LMFDB - Embedded newform 1200.2.h.j.1151.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,2,Mod(1151,1200)]

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([1, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1200.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1200.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$9.58204824255$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.4
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1200.1151
Dual form			1200.2.h.j.1151.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.41421i) q^{3} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})$	$q+(-1.00000 + 1.41421i) q^{3} +(-1.00000 - 2.82843i) q^{9} +4.89898 q^{11} -4.89898 q^{13} +3.46410i q^{17} +3.46410i q^{19} +6.00000 q^{23} +(5.00000 + 1.41421i) q^{27} +2.82843i q^{29} +3.46410i q^{31} +(-4.89898 + 6.92820i) q^{33} -4.89898 q^{37} +(4.89898 - 6.92820i) q^{39} -5.65685i q^{41} +8.48528i q^{43} -6.00000 q^{47} +7.00000 q^{49} +(-4.89898 - 3.46410i) q^{51} +10.3923i q^{53} +(-4.89898 - 3.46410i) q^{57} -4.89898 q^{59} -2.00000 q^{61} +8.48528i q^{67} +(-6.00000 + 8.48528i) q^{69} -9.79796 q^{71} -9.79796 q^{73} +10.3923i q^{79} +(-7.00000 + 5.65685i) q^{81} +6.00000 q^{83} +(-4.00000 - 2.82843i) q^{87} -5.65685i q^{89} +(-4.89898 - 3.46410i) q^{93} +(-4.89898 - 13.8564i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 4 * q^9	$ 4 q - 4 q^{3} - 4 q^{9} + 24 q^{23} + 20 q^{27} - 24 q^{47} + 28 q^{49} - 8 q^{61} - 24 q^{69} - 28 q^{81} + 24 q^{83} - 16 q^{87}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^9 + 24 * q^23 + 20 * q^27 - 24 * q^47 + 28 * q^49 - 8 * q^61 - 24 * q^69 - 28 * q^81 + 24 * q^83 - 16 * q^87

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.00000	+	1.41421i	−0.577350	+	0.816497i
$3$	−1.00000	+	1.41421i	−0.577350	+	0.816497i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−1.00000	−	2.82843i	−0.333333	−	0.942809i
$10$		0			0
$11$	4.89898			1.47710			0.738549	−	0.674200i	$-0.235511\pi$
$11$	4.89898			1.47710			0.738549	+	0.674200i	$0.235511\pi$
$12$		0			0
$13$	−4.89898			−1.35873			−0.679366	−	0.733799i	$-0.737745\pi$
$13$	−4.89898			−1.35873			−0.679366	+	0.733799i	$0.737745\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.46410i			0.840168i	0.907485	+	0.420084i	$0.137999\pi$
$17$			3.46410i			0.840168i	−0.907485	+	0.420084i	$0.862001\pi$
$18$		0			0
$19$			3.46410i			0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$			3.46410i			0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.00000			1.25109			0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000			1.25109			0.625543	+	0.780189i	$0.284877\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	5.00000	+	1.41421i	0.962250	+	0.272166i
$28$		0			0
$29$			2.82843i			0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$			2.82843i			0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$		0			0
$31$			3.46410i			0.622171i	0.950382	+	0.311086i	$0.100693\pi$
$31$			3.46410i			0.622171i	−0.950382	+	0.311086i	$0.899307\pi$
$32$		0			0
$33$	−4.89898	+	6.92820i	−0.852803	+	1.20605i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.89898			−0.805387			−0.402694	−	0.915335i	$-0.631926\pi$
$37$	−4.89898			−0.805387			−0.402694	+	0.915335i	$0.631926\pi$
$38$		0			0
$39$	4.89898	−	6.92820i	0.784465	−	1.10940i
$40$		0			0
$41$		−	5.65685i		−	0.883452i	−0.897150	−	0.441726i	$-0.854366\pi$
$41$		−	5.65685i		−	0.883452i	0.897150	−	0.441726i	$-0.145634\pi$
$42$		0			0
$43$			8.48528i			1.29399i	0.762493	+	0.646997i	$0.223975\pi$
$43$			8.48528i			1.29399i	−0.762493	+	0.646997i	$0.776025\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.00000			−0.875190			−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000			−0.875190			−0.437595	+	0.899172i	$0.644170\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−4.89898	−	3.46410i	−0.685994	−	0.485071i
$52$		0			0
$53$			10.3923i			1.42749i	0.700404	+	0.713746i	$0.253003\pi$
$53$			10.3923i			1.42749i	−0.700404	+	0.713746i	$0.746997\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−4.89898	−	3.46410i	−0.648886	−	0.458831i
$58$		0			0
$59$	−4.89898			−0.637793			−0.318896	−	0.947790i	$-0.603312\pi$
$59$	−4.89898			−0.637793			−0.318896	+	0.947790i	$0.603312\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$			8.48528i			1.03664i	0.855186	+	0.518321i	$0.173443\pi$
$67$			8.48528i			1.03664i	−0.855186	+	0.518321i	$0.826557\pi$
$68$		0			0
$69$	−6.00000	+	8.48528i	−0.722315	+	1.02151i
$70$		0			0
$71$	−9.79796			−1.16280			−0.581402	−	0.813617i	$-0.697496\pi$
$71$	−9.79796			−1.16280			−0.581402	+	0.813617i	$0.697496\pi$
$72$		0			0
$73$	−9.79796			−1.14676			−0.573382	−	0.819288i	$-0.694369\pi$
$73$	−9.79796			−1.14676			−0.573382	+	0.819288i	$0.694369\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$			10.3923i			1.16923i	0.811312	+	0.584613i	$0.198754\pi$
$79$			10.3923i			1.16923i	−0.811312	+	0.584613i	$0.801246\pi$
$80$		0			0
$81$	−7.00000	+	5.65685i	−0.777778	+	0.628539i
$82$		0			0
