LMFDB - Embedded newform 414.3.c.b.323.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,3,Mod(323,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("414.323");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	414.c (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:	$11.2806829445$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 19x^{6} - 88x^{5} + 301x^{4} - 1010x^{3} + 2713x^{2} - 7044x + 9558 $ x^8 - 2x^7 + 19x^6 - 88x^5 + 301x^4 - 1010x^3 + 2713x^2 - 7044*x + 9558
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			323.6
Root			$-0.919152 + 3.46316i$ of defining polynomial
Character	$\chi$	$=$	414.323
Dual form			414.3.c.b.323.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} -2.59976i q^{5} -3.01296 q^{7} -2.82843i q^{8} +O(q^{10})$	\(q+1.41421i q^{2} -2.00000 q^{4} -2.59976i q^{5} -3.01296 q^{7} -2.82843i q^{8} +3.67661 q^{10} +4.26097i q^{11} +7.88805 q^{13} -4.26097i q^{14} +4.00000 q^{16} +10.6277i q^{17} +36.1682 q^{19} +5.19951i q^{20} -6.02592 q^{22} +4.79583i q^{23} +18.2413 q^{25} +11.1554i q^{26} +6.02592 q^{28} +4.54166i q^{29} +33.1552 q^{31} +5.65685i q^{32} -15.0298 q^{34} +7.83296i q^{35} +21.7761 q^{37} +51.1496i q^{38} -7.35322 q^{40} -2.40221i q^{41} +27.0467 q^{43} -8.52194i q^{44} -6.78233 q^{46} +22.9998i q^{47} -39.9221 q^{49} +25.7971i q^{50} -15.7761 q^{52} +51.8594i q^{53} +11.0775 q^{55} +8.52194i q^{56} -6.42288 q^{58} -61.7308i q^{59} -49.6040 q^{61} +46.8886i q^{62} -8.00000 q^{64} -20.5070i q^{65} +70.2460 q^{67} -21.2554i q^{68} -11.0775 q^{70} +85.9618i q^{71} -7.23390 q^{73} +30.7961i q^{74} -72.3364 q^{76} -12.8381i q^{77} +12.2381 q^{79} -10.3990i q^{80} +3.39724 q^{82} -41.1149i q^{83} +27.6294 q^{85} +38.2498i q^{86} +12.0518 q^{88} -90.3910i q^{89} -23.7664 q^{91} -9.59166i q^{92} -32.5266 q^{94} -94.0285i q^{95} -98.6935 q^{97} -56.4583i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} + 16 q^{7}+O(q^{10}) $ 8 * q - 16 * q^4 + 16 * q^7	$ 8 q - 16 q^{4} + 16 q^{7} - 8 q^{10} - 8 q^{13} + 32 q^{16} - 48 q^{19} + 32 q^{22} - 32 q^{28} - 32 q^{31} - 8 q^{34} + 32 q^{37} + 16 q^{40} + 32 q^{43} + 80 q^{49} + 16 q^{52} + 32 q^{55} + 16 q^{58} + 48 q^{61} - 64 q^{64} - 16 q^{67} - 32 q^{70} - 432 q^{73} + 96 q^{76} - 416 q^{79} - 144 q^{82} + 584 q^{85} - 64 q^{88} + 368 q^{91} + 128 q^{94} - 8 q^{97}+O(q^{100}) $ 8 * q - 16 * q^4 + 16 * q^7 - 8 * q^10 - 8 * q^13 + 32 * q^16 - 48 * q^19 + 32 * q^22 - 32 * q^28 - 32 * q^31 - 8 * q^34 + 32 * q^37 + 16 * q^40 + 32 * q^43 + 80 * q^49 + 16 * q^52 + 32 * q^55 + 16 * q^58 + 48 * q^61 - 64 * q^64 - 16 * q^67 - 32 * q^70 - 432 * q^73 + 96 * q^76 - 416 * q^79 - 144 * q^82 + 584 * q^85 - 64 * q^88 + 368 * q^91 + 128 * q^94 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/414\mathbb{Z}\right)^\times$.

$n$	$47$	$235$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.41421i	0.707107i
$2$	1.41421i	0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	− 2.59976i	− 0.519951i	−0.965615	−	0.259976i	$-0.916285\pi$
$5$	− 2.59976i	− 0.519951i	0.965615	−	0.259976i	$-0.0837145\pi$
$6$	0	0
$7$	−3.01296	−0.430423	−0.215211	−	0.976567i	$-0.569044\pi$
$7$	−3.01296	−0.430423	−0.215211	+	0.976567i	$0.569044\pi$
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	3.67661	0.367661
$11$	4.26097i	0.387361i	0.981065	+	0.193680i	$0.0620425\pi$
$11$	4.26097i	0.387361i	−0.981065	+	0.193680i	$0.937958\pi$
$12$	0	0
$13$	7.88805	0.606773	0.303387	−	0.952868i	$-0.401883\pi$
$13$	7.88805	0.606773	0.303387	+	0.952868i	$0.401883\pi$
$14$	− 4.26097i	− 0.304355i
$15$	0	0
$16$	4.00000	0.250000
$17$	10.6277i	0.625158i	0.949892	+	0.312579i	$0.101193\pi$
$17$	10.6277i	0.625158i	−0.949892	+	0.312579i	$0.898807\pi$
$18$	0	0
$19$	36.1682	1.90359	0.951795	−	0.306736i	$-0.0992367\pi$
$19$	36.1682	1.90359	0.951795	+	0.306736i	$0.0992367\pi$
$20$	5.19951i	0.259976i
$21$	0	0
$22$	−6.02592	−0.273905
$23$	4.79583i	0.208514i
$23$	4.79583i	0.208514i
$24$	0	0
$25$	18.2413	0.729651
$26$	11.1554i	0.429053i
$27$	0	0
$28$	6.02592	0.215211
$29$	4.54166i	0.156609i	0.996929	+	0.0783045i	$0.0249506\pi$
$29$	4.54166i	0.156609i	−0.996929	+	0.0783045i	$0.975049\pi$
$30$	0	0
$31$	33.1552	1.06952	0.534762	−	0.845003i	$-0.320401\pi$
$31$	33.1552	1.06952	0.534762	+	0.845003i	$0.320401\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	−15.0298	−0.442054
$35$	7.83296i	0.223799i
$36$	0	0
$37$	21.7761	0.588543	0.294272	−	0.955722i	$-0.404923\pi$
$37$	21.7761	0.588543	0.294272	+	0.955722i	$0.404923\pi$
$38$	51.1496i	1.34604i
$39$	0	0
$40$	−7.35322	−0.183830
$41$	− 2.40221i	− 0.0585906i	−0.999571	−	0.0292953i	$-0.990674\pi$
$41$	− 2.40221i	− 0.0585906i	0.999571	−	0.0292953i	$-0.00932631\pi$
$42$	0	0
$43$	27.0467	0.628993	0.314497	−	0.949259i	$-0.398164\pi$
$43$	27.0467	0.628993	0.314497	+	0.949259i	$0.398164\pi$
$44$	− 8.52194i	− 0.193680i
$45$	0	0
$46$	−6.78233	−0.147442
$47$	22.9998i	0.489356i	0.969604	+	0.244678i	$0.0786823\pi$
$47$	22.9998i	0.489356i	−0.969604	+	0.244678i	$0.921318\pi$
$48$	0	0
$49$	−39.9221	−0.814736
$50$	25.7971i	0.515941i
$51$	0	0
$52$	−15.7761	−0.303387
$53$	51.8594i	0.978480i	0.872149	+	0.489240i	$0.162726\pi$
$53$	51.8594i	0.978480i	−0.872149	+	0.489240i	$0.837274\pi$
$54$	0	0
$55$	11.0775	0.201409
$56$	8.52194i	0.152177i
$57$	0	0
