LMFDB - Embedded newform 1840.2.m.f.1839.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1839,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1840.1839");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.m (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:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1839.8
Root			$-1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	1840.1839
Dual form			1840.2.m.f.1839.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{3} +(1.58114 + 1.58114i) q^{5} -2.82843i q^{7} +2.00000 q^{9} +O(q^{10})$	$q+2.23607 q^{3} +(1.58114 + 1.58114i) q^{5} -2.82843i q^{7} +2.00000 q^{9} +5.00000i q^{13} +(3.53553 + 3.53553i) q^{15} -3.16228 q^{17} +7.07107 q^{19} -6.32456i q^{21} +(2.23607 - 4.24264i) q^{23} +5.00000i q^{25} -2.23607 q^{27} +9.00000 q^{29} +6.70820i q^{31} +(4.47214 - 4.47214i) q^{35} +11.1803i q^{39} -3.00000 q^{41} +1.41421i q^{43} +(3.16228 + 3.16228i) q^{45} +6.70820 q^{47} -1.00000 q^{49} -7.07107 q^{51} +6.32456 q^{53} +15.8114 q^{57} -4.47214i q^{59} -9.48683i q^{61} -5.65685i q^{63} +(-7.90569 + 7.90569i) q^{65} -2.82843i q^{67} +(5.00000 - 9.48683i) q^{69} +6.70820i q^{71} -15.0000i q^{73} +11.1803i q^{75} -7.07107 q^{79} -11.0000 q^{81} -12.7279i q^{83} +(-5.00000 - 5.00000i) q^{85} +20.1246 q^{87} +3.16228i q^{89} +14.1421 q^{91} +15.0000i q^{93} +(11.1803 + 11.1803i) q^{95} -18.9737 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{9}+O(q^{10}) $ 8 * q + 16 * q^9	$ 8 q + 16 q^{9} + 72 q^{29} - 24 q^{41} - 8 q^{49} + 40 q^{69} - 88 q^{81} - 40 q^{85}+O(q^{100}) $ 8 * q + 16 * q^9 + 72 * q^29 - 24 * q^41 - 8 * q^49 + 40 * q^69 - 88 * q^81 - 40 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$	2.23607			1.29099			0.645497	−	0.763763i	$-0.276650\pi$
$3$	2.23607			1.29099			0.645497	+	0.763763i	$0.276650\pi$
$4$		0			0
$5$	1.58114	+	1.58114i	0.707107	+	0.707107i
$5$	1.58114	+	1.58114i	0.707107	+	0.707107i
$6$		0			0
$7$		−	2.82843i		−	1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$		−	2.82843i		−	1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$		0			0
$9$	2.00000			0.666667
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$			5.00000i			1.38675i	0.720577	+	0.693375i	$0.243877\pi$
$13$			5.00000i			1.38675i	−0.720577	+	0.693375i	$0.756123\pi$
$14$		0			0
$15$	3.53553	+	3.53553i	0.912871	+	0.912871i
$16$		0			0
$17$	−3.16228			−0.766965			−0.383482	−	0.923548i	$-0.625275\pi$
$17$	−3.16228			−0.766965			−0.383482	+	0.923548i	$0.625275\pi$
$18$		0			0
$19$	7.07107			1.62221			0.811107	−	0.584898i	$-0.198865\pi$
$19$	7.07107			1.62221			0.811107	+	0.584898i	$0.198865\pi$
$20$		0			0
$21$		−	6.32456i		−	1.38013i
$22$		0			0
$23$	2.23607	−	4.24264i	0.466252	−	0.884652i
$23$	2.23607	−	4.24264i	0.466252	−	0.884652i
$24$		0			0
$25$			5.00000i			1.00000i
$26$		0			0
$27$	−2.23607			−0.430331
$28$		0			0
$29$	9.00000			1.67126			0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000			1.67126			0.835629	+	0.549294i	$0.185103\pi$
$30$		0			0
$31$			6.70820i			1.20483i	0.798183	+	0.602414i	$0.205795\pi$
$31$			6.70820i			1.20483i	−0.798183	+	0.602414i	$0.794205\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.47214	−	4.47214i	0.755929	−	0.755929i
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$			11.1803i			1.79029i
$40$		0			0
$41$	−3.00000			−0.468521			−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000			−0.468521			−0.234261	+	0.972174i	$0.575267\pi$
$42$		0			0
$43$			1.41421i			0.215666i	0.994169	+	0.107833i	$0.0343911\pi$
$43$			1.41421i			0.215666i	−0.994169	+	0.107833i	$0.965609\pi$
$44$		0			0
$45$	3.16228	+	3.16228i	0.471405	+	0.471405i
$46$		0			0
$47$	6.70820			0.978492			0.489246	−	0.872146i	$-0.337272\pi$
$47$	6.70820			0.978492			0.489246	+	0.872146i	$0.337272\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−7.07107			−0.990148
$52$		0			0
$53$	6.32456			0.868744			0.434372	−	0.900733i	$-0.356970\pi$
$53$	6.32456			0.868744			0.434372	+	0.900733i	$0.356970\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	15.8114			2.09427
$58$		0			0
$59$		−	4.47214i		−	0.582223i	−0.956689	−	0.291111i	$-0.905975\pi$
$59$		−	4.47214i		−	0.582223i	0.956689	−	0.291111i	$-0.0940250\pi$
$60$		0			0
$61$		−	9.48683i		−	1.21466i	−0.794448	−	0.607332i	$-0.792240\pi$
$61$		−	9.48683i		−	1.21466i	0.794448	−	0.607332i	$-0.207760\pi$
$62$		0			0
$63$		−	5.65685i		−	0.712697i
$64$		0			0
$65$	−7.90569	+	7.90569i	−0.980581	+	0.980581i
$66$		0			0
$67$		−	2.82843i		−	0.345547i	−0.984962	−	0.172774i	$-0.944727\pi$
$67$		−	2.82843i		−	0.345547i	0.984962	−	0.172774i	$-0.0552729\pi$
$68$		0			0
$69$	5.00000	−	9.48683i	0.601929	−	1.14208i
$70$		0			0
$71$			6.70820i			0.796117i	0.917360	+	0.398059i	$0.130316\pi$
$71$			6.70820i			0.796117i	−0.917360	+	0.398059i	$0.869684\pi$
$72$		0			0
$73$		−	15.0000i		−	1.75562i	−0.479012	−	0.877809i	$-0.659005\pi$
$73$		−	15.0000i		−	1.75562i	0.479012	−	0.877809i	$-0.340995\pi$
$74$		0			0
$75$			11.1803i			1.29099i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−7.07107			−0.795557			−0.397779	−	0.917481i	$-0.630219\pi$
