367\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	336.805	+	336.805i	0.490254	+	0.490254i
$688$		0			0
$689$			19.0010i			0.0275777i
$690$		0			0
$691$	132.910			0.192345			0.0961724	−	0.995365i	$-0.469340\pi$
$691$	132.910			0.192345			0.0961724	+	0.995365i	$0.469340\pi$
$692$		0			0
$693$	−188.499	+	188.499i	−0.272005	+	0.272005i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	836.908	+	836.908i	1.20073	+	1.20073i
$698$		0			0
$699$			459.909i			0.657953i
$700$		0			0
$701$	158.758			0.226474			0.113237	−	0.993568i	$-0.463878\pi$
$701$	158.758			0.226474			0.113237	+	0.993568i	$0.463878\pi$
$702$		0			0
$703$	−53.7663	+	53.7663i	−0.0764812	+	0.0764812i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	309.787	+	309.787i	0.438171	+	0.438171i
$708$		0			0
$709$			674.514i			0.951360i	0.879618	+	0.475680i	$0.157798\pi$
$709$			674.514i			0.951360i	−0.879618	+	0.475680i	$0.842202\pi$
$710$		0			0
$711$	94.1816			0.132464
$712$		0			0
$713$	1096.32	−	1096.32i	1.53761	−	1.53761i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	21.6367	+	21.6367i	0.0301768	+	0.0301768i
$718$		0			0
$719$			103.485i			0.143929i	0.997407	+	0.0719643i	$0.0229268\pi$
$719$			103.485i			0.143929i	−0.997407	+	0.0719643i	$0.977073\pi$
$720$		0			0
$721$	881.271			1.22229
$722$		0			0
$723$	204.532	−	204.532i	0.282894	−	0.282894i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−658.392	−	658.392i	−0.905628	−	0.905628i	0.0902877	−	0.995916i	$-0.471221\pi$
$727$	−658.392	−	658.392i	−0.905628	−	0.905628i	−0.995916	+	0.0902877i	$0.971221\pi$
$728$		0			0
$729$			27.0000i			0.0370370i
$730$		0			0
$731$	317.060			0.433735
$732$		0			0
$733$	165.970	−	165.970i	0.226426	−	0.226426i	−0.584772	−	0.811198i	$-0.698816\pi$
$733$	165.970	−	165.970i	0.226426	−	0.226426i	0.811198	+	0.584772i	$0.198816\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	43.6821	+	43.6821i	0.0592702	+	0.0592702i
$738$		0			0
$739$		−	122.545i		−	0.165825i	−0.996557	−	0.0829126i	$-0.973578\pi$
$739$		−	122.545i		−	0.165825i	0.996557	−	0.0829126i	$-0.0264222\pi$
$740$		0			0
$741$	−165.211			−0.222957
$742$		0			0
$743$	541.485	−	541.485i	0.728782	−	0.728782i	−0.241595	−	0.970377i	$-0.577671\pi$
$743$	541.485	−	541.485i	0.728782	−	0.728782i	0.970377	+	0.241595i	$0.0776707\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	118.318	+	118.318i	0.158391	+	0.158391i
$748$		0			0
$749$			663.364i			0.885667i
$750$		0			0
$751$	495.212			0.659404			0.329702	−	0.944085i	$-0.393052\pi$
$751$	495.212			0.659404			0.329702	+	0.944085i	$0.393052\pi$
$752$		0			0
$753$	−32.3633	+	32.3633i	−0.0429791	+	0.0429791i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−565.547	−	565.547i	−0.747090	−	0.747090i	0.226842	−	0.973932i	$-0.427160\pi$
$757$	−565.547	−	565.547i	−0.747090	−	0.747090i	−0.973932	+	0.226842i	$0.927160\pi$
$758$		0			0
$759$			597.453i			0.787158i
$760$		0			0
$761$	−737.271			−0.968819			−0.484410	−	0.874841i	$-0.660966\pi$
$761$	−737.271			−0.968819			−0.484410	+	0.874841i	$0.660966\pi$
$762$		0			0