$58$	−6.42288	−0.110739
$59$	− 61.7308i	− 1.04628i	−0.852246	−	0.523142i	$-0.824760\pi$
$59$	− 61.7308i	− 1.04628i	0.852246	−	0.523142i	$-0.175240\pi$
$60$	0	0
$61$	−49.6040	−0.813181	−0.406591	−	0.913611i	$-0.633282\pi$
$61$	−49.6040	−0.813181	−0.406591	+	0.913611i	$0.633282\pi$
$62$	46.8886i	0.756268i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 20.5070i	− 0.315492i
$66$	0	0
$67$	70.2460	1.04845	0.524224	−	0.851581i	$-0.324356\pi$
$67$	70.2460	1.04845	0.524224	+	0.851581i	$0.324356\pi$
$68$	− 21.2554i	− 0.312579i
$69$	0	0
$70$	−11.0775	−0.158250
$71$	85.9618i	1.21073i	0.795948	+	0.605365i	$0.206973\pi$
$71$	85.9618i	1.21073i	−0.795948	+	0.605365i	$0.793027\pi$
$72$	0	0
$73$	−7.23390	−0.0990945	−0.0495473	−	0.998772i	$-0.515778\pi$
$73$	−7.23390	−0.0990945	−0.0495473	+	0.998772i	$0.515778\pi$
$74$	30.7961i	0.416163i
$75$	0	0
$76$	−72.3364	−0.951795
$77$	− 12.8381i	− 0.166729i
$78$	0	0
$79$	12.2381	0.154913	0.0774566	−	0.996996i	$-0.475320\pi$
$79$	12.2381	0.154913	0.0774566	+	0.996996i	$0.475320\pi$
$80$	− 10.3990i	− 0.129988i
$81$	0	0
$82$	3.39724	0.0414298
$83$	− 41.1149i	− 0.495360i	−0.968842	−	0.247680i	$-0.920332\pi$
$83$	− 41.1149i	− 0.495360i	0.968842	−	0.247680i	$-0.0796682\pi$
$84$	0	0
$85$	27.6294	0.325052
$86$	38.2498i	0.444765i
$87$	0	0
$88$	12.0518	0.136953
$89$	− 90.3910i	− 1.01563i	−0.861466	−	0.507814i	$-0.830454\pi$
$89$	− 90.3910i	− 1.01563i	0.861466	−	0.507814i	$-0.169546\pi$
$90$	0	0
$91$	−23.7664	−0.261169
$92$	− 9.59166i	− 0.104257i
$93$	0	0
$94$	−32.5266	−0.346027
$95$	− 94.0285i	− 0.989774i
$96$	0	0
$97$	−98.6935	−1.01746	−0.508730	−	0.860926i	$-0.669885\pi$
$97$	−98.6935	−1.01746	−0.508730	+	0.860926i	$0.669885\pi$
$98$	− 56.4583i	− 0.576105i
$99$	0	0
$100$	−36.4825	−0.364825
$101$	41.0011i	0.405952i	0.979184	+	0.202976i	$0.0650613\pi$
$101$	41.0011i	0.405952i	−0.979184	+	0.202976i	$0.934939\pi$
$102$	0	0
$103$	−60.6497	−0.588832	−0.294416	−	0.955677i	$-0.595125\pi$
$103$	−60.6497	−0.588832	−0.294416	+	0.955677i	$0.595125\pi$
$104$	− 22.3108i	− 0.214527i
$105$	0	0
$106$	−73.3403	−0.691890
$107$	− 46.1385i	− 0.431201i	−0.976482	−	0.215600i	$-0.930829\pi$
$107$	− 46.1385i	− 0.431201i	0.976482	−	0.215600i	$-0.0691708\pi$
$108$	0	0
$109$	143.340	1.31505	0.657524	−	0.753433i	$-0.271604\pi$
$109$	143.340	1.31505	0.657524	+	0.753433i	$0.271604\pi$
$110$	15.6659i	0.142417i
$111$	0	0
$112$	−12.0518	−0.107606
$113$	44.0074i	0.389446i	0.980858	+	0.194723i	$0.0623808\pi$
$113$	44.0074i	0.389446i	−0.980858	+	0.194723i	$0.937619\pi$
$114$	0	0
$115$	12.4680	0.108417
$116$	− 9.08332i	− 0.0783045i
$117$	0	0
$118$	87.3005	0.739835
$119$	− 32.0208i	− 0.269083i
$120$	0	0
$121$	102.844	0.849952
$122$	− 70.1507i	− 0.575006i
$123$	0	0
$124$	−66.3105	−0.534762
$125$	− 112.417i	− 0.899334i
$126$	0	0
$127$	41.0154	0.322956	0.161478	−	0.986876i	$-0.448374\pi$
$127$	41.0154	0.322956	0.161478	+	0.986876i	$0.448374\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	29.0013	0.223087
$131$	− 140.042i	− 1.06902i	−0.845161	−	0.534512i	$-0.820495\pi$
$131$	− 140.042i	− 1.06902i	0.845161	−	0.534512i	$-0.179505\pi$
$132$	0	0
$133$	−108.973	−0.819348
$134$	99.3428i	0.741364i
$135$	0	0
$136$	30.0597	0.221027
$137$	− 87.6154i	− 0.639528i	−0.947497	−	0.319764i	$-0.896396\pi$
$137$	− 87.6154i	− 0.639528i	0.947497	−	0.319764i	$-0.103604\pi$
$138$	0	0
$139$	117.559	0.845749	0.422875	−	0.906188i	$-0.361021\pi$
$139$	117.559	0.845749	0.422875	+	0.906188i	$0.361021\pi$
$140$	− 15.6659i	− 0.111899i
$141$	0	0
$142$	−121.568	−0.856116
$143$	33.6107i	0.235040i
$144$	0	0
$145$	11.8072	0.0814291
$146$	− 10.2303i	− 0.0700704i
$147$	0	0
$148$	−43.5522	−0.294272
$149$	− 44.3843i	− 0.297881i	−0.988846	−	0.148940i	$-0.952414\pi$
$149$	− 44.3843i	− 0.297881i	0.988846	−	0.148940i	$-0.0475863\pi$
$150$	0	0
$151$	−91.8270	−0.608126	−0.304063	−	0.952652i	$-0.598343\pi$
$151$	−91.8270	−0.608126	−0.304063	+	0.952652i	$0.598343\pi$
$152$	− 102.299i	− 0.673021i
$153$	0	0
$154$	18.1559	0.117895
$155$	− 86.1955i	− 0.556100i
$156$	0	0
$157$	−163.535	−1.04162	−0.520812	−	0.853671i	$-0.674371\pi$
$157$	−163.535	−1.04162	−0.520812	+	0.853671i	$0.674371\pi$
$158$	17.3073i	0.109540i
$159$	0	0
$160$	14.7064	0.0919152
$161$	− 14.4496i	− 0.0897494i
$162$	0	0
$163$	69.9897	0.429385	0.214692	−	0.976682i	$-0.431125\pi$
$163$	69.9897	0.429385	0.214692	+	0.976682i	$0.431125\pi$
$164$	4.80443i	0.0292953i
$165$	0	0
$166$	58.1452	0.350273
$167$	− 184.568i	− 1.10520i	−0.833447	−	0.552600i	$-0.813636\pi$
$167$	− 184.568i	− 1.10520i	0.833447	−	0.552600i	$-0.186364\pi$
$168$	0	0
$169$	−106.779	−0.631826
$170$	39.0739i	0.229846i
$171$	0	0
$172$	−54.0934	−0.314497
$173$	261.108i	1.50929i	0.656131	+	0.754647i	$0.272192\pi$
$173$	261.108i	1.50929i	−0.656131	+	0.754647i	$0.727808\pi$
$174$	0	0
$175$	−54.9602	−0.314058
$176$	17.0439i	0.0968402i
$177$	0	0
$178$	127.832	0.718158
$179$	93.5134i	0.522421i	0.965282	+	0.261211i	$0.0841217\pi$
$179$	93.5134i	0.522421i	−0.965282	+	0.261211i	$0.915878\pi$
$180$	0	0
$181$	−37.4683	−0.207007	−0.103504	−	0.994629i	$-0.533005\pi$
$181$	−37.4683	−0.207007	−0.103504	+	0.994629i	$0.533005\pi$
$182$	− 33.6107i	− 0.184674i
$183$	0	0
$184$	13.5647	0.0737210
$185$	− 56.6125i	− 0.306014i
$186$	0	0
$187$	−45.2843	−0.242162
$188$	− 45.9995i	− 0.244678i
$189$	0	0
