LMFDB - Embedded newform 960.4.b.a.671.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,4,Mod(671,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("960.671");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$960 = 2^{6} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	960.b (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:	$56.6418336055$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + 3472x^{12} + 2676096x^{8} + 573099300x^{4} + 31755240000$ x^16 + 3472x^12 + 2676096x^8 + 573099300*x^4 + 31755240000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{23}\cdot 3^{4}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			671.16
Root			$4.99314 + 4.99314i$ of defining polynomial
Character	$\chi$	$=$	960.671
Dual form			960.4.b.a.671.15

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.99314 + 1.43824i) q^{3} -5.00000 q^{5} -16.1856i q^{7} +(22.8629 + 14.3627i) q^{9} +15.1075i q^{11} -19.2259i q^{13} +(-24.9657 - 7.19122i) q^{15} +83.4642i q^{17} -38.2158 q^{19} +(23.2789 - 80.8171i) q^{21} +46.7087 q^{23} +25.0000 q^{25} +(93.5007 + 104.597i) q^{27} +192.661 q^{29} +132.947i q^{31} +(-21.7283 + 75.4341i) q^{33} +80.9281i q^{35} -89.0478i q^{37} +(27.6515 - 95.9975i) q^{39} -46.9335i q^{41} -8.02092 q^{43} +(-114.315 - 71.8135i) q^{45} +85.9514 q^{47} +81.0254 q^{49} +(-120.042 + 416.748i) q^{51} +372.581 q^{53} -75.5377i q^{55} +(-190.817 - 54.9636i) q^{57} -80.4063i q^{59} -154.416i q^{61} +(232.469 - 370.051i) q^{63} +96.1294i q^{65} +891.963 q^{67} +(233.223 + 67.1784i) q^{69} +1011.97 q^{71} -94.6740 q^{73} +(124.829 + 35.9561i) q^{75} +244.525 q^{77} +166.524i q^{79} +(316.425 + 656.746i) q^{81} +199.617i q^{83} -417.321i q^{85} +(961.984 + 277.094i) q^{87} +1345.95i q^{89} -311.183 q^{91} +(-191.211 + 663.824i) q^{93} +191.079 q^{95} +1240.10 q^{97} +(-216.985 + 345.402i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 80 q^{5} - 32 q^{9} + 248 q^{21} + 400 q^{25} - 192 q^{29} - 520 q^{33} + 160 q^{45} + 576 q^{49} - 960 q^{53} - 2056 q^{57} + 1992 q^{69} + 784 q^{73} - 4992 q^{77} - 3760 q^{81} + 4032 q^{93} + 1520 q^{97}+O(q^{100})$ 16 * q - 80 * q^5 - 32 * q^9 + 248 * q^21 + 400 * q^25 - 192 * q^29 - 520 * q^33 + 160 * q^45 + 576 * q^49 - 960 * q^53 - 2056 * q^57 + 1992 * q^69 + 784 * q^73 - 4992 * q^77 - 3760 * q^81 + 4032 * q^93 + 1520 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$ .

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$	4.99314	+	1.43824i	0.960930	+	0.276790i
$3$	4.99314	+	1.43824i	0.960930	+	0.276790i
$4$		0			0
$5$	−5.00000			−0.447214
$5$	−5.00000			−0.447214
$6$		0			0
$7$		−	16.1856i		−	0.873942i	−0.899476	−	0.436971i	$-0.856051\pi$
$7$		−	16.1856i		−	0.873942i	0.899476	−	0.436971i	$-0.143949\pi$
$8$		0			0
$9$	22.8629	+	14.3627i	0.846774	+	0.531952i
$10$		0			0
$11$			15.1075i			0.414100i	0.978330	+	0.207050i	$0.0663862\pi$
$11$			15.1075i			0.414100i	−0.978330	+	0.207050i	$0.933614\pi$
$12$		0			0
$13$		−	19.2259i		−	0.410177i	−0.978743	−	0.205088i	$-0.934252\pi$
$13$		−	19.2259i		−	0.410177i	0.978743	−	0.205088i	$-0.0657482\pi$
$14$		0			0
$15$	−24.9657	−	7.19122i	−0.429741	−	0.123784i
$16$		0			0
$17$			83.4642i			1.19077i	0.803442	+	0.595383i	$0.203000\pi$
$17$			83.4642i			1.19077i	−0.803442	+	0.595383i	$0.797000\pi$
$18$		0			0
$19$	−38.2158			−0.461437			−0.230718	−	0.973021i	$-0.574108\pi$
$19$	−38.2158			−0.461437			−0.230718	+	0.973021i	$0.574108\pi$
$20$		0			0
$21$	23.2789	−	80.8171i	0.241898	−	0.839797i
$22$		0			0
$23$	46.7087			0.423453			0.211727	−	0.977329i	$-0.432091\pi$
$23$	46.7087			0.423453			0.211727	+	0.977329i	$0.432091\pi$
$24$		0			0
$25$	25.0000			0.200000
$26$		0			0
$27$	93.5007	+	104.597i	0.666452	+	0.745548i
$28$		0			0
$29$	192.661			1.23366			0.616832	−	0.787095i	$-0.288416\pi$
$29$	192.661			1.23366			0.616832	+	0.787095i	$0.288416\pi$
$30$		0			0
$31$			132.947i			0.770259i	0.922862	+	0.385130i	$0.125843\pi$
$31$			132.947i			0.770259i	−0.922862	+	0.385130i	$0.874157\pi$
$32$		0			0
$33$	−21.7283	+	75.4341i	−0.114619	+	0.397921i
$34$		0			0
$35$			80.9281i			0.390839i
$36$		0			0
$37$		−	89.0478i		−	0.395659i	−0.980236	−	0.197829i	$-0.936611\pi$
$37$		−	89.0478i		−	0.395659i	0.980236	−	0.197829i	$-0.0633892\pi$
$38$		0			0
$39$	27.6515	−	95.9975i	0.113533	−	0.394151i
$40$		0			0
$41$		−	46.9335i		−	0.178775i	−0.995997	−	0.0893875i	$-0.971509\pi$
$41$		−	46.9335i		−	0.178775i	0.995997	−	0.0893875i	$-0.0284910\pi$
$42$		0			0
$43$	−8.02092			−0.0284460			−0.0142230	−	0.999899i	$-0.504527\pi$
$43$	−8.02092			−0.0284460			−0.0142230	+	0.999899i	$0.504527\pi$
$44$		0			0
$45$	−114.315	−	71.8135i	−0.378689	−	0.237896i
$46$		0			0
$47$	85.9514			0.266751			0.133376	−	0.991066i	$-0.457418\pi$
$47$	85.9514			0.266751			0.133376	+	0.991066i	$0.457418\pi$
$48$		0			0
$49$	81.0254			0.236226
$50$		0			0
$51$	−120.042	+	416.748i	−0.329592	+	1.14424i
$52$		0			0
$53$	372.581			0.965622			0.482811	−	0.875725i	$-0.339616\pi$
$53$	372.581			0.965622			0.482811	+	0.875725i	$0.339616\pi$
$54$		0			0
$55$		−	75.5377i		−	0.185191i
$56$		0			0
$57$	−190.817	−	54.9636i	−0.443409	−	0.127721i
$58$		0			0
$59$		−	80.4063i		−	0.177424i	−0.996057	−	0.0887119i	$-0.971725\pi$
$59$		−	80.4063i		−	0.177424i	0.996057	−	0.0887119i	$-0.0282750\pi$
$60$		0			0
$61$		−	154.416i		−	0.324114i	−0.986781	−	0.162057i	$-0.948187\pi$
$61$		−	154.416i		−	0.324114i	0.986781	−	0.162057i	$-0.0518129\pi$
$62$		0			0
$63$	232.469	−	370.051i	0.464895	−	0.740032i
$64$		0			0
$65$			96.1294i			0.183437i