$763$	1047.59	−	1047.59i	1.37299	−	1.37299i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−470.227	−	470.227i	−0.613073	−	0.613073i
$768$		0			0
$769$			1014.27i			1.31895i	0.751727	+	0.659474i	$0.229221\pi$
$769$			1014.27i			1.31895i	−0.751727	+	0.659474i	$0.770779\pi$
$770$		0			0
$771$	591.514			0.767204
$772$		0			0
$773$	−6.39491	+	6.39491i	−0.00827284	+	0.00827284i	−0.711231	−	0.702958i	$-0.751862\pi$
$773$	−6.39491	+	6.39491i	−0.00827284	+	0.00827284i	0.702958	+	0.711231i	$0.251862\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−221.121	−	221.121i	−0.284584	−	0.284584i
$778$		0			0
$779$			269.789i			0.346327i
$780$		0			0
$781$	336.545			0.430915
$782$		0			0
$783$	−76.0454	+	76.0454i	−0.0971206	+	0.0971206i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	565.806	+	565.806i	0.718940	+	0.718940i	0.968388	−	0.249448i	$-0.0802492\pi$
$787$	565.806	+	565.806i	0.718940	+	0.718940i	−0.249448	+	0.968388i	$0.580249\pi$
$788$		0			0
$789$		−	348.272i		−	0.441410i
$790$		0			0
$791$	1889.82			2.38915
$792$		0			0
$793$	−1093.87	+	1093.87i	−1.37941	+	1.37941i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	352.182	+	352.182i	0.441884	+	0.441884i	0.892645	−	0.450761i	$-0.148847\pi$
$797$	352.182	+	352.182i	0.441884	+	0.441884i	−0.450761	+	0.892645i	$0.648847\pi$
$798$		0			0
$799$			1549.63i			1.93947i
$800$		0			0
$801$	−124.182			−0.155033
$802$		0			0
$803$	−402.606	+	402.606i	−0.501377	+	0.501377i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−638.833	−	638.833i	−0.791615	−	0.791615i
$808$		0			0
$809$			563.728i			0.696820i	0.937342	+	0.348410i	$0.113278\pi$
$809$			563.728i			0.696820i	−0.937342	+	0.348410i	$0.886722\pi$
$810$		0			0
$811$	205.091			0.252886			0.126443	−	0.991974i	$-0.459644\pi$
$811$	205.091			0.252886			0.126443	+	0.991974i	$0.459644\pi$
$812$		0			0
$813$	−48.0250	+	48.0250i	−0.0590713	+	0.0590713i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	51.1043	+	51.1043i	0.0625512	+	0.0625512i
$818$		0			0
$819$		−	679.454i		−	0.829614i
$820$		0			0
$821$	867.576			1.05673			0.528365	−	0.849017i	$-0.322805\pi$
$821$	867.576			1.05673			0.528365	+	0.849017i	$0.322805\pi$
$822$		0			0
$823$	−80.8094	+	80.8094i	−0.0981889	+	0.0981889i	−0.754495	−	0.656306i	$-0.772118\pi$
$823$	−80.8094	+	80.8094i	−0.0981889	+	0.0981889i	0.656306	+	0.754495i	$0.272118\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	935.939	+	935.939i	1.13173	+	1.13173i	0.989890	+	0.141838i	$0.0453011\pi$
$827$	935.939	+	935.939i	1.13173	+	1.13173i	0.141838	+	0.989890i	$0.454699\pi$
$828$		0			0
$829$			1097.39i			1.32375i	0.749613	+	0.661877i	$0.230240\pi$
$829$			1097.39i			1.32375i	−0.749613	+	0.661877i	$0.769760\pi$
$830$		0			0
$831$	−170.455			−0.205120
$832$		0			0
$833$	−739.423	+	739.423i	−0.887663	+	0.887663i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	143.629	+	143.629i	0.171600	+	0.171600i
$838$		0			0
$839$		−	1047.18i		−	1.24813i	−0.781373	−	0.624065i	$-0.785480\pi$
$839$		−	1047.18i		−	1.24813i	0.781373	−	0.624065i	$-0.214520\pi$