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−4.00000	−	2.82843i	−0.428845	−	0.303239i
$88$		0			0
$89$		−	5.65685i		−	0.599625i	−0.953998	−	0.299813i	$-0.903076\pi$
$89$		−	5.65685i		−	0.599625i	0.953998	−	0.299813i	$-0.0969242\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−4.89898	−	3.46410i	−0.508001	−	0.359211i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$	−4.89898	−	13.8564i	−0.492366	−	1.39262i
$100$		0			0
$101$			14.1421i			1.40720i	0.710599	+	0.703598i	$0.248424\pi$
$101$			14.1421i			1.40720i	−0.710599	+	0.703598i	$0.751576\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	18.0000			1.74013			0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000			1.74013			0.870063	+	0.492941i	$0.164078\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$	4.89898	−	6.92820i	0.464991	−	0.657596i
$112$		0			0
$113$		−	3.46410i		−	0.325875i	−0.986636	−	0.162938i	$-0.947903\pi$
$113$		−	3.46410i		−	0.325875i	0.986636	−	0.162938i	$-0.0520969\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	4.89898	+	13.8564i	0.452911	+	1.28103i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	13.0000			1.18182
$122$		0			0
$123$	8.00000	+	5.65685i	0.721336	+	0.510061i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	16.9706i		−	1.50589i	−0.658081	−	0.752947i	$-0.728632\pi$
$127$		−	16.9706i		−	1.50589i	0.658081	−	0.752947i	$-0.271368\pi$
$128$		0			0
$129$	−12.0000	−	8.48528i	−1.05654	−	0.747087i
$130$		0			0
$131$	−4.89898			−0.428026			−0.214013	−	0.976831i	$-0.568653\pi$
$131$	−4.89898			−0.428026			−0.214013	+	0.976831i	$0.568653\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			3.46410i			0.295958i	0.988990	+	0.147979i	$0.0472768\pi$
$137$			3.46410i			0.295958i	−0.988990	+	0.147979i	$0.952723\pi$
$138$		0			0
$139$			17.3205i			1.46911i	0.678551	+	0.734553i	$0.262608\pi$
$139$			17.3205i			1.46911i	−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$	6.00000	−	8.48528i	0.505291	−	0.714590i
$142$		0			0
$143$	−24.0000			−2.00698
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−7.00000	+	9.89949i	−0.577350	+	0.816497i
$148$		0			0
$149$		−	2.82843i		−	0.231714i	−0.993266	−	0.115857i	$-0.963039\pi$
$149$		−	2.82843i		−	0.231714i	0.993266	−	0.115857i	$-0.0369614\pi$
$150$		0			0
$151$			3.46410i			0.281905i	0.990016	+	0.140952i	$0.0450164\pi$
$151$			3.46410i			0.281905i	−0.990016	+	0.140952i	$0.954984\pi$
$152$		0			0
$153$	9.79796	−	3.46410i	0.792118	−	0.280056i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	4.89898			0.390981			0.195491	−	0.980706i	$-0.437370\pi$
$157$	4.89898			0.390981			0.195491	+	0.980706i	$0.437370\pi$
$158$		0			0
$159$	−14.6969	−	10.3923i	−1.16554	−	0.824163i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	8.48528i		−	0.664619i	−0.943170	−	0.332309i	$-0.892172\pi$
$163$		−	8.48528i		−	0.664619i	0.943170	−	0.332309i	$-0.107828\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000			−0.464294			−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000			−0.464294			−0.232147	+	0.972681i	$0.574575\pi$
$168$		0			0
$169$	11.0000			0.846154
$170$		0			0
$171$	9.79796	−	3.46410i	0.749269	−	0.264906i
$172$		0			0
$173$		−	3.46410i		−	0.263371i	−0.991292	−	0.131685i	$-0.957961\pi$
$173$		−	3.46410i		−	0.263371i	0.991292	−	0.131685i	$-0.0420389\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	4.89898	−	6.92820i	0.368230	−	0.520756i
$178$		0			0
$179$	24.4949			1.83083			0.915417	−	0.402506i	$-0.131861\pi$
$179$	24.4949			1.83083			0.915417	+	0.402506i	$0.131861\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$	2.00000	−	2.82843i	0.147844	−	0.209083i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			16.9706i			1.24101i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	19.5959			1.41791			0.708955	−	0.705253i	$-0.249167\pi$
$191$	19.5959			1.41791			0.708955	+	0.705253i	$0.249167\pi$
$192$		0			0
$193$	19.5959			1.41055			0.705273	−	0.708936i	$-0.250825\pi$
$193$	19.5959			1.41055			0.705273	+	0.708936i	$0.250825\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	24.2487i		−	1.72765i	−0.503793	−	0.863825i	$-0.668062\pi$
$197$		−	24.2487i		−	1.72765i	0.503793	−	0.863825i	$-0.331938\pi$
$198$		0			0
$199$			10.3923i			0.736691i	0.929689	+	0.368345i	$0.120076\pi$
$199$			10.3923i			0.736691i	−0.929689	+	0.368345i	$0.879924\pi$
$200$		0			0
$201$	−12.0000	−	8.48528i	−0.846415	−	0.598506i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−6.00000	−	16.9706i	−0.417029	−	1.17954i
$208$		0			0
$209$			16.9706i			1.17388i
$210$		0			0
$211$			24.2487i			1.66935i	0.550743	+	0.834675i	$0.314345\pi$
$211$			24.2487i			1.66935i	−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$	9.79796	−	13.8564i	0.671345	−	0.949425i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	9.79796	−	13.8564i	0.662085	−	0.936329i
$220$		0			0
$221$		−	16.9706i		−	1.14156i
$222$		0			0
$223$		−	16.9706i		−	1.13643i	−0.822879	−	0.568216i	$-0.807634\pi$