$190$	132.976	0.699876
$191$	− 187.035i	− 0.979241i	−0.871936	−	0.489621i	$-0.837135\pi$
$191$	− 187.035i	− 0.979241i	0.871936	−	0.489621i	$-0.162865\pi$
$192$	0	0
$193$	−211.076	−1.09366	−0.546829	−	0.837244i	$-0.684165\pi$
$193$	−211.076	−1.09366	−0.546829	+	0.837244i	$0.684165\pi$
$194$	− 139.574i	− 0.719452i
$195$	0	0
$196$	79.8441	0.407368
$197$	19.4899i	0.0989336i	0.998776	+	0.0494668i	$0.0157522\pi$
$197$	19.4899i	0.0989336i	−0.998776	+	0.0494668i	$0.984248\pi$
$198$	0	0
$199$	122.418	0.615164	0.307582	−	0.951522i	$-0.400480\pi$
$199$	122.418	0.615164	0.307582	+	0.951522i	$0.400480\pi$
$200$	− 51.5941i	− 0.257971i
$201$	0	0
$202$	−57.9844	−0.287051
$203$	− 13.6838i	− 0.0674081i
$204$	0	0
$205$	−6.24517	−0.0304642
$206$	− 85.7717i	− 0.416367i
$207$	0	0
$208$	31.5522	0.151693
$209$	154.112i	0.737376i
$210$	0	0
$211$	−154.293	−0.731248	−0.365624	−	0.930763i	$-0.619144\pi$
$211$	−154.293	−0.731248	−0.365624	+	0.930763i	$0.619144\pi$
$212$	− 103.719i	− 0.489240i
$213$	0	0
$214$	65.2496	0.304905
$215$	− 70.3148i	− 0.327046i
$216$	0	0
$217$	−99.8954	−0.460348
$218$	202.714i	0.929880i
$219$	0	0
$220$	−22.1550	−0.100704
$221$	83.8318i	0.379329i
$222$	0	0
$223$	318.666	1.42899	0.714497	−	0.699638i	$-0.246655\pi$
$223$	318.666	1.42899	0.714497	+	0.699638i	$0.246655\pi$
$224$	− 17.0439i	− 0.0760887i
$225$	0	0
$226$	−62.2359	−0.275380
$227$	− 256.743i	− 1.13102i	−0.824740	−	0.565512i	$-0.808679\pi$
$227$	− 256.743i	− 1.13102i	0.824740	−	0.565512i	$-0.191321\pi$
$228$	0	0
$229$	−124.269	−0.542660	−0.271330	−	0.962486i	$-0.587464\pi$
$229$	−124.269	−0.542660	−0.271330	+	0.962486i	$0.587464\pi$
$230$	17.6324i	0.0766626i
$231$	0	0
$232$	12.8458	0.0553697
$233$	336.169i	1.44278i	0.692527	+	0.721392i	$0.256497\pi$
$233$	336.169i	1.44278i	−0.692527	+	0.721392i	$0.743503\pi$
$234$	0	0
$235$	59.7937	0.254441
$236$	123.462i	0.523142i
$237$	0	0
$238$	45.2843	0.190270
$239$	466.837i	1.95329i	0.214850	+	0.976647i	$0.431074\pi$
$239$	466.837i	1.95329i	−0.214850	+	0.976647i	$0.568926\pi$
$240$	0	0
$241$	−277.000	−1.14938	−0.574688	−	0.818373i	$-0.694877\pi$
$241$	−277.000	−1.14938	−0.574688	+	0.818373i	$0.694877\pi$
$242$	145.444i	0.601007i
$243$	0	0
$244$	99.2081	0.406591
$245$	103.788i	0.423623i
$246$	0	0
$247$	285.297	1.15505
$248$	− 93.7772i	− 0.378134i
$249$	0	0
$250$	158.981	0.635925
$251$	397.343i	1.58304i	0.611142	+	0.791521i	$0.290710\pi$
$251$	397.343i	1.58304i	−0.611142	+	0.791521i	$0.709290\pi$
$252$	0	0
$253$	−20.4349	−0.0807703
$254$	58.0045i	0.228364i
$255$	0	0
$256$	16.0000	0.0625000
$257$	344.898i	1.34201i	0.741451	+	0.671007i	$0.234138\pi$
$257$	344.898i	1.34201i	−0.741451	+	0.671007i	$0.765862\pi$
$258$	0	0
$259$	−65.6105	−0.253322
$260$	41.0140i	0.157746i
$261$	0	0
$262$	198.049	0.755914
$263$	37.5813i	0.142895i	0.997444	+	0.0714473i	$0.0227618\pi$
$263$	37.5813i	0.142895i	−0.997444	+	0.0714473i	$0.977238\pi$
$264$	0	0
$265$	134.822	0.508762
$266$	− 154.112i	− 0.579367i
$267$	0	0
$268$	−140.492	−0.524224
$269$	109.513i	0.407111i	0.979063	+	0.203556i	$0.0652498\pi$
$269$	109.513i	0.407111i	−0.979063	+	0.203556i	$0.934750\pi$
$270$	0	0
$271$	47.4483	0.175086	0.0875430	−	0.996161i	$-0.472098\pi$
$271$	47.4483	0.175086	0.0875430	+	0.996161i	$0.472098\pi$
$272$	42.5108i	0.156290i
$273$	0	0
$274$	123.907	0.452215
$275$	77.7255i	0.282638i
$276$	0	0
$277$	−392.580	−1.41725	−0.708627	−	0.705583i	$-0.750685\pi$
$277$	−392.580	−1.41725	−0.708627	+	0.705583i	$0.750685\pi$
$278$	166.254i	0.598035i
$279$	0	0
$280$	22.1550	0.0791249
$281$	− 287.955i	− 1.02475i	−0.858761	−	0.512376i	$-0.828766\pi$
$281$	− 287.955i	− 1.02475i	0.858761	−	0.512376i	$-0.171234\pi$
$282$	0	0
$283$	−310.411	−1.09686	−0.548430	−	0.836197i	$-0.684774\pi$
$283$	−310.411	−1.09686	−0.548430	+	0.836197i	$0.684774\pi$
$284$	− 171.924i	− 0.605365i
$285$	0	0
$286$	−47.5328	−0.166198
$287$	7.23777i	0.0252187i
$288$	0	0
$289$	176.052	0.609177
$290$	16.6979i	0.0575790i
$291$	0	0
$292$	14.4678	0.0495473
$293$	433.344i	1.47899i	0.673162	+	0.739495i	$0.264936\pi$
$293$	433.344i	1.47899i	−0.673162	+	0.739495i	$0.735064\pi$
$294$	0	0
$295$	−160.485	−0.544017
$296$	− 61.5921i	− 0.208081i
$297$	0	0
$298$	62.7688	0.210634
$299$	37.8298i	0.126521i
$300$	0	0
$301$	−81.4906	−0.270733
$302$	− 129.863i	− 0.430010i
$303$	0	0
$304$	144.673	0.475897
$305$	128.958i	0.422814i
$306$	0	0
$307$	490.791	1.59867	0.799334	−	0.600886i	$-0.205186\pi$
$307$	490.791	1.59867	0.799334	+	0.600886i	$0.205186\pi$
$308$	25.6763i	0.0833645i
$309$	0	0
$310$	121.899	0.393222
$311$	− 181.427i	− 0.583367i	−0.956515	−	0.291684i	$-0.905785\pi$
$311$	− 181.427i	− 0.583367i	0.956515	−	0.291684i	$-0.0942154\pi$
$312$	0	0
$313$	263.206	0.840913	0.420456	−	0.907313i	$-0.361870\pi$
$313$	263.206	0.840913	0.420456	+	0.907313i	$0.361870\pi$
$314$	− 231.274i	− 0.736540i
$315$	0	0
$316$	−24.4763	−0.0774566
$317$	482.244i	1.52127i	0.649177	+	0.760637i	$0.275113\pi$
$317$	482.244i	1.52127i	−0.649177	+	0.760637i	$0.724887\pi$
$318$	0	0
$319$	−19.3519	−0.0606642
$320$	20.7980i	0.0649939i
$321$	0	0
$322$	20.4349	0.0634624
$323$	384.385i	1.19005i
$324$	0	0
$325$	143.888	0.442732
$326$	98.9804i	0.303621i
$327$	0	0
$328$	−6.79448	−0.0207149
$329$	− 69.2973i	− 0.210630i
$330$	0	0
$331$	−126.745	−0.382917	−0.191458	−	0.981501i	$-0.561322\pi$