$79$	−7.07107			−0.795557			−0.397779	+	0.917481i	$0.630219\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$		−	12.7279i		−	1.39707i	−0.715575	−	0.698535i	$-0.753835\pi$
$83$		−	12.7279i		−	1.39707i	0.715575	−	0.698535i	$-0.246165\pi$
$84$		0			0
$85$	−5.00000	−	5.00000i	−0.542326	−	0.542326i
$86$		0			0
$87$	20.1246			2.15758
$88$		0			0
$89$			3.16228i			0.335201i	0.985855	+	0.167600i	$0.0536018\pi$
$89$			3.16228i			0.335201i	−0.985855	+	0.167600i	$0.946398\pi$
$90$		0			0
$91$	14.1421			1.48250
$92$		0			0
$93$			15.0000i			1.55543i
$94$		0			0
$95$	11.1803	+	11.1803i	1.14708	+	1.14708i
$96$		0			0
$97$	−18.9737			−1.92648			−0.963242	−	0.268635i	$-0.913427\pi$
$97$	−18.9737			−1.92648			−0.963242	+	0.268635i	$0.913427\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.0000			−1.19404			−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000			−1.19404			−0.597022	+	0.802225i	$0.703650\pi$
$102$		0			0
$103$		−	1.41421i		−	0.139347i	−0.997570	−	0.0696733i	$-0.977804\pi$
$103$		−	1.41421i		−	0.139347i	0.997570	−	0.0696733i	$-0.0221957\pi$
$104$		0			0
$105$	10.0000	−	10.0000i	0.975900	−	0.975900i
$106$		0			0
$107$			4.24264i			0.410152i	0.978746	+	0.205076i	$0.0657441\pi$
$107$			4.24264i			0.410152i	−0.978746	+	0.205076i	$0.934256\pi$
$108$		0			0
$109$			9.48683i			0.908674i	0.890830	+	0.454337i	$0.150124\pi$
$109$			9.48683i			0.908674i	−0.890830	+	0.454337i	$0.849876\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−12.6491			−1.18993			−0.594964	−	0.803752i	$-0.702834\pi$
$113$	−12.6491			−1.18993			−0.594964	+	0.803752i	$0.702834\pi$
$114$		0			0
$115$	10.2437	−	3.17267i	0.955233	−	0.295853i
$116$		0			0
$117$			10.0000i			0.924500i
$118$		0			0
$119$			8.94427i			0.819920i
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	−6.70820			−0.604858
$124$		0			0
$125$	−7.90569	+	7.90569i	−0.707107	+	0.707107i
$126$		0			0
$127$	−6.70820			−0.595257			−0.297628	−	0.954682i	$-0.596196\pi$
$127$	−6.70820			−0.595257			−0.297628	+	0.954682i	$0.596196\pi$
$128$		0			0
$129$			3.16228i			0.278423i
$130$		0			0
$131$			20.1246i			1.75830i	0.476549	+	0.879148i	$0.341887\pi$
$131$			20.1246i			1.75830i	−0.476549	+	0.879148i	$0.658113\pi$
$132$		0			0
$133$		−	20.0000i		−	1.73422i
$134$		0			0
$135$	−3.53553	−	3.53553i	−0.304290	−	0.304290i
$136$		0			0
$137$	−6.32456			−0.540343			−0.270172	−	0.962812i	$-0.587080\pi$
$137$	−6.32456			−0.540343			−0.270172	+	0.962812i	$0.587080\pi$
$138$		0			0
$139$			6.70820i			0.568982i	0.958679	+	0.284491i	$0.0918247\pi$
$139$			6.70820i			0.568982i	−0.958679	+	0.284491i	$0.908175\pi$
$140$		0			0
$141$	15.0000			1.26323
$142$		0			0
$143$		0			0
$144$		0			0
$145$	14.2302	+	14.2302i	1.18176	+	1.18176i
$146$		0			0
$147$	−2.23607			−0.184428
$148$		0			0
$149$			12.6491i			1.03626i	0.855303	+	0.518128i	$0.173371\pi$
$149$			12.6491i			1.03626i	−0.855303	+	0.518128i	$0.826629\pi$
$150$		0			0
$151$		−	20.1246i		−	1.63772i	−0.573995	−	0.818859i	$-0.694607\pi$
$151$		−	20.1246i		−	1.63772i	0.573995	−	0.818859i	$-0.305393\pi$
$152$		0			0
$153$	−6.32456			−0.511310
$154$		0			0
$155$	−10.6066	+	10.6066i	−0.851943	+	0.851943i
$156$		0			0
$157$	9.48683			0.757132			0.378566	−	0.925574i	$-0.376417\pi$
$157$	9.48683			0.757132			0.378566	+	0.925574i	$0.376417\pi$
$158$		0			0
$159$	14.1421			1.12154
$160$		0			0
$161$	−12.0000	−	6.32456i	−0.945732	−	0.498445i
$162$		0			0
$163$	−6.70820			−0.525427			−0.262714	−	0.964874i	$-0.584617\pi$
$163$	−6.70820			−0.525427			−0.262714	+	0.964874i	$0.584617\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.47214			0.346064			0.173032	−	0.984916i	$-0.444644\pi$
$167$	4.47214			0.346064			0.173032	+	0.984916i	$0.444644\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$		0			0
$171$	14.1421			1.08148
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$	14.1421			1.06904
$176$		0			0
$177$		−	10.0000i		−	0.751646i
$178$		0			0
$179$		−	2.23607i		−	0.167132i	−0.996502	−	0.0835658i	$-0.973369\pi$
$179$		−	2.23607i		−	0.167132i	0.996502	−	0.0835658i	$-0.0266309\pi$
$180$		0			0
$181$		−	18.9737i		−	1.41030i	−0.709057	−	0.705151i	$-0.750879\pi$
$181$		−	18.9737i		−	1.41030i	0.709057	−	0.705151i	$-0.249121\pi$
$182$		0			0
$183$		−	21.2132i		−	1.56813i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$			6.32456i			0.460044i
$190$		0			0
$191$	21.2132			1.53493			0.767467	−	0.641089i	$-0.221517\pi$
$191$	21.2132			1.53493			0.767467	+	0.641089i	$0.221517\pi$
$192$		0			0
$193$		−	5.00000i		−	0.359908i	−0.983675	−	0.179954i	$-0.942405\pi$
$193$		−	5.00000i		−	0.359908i	0.983675	−	0.179954i	$-0.0575949\pi$
$194$		0			0
$195$	−17.6777	+	17.6777i	−1.26592	+	1.26592i
$196$		0			0
$197$		−	15.0000i		−	1.06871i	−0.845262	−	0.534353i	$-0.820555\pi$
$197$		−	15.0000i		−	1.06871i	0.845262	−	0.534353i	$-0.179445\pi$
$198$		0			0
$199$	−14.1421			−1.00251			−0.501255	−	0.865300i	$-0.667128\pi$
$199$	−14.1421			−1.00251			−0.501255	+	0.865300i	$0.667128\pi$
$200$		0			0
$201$		−	6.32456i		−	0.446100i
$202$		0			0
$203$		−	25.4558i		−	1.78665i
$204$		0			0
$205$	−4.74342	−	4.74342i	−0.331295	−	0.331295i
$206$		0			0