$223$		−	16.9706i		−	1.13643i	0.822879	−	0.568216i	$-0.192366\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.00000			−0.398234			−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000			−0.398234			−0.199117	+	0.979976i	$0.563807\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	3.46410i		−	0.226941i	−0.993541	−	0.113470i	$-0.963803\pi$
$233$		−	3.46410i		−	0.226941i	0.993541	−	0.113470i	$-0.0361967\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−14.6969	−	10.3923i	−0.954669	−	0.675053i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	10.0000			0.644157			0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000			0.644157			0.322078	+	0.946713i	$0.395619\pi$
$242$		0			0
$243$	−1.00000	−	15.5563i	−0.0641500	−	0.997940i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	16.9706i		−	1.07981i
$248$		0			0
$249$	−6.00000	+	8.48528i	−0.380235	+	0.537733i
$250$		0			0
$251$	−14.6969			−0.927663			−0.463831	−	0.885924i	$-0.653526\pi$
$251$	−14.6969			−0.927663			−0.463831	+	0.885924i	$0.653526\pi$
$252$		0			0
$253$	29.3939			1.84798
$254$		0			0
$255$		0			0
$256$		0			0
$257$			3.46410i			0.216085i	0.994146	+	0.108042i	$0.0344582\pi$
$257$			3.46410i			0.216085i	−0.994146	+	0.108042i	$0.965542\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	8.00000	−	2.82843i	0.495188	−	0.175075i
$262$		0			0
$263$	6.00000			0.369976			0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000			0.369976			0.184988	+	0.982741i	$0.440775\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	8.00000	+	5.65685i	0.489592	+	0.346194i
$268$		0			0
$269$		−	19.7990i		−	1.20717i	−0.797300	−	0.603583i	$-0.793739\pi$
$269$		−	19.7990i		−	1.20717i	0.797300	−	0.603583i	$-0.206261\pi$
$270$		0			0
$271$		−	10.3923i		−	0.631288i	−0.948878	−	0.315644i	$-0.897780\pi$
$271$		−	10.3923i		−	0.631288i	0.948878	−	0.315644i	$-0.102220\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	14.6969			0.883053			0.441527	−	0.897248i	$-0.354437\pi$
$277$	14.6969			0.883053			0.441527	+	0.897248i	$0.354437\pi$
$278$		0			0
$279$	9.79796	−	3.46410i	0.586588	−	0.207390i
$280$		0			0
$281$		−	5.65685i		−	0.337460i	−0.985662	−	0.168730i	$-0.946033\pi$
$281$		−	5.65685i		−	0.337460i	0.985662	−	0.168730i	$-0.0539665\pi$
$282$		0			0
$283$			8.48528i			0.504398i	0.967675	+	0.252199i	$0.0811537\pi$
$283$			8.48528i			0.504398i	−0.967675	+	0.252199i	$0.918846\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	5.00000			0.294118
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	17.3205i		−	1.01187i	−0.862570	−	0.505937i	$-0.831147\pi$
$293$		−	17.3205i		−	1.01187i	0.862570	−	0.505937i	$-0.168853\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	24.4949	+	6.92820i	1.42134	+	0.402015i
$298$		0			0
$299$	−29.3939			−1.69989
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−20.0000	−	14.1421i	−1.14897	−	0.812444i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			8.48528i			0.484281i	0.970241	+	0.242140i	$0.0778494\pi$
$307$			8.48528i			0.484281i	−0.970241	+	0.242140i	$0.922151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	9.79796			0.555591			0.277796	−	0.960640i	$-0.410396\pi$
$311$	9.79796			0.555591			0.277796	+	0.960640i	$0.410396\pi$
$312$		0			0
$313$	9.79796			0.553813			0.276907	−	0.960897i	$-0.410691\pi$
$313$	9.79796			0.553813			0.276907	+	0.960897i	$0.410691\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			17.3205i			0.972817i	0.873732	+	0.486408i	$0.161693\pi$
$317$			17.3205i			0.972817i	−0.873732	+	0.486408i	$0.838307\pi$
$318$		0			0
$319$			13.8564i			0.775810i
$320$		0			0
$321$	−18.0000	+	25.4558i	−1.00466	+	1.42081i
$322$		0			0
$323$	−12.0000			−0.667698
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−2.00000	+	2.82843i	−0.110600	+	0.156412i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	17.3205i		−	0.952021i	−0.879440	−	0.476011i	$-0.842082\pi$
$331$		−	17.3205i		−	0.952021i	0.879440	−	0.476011i	$-0.157918\pi$
$332$		0			0
$333$	4.89898	+	13.8564i	0.268462	+	0.759326i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$	4.89898	+	3.46410i	0.266076	+	0.188144i
$340$		0			0
$341$			16.9706i			0.919007i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.00000			−0.322097			−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000			−0.322097			−0.161048	+	0.986947i	$0.551488\pi$
$348$		0			0
$349$	−14.0000			−0.749403			−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000			−0.749403			−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$	−24.4949	−	6.92820i	−1.30744	−	0.369800i
$352$		0			0
$353$		−	31.1769i		−	1.65938i	−0.558225	−	0.829690i	$-0.688517\pi$
$353$		−	31.1769i		−	1.65938i	0.558225	−	0.829690i	$-0.311483\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9.79796			−0.517116			−0.258558	−	0.965996i	$-0.583247\pi$
$359$	−9.79796			−0.517116			−0.258558	+	0.965996i	$0.583247\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$		0			0
$363$	−13.0000	+	18.3848i	−0.682323	+	0.964951i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	16.9706i		−	0.885856i	−0.896557	−	0.442928i	$-0.853940\pi$
$367$		−	16.9706i		−	0.885856i	0.896557	−	0.442928i	$-0.146060\pi$
$368$		0			0
