LMFDB - Embedded newform 1344.4.c.h.673.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(673,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1344.673");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.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:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 386x^{10} + 54793x^{8} + 3447408x^{6} + 90154296x^{4} + 707138208x^{2} + 525876624 $ x^12 + 386x^10 + 54793x^8 + 3447408x^6 + 90154296x^4 + 707138208*x^2 + 525876624
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			673.7
Root			$-3.41237i$ of defining polynomial
Character	$\chi$	$=$	1344.673
Dual form			1344.4.c.h.673.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -20.9786i q^{5} +7.00000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} -20.9786i q^{5} +7.00000 q^{7} -9.00000 q^{9} -10.4070i q^{11} -50.9189i q^{13} +62.9357 q^{15} +19.1451 q^{17} -24.3954i q^{19} +21.0000i q^{21} -101.598 q^{23} -315.100 q^{25} -27.0000i q^{27} -179.241i q^{29} +264.163 q^{31} +31.2211 q^{33} -146.850i q^{35} -328.492i q^{37} +152.757 q^{39} -89.3686 q^{41} +124.343i q^{43} +188.807i q^{45} +446.379 q^{47} +49.0000 q^{49} +57.4353i q^{51} +384.874i q^{53} -218.325 q^{55} +73.1863 q^{57} +94.7027i q^{59} +376.701i q^{61} -63.0000 q^{63} -1068.21 q^{65} -338.575i q^{67} -304.795i q^{69} +268.469 q^{71} -634.633 q^{73} -945.300i q^{75} -72.8493i q^{77} -1134.80 q^{79} +81.0000 q^{81} +589.208i q^{83} -401.637i q^{85} +537.724 q^{87} -459.183 q^{89} -356.433i q^{91} +792.488i q^{93} -511.781 q^{95} -1365.75 q^{97} +93.6634i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 84 q^{7} - 108 q^{9}+O(q^{10}) $ 12 * q + 84 * q^7 - 108 * q^9	$ 12 q + 84 q^{7} - 108 q^{9} + 24 q^{15} + 24 q^{17} - 80 q^{23} - 564 q^{25} + 640 q^{31} - 408 q^{33} - 120 q^{39} + 1416 q^{41} + 1536 q^{47} + 588 q^{49} - 1392 q^{55} - 336 q^{57} - 756 q^{63} - 2880 q^{65} - 1392 q^{71} + 2472 q^{73} + 544 q^{79} + 972 q^{81} - 720 q^{87} + 888 q^{89} - 2368 q^{95} - 2712 q^{97}+O(q^{100}) $ 12 * q + 84 * q^7 - 108 * q^9 + 24 * q^15 + 24 * q^17 - 80 * q^23 - 564 * q^25 + 640 * q^31 - 408 * q^33 - 120 * q^39 + 1416 * q^41 + 1536 * q^47 + 588 * q^49 - 1392 * q^55 - 336 * q^57 - 756 * q^63 - 2880 * q^65 - 1392 * q^71 + 2472 * q^73 + 544 * q^79 + 972 * q^81 - 720 * q^87 + 888 * q^89 - 2368 * q^95 - 2712 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0	0
$5$	− 20.9786i	− 1.87638i	−0.346121	−	0.938190i	$-0.612501\pi$
$5$	− 20.9786i	− 1.87638i	0.346121	−	0.938190i	$-0.387499\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	− 10.4070i	− 0.285258i	−0.989776	−	0.142629i	$-0.954444\pi$
$11$	− 10.4070i	− 0.285258i	0.989776	−	0.142629i	$-0.0455556\pi$
$12$	0	0
$13$	− 50.9189i	− 1.08634i	−0.839624	−	0.543168i	$-0.817225\pi$
$13$	− 50.9189i	− 1.08634i	0.839624	−	0.543168i	$-0.182775\pi$
$14$	0	0
$15$	62.9357	1.08333
$16$	0	0
$17$	19.1451	0.273139	0.136570	−	0.990630i	$-0.456392\pi$
$17$	19.1451	0.273139	0.136570	+	0.990630i	$0.456392\pi$
$18$	0	0
$19$	− 24.3954i	− 0.294563i	−0.989095	−	0.147281i	$-0.952948\pi$
$19$	− 24.3954i	− 0.294563i	0.989095	−	0.147281i	$-0.0470523\pi$
$20$	0	0
$21$	21.0000i	0.218218i
$22$	0	0
$23$	−101.598	−0.921074	−0.460537	−	0.887641i	$-0.652343\pi$
$23$	−101.598	−0.921074	−0.460537	+	0.887641i	$0.652343\pi$
$24$	0	0
$25$	−315.100	−2.52080
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	− 179.241i	− 1.14773i	−0.818949	−	0.573866i	$-0.805443\pi$
$29$	− 179.241i	− 1.14773i	0.818949	−	0.573866i	$-0.194557\pi$
$30$	0	0
$31$	264.163	1.53048	0.765242	−	0.643743i	$-0.222619\pi$
$31$	264.163	1.53048	0.765242	+	0.643743i	$0.222619\pi$
$32$	0	0
$33$	31.2211	0.164694
$34$	0	0
$35$	− 146.850i	− 0.709205i
$36$	0	0
$37$	− 328.492i	− 1.45956i	−0.683682	−	0.729780i	$-0.739623\pi$
$37$	− 328.492i	− 1.45956i	0.683682	−	0.729780i	$-0.260377\pi$
$38$	0	0
$39$	152.757	0.627197
$40$	0	0
$41$	−89.3686	−0.340415	−0.170208	−	0.985408i	$-0.554444\pi$
$41$	−89.3686	−0.340415	−0.170208	+	0.985408i	$0.554444\pi$
$42$	0	0
$43$	124.343i	0.440979i	0.975389	+	0.220490i	$0.0707655\pi$
$43$	124.343i	0.440979i	−0.975389	+	0.220490i	$0.929234\pi$
$44$	0	0
$45$	188.807i	0.625460i
$46$	0	0
$47$	446.379	1.38534	0.692672	−	0.721253i	$-0.256434\pi$
$47$	446.379	1.38534	0.692672	+	0.721253i	$0.256434\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	57.4353i	0.157697i
$52$	0	0
$53$	384.874i	0.997481i	0.866751	+	0.498740i	$0.166204\pi$
$53$	384.874i	0.997481i	−0.866751	+	0.498740i	$0.833796\pi$
$54$	0	0
$55$	−218.325	−0.535253
$56$	0	0
$57$	73.1863	0.170066
$58$	0	0
$59$	94.7027i	0.208970i	0.994526	+	0.104485i	$0.0333194\pi$