$331$	−126.745	−0.382917	−0.191458	+	0.981501i	$0.561322\pi$
$332$	82.2298i	0.247680i
$333$	0	0
$334$	261.019	0.781494
$335$	− 182.622i	− 0.545141i
$336$	0	0
$337$	−253.149	−0.751185	−0.375593	−	0.926785i	$-0.622561\pi$
$337$	−253.149	−0.751185	−0.375593	+	0.926785i	$0.622561\pi$
$338$	− 151.008i	− 0.446769i
$339$	0	0
$340$	−55.2588	−0.162526
$341$	141.273i	0.414292i
$342$	0	0
$343$	267.919	0.781104
$344$	− 76.4996i	− 0.222383i
$345$	0	0
$346$	−369.262	−1.06723
$347$	− 220.063i	− 0.634187i	−0.948394	−	0.317093i	$-0.897293\pi$
$347$	− 220.063i	− 0.634187i	0.948394	−	0.317093i	$-0.102707\pi$
$348$	0	0
$349$	−94.1450	−0.269757	−0.134878	−	0.990862i	$-0.543064\pi$
$349$	−94.1450	−0.269757	−0.134878	+	0.990862i	$0.543064\pi$
$350$	− 77.7255i	− 0.222073i
$351$	0	0
$352$	−24.1037	−0.0684764
$353$	− 572.508i	− 1.62184i	−0.585159	−	0.810918i	$-0.698968\pi$
$353$	− 572.508i	− 1.62184i	0.585159	−	0.810918i	$-0.301032\pi$
$354$	0	0
$355$	223.480	0.629521
$356$	180.782i	0.507814i
$357$	0	0
$358$	−132.248	−0.369408
$359$	− 708.506i	− 1.97356i	−0.162079	−	0.986778i	$-0.551820\pi$
$359$	− 708.506i	− 1.97356i	0.162079	−	0.986778i	$-0.448180\pi$
$360$	0	0
$361$	947.139	2.62365
$362$	− 52.9882i	− 0.146376i
$363$	0	0
$364$	47.5328	0.130585
$365$	18.8064i	0.0515243i
$366$	0	0
$367$	−171.922	−0.468453	−0.234226	−	0.972182i	$-0.575256\pi$
$367$	−171.922	−0.468453	−0.234226	+	0.972182i	$0.575256\pi$
$368$	19.1833i	0.0521286i
$369$	0	0
$370$	80.0622	0.216384
$371$	− 156.250i	− 0.421160i
$372$	0	0
$373$	−452.033	−1.21188	−0.605942	−	0.795509i	$-0.707204\pi$
$373$	−452.033	−1.21188	−0.605942	+	0.795509i	$0.707204\pi$
$374$	− 64.0416i	− 0.171234i
$375$	0	0
$376$	65.0531	0.173014
$377$	35.8249i	0.0950262i
$378$	0	0
$379$	171.883	0.453517	0.226759	−	0.973951i	$-0.427187\pi$
$379$	171.883	0.453517	0.226759	+	0.973951i	$0.427187\pi$
$380$	188.057i	0.494887i
$381$	0	0
$382$	264.508	0.692428
$383$	584.883i	1.52711i	0.645744	+	0.763554i	$0.276547\pi$
$383$	584.883i	1.52711i	−0.645744	+	0.763554i	$0.723453\pi$
$384$	0	0
$385$	−33.3760	−0.0866909
$386$	− 298.507i	− 0.773333i
$387$	0	0
$388$	197.387	0.508730
$389$	− 105.667i	− 0.271638i	−0.990734	−	0.135819i	$-0.956633\pi$
$389$	− 105.667i	− 0.271638i	0.990734	−	0.135819i	$-0.0433665\pi$
$390$	0	0
$391$	−50.9686	−0.130355
$392$	112.917i	0.288053i
$393$	0	0
$394$	−27.5629	−0.0699567
$395$	− 31.8162i	− 0.0805473i
$396$	0	0
$397$	−189.463	−0.477237	−0.238619	−	0.971113i	$-0.576695\pi$
$397$	−189.463	−0.477237	−0.238619	+	0.971113i	$0.576695\pi$
$398$	173.125i	0.434986i
$399$	0	0
$400$	72.9651	0.182413
$401$	− 430.462i	− 1.07347i	−0.843751	−	0.536735i	$-0.819657\pi$
$401$	− 430.462i	− 1.07347i	0.843751	−	0.536735i	$-0.180343\pi$
$402$	0	0
$403$	261.530	0.648958
$404$	− 82.0023i	− 0.202976i
$405$	0	0
$406$	19.3519	0.0476647
$407$	92.7873i	0.227979i
$408$	0	0
$409$	−344.934	−0.843359	−0.421680	−	0.906745i	$-0.638559\pi$
$409$	−344.934	−0.843359	−0.421680	+	0.906745i	$0.638559\pi$
$410$	− 8.83200i	− 0.0215415i
$411$	0	0
$412$	121.299	0.294416
$413$	185.992i	0.450345i
$414$	0	0
$415$	−106.889	−0.257563
$416$	44.6215i	0.107263i
$417$	0	0
$418$	−217.947	−0.521404
$419$	− 396.227i	− 0.945650i	−0.881156	−	0.472825i	$-0.843234\pi$
$419$	− 396.227i	− 0.945650i	0.881156	−	0.472825i	$-0.156766\pi$
$420$	0	0
$421$	−552.518	−1.31239	−0.656197	−	0.754590i	$-0.727836\pi$
$421$	−552.518	−1.31239	−0.656197	+	0.754590i	$0.727836\pi$
$422$	− 218.204i	− 0.517070i
$423$	0	0
$424$	146.681	0.345945
$425$	193.863i	0.456147i
$426$	0	0
$427$	149.455	0.350012
$428$	92.2769i	0.215600i
$429$	0	0
$430$	99.4402	0.231256
$431$	− 570.680i	− 1.32408i	−0.749467	−	0.662042i	$-0.769690\pi$
$431$	− 570.680i	− 1.32408i	0.749467	−	0.662042i	$-0.230310\pi$
$432$	0	0
$433$	−532.584	−1.22999	−0.614994	−	0.788532i	$-0.710841\pi$
$433$	−532.584	−1.22999	−0.614994	+	0.788532i	$0.710841\pi$
$434$	− 141.273i	− 0.325515i
$435$	0	0
$436$	−286.681	−0.657524
$437$	173.457i	0.396926i
$438$	0	0
$439$	361.798	0.824141	0.412070	−	0.911152i	$-0.364806\pi$
$439$	361.798	0.824141	0.412070	+	0.911152i	$0.364806\pi$
$440$	− 31.3318i	− 0.0712087i
$441$	0	0
$442$	−118.556	−0.268226
$443$	− 72.9417i	− 0.164654i	−0.996605	−	0.0823270i	$-0.973765\pi$
$443$	− 72.9417i	− 0.164654i	0.996605	−	0.0823270i	$-0.0262352\pi$
$444$	0	0
$445$	−234.994	−0.528077
$446$	450.662i	1.01045i
$447$	0	0
$448$	24.1037	0.0538029
$449$	242.942i	0.541073i	0.962710	+	0.270536i	$0.0872010\pi$
$449$	242.942i	0.541073i	−0.962710	+	0.270536i	$0.912799\pi$
$450$	0	0
$451$	10.2358	0.0226957
$452$	− 88.0149i	− 0.194723i
$453$	0	0
$454$	363.089	0.799755
$455$	61.7868i	0.135795i
$456$	0	0
$457$	−244.855	−0.535787	−0.267894	−	0.963448i	$-0.586328\pi$
$457$	−244.855	−0.535787	−0.267894	+	0.963448i	$0.586328\pi$
$458$	− 175.743i	− 0.383719i
$459$	0	0
$460$	−24.9360	−0.0542087
$461$	− 384.660i	− 0.834404i	−0.908814	−	0.417202i	$-0.863011\pi$
$461$	− 384.660i	− 0.834404i	0.908814	−	0.417202i	$-0.136989\pi$
$462$	0	0
$463$	4.90140	0.0105862	0.00529309	−	0.999986i	$-0.498315\pi$
$463$	4.90140	0.0105862	0.00529309	+	0.999986i	$0.498315\pi$
$464$	18.1666i	0.0391523i
$465$	0	0
$466$	−475.414	−1.02020
$467$	− 627.229i	− 1.34310i	−0.740958	−	0.671552i	$-0.765628\pi$
$467$	− 627.229i	− 1.34310i	0.740958	−	0.671552i	$-0.234372\pi$