$369$	−16.0000	+	5.65685i	−0.832927	+	0.294484i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−14.6969			−0.760979			−0.380489	−	0.924785i	$-0.624244\pi$
$373$	−14.6969			−0.760979			−0.380489	+	0.924785i	$0.624244\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	13.8564i		−	0.713641i
$378$		0			0
$379$		−	10.3923i		−	0.533817i	−0.963722	−	0.266908i	$-0.913998\pi$
$379$		−	10.3923i		−	0.533817i	0.963722	−	0.266908i	$-0.0860021\pi$
$380$		0			0
$381$	24.0000	+	16.9706i	1.22956	+	0.869428i
$382$		0			0
$383$	6.00000			0.306586			0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000			0.306586			0.153293	+	0.988181i	$0.451012\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	24.0000	−	8.48528i	1.21999	−	0.431331i
$388$		0			0
$389$		−	2.82843i		−	0.143407i	−0.997426	−	0.0717035i	$-0.977156\pi$
$389$		−	2.82843i		−	0.143407i	0.997426	−	0.0717035i	$-0.0228435\pi$
$390$		0			0
$391$			20.7846i			1.05112i
$392$		0			0
$393$	4.89898	−	6.92820i	0.247121	−	0.349482i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	4.89898			0.245873			0.122936	−	0.992415i	$-0.460769\pi$
$397$	4.89898			0.245873			0.122936	+	0.992415i	$0.460769\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			22.6274i			1.12996i	0.825105	+	0.564980i	$0.191116\pi$
$401$			22.6274i			1.12996i	−0.825105	+	0.564980i	$0.808884\pi$
$402$		0			0
$403$		−	16.9706i		−	0.845364i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−24.0000			−1.18964
$408$		0			0
$409$	−10.0000			−0.494468			−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000			−0.494468			−0.247234	+	0.968956i	$0.579522\pi$
$410$		0			0
$411$	−4.89898	−	3.46410i	−0.241649	−	0.170872i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−24.4949	−	17.3205i	−1.19952	−	0.848189i
$418$		0			0
$419$	−14.6969			−0.717992			−0.358996	−	0.933339i	$-0.616881\pi$
$419$	−14.6969			−0.717992			−0.358996	+	0.933339i	$0.616881\pi$
$420$		0			0
$421$	−26.0000			−1.26716			−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$		0			0
$423$	6.00000	+	16.9706i	0.291730	+	0.825137i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	24.0000	−	33.9411i	1.15873	−	1.63869i
$430$		0			0
$431$	−19.5959			−0.943902			−0.471951	−	0.881625i	$-0.656450\pi$
$431$	−19.5959			−0.943902			−0.471951	+	0.881625i	$0.656450\pi$
$432$		0			0
$433$	−19.5959			−0.941720			−0.470860	−	0.882208i	$-0.656056\pi$
$433$	−19.5959			−0.941720			−0.470860	+	0.882208i	$0.656056\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			20.7846i			0.994263i
$438$		0			0
$439$		−	17.3205i		−	0.826663i	−0.910581	−	0.413331i	$-0.864365\pi$
$439$		−	17.3205i		−	0.826663i	0.910581	−	0.413331i	$-0.135635\pi$
$440$		0			0
$441$	−7.00000	−	19.7990i	−0.333333	−	0.942809i
$442$		0			0
$443$	30.0000			1.42534			0.712672	−	0.701498i	$-0.247485\pi$
$443$	30.0000			1.42534			0.712672	+	0.701498i	$0.247485\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	4.00000	+	2.82843i	0.189194	+	0.133780i
$448$		0			0
$449$			22.6274i			1.06785i	0.845531	+	0.533927i	$0.179284\pi$
$449$			22.6274i			1.06785i	−0.845531	+	0.533927i	$0.820716\pi$
$450$		0			0
$451$		−	27.7128i		−	1.30495i
$452$		0			0
$453$	−4.89898	−	3.46410i	−0.230174	−	0.162758i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−9.79796			−0.458329			−0.229165	−	0.973388i	$-0.573599\pi$
$457$	−9.79796			−0.458329			−0.229165	+	0.973388i	$0.573599\pi$
$458$		0			0
$459$	−4.89898	+	17.3205i	−0.228665	+	0.808452i
$460$		0			0
$461$		−	36.7696i		−	1.71253i	−0.516538	−	0.856264i	$-0.672779\pi$
$461$		−	36.7696i		−	1.71253i	0.516538	−	0.856264i	$-0.327221\pi$
$462$		0			0
$463$		−	16.9706i		−	0.788689i	−0.918963	−	0.394344i	$-0.870972\pi$
$463$		−	16.9706i		−	0.788689i	0.918963	−	0.394344i	$-0.129028\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	18.0000			0.832941			0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000			0.832941			0.416470	+	0.909149i	$0.363267\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−4.89898	+	6.92820i	−0.225733	+	0.319235i
$472$		0			0
$473$			41.5692i			1.91135i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	29.3939	−	10.3923i	1.34585	−	0.475831i
$478$		0			0
$479$	19.5959			0.895360			0.447680	−	0.894194i	$-0.352250\pi$
$479$	19.5959			0.895360			0.447680	+	0.894194i	$0.352250\pi$
$480$		0			0
$481$	24.0000			1.09431
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$			33.9411i			1.53802i	0.639237	+	0.769010i	$0.279250\pi$
$487$			33.9411i			1.53802i	−0.639237	+	0.769010i	$0.720750\pi$
$488$		0			0
$489$	12.0000	+	8.48528i	0.542659	+	0.383718i
$490$		0			0
$491$	24.4949			1.10544			0.552720	−	0.833367i	$-0.313590\pi$
$491$	24.4949			1.10544			0.552720	+	0.833367i	$0.313590\pi$
$492$		0			0
$493$	−9.79796			−0.441278
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			3.46410i			0.155074i	0.996989	+	0.0775372i	$0.0247057\pi$
$499$			3.46410i			0.155074i	−0.996989	+	0.0775372i	$0.975294\pi$
$500$		0			0
$501$	6.00000	−	8.48528i	0.268060	−	0.379094i
$502$		0			0
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−11.0000	+	15.5563i	−0.488527	+	0.690882i
$508$		0			0