$59$	94.7027i	0.208970i	−0.994526	+	0.104485i	$0.966681\pi$
$60$	0	0
$61$	376.701i	0.790682i	0.918534	+	0.395341i	$0.129374\pi$
$61$	376.701i	0.790682i	−0.918534	+	0.395341i	$0.870626\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	−1068.21	−2.03838
$66$	0	0
$67$	− 338.575i	− 0.617365i	−0.951165	−	0.308683i	$-0.900112\pi$
$67$	− 338.575i	− 0.617365i	0.951165	−	0.308683i	$-0.0998881\pi$
$68$	0	0
$69$	− 304.795i	− 0.531782i
$70$	0	0
$71$	268.469	0.448752	0.224376	−	0.974503i	$-0.427966\pi$
$71$	268.469	0.448752	0.224376	+	0.974503i	$0.427966\pi$
$72$	0	0
$73$	−634.633	−1.01751	−0.508755	−	0.860911i	$-0.669894\pi$
$73$	−634.633	−1.01751	−0.508755	+	0.860911i	$0.669894\pi$
$74$	0	0
$75$	− 945.300i	− 1.45538i
$76$	0	0
$77$	− 72.8493i	− 0.107817i
$78$	0	0
$79$	−1134.80	−1.61614	−0.808068	−	0.589089i	$-0.799487\pi$
$79$	−1134.80	−1.61614	−0.808068	+	0.589089i	$0.799487\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	589.208i	0.779204i	0.920983	+	0.389602i	$0.127387\pi$
$83$	589.208i	0.779204i	−0.920983	+	0.389602i	$0.872613\pi$
$84$	0	0
$85$	− 401.637i	− 0.512513i
$86$	0	0
$87$	537.724	0.662644
$88$	0	0
$89$	−459.183	−0.546891	−0.273446	−	0.961887i	$-0.588163\pi$
$89$	−459.183	−0.546891	−0.273446	+	0.961887i	$0.588163\pi$
$90$	0	0
$91$	− 356.433i	− 0.410597i
$92$	0	0
$93$	792.488i	0.883625i
$94$	0	0
$95$	−511.781	−0.552712
$96$	0	0
$97$	−1365.75	−1.42960	−0.714801	−	0.699328i	$-0.753483\pi$
$97$	−1365.75	−1.42960	−0.714801	+	0.699328i	$0.753483\pi$
$98$	0	0
$99$	93.6634i	0.0950861i
$100$	0	0
$101$	1558.33i	1.53524i	0.640905	+	0.767620i	$0.278559\pi$
$101$	1558.33i	1.53524i	−0.640905	+	0.767620i	$0.721441\pi$
$102$	0	0
$103$	−1257.71	−1.20316	−0.601581	−	0.798812i	$-0.705462\pi$
$103$	−1257.71	−1.20316	−0.601581	+	0.798812i	$0.705462\pi$
$104$	0	0
$105$	440.550	0.409460
$106$	0	0
$107$	− 703.204i	− 0.635339i	−0.948201	−	0.317670i	$-0.897100\pi$
$107$	− 703.204i	− 0.635339i	0.948201	−	0.317670i	$-0.102900\pi$
$108$	0	0
$109$	− 904.817i	− 0.795099i	−0.917581	−	0.397549i	$-0.869861\pi$
$109$	− 904.817i	− 0.795099i	0.917581	−	0.397549i	$-0.130139\pi$
$110$	0	0
$111$	985.475	0.842677
$112$	0	0
$113$	2157.08	1.79576	0.897882	−	0.440236i	$-0.145105\pi$
$113$	2157.08	1.79576	0.897882	+	0.440236i	$0.145105\pi$
$114$	0	0
$115$	2131.39i	1.72828i
$116$	0	0
$117$	458.270i	0.362112i
$118$	0	0
$119$	134.016	0.103237
$120$	0	0
$121$	1222.69	0.918628
$122$	0	0
$123$	− 268.106i	− 0.196539i
$124$	0	0
$125$	3988.03i	2.85360i
$126$	0	0
$127$	−1544.72	−1.07931	−0.539654	−	0.841887i	$-0.681445\pi$
$127$	−1544.72	−1.07931	−0.539654	+	0.841887i	$0.681445\pi$
$128$	0	0
$129$	−373.029	−0.254600
$130$	0	0
$131$	28.1940i	0.0188040i	0.999956	+	0.00940199i	$0.00299279\pi$
$131$	28.1940i	0.0188040i	−0.999956	+	0.00940199i	$0.997007\pi$
$132$	0	0
$133$	− 170.768i	− 0.111334i
$134$	0	0
$135$	−566.421	−0.361109
$136$	0	0
$137$	1141.37	0.711778	0.355889	−	0.934528i	$-0.384178\pi$
$137$	1141.37	0.711778	0.355889	+	0.934528i	$0.384178\pi$
$138$	0	0
$139$	− 1977.61i	− 1.20675i	−0.797456	−	0.603377i	$-0.793821\pi$
$139$	− 1977.61i	− 1.20675i	0.797456	−	0.603377i	$-0.206179\pi$
$140$	0	0
$141$	1339.14i	0.799828i
$142$	0	0
$143$	−529.915	−0.309886
$144$	0	0
$145$	−3760.22	−2.15358
$146$	0	0
$147$	147.000i	0.0824786i
$148$	0	0
$149$	2276.30i	1.25156i	0.780001	+	0.625778i	$0.215218\pi$
$149$	2276.30i	1.25156i	−0.780001	+	0.625778i	$0.784782\pi$
$150$	0	0
$151$	−3060.92	−1.64963	−0.824815	−	0.565403i	$-0.808721\pi$
$151$	−3060.92	−1.64963	−0.824815	+	0.565403i	$0.808721\pi$
$152$	0	0
$153$	−172.306	−0.0910464
$154$	0	0
$155$	− 5541.75i	− 2.87177i
$156$	0	0
$157$	− 128.085i	− 0.0651102i	−0.999470	−	0.0325551i	$-0.989636\pi$
$157$	− 128.085i	− 0.0651102i	0.999470	−	0.0325551i	$-0.0103644\pi$
$158$	0	0
$159$	−1154.62	−0.575896
$160$	0	0
$161$	−711.188	−0.348133
$162$	0	0
$163$	1544.81i	0.742325i	0.928568	+	0.371163i	$0.121041\pi$
$163$	1544.81i	0.742325i	−0.928568	+	0.371163i	$0.878959\pi$
$164$	0	0
$165$	− 654.974i	− 0.309028i
$166$	0	0
$167$	−3135.14	−1.45272	−0.726359	−	0.687315i	$-0.758789\pi$
$167$	−3135.14	−1.45272	−0.726359	+	0.687315i	$0.758789\pi$
$168$	0	0
$169$	−395.738	−0.180126
$170$	0	0
$171$	219.559i	0.0981876i
$172$	0	0
$173$	− 1895.37i	− 0.832963i	−0.909144	−	0.416481i	$-0.863263\pi$
$173$	− 1895.37i	− 0.832963i	0.909144	−	0.416481i	$-0.136737\pi$
$174$	0	0
$175$	−2205.70	−0.952773
$176$	0	0
$177$	−284.108	−0.120649
$178$	0	0
$179$	1200.50i	0.501284i	0.968080	+	0.250642i	$0.0806417\pi$
$179$	1200.50i	0.501284i	−0.968080	+	0.250642i	$0.919358\pi$
$180$	0	0
$181$	− 2505.09i	− 1.02874i	−0.857568	−	0.514371i	$-0.828026\pi$
$181$	− 2505.09i	− 1.02874i	0.857568	−	0.514371i	$-0.171974\pi$