$66$		0			0
$67$	891.963			1.62643			0.813213	−	0.581966i	$-0.197716\pi$
$67$	891.963			1.62643			0.813213	+	0.581966i	$0.197716\pi$
$68$		0			0
$69$	233.223	+	67.1784i	0.406909	+	0.117208i
$70$		0			0
$71$	1011.97			1.69153			0.845765	−	0.533556i	$-0.179145\pi$
$71$	1011.97			1.69153			0.845765	+	0.533556i	$0.179145\pi$
$72$		0			0
$73$	−94.6740			−0.151791			−0.0758956	−	0.997116i	$-0.524182\pi$
$73$	−94.6740			−0.151791			−0.0758956	+	0.997116i	$0.524182\pi$
$74$		0			0
$75$	124.829	+	35.9561i	0.192186	+	0.0553580i
$76$		0			0
$77$	244.525			0.361899
$78$		0			0
$79$			166.524i			0.237157i	0.992945	+	0.118578i	$0.0378337\pi$
$79$			166.524i			0.237157i	−0.992945	+	0.118578i	$0.962166\pi$
$80$		0			0
$81$	316.425	+	656.746i	0.434054	+	0.900887i
$82$		0			0
$83$			199.617i			0.263986i	0.991251	+	0.131993i	$0.0421376\pi$
$83$			199.617i			0.263986i	−0.991251	+	0.131993i	$0.957862\pi$
$84$		0			0
$85$		−	417.321i		−	0.532527i
$86$		0			0
$87$	961.984	+	277.094i	1.18547	+	0.341466i
$88$		0			0
$89$			1345.95i			1.60303i	0.597972	+	0.801517i	$0.295973\pi$
$89$			1345.95i			1.60303i	−0.597972	+	0.801517i	$0.704027\pi$
$90$		0			0
$91$	−311.183			−0.358471
$92$		0			0
$93$	−191.211	+	663.824i	−0.213200	+	0.740166i
$94$		0			0
$95$	191.079			0.206361
$96$		0			0
$97$	1240.10			1.29808			0.649038	−	0.760756i	$-0.275172\pi$
$97$	1240.10			1.29808			0.649038	+	0.760756i	$0.275172\pi$
$98$		0			0
$99$	−216.985	+	345.402i	−0.220281	+	0.350649i
$100$		0			0
$101$	−1047.31			−1.03180			−0.515899	−	0.856649i	$-0.672542\pi$
$101$	−1047.31			−1.03180			−0.515899	+	0.856649i	$0.672542\pi$
$102$		0			0
$103$		−	1179.34i		−	1.12819i	−0.825710	−	0.564095i	$-0.809225\pi$
$103$		−	1179.34i		−	1.12819i	0.825710	−	0.564095i	$-0.190775\pi$
$104$		0			0
$105$	−116.394	+	404.086i	−0.108180	+	0.375569i
$106$		0			0
$107$		−	1881.45i		−	1.69987i	−0.526885	−	0.849937i	$-0.676640\pi$
$107$		−	1881.45i		−	1.69987i	0.526885	−	0.849937i	$-0.323360\pi$
$108$		0			0
$109$			1186.74i			1.04284i	0.853301	+	0.521418i	$0.174597\pi$
$109$			1186.74i			1.04284i	−0.853301	+	0.521418i	$0.825403\pi$
$110$		0			0
$111$	128.072	−	444.628i	0.109514	−	0.380200i
$112$		0			0
$113$			1690.98i			1.40773i	0.710333	+	0.703866i	$0.248544\pi$
$113$			1690.98i			1.40773i	−0.710333	+	0.703866i	$0.751456\pi$
$114$		0			0
$115$	−233.543			−0.189374
$116$		0			0
$117$	276.136	−	439.559i	0.218194	−	0.347327i
$118$		0			0
$119$	1350.92			1.04066
$120$		0			0
$121$	1102.76			0.828522
$122$		0			0
$123$	67.5017	−	234.345i	0.0494831	−	0.171790i
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$		−	295.964i		−	0.206792i	−0.994640	−	0.103396i	$-0.967029\pi$
$127$		−	295.964i		−	0.206792i	0.994640	−	0.103396i	$-0.0329709\pi$
$128$		0			0
$129$	−40.0496	−	11.5360i	−0.0273347	−	0.00787358i
$130$		0			0
$131$		−	330.510i		−	0.220434i	−0.993908	−	0.110217i	$-0.964845\pi$
$131$		−	330.510i		−	0.220434i	0.993908	−	0.110217i	$-0.0351545\pi$
$132$		0			0
$133$			618.546i			0.403269i
$134$		0			0
$135$	−467.503	−	522.987i	−0.298047	−	0.333419i
$136$		0			0
$137$			2370.48i			1.47828i	0.673553	+	0.739139i	$0.264767\pi$
$137$			2370.48i			1.47828i	−0.673553	+	0.739139i	$0.735233\pi$
$138$		0			0
$139$	−2296.01			−1.40105			−0.700523	−	0.713630i	$-0.747050\pi$
$139$	−2296.01			−1.40105			−0.700523	+	0.713630i	$0.747050\pi$
$140$		0			0
$141$	429.168	+	123.619i	0.256329	+	0.0738341i
$142$		0			0
$143$	290.456			0.169854
$144$		0			0
$145$	−963.305			−0.551711
$146$		0			0
$147$	404.571	+	116.534i	0.226996	+	0.0653849i
$148$		0			0
$149$	−2564.14			−1.40982			−0.704908	−	0.709299i	$-0.749012\pi$
$149$	−2564.14			−1.40982			−0.704908	+	0.709299i	$0.749012\pi$
$150$		0			0
$151$			1965.29i			1.05916i	0.848260	+	0.529581i	$0.177651\pi$
$151$			1965.29i			1.05916i	−0.848260	+	0.529581i	$0.822349\pi$
$152$		0			0
$153$	−1198.77	+	1908.23i	−0.633431	+	1.00831i
$154$		0			0
$155$		−	664.736i		−	0.344470i
$156$		0			0
$157$		−	615.854i		−	0.313061i	−0.987673	−	0.156530i	$-0.949969\pi$
$157$		−	615.854i		−	0.313061i	0.987673	−	0.156530i	$-0.0500309\pi$
$158$		0			0
$159$	1860.35	+	535.862i	0.927895	+	0.267274i
$160$		0			0
$161$		−	756.009i		−	0.370074i
$162$		0			0
$163$	104.709			0.0503158			0.0251579	−	0.999683i	$-0.491991\pi$
$163$	104.709			0.0503158			0.0251579	+	0.999683i	$0.491991\pi$
$164$		0			0
$165$	108.642	−	377.170i	0.0512590	−	0.177956i
$166$		0			0
$167$	2024.38			0.938029			0.469015	−	0.883190i	$-0.344609\pi$
$167$	2024.38			0.938029			0.469015	+	0.883190i	$0.344609\pi$
$168$		0			0
$169$	1827.37			0.831755
$170$		0			0
$171$	−873.724	−	548.882i	−0.390733	−	0.245462i
$172$		0			0
$173$	−2990.04			−1.31404			−0.657020	−	0.753873i	$-0.728183\pi$
$173$	−2990.04			−1.31404			−0.657020	+	0.753873i	$0.728183\pi$
$174$		0			0
$175$		−	404.641i		−	0.174788i
$176$		0			0
$177$	115.644	−	401.480i	0.0491092	−	0.170492i
$178$		0			0
$179$			563.495i			0.235294i	0.993055	+	0.117647i	$0.0375351\pi$
$179$			563.495i			0.235294i	−0.993055	+	0.117647i	$0.962465\pi$
$180$		0			0
$181$			1683.92i			0.691518i	0.938323	+	0.345759i	$0.112378\pi$
$181$			1683.92i			0.691518i	−0.938323	+	0.345759i	$0.887622\pi$
$182$		0			0
$183$	222.088	−	771.021i	0.0897116	−	0.311451i
$184$		0			0
$185$			445.239i			0.176944i
$186$		0			0
$187$	−1260.94			−0.493096
$188$		0			0
$189$	1692.98	−	1513.37i	0.651565	−	0.582441i
$190$		0			0
$191$	4690.57			1.77695			0.888476	−	0.458922i	$-0.151764\pi$
$191$	4690.57			1.77695			0.888476	+	0.458922i	$0.151764\pi$