$840$		0			0
$841$	412.637			0.490650
$842$		0			0
$843$	−636.681	+	636.681i	−0.755256	+	0.755256i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	327.738	+	327.738i	0.386940	+	0.386940i
$848$		0			0
$849$		−	375.817i		−	0.442659i
$850$		0			0
$851$	−700.849			−0.823559
$852$		0			0
$853$	−902.612	+	902.612i	−1.05816	+	1.05816i	−0.0599610	+	0.998201i	$0.519098\pi$
$853$	−902.612	+	902.612i	−1.05816	+	1.05816i	−0.998201	+	0.0599610i	$0.980902\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	928.454	+	928.454i	1.08338	+	1.08338i	0.996192	+	0.0871848i	$0.0277871\pi$
$857$	928.454	+	928.454i	1.08338	+	1.08338i	0.0871848	+	0.996192i	$0.472213\pi$
$858$		0			0
$859$		−	581.151i		−	0.676544i	−0.941048	−	0.338272i	$-0.890158\pi$
$859$		−	581.151i		−	0.676544i	0.941048	−	0.338272i	$-0.109842\pi$
$860$		0			0
$861$	−1109.54			−1.28867
$862$		0			0
$863$	317.271	−	317.271i	0.367638	−	0.367638i	−0.498977	−	0.866615i	$-0.666291\pi$
$863$	317.271	−	317.271i	0.367638	−	0.367638i	0.866615	+	0.498977i	$0.166291\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	82.5028	+	82.5028i	0.0951589	+	0.0951589i
$868$		0			0
$869$			273.031i			0.314189i
$870$		0			0
$871$	−157.454			−0.180774
$872$		0			0
$873$	162.371	−	162.371i	0.185992	−	0.185992i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1109.75	+	1109.75i	1.26540	+	1.26540i	0.948439	+	0.316958i	$0.102662\pi$
$877$	1109.75	+	1109.75i	1.26540	+	1.26540i	0.316958	+	0.948439i	$0.397338\pi$
$878$		0			0
$879$			48.0612i			0.0546772i
$880$		0			0
$881$	119.455			0.135590			0.0677952	−	0.997699i	$-0.478404\pi$
$881$	119.455			0.135590			0.0677952	+	0.997699i	$0.478404\pi$
$882$		0			0
$883$	−252.619	+	252.619i	−0.286091	+	0.286091i	−0.835532	−	0.549441i	$-0.814841\pi$
$883$	−252.619	+	252.619i	−0.286091	+	0.286091i	0.549441	+	0.835532i	$0.314841\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−516.347	−	516.347i	−0.582128	−	0.582128i	0.353360	−	0.935488i	$-0.385039\pi$
$887$	−516.347	−	516.347i	−0.582128	−	0.582128i	−0.935488	+	0.353360i	$0.885039\pi$
$888$		0			0
$889$		−	307.818i		−	0.346252i
$890$		0			0
$891$	−78.2724			−0.0878479
$892$		0			0
$893$	−249.773	+	249.773i	−0.279701	+	0.279701i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1076.77	−	1076.77i	−1.20041	−	1.20041i
$898$		0			0
$899$			809.060i			0.899956i
$900$		0			0
$901$	16.1816			0.0179596
$902$		0			0
$903$	−210.174	+	210.174i	−0.232751	+	0.232751i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−983.160	−	983.160i	−1.08397	−	1.08397i	−0.996135	−	0.0878342i	$-0.972005\pi$
$907$	−983.160	−	983.160i	−1.08397	−	1.08397i	−0.0878342	−	0.996135i	$-0.527995\pi$
$908$		0			0
$909$			128.636i			0.141513i
$910$		0			0
$911$	1698.00			1.86389			0.931943	−	0.362605i	$-0.118113\pi$
$911$	1698.00			1.86389			0.931943	+	0.362605i	$0.118113\pi$
$912$		0			0
$913$	−343.001	+	343.001i	−0.375686	+	0.375686i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	782.241	+	782.241i	0.853043	+	0.853043i
$918$		0			0
$919$		−	581.272i		−	0.632505i	−0.948675	−	0.316253i	$-0.897575\pi$