$182$	0	0
$183$	−1130.10	−0.456501
$184$	0	0
$185$	−6891.28	−2.73869
$186$	0	0
$187$	− 199.244i	− 0.0779152i
$188$	0	0
$189$	− 189.000i	− 0.0727393i
$190$	0	0
$191$	4889.67	1.85238	0.926188	−	0.377061i	$-0.123065\pi$
$191$	4889.67	1.85238	0.926188	+	0.377061i	$0.123065\pi$
$192$	0	0
$193$	30.9625	0.0115478	0.00577391	−	0.999983i	$-0.498162\pi$
$193$	30.9625	0.0115478	0.00577391	+	0.999983i	$0.498162\pi$
$194$	0	0
$195$	− 3204.62i	− 1.17686i
$196$	0	0
$197$	− 2367.06i	− 0.856070i	−0.903762	−	0.428035i	$-0.859206\pi$
$197$	− 2367.06i	− 0.856070i	0.903762	−	0.428035i	$-0.140794\pi$
$198$	0	0
$199$	1509.90	0.537858	0.268929	−	0.963160i	$-0.413330\pi$
$199$	1509.90	0.537858	0.268929	+	0.963160i	$0.413330\pi$
$200$	0	0
$201$	1015.72	0.356436
$202$	0	0
$203$	− 1254.69i	− 0.433802i
$204$	0	0
$205$	1874.82i	0.638748i
$206$	0	0
$207$	914.384	0.307025
$208$	0	0
$209$	−253.884	−0.0840265
$210$	0	0
$211$	− 5169.66i	− 1.68670i	−0.537362	−	0.843352i	$-0.680579\pi$
$211$	− 5169.66i	− 1.68670i	0.537362	−	0.843352i	$-0.319421\pi$
$212$	0	0
$213$	805.407i	0.259087i
$214$	0	0
$215$	2608.54	0.827445
$216$	0	0
$217$	1849.14	0.578469
$218$	0	0
$219$	− 1903.90i	− 0.587460i
$220$	0	0
$221$	− 974.848i	− 0.296721i
$222$	0	0
$223$	1702.47	0.511237	0.255619	−	0.966778i	$-0.417721\pi$
$223$	1702.47	0.511237	0.255619	+	0.966778i	$0.417721\pi$
$224$	0	0
$225$	2835.90	0.840267
$226$	0	0
$227$	2199.68i	0.643162i	0.946882	+	0.321581i	$0.104214\pi$
$227$	2199.68i	0.643162i	−0.946882	+	0.321581i	$0.895786\pi$
$228$	0	0
$229$	− 1970.83i	− 0.568718i	−0.958718	−	0.284359i	$-0.908219\pi$
$229$	− 1970.83i	− 0.568718i	0.958718	−	0.284359i	$-0.0917807\pi$
$230$	0	0
$231$	218.548	0.0622484
$232$	0	0
$233$	−5697.11	−1.60185	−0.800923	−	0.598767i	$-0.795657\pi$
$233$	−5697.11	−1.60185	−0.800923	+	0.598767i	$0.795657\pi$
$234$	0	0
$235$	− 9364.40i	− 2.59943i
$236$	0	0
$237$	− 3404.40i	− 0.933077i
$238$	0	0
$239$	−95.7576	−0.0259165	−0.0129582	−	0.999916i	$-0.504125\pi$
$239$	−95.7576	−0.0259165	−0.0129582	+	0.999916i	$0.504125\pi$
$240$	0	0
$241$	−5641.06	−1.50777	−0.753886	−	0.657006i	$-0.771823\pi$
$241$	−5641.06	−1.50777	−0.753886	+	0.657006i	$0.771823\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	− 1027.95i	− 0.268054i
$246$	0	0
$247$	−1242.19	−0.319994
$248$	0	0
$249$	−1767.62	−0.449874
$250$	0	0
$251$	− 4647.92i	− 1.16882i	−0.811458	−	0.584411i	$-0.801326\pi$
$251$	− 4647.92i	− 1.16882i	0.811458	−	0.584411i	$-0.198674\pi$
$252$	0	0
$253$	1057.34i	0.262744i
$254$	0	0
$255$	1204.91	0.295900
$256$	0	0
$257$	−4602.82	−1.11718	−0.558591	−	0.829443i	$-0.688658\pi$
$257$	−4602.82	−1.11718	−0.558591	+	0.829443i	$0.688658\pi$
$258$	0	0
$259$	− 2299.44i	− 0.551662i
$260$	0	0
$261$	1613.17i	0.382578i
$262$	0	0
$263$	−4475.15	−1.04924	−0.524619	−	0.851337i	$-0.675792\pi$
$263$	−4475.15	−1.04924	−0.524619	+	0.851337i	$0.675792\pi$
$264$	0	0
$265$	8074.10	1.87165
$266$	0	0
$267$	− 1377.55i	− 0.315748i
$268$	0	0
$269$	7509.73i	1.70214i	0.525051	+	0.851071i	$0.324046\pi$
$269$	7509.73i	1.70214i	−0.525051	+	0.851071i	$0.675954\pi$
$270$	0	0
$271$	2967.44	0.665162	0.332581	−	0.943075i	$-0.392080\pi$
$271$	2967.44	0.665162	0.332581	+	0.943075i	$0.392080\pi$
$272$	0	0
$273$	1069.30	0.237058
$274$	0	0
$275$	3279.26i	0.719079i
$276$	0	0
$277$	2729.73i	0.592106i	0.955171	+	0.296053i	$0.0956705\pi$
$277$	2729.73i	0.592106i	−0.955171	+	0.296053i	$0.904329\pi$
$278$	0	0
$279$	−2377.46	−0.510161
$280$	0	0
$281$	−7993.52	−1.69699	−0.848494	−	0.529205i	$-0.822490\pi$
$281$	−7993.52	−1.69699	−0.848494	+	0.529205i	$0.822490\pi$
$282$	0	0
$283$	− 3955.08i	− 0.830761i	−0.909648	−	0.415380i	$-0.863648\pi$
$283$	− 3955.08i	− 0.830761i	0.909648	−	0.415380i	$-0.136352\pi$
$284$	0	0
$285$	− 1535.34i	− 0.319108i
$286$	0	0
$287$	−625.580	−0.128665
$288$	0	0
$289$	−4546.47	−0.925395
$290$	0	0
$291$	− 4097.26i	− 0.825381i
$292$	0	0
$293$	− 5416.11i	− 1.07991i	−0.841695	−	0.539953i	$-0.818442\pi$
$293$	− 5416.11i	− 1.07991i	0.841695	−	0.539953i	$-0.181558\pi$
$294$	0	0
$295$	1986.73	0.392107
$296$	0	0
$297$	−280.990	−0.0548980
$298$	0	0
$299$	5173.28i	1.00060i
$300$	0	0
$301$	870.400i	0.166675i
$302$	0	0
$303$	−4674.98	−0.886371
$304$	0	0
$305$	7902.64	1.48362
$306$	0	0
$307$	240.932i	0.0447907i	0.999749	+	0.0223953i	$0.00712925\pi$
$307$	240.932i	0.0447907i	−0.999749	+	0.0223953i	$0.992871\pi$
$308$	0	0
$309$	− 3773.12i	− 0.694646i
$310$	0	0
$311$	7215.09	1.31553	0.657765	−	0.753223i	$-0.271502\pi$
$311$	7215.09	1.31553	0.657765	+	0.753223i	$0.271502\pi$
$312$	0	0
$313$	−723.443	−0.130643	−0.0653217	−	0.997864i	$-0.520807\pi$
$313$	−723.443	−0.130643	−0.0653217	+	0.997864i	$0.520807\pi$