$192$		0			0
$193$	−3598.43			−1.34208			−0.671038	−	0.741423i	$-0.734151\pi$
$193$	−3598.43			−1.34208			−0.671038	+	0.741423i	$0.734151\pi$
$194$		0			0
$195$	−138.257	+	479.988i	−0.0507734	+	0.176270i
$196$		0			0
$197$	−2431.31			−0.879306			−0.439653	−	0.898168i	$-0.644899\pi$
$197$	−2431.31			−0.879306			−0.439653	+	0.898168i	$0.644899\pi$
$198$		0			0
$199$		−	3011.78i		−	1.07286i	−0.843944	−	0.536431i	$-0.819772\pi$
$199$		−	3011.78i		−	1.07286i	0.843944	−	0.536431i	$-0.180228\pi$
$200$		0			0
$201$	4453.70	+	1282.86i	1.56288	+	0.450179i
$202$		0			0
$203$		−	3118.34i		−	1.07815i
$204$		0			0
$205$			234.667i			0.0799506i
$206$		0			0
$207$	1067.90	+	670.863i	0.358570	+	0.225257i
$208$		0			0
$209$		−	577.346i		−	0.191081i
$210$		0			0
$211$	3830.17			1.24967			0.624833	−	0.780758i	$-0.285167\pi$
$211$	3830.17			1.24967			0.624833	+	0.780758i	$0.285167\pi$
$212$		0			0
$213$	5052.90	+	1455.46i	1.62544	+	0.468199i
$214$		0			0
$215$	40.1046			0.0127215
$216$		0			0
$217$	2151.84			0.673162
$218$		0			0
$219$	−472.721	−	136.164i	−0.145861	−	0.0420143i
$220$		0			0
$221$	1604.67			0.488425
$222$		0			0
$223$		−	4304.09i		−	1.29248i	−0.763134	−	0.646240i	$-0.776340\pi$
$223$		−	4304.09i		−	1.29248i	0.763134	−	0.646240i	$-0.223660\pi$
$224$		0			0
$225$	571.573	+	359.068i	0.169355	+	0.106390i
$226$		0			0
$227$			2960.94i			0.865747i	0.901455	+	0.432874i	$0.142500\pi$
$227$			2960.94i			0.865747i	−0.901455	+	0.432874i	$0.857500\pi$
$228$		0			0
$229$		−	6091.99i		−	1.75795i	−0.476870	−	0.878974i	$-0.658229\pi$
$229$		−	6091.99i		−	1.75795i	0.476870	−	0.878974i	$-0.341771\pi$
$230$		0			0
$231$	1220.95	+	351.687i	0.347760	+	0.100170i
$232$		0			0
$233$			5739.74i			1.61383i	0.590667	+	0.806915i	$0.298865\pi$
$233$			5739.74i			1.61383i	−0.590667	+	0.806915i	$0.701135\pi$
$234$		0			0
$235$	−429.757			−0.119295
$236$		0			0
$237$	−239.502	+	831.477i	−0.0656427	+	0.227891i
$238$		0			0
$239$	385.738			0.104399			0.0521994	−	0.998637i	$-0.483377\pi$
$239$	385.738			0.104399			0.0521994	+	0.998637i	$0.483377\pi$
$240$		0			0
$241$	834.546			0.223061			0.111531	−	0.993761i	$-0.464425\pi$
$241$	834.546			0.223061			0.111531	+	0.993761i	$0.464425\pi$
$242$		0			0
$243$	635.395	+	3734.32i	0.167739	+	0.985831i
$244$		0			0
$245$	−405.127			−0.105643
$246$		0			0
$247$			734.731i			0.189271i
$248$		0			0
$249$	−287.098	+	996.717i	−0.0730687	+	0.253672i
$250$		0			0
$251$		−	3188.54i		−	0.801828i	−0.916116	−	0.400914i	$-0.868693\pi$
$251$		−	3188.54i		−	0.801828i	0.916116	−	0.400914i	$-0.131307\pi$
$252$		0			0
$253$			705.653i			0.175352i
$254$		0			0
$255$	600.209	−	2083.74i	0.147398	−	0.511722i
$256$		0			0
$257$		−	3900.85i		−	0.946803i	−0.880847	−	0.473401i	$-0.843026\pi$
$257$		−	3900.85i		−	0.946803i	0.880847	−	0.473401i	$-0.156974\pi$
$258$		0			0
$259$	−1441.29			−0.345783
$260$		0			0
$261$	4404.79	+	2767.13i	1.04464	+	0.656250i
$262$		0			0
$263$	−3647.80			−0.855258			−0.427629	−	0.903954i	$-0.640651\pi$
$263$	−3647.80			−0.855258			−0.427629	+	0.903954i	$0.640651\pi$
$264$		0			0
$265$	−1862.91			−0.431839
$266$		0			0
$267$	−1935.80	+	6720.50i	−0.443704	+	1.54040i
$268$		0			0
$269$	−837.038			−0.189722			−0.0948608	−	0.995491i	$-0.530241\pi$
$269$	−837.038			−0.189722			−0.0948608	+	0.995491i	$0.530241\pi$
$270$		0			0
$271$			6576.31i			1.47410i	0.675836	+	0.737052i	$0.263783\pi$
$271$			6576.31i			1.47410i	−0.675836	+	0.737052i	$0.736217\pi$
$272$		0			0
$273$	−1553.78	−	447.557i	−0.344465	−	0.0992211i
$274$		0			0
$275$			377.688i			0.0828199i
$276$		0			0
$277$			2414.09i			0.523641i	0.965117	+	0.261821i	$0.0843229\pi$
$277$			2414.09i			0.523641i	−0.965117	+	0.261821i	$0.915677\pi$
$278$		0			0
$279$	−1909.48	+	3039.56i	−0.409741	+	0.652236i
$280$		0			0
$281$			491.109i			0.104260i	0.998640	+	0.0521301i	$0.0166011\pi$
$281$			491.109i			0.104260i	−0.998640	+	0.0521301i	$0.983399\pi$
$282$		0			0
$283$	9271.34			1.94743			0.973717	−	0.227761i	$-0.0731406\pi$
$283$	9271.34			1.94743			0.973717	+	0.227761i	$0.0731406\pi$
$284$		0			0
$285$	954.083	+	274.818i	0.198298	+	0.0571186i
$286$		0			0
$287$	−759.648			−0.156239
$288$		0			0
$289$	−2053.27			−0.417926
$290$		0			0
$291$	6192.01	+	1783.57i	1.24736	+	0.359295i
$292$		0			0
$293$	687.236			0.137026			0.0685132	−	0.997650i	$-0.478174\pi$
$293$	687.236			0.137026			0.0685132	+	0.997650i	$0.478174\pi$
$294$		0			0
$295$			402.031i			0.0793463i
$296$		0			0
$297$	−1580.21	+	1412.57i	−0.308731	+	0.275978i
$298$		0			0
$299$		−	898.015i		−	0.173691i
$300$		0			0
$301$			129.824i			0.0248602i
$302$		0			0
$303$	−5229.38	−	1506.29i	−0.991486	−	0.285591i
$304$		0			0
$305$			772.081i			0.144948i
$306$		0			0
$307$	−1299.44			−0.241573			−0.120787	−	0.992678i	$-0.538542\pi$
$307$	−1299.44			−0.241573			−0.120787	+	0.992678i	$0.538542\pi$
$308$		0			0
$309$	1696.17	−	5888.60i	0.312272	−	1.08411i
$310$		0			0
$311$	−5227.97			−0.953219			−0.476610	−	0.879115i	$-0.658134\pi$
$311$	−5227.97			−0.953219			−0.476610	+	0.879115i	$0.658134\pi$
$312$		0			0
$313$	−2260.72			−0.408255			−0.204127	−	0.978944i	$-0.565436\pi$
$313$	−2260.72			−0.408255			−0.204127	+	0.978944i	$0.565436\pi$
$314$		0			0
$315$	−1162.35	+	1850.25i	−0.207907	+	0.330952i
$316$		0			0
$317$	363.429			0.0643918			0.0321959	−	0.999482i	$-0.489750\pi$
$317$	363.429			0.0643918			0.0321959	+	0.999482i	$0.489750\pi$
$318$		0			0
$319$			2910.63i			0.510860i
$320$		0			0
$321$	2705.98	−	9394.33i	0.470508	−	1.63346i
$322$		0			0
$323$		−	3189.65i		−	0.549463i