$509$			36.7696i			1.62978i	0.579614	+	0.814891i	$0.303203\pi$
$509$			36.7696i			1.62978i	−0.579614	+	0.814891i	$0.696797\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−4.89898	+	17.3205i	−0.216295	+	0.764719i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−29.3939			−1.29274
$518$		0			0
$519$	4.89898	+	3.46410i	0.215041	+	0.152057i
$520$		0			0
$521$			5.65685i			0.247831i	0.992293	+	0.123916i	$0.0395452\pi$
$521$			5.65685i			0.247831i	−0.992293	+	0.123916i	$0.960455\pi$
$522$		0			0
$523$			42.4264i			1.85518i	0.373603	+	0.927589i	$0.378122\pi$
$523$			42.4264i			1.85518i	−0.373603	+	0.927589i	$0.621878\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.0000			−0.522728
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$	4.89898	+	13.8564i	0.212598	+	0.601317i
$532$		0			0
$533$			27.7128i			1.20038i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−24.4949	+	34.6410i	−1.05703	+	1.49487i
$538$		0			0
$539$	34.2929			1.47710
$540$		0			0
$541$	−34.0000			−1.46177			−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000			−1.46177			−0.730887	+	0.682498i	$0.760893\pi$
$542$		0			0
$543$	10.0000	−	14.1421i	0.429141	−	0.606897i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	25.4558i		−	1.08841i	−0.838951	−	0.544207i	$-0.816831\pi$
$547$		−	25.4558i		−	1.08841i	0.838951	−	0.544207i	$-0.183169\pi$
$548$		0			0
$549$	2.00000	+	5.65685i	0.0853579	+	0.241429i
$550$		0			0
$551$	−9.79796			−0.417407
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$			3.46410i			0.146779i	0.997303	+	0.0733893i	$0.0233816\pi$
$557$			3.46410i			0.146779i	−0.997303	+	0.0733893i	$0.976618\pi$
$558$		0			0
$559$		−	41.5692i		−	1.75819i
$560$		0			0
$561$	−24.0000	−	16.9706i	−1.01328	−	0.716498i
$562$		0			0
$563$	−18.0000			−0.758610			−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000			−0.758610			−0.379305	+	0.925272i	$0.623837\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	28.2843i		−	1.18574i	−0.805299	−	0.592869i	$-0.797995\pi$
$569$		−	28.2843i		−	1.18574i	0.805299	−	0.592869i	$-0.202005\pi$
$570$		0			0
$571$			38.1051i			1.59465i	0.603550	+	0.797325i	$0.293752\pi$
$571$			38.1051i			1.59465i	−0.603550	+	0.797325i	$0.706248\pi$
$572$		0			0
$573$	−19.5959	+	27.7128i	−0.818631	+	1.15772i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	19.5959			0.815789			0.407894	−	0.913029i	$-0.366263\pi$
$577$	19.5959			0.815789			0.407894	+	0.913029i	$0.366263\pi$
$578$		0			0
$579$	−19.5959	+	27.7128i	−0.814379	+	1.15171i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			50.9117i			2.10855i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	18.0000			0.742940			0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000			0.742940			0.371470	+	0.928445i	$0.378854\pi$
$588$		0			0
$589$	−12.0000			−0.494451
$590$		0			0
$591$	34.2929	+	24.2487i	1.41062	+	0.997459i
$592$		0			0
$593$			24.2487i			0.995775i	0.867242	+	0.497888i	$0.165891\pi$
$593$			24.2487i			0.995775i	−0.867242	+	0.497888i	$0.834109\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−14.6969	−	10.3923i	−0.601506	−	0.425329i
$598$		0			0
$599$	9.79796			0.400334			0.200167	−	0.979762i	$-0.435852\pi$
$599$	9.79796			0.400334			0.200167	+	0.979762i	$0.435852\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$	24.0000	−	8.48528i	0.977356	−	0.345547i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			16.9706i			0.688814i	0.938820	+	0.344407i	$0.111920\pi$
$607$			16.9706i			0.688814i	−0.938820	+	0.344407i	$0.888080\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	29.3939			1.18915
$612$		0			0
$613$	−14.6969			−0.593604			−0.296802	−	0.954939i	$-0.595920\pi$
$613$	−14.6969			−0.593604			−0.296802	+	0.954939i	$0.595920\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	24.2487i		−	0.976216i	−0.872783	−	0.488108i	$-0.837687\pi$
$617$		−	24.2487i		−	0.976216i	0.872783	−	0.488108i	$-0.162313\pi$
$618$		0			0
$619$		−	10.3923i		−	0.417702i	−0.977947	−	0.208851i	$-0.933028\pi$
$619$		−	10.3923i		−	0.417702i	0.977947	−	0.208851i	$-0.0669724\pi$
$620$		0			0
$621$	30.0000	+	8.48528i	1.20386	+	0.340503i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−24.0000	−	16.9706i	−0.958468	−	0.677739i
$628$		0			0
$629$		−	16.9706i		−	0.676661i
$630$		0			0
$631$		−	38.1051i		−	1.51694i	−0.651707	−	0.758470i	$-0.725947\pi$
$631$		−	38.1051i		−	1.51694i	0.651707	−	0.758470i	$-0.274053\pi$
$632$		0			0
$633$	−34.2929	−	24.2487i	−1.36302	−	0.963800i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−34.2929			−1.35873
$638$		0			0
$639$	9.79796	+	27.7128i	0.387601	+	1.09630i
$640$		0			0
$641$		−	22.6274i		−	0.893729i	−0.894602	−	0.446865i	$-0.852541\pi$
$641$		−	22.6274i		−	0.893729i	0.894602	−	0.446865i	$-0.147459\pi$
$642$		0			0
$643$		−	8.48528i		−	0.334627i	−0.985904	−	0.167313i	$-0.946491\pi$
$643$		−	8.48528i		−	0.334627i	0.985904	−	0.167313i	$-0.0535092\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	42.0000			1.65119			0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000			1.65119			0.825595	+	0.564263i	$0.190840\pi$