$314$	0	0
$315$	1321.65i	0.236402i
$316$	0	0
$317$	− 2823.83i	− 0.500321i	−0.968204	−	0.250161i	$-0.919517\pi$
$317$	− 2823.83i	− 0.500321i	0.968204	−	0.250161i	$-0.0804835\pi$
$318$	0	0
$319$	−1865.37	−0.327400
$320$	0	0
$321$	2109.61	0.366813
$322$	0	0
$323$	− 467.053i	− 0.0804567i
$324$	0	0
$325$	16044.6i	2.73844i
$326$	0	0
$327$	2714.45	0.459050
$328$	0	0
$329$	3124.66	0.523611
$330$	0	0
$331$	10663.4i	1.77073i	0.464898	+	0.885364i	$0.346091\pi$
$331$	10663.4i	1.77073i	−0.464898	+	0.885364i	$0.653909\pi$
$332$	0	0
$333$	2956.43i	0.486520i
$334$	0	0
$335$	−7102.81	−1.15841
$336$	0	0
$337$	6999.28	1.13138	0.565690	−	0.824618i	$-0.308610\pi$
$337$	6999.28	1.13138	0.565690	+	0.824618i	$0.308610\pi$
$338$	0	0
$339$	6471.25i	1.03678i
$340$	0	0
$341$	− 2749.15i	− 0.436583i
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−6394.16	−0.997825
$346$	0	0
$347$	3111.79i	0.481411i	0.970598	+	0.240706i	$0.0773788\pi$
$347$	3111.79i	0.481411i	−0.970598	+	0.240706i	$0.922621\pi$
$348$	0	0
$349$	− 2210.72i	− 0.339075i	−0.985524	−	0.169537i	$-0.945773\pi$
$349$	− 2210.72i	− 0.339075i	0.985524	−	0.169537i	$-0.0542274\pi$
$350$	0	0
$351$	−1374.81	−0.209066
$352$	0	0
$353$	−6372.88	−0.960889	−0.480445	−	0.877025i	$-0.659525\pi$
$353$	−6372.88	−0.960889	−0.480445	+	0.877025i	$0.659525\pi$
$354$	0	0
$355$	− 5632.10i	− 0.842030i
$356$	0	0
$357$	402.047i	0.0596039i
$358$	0	0
$359$	−9991.87	−1.46894	−0.734472	−	0.678639i	$-0.762570\pi$
$359$	−9991.87	−1.46894	−0.734472	+	0.678639i	$0.762570\pi$
$360$	0	0
$361$	6263.86	0.913233
$362$	0	0
$363$	3668.08i	0.530370i
$364$	0	0
$365$	13313.7i	1.90923i
$366$	0	0
$367$	1731.20	0.246234	0.123117	−	0.992392i	$-0.460711\pi$
$367$	1731.20	0.246234	0.123117	+	0.992392i	$0.460711\pi$
$368$	0	0
$369$	804.317	0.113472
$370$	0	0
$371$	2694.12i	0.377012i
$372$	0	0
$373$	− 1327.90i	− 0.184333i	−0.995744	−	0.0921666i	$-0.970621\pi$
$373$	− 1327.90i	− 0.184333i	0.995744	−	0.0921666i	$-0.0293792\pi$
$374$	0	0
$375$	−11964.1	−1.64753
$376$	0	0
$377$	−9126.77	−1.24682
$378$	0	0
$379$	− 2160.90i	− 0.292871i	−0.989220	−	0.146436i	$-0.953220\pi$
$379$	− 2160.90i	− 0.292871i	0.989220	−	0.146436i	$-0.0467801\pi$
$380$	0	0
$381$	− 4634.17i	− 0.623139i
$382$	0	0
$383$	6004.88	0.801136	0.400568	−	0.916267i	$-0.368813\pi$
$383$	6004.88	0.801136	0.400568	+	0.916267i	$0.368813\pi$
$384$	0	0
$385$	−1528.27	−0.202307
$386$	0	0
$387$	− 1119.09i	− 0.146993i
$388$	0	0
$389$	1246.72i	0.162497i	0.996694	+	0.0812487i	$0.0258908\pi$
$389$	1246.72i	0.162497i	−0.996694	+	0.0812487i	$0.974109\pi$
$390$	0	0
$391$	−1945.11	−0.251582
$392$	0	0
$393$	−84.5820	−0.0108565
$394$	0	0
$395$	23806.4i	3.03249i
$396$	0	0
$397$	13520.7i	1.70927i	0.519225	+	0.854637i	$0.326221\pi$
$397$	13520.7i	1.70927i	−0.519225	+	0.854637i	$0.673779\pi$
$398$	0	0
$399$	512.304	0.0642789
$400$	0	0
$401$	−3991.79	−0.497108	−0.248554	−	0.968618i	$-0.579955\pi$
$401$	−3991.79	−0.497108	−0.248554	+	0.968618i	$0.579955\pi$
$402$	0	0
$403$	− 13450.9i	− 1.66262i
$404$	0	0
$405$	− 1699.26i	− 0.208487i
$406$	0	0
$407$	−3418.63	−0.416351
$408$	0	0
$409$	−1723.34	−0.208346	−0.104173	−	0.994559i	$-0.533220\pi$
$409$	−1723.34	−0.208346	−0.104173	+	0.994559i	$0.533220\pi$
$410$	0	0
$411$	3424.10i	0.410945i
$412$	0	0
$413$	662.919i	0.0789833i
$414$	0	0
$415$	12360.7	1.46208
$416$	0	0
$417$	5932.83	0.696720
$418$	0	0
$419$	− 6902.47i	− 0.804791i	−0.915466	−	0.402396i	$-0.868178\pi$
$419$	− 6902.47i	− 0.804791i	0.915466	−	0.402396i	$-0.131822\pi$
$420$	0	0
$421$	− 11855.5i	− 1.37245i	−0.727387	−	0.686227i	$-0.759265\pi$
$421$	− 11855.5i	− 1.37245i	0.727387	−	0.686227i	$-0.240735\pi$
$422$	0	0
$423$	−4017.41	−0.461781
$424$	0	0
$425$	−6032.62	−0.688530
$426$	0	0
$427$	2636.91i	0.298850i
$428$	0	0
$429$	− 1589.75i	− 0.178913i
$430$	0	0
$431$	13678.4	1.52869	0.764345	−	0.644808i	$-0.223063\pi$
$431$	13678.4	1.52869	0.764345	+	0.644808i	$0.223063\pi$
$432$	0	0
$433$	11620.4	1.28970	0.644849	−	0.764310i	$-0.276920\pi$
$433$	11620.4	1.28970	0.644849	+	0.764310i	$0.276920\pi$
$434$	0	0
$435$	− 11280.7i	− 1.24337i
$436$	0	0
$437$	2478.53i	0.271314i
$438$	0	0
$439$	−8292.22	−0.901518	−0.450759	−	0.892646i	$-0.648847\pi$
$439$	−8292.22	−0.901518	−0.450759	+	0.892646i	$0.648847\pi$
$440$	0	0
$441$	−441.000	−0.0476190
$442$	0	0
$443$	14991.6i	1.60784i	0.594736	+	0.803921i	$0.297256\pi$
$443$	14991.6i	1.60784i	−0.594736	+	0.803921i	$0.702744\pi$
$444$	0	0
$445$	9633.01i	1.02618i
$446$	0	0
$447$	−6828.91	−0.722586
$448$	0	0
$449$	−2614.56	−0.274807	−0.137404	−	0.990515i	$-0.543876\pi$
$449$	−2614.56	−0.274807	−0.137404	+	0.990515i	$0.543876\pi$
$450$	0	0
$451$	930.062i	0.0971063i