$324$		0			0
$325$		−	480.647i		−	0.0820354i
$326$		0			0
$327$	−1706.82	+	5925.56i	−0.288647	+	1.00209i
$328$		0			0
$329$		−	1391.18i		−	0.233125i
$330$		0			0
$331$	−601.117			−0.0998199			−0.0499100	−	0.998754i	$-0.515893\pi$
$331$	−601.117			−0.0998199			−0.0499100	+	0.998754i	$0.515893\pi$
$332$		0			0
$333$	1278.97	−	2035.89i	0.210471	−	0.335034i
$334$		0			0
$335$	−4459.81			−0.727360
$336$		0			0
$337$	4758.64			0.769198			0.384599	−	0.923084i	$-0.374340\pi$
$337$	4758.64			0.769198			0.384599	+	0.923084i	$0.374340\pi$
$338$		0			0
$339$	−2432.03	+	8443.28i	−0.389646	+	1.35273i
$340$		0			0
$341$	−2008.51			−0.318964
$342$		0			0
$343$		−	6863.12i		−	1.08039i
$344$		0			0
$345$	−1166.11	−	335.892i	−0.181975	−	0.0524169i
$346$		0			0
$347$		−	8175.86i		−	1.26485i	−0.774622	−	0.632425i	$-0.782060\pi$
$347$		−	8175.86i		−	1.26485i	0.774622	−	0.632425i	$-0.217940\pi$
$348$		0			0
$349$		−	8651.37i		−	1.32693i	−0.748209	−	0.663463i	$-0.769086\pi$
$349$		−	8651.37i		−	1.32693i	0.748209	−	0.663463i	$-0.230914\pi$
$350$		0			0
$351$	2010.98	−	1797.63i	0.305806	−	0.273363i
$352$		0			0
$353$		−	9940.40i		−	1.49879i	−0.662121	−	0.749397i	$-0.730344\pi$
$353$		−	9940.40i		−	1.49879i	0.662121	−	0.749397i	$-0.269656\pi$
$354$		0			0
$355$	−5059.84			−0.756475
$356$		0			0
$357$	6745.33	+	1942.95i	1.00000	+	0.288045i
$358$		0			0
$359$	−2871.55			−0.422157			−0.211079	−	0.977469i	$-0.567698\pi$
$359$	−2871.55			−0.422157			−0.211079	+	0.977469i	$0.567698\pi$
$360$		0			0
$361$	−5398.56			−0.787076
$362$		0			0
$363$	5506.25	+	1586.04i	0.796152	+	0.229327i
$364$		0			0
$365$	473.370			0.0678831
$366$		0			0
$367$		−	1849.45i		−	0.263054i	−0.991313	−	0.131527i	$-0.958012\pi$
$367$		−	1849.45i		−	0.263054i	0.991313	−	0.131527i	$-0.0419879\pi$
$368$		0			0
$369$	674.091	−	1073.04i	0.0950997	−	0.151382i
$370$		0			0
$371$		−	6030.46i		−	0.843897i
$372$		0			0
$373$			10455.0i			1.45132i	0.688055	+	0.725658i	$0.258465\pi$
$373$			10455.0i			1.45132i	−0.688055	+	0.725658i	$0.741535\pi$
$374$		0			0
$375$	−624.143	−	179.780i	−0.0859482	−	0.0247569i
$376$		0			0
$377$		−	3704.08i		−	0.506020i
$378$		0			0
$379$	−6285.78			−0.851923			−0.425962	−	0.904741i	$-0.640064\pi$
$379$	−6285.78			−0.851923			−0.425962	+	0.904741i	$0.640064\pi$
$380$		0			0
$381$	425.669	−	1477.79i	0.0572380	−	0.198713i
$382$		0			0
$383$	−1707.73			−0.227836			−0.113918	−	0.993490i	$-0.536340\pi$
$383$	−1707.73			−0.227836			−0.113918	+	0.993490i	$0.536340\pi$
$384$		0			0
$385$	−1222.63			−0.161846
$386$		0			0
$387$	−183.382	−	115.202i	−0.0240874	−	0.0151319i
$388$		0			0
$389$	−14594.1			−1.90218			−0.951091	−	0.308910i	$-0.900036\pi$
$389$	−14594.1			−1.90218			−0.951091	+	0.308910i	$0.900036\pi$
$390$		0			0
$391$			3898.50i			0.504234i
$392$		0			0
$393$	475.354	−	1650.28i	0.0610138	−	0.211821i
$394$		0			0
$395$		−	832.619i		−	0.106060i
$396$		0			0
$397$			3301.41i			0.417363i	0.977984	+	0.208681i	$0.0669172\pi$
$397$			3301.41i			0.417363i	−0.977984	+	0.208681i	$0.933083\pi$
$398$		0			0
$399$	−889.620	+	3088.49i	−0.111621	+	0.387513i
$400$		0			0
$401$		−	7988.67i		−	0.994850i	−0.867507	−	0.497425i	$-0.834279\pi$
$401$		−	7988.67i		−	0.994850i	0.867507	−	0.497425i	$-0.165721\pi$
$402$		0			0
$403$	2556.03			0.315942
$404$		0			0
$405$	−1582.13	−	3283.73i	−0.194115	−	0.402889i
$406$		0			0
$407$	1345.29			0.163842
$408$		0			0
$409$	1717.45			0.207634			0.103817	−	0.994596i	$-0.466894\pi$
$409$	1717.45			0.207634			0.103817	+	0.994596i	$0.466894\pi$
$410$		0			0
$411$	−3409.33	+	11836.2i	−0.409172	+	1.42052i
$412$		0			0
$413$	−1301.43			−0.155058
$414$		0			0
$415$		−	998.086i		−	0.118058i
$416$		0			0
$417$	−11464.3	−	3302.23i	−1.34631	−	0.387796i
$418$		0			0
$419$		−	7851.09i		−	0.915395i	−0.889108	−	0.457698i	$-0.848674\pi$
$419$		−	7851.09i		−	0.915395i	0.889108	−	0.457698i	$-0.151326\pi$
$420$		0			0
$421$			5132.10i			0.594117i	0.954859	+	0.297059i	$0.0960057\pi$
$421$			5132.10i			0.594117i	−0.954859	+	0.297059i	$0.903994\pi$
$422$		0			0
$423$	1965.10	+	1234.49i	0.225878	+	0.141899i
$424$		0			0
$425$			2086.60i			0.238153i
$426$		0			0
$427$	−2499.32			−0.283257
$428$		0			0
$429$	1450.29	+	417.746i	0.163218	+	0.0470139i
$430$		0			0
$431$	−7312.72			−0.817265			−0.408633	−	0.912699i	$-0.633994\pi$
$431$	−7312.72			−0.817265			−0.408633	+	0.912699i	$0.633994\pi$
$432$		0			0
$433$	−4271.64			−0.474092			−0.237046	−	0.971498i	$-0.576179\pi$
$433$	−4271.64			−0.474092			−0.237046	+	0.971498i	$0.576179\pi$
$434$		0			0
$435$	−4809.92	−	1385.47i	−0.530156	−	0.152708i
$436$		0			0
$437$	−1785.01			−0.195397
$438$		0			0
$439$		−	3462.96i		−	0.376487i	−0.982122	−	0.188244i	$-0.939721\pi$
$439$		−	3462.96i		−	0.376487i	0.982122	−	0.188244i	$-0.0602794\pi$
$440$		0			0
$441$	1852.48	+	1163.74i	0.200030	+	0.125661i
$442$		0			0
$443$		−	8898.08i		−	0.954312i	−0.878819	−	0.477156i	$-0.841668\pi$
$443$		−	8898.08i		−	0.954312i	0.878819	−	0.477156i	$-0.158332\pi$
$444$		0			0
$445$		−	6729.73i		−	0.716898i
$446$		0			0
$447$	−12803.1	−	3687.86i	−1.35473	−	0.390223i
$448$		0			0
$449$		−	14845.1i		−	1.56032i	−0.625581	−	0.780159i	$-0.715138\pi$
$449$		−	14845.1i		−	1.56032i	0.625581	−	0.780159i	$-0.284862\pi$
$450$		0			0
$451$	709.049			0.0740306
$452$		0			0
$453$	−2826.57	+	9812.99i	−0.293165	+	1.01778i
$454$		0			0
$455$	1555.91			0.160313
$456$		0			0
$457$	6834.39			0.699561			0.349780	−	0.936832i	$-0.386256\pi$
$457$	6834.39			0.699561			0.349780	+	0.936832i	$0.386256\pi$
$458$		0			0
