LMFDB - Embedded newform 1200.3.c.m.449.1

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 138x^{12} + 3393x^{8} + 15208x^{4} + 1296 $ x^16 + 138x^12 + 3393x^8 + 15208*x^4 + 1296
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-2.27869 - 2.27869i$ of defining polynomial
Character	$\chi$	$=$	1200.449
Dual form			1200.3.c.m.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.98579 - 0.291610i) q^{3} +4.46268i q^{7} +(8.82993 + 1.74137i) q^{9} +O(q^{10})$	$q+(-2.98579 - 0.291610i) q^{3} +4.46268i q^{7} +(8.82993 + 1.74137i) q^{9} +17.8696i q^{11} -11.0107i q^{13} +0.794055 q^{17} +26.5852 q^{19} +(1.30136 - 13.3246i) q^{21} +14.9276 q^{23} +(-25.8565 - 7.77428i) q^{27} -5.58545i q^{29} -53.1074 q^{31} +(5.21095 - 53.3549i) q^{33} -51.7565i q^{37} +(-3.21084 + 32.8758i) q^{39} +67.8236i q^{41} +40.8243i q^{43} -12.3483 q^{47} +29.0845 q^{49} +(-2.37088 - 0.231554i) q^{51} -37.0351 q^{53} +(-79.3779 - 7.75251i) q^{57} -61.0932i q^{59} +97.8289 q^{61} +(-7.77119 + 39.4051i) q^{63} +3.02541i q^{67} +(-44.5708 - 4.35305i) q^{69} +57.0787i q^{71} -31.4690i q^{73} -79.7461 q^{77} -2.16053 q^{79} +(74.9352 + 30.7524i) q^{81} -13.0710 q^{83} +(-1.62877 + 16.6770i) q^{87} +173.692i q^{89} +49.1374 q^{91} +(158.568 + 15.4866i) q^{93} +91.6381i q^{97} +(-31.1176 + 157.787i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 40 q^{9}+O(q^{10}) $ 16 * q - 40 * q^9	$ 16 q - 40 q^{9} - 16 q^{19} + 56 q^{21} - 240 q^{31} - 144 q^{39} - 128 q^{49} - 128 q^{51} + 16 q^{61} - 200 q^{69} + 176 q^{79} + 448 q^{81} - 1120 q^{91} + 64 q^{99}+O(q^{100}) $ 16 * q - 40 * q^9 - 16 * q^19 + 56 * q^21 - 240 * q^31 - 144 * q^39 - 128 * q^49 - 128 * q^51 + 16 * q^61 - 200 * q^69 + 176 * q^79 + 448 * q^81 - 1120 * q^91 + 64 * 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
$3$	−2.98579	−	0.291610i	−0.995265	−	0.0972033i
$3$	−2.98579	−	0.291610i	−0.995265	−	0.0972033i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			4.46268i			0.637525i	0.947835	+	0.318763i	$0.103267\pi$
$7$			4.46268i			0.637525i	−0.947835	+	0.318763i	$0.896733\pi$
$8$		0			0
$9$	8.82993	+	1.74137i	0.981103	+	0.193486i
$10$		0			0
$11$			17.8696i			1.62451i	0.583305	+	0.812253i	$0.301759\pi$
$11$			17.8696i			1.62451i	−0.583305	+	0.812253i	$0.698241\pi$
$12$		0			0
$13$		−	11.0107i		−	0.846980i	−0.905901	−	0.423490i	$-0.860805\pi$
$13$		−	11.0107i		−	0.846980i	0.905901	−	0.423490i	$-0.139195\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.794055			0.0467091			0.0233546	−	0.999727i	$-0.492565\pi$
$17$	0.794055			0.0467091			0.0233546	+	0.999727i	$0.492565\pi$
$18$		0			0
$19$	26.5852			1.39922			0.699611	−	0.714524i	$-0.253357\pi$
$19$	26.5852			1.39922			0.699611	+	0.714524i	$0.253357\pi$
$20$		0			0
$21$	1.30136	−	13.3246i	0.0619696	−	0.634506i
$22$		0			0
$23$	14.9276			0.649027			0.324514	−	0.945881i	$-0.394799\pi$
$23$	14.9276			0.649027			0.324514	+	0.945881i	$0.394799\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−25.8565	−	7.77428i	−0.957650	−	0.287936i
$28$		0			0
$29$		−	5.58545i		−	0.192602i	−0.995352	−	0.0963008i	$-0.969299\pi$
$29$		−	5.58545i		−	0.192602i	0.995352	−	0.0963008i	$-0.0307011\pi$
$30$		0			0
$31$	−53.1074			−1.71314			−0.856571	−	0.516030i	$-0.827409\pi$
$31$	−53.1074			−1.71314			−0.856571	+	0.516030i	$0.827409\pi$
$32$		0			0
$33$	5.21095	−	53.3549i	0.157907	−	1.61681i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	51.7565i		−	1.39882i	−0.714719	−	0.699412i	$-0.753445\pi$
$37$		−	51.7565i		−	1.39882i	0.714719	−	0.699412i	$-0.246555\pi$
$38$		0			0
$39$	−3.21084	+	32.8758i	−0.0823293	+	0.842969i
$40$		0			0
$41$			67.8236i			1.65423i	0.562030	+	0.827117i	$0.310020\pi$
$41$			67.8236i			1.65423i	−0.562030	+	0.827117i	$0.689980\pi$
$42$		0			0
$43$			40.8243i			0.949403i	0.880147	+	0.474701i	$0.157444\pi$
$43$			40.8243i			0.949403i	−0.880147	+	0.474701i	$0.842556\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−12.3483			−0.262729			−0.131365	−	0.991334i	$-0.541936\pi$
$47$	−12.3483			−0.262729			−0.131365	+	0.991334i	$0.541936\pi$
$48$		0			0
$49$	29.0845			0.593562
$50$		0			0
$51$	−2.37088	−	0.231554i	−0.0464879	−	0.00454028i
$52$		0			0
$53$	−37.0351			−0.698775			−0.349387	−	0.936978i	$-0.613610\pi$
$53$	−37.0351			−0.698775			−0.349387	+	0.936978i	$0.613610\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−79.3779	−	7.75251i	−1.39260	−	0.136009i
$58$		0			0
$59$		−	61.0932i		−	1.03548i	−0.855539	−	0.517739i	$-0.826774\pi$
$59$		−	61.0932i		−	1.03548i	0.855539	−	0.517739i	$-0.173226\pi$
$60$		0			0
$61$	97.8289			1.60375			0.801876	−	0.597490i	$-0.203835\pi$
$61$	97.8289			1.60375			0.801876	+	0.597490i	$0.203835\pi$
$62$		0			0
$63$	−7.77119	+	39.4051i	−0.123352	+	0.625478i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			3.02541i			0.0451553i	0.999745	+	0.0225777i	$0.00718731\pi$
$67$			3.02541i			0.0451553i	−0.999745	+	0.0225777i	$0.992813\pi$
$68$		0			0
$69$	−44.5708	−	4.35305i	−0.645954	−	0.0630876i
$70$		0			0
$71$			57.0787i			0.803926i	0.915656	+	0.401963i	$0.131672\pi$
$71$			57.0787i			0.803926i	−0.915656	+	0.401963i	$0.868328\pi$
$72$		0			0
$73$		−	31.4690i		−	0.431082i	−0.976495	−	0.215541i	$-0.930848\pi$
$73$		−	31.4690i		−	0.431082i	0.976495	−	0.215541i	$-0.0691515\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−79.7461			−1.03566
$78$		0			0
$79$	−2.16053			−0.0273485			−0.0136743	−	0.999907i	$-0.504353\pi$
$79$	−2.16053			−0.0273485			−0.0136743	+	0.999907i	$0.504353\pi$
$80$		0			0