$648$		0			0
$649$	−24.0000			−0.942082
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	31.1769i		−	1.22005i	−0.792383	−	0.610023i	$-0.791160\pi$
$653$		−	31.1769i		−	1.22005i	0.792383	−	0.610023i	$-0.208840\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	9.79796	+	27.7128i	0.382255	+	1.08118i
$658$		0			0
$659$	−14.6969			−0.572511			−0.286256	−	0.958153i	$-0.592411\pi$
$659$	−14.6969			−0.572511			−0.286256	+	0.958153i	$0.592411\pi$
$660$		0			0
$661$	22.0000			0.855701			0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000			0.855701			0.427850	+	0.903850i	$0.359271\pi$
$662$		0			0
$663$	24.0000	+	16.9706i	0.932083	+	0.659082i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			16.9706i			0.657103i
$668$		0			0
$669$	24.0000	+	16.9706i	0.927894	+	0.656120i
$670$		0			0
$671$	−9.79796			−0.378246
$672$		0			0
$673$	−39.1918			−1.51073			−0.755367	−	0.655302i	$-0.772541\pi$
$673$	−39.1918			−1.51073			−0.755367	+	0.655302i	$0.772541\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	10.3923i		−	0.399409i	−0.979856	−	0.199704i	$-0.936002\pi$
$677$		−	10.3923i		−	0.399409i	0.979856	−	0.199704i	$-0.0639982\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	6.00000	−	8.48528i	0.229920	−	0.325157i
$682$		0			0
$683$	6.00000			0.229584			0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000			0.229584			0.114792	+	0.993390i	$0.463380\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−10.0000	+	14.1421i	−0.381524	+	0.539556i
$688$		0			0
$689$		−	50.9117i		−	1.93958i
$690$		0			0
$691$		−	31.1769i		−	1.18603i	−0.805193	−	0.593013i	$-0.797938\pi$
$691$		−	31.1769i		−	1.18603i	0.805193	−	0.593013i	$-0.202062\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	19.5959			0.742248
$698$		0			0
$699$	4.89898	+	3.46410i	0.185296	+	0.131024i
$700$		0			0
$701$			19.7990i			0.747798i	0.927470	+	0.373899i	$0.121979\pi$
$701$			19.7990i			0.747798i	−0.927470	+	0.373899i	$0.878021\pi$
$702$		0			0
$703$		−	16.9706i		−	0.640057i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−38.0000			−1.42712			−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000			−1.42712			−0.713560	+	0.700594i	$0.752918\pi$
$710$		0			0
$711$	29.3939	−	10.3923i	1.10236	−	0.389742i
$712$		0			0
$713$			20.7846i			0.778390i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−19.5959			−0.730804			−0.365402	−	0.930850i	$-0.619069\pi$
$719$	−19.5959			−0.730804			−0.365402	+	0.930850i	$0.619069\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−10.0000	+	14.1421i	−0.371904	+	0.525952i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	23.0000	+	14.1421i	0.851852	+	0.523783i
$730$		0			0
$731$	−29.3939			−1.08717
$732$		0			0
$733$	−44.0908			−1.62853			−0.814266	−	0.580492i	$-0.802860\pi$
$733$	−44.0908			−1.62853			−0.814266	+	0.580492i	$0.802860\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			41.5692i			1.53122i
$738$		0			0
$739$		−	24.2487i		−	0.892003i	−0.895032	−	0.446002i	$-0.852848\pi$
$739$		−	24.2487i		−	0.892003i	0.895032	−	0.446002i	$-0.147152\pi$
$740$		0			0
$741$	24.0000	+	16.9706i	0.881662	+	0.623429i
$742$		0			0
$743$	6.00000			0.220119			0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000			0.220119			0.110059	+	0.993925i	$0.464896\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−6.00000	−	16.9706i	−0.219529	−	0.620920i
$748$		0			0
$749$		0			0
$750$		0			0
$751$			31.1769i			1.13766i	0.822455	+	0.568831i	$0.192604\pi$
$751$			31.1769i			1.13766i	−0.822455	+	0.568831i	$0.807396\pi$
$752$		0			0
$753$	14.6969	−	20.7846i	0.535586	−	0.757433i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	14.6969			0.534169			0.267085	−	0.963673i	$-0.413940\pi$
$757$	14.6969			0.534169			0.267085	+	0.963673i	$0.413940\pi$
$758$		0			0
$759$	−29.3939	+	41.5692i	−1.06693	+	1.50887i
$760$		0			0
$761$			39.5980i			1.43543i	0.696339	+	0.717713i	$0.254811\pi$
$761$			39.5980i			1.43543i	−0.696339	+	0.717713i	$0.745189\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	24.0000			0.866590
$768$		0			0
$769$	22.0000			0.793340			0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000			0.793340			0.396670	+	0.917961i	$0.370166\pi$
$770$		0			0
$771$	−4.89898	−	3.46410i	−0.176432	−	0.124757i
$772$		0			0
$773$			24.2487i			0.872166i	0.899907	+	0.436083i	$0.143635\pi$
$773$			24.2487i			0.872166i	−0.899907	+	0.436083i	$0.856365\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	19.5959			0.702097
$780$		0			0
$781$	−48.0000			−1.71758
$782$		0			0
$783$	−4.00000	+	14.1421i	−0.142948	+	0.505399i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	25.4558i		−	0.907403i	−0.891154	−	0.453701i	$-0.850103\pi$
$787$		−	25.4558i		−	0.907403i	0.891154	−	0.453701i	$-0.149897\pi$
$788$		0			0
$789$	−6.00000	+	8.48528i	−0.213606	+	0.302084i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	9.79796			0.347936
$794$		0			0
$795$		0			0
$796$		0			0
$797$			17.3205i			0.613524i	0.951786	+	0.306762i	$0.0992455\pi$
$797$			17.3205i			0.613524i	−0.951786	+	0.306762i	$0.900754\pi$
$798$		0			0