$207$	4.47214	−	8.48528i	0.310835	−	0.589768i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	26.8328i		−	1.84725i	−0.383301	−	0.923624i	$-0.625213\pi$
$211$		−	26.8328i		−	1.84725i	0.383301	−	0.923624i	$-0.374787\pi$
$212$		0			0
$213$			15.0000i			1.02778i
$214$		0			0
$215$	−2.23607	+	2.23607i	−0.152499	+	0.152499i
$216$		0			0
$217$	18.9737			1.28802
$218$		0			0
$219$		−	33.5410i		−	2.26649i
$220$		0			0
$221$		−	15.8114i		−	1.06359i
$222$		0			0
$223$	13.4164			0.898429			0.449215	−	0.893424i	$-0.351704\pi$
$223$	13.4164			0.898429			0.449215	+	0.893424i	$0.351704\pi$
$224$		0			0
$225$			10.0000i			0.666667i
$226$		0			0
$227$			16.9706i			1.12638i	0.826329	+	0.563188i	$0.190425\pi$
$227$			16.9706i			1.12638i	−0.826329	+	0.563188i	$0.809575\pi$
$228$		0			0
$229$			28.4605i			1.88072i	0.340177	+	0.940361i	$0.389513\pi$
$229$			28.4605i			1.88072i	−0.340177	+	0.940361i	$0.610487\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			15.0000i			0.982683i	0.870967	+	0.491341i	$0.163493\pi$
$233$			15.0000i			0.982683i	−0.870967	+	0.491341i	$0.836507\pi$
$234$		0			0
$235$	10.6066	+	10.6066i	0.691898	+	0.691898i
$236$		0			0
$237$	−15.8114			−1.02706
$238$		0			0
$239$			6.70820i			0.433918i	0.976181	+	0.216959i	$0.0696137\pi$
$239$			6.70820i			0.433918i	−0.976181	+	0.216959i	$0.930386\pi$
$240$		0			0
$241$		−	9.48683i		−	0.611101i	−0.952176	−	0.305550i	$-0.901160\pi$
$241$		−	9.48683i		−	0.611101i	0.952176	−	0.305550i	$-0.0988404\pi$
$242$		0			0
$243$	−17.8885			−1.14755
$244$		0			0
$245$	−1.58114	−	1.58114i	−0.101015	−	0.101015i
$246$		0			0
$247$			35.3553i			2.24961i
$248$		0			0
$249$		−	28.4605i		−	1.80361i
$250$		0			0
$251$	−21.2132			−1.33897			−0.669483	−	0.742828i	$-0.733484\pi$
$251$	−21.2132			−1.33897			−0.669483	+	0.742828i	$0.733484\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−11.1803	−	11.1803i	−0.700140	−	0.700140i
$256$		0			0
$257$		−	15.0000i		−	0.935674i	−0.883815	−	0.467837i	$-0.845033\pi$
$257$		−	15.0000i		−	0.935674i	0.883815	−	0.467837i	$-0.154967\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	18.0000			1.11417
$262$		0			0
$263$		−	12.7279i		−	0.784837i	−0.919787	−	0.392419i	$-0.871638\pi$
$263$		−	12.7279i		−	0.784837i	0.919787	−	0.392419i	$-0.128362\pi$
$264$		0			0
$265$	10.0000	+	10.0000i	0.614295	+	0.614295i
$266$		0			0
$267$			7.07107i			0.432742i
$268$		0			0
$269$	−9.00000			−0.548740			−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000			−0.548740			−0.274370	+	0.961624i	$0.588469\pi$
$270$		0			0
$271$			13.4164i			0.814989i	0.913208	+	0.407494i	$0.133597\pi$
$271$			13.4164i			0.814989i	−0.913208	+	0.407494i	$0.866403\pi$
$272$		0			0
$273$	31.6228			1.91390
$274$		0			0
$275$		0			0
$276$		0			0
$277$			15.0000i			0.901263i	0.892710	+	0.450631i	$0.148801\pi$
$277$			15.0000i			0.901263i	−0.892710	+	0.450631i	$0.851199\pi$
$278$		0			0
$279$			13.4164i			0.803219i
$280$		0			0
$281$			6.32456i			0.377291i	0.982045	+	0.188646i	$0.0604098\pi$
$281$			6.32456i			0.377291i	−0.982045	+	0.188646i	$0.939590\pi$
$282$		0			0
$283$		−	15.5563i		−	0.924729i	−0.886690	−	0.462364i	$-0.847001\pi$
$283$		−	15.5563i		−	0.924729i	0.886690	−	0.462364i	$-0.152999\pi$
$284$		0			0
$285$	25.0000	+	25.0000i	1.48087	+	1.48087i
$286$		0			0
$287$			8.48528i			0.500870i
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$	−42.4264			−2.48708
$292$		0			0
$293$	12.6491			0.738969			0.369484	−	0.929237i	$-0.379534\pi$
$293$	12.6491			0.738969			0.369484	+	0.929237i	$0.379534\pi$
$294$		0			0
$295$	7.07107	−	7.07107i	0.411693	−	0.411693i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	21.2132	+	11.1803i	1.22679	+	0.646576i
$300$		0			0
$301$	4.00000			0.230556
$302$		0			0
$303$	−26.8328			−1.54150
$304$		0			0
$305$	15.0000	−	15.0000i	0.858898	−	0.858898i
$306$		0			0
$307$	−13.4164			−0.765715			−0.382857	−	0.923807i	$-0.625060\pi$
$307$	−13.4164			−0.765715			−0.382857	+	0.923807i	$0.625060\pi$
$308$		0			0
$309$		−	3.16228i		−	0.179896i
$310$		0			0
$311$			24.5967i			1.39475i	0.716705	+	0.697377i	$0.245650\pi$
$311$			24.5967i			1.39475i	−0.716705	+	0.697377i	$0.754350\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$	8.94427	−	8.94427i	0.503953	−	0.503953i
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$			9.48683i			0.529503i
$322$		0			0
$323$	−22.3607			−1.24418
$324$		0			0
$325$	−25.0000			−1.38675
$326$		0			0
$327$			21.2132i			1.17309i
$328$		0			0
$329$		−	18.9737i		−	1.04605i
$330$		0			0
$331$		−	20.1246i		−	1.10615i	−0.833132	−	0.553074i	$-0.813455\pi$
$331$		−	20.1246i		−	1.10615i	0.833132	−	0.553074i	$-0.186545\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.47214	−	4.47214i	0.244339	−	0.244339i
$336$		0			0
$337$	−9.48683			−0.516781			−0.258390	−	0.966041i	$-0.583192\pi$
$337$	−9.48683			−0.516781			−0.258390	+	0.966041i	$0.583192\pi$
$338$		0			0
$339$	−28.2843			−1.53619
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	16.9706i		−	0.916324i
$344$		0			0
$345$	22.9057	−	7.09431i	1.23320	−	0.381945i
$346$		0			0
$347$	13.4164			0.720231			0.360115	−	0.932908i	$-0.382737\pi$
$347$	13.4164			0.720231			0.360115	+	0.932908i	$0.382737\pi$
$348$		0			0