$459$	−8730.14	+	7803.96i	−0.887773	+	0.793589i
$460$		0			0
$461$	−18191.3			−1.83786			−0.918932	−	0.394417i	$-0.870947\pi$
$461$	−18191.3			−1.83786			−0.918932	+	0.394417i	$0.870947\pi$
$462$		0			0
$463$			13035.5i			1.30844i	0.756303	+	0.654222i	$0.227004\pi$
$463$			13035.5i			1.30844i	−0.756303	+	0.654222i	$0.772996\pi$
$464$		0			0
$465$	956.053	−	3319.12i	0.0953460	−	0.331012i
$466$		0			0
$467$		−	16137.5i		−	1.59904i	−0.600637	−	0.799522i	$-0.705086\pi$
$467$		−	16137.5i		−	1.59904i	0.600637	−	0.799522i	$-0.294914\pi$
$468$		0			0
$469$		−	14437.0i		−	1.42140i
$470$		0			0
$471$	885.749	−	3075.05i	0.0866521	−	0.300830i
$472$		0			0
$473$		−	121.176i		−	0.0117795i
$474$		0			0
$475$	−955.394			−0.0922873
$476$		0			0
$477$	8518.29	+	5351.27i	0.817664	+	0.513664i
$478$		0			0
$479$	−11024.0			−1.05157			−0.525783	−	0.850619i	$-0.676228\pi$
$479$	−11024.0			−1.05157			−0.525783	+	0.850619i	$0.676228\pi$
$480$		0			0
$481$	−1712.02			−0.162290
$482$		0			0
$483$	1087.33	−	3774.86i	0.102433	−	0.355615i
$484$		0			0
$485$	−6200.52			−0.580517
$486$		0			0
$487$			7808.76i			0.726589i	0.931674	+	0.363294i	$0.118348\pi$
$487$			7808.76i			0.726589i	−0.931674	+	0.363294i	$0.881652\pi$
$488$		0			0
$489$	522.829	+	150.598i	0.0483500	+	0.0139269i
$490$		0			0
$491$		−	5622.56i		−	0.516787i	−0.966040	−	0.258393i	$-0.916807\pi$
$491$		−	5622.56i		−	0.516787i	0.966040	−	0.258393i	$-0.0831931\pi$
$492$		0			0
$493$			16080.3i			1.46901i
$494$		0			0
$495$	1084.93	−	1727.01i	0.0985127	−	0.156815i
$496$		0			0
$497$		−	16379.4i		−	1.47830i
$498$		0			0
$499$	−17202.7			−1.54329			−0.771643	−	0.636056i	$-0.780565\pi$
$499$	−17202.7			−1.54329			−0.771643	+	0.636056i	$0.780565\pi$
$500$		0			0
$501$	10108.0	+	2911.55i	0.901381	+	0.259637i
$502$		0			0
$503$	−10628.9			−0.942189			−0.471095	−	0.882083i	$-0.656141\pi$
$503$	−10628.9			−0.942189			−0.471095	+	0.882083i	$0.656141\pi$
$504$		0			0
$505$	5236.57			0.461434
$506$		0			0
$507$	9124.29	+	2628.20i	0.799259	+	0.230222i
$508$		0			0
$509$	−1796.29			−0.156422			−0.0782112	−	0.996937i	$-0.524921\pi$
$509$	−1796.29			−0.156422			−0.0782112	+	0.996937i	$0.524921\pi$
$510$		0			0
$511$			1532.36i			0.132657i
$512$		0			0
$513$	−3573.20	−	3997.27i	−0.307526	−	0.344023i
$514$		0			0
$515$			5896.69i			0.504542i
$516$		0			0
$517$			1298.51i			0.110462i
$518$		0			0
$519$	−14929.7	−	4300.41i	−1.26270	−	0.363713i
$520$		0			0
$521$			7397.00i			0.622013i	0.950408	+	0.311006i	$0.100666\pi$
$521$			7397.00i			0.622013i	−0.950408	+	0.311006i	$0.899334\pi$
$522$		0			0
$523$	14977.7			1.25225			0.626126	−	0.779722i	$-0.284640\pi$
$523$	14977.7			1.25225			0.626126	+	0.779722i	$0.284640\pi$
$524$		0			0
$525$	581.972	−	2020.43i	0.0483797	−	0.167959i
$526$		0			0
$527$	−11096.3			−0.917199
$528$		0			0
$529$	−9985.30			−0.820687
$530$		0			0
$531$	1154.85	−	1838.32i	0.0943810	−	0.150238i
$532$		0			0
$533$	−902.337			−0.0733294
$534$		0			0
$535$			9407.24i			0.760206i
$536$		0			0
$537$	−810.443	+	2813.61i	−0.0651270	+	0.226101i
$538$		0			0
$539$			1224.09i			0.0978210i
$540$		0			0
$541$			2468.07i			0.196138i	0.995180	+	0.0980691i	$0.0312666\pi$
$541$			2468.07i			0.196138i	−0.995180	+	0.0980691i	$0.968733\pi$
$542$		0			0
$543$	−2421.89	+	8408.04i	−0.191405	+	0.664500i
$544$		0			0
$545$		−	5933.70i		−	0.466371i
$546$		0			0
$547$	19532.2			1.52676			0.763380	−	0.645950i	$-0.223538\pi$
$547$	19532.2			1.52676			0.763380	+	0.645950i	$0.223538\pi$
$548$		0			0
$549$	2217.83	−	3530.40i	0.172413	−	0.274452i
$550$		0			0
$551$	−7362.69			−0.569258
$552$		0			0
$553$	2695.29			0.207261
$554$		0			0
$555$	−640.362	+	2223.14i	−0.0489763	+	0.170031i
$556$		0			0
$557$	−18683.3			−1.42125			−0.710625	−	0.703571i	$-0.751588\pi$
$557$	−18683.3			−1.42125			−0.710625	+	0.703571i	$0.751588\pi$
$558$		0			0
$559$			154.209i			0.0116679i
$560$		0			0
$561$	−6296.04	−	1813.54i	−0.473831	−	0.136484i
$562$		0			0
$563$		−	8115.47i		−	0.607507i	−0.952751	−	0.303753i	$-0.901760\pi$
$563$		−	8115.47i		−	0.607507i	0.952751	−	0.303753i	$-0.0982399\pi$
$564$		0			0
$565$		−	8454.88i		−	0.629557i
$566$		0			0
$567$	10629.9	−	5121.54i	0.787323	−	0.379338i
$568$		0			0
$569$			18015.5i			1.32732i	0.748032	+	0.663662i	$0.230999\pi$
$569$			18015.5i			1.32732i	−0.748032	+	0.663662i	$0.769001\pi$
$570$		0			0
$571$	7359.90			0.539408			0.269704	−	0.962943i	$-0.413074\pi$
$571$	7359.90			0.539408			0.269704	+	0.962943i	$0.413074\pi$
$572$		0			0
$573$	23420.7	+	6746.19i	1.70753	+	0.491843i
$574$		0			0
$575$	1167.72			0.0846907
$576$		0			0
$577$	−8867.74			−0.639807			−0.319904	−	0.947450i	$-0.603651\pi$
$577$	−8867.74			−0.639807			−0.319904	+	0.947450i	$0.603651\pi$
$578$		0			0
$579$	−17967.5	−	5175.42i	−1.28964	−	0.371473i
$580$		0			0
$581$	3230.93			0.230708
$582$		0			0
$583$			5628.78i			0.399863i
$584$		0			0
$585$	−1380.68	+	2197.80i	−0.0975795	+	0.155329i
$586$		0			0
$587$			12971.5i			0.912082i	0.889959	+	0.456041i	$0.150733\pi$
$587$			12971.5i			0.912082i	−0.889959	+	0.456041i	$0.849267\pi$
$588$		0			0
$589$		−	5080.68i		−	0.355426i
$590$		0			0
$591$	−12139.8	−	3496.81i	−0.844952	−	0.243383i
$592$		0			0
$593$		−	12494.6i		−	0.865248i	−0.901575	−	0.432624i	$-0.857588\pi$
$593$		−	12494.6i		−	0.865248i	0.901575	−	0.432624i	$-0.142412\pi$
$594$		0			0
$595$	−6754.60			−0.465398
$596$		0			0
$597$	4331.68	−	15038.3i	0.296958	−	1.03095i
$598$		0			0
$599$	12126.1			0.827146			0.413573	−	0.910471i	$-0.364281\pi$