$81$	74.9352	+	30.7524i	0.925126	+	0.379660i
$82$		0			0
$83$	−13.0710			−0.157482			−0.0787411	−	0.996895i	$-0.525090\pi$
$83$	−13.0710			−0.157482			−0.0787411	+	0.996895i	$0.525090\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.62877	+	16.6770i	−0.0187215	+	0.191690i
$88$		0			0
$89$			173.692i			1.95160i	0.218667	+	0.975800i	$0.429829\pi$
$89$			173.692i			1.95160i	−0.218667	+	0.975800i	$0.570171\pi$
$90$		0			0
$91$	49.1374			0.539971
$92$		0			0
$93$	158.568	+	15.4866i	1.70503	+	0.166523i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			91.6381i			0.944722i	0.881405	+	0.472361i	$0.156598\pi$
$97$			91.6381i			0.944722i	−0.881405	+	0.472361i	$0.843402\pi$
$98$		0			0
$99$	−31.1176	+	157.787i	−0.314319	+	1.59381i
$100$		0			0
$101$			116.353i			1.15201i	0.817446	+	0.576005i	$0.195389\pi$
$101$			116.353i			1.15201i	−0.817446	+	0.576005i	$0.804611\pi$
$102$		0			0
$103$			182.071i			1.76768i	0.467786	+	0.883842i	$0.345052\pi$
$103$			182.071i			1.76768i	−0.467786	+	0.883842i	$0.654948\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−102.828			−0.961013			−0.480506	−	0.876991i	$-0.659547\pi$
$107$	−102.828			−0.961013			−0.480506	+	0.876991i	$0.659547\pi$
$108$		0			0
$109$	−75.4257			−0.691979			−0.345990	−	0.938238i	$-0.612457\pi$
$109$	−75.4257			−0.691979			−0.345990	+	0.938238i	$0.612457\pi$
$110$		0			0
$111$	−15.0927	+	154.534i	−0.135970	+	1.39220i
$112$		0			0
$113$	−141.923			−1.25596			−0.627980	−	0.778230i	$-0.716118\pi$
$113$	−141.923			−1.25596			−0.627980	+	0.778230i	$0.716118\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	19.1738	−	97.2241i	0.163879	−	0.830975i
$118$		0			0
$119$			3.54361i			0.0297782i
$120$		0			0
$121$	−198.322			−1.63902
$122$		0			0
$123$	19.7780	−	202.507i	0.160797	−	1.64640i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			112.920i			0.889132i	0.895746	+	0.444566i	$0.146642\pi$
$127$			112.920i			0.889132i	−0.895746	+	0.444566i	$0.853358\pi$
$128$		0			0
$129$	11.9048	−	121.893i	0.0922851	−	0.944907i
$130$		0			0
$131$			107.233i			0.818570i	0.912407	+	0.409285i	$0.134222\pi$
$131$			107.233i			0.818570i	−0.912407	+	0.409285i	$0.865778\pi$
$132$		0			0
$133$			118.641i			0.892039i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	67.7434			0.494477			0.247239	−	0.968955i	$-0.420477\pi$
$137$	67.7434			0.494477			0.247239	+	0.968955i	$0.420477\pi$
$138$		0			0
$139$	−141.204			−1.01586			−0.507928	−	0.861399i	$-0.669589\pi$
$139$	−141.204			−1.01586			−0.507928	+	0.861399i	$0.669589\pi$
$140$		0			0
$141$	36.8694	+	3.60088i	0.261485	+	0.0255382i
$142$		0			0
$143$	196.757			1.37593
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−86.8404	−	8.48134i	−0.590751	−	0.0576962i
$148$		0			0
$149$		−	55.0831i		−	0.369685i	−0.982768	−	0.184843i	$-0.940822\pi$
$149$		−	55.0831i		−	0.369685i	0.982768	−	0.184843i	$-0.0591775\pi$
$150$		0			0
$151$	−56.1302			−0.371723			−0.185862	−	0.982576i	$-0.559508\pi$
$151$	−56.1302			−0.371723			−0.185862	+	0.982576i	$0.559508\pi$
$152$		0			0
$153$	7.01145	+	1.38275i	0.0458265	+	0.00903756i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			274.266i			1.74692i	0.486896	+	0.873460i	$0.338129\pi$
$157$			274.266i			1.74692i	−0.486896	+	0.873460i	$0.661871\pi$
$158$		0			0
$159$	110.579	+	10.7998i	0.695466	+	0.0679232i
$160$		0			0
$161$			66.6172i			0.413771i
$162$		0			0
$163$		−	260.316i		−	1.59703i	−0.601974	−	0.798516i	$-0.705619\pi$
$163$		−	260.316i		−	1.59703i	0.601974	−	0.798516i	$-0.294381\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−179.949			−1.07754			−0.538769	−	0.842453i	$-0.681111\pi$
$167$	−179.949			−1.07754			−0.538769	+	0.842453i	$0.681111\pi$
$168$		0			0
$169$	47.7635			0.282624
$170$		0			0
$171$	234.745	+	46.2948i	1.37278	+	0.270730i
$172$		0			0
$173$	111.265			0.643148			0.321574	−	0.946884i	$-0.395788\pi$
$173$	111.265			0.643148			0.321574	+	0.946884i	$0.395788\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−17.8154	+	182.412i	−0.100652	+	1.03057i
$178$		0			0
$179$		−	77.8475i		−	0.434902i	−0.976071	−	0.217451i	$-0.930226\pi$
$179$		−	77.8475i		−	0.434902i	0.976071	−	0.217451i	$-0.0697743\pi$
$180$		0			0
$181$	−238.852			−1.31963			−0.659813	−	0.751430i	$-0.729364\pi$
$181$	−238.852			−1.31963			−0.659813	+	0.751430i	$0.729364\pi$
$182$		0			0
$183$	−292.097	−	28.5279i	−1.59616	−	0.155890i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			14.1894i			0.0758793i
$188$		0			0
$189$	34.6941	−	115.389i	0.183567	−	0.610526i
$190$		0			0
$191$		−	177.248i		−	0.928002i	−0.885835	−	0.464001i	$-0.846413\pi$
$191$		−	177.248i		−	0.928002i	0.885835	−	0.464001i	$-0.153587\pi$
$192$		0			0
$193$			284.254i			1.47282i	0.676536	+	0.736409i	$0.263480\pi$
$193$			284.254i			1.47282i	−0.676536	+	0.736409i	$0.736520\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−244.341			−1.24031			−0.620156	−	0.784479i	$-0.712931\pi$
$197$	−244.341			−1.24031			−0.620156	+	0.784479i	$0.712931\pi$
$198$		0			0
$199$	74.0122			0.371921			0.185960	−	0.982557i	$-0.440460\pi$
$199$	74.0122			0.371921			0.185960	+	0.982557i	$0.440460\pi$
$200$		0			0
$201$	0.882239	−	9.03324i	0.00438925	−	0.0449415i
$202$		0			0
$203$	24.9260			0.122788
$204$		0			0
$205$		0			0
$206$		0			0
$207$	131.810	+	25.9946i	0.636763	+	0.125578i
$208$		0			0
$209$			475.066i			2.27304i
$210$		0			0
$211$	31.0682			0.147243			0.0736214	−	0.997286i	$-0.476544\pi$
$211$	31.0682			0.147243			0.0736214	+	0.997286i	$0.476544\pi$
$212$		0			0