$452$	0	0
$453$	− 9182.76i	− 0.952414i
$454$	0	0
$455$	−7477.44	−0.770435
$456$	0	0
$457$	16248.2	1.66315	0.831573	−	0.555416i	$-0.187441\pi$
$457$	16248.2	1.66315	0.831573	+	0.555416i	$0.187441\pi$
$458$	0	0
$459$	− 516.918i	− 0.0525657i
$460$	0	0
$461$	− 7665.84i	− 0.774477i	−0.921980	−	0.387238i	$-0.873429\pi$
$461$	− 7665.84i	− 0.774477i	0.921980	−	0.387238i	$-0.126571\pi$
$462$	0	0
$463$	15029.1	1.50855	0.754277	−	0.656557i	$-0.227988\pi$
$463$	15029.1	1.50855	0.754277	+	0.656557i	$0.227988\pi$
$464$	0	0
$465$	16625.3	1.65802
$466$	0	0
$467$	5847.67i	0.579439i	0.957112	+	0.289719i	$0.0935620\pi$
$467$	5847.67i	0.579439i	−0.957112	+	0.289719i	$0.906438\pi$
$468$	0	0
$469$	− 2370.02i	− 0.233342i
$470$	0	0
$471$	384.255	0.0375914
$472$	0	0
$473$	1294.04	0.125793
$474$	0	0
$475$	7687.00i	0.742534i
$476$	0	0
$477$	− 3463.86i	− 0.332494i
$478$	0	0
$479$	6500.50	0.620074	0.310037	−	0.950725i	$-0.399659\pi$
$479$	6500.50	0.620074	0.310037	+	0.950725i	$0.399659\pi$
$480$	0	0
$481$	−16726.4	−1.58557
$482$	0	0
$483$	− 2133.56i	− 0.200995i
$484$	0	0
$485$	28651.6i	2.68248i
$486$	0	0
$487$	−4298.52	−0.399968	−0.199984	−	0.979799i	$-0.564089\pi$
$487$	−4298.52	−0.399968	−0.199984	+	0.979799i	$0.564089\pi$
$488$	0	0
$489$	−4634.44	−0.428582
$490$	0	0
$491$	1928.34i	0.177240i	0.996066	+	0.0886198i	$0.0282456\pi$
$491$	1928.34i	0.177240i	−0.996066	+	0.0886198i	$0.971754\pi$
$492$	0	0
$493$	− 3431.59i	− 0.313491i
$494$	0	0
$495$	1964.92	0.178418
$496$	0	0
$497$	1879.28	0.169612
$498$	0	0
$499$	7456.37i	0.668923i	0.942409	+	0.334462i	$0.108554\pi$
$499$	7456.37i	0.668923i	−0.942409	+	0.334462i	$0.891446\pi$
$500$	0	0
$501$	− 9405.41i	− 0.838728i
$502$	0	0
$503$	18407.6	1.63172	0.815860	−	0.578250i	$-0.196264\pi$
$503$	18407.6	1.63172	0.815860	+	0.578250i	$0.196264\pi$
$504$	0	0
$505$	32691.4	2.88069
$506$	0	0
$507$	− 1187.21i	− 0.103996i
$508$	0	0
$509$	− 2154.61i	− 0.187625i	−0.995590	−	0.0938127i	$-0.970095\pi$
$509$	− 2154.61i	− 0.187625i	0.995590	−	0.0938127i	$-0.0299055\pi$
$510$	0	0
$511$	−4442.43	−0.384583
$512$	0	0
$513$	−658.677	−0.0566887
$514$	0	0
$515$	26384.9i	2.25759i
$516$	0	0
$517$	− 4645.49i	− 0.395181i
$518$	0	0
$519$	5686.12	0.480911
$520$	0	0
$521$	6052.25	0.508933	0.254467	−	0.967082i	$-0.418100\pi$
$521$	6052.25	0.508933	0.254467	+	0.967082i	$0.418100\pi$
$522$	0	0
$523$	− 11824.5i	− 0.988618i	−0.869286	−	0.494309i	$-0.835421\pi$
$523$	− 11824.5i	− 0.988618i	0.869286	−	0.494309i	$-0.164579\pi$
$524$	0	0
$525$	− 6617.10i	− 0.550084i
$526$	0	0
$527$	5057.42	0.418035
$528$	0	0
$529$	−1844.79	−0.151623
$530$	0	0
$531$	− 852.324i	− 0.0696567i
$532$	0	0
$533$	4550.55i	0.369805i
$534$	0	0
$535$	−14752.2	−1.19214
$536$	0	0
$537$	−3601.51	−0.289416
$538$	0	0
$539$	− 509.945i	− 0.0407512i
$540$	0	0
$541$	− 21393.6i	− 1.70015i	−0.526660	−	0.850076i	$-0.676556\pi$
$541$	− 21393.6i	− 1.70015i	0.526660	−	0.850076i	$-0.323444\pi$
$542$	0	0
$543$	7515.28	0.593944
$544$	0	0
$545$	−18981.8	−1.49191
$546$	0	0
$547$	− 4427.67i	− 0.346094i	−0.984914	−	0.173047i	$-0.944639\pi$
$547$	− 4427.67i	− 0.346094i	0.984914	−	0.173047i	$-0.0553613\pi$
$548$	0	0
$549$	− 3390.31i	− 0.263561i
$550$	0	0
$551$	−4372.67	−0.338080
$552$	0	0
$553$	−7943.59	−0.610842
$554$	0	0
$555$	− 20673.9i	− 1.58118i
$556$	0	0
$557$	− 12901.0i	− 0.981390i	−0.871331	−	0.490695i	$-0.836743\pi$
$557$	− 12901.0i	− 0.981390i	0.871331	−	0.490695i	$-0.163257\pi$
$558$	0	0
$559$	6331.41	0.479052
$560$	0	0
$561$	597.731	0.0449844
$562$	0	0
$563$	− 12560.1i	− 0.940225i	−0.882607	−	0.470112i	$-0.844213\pi$
$563$	− 12560.1i	− 0.940225i	0.882607	−	0.470112i	$-0.155787\pi$
$564$	0	0
$565$	− 45252.5i	− 3.36954i
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	−7389.66	−0.544448	−0.272224	−	0.962234i	$-0.587759\pi$
$569$	−7389.66	−0.544448	−0.272224	+	0.962234i	$0.587759\pi$
$570$	0	0
$571$	26490.4i	1.94148i	0.240124	+	0.970742i	$0.422812\pi$
$571$	26490.4i	1.94148i	−0.240124	+	0.970742i	$0.577188\pi$
$572$	0	0
$573$	14669.0i	1.06947i
$574$	0	0
$575$	32013.6	2.32184
$576$	0	0
$577$	−14125.9	−1.01919	−0.509593	−	0.860415i	$-0.670204\pi$
$577$	−14125.9	−1.01919	−0.509593	+	0.860415i	$0.670204\pi$
$578$	0	0
$579$	92.8874i	0.00666713i
$580$	0	0
$581$	4124.45i	0.294511i
$582$	0	0
$583$	4005.40	0.284540
$584$	0	0
$585$	9613.85	0.679460
$586$	0	0
$587$	6882.21i	0.483917i	0.970287	+	0.241959i	$0.0777898\pi$
$587$	6882.21i	0.483917i	−0.970287	+	0.241959i	$0.922210\pi$
$588$	0	0
$589$	− 6444.36i	− 0.450824i
$590$	0	0
$591$	7101.17	0.494252
$592$	0	0
$593$	20160.3	1.39609	0.698047	−	0.716052i	$-0.254052\pi$
$593$	20160.3	1.39609	0.698047	+	0.716052i	$0.254052\pi$
$594$	0	0