$468$	0	0
$469$	−211.648	−0.451276
$470$	84.5611i	0.179917i
$471$	0	0
$472$	−174.601	−0.369917
$473$	115.245i	0.243647i
$474$	0	0
$475$	659.754	1.38896
$476$	64.0416i	0.134541i
$477$	0	0
$478$	−660.208	−1.38119
$479$	137.985i	0.288070i	0.989573	+	0.144035i	$0.0460078\pi$
$479$	137.985i	0.288070i	−0.989573	+	0.144035i	$0.953992\pi$
$480$	0	0
$481$	171.771	0.357112
$482$	− 391.737i	− 0.812732i
$483$	0	0
$484$	−205.688	−0.424976
$485$	256.579i	0.529029i
$486$	0	0
$487$	−433.985	−0.891140	−0.445570	−	0.895247i	$-0.646999\pi$
$487$	−433.985	−0.891140	−0.445570	+	0.895247i	$0.646999\pi$
$488$	140.301i	0.287503i
$489$	0	0
$490$	−146.778	−0.299547
$491$	− 29.8919i	− 0.0608796i	−0.999537	−	0.0304398i	$-0.990309\pi$
$491$	− 29.8919i	− 0.0608796i	0.999537	−	0.0304398i	$-0.00969078\pi$
$492$	0	0
$493$	−48.2674	−0.0979055
$494$	403.470i	0.816741i
$495$	0	0
$496$	132.621	0.267381
$497$	− 259.000i	− 0.521126i
$498$	0	0
$499$	257.354	0.515740	0.257870	−	0.966180i	$-0.416979\pi$
$499$	257.354	0.515740	0.257870	+	0.966180i	$0.416979\pi$
$500$	224.833i	0.449667i
$501$	0	0
$502$	−561.928	−1.11938
$503$	− 457.589i	− 0.909719i	−0.890563	−	0.454859i	$-0.849689\pi$
$503$	− 457.589i	− 0.909719i	0.890563	−	0.454859i	$-0.150311\pi$
$504$	0	0
$505$	106.593	0.211075
$506$	− 28.8993i	− 0.0571132i
$507$	0	0
$508$	−82.0307	−0.161478
$509$	− 412.356i	− 0.810130i	−0.914288	−	0.405065i	$-0.867249\pi$
$509$	− 412.356i	− 0.810130i	0.914288	−	0.405065i	$-0.132751\pi$
$510$	0	0
$511$	21.7955	0.0426526
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	−487.759	−0.948948
$515$	157.675i	0.306164i
$516$	0	0
$517$	−98.0012	−0.189558
$518$	− 92.7873i	− 0.179126i
$519$	0	0
$520$	−58.0026	−0.111543
$521$	− 742.082i	− 1.42434i	−0.702006	−	0.712171i	$-0.747712\pi$
$521$	− 742.082i	− 1.42434i	0.702006	−	0.712171i	$-0.252288\pi$
$522$	0	0
$523$	705.148	1.34828	0.674138	−	0.738606i	$-0.264515\pi$
$523$	705.148	1.34828	0.674138	+	0.738606i	$0.264515\pi$
$524$	280.084i	0.534512i
$525$	0	0
$526$	−53.1480	−0.101042
$527$	352.364i	0.668622i
$528$	0	0
$529$	−23.0000	−0.0434783
$530$	190.667i	0.359749i
$531$	0	0
$532$	217.947	0.409674
$533$	− 18.9488i	− 0.0355512i
$534$	0	0
$535$	−119.949	−0.224203
$536$	− 198.686i	− 0.370682i
$537$	0	0
$538$	−154.875	−0.287871
$539$	− 170.107i	− 0.315597i
$540$	0	0
$541$	−438.563	−0.810653	−0.405326	−	0.914172i	$-0.632842\pi$
$541$	−438.563	−0.810653	−0.405326	+	0.914172i	$0.632842\pi$
$542$	67.1020i	0.123804i
$543$	0	0
$544$	−60.1193	−0.110513
$545$	− 372.650i	− 0.683761i
$546$	0	0
$547$	−1075.39	−1.96598	−0.982989	−	0.183665i	$-0.941204\pi$
$547$	−1075.39	−1.96598	−0.982989	+	0.183665i	$0.941204\pi$
$548$	175.231i	0.319764i
$549$	0	0
$550$	−109.920	−0.199855
$551$	164.264i	0.298119i
$552$	0	0
$553$	−36.8730	−0.0666782
$554$	− 555.191i	− 1.00215i
$555$	0	0
$556$	−235.118	−0.422875
$557$	− 348.278i	− 0.625275i	−0.949872	−	0.312638i	$-0.898787\pi$
$557$	− 348.278i	− 0.625275i	0.949872	−	0.312638i	$-0.101213\pi$
$558$	0	0
$559$	213.346	0.381656
$560$	31.3318i	0.0559497i
$561$	0	0
$562$	407.230	0.724608
$563$	432.266i	0.767790i	0.923377	+	0.383895i	$0.125417\pi$
$563$	432.266i	0.767790i	−0.923377	+	0.383895i	$0.874583\pi$
$564$	0	0
$565$	114.409	0.202493
$566$	− 438.988i	− 0.775597i
$567$	0	0
$568$	243.137	0.428058
$569$	− 858.829i	− 1.50937i	−0.656090	−	0.754683i	$-0.727791\pi$
$569$	− 858.829i	− 1.50937i	0.656090	−	0.754683i	$-0.272209\pi$
$570$	0	0
$571$	−704.507	−1.23381	−0.616906	−	0.787037i	$-0.711614\pi$
$571$	−704.507	−1.23381	−0.616906	+	0.787037i	$0.711614\pi$
$572$	− 67.2215i	− 0.117520i
$573$	0	0
$574$	−10.2358	−0.0178323
$575$	87.4821i	0.152143i
$576$	0	0
$577$	505.398	0.875907	0.437954	−	0.898998i	$-0.355703\pi$
$577$	505.398	0.875907	0.437954	+	0.898998i	$0.355703\pi$
$578$	248.975i	0.430753i
$579$	0	0
$580$	−23.6144	−0.0407145
$581$	123.878i	0.213214i
$582$	0	0
$583$	−220.971	−0.379025
$584$	20.4606i	0.0350352i
$585$	0	0
$586$	−612.841	−1.04580
$587$	− 166.163i	− 0.283072i	−0.989933	−	0.141536i	$-0.954796\pi$
$587$	− 166.163i	− 0.283072i	0.989933	−	0.141536i	$-0.0452041\pi$
$588$	0	0
$589$	1199.17	2.03593
$590$	− 226.960i	− 0.384678i
$591$	0	0
$592$	87.1044	0.147136
$593$	586.384i	0.988842i	0.869222	+	0.494421i	$0.164620\pi$
$593$	586.384i	0.988842i	−0.869222	+	0.494421i	$0.835380\pi$
$594$	0	0
$595$	−83.2463	−0.139910
$596$	88.7685i	0.148940i
$597$	0	0
$598$	−53.4994	−0.0894638
$599$	491.326i	0.820244i	0.912031	+	0.410122i	$0.134514\pi$
$599$	491.326i	0.820244i	−0.912031	+	0.410122i	$0.865486\pi$
$600$	0	0
$601$	303.138	0.504389	0.252195	−	0.967677i	$-0.418848\pi$
$601$	303.138	0.504389	0.252195	+	0.967677i	$0.418848\pi$
$602$	− 115.245i	− 0.191437i
$603$	0	0
$604$	183.654	0.304063
$605$	− 267.370i	− 0.441933i
$606$	0	0
$607$	−742.449	−1.22314	−0.611572	−	0.791189i	$-0.709463\pi$
$607$	−742.449	−1.22314	−0.611572	+	0.791189i	$0.709463\pi$
$608$	204.598i	0.336510i
$609$	0	0
$610$	−182.375	−0.298975
$611$	181.423i	0.296928i
$612$	0	0
$613$	680.481	1.11008	0.555041	−	0.831823i	$-0.312702\pi$
$613$	680.481	1.11008	0.555041	+	0.831823i	$0.312702\pi$
$614$	694.084i	1.13043i
$615$	0	0
$616$	−36.3117	−0.0589476
$617$	441.967i	0.716317i	0.933661	+	0.358158i	$0.116595\pi$
$617$	441.967i	0.716317i	−0.933661	+	0.358158i	$0.883405\pi$