$799$		−	20.7846i		−	0.735307i
$800$		0			0
$801$	−16.0000	+	5.65685i	−0.565332	+	0.199875i
$802$		0			0
$803$	−48.0000			−1.69388
$804$		0			0
$805$		0			0
$806$		0			0
$807$	28.0000	+	19.7990i	0.985647	+	0.696957i
$808$		0			0
$809$		−	5.65685i		−	0.198884i	−0.995043	−	0.0994422i	$-0.968294\pi$
$809$		−	5.65685i		−	0.198884i	0.995043	−	0.0994422i	$-0.0317058\pi$
$810$		0			0
$811$		−	17.3205i		−	0.608205i	−0.952639	−	0.304103i	$-0.901643\pi$
$811$		−	17.3205i		−	0.608205i	0.952639	−	0.304103i	$-0.0983566\pi$
$812$		0			0
$813$	14.6969	+	10.3923i	0.515444	+	0.364474i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−29.3939			−1.02836
$818$		0			0
$819$		0			0
$820$		0			0
$821$			2.82843i			0.0987128i	0.998781	+	0.0493564i	$0.0157170\pi$
$821$			2.82843i			0.0987128i	−0.998781	+	0.0493564i	$0.984283\pi$
$822$		0			0
$823$			33.9411i			1.18311i	0.806263	+	0.591557i	$0.201486\pi$
$823$			33.9411i			1.18311i	−0.806263	+	0.591557i	$0.798514\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−54.0000			−1.87776			−0.938882	−	0.344239i	$-0.888137\pi$
$827$	−54.0000			−1.87776			−0.938882	+	0.344239i	$0.888137\pi$
$828$		0			0
$829$	−46.0000			−1.59765			−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000			−1.59765			−0.798823	+	0.601566i	$0.794544\pi$
$830$		0			0
$831$	−14.6969	+	20.7846i	−0.509831	+	0.721010i
$832$		0			0
$833$			24.2487i			0.840168i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−4.89898	+	17.3205i	−0.169334	+	0.598684i
$838$		0			0
$839$	−9.79796			−0.338263			−0.169132	−	0.985593i	$-0.554096\pi$
$839$	−9.79796			−0.338263			−0.169132	+	0.985593i	$0.554096\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		0			0
$843$	8.00000	+	5.65685i	0.275535	+	0.194832i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−12.0000	−	8.48528i	−0.411839	−	0.291214i
$850$		0			0
$851$	−29.3939			−1.00761
$852$		0			0
$853$	44.0908			1.50964			0.754820	−	0.655932i	$-0.227724\pi$
$853$	44.0908			1.50964			0.754820	+	0.655932i	$0.227724\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	24.2487i		−	0.828320i	−0.910204	−	0.414160i	$-0.864075\pi$
$857$		−	24.2487i		−	0.828320i	0.910204	−	0.414160i	$-0.135925\pi$
$858$		0			0
$859$		−	38.1051i		−	1.30013i	−0.759879	−	0.650065i	$-0.774742\pi$
$859$		−	38.1051i		−	1.30013i	0.759879	−	0.650065i	$-0.225258\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	6.00000			0.204242			0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000			0.204242			0.102121	+	0.994772i	$0.467437\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−5.00000	+	7.07107i	−0.169809	+	0.240146i
$868$		0			0
$869$			50.9117i			1.72706i
$870$		0			0
$871$		−	41.5692i		−	1.40852i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	24.4949			0.827134			0.413567	−	0.910474i	$-0.364283\pi$
$877$	24.4949			0.827134			0.413567	+	0.910474i	$0.364283\pi$
$878$		0			0
$879$	24.4949	+	17.3205i	0.826192	+	0.584206i
$880$		0			0
$881$		−	22.6274i		−	0.762337i	−0.924506	−	0.381169i	$-0.875522\pi$
$881$		−	22.6274i		−	0.762337i	0.924506	−	0.381169i	$-0.124478\pi$
$882$		0			0
$883$		−	42.4264i		−	1.42776i	−0.700267	−	0.713881i	$-0.746936\pi$
$883$		−	42.4264i		−	1.42776i	0.700267	−	0.713881i	$-0.253064\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	42.0000			1.41022			0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000			1.41022			0.705111	+	0.709097i	$0.250897\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−34.2929	+	27.7128i	−1.14885	+	0.928414i
$892$		0			0
$893$		−	20.7846i		−	0.695530i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	29.3939	−	41.5692i	0.981433	−	1.38796i
$898$		0			0
$899$	−9.79796			−0.326780
$900$		0			0
$901$	−36.0000			−1.19933
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	8.48528i		−	0.281749i	−0.990027	−	0.140875i	$-0.955009\pi$
$907$		−	8.48528i		−	0.281749i	0.990027	−	0.140875i	$-0.0449914\pi$
$908$		0			0
$909$	40.0000	−	14.1421i	1.32672	−	0.469065i
$910$		0			0
$911$	−19.5959			−0.649242			−0.324621	−	0.945844i	$-0.605237\pi$
$911$	−19.5959			−0.649242			−0.324621	+	0.945844i	$0.605237\pi$
$912$		0			0
$913$	29.3939			0.972795
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	3.46410i		−	0.114270i	−0.998366	−	0.0571351i	$-0.981803\pi$
$919$		−	3.46410i		−	0.114270i	0.998366	−	0.0571351i	$-0.0181966\pi$
$920$		0			0
$921$	−12.0000	−	8.48528i	−0.395413	−	0.279600i
$922$		0			0
$923$	48.0000			1.57994
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			22.6274i			0.742381i	0.928557	+	0.371191i	$0.121050\pi$
$929$			22.6274i			0.742381i	−0.928557	+	0.371191i	$0.878950\pi$
$930$		0			0
$931$			24.2487i			0.794719i
$932$		0			0
$933$	−9.79796	+	13.8564i	−0.320771	+	0.453638i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−48.9898			−1.60043			−0.800213	−	0.599715i	$-0.795280\pi$
$937$	−48.9898			−1.60043			−0.800213	+	0.599715i	$0.795280\pi$
$938$		0			0
$939$	−9.79796	+	13.8564i	−0.319744	+	0.452187i
$940$		0			0
$941$		−	2.82843i		−	0.0922041i	−0.998937	−	0.0461020i	$-0.985320\pi$
$941$		−	2.82843i		−	0.0922041i	0.998937	−	0.0461020i	$-0.0146799\pi$