$213$	16.6447	−	170.425i	0.0781443	−	0.800119i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	237.001i		−	1.09217i
$218$		0			0
$219$	−9.17668	+	93.9600i	−0.0419026	+	0.429041i
$220$		0			0
$221$		−	8.74314i		−	0.0395617i
$222$		0			0
$223$		−	100.432i		−	0.450366i	−0.974316	−	0.225183i	$-0.927702\pi$
$223$		−	100.432i		−	0.450366i	0.974316	−	0.225183i	$-0.0722980\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	349.117			1.53796			0.768980	−	0.639273i	$-0.220764\pi$
$227$	349.117			1.53796			0.768980	+	0.639273i	$0.220764\pi$
$228$		0			0
$229$	165.007			0.720555			0.360278	−	0.932845i	$-0.382682\pi$
$229$	165.007			0.720555			0.360278	+	0.932845i	$0.382682\pi$
$230$		0			0
$231$	238.105	+	23.2548i	1.03076	+	0.100670i
$232$		0			0
$233$	−124.800			−0.535620			−0.267810	−	0.963472i	$-0.586300\pi$
$233$	−124.800			−0.535620			−0.267810	+	0.963472i	$0.586300\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	6.45091	+	0.630033i	0.0272190	+	0.00265837i
$238$		0			0
$239$		−	61.8321i		−	0.258712i	−0.991598	−	0.129356i	$-0.958709\pi$
$239$		−	61.8321i		−	0.258712i	0.991598	−	0.129356i	$-0.0412910\pi$
$240$		0			0
$241$	2.25555			0.00935913			0.00467956	−	0.999989i	$-0.498510\pi$
$241$	2.25555			0.00935913			0.00467956	+	0.999989i	$0.498510\pi$
$242$		0			0
$243$	−214.773	−	113.672i	−0.883841	−	0.467787i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	292.723i		−	1.18511i
$248$		0			0
$249$	39.0274	+	3.81164i	0.156736	+	0.0153078i
$250$		0			0
$251$		−	108.993i		−	0.434234i	−0.976146	−	0.217117i	$-0.930335\pi$
$251$		−	108.993i		−	0.434234i	0.976146	−	0.217117i	$-0.0696654\pi$
$252$		0			0
$253$			266.750i			1.05435i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	31.4664			0.122437			0.0612187	−	0.998124i	$-0.480501\pi$
$257$	31.4664			0.122437			0.0612187	+	0.998124i	$0.480501\pi$
$258$		0			0
$259$	230.972			0.891785
$260$		0			0
$261$	9.72636	−	49.3191i	0.0372657	−	0.188962i
$262$		0			0
$263$	235.190			0.894260			0.447130	−	0.894469i	$-0.352446\pi$
$263$	235.190			0.894260			0.447130	+	0.894469i	$0.352446\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	50.6504	−	518.609i	0.189702	−	1.94236i
$268$		0			0
$269$		−	221.066i		−	0.821808i	−0.911679	−	0.410904i	$-0.865213\pi$
$269$		−	221.066i		−	0.821808i	0.911679	−	0.410904i	$-0.134787\pi$
$270$		0			0
$271$	−268.830			−0.991991			−0.495995	−	0.868325i	$-0.665197\pi$
$271$	−268.830			−0.991991			−0.495995	+	0.868325i	$0.665197\pi$
$272$		0			0
$273$	−146.714	−	14.3290i	−0.537414	−	0.0524870i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	144.697i		−	0.522370i	−0.965289	−	0.261185i	$-0.915887\pi$
$277$		−	144.697i		−	0.522370i	0.965289	−	0.261185i	$-0.0841133\pi$
$278$		0			0
$279$	−468.934	−	92.4798i	−1.68077	−	0.331469i
$280$		0			0
$281$			189.426i			0.674112i	0.941485	+	0.337056i	$0.109431\pi$
$281$			189.426i			0.674112i	−0.941485	+	0.337056i	$0.890569\pi$
$282$		0			0
$283$			420.621i			1.48629i	0.669129	+	0.743146i	$0.266667\pi$
$283$			420.621i			1.48629i	−0.669129	+	0.743146i	$0.733333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−302.675			−1.05462
$288$		0			0
$289$	−288.369			−0.997818
$290$		0			0
$291$	26.7226	−	273.612i	0.0918302	−	0.940249i
$292$		0			0
$293$	283.828			0.968696			0.484348	−	0.874875i	$-0.339057\pi$
$293$	283.828			0.968696			0.484348	+	0.874875i	$0.339057\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	138.923	−	462.045i	0.467754	−	1.55571i
$298$		0			0
$299$		−	164.364i		−	0.549713i
$300$		0			0
$301$	−182.186			−0.605268
$302$		0			0
$303$	33.9297	−	347.406i	0.111979	−	1.14656i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			167.979i			0.547162i	0.961849	+	0.273581i	$0.0882081\pi$
$307$			167.979i			0.547162i	−0.961849	+	0.273581i	$0.911792\pi$
$308$		0			0
$309$	53.0938	−	543.628i	0.171825	−	1.75931i
$310$		0			0
$311$		−	192.970i		−	0.620482i	−0.950658	−	0.310241i	$-0.899590\pi$
$311$		−	192.970i		−	0.620482i	0.950658	−	0.310241i	$-0.100410\pi$
$312$		0			0
$313$			161.056i			0.514555i	0.966338	+	0.257278i	$0.0828255\pi$
$313$			161.056i			0.514555i	−0.966338	+	0.257278i	$0.917175\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	514.761			1.62385			0.811926	−	0.583761i	$-0.198419\pi$
$317$	514.761			1.62385			0.811926	+	0.583761i	$0.198419\pi$
$318$		0			0
$319$	99.8096			0.312883
$320$		0			0
$321$	307.024	+	29.9858i	0.956462	+	0.0934136i
$322$		0			0
$323$	21.1101			0.0653564
$324$		0			0
$325$		0			0
$326$		0			0
$327$	225.206	+	21.9949i	0.688702	+	0.0672627i
$328$		0			0
$329$		−	55.1064i		−	0.167497i
$330$		0			0
$331$	−63.7466			−0.192588			−0.0962939	−	0.995353i	$-0.530699\pi$
$331$	−63.7466			−0.192588			−0.0962939	+	0.995353i	$0.530699\pi$
$332$		0			0
$333$	90.1274	−	457.006i	0.270653	−	1.37239i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			64.2444i			0.190636i	0.995447	+	0.0953181i	$0.0303868\pi$
$337$			64.2444i			0.190636i	−0.995447	+	0.0953181i	$0.969613\pi$
$338$		0			0
$339$	423.754	+	41.3863i	1.25001	+	0.122083i
$340$		0			0
$341$		−	949.006i		−	2.78301i
$342$		0			0
$343$			348.466i			1.01594i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−66.1416			−0.190610			−0.0953049	−	0.995448i	$-0.530383\pi$
$347$	−66.1416			−0.190610			−0.0953049	+	0.995448i	$0.530383\pi$
$348$		0			0
$349$	188.752			0.540837			0.270419	−	0.962743i	$-0.412838\pi$
$349$	188.752			0.540837			0.270419	+	0.962743i	$0.412838\pi$
$350$		0			0
$351$	−85.6006	+	284.700i	−0.243876	+	0.811110i
$352$		0			0