$618$	0	0
$619$	658.098	1.06316	0.531582	−	0.847007i	$-0.321598\pi$
$619$	658.098	1.06316	0.531582	+	0.847007i	$0.321598\pi$
$620$	172.391i	0.278050i
$621$	0	0
$622$	256.577	0.412503
$623$	272.344i	0.437150i
$624$	0	0
$625$	163.776	0.262041
$626$	372.229i	0.594615i
$627$	0	0
$628$	327.070	0.520812
$629$	231.430i	0.367933i
$630$	0	0
$631$	435.134	0.689595	0.344797	−	0.938677i	$-0.387948\pi$
$631$	435.134	0.689595	0.344797	+	0.938677i	$0.387948\pi$
$632$	− 34.6147i	− 0.0547701i
$633$	0	0
$634$	−681.996	−1.07570
$635$	− 106.630i	− 0.167921i
$636$	0	0
$637$	−314.907	−0.494360
$638$	− 27.3677i	− 0.0428961i
$639$	0	0
$640$	−29.4129	−0.0459576
$641$	− 1119.66i	− 1.74673i	−0.487064	−	0.873367i	$-0.661932\pi$
$641$	− 1119.66i	− 1.74673i	0.487064	−	0.873367i	$-0.338068\pi$
$642$	0	0
$643$	22.1423	0.0344359	0.0172179	−	0.999852i	$-0.494519\pi$
$643$	22.1423	0.0344359	0.0172179	+	0.999852i	$0.494519\pi$
$644$	28.8993i	0.0448747i
$645$	0	0
$646$	−543.602	−0.841489
$647$	− 665.522i	− 1.02863i	−0.857602	−	0.514314i	$-0.828047\pi$
$647$	− 665.522i	− 1.02863i	0.857602	−	0.514314i	$-0.171953\pi$
$648$	0	0
$649$	263.033	0.405289
$650$	203.488i	0.313059i
$651$	0	0
$652$	−139.979	−0.214692
$653$	1055.53i	1.61644i	0.588882	+	0.808219i	$0.299568\pi$
$653$	1055.53i	1.61644i	−0.588882	+	0.808219i	$0.700432\pi$
$654$	0	0
$655$	−364.075	−0.555840
$656$	− 9.60885i	− 0.0146476i
$657$	0	0
$658$	98.0012	0.148938
$659$	− 17.0709i	− 0.0259042i	−0.999916	−	0.0129521i	$-0.995877\pi$
$659$	− 17.0709i	− 0.0259042i	0.999916	−	0.0129521i	$-0.00412290\pi$
$660$	0	0
$661$	539.521	0.816220	0.408110	−	0.912933i	$-0.366188\pi$
$661$	539.521	0.816220	0.408110	+	0.912933i	$0.366188\pi$
$662$	− 179.245i	− 0.270763i
$663$	0	0
$664$	−116.290	−0.175136
$665$	283.304i	0.426021i
$666$	0	0
$667$	−21.7810	−0.0326552
$668$	369.136i	0.552600i
$669$	0	0
$670$	258.267	0.385473
$671$	− 211.361i	− 0.314994i
$672$	0	0
$673$	1034.19	1.53669	0.768346	−	0.640035i	$-0.221080\pi$
$673$	1034.19	1.53669	0.768346	+	0.640035i	$0.221080\pi$
$674$	− 358.007i	− 0.531168i
$675$	0	0
$676$	213.557	0.315913
$677$	− 838.801i	− 1.23900i	−0.784998	−	0.619498i	$-0.787336\pi$
$677$	− 838.801i	− 1.23900i	0.784998	−	0.619498i	$-0.212664\pi$
$678$	0	0
$679$	297.360	0.437938
$680$	− 78.1478i	− 0.114923i
$681$	0	0
$682$	−199.791	−0.292948
$683$	131.516i	0.192556i	0.995354	+	0.0962782i	$0.0306939\pi$
$683$	131.516i	0.192556i	−0.995354	+	0.0962782i	$0.969306\pi$
$684$	0	0
$685$	−227.779	−0.332524
$686$	378.894i	0.552324i
$687$	0	0
$688$	108.187	0.157248
$689$	409.070i	0.593715i
$690$	0	0
$691$	−610.066	−0.882874	−0.441437	−	0.897292i	$-0.645531\pi$
$691$	−610.066	−0.882874	−0.441437	+	0.897292i	$0.645531\pi$
$692$	− 522.216i	− 0.754647i
$693$	0	0
$694$	311.216	0.448438
$695$	− 305.625i	− 0.439748i
$696$	0	0
$697$	25.5300	0.0366284
$698$	− 133.141i	− 0.190747i
$699$	0	0
$700$	109.920	0.157029
$701$	907.311i	1.29431i	0.762359	+	0.647155i	$0.224041\pi$
$701$	907.311i	1.29431i	−0.762359	+	0.647155i	$0.775959\pi$
$702$	0	0
$703$	787.602	1.12034
$704$	− 34.0878i	− 0.0484201i
$705$	0	0
$706$	809.649	1.14681
$707$	− 123.535i	− 0.174731i
$708$	0	0
$709$	−1090.50	−1.53808	−0.769039	−	0.639202i	$-0.779265\pi$
$709$	−1090.50	−1.53808	−0.769039	+	0.639202i	$0.779265\pi$
$710$	316.048i	0.445138i
$711$	0	0
$712$	−255.664	−0.359079
$713$	159.007i	0.223011i
$714$	0	0
$715$	87.3797	0.122209
$716$	− 187.027i	− 0.261211i
$717$	0	0
$718$	1001.98	1.39551
$719$	− 6.04170i	− 0.00840292i	−0.999991	−	0.00420146i	$-0.998663\pi$
$719$	− 6.04170i	− 0.00840292i	0.999991	−	0.00420146i	$-0.00133737\pi$
$720$	0	0
$721$	182.735	0.253447
$722$	1339.46i	1.85520i
$723$	0	0
$724$	74.9367	0.103504
$725$	82.8457i	0.114270i
$726$	0	0
$727$	1227.14	1.68796	0.843978	−	0.536377i	$-0.180208\pi$
$727$	1227.14	1.68796	0.843978	+	0.536377i	$0.180208\pi$
$728$	67.2215i	0.0923372i
$729$	0	0
$730$	−26.5962	−0.0364332
$731$	287.444i	0.393220i
$732$	0	0
$733$	176.585	0.240907	0.120454	−	0.992719i	$-0.461565\pi$
$733$	176.585	0.240907	0.120454	+	0.992719i	$0.461565\pi$
$734$	− 243.135i	− 0.331246i
$735$	0	0
$736$	−27.1293	−0.0368605
$737$	299.316i	0.406127i
$738$	0	0
$739$	319.074	0.431765	0.215882	−	0.976419i	$-0.430737\pi$
$739$	319.074	0.431765	0.215882	+	0.976419i	$0.430737\pi$
$740$	113.225i	0.153007i
$741$	0	0
$742$	220.971	0.297805
$743$	831.630i	1.11929i	0.828734	+	0.559643i	$0.189062\pi$
$743$	831.630i	1.11929i	−0.828734	+	0.559643i	$0.810938\pi$
$744$	0	0
$745$	−115.388	−0.154884
$746$	− 639.271i	− 0.856932i
$747$	0	0
$748$	90.5686	0.121081
$749$	139.013i	0.185599i
$750$	0	0
$751$	709.125	0.944241	0.472121	−	0.881534i	$-0.343489\pi$
$751$	709.125	0.944241	0.472121	+	0.881534i	$0.343489\pi$
$752$	91.9990i	0.122339i
$753$	0	0
$754$	−50.6640	−0.0671936
$755$	238.728i	0.316196i
$756$	0	0
$757$	422.022	0.557493	0.278746	−	0.960365i	$-0.410081\pi$
$757$	422.022	0.557493	0.278746	+	0.960365i	$0.410081\pi$
$758$	243.079i	0.320685i
$759$	0	0
$760$	−265.953	−0.349938
$761$	1006.40i	1.32247i	0.750179	+	0.661235i	$0.229967\pi$
$761$	1006.40i	1.32247i	−0.750179	+	0.661235i	$0.770033\pi$
$762$	0	0
$763$	−431.879	−0.566027
$764$	374.070i	0.489621i
$765$	0	0
$766$	−827.149	−1.07983
$767$	− 486.935i	− 0.634857i
$768$	0	0
$769$	1181.97	1.53702	0.768508	−	0.639840i	$-0.221000\pi$