$599$	12126.1			0.827146			0.413573	+	0.910471i	$0.364281\pi$
$600$		0			0
$601$	−2638.14			−0.179055			−0.0895276	−	0.995984i	$-0.528536\pi$
$601$	−2638.14			−0.179055			−0.0895276	+	0.995984i	$0.528536\pi$
$602$		0			0
$603$	20392.9	+	12811.0i	1.37722	+	0.865181i
$604$		0			0
$605$	−5513.81			−0.370526
$606$		0			0
$607$		−	6958.98i		−	0.465331i	−0.972557	−	0.232666i	$-0.925255\pi$
$607$		−	6958.98i		−	0.465331i	0.972557	−	0.232666i	$-0.0747448\pi$
$608$		0			0
$609$	4484.93	−	15570.3i	0.298421	−	1.03603i
$610$		0			0
$611$		−	1652.49i		−	0.109415i
$612$		0			0
$613$			15348.6i			1.01129i	0.862741	+	0.505646i	$0.168746\pi$
$613$			15348.6i			1.01129i	−0.862741	+	0.505646i	$0.831254\pi$
$614$		0			0
$615$	−337.509	+	1171.73i	−0.0221295	+	0.0768270i
$616$		0			0
$617$			25707.3i			1.67737i	0.544615	+	0.838686i	$0.316676\pi$
$617$			25707.3i			1.67737i	−0.544615	+	0.838686i	$0.683324\pi$
$618$		0			0
$619$	−27355.0			−1.77624			−0.888118	−	0.459615i	$-0.847987\pi$
$619$	−27355.0			−1.77624			−0.888118	+	0.459615i	$0.847987\pi$
$620$		0			0
$621$	4367.29	+	4885.61i	0.282212	+	0.315705i
$622$		0			0
$623$	21785.0			1.40096
$624$		0			0
$625$	625.000			0.0400000
$626$		0			0
$627$	830.364	−	2882.77i	0.0528892	−	0.183615i
$628$		0			0
$629$	7432.30			0.471137
$630$		0			0
$631$		−	20107.2i		−	1.26855i	−0.773108	−	0.634275i	$-0.781299\pi$
$631$		−	20107.2i		−	1.26855i	0.773108	−	0.634275i	$-0.218701\pi$
$632$		0			0
$633$	19124.6	+	5508.71i	1.20084	+	0.345895i
$634$		0			0
$635$			1479.82i			0.0924802i
$636$		0			0
$637$		−	1557.78i		−	0.0968943i
$638$		0			0
$639$	23136.6	+	14534.6i	1.43234	+	0.899813i
$640$		0			0
$641$		−	7562.85i		−	0.466013i	−0.972475	−	0.233007i	$-0.925144\pi$
$641$		−	7562.85i		−	0.466013i	0.972475	−	0.233007i	$-0.0748564\pi$
$642$		0			0
$643$	6839.33			0.419466			0.209733	−	0.977759i	$-0.432740\pi$
$643$	6839.33			0.419466			0.209733	+	0.977759i	$0.432740\pi$
$644$		0			0
$645$	200.248	+	57.6802i	0.0122244	+	0.00352117i
$646$		0			0
$647$	−2348.57			−0.142708			−0.0713539	−	0.997451i	$-0.522732\pi$
$647$	−2348.57			−0.142708			−0.0713539	+	0.997451i	$0.522732\pi$
$648$		0			0
$649$	1214.74			0.0734711
$650$		0			0
$651$	10744.4	+	3094.86i	0.646862	+	0.186324i
$652$		0			0
$653$	3786.26			0.226903			0.113452	−	0.993544i	$-0.463809\pi$
$653$	3786.26			0.226903			0.113452	+	0.993544i	$0.463809\pi$
$654$		0			0
$655$			1652.55i			0.0985809i
$656$		0			0
$657$	−2164.52	−	1359.77i	−0.128533	−	0.0807456i
$658$		0			0
$659$			13627.8i			0.805562i	0.915296	+	0.402781i	$0.131956\pi$
$659$			13627.8i			0.805562i	−0.915296	+	0.402781i	$0.868044\pi$
$660$		0			0
$661$			15895.8i			0.935363i	0.883897	+	0.467681i	$0.154911\pi$
$661$			15895.8i			0.935363i	−0.883897	+	0.467681i	$0.845089\pi$
$662$		0			0
$663$	8012.35	+	2307.91i	0.469342	+	0.135191i
$664$		0			0
$665$		−	3092.73i		−	0.180347i
$666$		0			0
$667$	8998.94			0.522399
$668$		0			0
$669$	6190.33	−	21490.9i	0.357746	−	1.24198i
$670$		0			0
$671$	2332.85			0.134215
$672$		0			0
$673$	31175.0			1.78560			0.892800	−	0.450453i	$-0.148738\pi$
$673$	31175.0			1.78560			0.892800	+	0.450453i	$0.148738\pi$
$674$		0			0
$675$	2337.52	+	2614.94i	0.133290	+	0.149110i
$676$		0			0
$677$	−6503.22			−0.369186			−0.184593	−	0.982815i	$-0.559097\pi$
$677$	−6503.22			−0.369186			−0.184593	+	0.982815i	$0.559097\pi$
$678$		0			0
$679$		−	20071.9i		−	1.13444i
$680$		0			0
$681$	−4258.55	+	14784.4i	−0.239630	+	0.831923i
$682$		0			0
$683$		−	8888.28i		−	0.497951i	−0.968510	−	0.248975i	$-0.919906\pi$
$683$		−	8888.28i		−	0.497951i	0.968510	−	0.248975i	$-0.0800939\pi$
$684$		0			0
$685$		−	11852.4i		−	0.661106i
$686$		0			0
$687$	8761.77	−	30418.2i	0.486583	−	1.68927i
$688$		0			0
$689$		−	7163.20i		−	0.396076i
$690$		0			0
$691$	16298.8			0.897301			0.448651	−	0.893707i	$-0.351905\pi$
$691$	16298.8			0.897301			0.448651	+	0.893707i	$0.351905\pi$
$692$		0			0
$693$	5590.55	+	3512.04i	0.306447	+	0.192513i
$694$		0			0
$695$	11480.1			0.626567
$696$		0			0
$697$	3917.26			0.212879
$698$		0			0
$699$	−8255.14	+	28659.3i	−0.446692	+	1.55078i
$700$		0			0
$701$	24267.7			1.30753			0.653764	−	0.756698i	$-0.273189\pi$
$701$	24267.7			1.30753			0.653764	+	0.756698i	$0.273189\pi$
$702$		0			0
$703$			3403.03i			0.182571i
$704$		0			0
$705$	−2145.84	−	618.095i	−0.114634	−	0.0330196i
$706$		0			0
$707$			16951.4i			0.901731i
$708$		0			0
$709$		−	36767.6i		−	1.94758i	−0.227447	−	0.973790i	$-0.573038\pi$
$709$		−	36767.6i		−	1.94758i	0.227447	−	0.973790i	$-0.426962\pi$
$710$		0			0
$711$	−2391.73	+	3807.22i	−0.126156	+	0.200818i
$712$		0			0
$713$			6209.79i			0.326169i
$714$		0			0
$715$	−1452.28			−0.0759610
$716$		0			0
$717$	1926.04	+	554.785i	0.100320	+	0.0288965i
$718$		0			0
$719$	14128.1			0.732806			0.366403	−	0.930456i	$-0.380589\pi$
$719$	14128.1			0.732806			0.366403	+	0.930456i	$0.380589\pi$
$720$		0			0
$721$	−19088.3			−0.985972
$722$		0			0
$723$	4167.00	+	1200.28i	0.214347	+	0.0617412i
$724$		0			0
$725$	4816.53			0.246733
$726$		0			0
$727$			30461.0i			1.55397i	0.629518	+	0.776986i	$0.283252\pi$
$727$			30461.0i			1.55397i	−0.629518	+	0.776986i	$0.716748\pi$
$728$		0			0
$729$	−2198.25	+	19559.9i	−0.111683	+	0.993744i
$730$		0			0
$731$		−	669.460i		−	0.0338726i
$732$		0			0
$733$			15628.1i			0.787498i	0.919218	+	0.393749i	$0.128822\pi$
$733$			15628.1i			0.787498i	−0.919218	+	0.393749i	$0.871178\pi$
$734$		0			0
$735$	−2022.86	−	582.671i	−0.101516	−	0.0292410i
$736$		0			0
$737$			13475.4i			0.673502i
$738$		0			0