$353$	−435.927			−1.23492			−0.617461	−	0.786602i	$-0.711839\pi$
$353$	−435.927			−1.23492			−0.617461	+	0.786602i	$0.711839\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	1.03335	−	10.5805i	0.00289454	−	0.0296372i
$358$		0			0
$359$			188.255i			0.524386i	0.965015	+	0.262193i	$0.0844457\pi$
$359$			188.255i			0.524386i	−0.965015	+	0.262193i	$0.915554\pi$
$360$		0			0
$361$	345.773			0.957820
$362$		0			0
$363$	592.148	+	57.8326i	1.63126	+	0.159318i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			276.868i			0.754409i	0.926130	+	0.377205i	$0.123115\pi$
$367$			276.868i			0.754409i	−0.926130	+	0.377205i	$0.876885\pi$
$368$		0			0
$369$	−118.106	+	598.877i	−0.320071	+	1.62297i
$370$		0			0
$371$		−	165.275i		−	0.445486i
$372$		0			0
$373$		−	510.098i		−	1.36756i	−0.729691	−	0.683778i	$-0.760336\pi$
$373$		−	510.098i		−	1.36756i	0.729691	−	0.683778i	$-0.239664\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−61.4999			−0.163130
$378$		0			0
$379$	−298.461			−0.787497			−0.393748	−	0.919218i	$-0.628822\pi$
$379$	−298.461			−0.787497			−0.393748	+	0.919218i	$0.628822\pi$
$380$		0			0
$381$	32.9285	−	337.155i	0.0864266	−	0.884922i
$382$		0			0
$383$	−500.548			−1.30691			−0.653457	−	0.756963i	$-0.726682\pi$
$383$	−500.548			−1.30691			−0.653457	+	0.756963i	$0.726682\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−71.0904	+	360.476i	−0.183696	+	0.931462i
$388$		0			0
$389$		−	465.726i		−	1.19724i	−0.801033	−	0.598620i	$-0.795716\pi$
$389$		−	465.726i		−	1.19724i	0.801033	−	0.598620i	$-0.204284\pi$
$390$		0			0
$391$	11.8534			0.0303155
$392$		0			0
$393$	31.2701	−	320.175i	0.0795678	−	0.814694i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			344.998i			0.869013i	0.900669	+	0.434507i	$0.143077\pi$
$397$			344.998i			0.869013i	−0.900669	+	0.434507i	$0.856923\pi$
$398$		0			0
$399$	34.5969	−	354.238i	0.0867091	−	0.887815i
$400$		0			0
$401$		−	428.755i		−	1.06922i	−0.845100	−	0.534608i	$-0.820459\pi$
$401$		−	428.755i		−	1.06922i	0.845100	−	0.534608i	$-0.179541\pi$
$402$		0			0
$403$			584.752i			1.45100i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	924.866			2.27240
$408$		0			0
$409$	349.445			0.854389			0.427194	−	0.904160i	$-0.359502\pi$
$409$	349.445			0.854389			0.427194	+	0.904160i	$0.359502\pi$
$410$		0			0
$411$	−202.268	−	19.7547i	−0.492136	−	0.0480648i
$412$		0			0
$413$	272.639			0.660143
$414$		0			0
$415$		0			0
$416$		0			0
$417$	421.606	+	41.1765i	1.01105	+	0.0987447i
$418$		0			0
$419$		−	225.745i		−	0.538770i	−0.963033	−	0.269385i	$-0.913180\pi$
$419$		−	225.745i		−	0.538770i	0.963033	−	0.269385i	$-0.0868205\pi$
$420$		0			0
$421$	−138.931			−0.330003			−0.165002	−	0.986293i	$-0.552763\pi$
$421$	−138.931			−0.330003			−0.165002	+	0.986293i	$0.552763\pi$
$422$		0			0
$423$	−109.034	−	21.5030i	−0.257765	−	0.0508345i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			436.579i			1.02243i
$428$		0			0
$429$	−587.477	−	57.3764i	−1.36941	−	0.133745i
$430$		0			0
$431$		−	289.802i		−	0.672394i	−0.941792	−	0.336197i	$-0.890859\pi$
$431$		−	289.802i		−	0.672394i	0.941792	−	0.336197i	$-0.109141\pi$
$432$		0			0
$433$			53.8425i			0.124348i	0.998065	+	0.0621738i	$0.0198033\pi$
$433$			53.8425i			0.124348i	−0.998065	+	0.0621738i	$0.980197\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	396.854			0.908133
$438$		0			0
$439$	333.179			0.758949			0.379474	−	0.925202i	$-0.376105\pi$
$439$	333.179			0.758949			0.379474	+	0.925202i	$0.376105\pi$
$440$		0			0
$441$	256.814	+	50.6471i	0.582345	+	0.114846i
$442$		0			0
$443$	−652.015			−1.47182			−0.735909	−	0.677081i	$-0.763245\pi$
$443$	−652.015			−1.47182			−0.735909	+	0.677081i	$0.763245\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−16.0628	+	164.467i	−0.0359346	+	0.367935i
$448$		0			0
$449$			115.054i			0.256245i	0.991758	+	0.128122i	$0.0408950\pi$
$449$			115.054i			0.256245i	−0.991758	+	0.128122i	$0.959105\pi$
$450$		0			0
$451$	−1211.98			−2.68731
$452$		0			0
$453$	167.593	+	16.3681i	0.369963	+	0.0361327i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			138.318i			0.302666i	0.988483	+	0.151333i	$0.0483566\pi$
$457$			138.318i			0.302666i	−0.988483	+	0.151333i	$0.951643\pi$
$458$		0			0
$459$	−20.5315	−	6.17321i	−0.0447310	−	0.0134493i
$460$		0			0
$461$			300.475i			0.651789i	0.945406	+	0.325894i	$0.105665\pi$
$461$			300.475i			0.651789i	−0.945406	+	0.325894i	$0.894335\pi$
$462$		0			0
$463$			91.5721i			0.197780i	0.995098	+	0.0988899i	$0.0315292\pi$
$463$			91.5721i			0.197780i	−0.995098	+	0.0988899i	$0.968471\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	308.136			0.659821			0.329910	−	0.944012i	$-0.392981\pi$
$467$	308.136			0.659821			0.329910	+	0.944012i	$0.392981\pi$
$468$		0			0
$469$	−13.5014			−0.0287877
$470$		0			0
$471$	79.9788	−	818.903i	0.169806	−	1.73865i
$472$		0			0
$473$	−729.513			−1.54231
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−327.017	−	64.4919i	−0.685570	−	0.135203i
$478$		0			0
$479$			218.332i			0.455808i	0.973684	+	0.227904i	$0.0731873\pi$
$479$			218.332i			0.455808i	−0.973684	+	0.227904i	$0.926813\pi$
$480$		0			0
$481$	−569.877			−1.18478
$482$		0			0
$483$	19.4262	−	198.905i	0.0402200	−	0.411812i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	538.368i		−	1.10548i	−0.833354	−	0.552739i	$-0.813583\pi$
$487$		−	538.368i		−	1.10548i	0.833354	−	0.552739i	$-0.186417\pi$
$488$		0			0
$489$	−75.9108	+	777.250i	−0.155237	+	1.58947i
$490$		0			0
$491$			418.223i			0.851777i	0.904776	+	0.425889i	$0.140038\pi$