$349$	9.00000			0.481759			0.240879	−	0.970555i	$-0.422564\pi$
$349$	9.00000			0.481759			0.240879	+	0.970555i	$0.422564\pi$
$350$		0			0
$351$		−	11.1803i		−	0.596762i
$352$		0			0
$353$		−	15.0000i		−	0.798369i	−0.916871	−	0.399185i	$-0.869293\pi$
$353$		−	15.0000i		−	0.798369i	0.916871	−	0.399185i	$-0.130707\pi$
$354$		0			0
$355$	−10.6066	+	10.6066i	−0.562940	+	0.562940i
$356$		0			0
$357$			20.0000i			1.05851i
$358$		0			0
$359$	21.2132			1.11959			0.559795	−	0.828631i	$-0.310880\pi$
$359$	21.2132			1.11959			0.559795	+	0.828631i	$0.310880\pi$
$360$		0			0
$361$	31.0000			1.63158
$362$		0			0
$363$	−24.5967			−1.29099
$364$		0			0
$365$	23.7171	−	23.7171i	1.24141	−	1.24141i
$366$		0			0
$367$		−	24.0416i		−	1.25496i	−0.778632	−	0.627481i	$-0.784086\pi$
$367$		−	24.0416i		−	1.25496i	0.778632	−	0.627481i	$-0.215914\pi$
$368$		0			0
$369$	−6.00000			−0.312348
$370$		0			0
$371$		−	17.8885i		−	0.928727i
$372$		0			0
$373$	−28.4605			−1.47363			−0.736814	−	0.676095i	$-0.763671\pi$
$373$	−28.4605			−1.47363			−0.736814	+	0.676095i	$0.763671\pi$
$374$		0			0
$375$	−17.6777	+	17.6777i	−0.912871	+	0.912871i
$376$		0			0
$377$			45.0000i			2.31762i
$378$		0			0
$379$	−28.2843			−1.45287			−0.726433	−	0.687238i	$-0.758823\pi$
$379$	−28.2843			−1.45287			−0.726433	+	0.687238i	$0.758823\pi$
$380$		0			0
$381$	−15.0000			−0.768473
$382$		0			0
$383$			12.7279i			0.650366i	0.945651	+	0.325183i	$0.105426\pi$
$383$			12.7279i			0.650366i	−0.945651	+	0.325183i	$0.894574\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			2.82843i			0.143777i
$388$		0			0
$389$			25.2982i			1.28267i	0.767261	+	0.641335i	$0.221619\pi$
$389$			25.2982i			1.28267i	−0.767261	+	0.641335i	$0.778381\pi$
$390$		0			0
$391$	−7.07107	+	13.4164i	−0.357599	+	0.678497i
$392$		0			0
$393$			45.0000i			2.26995i
$394$		0			0
$395$	−11.1803	−	11.1803i	−0.562544	−	0.562544i
$396$		0			0
$397$		−	15.0000i		−	0.752828i	−0.926451	−	0.376414i	$-0.877157\pi$
$397$		−	15.0000i		−	0.752828i	0.926451	−	0.376414i	$-0.122843\pi$
$398$		0			0
$399$		−	44.7214i		−	2.23887i
$400$		0			0
$401$			22.1359i			1.10542i	0.833375	+	0.552708i	$0.186406\pi$
$401$			22.1359i			1.10542i	−0.833375	+	0.552708i	$0.813594\pi$
$402$		0			0
$403$	−33.5410			−1.67080
$404$		0			0
$405$	−17.3925	−	17.3925i	−0.864242	−	0.864242i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	9.00000			0.445021			0.222511	−	0.974930i	$-0.428575\pi$
$409$	9.00000			0.445021			0.222511	+	0.974930i	$0.428575\pi$
$410$		0			0
$411$	−14.1421			−0.697580
$412$		0			0
$413$	−12.6491			−0.622422
$414$		0			0
$415$	20.1246	−	20.1246i	0.987878	−	0.987878i
$416$		0			0
$417$			15.0000i			0.734553i
$418$		0			0
$419$	−21.2132			−1.03633			−0.518166	−	0.855280i	$-0.673385\pi$
$419$	−21.2132			−1.03633			−0.518166	+	0.855280i	$0.673385\pi$
$420$		0			0
$421$			18.9737i			0.924720i	0.886692	+	0.462360i	$0.152997\pi$
$421$			18.9737i			0.924720i	−0.886692	+	0.462360i	$0.847003\pi$
$422$		0			0
$423$	13.4164			0.652328
$424$		0			0
$425$		−	15.8114i		−	0.766965i
$426$		0			0
$427$	−26.8328			−1.29853
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−21.2132			−1.02180			−0.510902	−	0.859639i	$-0.670689\pi$
$431$	−21.2132			−1.02180			−0.510902	+	0.859639i	$0.670689\pi$
$432$		0			0
$433$	9.48683			0.455908			0.227954	−	0.973672i	$-0.426796\pi$
$433$	9.48683			0.455908			0.227954	+	0.973672i	$0.426796\pi$
$434$		0			0
$435$	31.8198	+	31.8198i	1.52564	+	1.52564i
$436$		0			0
$437$	15.8114	−	30.0000i	0.756361	−	1.43509i
$438$		0			0
$439$		−	20.1246i		−	0.960495i	−0.877133	−	0.480248i	$-0.840547\pi$
$439$		−	20.1246i		−	0.960495i	0.877133	−	0.480248i	$-0.159453\pi$
$440$		0			0
$441$	−2.00000			−0.0952381
$442$		0			0
$443$	29.0689			1.38110			0.690552	−	0.723283i	$-0.257368\pi$
$443$	29.0689			1.38110			0.690552	+	0.723283i	$0.257368\pi$
$444$		0			0
$445$	−5.00000	+	5.00000i	−0.237023	+	0.237023i
$446$		0			0
$447$			28.2843i			1.33780i
$448$		0			0
$449$	24.0000			1.13263			0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000			1.13263			0.566315	+	0.824189i	$0.308369\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		−	45.0000i		−	2.11428i
$454$		0			0
$455$	22.3607	+	22.3607i	1.04828	+	1.04828i
$456$		0			0
$457$	−9.48683			−0.443775			−0.221888	−	0.975072i	$-0.571222\pi$
$457$	−9.48683			−0.443775			−0.221888	+	0.975072i	$0.571222\pi$
$458$		0			0
$459$	7.07107			0.330049
$460$		0			0
$461$	3.00000			0.139724			0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000			0.139724			0.0698620	+	0.997557i	$0.477744\pi$
$462$		0			0
$463$	−40.2492			−1.87054			−0.935270	−	0.353935i	$-0.884843\pi$
$463$	−40.2492			−1.87054			−0.935270	+	0.353935i	$0.884843\pi$
$464$		0			0
$465$	−23.7171	+	23.7171i	−1.09985	+	1.09985i
$466$		0			0
$467$		−	25.4558i		−	1.17796i	−0.808149	−	0.588978i	$-0.799530\pi$
$467$		−	25.4558i		−	1.17796i	0.808149	−	0.588978i	$-0.200470\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$		0			0
$471$	21.2132			0.977453
$472$		0			0
$473$		0			0
$474$		0			0
$475$			35.3553i			1.62221i
$476$		0			0
$477$	12.6491			0.579163
$478$		0			0
$479$	21.2132			0.969256			0.484628	−	0.874720i	$-0.338955\pi$