$595$	− 2811.46i	− 0.193712i
$596$	0	0
$597$	4529.69i	0.310533i
$598$	0	0
$599$	10896.5	0.743268	0.371634	−	0.928379i	$-0.378798\pi$
$599$	10896.5	0.743268	0.371634	+	0.928379i	$0.378798\pi$
$600$	0	0
$601$	4724.07	0.320631	0.160315	−	0.987066i	$-0.448749\pi$
$601$	4724.07	0.320631	0.160315	+	0.987066i	$0.448749\pi$
$602$	0	0
$603$	3047.17i	0.205788i
$604$	0	0
$605$	− 25650.4i	− 1.72369i
$606$	0	0
$607$	12422.1	0.830636	0.415318	−	0.909676i	$-0.363670\pi$
$607$	12422.1	0.830636	0.415318	+	0.909676i	$0.363670\pi$
$608$	0	0
$609$	3764.06	0.250456
$610$	0	0
$611$	− 22729.2i	− 1.50495i
$612$	0	0
$613$	5228.08i	0.344470i	0.985056	+	0.172235i	$0.0550989\pi$
$613$	5228.08i	0.344470i	−0.985056	+	0.172235i	$0.944901\pi$
$614$	0	0
$615$	−5624.47	−0.368781
$616$	0	0
$617$	17595.6	1.14809	0.574046	−	0.818823i	$-0.305373\pi$
$617$	17595.6	1.14809	0.574046	+	0.818823i	$0.305373\pi$
$618$	0	0
$619$	− 21111.0i	− 1.37080i	−0.728168	−	0.685399i	$-0.759628\pi$
$619$	− 21111.0i	− 1.37080i	0.728168	−	0.685399i	$-0.240372\pi$
$620$	0	0
$621$	2743.15i	0.177261i
$622$	0	0
$623$	−3214.28	−0.206706
$624$	0	0
$625$	44275.5	2.83363
$626$	0	0
$627$	− 761.653i	− 0.0485127i
$628$	0	0
$629$	− 6289.01i	− 0.398663i
$630$	0	0
$631$	11648.1	0.734872	0.367436	−	0.930049i	$-0.380236\pi$
$631$	11648.1	0.734872	0.367436	+	0.930049i	$0.380236\pi$
$632$	0	0
$633$	15509.0	0.973819
$634$	0	0
$635$	32406.1i	2.02519i
$636$	0	0
$637$	− 2495.03i	− 0.155191i
$638$	0	0
$639$	−2416.22	−0.149584
$640$	0	0
$641$	−6913.84	−0.426022	−0.213011	−	0.977050i	$-0.568327\pi$
$641$	−6913.84	−0.426022	−0.213011	+	0.977050i	$0.568327\pi$
$642$	0	0
$643$	13390.6i	0.821263i	0.911801	+	0.410631i	$0.134692\pi$
$643$	13390.6i	0.821263i	−0.911801	+	0.410631i	$0.865308\pi$
$644$	0	0
$645$	7825.61i	0.477725i
$646$	0	0
$647$	12474.0	0.757966	0.378983	−	0.925404i	$-0.376274\pi$
$647$	12474.0	0.757966	0.378983	+	0.925404i	$0.376274\pi$
$648$	0	0
$649$	985.574	0.0596104
$650$	0	0
$651$	5547.41i	0.333979i
$652$	0	0
$653$	− 17114.5i	− 1.02564i	−0.858497	−	0.512819i	$-0.828601\pi$
$653$	− 17114.5i	− 1.02564i	0.858497	−	0.512819i	$-0.171399\pi$
$654$	0	0
$655$	591.470	0.0352834
$656$	0	0
$657$	5711.70	0.339170
$658$	0	0
$659$	28563.2i	1.68841i	0.536018	+	0.844207i	$0.319928\pi$
$659$	28563.2i	1.68841i	−0.536018	+	0.844207i	$0.680072\pi$
$660$	0	0
$661$	18291.7i	1.07635i	0.842834	+	0.538173i	$0.180885\pi$
$661$	18291.7i	1.07635i	−0.842834	+	0.538173i	$0.819115\pi$
$662$	0	0
$663$	2924.54	0.171312
$664$	0	0
$665$	−3582.47	−0.208905
$666$	0	0
$667$	18210.6i	1.05715i
$668$	0	0
$669$	5107.41i	0.295163i
$670$	0	0
$671$	3920.34	0.225549
$672$	0	0
$673$	−18737.9	−1.07324	−0.536622	−	0.843823i	$-0.680300\pi$
$673$	−18737.9	−1.07324	−0.536622	+	0.843823i	$0.680300\pi$
$674$	0	0
$675$	8507.70i	0.485128i
$676$	0	0
$677$	− 22789.1i	− 1.29373i	−0.762603	−	0.646866i	$-0.776079\pi$
$677$	− 22789.1i	− 1.29373i	0.762603	−	0.646866i	$-0.223921\pi$
$678$	0	0
$679$	−9560.28	−0.540339
$680$	0	0
$681$	−6599.03	−0.371330
$682$	0	0
$683$	− 21025.9i	− 1.17794i	−0.808155	−	0.588970i	$-0.799534\pi$
$683$	− 21025.9i	− 1.17794i	0.808155	−	0.588970i	$-0.200466\pi$
$684$	0	0
$685$	− 23944.2i	− 1.33557i
$686$	0	0
$687$	5912.50	0.328349
$688$	0	0
$689$	19597.4	1.08360
$690$	0	0
$691$	− 26862.7i	− 1.47888i	−0.673223	−	0.739440i	$-0.735091\pi$
$691$	− 26862.7i	− 1.47888i	0.673223	−	0.739440i	$-0.264909\pi$
$692$	0	0
$693$	655.643i	0.0359392i
$694$	0	0
$695$	−41487.4	−2.26433
$696$	0	0
$697$	−1710.97	−0.0929808
$698$	0	0
$699$	− 17091.3i	− 0.924826i
$700$	0	0
$701$	31268.5i	1.68473i	0.538908	+	0.842365i	$0.318837\pi$
$701$	31268.5i	1.68473i	−0.538908	+	0.842365i	$0.681163\pi$
$702$	0	0
$703$	−8013.70	−0.429932
$704$	0	0
$705$	28093.2	1.50078
$706$	0	0
$707$	10908.3i	0.580266i
$708$	0	0
$709$	− 26727.7i	− 1.41577i	−0.706328	−	0.707884i	$-0.749650\pi$
$709$	− 26727.7i	− 1.41577i	0.706328	−	0.707884i	$-0.250350\pi$
$710$	0	0
$711$	10213.2	0.538712
$712$	0	0
$713$	−26838.5	−1.40969
$714$	0	0
$715$	11116.9i	0.581464i
$716$	0	0
$717$	− 287.273i	− 0.0149629i
$718$	0	0
$719$	20723.9	1.07493	0.537463	−	0.843287i	$-0.319383\pi$
$719$	20723.9	1.07493	0.537463	+	0.843287i	$0.319383\pi$
$720$	0	0
$721$	−8803.95	−0.454752
$722$	0	0
$723$	− 16923.2i	− 0.870512i
$724$	0	0
$725$	56478.9i	2.89321i
$726$	0	0
$727$	−18312.1	−0.934194	−0.467097	−	0.884206i	$-0.654700\pi$
$727$	−18312.1	−0.934194	−0.467097	+	0.884206i	$0.654700\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	2380.56i	0.120449i
$732$	0	0
$733$	− 3140.53i	− 0.158251i	−0.996865	−	0.0791255i	$-0.974787\pi$
$733$	− 3140.53i	− 0.158251i	0.996865	−	0.0791255i	$-0.0252128\pi$
$734$	0	0