$491$			418.223i			0.851777i	−0.904776	+	0.425889i	$0.859962\pi$
$492$		0			0
$493$		−	4.43515i		−	0.00899625i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−254.724			−0.512523
$498$		0			0
$499$	−732.199			−1.46733			−0.733667	−	0.679510i	$-0.762193\pi$
$499$	−732.199			−1.46733			−0.733667	+	0.679510i	$0.762193\pi$
$500$		0			0
$501$	537.291	+	52.4749i	1.07244	+	0.104740i
$502$		0			0
$503$	338.339			0.672641			0.336321	−	0.941748i	$-0.390817\pi$
$503$	338.339			0.672641			0.336321	+	0.941748i	$0.390817\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−142.612	−	13.9283i	−0.281286	−	0.0274720i
$508$		0			0
$509$		−	864.760i		−	1.69894i	−0.527637	−	0.849470i	$-0.676922\pi$
$509$		−	864.760i		−	1.69894i	0.527637	−	0.849470i	$-0.323078\pi$
$510$		0			0
$511$	140.436			0.274826
$512$		0			0
$513$	−687.401	−	206.681i	−1.33996	−	0.402887i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	220.659i		−	0.426806i
$518$		0			0
$519$	−332.213	−	32.4459i	−0.640103	−	0.0625162i
$520$		0			0
$521$			294.647i			0.565541i	0.959188	+	0.282770i	$0.0912534\pi$
$521$			294.647i			0.565541i	−0.959188	+	0.282770i	$0.908747\pi$
$522$		0			0
$523$			613.850i			1.17371i	0.809692	+	0.586855i	$0.199634\pi$
$523$			613.850i			1.17371i	−0.809692	+	0.586855i	$0.800366\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−42.1702			−0.0800193
$528$		0			0
$529$	−306.166			−0.578763
$530$		0			0
$531$	106.386	−	539.448i	0.200351	−	1.01591i
$532$		0			0
$533$	746.788			1.40110
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−22.7011	+	232.437i	−0.0422740	+	0.432843i
$538$		0			0
$539$			519.728i			0.964245i
$540$		0			0
$541$	188.436			0.348311			0.174156	−	0.984718i	$-0.444280\pi$
$541$	188.436			0.348311			0.174156	+	0.984718i	$0.444280\pi$
$542$		0			0
$543$	713.163	+	69.6517i	1.31338	+	0.128272i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			637.744i			1.16589i	0.812510	+	0.582947i	$0.198101\pi$
$547$			637.744i			1.16589i	−0.812510	+	0.582947i	$0.801899\pi$
$548$		0			0
$549$	863.822	+	170.357i	1.57345	+	0.310304i
$550$		0			0
$551$		−	148.490i		−	0.269492i
$552$		0			0
$553$		−	9.64176i		−	0.0174354i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	77.2307			0.138655			0.0693274	−	0.997594i	$-0.477915\pi$
$557$	77.2307			0.138655			0.0693274	+	0.997594i	$0.477915\pi$
$558$		0			0
$559$	449.506			0.804125
$560$		0			0
$561$	4.13778	−	42.3667i	0.00737572	−	0.0755200i
$562$		0			0
$563$	257.584			0.457521			0.228760	−	0.973483i	$-0.426533\pi$
$563$	257.584			0.457521			0.228760	+	0.973483i	$0.426533\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−137.238	+	334.412i	−0.242042	+	0.589791i
$568$		0			0
$569$		−	307.483i		−	0.540392i	−0.962805	−	0.270196i	$-0.912911\pi$
$569$		−	307.483i		−	0.540392i	0.962805	−	0.270196i	$-0.0870886\pi$
$570$		0			0
$571$	384.100			0.672679			0.336340	−	0.941741i	$-0.390811\pi$
$571$	384.100			0.672679			0.336340	+	0.941741i	$0.390811\pi$
$572$		0			0
$573$	−51.6874	+	529.227i	−0.0902049	+	0.923608i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			235.337i			0.407863i	0.978985	+	0.203932i	$0.0653720\pi$
$577$			235.337i			0.407863i	−0.978985	+	0.203932i	$0.934628\pi$
$578$		0			0
$579$	82.8913	−	848.724i	0.143163	−	1.46584i
$580$		0			0
$581$		−	58.3317i		−	0.100399i
$582$		0			0
$583$		−	661.801i		−	1.13516i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	230.275			0.392292			0.196146	−	0.980575i	$-0.437157\pi$
$587$	230.275			0.392292			0.196146	+	0.980575i	$0.437157\pi$
$588$		0			0
$589$	−1411.87			−2.39706
$590$		0			0
$591$	729.553	+	71.2524i	1.23444	+	0.120562i
$592$		0			0
$593$	1042.09			1.75731			0.878656	−	0.477455i	$-0.158441\pi$
$593$	1042.09			1.75731			0.878656	+	0.477455i	$0.158441\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−220.985	−	21.5827i	−0.370160	−	0.0361519i
$598$		0			0
$599$		−	136.181i		−	0.227348i	−0.993518	−	0.113674i	$-0.963738\pi$
$599$		−	136.181i		−	0.227348i	0.993518	−	0.113674i	$-0.0362619\pi$
$600$		0			0
$601$	136.625			0.227329			0.113665	−	0.993519i	$-0.463741\pi$
$601$	136.625			0.227329			0.113665	+	0.993519i	$0.463741\pi$
$602$		0			0
$603$	−5.26837	+	26.7141i	−0.00873693	+	0.0443020i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	426.666i		−	0.702910i	−0.936205	−	0.351455i	$-0.885687\pi$
$607$		−	426.666i		−	0.702910i	0.936205	−	0.351455i	$-0.114313\pi$
$608$		0			0
$609$	−74.4240	−	7.26868i	−0.122207	−	0.0119354i
$610$		0			0
$611$			135.964i			0.222527i
$612$		0			0
$613$			665.317i			1.08535i	0.839944	+	0.542673i	$0.182588\pi$
$613$			665.317i			1.08535i	−0.839944	+	0.542673i	$0.817412\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	826.628			1.33975			0.669877	−	0.742473i	$-0.266347\pi$
$617$	826.628			1.33975			0.669877	+	0.742473i	$0.266347\pi$
$618$		0			0
$619$	19.8176			0.0320155			0.0160078	−	0.999872i	$-0.494904\pi$
$619$	19.8176			0.0320155			0.0160078	+	0.999872i	$0.494904\pi$
$620$		0			0
$621$	−385.977	−	116.052i	−0.621541	−	0.186879i
$622$		0			0
$623$	−775.133			−1.24419
$624$		0			0
$625$		0			0
$626$		0			0
$627$	138.534	−	1418.45i	0.220948	−	2.26228i
$628$		0			0
$629$		−	41.0975i		−	0.0653378i
$630$		0			0
$631$	728.156			1.15397			0.576986	−	0.816754i	$-0.304229\pi$
$631$	728.156			1.15397			0.576986	+	0.816754i	$0.304229\pi$
$632$		0			0
$633$	−92.7633	−	9.05981i	−0.146546	−	0.0143125i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	320.242i		−	0.502735i
$638$		0			0
$639$	−99.3955	+	504.001i	−0.155548	+	0.788734i