$479$	21.2132			0.969256			0.484628	+	0.874720i	$0.338955\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−26.8328	−	14.1421i	−1.22094	−	0.643489i
$484$		0			0
$485$	−30.0000	−	30.0000i	−1.36223	−	1.36223i
$486$		0			0
$487$	6.70820			0.303978			0.151989	−	0.988382i	$-0.451432\pi$
$487$	6.70820			0.303978			0.151989	+	0.988382i	$0.451432\pi$
$488$		0			0
$489$	−15.0000			−0.678323
$490$		0			0
$491$		−	15.6525i		−	0.706386i	−0.935550	−	0.353193i	$-0.885096\pi$
$491$		−	15.6525i		−	0.706386i	0.935550	−	0.353193i	$-0.114904\pi$
$492$		0			0
$493$	−28.4605			−1.28180
$494$		0			0
$495$		0			0
$496$		0			0
$497$	18.9737			0.851085
$498$		0			0
$499$		−	20.1246i		−	0.900901i	−0.892801	−	0.450451i	$-0.851263\pi$
$499$		−	20.1246i		−	0.900901i	0.892801	−	0.450451i	$-0.148737\pi$
$500$		0			0
$501$	10.0000			0.446767
$502$		0			0
$503$		−	33.9411i		−	1.51336i	−0.653785	−	0.756680i	$-0.726820\pi$
$503$		−	33.9411i		−	1.51336i	0.653785	−	0.756680i	$-0.273180\pi$
$504$		0			0
$505$	−18.9737	−	18.9737i	−0.844317	−	0.844317i
$506$		0			0
$507$	−26.8328			−1.19169
$508$		0			0
$509$	−21.0000			−0.930809			−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000			−0.930809			−0.465404	+	0.885098i	$0.654091\pi$
$510$		0			0
$511$	−42.4264			−1.87683
$512$		0			0
$513$	−15.8114			−0.698090
$514$		0			0
$515$	2.23607	−	2.23607i	0.0985329	−	0.0985329i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			44.2719i			1.93959i	0.243928	+	0.969793i	$0.421564\pi$
$521$			44.2719i			1.93959i	−0.243928	+	0.969793i	$0.578436\pi$
$522$		0			0
$523$			22.6274i			0.989428i	0.869056	+	0.494714i	$0.164727\pi$
$523$			22.6274i			0.989428i	−0.869056	+	0.494714i	$0.835273\pi$
$524$		0			0
$525$	31.6228			1.38013
$526$		0			0
$527$		−	21.2132i		−	0.924062i
$528$		0			0
$529$	−13.0000	−	18.9737i	−0.565217	−	0.824942i
$530$		0			0
$531$		−	8.94427i		−	0.388148i
$532$		0			0
$533$		−	15.0000i		−	0.649722i
$534$		0			0
$535$	−6.70820	+	6.70820i	−0.290021	+	0.290021i
$536$		0			0
$537$		−	5.00000i		−	0.215766i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−3.00000			−0.128980			−0.0644900	−	0.997918i	$-0.520542\pi$
$541$	−3.00000			−0.128980			−0.0644900	+	0.997918i	$0.520542\pi$
$542$		0			0
$543$		−	42.4264i		−	1.82069i
$544$		0			0
$545$	−15.0000	+	15.0000i	−0.642529	+	0.642529i
$546$		0			0
$547$	−20.1246			−0.860466			−0.430233	−	0.902718i	$-0.641569\pi$
$547$	−20.1246			−0.860466			−0.430233	+	0.902718i	$0.641569\pi$
$548$		0			0
$549$		−	18.9737i		−	0.809776i
$550$		0			0
$551$	63.6396			2.71114
$552$		0			0
$553$			20.0000i			0.850487i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	22.1359			0.937930			0.468965	−	0.883217i	$-0.344627\pi$
$557$	22.1359			0.937930			0.468965	+	0.883217i	$0.344627\pi$
$558$		0			0
$559$	−7.07107			−0.299074
$560$		0			0
$561$		0			0
$562$		0			0
$563$			33.9411i			1.43045i	0.698895	+	0.715224i	$0.253675\pi$
$563$			33.9411i			1.43045i	−0.698895	+	0.715224i	$0.746325\pi$
$564$		0			0
$565$	−20.0000	−	20.0000i	−0.841406	−	0.841406i
$566$		0			0
$567$			31.1127i			1.30661i
$568$		0			0
$569$		−	22.1359i		−	0.927987i	−0.885839	−	0.463994i	$-0.846416\pi$
$569$		−	22.1359i		−	0.927987i	0.885839	−	0.463994i	$-0.153584\pi$
$570$		0			0
$571$	35.3553			1.47957			0.739787	−	0.672841i	$-0.234926\pi$
$571$	35.3553			1.47957			0.739787	+	0.672841i	$0.234926\pi$
$572$		0			0
$573$	47.4342			1.98159
$574$		0			0
$575$	21.2132	+	11.1803i	0.884652	+	0.466252i
$576$		0			0
$577$		−	25.0000i		−	1.04076i	−0.853934	−	0.520382i	$-0.825790\pi$
$577$		−	25.0000i		−	1.04076i	0.853934	−	0.520382i	$-0.174210\pi$
$578$		0			0
$579$		−	11.1803i		−	0.464639i
$580$		0			0
$581$	−36.0000			−1.49353
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−15.8114	+	15.8114i	−0.653720	+	0.653720i
$586$		0			0
$587$	11.1803			0.461462			0.230731	−	0.973018i	$-0.425888\pi$
$587$	11.1803			0.461462			0.230731	+	0.973018i	$0.425888\pi$
$588$		0			0
$589$			47.4342i			1.95449i
$590$		0			0
$591$		−	33.5410i		−	1.37969i
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$	−14.1421	+	14.1421i	−0.579771	+	0.579771i
$596$		0			0
$597$	−31.6228			−1.29423
$598$		0			0
$599$			13.4164i			0.548180i	0.961704	+	0.274090i	$0.0883765\pi$
$599$			13.4164i			0.548180i	−0.961704	+	0.274090i	$0.911623\pi$
$600$		0			0
$601$	33.0000			1.34610			0.673049	−	0.739598i	$-0.264984\pi$
$601$	33.0000			1.34610			0.673049	+	0.739598i	$0.264984\pi$
$602$		0			0
$603$		−	5.65685i		−	0.230365i
$604$		0			0
$605$	−17.3925	−	17.3925i	−0.707107	−	0.707107i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		−	56.9210i		−	2.30656i
$610$		0			0
$611$			33.5410i			1.35692i
$612$		0			0
$613$	9.48683			0.383170			0.191585	−	0.981476i	$-0.438637\pi$
$613$	9.48683			0.383170			0.191585	+	0.981476i	$0.438637\pi$
$614$		0			0
$615$	−10.6066	−	10.6066i	−0.427699	−	0.427699i
$616$		0			0
$617$	−22.1359			−0.891160			−0.445580	−	0.895242i	$-0.647002\pi$
$617$	−22.1359			−0.891160			−0.445580	+	0.895242i	$0.647002\pi$
$618$		0			0
$619$	−14.1421			−0.568420			−0.284210	−	0.958762i	$-0.591731\pi$
$619$	−14.1421			−0.568420			−0.284210	+	0.958762i	$0.591731\pi$