$735$	3083.85	0.154761
$736$	0	0
$737$	−3523.56	−0.176109
$738$	0	0
$739$	678.083i	0.0337533i	0.999858	+	0.0168766i	$0.00537226\pi$
$739$	678.083i	0.0337533i	−0.999858	+	0.0168766i	$0.994628\pi$
$740$	0	0
$741$	− 3726.57i	− 0.184749i
$742$	0	0
$743$	2610.32	0.128887	0.0644436	−	0.997921i	$-0.479473\pi$
$743$	2610.32	0.128887	0.0644436	+	0.997921i	$0.479473\pi$
$744$	0	0
$745$	47753.5	2.34839
$746$	0	0
$747$	− 5302.87i	− 0.259735i
$748$	0	0
$749$	− 4922.43i	− 0.240136i
$750$	0	0
$751$	−34275.1	−1.66540	−0.832700	−	0.553724i	$-0.813206\pi$
$751$	−34275.1	−1.66540	−0.832700	+	0.553724i	$0.813206\pi$
$752$	0	0
$753$	13943.8	0.674820
$754$	0	0
$755$	64213.7i	3.09533i
$756$	0	0
$757$	12274.8i	0.589347i	0.955598	+	0.294673i	$0.0952108\pi$
$757$	12274.8i	0.589347i	−0.955598	+	0.294673i	$0.904789\pi$
$758$	0	0
$759$	−3172.01	−0.151695
$760$	0	0
$761$	−10216.4	−0.486656	−0.243328	−	0.969944i	$-0.578239\pi$
$761$	−10216.4	−0.486656	−0.243328	+	0.969944i	$0.578239\pi$
$762$	0	0
$763$	− 6333.72i	− 0.300519i
$764$	0	0
$765$	3614.73i	0.170838i
$766$	0	0
$767$	4822.16	0.227012
$768$	0	0
$769$	−3611.71	−0.169365	−0.0846823	−	0.996408i	$-0.526988\pi$
$769$	−3611.71	−0.169365	−0.0846823	+	0.996408i	$0.526988\pi$
$770$	0	0
$771$	− 13808.5i	− 0.645006i
$772$	0	0
$773$	28663.3i	1.33370i	0.745194	+	0.666848i	$0.232357\pi$
$773$	28663.3i	1.33370i	−0.745194	+	0.666848i	$0.767643\pi$
$774$	0	0
$775$	−83237.6	−3.85804
$776$	0	0
$777$	6898.33	0.318502
$778$	0	0
$779$	2180.19i	0.100274i
$780$	0	0
$781$	− 2793.97i	− 0.128010i
$782$	0	0
$783$	−4839.51	−0.220881
$784$	0	0
$785$	−2687.04	−0.122172
$786$	0	0
$787$	− 17764.4i	− 0.804613i	−0.915505	−	0.402307i	$-0.868209\pi$
$787$	− 17764.4i	− 0.804613i	0.915505	−	0.402307i	$-0.131791\pi$
$788$	0	0
$789$	− 13425.4i	− 0.605778i
$790$	0	0
$791$	15099.6	0.678735
$792$	0	0
$793$	19181.2	0.858947
$794$	0	0
$795$	24222.3i	1.08060i
$796$	0	0
$797$	34572.4i	1.53653i	0.640129	+	0.768267i	$0.278881\pi$
$797$	34572.4i	1.53653i	−0.640129	+	0.768267i	$0.721119\pi$
$798$	0	0
$799$	8545.98	0.378392
$800$	0	0
$801$	4132.65	0.182297
$802$	0	0
$803$	6604.65i	0.290253i
$804$	0	0
$805$	14919.7i	0.653230i
$806$	0	0
$807$	−22529.2	−0.982732
$808$	0	0
$809$	−7092.56	−0.308234	−0.154117	−	0.988053i	$-0.549253\pi$
$809$	−7092.56	−0.308234	−0.154117	+	0.988053i	$0.549253\pi$
$810$	0	0
$811$	43146.0i	1.86814i	0.357090	+	0.934070i	$0.383769\pi$
$811$	43146.0i	1.86814i	−0.357090	+	0.934070i	$0.616231\pi$
$812$	0	0
$813$	8902.31i	0.384032i
$814$	0	0
$815$	32407.9	1.39288
$816$	0	0
$817$	3033.40	0.129896
$818$	0	0
$819$	3207.89i	0.136866i
$820$	0	0
$821$	− 7634.01i	− 0.324518i	−0.986748	−	0.162259i	$-0.948122\pi$
$821$	− 7634.01i	− 0.324518i	0.986748	−	0.162259i	$-0.0518779\pi$
$822$	0	0
$823$	−8563.47	−0.362702	−0.181351	−	0.983418i	$-0.558047\pi$
$823$	−8563.47	−0.362702	−0.181351	+	0.983418i	$0.558047\pi$
$824$	0	0
$825$	−9837.78	−0.415160
$826$	0	0
$827$	− 26203.9i	− 1.10181i	−0.834567	−	0.550906i	$-0.814282\pi$
$827$	− 26203.9i	− 1.10181i	0.834567	−	0.550906i	$-0.185718\pi$
$828$	0	0
$829$	− 22349.0i	− 0.936324i	−0.883643	−	0.468162i	$-0.844916\pi$
$829$	− 22349.0i	− 0.936324i	0.883643	−	0.468162i	$-0.155084\pi$
$830$	0	0
$831$	−8189.18	−0.341853
$832$	0	0
$833$	938.110	0.0390199
$834$	0	0
$835$	65770.6i	2.72585i
$836$	0	0
$837$	− 7132.39i	− 0.294542i
$838$	0	0
$839$	−33036.4	−1.35941	−0.679705	−	0.733486i	$-0.737892\pi$
$839$	−33036.4	−1.35941	−0.679705	+	0.733486i	$0.737892\pi$
$840$	0	0
$841$	−7738.40	−0.317291
$842$	0	0
$843$	− 23980.6i	− 0.979756i
$844$	0	0
$845$	8302.01i	0.337986i
$846$	0	0
$847$	8558.85	0.347209
$848$	0	0
$849$	11865.3	0.479640
$850$	0	0
$851$	33374.2i	1.34436i
$852$	0	0
$853$	− 25570.3i	− 1.02639i	−0.858272	−	0.513196i	$-0.828461\pi$
$853$	− 25570.3i	− 1.02639i	0.858272	−	0.513196i	$-0.171539\pi$
$854$	0	0
$855$	4606.03	0.184237
$856$	0	0
$857$	47355.1	1.88754	0.943768	−	0.330607i	$-0.107254\pi$
$857$	47355.1	1.88754	0.943768	+	0.330607i	$0.107254\pi$
$858$	0	0
$859$	− 13079.5i	− 0.519518i	−0.965673	−	0.259759i	$-0.916357\pi$
$859$	− 13079.5i	− 0.519518i	0.965673	−	0.259759i	$-0.0836431\pi$
$860$	0	0
$861$	− 1876.74i	− 0.0742847i
$862$	0	0
$863$	11909.5	0.469761	0.234881	−	0.972024i	$-0.424530\pi$
$863$	11909.5	0.469761	0.234881	+	0.972024i	$0.424530\pi$
$864$	0	0
$865$	−39762.2	−1.56295
$866$	0	0
$867$	− 13639.4i	− 0.534277i
$868$	0	0
$869$	11809.9i	0.461016i
$870$	0	0
$871$	−17239.9	−0.670666
$872$	0	0
$873$	12291.8	0.476534
$874$	0	0
$875$	27916.2i	1.07856i
$876$	0	0
$877$	− 42127.8i	− 1.62207i	−0.584998	−	0.811035i	$-0.698905\pi$
$877$	− 42127.8i	− 1.62207i	0.584998	−	0.811035i	$-0.301095\pi$