$640$		0			0
$641$			438.969i			0.684819i	0.939551	+	0.342410i	$0.111243\pi$
$641$			438.969i			0.684819i	−0.939551	+	0.342410i	$0.888757\pi$
$642$		0			0
$643$		−	483.535i		−	0.751998i	−0.926620	−	0.375999i	$-0.877300\pi$
$643$		−	483.535i		−	0.751998i	0.926620	−	0.375999i	$-0.122700\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−575.687			−0.889780			−0.444890	−	0.895585i	$-0.646757\pi$
$647$	−575.687			−0.889780			−0.444890	+	0.895585i	$0.646757\pi$
$648$		0			0
$649$	1091.71			1.68214
$650$		0			0
$651$	−69.1119	+	707.636i	−0.106163	+	1.08700i
$652$		0			0
$653$	407.885			0.624633			0.312317	−	0.949978i	$-0.398895\pi$
$653$	407.885			0.624633			0.312317	+	0.949978i	$0.398895\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	54.7993	−	277.869i	0.0834084	−	0.422936i
$658$		0			0
$659$		−	187.060i		−	0.283854i	−0.989877	−	0.141927i	$-0.954670\pi$
$659$		−	187.060i		−	0.283854i	0.989877	−	0.141927i	$-0.0453298\pi$
$660$		0			0
$661$	483.965			0.732171			0.366085	−	0.930581i	$-0.380698\pi$
$661$	483.965			0.732171			0.366085	+	0.930581i	$0.380698\pi$
$662$		0			0
$663$	−2.54959	+	26.1052i	−0.00384553	+	0.0393744i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	83.3775i		−	0.125004i
$668$		0			0
$669$	−29.2869	+	299.868i	−0.0437771	+	0.448233i
$670$		0			0
$671$			1748.16i			2.60531i
$672$		0			0
$673$			480.449i			0.713891i	0.934125	+	0.356945i	$0.116182\pi$
$673$			480.449i			0.713891i	−0.934125	+	0.356945i	$0.883818\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1245.12			−1.83918			−0.919589	−	0.392881i	$-0.871478\pi$
$677$	−1245.12			−1.83918			−0.919589	+	0.392881i	$0.871478\pi$
$678$		0			0
$679$	−408.951			−0.602284
$680$		0			0
$681$	−1042.39	−	101.806i	−1.53068	−	0.149495i
$682$		0			0
$683$	−118.409			−0.173366			−0.0866832	−	0.996236i	$-0.527627\pi$
$683$	−118.409			−0.173366			−0.0866832	+	0.996236i	$0.527627\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−492.677	−	48.1177i	−0.717143	−	0.0700404i
$688$		0			0
$689$			407.784i			0.591848i
$690$		0			0
$691$	481.257			0.696465			0.348232	−	0.937408i	$-0.386782\pi$
$691$	481.257			0.696465			0.348232	+	0.937408i	$0.386782\pi$
$692$		0			0
$693$	−704.152	−	138.868i	−1.01609	−	0.200387i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			53.8556i			0.0772678i
$698$		0			0
$699$	372.626	+	36.3928i	0.533084	+	0.0520641i
$700$		0			0
$701$		−	644.879i		−	0.919941i	−0.887934	−	0.459971i	$-0.847860\pi$
$701$		−	644.879i		−	0.919941i	0.887934	−	0.459971i	$-0.152140\pi$
$702$		0			0
$703$		−	1375.96i		−	1.95726i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−519.246			−0.734436
$708$		0			0
$709$	534.801			0.754303			0.377151	−	0.926152i	$-0.376904\pi$
$709$	534.801			0.754303			0.377151	+	0.926152i	$0.376904\pi$
$710$		0			0
$711$	−19.0774	−	3.76230i	−0.0268317	−	0.00529156i
$712$		0			0
$713$	−792.767			−1.11188
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−18.0309	+	184.618i	−0.0251476	+	0.257487i
$718$		0			0
$719$			891.507i			1.23993i	0.784631	+	0.619963i	$0.212852\pi$
$719$			891.507i			1.23993i	−0.784631	+	0.619963i	$0.787148\pi$
$720$		0			0
$721$	−812.526			−1.12694
$722$		0			0
$723$	−6.73461	−	0.657741i	−0.00931481	−	0.000909739i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	720.024i		−	0.990405i	−0.868778	−	0.495202i	$-0.835094\pi$
$727$		−	720.024i		−	0.990405i	0.868778	−	0.495202i	$-0.164906\pi$
$728$		0			0
$729$	608.121	+	402.032i	0.834185	+	0.551484i
$730$		0			0
$731$			32.4167i			0.0443458i
$732$		0			0
$733$			953.335i			1.30059i	0.759681	+	0.650296i	$0.225355\pi$
$733$			953.335i			1.30059i	−0.759681	+	0.650296i	$0.774645\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−54.0627			−0.0733552
$738$		0			0
$739$	615.723			0.833185			0.416592	−	0.909093i	$-0.363224\pi$
$739$	615.723			0.833185			0.416592	+	0.909093i	$0.363224\pi$
$740$		0			0
$741$	−85.3609	+	874.010i	−0.115197	+	1.17950i
$742$		0			0
$743$	−377.743			−0.508403			−0.254201	−	0.967151i	$-0.581813\pi$
$743$	−377.743			−0.508403			−0.254201	+	0.967151i	$0.581813\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−115.416	−	22.7615i	−0.154506	−	0.0304706i
$748$		0			0
$749$		−	458.890i		−	0.612670i
$750$		0			0
$751$	−735.578			−0.979465			−0.489733	−	0.871873i	$-0.662906\pi$
$751$	−735.578			−0.979465			−0.489733	+	0.871873i	$0.662906\pi$
$752$		0			0
$753$	−31.7834	+	325.430i	−0.0422090	+	0.432178i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	921.601i		−	1.21744i	−0.793386	−	0.608719i	$-0.791684\pi$
$757$		−	921.601i		−	1.21744i	0.793386	−	0.608719i	$-0.208316\pi$
$758$		0			0
$759$	77.7871	−	796.462i	0.102486	−	1.04936i
$760$		0			0
$761$		−	240.158i		−	0.315582i	−0.987473	−	0.157791i	$-0.949563\pi$
$761$		−	240.158i		−	0.315582i	0.987473	−	0.157791i	$-0.0504372\pi$
$762$		0			0
$763$		−	336.601i		−	0.441154i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−672.682			−0.877029
$768$		0			0
$769$	4.79672			0.00623760			0.00311880	−	0.999995i	$-0.499007\pi$
$769$	4.79672			0.00623760			0.00311880	+	0.999995i	$0.499007\pi$
$770$		0			0
$771$	−93.9522	−	9.17592i	−0.121858	−	0.0119013i
$772$		0			0
$773$	140.044			0.181170			0.0905850	−	0.995889i	$-0.471126\pi$
$773$	140.044			0.181170			0.0905850	+	0.995889i	$0.471126\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−689.636	−	67.3539i	−0.887562	−	0.0866845i
$778$		0			0
$779$			1803.10i			2.31464i
$780$		0			0
$781$	−1019.97			−1.30598
$782$		0			0