$620$		0			0
$621$	−5.00000	+	9.48683i	−0.200643	+	0.380693i
$622$		0			0
$623$	8.94427			0.358345
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−7.07107			−0.281495			−0.140747	−	0.990046i	$-0.544951\pi$
$631$	−7.07107			−0.281495			−0.140747	+	0.990046i	$0.544951\pi$
$632$		0			0
$633$		−	60.0000i		−	2.38479i
$634$		0			0
$635$	−10.6066	−	10.6066i	−0.420910	−	0.420910i
$636$		0			0
$637$		−	5.00000i		−	0.198107i
$638$		0			0
$639$			13.4164i			0.530745i
$640$		0			0
$641$			3.16228i			0.124902i	0.998048	+	0.0624512i	$0.0198918\pi$
$641$			3.16228i			0.124902i	−0.998048	+	0.0624512i	$0.980108\pi$
$642$		0			0
$643$		−	15.5563i		−	0.613483i	−0.951793	−	0.306741i	$-0.900761\pi$
$643$		−	15.5563i		−	0.613483i	0.951793	−	0.306741i	$-0.0992386\pi$
$644$		0			0
$645$	−5.00000	+	5.00000i	−0.196875	+	0.196875i
$646$		0			0
$647$	33.5410			1.31863			0.659317	−	0.751865i	$-0.270846\pi$
$647$	33.5410			1.31863			0.659317	+	0.751865i	$0.270846\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	42.4264			1.66282
$652$		0			0
$653$		−	15.0000i		−	0.586995i	−0.955960	−	0.293498i	$-0.905181\pi$
$653$		−	15.0000i		−	0.586995i	0.955960	−	0.293498i	$-0.0948193\pi$
$654$		0			0
$655$	−31.8198	+	31.8198i	−1.24330	+	1.24330i
$656$		0			0
$657$		−	30.0000i		−	1.17041i
$658$		0			0
$659$	21.2132			0.826349			0.413175	−	0.910652i	$-0.364420\pi$
$659$	21.2132			0.826349			0.413175	+	0.910652i	$0.364420\pi$
$660$		0			0
$661$			18.9737i			0.737990i	0.929431	+	0.368995i	$0.120298\pi$
$661$			18.9737i			0.737990i	−0.929431	+	0.368995i	$0.879702\pi$
$662$		0			0
$663$		−	35.3553i		−	1.37309i
$664$		0			0
$665$	31.6228	−	31.6228i	1.22628	−	1.22628i
$666$		0			0
$667$	20.1246	−	38.1838i	0.779228	−	1.47848i
$668$		0			0
$669$	30.0000			1.15987
$670$		0			0
$671$		0			0
$672$		0			0
$673$			35.0000i			1.34915i	0.738206	+	0.674575i	$0.235673\pi$
$673$			35.0000i			1.34915i	−0.738206	+	0.674575i	$0.764327\pi$
$674$		0			0
$675$		−	11.1803i		−	0.430331i
$676$		0			0
$677$	−44.2719			−1.70151			−0.850753	−	0.525565i	$-0.823854\pi$
$677$	−44.2719			−1.70151			−0.850753	+	0.525565i	$0.823854\pi$
$678$		0			0
$679$			53.6656i			2.05950i
$680$		0			0
$681$			37.9473i			1.45414i
$682$		0			0
$683$	42.4853			1.62565			0.812827	−	0.582505i	$-0.197927\pi$
$683$	42.4853			1.62565			0.812827	+	0.582505i	$0.197927\pi$
$684$		0			0
$685$	−10.0000	−	10.0000i	−0.382080	−	0.382080i
$686$		0			0
$687$			63.6396i			2.42800i
$688$		0			0
$689$			31.6228i			1.20473i
$690$		0			0
$691$		−	13.4164i		−	0.510384i	−0.966890	−	0.255192i	$-0.917861\pi$
$691$		−	13.4164i		−	0.510384i	0.966890	−	0.255192i	$-0.0821387\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−10.6066	+	10.6066i	−0.402331	+	0.402331i
$696$		0			0
$697$	9.48683			0.359339
$698$		0			0
$699$			33.5410i			1.26864i
$700$		0			0
$701$		−	31.6228i		−	1.19438i	−0.802101	−	0.597188i	$-0.796285\pi$
$701$		−	31.6228i		−	1.19438i	0.802101	−	0.597188i	$-0.203715\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	23.7171	+	23.7171i	0.893237	+	0.893237i
$706$		0			0
$707$			33.9411i			1.27649i
$708$		0			0
$709$		−	18.9737i		−	0.712571i	−0.934377	−	0.356285i	$-0.884043\pi$
$709$		−	18.9737i		−	0.712571i	0.934377	−	0.356285i	$-0.115957\pi$
$710$		0			0
$711$	−14.1421			−0.530372
$712$		0			0
$713$	28.4605	+	15.0000i	1.06585	+	0.561754i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			15.0000i			0.560185i
$718$		0			0
$719$		−	22.3607i		−	0.833913i	−0.908927	−	0.416956i	$-0.863097\pi$
$719$		−	22.3607i		−	0.833913i	0.908927	−	0.416956i	$-0.136903\pi$
$720$		0			0
$721$	−4.00000			−0.148968
$722$		0			0
$723$		−	21.2132i		−	0.788928i
$724$		0			0
$725$			45.0000i			1.67126i
$726$		0			0
$727$		−	18.3848i		−	0.681854i	−0.940090	−	0.340927i	$-0.889259\pi$
$727$		−	18.3848i		−	0.681854i	0.940090	−	0.340927i	$-0.110741\pi$
$728$		0			0
$729$	−7.00000			−0.259259
$730$		0			0
$731$		−	4.47214i		−	0.165408i
$732$		0			0
$733$	−9.48683			−0.350404			−0.175202	−	0.984532i	$-0.556058\pi$
$733$	−9.48683			−0.350404			−0.175202	+	0.984532i	$0.556058\pi$
$734$		0			0
$735$	−3.53553	−	3.53553i	−0.130410	−	0.130410i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	33.5410i		−	1.23383i	−0.787031	−	0.616913i	$-0.788383\pi$
$739$		−	33.5410i		−	1.23383i	0.787031	−	0.616913i	$-0.211617\pi$
$740$		0			0
$741$			79.0569i			2.90423i
$742$		0			0
$743$		−	29.6985i		−	1.08953i	−0.838588	−	0.544766i	$-0.816619\pi$
$743$		−	29.6985i		−	1.08953i	0.838588	−	0.544766i	$-0.183381\pi$
$744$		0			0
$745$	−20.0000	+	20.0000i	−0.732743	+	0.732743i
$746$		0			0
$747$		−	25.4558i		−	0.931381i
$748$		0			0
$749$	12.0000			0.438470
$750$		0			0
$751$	−35.3553			−1.29013			−0.645067	−	0.764126i	$-0.723171\pi$
$751$	−35.3553			−1.29013			−0.645067	+	0.764126i	$0.723171\pi$
$752$		0			0
$753$	−47.4342			−1.72860
$754$		0			0
$755$	31.8198	−	31.8198i	1.15804	−	1.15804i
$756$		0			0
$757$	18.9737			0.689610			0.344805	−	0.938674i	$-0.387945\pi$
$757$	18.9737			0.689610			0.344805	+	0.938674i	$0.387945\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	27.0000			0.978749			0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000			0.978749			0.489375	+	0.872074i	$0.337225\pi$