$942$		0			0
$943$		−	33.9411i		−	1.10528i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−54.0000			−1.75476			−0.877382	−	0.479792i	$-0.840712\pi$
$947$	−54.0000			−1.75476			−0.877382	+	0.479792i	$0.840712\pi$
$948$		0			0
$949$	48.0000			1.55815
$950$		0			0
$951$	−24.4949	−	17.3205i	−0.794301	−	0.561656i
$952$		0			0
$953$		−	58.8897i		−	1.90763i	−0.300402	−	0.953813i	$-0.597121\pi$
$953$		−	58.8897i		−	1.90763i	0.300402	−	0.953813i	$-0.402879\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−19.5959	−	13.8564i	−0.633446	−	0.447914i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$	−18.0000	−	50.9117i	−0.580042	−	1.64061i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			33.9411i			1.09147i	0.837957	+	0.545737i	$0.183750\pi$
$967$			33.9411i			1.09147i	−0.837957	+	0.545737i	$0.816250\pi$
$968$		0			0
$969$	12.0000	−	16.9706i	0.385496	−	0.545173i
$970$		0			0
$971$	4.89898			0.157216			0.0786079	−	0.996906i	$-0.474952\pi$
$971$	4.89898			0.157216			0.0786079	+	0.996906i	$0.474952\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	51.9615i		−	1.66240i	−0.555976	−	0.831198i	$-0.687655\pi$
$977$		−	51.9615i		−	1.66240i	0.555976	−	0.831198i	$-0.312345\pi$
$978$		0			0
$979$		−	27.7128i		−	0.885705i
$980$		0			0
$981$	−2.00000	−	5.65685i	−0.0638551	−	0.180609i
$982$		0			0
$983$	6.00000			0.191370			0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000			0.191370			0.0956851	+	0.995412i	$0.469496\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			50.9117i			1.61890i
$990$		0			0
$991$		−	38.1051i		−	1.21045i	−0.796055	−	0.605224i	$-0.793083\pi$
$991$		−	38.1051i		−	1.21045i	0.796055	−	0.605224i	$-0.206917\pi$
$992$		0			0
$993$	24.4949	+	17.3205i	0.777322	+	0.549650i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	53.8888			1.70667			0.853337	−	0.521359i	$-0.174575\pi$
$997$	53.8888			1.70667			0.853337	+	0.521359i	$0.174575\pi$
$998$		0			0
$999$	−24.4949	−	6.92820i	−0.774984	−	0.219199i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.2.h.j.1151.4		4
3.2	odd	2		1200.2.h.n.1151.3		4
4.3	odd	2		1200.2.h.n.1151.1		4
5.2	odd	4		240.2.o.b.239.7	yes	8
5.3	odd	4		240.2.o.b.239.2	yes	8
5.4	even	2		1200.2.h.n.1151.2		4
12.11	even	2	inner	1200.2.h.j.1151.2		4
15.2	even	4		240.2.o.b.239.6	yes	8
15.8	even	4		240.2.o.b.239.3	yes	8
15.14	odd	2	inner	1200.2.h.j.1151.1		4
20.3	even	4		240.2.o.b.239.8	yes	8
20.7	even	4		240.2.o.b.239.1	✓	8
20.19	odd	2	inner	1200.2.h.j.1151.3		4
40.3	even	4		960.2.o.d.959.1		8
40.13	odd	4		960.2.o.d.959.7		8
40.27	even	4		960.2.o.d.959.8		8
40.37	odd	4		960.2.o.d.959.2		8
60.23	odd	4		240.2.o.b.239.5	yes	8
60.47	odd	4		240.2.o.b.239.4	yes	8
60.59	even	2		1200.2.h.n.1151.4		4
120.53	even	4		960.2.o.d.959.6		8
120.77	even	4		960.2.o.d.959.3		8
120.83	odd	4		960.2.o.d.959.4		8
120.107	odd	4		960.2.o.d.959.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.o.b.239.1	✓	8	20.7	even	4
240.2.o.b.239.2	yes	8	5.3	odd	4
240.2.o.b.239.3	yes	8	15.8	even	4
240.2.o.b.239.4	yes	8	60.47	odd	4
240.2.o.b.239.5	yes	8	60.23	odd	4
240.2.o.b.239.6	yes	8	15.2	even	4
240.2.o.b.239.7	yes	8	5.2	odd	4
240.2.o.b.239.8	yes	8	20.3	even	4
960.2.o.d.959.1		8	40.3	even	4
960.2.o.d.959.2		8	40.37	odd	4
960.2.o.d.959.3		8	120.77	even	4
960.2.o.d.959.4		8	120.83	odd	4
960.2.o.d.959.5		8	120.107	odd	4
960.2.o.d.959.6		8	120.53	even	4
960.2.o.d.959.7		8	40.13	odd	4
960.2.o.d.959.8		8	40.27	even	4
1200.2.h.j.1151.1		4	15.14	odd	2	inner
1200.2.h.j.1151.2		4	12.11	even	2	inner
1200.2.h.j.1151.3		4	20.19	odd	2	inner
1200.2.h.j.1151.4		4	1.1	even	1	trivial
1200.2.h.n.1151.1		4	4.3	odd	2
1200.2.h.n.1151.2		4	5.4	even	2
1200.2.h.n.1151.3		4	3.2	odd	2
1200.2.h.n.1151.4		4	60.59	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1200.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.41421i) q^{3} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})\)	\(q+(-1.00000 + 1.41421i) q^{3} +(-1.00000 - 2.82843i) q^{9} +4.89898 q^{11} -4.89898 q^{13} +3.46410i q^{17} +3.46410i q^{19} +6.00000 q^{23} +(5.00000 + 1.41421i) q^{27} +2.82843i q^{29} +3.46410i q^{31} +(-4.89898 + 6.92820i) q^{33} -4.89898 q^{37} +(4.89898 - 6.92820i) q^{39} -5.65685i q^{41} +8.48528i q^{43} -6.00000 q^{47} +7.00000 q^{49} +(-4.89898 - 3.46410i) q^{51} +10.3923i q^{53} +(-4.89898 - 3.46410i) q^{57} -4.89898 q^{59} -2.00000 q^{61} +8.48528i q^{67} +(-6.00000 + 8.48528i) q^{69} -9.79796 q^{71} -9.79796 q^{73} +10.3923i q^{79} +(-7.00000 + 5.65685i) q^{81} +6.00000 q^{83} +(-4.00000 - 2.82843i) q^{87} -5.65685i q^{89} +(-4.89898 - 3.46410i) q^{93} +(-4.89898 - 13.8564i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 4 * q^9	\( 4 q - 4 q^{3} - 4 q^{9} + 24 q^{23} + 20 q^{27} - 24 q^{47} + 28 q^{49} - 8 q^{61} - 24 q^{69} - 28 q^{81} + 24 q^{83} - 16 q^{87}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^9 + 24 * q^23 + 20 * q^27 - 24 * q^47 + 28 * q^49 - 8 * q^61 - 24 * q^69 - 28 * q^81 + 24 * q^83 - 16 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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