$919$		−	581.272i		−	0.632505i	0.948675	−	0.316253i	$-0.102425\pi$
$920$		0			0
$921$	26.7276			0.0290201
$922$		0			0
$923$	−606.545	+	606.545i	−0.657145	+	0.657145i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	182.969	+	182.969i	0.197378	+	0.197378i
$928$		0			0
$929$		−	1328.57i		−	1.43011i	−0.699067	−	0.715056i	$-0.746401\pi$
$929$		−	1328.57i		−	1.43011i	0.699067	−	0.715056i	$-0.253599\pi$
$930$		0			0
$931$	−238.363			−0.256029
$932$		0			0
$933$	−79.3485	+	79.3485i	−0.0850466	+	0.0850466i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−626.487	−	626.487i	−0.668609	−	0.668609i	0.288785	−	0.957394i	$-0.406749\pi$
$937$	−626.487	−	626.487i	−0.668609	−	0.668609i	−0.957394	+	0.288785i	$0.906749\pi$
$938$		0			0
$939$			62.3326i			0.0663819i
$940$		0			0
$941$	762.422			0.810226			0.405113	−	0.914267i	$-0.367232\pi$
$941$	762.422			0.810226			0.405113	+	0.914267i	$0.367232\pi$
$942$		0			0
$943$	−1758.36	+	1758.36i	−1.86465	+	1.86465i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	614.983	+	614.983i	0.649401	+	0.649401i	0.952848	−	0.303447i	$-0.0981375\pi$
$947$	614.983	+	614.983i	0.649401	+	0.649401i	−0.303447	+	0.952848i	$0.598138\pi$
$948$		0			0
$949$		−	1451.21i		−	1.52920i
$950$		0			0
$951$	−1055.88			−1.11028
$952$		0			0
$953$	−328.454	+	328.454i	−0.344653	+	0.344653i	−0.858113	−	0.513460i	$-0.828363\pi$
$953$	−328.454	+	328.454i	−0.344653	+	0.344653i	0.513460	+	0.858113i	$0.328363\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−220.454	−	220.454i	−0.230360	−	0.230360i
$958$		0			0
$959$			1107.58i			1.15493i
$960$		0			0
$961$	567.092			0.590106
$962$		0			0
$963$	−137.728	+	137.728i	−0.143019	+	0.143019i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−172.009	−	172.009i	−0.177879	−	0.177879i	0.612551	−	0.790431i	$-0.290143\pi$
$967$	−172.009	−	172.009i	−0.177879	−	0.177879i	−0.790431	+	0.612551i	$0.790143\pi$
$968$		0			0
$969$			140.697i			0.145198i
$970$		0			0
$971$	1273.82			1.31186			0.655931	−	0.754821i	$-0.272276\pi$
$971$	1273.82			1.31186			0.655931	+	0.754821i	$0.272276\pi$
$972$		0			0
$973$	1103.85	−	1103.85i	1.13448	−	1.13448i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1013.86	−	1013.86i	−1.03773	−	1.03773i	−0.999260	−	0.0384719i	$-0.987751\pi$
$977$	−1013.86	−	1013.86i	−1.03773	−	1.03773i	−0.0384719	−	0.999260i	$-0.512249\pi$
$978$		0			0
$979$		−	360.000i		−	0.367722i
$980$		0			0
$981$	435.000			0.443425
$982$		0			0
$983$	856.332	−	856.332i	0.871141	−	0.871141i	−0.121456	−	0.992597i	$-0.538756\pi$
$983$	856.332	−	856.332i	0.871141	−	0.871141i	0.992597	+	0.121456i	$0.0387563\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1027.23	−	1027.23i	−1.04076	−	1.04076i
$988$		0			0
$989$			666.150i			0.673559i
$990$		0			0
$991$	−688.605			−0.694859			−0.347429	−	0.937706i	$-0.612945\pi$
$991$	−688.605			−0.694859			−0.347429	+	0.937706i	$0.612945\pi$
$992$		0			0
$993$	−362.449	+	362.449i	−0.365005	+	0.365005i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1260.82	−	1260.82i	−1.26461	−	1.26461i	−0.948833	−	0.315778i	$-0.897734\pi$