$762$		0			0
$763$	26.8328			0.971413
$764$		0			0
$765$	−10.0000	−	10.0000i	−0.361551	−	0.361551i
$766$		0			0
$767$	22.3607			0.807397
$768$		0			0
$769$		−	9.48683i		−	0.342104i	−0.985262	−	0.171052i	$-0.945283\pi$
$769$		−	9.48683i		−	0.342104i	0.985262	−	0.171052i	$-0.0547166\pi$
$770$		0			0
$771$		−	33.5410i		−	1.20795i
$772$		0			0
$773$	−34.7851			−1.25113			−0.625566	−	0.780171i	$-0.715132\pi$
$773$	−34.7851			−1.25113			−0.625566	+	0.780171i	$0.715132\pi$
$774$		0			0
$775$	−33.5410			−1.20483
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−21.2132			−0.760042
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−20.1246			−0.719195
$784$		0			0
$785$	15.0000	+	15.0000i	0.535373	+	0.535373i
$786$		0			0
$787$			11.3137i			0.403290i	0.979459	+	0.201645i	$0.0646288\pi$
$787$			11.3137i			0.403290i	−0.979459	+	0.201645i	$0.935371\pi$
$788$		0			0
$789$		−	28.4605i		−	1.01322i
$790$		0			0
$791$			35.7771i			1.27209i
$792$		0			0
$793$	47.4342			1.68444
$794$		0			0
$795$	22.3607	+	22.3607i	0.793052	+	0.793052i
$796$		0			0
$797$	−53.7587			−1.90423			−0.952116	−	0.305738i	$-0.901097\pi$
$797$	−53.7587			−1.90423			−0.952116	+	0.305738i	$0.901097\pi$
$798$		0			0
$799$	−21.2132			−0.750469
$800$		0			0
$801$			6.32456i			0.223467i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−8.97367	−	28.9737i	−0.316280	−	1.02119i
$806$		0			0
$807$	−20.1246			−0.708420
$808$		0			0
$809$	24.0000			0.843795			0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000			0.843795			0.421898	+	0.906644i	$0.361364\pi$
$810$		0			0
$811$			46.9574i			1.64890i	0.565936	+	0.824449i	$0.308515\pi$
$811$			46.9574i			1.64890i	−0.565936	+	0.824449i	$0.691485\pi$
$812$		0			0
$813$			30.0000i			1.05215i
$814$		0			0
$815$	−10.6066	−	10.6066i	−0.371533	−	0.371533i
$816$		0			0
$817$			10.0000i			0.349856i
$818$		0			0
$819$	28.2843			0.988332
$820$		0			0
$821$	18.0000			0.628204			0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000			0.628204			0.314102	+	0.949389i	$0.398297\pi$
$822$		0			0
$823$	−33.5410			−1.16917			−0.584583	−	0.811334i	$-0.698742\pi$
$823$	−33.5410			−1.16917			−0.584583	+	0.811334i	$0.698742\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	25.4558i		−	0.885186i	−0.896723	−	0.442593i	$-0.854059\pi$
$827$		−	25.4558i		−	0.885186i	0.896723	−	0.442593i	$-0.145941\pi$
$828$		0			0
$829$	24.0000			0.833554			0.416777	−	0.909009i	$-0.363160\pi$
$829$	24.0000			0.833554			0.416777	+	0.909009i	$0.363160\pi$
$830$		0			0
$831$			33.5410i			1.16353i
$832$		0			0
$833$	3.16228			0.109566
$834$		0			0
$835$	7.07107	+	7.07107i	0.244704	+	0.244704i
$836$		0			0
$837$		−	15.0000i		−	0.518476i
$838$		0			0
$839$	−42.4264			−1.46472			−0.732361	−	0.680916i	$-0.761582\pi$
$839$	−42.4264			−1.46472			−0.732361	+	0.680916i	$0.761582\pi$
$840$		0			0
$841$	52.0000			1.79310
$842$		0			0
$843$			14.1421i			0.487081i
$844$		0			0
$845$	−18.9737	−	18.9737i	−0.652714	−	0.652714i
$846$		0			0
$847$			31.1127i			1.06904i
$848$		0			0
$849$		−	34.7851i		−	1.19382i
$850$		0			0
$851$		0			0
$852$		0			0
$853$			30.0000i			1.02718i	0.858036	+	0.513590i	$0.171685\pi$
$853$			30.0000i			1.02718i	−0.858036	+	0.513590i	$0.828315\pi$
$854$		0			0
$855$	22.3607	+	22.3607i	0.764719	+	0.764719i
$856$		0			0
$857$			15.0000i			0.512390i	0.966625	+	0.256195i	$0.0824690\pi$
$857$			15.0000i			0.512390i	−0.966625	+	0.256195i	$0.917531\pi$
$858$		0			0
$859$			46.9574i			1.60217i	0.598553	+	0.801083i	$0.295743\pi$
$859$			46.9574i			1.60217i	−0.598553	+	0.801083i	$0.704257\pi$
$860$		0			0
$861$			18.9737i			0.646621i
$862$		0			0
$863$	6.70820			0.228350			0.114175	−	0.993461i	$-0.463578\pi$
$863$	6.70820			0.228350			0.114175	+	0.993461i	$0.463578\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−15.6525			−0.531586
$868$		0			0
$869$		0			0
$870$		0			0
$871$	14.1421			0.479188
$872$		0			0
$873$	−37.9473			−1.28432
$874$		0			0
$875$	22.3607	+	22.3607i	0.755929	+	0.755929i
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$	28.2843			0.954005
$880$		0			0
$881$		−	22.1359i		−	0.745779i	−0.927876	−	0.372889i	$-0.878367\pi$
$881$		−	22.1359i		−	0.745779i	0.927876	−	0.372889i	$-0.121633\pi$
$882$		0			0
$883$	26.8328			0.902996			0.451498	−	0.892272i	$-0.350890\pi$
$883$	26.8328			0.902996			0.451498	+	0.892272i	$0.350890\pi$
$884$		0			0
$885$	15.8114	−	15.8114i	0.531494	−	0.531494i
$886$		0			0
$887$	46.9574			1.57668			0.788338	−	0.615242i	$-0.210942\pi$
$887$	46.9574			1.57668			0.788338	+	0.615242i	$0.210942\pi$
$888$		0			0
$889$			18.9737i			0.636356i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	47.4342			1.58732
$894$		0			0
$895$	3.53553	−	3.53553i	0.118180	−	0.118180i
$896$		0			0
$897$	47.4342	+	25.0000i	1.58378	+	0.834726i
$898$		0			0
$899$			60.3738i			2.01358i
$900$		0			0
$901$	−20.0000			−0.666297
$902$		0			0
$903$	8.94427			0.297647
$904$		0			0
$905$	30.0000	−	30.0000i	0.997234	−	0.997234i
$906$		0			0
$907$		−	18.3848i		−	0.610456i	−0.952279	−	0.305228i	$-0.901267\pi$