$769$	1181.97	1.53702	0.768508	+	0.639840i	$0.221000\pi$
$770$	− 47.2008i	− 0.0612997i
$771$	0	0
$772$	422.152	0.546829
$773$	− 233.295i	− 0.301805i	−0.988549	−	0.150903i	$-0.951782\pi$
$773$	− 233.295i	− 0.301805i	0.988549	−	0.150903i	$-0.0482180\pi$
$774$	0	0
$775$	604.794	0.780379
$776$	279.147i	0.359726i
$777$	0	0
$778$	149.436	0.192077
$779$	− 86.8837i	− 0.111532i
$780$	0	0
$781$	−366.281	−0.468989
$782$	− 72.0805i	− 0.0921746i
$783$	0	0
$784$	−159.688	−0.203684
$785$	425.151i	0.541594i
$786$	0	0
$787$	63.6472	0.0808732	0.0404366	−	0.999182i	$-0.487125\pi$
$787$	63.6472	0.0808732	0.0404366	+	0.999182i	$0.487125\pi$
$788$	− 38.9799i	− 0.0494668i
$789$	0	0
$790$	44.9949	0.0569555
$791$	− 132.593i	− 0.167627i
$792$	0	0
$793$	−391.279	−0.493416
$794$	− 267.942i	− 0.337458i
$795$	0	0
$796$	−244.835	−0.307582
$797$	901.271i	1.13083i	0.824807	+	0.565415i	$0.191284\pi$
$797$	901.271i	1.13083i	−0.824807	+	0.565415i	$0.808716\pi$
$798$	0	0
$799$	−244.434	−0.305925
$800$	103.188i	0.128985i
$801$	0	0
$802$	608.765	0.759059
$803$	− 30.8234i	− 0.0383853i
$804$	0	0
$805$	−37.5656	−0.0466653
$806$	369.860i	0.458883i
$807$	0	0
$808$	115.969	0.143526
$809$	− 728.266i	− 0.900206i	−0.892977	−	0.450103i	$-0.851387\pi$
$809$	− 728.266i	− 0.900206i	0.892977	−	0.450103i	$-0.148613\pi$
$810$	0	0
$811$	−1489.78	−1.83697	−0.918484	−	0.395457i	$-0.870586\pi$
$811$	−1489.78	−1.83697	−0.918484	+	0.395457i	$0.870586\pi$
$812$	27.3677i	0.0337041i
$813$	0	0
$814$	−131.221	−0.161205
$815$	− 181.956i	− 0.223259i
$816$	0	0
$817$	978.230	1.19734
$818$	− 487.810i	− 0.596345i
$819$	0	0
$820$	12.4903	0.0152321
$821$	− 1480.62i	− 1.80343i	−0.432332	−	0.901715i	$-0.642309\pi$
$821$	− 1480.62i	− 1.80343i	0.432332	−	0.901715i	$-0.357691\pi$
$822$	0	0
$823$	−224.792	−0.273137	−0.136569	−	0.990631i	$-0.543607\pi$
$823$	−224.792	−0.273137	−0.136569	+	0.990631i	$0.543607\pi$
$824$	171.543i	0.208184i
$825$	0	0
$826$	−263.033	−0.318442
$827$	1335.21i	1.61452i	0.590194	+	0.807262i	$0.299051\pi$
$827$	1335.21i	1.61452i	−0.590194	+	0.807262i	$0.700949\pi$
$828$	0	0
$829$	−1641.94	−1.98062	−0.990311	−	0.138867i	$-0.955654\pi$
$829$	−1641.94	−1.98062	−0.990311	+	0.138867i	$0.955654\pi$
$830$	− 151.163i	− 0.182125i
$831$	0	0
$832$	−63.1044	−0.0758466
$833$	− 424.280i	− 0.509339i
$834$	0	0
$835$	−479.832	−0.574650
$836$	− 308.223i	− 0.368688i
$837$	0	0
$838$	560.350	0.668675
$839$	− 368.360i	− 0.439046i	−0.975607	−	0.219523i	$-0.929550\pi$
$839$	− 368.360i	− 0.439046i	0.975607	−	0.219523i	$-0.0704501\pi$
$840$	0	0
$841$	820.373	0.975474
$842$	− 781.378i	− 0.928002i
$843$	0	0
$844$	308.587	0.365624
$845$	277.598i	0.328519i
$846$	0	0
$847$	−309.865	−0.365839
$848$	207.438i	0.244620i
$849$	0	0
$850$	−274.163	−0.322545
$851$	104.435i	0.122720i
$852$	0	0
$853$	435.840	0.510950	0.255475	−	0.966816i	$-0.417768\pi$
$853$	435.840	0.510950	0.255475	+	0.966816i	$0.417768\pi$
$854$	211.361i	0.247496i
$855$	0	0
$856$	−130.499	−0.152452
$857$	− 105.198i	− 0.122751i	−0.998115	−	0.0613756i	$-0.980451\pi$
$857$	− 105.198i	− 0.122751i	0.998115	−	0.0613756i	$-0.0195487\pi$
$858$	0	0
$859$	−30.7345	−0.0357794	−0.0178897	−	0.999840i	$-0.505695\pi$
$859$	−30.7345	−0.0357794	−0.0178897	+	0.999840i	$0.505695\pi$
$860$	140.630i	0.163523i
$861$	0	0
$862$	807.064	0.936269
$863$	12.1204i	0.0140445i	0.999975	+	0.00702224i	$0.00223527\pi$
$863$	12.1204i	0.0140445i	−0.999975	+	0.00702224i	$0.997765\pi$
$864$	0	0
$865$	678.817	0.784760
$866$	− 753.188i	− 0.869732i
$867$	0	0
$868$	199.791	0.230174
$869$	52.1463i	0.0600073i
$870$	0	0
$871$	554.104	0.636170
$872$	− 405.428i	− 0.464940i
$873$	0	0
$874$	−245.305	−0.280669
$875$	338.707i	0.387094i
$876$	0	0
$877$	−1560.48	−1.77934	−0.889672	−	0.456600i	$-0.849067\pi$
$877$	−1560.48	−1.77934	−0.889672	+	0.456600i	$0.849067\pi$
$878$	511.659i	0.582755i
$879$	0	0
$880$	44.3099	0.0503522
$881$	391.438i	0.444311i	0.975011	+	0.222156i	$0.0713093\pi$
$881$	391.438i	0.444311i	−0.975011	+	0.222156i	$0.928691\pi$
$882$	0	0
$883$	−1360.03	−1.54024	−0.770120	−	0.637899i	$-0.779804\pi$
$883$	−1360.03	−1.54024	−0.770120	+	0.637899i	$0.779804\pi$
$884$	− 167.664i	− 0.189665i
$885$	0	0
$886$	103.155	0.116428
$887$	499.999i	0.563697i	0.959459	+	0.281849i	$0.0909476\pi$
$887$	499.999i	0.563697i	−0.959459	+	0.281849i	$0.909052\pi$
$888$	0	0
$889$	−123.578	−0.139007
$890$	− 332.332i	− 0.373407i
$891$	0	0
$892$	−637.332	−0.714497
$893$	831.860i	0.931534i
$894$	0	0
$895$	243.112	0.271634
$896$	34.0878i	0.0380444i
$897$	0	0
$898$	−343.571	−0.382596
$899$	150.580i	0.167497i
$900$	0	0
$901$	−551.146	−0.611705
$902$	14.4755i	0.0160483i
$903$	0	0
$904$	124.472	0.137690
$905$	97.4086i	0.107634i
$906$	0	0
$907$	−195.684	−0.215749	−0.107874	−	0.994165i	$-0.534404\pi$
$907$	−195.684	−0.215749	−0.107874	+	0.994165i	$0.534404\pi$
$908$	513.485i	0.565512i
$909$	0	0
$910$	−87.3797	−0.0960217
$911$	280.762i	0.308191i	0.988056	+	0.154095i	$0.0492463\pi$
$911$	280.762i	0.308191i	−0.988056	+	0.154095i	$0.950754\pi$
$912$	0	0
$913$	175.189	0.191883
$914$	− 346.277i	− 0.378859i
$915$	0	0
$916$	248.538	0.271330
$917$	421.941i	0.460132i
$918$	0	0
$919$	−1682.42	−1.83071	−0.915354	−	0.402649i	$-0.868090\pi$
$919$	−1682.42	−1.83071	−0.915354	+	0.402649i	$0.868090\pi$
$920$	− 35.2648i	− 0.0383313i