$783$	−43.4228	+	144.420i	−0.0554570	+	0.184445i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			649.685i			0.825520i	0.910840	+	0.412760i	$0.135435\pi$
$787$			649.685i			0.825520i	−0.910840	+	0.412760i	$0.864565\pi$
$788$		0			0
$789$	−702.230	−	68.5838i	−0.890025	−	0.0869250i
$790$		0			0
$791$		−	633.358i		−	0.800706i
$792$		0			0
$793$		−	1077.17i		−	1.35835i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−547.802			−0.687329			−0.343665	−	0.939092i	$-0.611668\pi$
$797$	−547.802			−0.687329			−0.343665	+	0.939092i	$0.611668\pi$
$798$		0			0
$799$	−9.80522			−0.0122719
$800$		0			0
$801$	−302.463	+	1533.69i	−0.377607	+	1.91472i
$802$		0			0
$803$	562.338			0.700296
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−64.4652	+	660.059i	−0.0798825	+	0.817916i
$808$		0			0
$809$			719.685i			0.889598i	0.895630	+	0.444799i	$0.146725\pi$
$809$			719.685i			0.889598i	−0.895630	+	0.444799i	$0.853275\pi$
$810$		0			0
$811$	775.519			0.956251			0.478125	−	0.878292i	$-0.341316\pi$
$811$	775.519			0.956251			0.478125	+	0.878292i	$0.341316\pi$
$812$		0			0
$813$	802.669	+	78.3934i	0.987293	+	0.0964248i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			1085.32i			1.32842i
$818$		0			0
$819$	433.879	+	85.5666i	0.529767	+	0.104477i
$820$		0			0
$821$			1383.11i			1.68466i	0.538960	+	0.842331i	$0.318817\pi$
$821$			1383.11i			1.68466i	−0.538960	+	0.842331i	$0.681183\pi$
$822$		0			0
$823$		−	536.474i		−	0.651852i	−0.945395	−	0.325926i	$-0.894324\pi$
$823$		−	536.474i		−	0.651852i	0.945395	−	0.325926i	$-0.105676\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1652.28			−1.99792			−0.998961	−	0.0455686i	$-0.985490\pi$
$827$	−1652.28			−1.99792			−0.998961	+	0.0455686i	$0.985490\pi$
$828$		0			0
$829$	999.416			1.20557			0.602784	−	0.797904i	$-0.294058\pi$
$829$	999.416			1.20557			0.602784	+	0.797904i	$0.294058\pi$
$830$		0			0
$831$	−42.1950	+	432.034i	−0.0507761	+	0.519897i
$832$		0			0
$833$	23.0947			0.0277247
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1373.17	+	412.872i	1.64059	+	0.493276i
$838$		0			0
$839$		−	1394.04i		−	1.66155i	−0.556611	−	0.830773i	$-0.687899\pi$
$839$		−	1394.04i		−	1.66155i	0.556611	−	0.830773i	$-0.312101\pi$
$840$		0			0
$841$	809.803			0.962905
$842$		0			0
$843$	55.2384	−	565.586i	0.0655260	−	0.670920i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	885.045i		−	1.04492i
$848$		0			0
$849$	122.657	−	1255.89i	0.144473	−	1.47925i
$850$		0			0
$851$		−	772.602i		−	0.907875i
$852$		0			0
$853$		−	1293.04i		−	1.51587i	−0.652330	−	0.757935i	$-0.726208\pi$
$853$		−	1293.04i		−	1.51587i	0.652330	−	0.757935i	$-0.273792\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1513.48			−1.76602			−0.883008	−	0.469357i	$-0.844486\pi$
$857$	−1513.48			−1.76602			−0.883008	+	0.469357i	$0.844486\pi$
$858$		0			0
$859$	767.814			0.893846			0.446923	−	0.894572i	$-0.352520\pi$
$859$	767.814			0.893846			0.446923	+	0.894572i	$0.352520\pi$
$860$		0			0
$861$	903.724	+	88.2629i	1.04962	+	0.102512i
$862$		0			0
$863$	−29.2279			−0.0338677			−0.0169339	−	0.999857i	$-0.505390\pi$
$863$	−29.2279			−0.0338677			−0.0169339	+	0.999857i	$0.505390\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	861.012	+	84.0914i	0.993093	+	0.0969913i
$868$		0			0
$869$		−	38.6078i		−	0.0444279i
$870$		0			0
$871$	33.3120			0.0382457
$872$		0			0
$873$	−159.576	+	809.157i	−0.182791	+	0.926870i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			1070.98i			1.22119i	0.791943	+	0.610595i	$0.209070\pi$
$877$			1070.98i			1.22119i	−0.791943	+	0.610595i	$0.790930\pi$
$878$		0			0
$879$	−847.452	−	82.7671i	−0.964109	−	0.0941605i
$880$		0			0
$881$			489.535i			0.555659i	0.960630	+	0.277829i	$0.0896150\pi$
$881$			489.535i			0.555659i	−0.960630	+	0.277829i	$0.910385\pi$
$882$		0			0
$883$			742.175i			0.840515i	0.907405	+	0.420257i	$0.138060\pi$
$883$			742.175i			0.840515i	−0.907405	+	0.420257i	$0.861940\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1441.82			1.62550			0.812751	−	0.582612i	$-0.197969\pi$
$887$	1441.82			1.62550			0.812751	+	0.582612i	$0.197969\pi$
$888$		0			0
$889$	−503.924			−0.566844
$890$		0			0
$891$	−549.533	+	1339.06i	−0.616759	+	1.50287i
$892$		0			0
$893$	−328.282			−0.367617
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−47.9303	+	490.758i	−0.0534340	+	0.547110i
$898$		0			0
$899$			296.629i			0.329954i
$900$		0			0
$901$	−29.4079			−0.0326392
$902$		0			0
$903$	543.969	+	53.1272i	0.602402	+	0.0588341i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	352.397i		−	0.388531i	−0.980949	−	0.194265i	$-0.937768\pi$
$907$		−	352.397i		−	0.388531i	0.980949	−	0.194265i	$-0.0622323\pi$
$908$		0			0
$909$	−202.614	+	1027.39i	−0.222898	+	1.13024i
$910$		0			0
$911$			184.505i			0.202531i	0.994859	+	0.101265i	$0.0322891\pi$
$911$			184.505i			0.202531i	−0.994859	+	0.101265i	$0.967711\pi$
$912$		0			0
$913$		−	233.574i		−	0.255831i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−478.545			−0.521859
$918$		0			0
$919$	666.062			0.724769			0.362384	−	0.932029i	$-0.381963\pi$
$919$	666.062			0.724769			0.362384	+	0.932029i	$0.381963\pi$
$920$		0			0
$921$	48.9843	−	501.550i	0.0531860	−	0.544571i
$922$		0			0
$923$	628.479			0.680909
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−317.055	+	1607.68i	−0.342022	+	1.73428i
$928$		0			0
$929$			957.310i			1.03047i	0.857048	+	0.515237i	$0.172296\pi$
$929$			957.310i			1.03047i	−0.857048	+	0.515237i	$0.827704\pi$
$930$		0			0
$931$	773.218			0.830524
$932$		0			0