$907$		−	18.3848i		−	0.610456i	0.952279	−	0.305228i	$-0.0987328\pi$
$908$		0			0
$909$	−24.0000			−0.796030
$910$		0			0
$911$	21.2132			0.702825			0.351412	−	0.936221i	$-0.385702\pi$
$911$	21.2132			0.702825			0.351412	+	0.936221i	$0.385702\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	33.5410	−	33.5410i	1.10883	−	1.10883i
$916$		0			0
$917$	56.9210			1.87970
$918$		0			0
$919$	56.5685			1.86602			0.933012	−	0.359845i	$-0.117171\pi$
$919$	56.5685			1.86602			0.933012	+	0.359845i	$0.117171\pi$
$920$		0			0
$921$	−30.0000			−0.988534
$922$		0			0
$923$	−33.5410			−1.10402
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	2.82843i		−	0.0928977i
$928$		0			0
$929$	−9.00000			−0.295280			−0.147640	−	0.989041i	$-0.547168\pi$
$929$	−9.00000			−0.295280			−0.147640	+	0.989041i	$0.547168\pi$
$930$		0			0
$931$	−7.07107			−0.231745
$932$		0			0
$933$			55.0000i			1.80062i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	37.9473			1.23969			0.619843	−	0.784726i	$-0.287196\pi$
$937$	37.9473			1.23969			0.619843	+	0.784726i	$0.287196\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	15.8114i		−	0.515437i	−0.966220	−	0.257718i	$-0.917029\pi$
$941$		−	15.8114i		−	0.515437i	0.966220	−	0.257718i	$-0.0829706\pi$
$942$		0			0
$943$	−6.70820	+	12.7279i	−0.218449	+	0.414478i
$944$		0			0
$945$	−10.0000	+	10.0000i	−0.325300	+	0.325300i
$946$		0			0
$947$	−6.70820			−0.217987			−0.108994	−	0.994042i	$-0.534763\pi$
$947$	−6.70820			−0.217987			−0.108994	+	0.994042i	$0.534763\pi$
$948$		0			0
$949$	75.0000			2.43460
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−31.6228			−1.02436			−0.512181	−	0.858877i	$-0.671162\pi$
$953$	−31.6228			−1.02436			−0.512181	+	0.858877i	$0.671162\pi$
$954$		0			0
$955$	33.5410	+	33.5410i	1.08536	+	1.08536i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			17.8885i			0.577651i
$960$		0			0
$961$	−14.0000			−0.451613
$962$		0			0
$963$			8.48528i			0.273434i
$964$		0			0
$965$	7.90569	−	7.90569i	0.254493	−	0.254493i
$966$		0			0
$967$	60.3738			1.94149			0.970746	−	0.240109i	$-0.0771833\pi$
$967$	60.3738			1.94149			0.970746	+	0.240109i	$0.0771833\pi$
$968$		0			0
$969$	−50.0000			−1.60623
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$	18.9737			0.608268
$974$		0			0
$975$	−55.9017			−1.79029
$976$		0			0
$977$	31.6228			1.01170			0.505851	−	0.862621i	$-0.331179\pi$
$977$	31.6228			1.01170			0.505851	+	0.862621i	$0.331179\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$			18.9737i			0.605783i
$982$		0			0
$983$		−	8.48528i		−	0.270638i	−0.990802	−	0.135319i	$-0.956794\pi$
$983$		−	8.48528i		−	0.270638i	0.990802	−	0.135319i	$-0.0432060\pi$
$984$		0			0
$985$	23.7171	−	23.7171i	0.755689	−	0.755689i
$986$		0			0
$987$		−	42.4264i		−	1.35045i
$988$		0			0
$989$	6.00000	+	3.16228i	0.190789	+	0.100555i
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		−	45.0000i		−	1.42803i
$994$		0			0
$995$	−22.3607	−	22.3607i	−0.708881	−	0.708881i
$996$		0			0
$997$		−	10.0000i		−	0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$		−	10.0000i		−	0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.m.f.1839.8	yes	8
4.3	odd	2	inner	1840.2.m.f.1839.4	yes	8
5.4	even	2	inner	1840.2.m.f.1839.2	yes	8
20.19	odd	2	inner	1840.2.m.f.1839.6	yes	8
23.22	odd	2	inner	1840.2.m.f.1839.5	yes	8
92.91	even	2	inner	1840.2.m.f.1839.1	✓	8
115.114	odd	2	inner	1840.2.m.f.1839.3	yes	8
460.459	even	2	inner	1840.2.m.f.1839.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.2.m.f.1839.1	✓	8	92.91	even	2	inner
1840.2.m.f.1839.2	yes	8	5.4	even	2	inner
1840.2.m.f.1839.3	yes	8	115.114	odd	2	inner
1840.2.m.f.1839.4	yes	8	4.3	odd	2	inner
1840.2.m.f.1839.5	yes	8	23.22	odd	2	inner
1840.2.m.f.1839.6	yes	8	20.19	odd	2	inner
1840.2.m.f.1839.7	yes	8	460.459	even	2	inner
1840.2.m.f.1839.8	yes	8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.23607 q^{3} +(1.58114 + 1.58114i) q^{5} -2.82843i q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+2.23607 q^{3} +(1.58114 + 1.58114i) q^{5} -2.82843i q^{7} +2.00000 q^{9} +5.00000i q^{13} +(3.53553 + 3.53553i) q^{15} -3.16228 q^{17} +7.07107 q^{19} -6.32456i q^{21} +(2.23607 - 4.24264i) q^{23} +5.00000i q^{25} -2.23607 q^{27} +9.00000 q^{29} +6.70820i q^{31} +(4.47214 - 4.47214i) q^{35} +11.1803i q^{39} -3.00000 q^{41} +1.41421i q^{43} +(3.16228 + 3.16228i) q^{45} +6.70820 q^{47} -1.00000 q^{49} -7.07107 q^{51} +6.32456 q^{53} +15.8114 q^{57} -4.47214i q^{59} -9.48683i q^{61} -5.65685i q^{63} +(-7.90569 + 7.90569i) q^{65} -2.82843i q^{67} +(5.00000 - 9.48683i) q^{69} +6.70820i q^{71} -15.0000i q^{73} +11.1803i q^{75} -7.07107 q^{79} -11.0000 q^{81} -12.7279i q^{83} +(-5.00000 - 5.00000i) q^{85} +20.1246 q^{87} +3.16228i q^{89} +14.1421 q^{91} +15.0000i q^{93} +(11.1803 + 11.1803i) q^{95} -18.9737 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{9}+O(q^{10}) \) 8 * q + 16 * q^9	\( 8 q + 16 q^{9} + 72 q^{29} - 24 q^{41} - 8 q^{49} + 40 q^{69} - 88 q^{81} - 40 q^{85}+O(q^{100}) \) 8 * q + 16 * q^9 + 72 * q^29 - 24 * q^41 - 8 * q^49 + 40 * q^69 - 88 * q^81 - 40 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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