$997$	−1260.82	−	1260.82i	−1.26461	−	1.26461i	−0.315778	−	0.948833i	$-0.602266\pi$
$998$		0			0
$999$		−	91.8184i		−	0.0919103i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.bg.a.1057.2		4
4.3	odd	2		150.3.f.c.7.1	yes	4
5.2	odd	4		1200.3.bg.p.193.1		4
5.3	odd	4	inner	1200.3.bg.a.193.2		4
5.4	even	2		1200.3.bg.p.1057.1		4
12.11	even	2		450.3.g.g.307.2		4
20.3	even	4		150.3.f.c.43.1	yes	4
20.7	even	4		150.3.f.a.43.2	yes	4
20.19	odd	2		150.3.f.a.7.2	✓	4
60.23	odd	4		450.3.g.g.343.2		4
60.47	odd	4		450.3.g.h.343.1		4
60.59	even	2		450.3.g.h.307.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.f.a.7.2	✓	4	20.19	odd	2
150.3.f.a.43.2	yes	4	20.7	even	4
150.3.f.c.7.1	yes	4	4.3	odd	2
150.3.f.c.43.1	yes	4	20.3	even	4
450.3.g.g.307.2		4	12.11	even	2
450.3.g.g.343.2		4	60.23	odd	4
450.3.g.h.307.1		4	60.59	even	2
450.3.g.h.343.1		4	60.47	odd	4
1200.3.bg.a.193.2		4	5.3	odd	4	inner
1200.3.bg.a.1057.2		4	1.1	even	1	trivial
1200.3.bg.p.193.1		4	5.2	odd	4
1200.3.bg.p.1057.1		4	5.4	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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.bg (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 1.22474i) q^{3} +(-7.22474 - 7.22474i) q^{7} -3.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 - 1.22474i) q^{3} +(-7.22474 - 7.22474i) q^{7} -3.00000i q^{9} +8.69694 q^{11} +(-15.6742 + 15.6742i) q^{13} +(13.3485 + 13.3485i) q^{17} +4.30306i q^{19} -17.6969 q^{21} +(-28.0454 + 28.0454i) q^{23} +(-3.67423 - 3.67423i) q^{27} -20.6969i q^{29} -39.0908 q^{31} +(10.6515 - 10.6515i) q^{33} +(12.4949 + 12.4949i) q^{37} +38.3939i q^{39} +62.6969 q^{41} +(11.8763 - 11.8763i) q^{43} +(58.0454 + 58.0454i) q^{47} +55.3939i q^{49} +32.6969 q^{51} +(0.606123 - 0.606123i) q^{53} +(5.27015 + 5.27015i) q^{57} +30.0000i q^{59} +69.7878 q^{61} +(-21.6742 + 21.6742i) q^{63} +(5.02270 + 5.02270i) q^{67} +68.6969i q^{69} +38.6969 q^{71} +(-46.2929 + 46.2929i) q^{73} +(-62.8332 - 62.8332i) q^{77} +31.3939i q^{79} -9.00000 q^{81} +(-39.4393 + 39.4393i) q^{83} +(-25.3485 - 25.3485i) q^{87} -41.3939i q^{89} +226.485 q^{91} +(-47.8763 + 47.8763i) q^{93} +(54.1237 + 54.1237i) q^{97} -26.0908i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 24 q^{7}+O(q^{10}) \) 4 * q - 24 * q^7	\( 4 q - 24 q^{7} - 24 q^{11} - 48 q^{13} + 24 q^{17} - 12 q^{21} - 24 q^{23} + 20 q^{31} + 72 q^{33} - 48 q^{37} + 192 q^{41} + 72 q^{43} + 144 q^{47} + 72 q^{51} + 120 q^{53} - 72 q^{57} + 44 q^{61} - 72 q^{63} - 24 q^{67} + 96 q^{71} - 48 q^{73} + 72 q^{77} - 36 q^{81} + 48 q^{83} - 72 q^{87} + 612 q^{91} - 216 q^{93} + 192 q^{97}+O(q^{100}) \) 4 * q - 24 * q^7 - 24 * q^11 - 48 * q^13 + 24 * q^17 - 12 * q^21 - 24 * q^23 + 20 * q^31 + 72 * q^33 - 48 * q^37 + 192 * q^41 + 72 * q^43 + 144 * q^47 + 72 * q^51 + 120 * q^53 - 72 * q^57 + 44 * q^61 - 72 * q^63 - 24 * q^67 + 96 * q^71 - 48 * q^73 + 72 * q^77 - 36 * q^81 + 48 * q^83 - 72 * q^87 + 612 * q^91 - 216 * q^93 + 192 * q^97

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