$933$	−56.2720	+	576.169i	−0.0603130	+	0.617544i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	1108.63i		−	1.18317i	−0.806243	−	0.591585i	$-0.798502\pi$
$937$		−	1108.63i		−	1.18317i	0.806243	−	0.591585i	$-0.201498\pi$
$938$		0			0
$939$	46.9655	−	480.879i	0.0500165	−	0.512119i
$940$		0			0
$941$		−	50.4523i		−	0.0536157i	−0.999641	−	0.0268078i	$-0.991466\pi$
$941$		−	50.4523i		−	0.0536157i	0.999641	−	0.0268078i	$-0.00853422\pi$
$942$		0			0
$943$			1012.45i			1.07364i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	883.676			0.933132			0.466566	−	0.884486i	$-0.345491\pi$
$947$	883.676			0.933132			0.466566	+	0.884486i	$0.345491\pi$
$948$		0			0
$949$	−346.497			−0.365118
$950$		0			0
$951$	−1536.97	−	150.109i	−1.61616	−	0.157844i
$952$		0			0
$953$	1238.78			1.29988			0.649938	−	0.759987i	$-0.274795\pi$
$953$	1238.78			1.29988			0.649938	+	0.759987i	$0.274795\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−298.011	−	29.1055i	−0.311401	−	0.0304132i
$958$		0			0
$959$			302.317i			0.315242i
$960$		0			0
$961$	1859.39			1.93485
$962$		0			0
$963$	−907.967	−	179.063i	−0.942852	−	0.185943i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1221.47i			1.26315i	0.775314	+	0.631575i	$0.217591\pi$
$967$			1221.47i			1.26315i	−0.775314	+	0.631575i	$0.782409\pi$
$968$		0			0
$969$	−63.0304	−	6.15592i	−0.0650469	−	0.00635286i
$970$		0			0
$971$			1142.08i			1.17619i	0.808792	+	0.588095i	$0.200122\pi$
$971$			1142.08i			1.17619i	−0.808792	+	0.588095i	$0.799878\pi$
$972$		0			0
$973$		−	630.148i		−	0.647634i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	364.738			0.373324			0.186662	−	0.982424i	$-0.440233\pi$
$977$	364.738			0.373324			0.186662	+	0.982424i	$0.440233\pi$
$978$		0			0
$979$	−3103.81			−3.17039
$980$		0			0
$981$	−666.004	−	131.344i	−0.678903	−	0.133888i
$982$		0			0
$983$	40.0459			0.0407384			0.0203692	−	0.999793i	$-0.493516\pi$
$983$	40.0459			0.0407384			0.0203692	+	0.999793i	$0.493516\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−16.0696	+	164.536i	−0.0162812	+	0.166703i
$988$		0			0
$989$			609.410i			0.616188i
$990$		0			0
$991$	910.307			0.918574			0.459287	−	0.888288i	$-0.348105\pi$
$991$	910.307			0.918574			0.459287	+	0.888288i	$0.348105\pi$
$992$		0			0
$993$	190.334	+	18.5891i	0.191676	+	0.0187202i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	358.437i		−	0.359516i	−0.983711	−	0.179758i	$-0.942469\pi$
$997$		−	358.437i		−	0.359516i	0.983711	−	0.179758i	$-0.0575314\pi$
$998$		0			0
$999$	−402.369	+	1338.24i	−0.402772	+	1.33958i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.c.m.449.1		16
3.2	odd	2	inner	1200.3.c.m.449.15		16
4.3	odd	2		600.3.c.d.449.16		16
5.2	odd	4		240.3.l.d.161.4		8
5.3	odd	4		1200.3.l.x.401.5		8
5.4	even	2	inner	1200.3.c.m.449.16		16
12.11	even	2		600.3.c.d.449.2		16
15.2	even	4		240.3.l.d.161.3		8
15.8	even	4		1200.3.l.x.401.6		8
15.14	odd	2	inner	1200.3.c.m.449.2		16
20.3	even	4		600.3.l.f.401.4		8
20.7	even	4		120.3.l.a.41.5	✓	8
20.19	odd	2		600.3.c.d.449.1		16
40.27	even	4		960.3.l.h.641.4		8
40.37	odd	4		960.3.l.g.641.5		8
60.23	odd	4		600.3.l.f.401.3		8
60.47	odd	4		120.3.l.a.41.6	yes	8
60.59	even	2		600.3.c.d.449.15		16
120.77	even	4		960.3.l.g.641.6		8
120.107	odd	4		960.3.l.h.641.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.l.a.41.5	✓	8	20.7	even	4
120.3.l.a.41.6	yes	8	60.47	odd	4
240.3.l.d.161.3		8	15.2	even	4
240.3.l.d.161.4		8	5.2	odd	4
600.3.c.d.449.1		16	20.19	odd	2
600.3.c.d.449.2		16	12.11	even	2
600.3.c.d.449.15		16	60.59	even	2
600.3.c.d.449.16		16	4.3	odd	2
600.3.l.f.401.3		8	60.23	odd	4
600.3.l.f.401.4		8	20.3	even	4
960.3.l.g.641.5		8	40.37	odd	4
960.3.l.g.641.6		8	120.77	even	4
960.3.l.h.641.3		8	120.107	odd	4
960.3.l.h.641.4		8	40.27	even	4
1200.3.c.m.449.1		16	1.1	even	1	trivial
1200.3.c.m.449.2		16	15.14	odd	2	inner
1200.3.c.m.449.15		16	3.2	odd	2	inner
1200.3.c.m.449.16		16	5.4	even	2	inner
1200.3.l.x.401.5		8	5.3	odd	4
1200.3.l.x.401.6		8	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)

\(f(q)\)	\(=\)	\(q+(-2.98579 - 0.291610i) q^{3} +4.46268i q^{7} +(8.82993 + 1.74137i) q^{9} +O(q^{10})\)	\(q+(-2.98579 - 0.291610i) q^{3} +4.46268i q^{7} +(8.82993 + 1.74137i) q^{9} +17.8696i q^{11} -11.0107i q^{13} +0.794055 q^{17} +26.5852 q^{19} +(1.30136 - 13.3246i) q^{21} +14.9276 q^{23} +(-25.8565 - 7.77428i) q^{27} -5.58545i q^{29} -53.1074 q^{31} +(5.21095 - 53.3549i) q^{33} -51.7565i q^{37} +(-3.21084 + 32.8758i) q^{39} +67.8236i q^{41} +40.8243i q^{43} -12.3483 q^{47} +29.0845 q^{49} +(-2.37088 - 0.231554i) q^{51} -37.0351 q^{53} +(-79.3779 - 7.75251i) q^{57} -61.0932i q^{59} +97.8289 q^{61} +(-7.77119 + 39.4051i) q^{63} +3.02541i q^{67} +(-44.5708 - 4.35305i) q^{69} +57.0787i q^{71} -31.4690i q^{73} -79.7461 q^{77} -2.16053 q^{79} +(74.9352 + 30.7524i) q^{81} -13.0710 q^{83} +(-1.62877 + 16.6770i) q^{87} +173.692i q^{89} +49.1374 q^{91} +(158.568 + 15.4866i) q^{93} +91.6381i q^{97} +(-31.1176 + 157.787i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 40 q^{9}+O(q^{10}) \) 16 * q - 40 * q^9	\( 16 q - 40 q^{9} - 16 q^{19} + 56 q^{21} - 240 q^{31} - 144 q^{39} - 128 q^{49} - 128 q^{51} + 16 q^{61} - 200 q^{69} + 176 q^{79} + 448 q^{81} - 1120 q^{91} + 64 q^{99}+O(q^{100}) \) 16 * q - 40 * q^9 - 16 * q^19 + 56 * q^21 - 240 * q^31 - 144 * q^39 - 128 * q^49 - 128 * q^51 + 16 * q^61 - 200 * q^69 + 176 * q^79 + 448 * q^81 - 1120 * q^91 + 64 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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