$739$	−27037.9			−1.34588			−0.672940	−	0.739697i	$-0.734969\pi$
$739$	−27037.9			−1.34588			−0.672940	+	0.739697i	$0.734969\pi$
$740$		0			0
$741$	−1056.72	+	3668.62i	−0.0523882	+	0.181876i
$742$		0			0
$743$	1697.21			0.0838017			0.0419009	−	0.999122i	$-0.486659\pi$
$743$	1697.21			0.0838017			0.0419009	+	0.999122i	$0.486659\pi$
$744$		0			0
$745$	12820.7			0.630489
$746$		0			0
$747$	−2867.04	+	4563.83i	−0.140428	+	0.223537i
$748$		0			0
$749$	−30452.4			−1.48559
$750$		0			0
$751$		−	29710.1i		−	1.44359i	−0.692107	−	0.721795i	$-0.743317\pi$
$751$		−	29710.1i		−	1.44359i	0.692107	−	0.721795i	$-0.256683\pi$
$752$		0			0
$753$	4585.90	−	15920.8i	0.221938	−	0.770501i
$754$		0			0
$755$		−	9826.47i		−	0.473671i
$756$		0			0
$757$		−	24486.7i		−	1.17567i	−0.808980	−	0.587836i	$-0.799980\pi$
$757$		−	24486.7i		−	1.17567i	0.808980	−	0.587836i	$-0.200020\pi$
$758$		0			0
$759$	−1014.90	+	3523.42i	−0.0485357	+	0.168501i
$760$		0			0
$761$			27354.7i			1.30303i	0.758635	+	0.651515i	$0.225866\pi$
$761$			27354.7i			1.30303i	−0.758635	+	0.651515i	$0.774134\pi$
$762$		0			0
$763$	19208.1			0.911378
$764$		0			0
$765$	5993.86	−	9541.17i	0.283279	−	0.450930i
$766$		0			0
$767$	−1545.88			−0.0727751
$768$		0			0
$769$	−19052.5			−0.893432			−0.446716	−	0.894676i	$-0.647407\pi$
$769$	−19052.5			−0.893432			−0.446716	+	0.894676i	$0.647407\pi$
$770$		0			0
$771$	5610.37	−	19477.5i	0.262066	−	0.909811i
$772$		0			0
$773$	4302.43			0.200191			0.100095	−	0.994978i	$-0.468085\pi$
$773$	4302.43			0.200191			0.100095	+	0.994978i	$0.468085\pi$
$774$		0			0
$775$			3323.68i			0.154052i
$776$		0			0
$777$	−7196.59	−	2072.93i	−0.332273	−	0.0957092i
$778$		0			0
$779$			1793.60i			0.0824933i
$780$		0			0
$781$			15288.4i			0.700462i
$782$		0			0
$783$	18013.9	+	20151.9i	0.822178	+	0.919755i
$784$		0			0
$785$			3079.27i			0.140005i
$786$		0			0
$787$	−18936.4			−0.857701			−0.428851	−	0.903375i	$-0.641081\pi$
$787$	−18936.4			−0.857701			−0.428851	+	0.903375i	$0.641081\pi$
$788$		0			0
$789$	−18214.0	−	5246.42i	−0.821844	−	0.236727i
$790$		0			0
$791$	27369.5			1.23028
$792$		0			0
$793$	−2968.78			−0.132944
$794$		0			0
$795$	−9301.75	−	2679.31i	−0.414967	−	0.119529i
$796$		0			0
$797$	26499.5			1.17774			0.588871	−	0.808227i	$-0.299572\pi$
$797$	26499.5			1.17774			0.588871	+	0.808227i	$0.299572\pi$
$798$		0			0
$799$			7173.87i			0.317638i
$800$		0			0
$801$	−19331.4	+	30772.2i	−0.852737	+	1.35741i
$802$		0			0
$803$		−	1430.29i		−	0.0628566i
$804$		0			0
$805$			3780.05i			0.165502i
$806$		0			0
$807$	−4179.45	−	1203.86i	−0.182309	−	0.0525131i
$808$		0			0
$809$			17982.5i			0.781496i	0.920498	+	0.390748i	$0.127784\pi$
$809$			17982.5i			0.781496i	−0.920498	+	0.390748i	$0.872216\pi$
$810$		0			0
$811$	25868.5			1.12005			0.560027	−	0.828474i	$-0.310791\pi$
$811$	25868.5			1.12005			0.560027	+	0.828474i	$0.310791\pi$
$812$		0			0
$813$	−9458.33	+	32836.4i	−0.408018	+	1.41651i
$814$		0			0
$815$	−523.547			−0.0225019
$816$		0			0
$817$	306.526			0.0131260
$818$		0			0
$819$	−7114.55	−	4469.43i	−0.303544	−	0.190689i
$820$		0			0
$821$	9280.65			0.394515			0.197258	−	0.980352i	$-0.436796\pi$
$821$	9280.65			0.394515			0.197258	+	0.980352i	$0.436796\pi$
$822$		0			0
$823$		−	260.648i		−	0.0110396i	−0.999985	−	0.00551981i	$-0.998243\pi$
$823$		−	260.648i		−	0.0110396i	0.999985	−	0.00551981i	$-0.00175702\pi$
$824$		0			0
$825$	−543.208	+	1885.85i	−0.0229237	+	0.0795842i
$826$		0			0
$827$		−	29070.2i		−	1.22233i	−0.791502	−	0.611166i	$-0.790701\pi$
$827$		−	29070.2i		−	1.22233i	0.791502	−	0.611166i	$-0.209299\pi$
$828$		0			0
$829$		−	33065.3i		−	1.38529i	−0.721279	−	0.692644i	$-0.756446\pi$
$829$		−	33065.3i		−	1.38529i	0.721279	−	0.692644i	$-0.243554\pi$
$830$		0			0
$831$	−3472.05	+	12053.9i	−0.144939	+	0.503183i
$832$		0			0
$833$			6762.72i			0.281290i
$834$		0			0
$835$	−10121.9			−0.419499
$836$		0			0
$837$	−13905.9	+	12430.7i	−0.574265	+	0.513341i
$838$		0			0
$839$	−32192.6			−1.32469			−0.662344	−	0.749200i	$-0.730438\pi$
$839$	−32192.6			−1.32469			−0.662344	+	0.749200i	$0.730438\pi$
$840$		0			0
$841$	12729.3			0.521927
$842$		0			0
$843$	−706.335	+	2452.18i	−0.0288582	+	0.100187i
$844$		0			0
$845$	−9136.83			−0.371972
$846$		0			0
$847$		−	17848.9i		−	0.724080i
$848$		0			0
$849$	46293.1	+	13334.4i	1.87135	+	0.539030i
$850$		0			0
$851$		−	4159.30i		−	0.167543i
$852$		0			0
$853$			6772.88i			0.271863i	0.990718	+	0.135931i	$0.0434027\pi$
$853$			6772.88i			0.271863i	−0.990718	+	0.135931i	$0.956597\pi$
$854$		0			0
$855$	4368.62	+	2744.41i	0.174741	+	0.109774i
$856$		0			0
$857$			20970.9i			0.835883i	0.908474	+	0.417942i	$0.137248\pi$
$857$			20970.9i			0.835883i	−0.908474	+	0.417942i	$0.862752\pi$
$858$		0			0
$859$	−15056.3			−0.598039			−0.299019	−	0.954247i	$-0.596660\pi$
$859$	−15056.3			−0.598039			−0.299019	+	0.954247i	$0.596660\pi$
$860$		0			0
$861$	−3793.03	−	1092.56i	−0.150135	−	0.0432454i
$862$		0			0
$863$	31506.7			1.24276			0.621378	−	0.783511i	$-0.286573\pi$
$863$	31506.7			1.24276			0.621378	+	0.783511i	$0.286573\pi$
$864$		0			0
$865$	14950.2			0.587656
$866$		0			0
$867$	−10252.3	−	2953.10i	−0.401598	−	0.115678i
$868$		0			0
$869$	−2515.77			−0.0982066
$870$		0			0
$871$		−	17148.8i		−	0.667122i
$872$		0			0
$873$	28352.4	+	17811.2i	1.09918	+	0.690514i
$874$		0			0
$875$			2023.20i			0.0781677i
$876$		0			0
$877$			27344.6i			1.05286i	0.850218	+	0.526431i	$0.176470\pi$
$877$			27344.6i			1.05286i	−0.850218	+	0.526431i	$0.823530\pi$
$878$		0			0
$879$	3431.46	+	988.412i	0.131673	+	0.0379276i