$921$	0	0
$922$	543.992	0.590013
$923$	678.071i	0.734638i
$924$	0	0
$925$	397.224	0.429431
$926$	6.93163i	0.00748556i
$927$	0	0
$928$	−25.6915	−0.0276848
$929$	− 936.090i	− 1.00763i	−0.863811	−	0.503816i	$-0.831929\pi$
$929$	− 936.090i	− 1.00763i	0.863811	−	0.503816i	$-0.168071\pi$
$930$	0	0
$931$	−1443.91	−1.55092
$932$	− 672.337i	− 0.721392i
$933$	0	0
$934$	887.036	0.949718
$935$	117.728i	0.125912i
$936$	0	0
$937$	228.541	0.243907	0.121954	−	0.992536i	$-0.461084\pi$
$937$	228.541	0.243907	0.121954	+	0.992536i	$0.461084\pi$
$938$	− 299.316i	− 0.319100i
$939$	0	0
$940$	−119.587	−0.127221
$941$	− 1417.97i	− 1.50688i	−0.657519	−	0.753438i	$-0.728394\pi$
$941$	− 1417.97i	− 1.50688i	0.657519	−	0.753438i	$-0.271606\pi$
$942$	0	0
$943$	11.5206	0.0122170
$944$	− 246.923i	− 0.261571i
$945$	0	0
$946$	−162.981	−0.172285
$947$	1055.59i	1.11467i	0.830287	+	0.557336i	$0.188177\pi$
$947$	1055.59i	1.11467i	−0.830287	+	0.557336i	$0.811823\pi$
$948$	0	0
$949$	−57.0614	−0.0601279
$950$	933.033i	0.982140i
$951$	0	0
$952$	−90.5686	−0.0951350
$953$	47.8578i	0.0502180i	0.999685	+	0.0251090i	$0.00799328\pi$
$953$	47.8578i	0.0502180i	−0.999685	+	0.0251090i	$0.992007\pi$
$954$	0	0
$955$	−486.246	−0.509158
$956$	− 933.674i	− 0.976647i
$957$	0	0
$958$	−195.141	−0.203696
$959$	263.982i	0.275268i
$960$	0	0
$961$	138.270	0.143881
$962$	242.921i	0.252516i
$963$	0	0
$964$	553.999	0.574688
$965$	548.746i	0.568649i
$966$	0	0
$967$	−1287.05	−1.33098	−0.665488	−	0.746408i	$-0.731777\pi$
$967$	−1287.05	−1.33098	−0.665488	+	0.746408i	$0.731777\pi$
$968$	− 290.887i	− 0.300503i
$969$	0	0
$970$	−362.858	−0.374080
$971$	− 894.059i	− 0.920761i	−0.887722	−	0.460381i	$-0.847713\pi$
$971$	− 894.059i	− 0.920761i	0.887722	−	0.460381i	$-0.152287\pi$
$972$	0	0
$973$	−354.201	−0.364030
$974$	− 613.748i	− 0.630131i
$975$	0	0
$976$	−198.416	−0.203295
$977$	− 777.919i	− 0.796232i	−0.917335	−	0.398116i	$-0.869664\pi$
$977$	− 777.919i	− 0.796232i	0.917335	−	0.398116i	$-0.130336\pi$
$978$	0	0
$979$	385.153	0.393415
$980$	− 207.575i	− 0.211811i
$981$	0	0
$982$	42.2735	0.0430484
$983$	− 340.370i	− 0.346256i	−0.984899	−	0.173128i	$-0.944613\pi$
$983$	− 340.370i	− 0.346256i	0.984899	−	0.173128i	$-0.0553875\pi$
$984$	0	0
$985$	50.6691	0.0514407
$986$	− 68.2604i	− 0.0692296i
$987$	0	0
$988$	−570.593	−0.577523
$989$	129.711i	0.131154i
$990$	0	0
$991$	−1254.27	−1.26566	−0.632828	−	0.774292i	$-0.718106\pi$
$991$	−1254.27	−1.26566	−0.632828	+	0.774292i	$0.718106\pi$
$992$	187.554i	0.189067i
$993$	0	0
$994$	366.281	0.368492
$995$	− 318.256i	− 0.319855i
$996$	0	0
$997$	558.080	0.559760	0.279880	−	0.960035i	$-0.409705\pi$
$997$	558.080	0.559760	0.279880	+	0.960035i	$0.409705\pi$
$998$	363.954i	0.364683i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.3.c.b.323.6	yes	8
3.2	odd	2	inner	414.3.c.b.323.3	✓	8
4.3	odd	2		3312.3.g.b.737.4		8
12.11	even	2		3312.3.g.b.737.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.3.c.b.323.3	✓	8	3.2	odd	2	inner
414.3.c.b.323.6	yes	8	1.1	even	1	trivial
3312.3.g.b.737.4		8	4.3	odd	2
3312.3.g.b.737.5		8	12.11	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:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	414.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -2.59976i q^{5} -3.01296 q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} -2.59976i q^{5} -3.01296 q^{7} -2.82843i q^{8} +3.67661 q^{10} +4.26097i q^{11} +7.88805 q^{13} -4.26097i q^{14} +4.00000 q^{16} +10.6277i q^{17} +36.1682 q^{19} +5.19951i q^{20} -6.02592 q^{22} +4.79583i q^{23} +18.2413 q^{25} +11.1554i q^{26} +6.02592 q^{28} +4.54166i q^{29} +33.1552 q^{31} +5.65685i q^{32} -15.0298 q^{34} +7.83296i q^{35} +21.7761 q^{37} +51.1496i q^{38} -7.35322 q^{40} -2.40221i q^{41} +27.0467 q^{43} -8.52194i q^{44} -6.78233 q^{46} +22.9998i q^{47} -39.9221 q^{49} +25.7971i q^{50} -15.7761 q^{52} +51.8594i q^{53} +11.0775 q^{55} +8.52194i q^{56} -6.42288 q^{58} -61.7308i q^{59} -49.6040 q^{61} +46.8886i q^{62} -8.00000 q^{64} -20.5070i q^{65} +70.2460 q^{67} -21.2554i q^{68} -11.0775 q^{70} +85.9618i q^{71} -7.23390 q^{73} +30.7961i q^{74} -72.3364 q^{76} -12.8381i q^{77} +12.2381 q^{79} -10.3990i q^{80} +3.39724 q^{82} -41.1149i q^{83} +27.6294 q^{85} +38.2498i q^{86} +12.0518 q^{88} -90.3910i q^{89} -23.7664 q^{91} -9.59166i q^{92} -32.5266 q^{94} -94.0285i q^{95} -98.6935 q^{97} -56.4583i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 16 q^{7}+O(q^{10}) \) 8 * q - 16 * q^4 + 16 * q^7	\( 8 q - 16 q^{4} + 16 q^{7} - 8 q^{10} - 8 q^{13} + 32 q^{16} - 48 q^{19} + 32 q^{22} - 32 q^{28} - 32 q^{31} - 8 q^{34} + 32 q^{37} + 16 q^{40} + 32 q^{43} + 80 q^{49} + 16 q^{52} + 32 q^{55} + 16 q^{58} + 48 q^{61} - 64 q^{64} - 16 q^{67} - 32 q^{70} - 432 q^{73} + 96 q^{76} - 416 q^{79} - 144 q^{82} + 584 q^{85} - 64 q^{88} + 368 q^{91} + 128 q^{94} - 8 q^{97}+O(q^{100}) \) 8 * q - 16 * q^4 + 16 * q^7 - 8 * q^10 - 8 * q^13 + 32 * q^16 - 48 * q^19 + 32 * q^22 - 32 * q^28 - 32 * q^31 - 8 * q^34 + 32 * q^37 + 16 * q^40 + 32 * q^43 + 80 * q^49 + 16 * q^52 + 32 * q^55 + 16 * q^58 + 48 * q^61 - 64 * q^64 - 16 * q^67 - 32 * q^70 - 432 * q^73 + 96 * q^76 - 416 * q^79 - 144 * q^82 + 584 * q^85 - 64 * q^88 + 368 * q^91 + 128 * q^94 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more