$880$		0			0
$881$			9957.73i			0.380799i	0.981707	+	0.190400i	$0.0609784\pi$
$881$			9957.73i			0.380799i	−0.981707	+	0.190400i	$0.939022\pi$
$882$		0			0
$883$	41428.0			1.57889			0.789447	−	0.613819i	$-0.210368\pi$
$883$	41428.0			1.57889			0.789447	+	0.613819i	$0.210368\pi$
$884$		0			0
$885$	−578.219	+	2007.40i	−0.0219623	+	0.0762463i
$886$		0			0
$887$	32204.1			1.21906			0.609531	−	0.792762i	$-0.291358\pi$
$887$	32204.1			1.21906			0.609531	+	0.792762i	$0.291358\pi$
$888$		0			0
$889$	−4790.37			−0.180724
$890$		0			0
$891$	−9921.82	+	4780.41i	−0.373057	+	0.179742i
$892$		0			0
$893$	−3284.70			−0.123089
$894$		0			0
$895$		−	2817.47i		−	0.105227i
$896$		0			0
$897$	1291.56	−	4483.92i	0.0480759	−	0.166905i
$898$		0			0
$899$			25613.8i			0.950241i
$900$		0			0
$901$			31097.2i			1.14983i
$902$		0			0
$903$	−186.718	+	648.228i	−0.00688105	+	0.0238889i
$904$		0			0
$905$		−	8419.59i		−	0.309256i
$906$		0			0
$907$	−13255.9			−0.485288			−0.242644	−	0.970115i	$-0.578015\pi$
$907$	−13255.9			−0.485288			−0.242644	+	0.970115i	$0.578015\pi$
$908$		0			0
$909$	−23944.6	−	15042.3i	−0.873700	−	0.548867i
$910$		0			0
$911$	15092.0			0.548868			0.274434	−	0.961606i	$-0.411509\pi$
$911$	15092.0			0.548868			0.274434	+	0.961606i	$0.411509\pi$
$912$		0			0
$913$	−3015.72			−0.109316
$914$		0			0
$915$	−1110.44	+	3855.11i	−0.0401202	+	0.139285i
$916$		0			0
$917$	−5349.51			−0.192646
$918$		0			0
$919$		−	47179.7i		−	1.69349i	−0.532000	−	0.846744i	$-0.678559\pi$
$919$		−	47179.7i		−	1.69349i	0.532000	−	0.846744i	$-0.321441\pi$
$920$		0			0
$921$	−6488.29	−	1868.91i	−0.232135	−	0.0668651i
$922$		0			0
$923$		−	19456.0i		−	0.693826i
$924$		0			0
$925$		−	2226.19i		−	0.0791317i
$926$		0			0
$927$	16938.5	−	26963.1i	0.600143	−	0.955322i
$928$		0			0
$929$			33295.7i			1.17589i	0.808902	+	0.587943i	$0.200062\pi$
$929$			33295.7i			1.17589i	−0.808902	+	0.587943i	$0.799938\pi$
$930$		0			0
$931$	−3096.45			−0.109003
$932$		0			0
$933$	−26104.0	−	7519.10i	−0.915977	−	0.263842i
$934$		0			0
$935$	6304.69			0.220519
$936$		0			0
$937$	6092.97			0.212432			0.106216	−	0.994343i	$-0.466127\pi$
$937$	6092.97			0.212432			0.106216	+	0.994343i	$0.466127\pi$
$938$		0			0
$939$	−11288.1	−	3251.47i	−0.392304	−	0.113001i
$940$		0			0
$941$	−1728.53			−0.0598814			−0.0299407	−	0.999552i	$-0.509532\pi$
$941$	−1728.53			−0.0598814			−0.0299407	+	0.999552i	$0.509532\pi$
$942$		0			0
$943$		−	2192.20i		−	0.0757029i
$944$		0			0
$945$	−8464.88	+	7566.84i	−0.291389	+	0.260475i
$946$		0			0
$947$			4971.37i			0.170589i	0.996356	+	0.0852945i	$0.0271831\pi$
$947$			4971.37i			0.170589i	−0.996356	+	0.0852945i	$0.972817\pi$
$948$		0			0
$949$			1820.19i			0.0622612i
$950$		0			0
$951$	1814.65	+	522.699i	0.0618760	+	0.0178230i
$952$		0			0
$953$			13671.0i			0.464688i	0.972634	+	0.232344i	$0.0746395\pi$
$953$			13671.0i			0.464688i	−0.972634	+	0.232344i	$0.925361\pi$
$954$		0			0
$955$	−23452.9			−0.794677
$956$		0			0
$957$	−4186.20	+	14533.2i	−0.141401	+	0.490901i
$958$		0			0
$959$	38367.7			1.29193
$960$		0			0
$961$	12116.0			0.406701
$962$		0			0
$963$	27022.7	−	43015.4i	0.904251	−	1.43941i
$964$		0			0
$965$	17992.2			0.600195
$966$		0			0
$967$			39411.1i			1.31063i	0.755357	+	0.655314i	$0.227464\pi$
$967$			39411.1i			1.31063i	−0.755357	+	0.655314i	$0.772536\pi$
$968$		0			0
$969$	4587.49	−	15926.4i	0.152086	−	0.527996i
$970$		0			0
$971$		−	42243.7i		−	1.39615i	−0.716023	−	0.698076i	$-0.754040\pi$
$971$		−	42243.7i		−	1.39615i	0.716023	−	0.698076i	$-0.245960\pi$
$972$		0			0
$973$			37162.4i			1.22443i
$974$		0			0
$975$	691.287	−	2399.94i	0.0227066	−	0.0788303i
$976$		0			0
$977$			4702.62i			0.153992i	0.997031	+	0.0769960i	$0.0245329\pi$
$977$			4702.62i			0.153992i	−0.997031	+	0.0769960i	$0.975467\pi$
$978$		0			0
$979$	−20333.9			−0.663815
$980$		0			0
$981$	−17044.8	+	27132.3i	−0.554739	+	0.883047i
$982$		0			0
$983$	41600.6			1.34980			0.674900	−	0.737910i	$-0.264187\pi$
$983$	41600.6			1.34980			0.674900	+	0.737910i	$0.264187\pi$
$984$		0			0
$985$	12156.5			0.393238
$986$		0			0
$987$	2000.85	−	6946.35i	0.0645267	−	0.224017i
$988$		0			0
$989$	−374.647			−0.0120456
$990$		0			0
$991$		−	1313.39i		−	0.0421003i	−0.999778	−	0.0210501i	$-0.993299\pi$
$991$		−	1313.39i		−	0.0421003i	0.999778	−	0.0210501i	$-0.00670096\pi$
$992$		0			0
$993$	−3001.46	−	864.553i	−0.0959200	−	0.0276292i
$994$		0			0
$995$			15058.9i			0.479799i
$996$		0			0
$997$			34690.2i			1.10195i	0.834520	+	0.550977i	$0.185745\pi$
$997$			34690.2i			1.10195i	−0.834520	+	0.550977i	$0.814255\pi$
$998$		0			0
$999$	9314.17	−	8326.03i	0.294982	−	0.263688i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.4.b.a.671.16	yes	16
3.2	odd	2		960.4.b.b.671.15	yes	16
4.3	odd	2	inner	960.4.b.a.671.1	✓	16
8.3	odd	2		960.4.b.b.671.16	yes	16
8.5	even	2		960.4.b.b.671.1	yes	16
12.11	even	2		960.4.b.b.671.2	yes	16
24.5	odd	2	inner	960.4.b.a.671.2	yes	16
24.11	even	2	inner	960.4.b.a.671.15	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.4.b.a.671.1	✓	16	4.3	odd	2	inner
960.4.b.a.671.2	yes	16	24.5	odd	2	inner
960.4.b.a.671.15	yes	16	24.11	even	2	inner
960.4.b.a.671.16	yes	16	1.1	even	1	trivial
960.4.b.b.671.1	yes	16	8.5	even	2
960.4.b.b.671.2	yes	16	12.11	even	2
960.4.b.b.671.15	yes	16	3.2	odd	2
960.4.b.b.671.16	yes	16	8.3	odd	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	960.4.b.a.671.16

Level	$960$
Weight	$4$
Character	960.671
Analytic conductor	$56.642$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$