$878$	0	0
$879$	16248.3	0.623484
$880$	0	0
$881$	15490.5	0.592380	0.296190	−	0.955129i	$-0.404284\pi$
$881$	15490.5	0.592380	0.296190	+	0.955129i	$0.404284\pi$
$882$	0	0
$883$	13891.7i	0.529437i	0.964326	+	0.264719i	$0.0852791\pi$
$883$	13891.7i	0.529437i	−0.964326	+	0.264719i	$0.914721\pi$
$884$	0	0
$885$	5960.18i	0.226383i
$886$	0	0
$887$	28588.2	1.08218	0.541092	−	0.840963i	$-0.318011\pi$
$887$	28588.2	1.08218	0.541092	+	0.840963i	$0.318011\pi$
$888$	0	0
$889$	−10813.1	−0.407940
$890$	0	0
$891$	− 842.970i	− 0.0316954i
$892$	0	0
$893$	− 10889.6i	− 0.408071i
$894$	0	0
$895$	25184.8	0.940599
$896$	0	0
$897$	−15519.8	−0.577694
$898$	0	0
$899$	− 47348.8i	− 1.75659i
$900$	0	0
$901$	7368.45i	0.272451i
$902$	0	0
$903$	−2611.20	−0.0962296
$904$	0	0
$905$	−52553.3	−1.93031
$906$	0	0
$907$	− 37104.2i	− 1.35835i	−0.733975	−	0.679177i	$-0.762337\pi$
$907$	− 37104.2i	− 1.35835i	0.733975	−	0.679177i	$-0.237663\pi$
$908$	0	0
$909$	− 14024.9i	− 0.511747i
$910$	0	0
$911$	42276.6	1.53753	0.768763	−	0.639534i	$-0.220873\pi$
$911$	42276.6	1.53753	0.768763	+	0.639534i	$0.220873\pi$
$912$	0	0
$913$	6131.91	0.222274
$914$	0	0
$915$	23707.9i	0.856568i
$916$	0	0
$917$	197.358i	0.00710724i
$918$	0	0
$919$	37225.8	1.33620	0.668099	−	0.744072i	$-0.267108\pi$
$919$	37225.8	1.33620	0.668099	+	0.744072i	$0.267108\pi$
$920$	0	0
$921$	−722.797	−0.0258599
$922$	0	0
$923$	− 13670.2i	− 0.487496i
$924$	0	0
$925$	103508.i	3.67926i
$926$	0	0
$927$	11319.4	0.401054
$928$	0	0
$929$	−52977.0	−1.87096	−0.935479	−	0.353382i	$-0.885032\pi$
$929$	−52977.0	−1.87096	−0.935479	+	0.353382i	$0.885032\pi$
$930$	0	0
$931$	− 1195.38i	− 0.0420804i
$932$	0	0
$933$	21645.3i	0.759522i
$934$	0	0
$935$	−4179.85	−0.146199
$936$	0	0
$937$	−6236.65	−0.217441	−0.108721	−	0.994072i	$-0.534675\pi$
$937$	−6236.65	−0.217441	−0.108721	+	0.994072i	$0.534675\pi$
$938$	0	0
$939$	− 2170.33i	− 0.0754270i
$940$	0	0
$941$	− 15813.3i	− 0.547820i	−0.961755	−	0.273910i	$-0.911683\pi$
$941$	− 15813.3i	− 0.547820i	0.961755	−	0.273910i	$-0.0883171\pi$
$942$	0	0
$943$	9079.69	0.313548
$944$	0	0
$945$	−3964.95	−0.136487
$946$	0	0
$947$	− 35736.3i	− 1.22626i	−0.789980	−	0.613132i	$-0.789909\pi$
$947$	− 35736.3i	− 1.22626i	0.789980	−	0.613132i	$-0.210091\pi$
$948$	0	0
$949$	32314.9i	1.10536i
$950$	0	0
$951$	8471.48	0.288861
$952$	0	0
$953$	−29928.6	−1.01729	−0.508647	−	0.860975i	$-0.669854\pi$
$953$	−29928.6	−1.01729	−0.508647	+	0.860975i	$0.669854\pi$
$954$	0	0
$955$	− 102578.i	− 3.47576i
$956$	0	0
$957$	− 5596.11i	− 0.189025i
$958$	0	0
$959$	7989.57	0.269027
$960$	0	0
$961$	39990.9	1.34238
$962$	0	0
$963$	6328.83i	0.211780i
$964$	0	0
$965$	− 649.548i	− 0.0216681i
$966$	0	0
$967$	43498.1	1.44654	0.723270	−	0.690565i	$-0.242638\pi$
$967$	43498.1	1.44654	0.723270	+	0.690565i	$0.242638\pi$
$968$	0	0
$969$	1401.16	0.0464517
$970$	0	0
$971$	− 15759.7i	− 0.520859i	−0.965493	−	0.260430i	$-0.916136\pi$
$971$	− 15759.7i	− 0.520859i	0.965493	−	0.260430i	$-0.0838642\pi$
$972$	0	0
$973$	− 13843.3i	− 0.456110i
$974$	0	0
$975$	−48133.7	−1.58104
$976$	0	0
$977$	−15388.1	−0.503900	−0.251950	−	0.967740i	$-0.581072\pi$
$977$	−15388.1	−0.503900	−0.251950	+	0.967740i	$0.581072\pi$
$978$	0	0
$979$	4778.74i	0.156005i
$980$	0	0
$981$	8143.35i	0.265033i
$982$	0	0
$983$	−25215.2	−0.818147	−0.409074	−	0.912501i	$-0.634148\pi$
$983$	−25215.2	−0.818147	−0.409074	+	0.912501i	$0.634148\pi$
$984$	0	0
$985$	−49657.4	−1.60631
$986$	0	0
$987$	9373.97i	0.302307i
$988$	0	0
$989$	− 12633.0i	− 0.406175i
$990$	0	0
$991$	37192.6	1.19219	0.596096	−	0.802913i	$-0.296718\pi$
$991$	37192.6	1.19219	0.596096	+	0.802913i	$0.296718\pi$
$992$	0	0
$993$	−31990.1	−1.02233
$994$	0	0
$995$	− 31675.5i	− 1.00923i
$996$	0	0
$997$	4547.23i	0.144446i	0.997389	+	0.0722228i	$0.0230093\pi$
$997$	4547.23i	0.144446i	−0.997389	+	0.0722228i	$0.976991\pi$
$998$	0	0
$999$	−8869.28	−0.280892

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.c.h.673.7	yes	12
4.3	odd	2		1344.4.c.e.673.1	✓	12
8.3	odd	2		1344.4.c.e.673.12	yes	12
8.5	even	2	inner	1344.4.c.h.673.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.c.e.673.1	✓	12	4.3	odd	2
1344.4.c.e.673.12	yes	12	8.3	odd	2
1344.4.c.h.673.6	yes	12	8.5	even	2	inner
1344.4.c.h.673.7	yes	12	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -20.9786i q^{5} +7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} -20.9786i q^{5} +7.00000 q^{7} -9.00000 q^{9} -10.4070i q^{11} -50.9189i q^{13} +62.9357 q^{15} +19.1451 q^{17} -24.3954i q^{19} +21.0000i q^{21} -101.598 q^{23} -315.100 q^{25} -27.0000i q^{27} -179.241i q^{29} +264.163 q^{31} +31.2211 q^{33} -146.850i q^{35} -328.492i q^{37} +152.757 q^{39} -89.3686 q^{41} +124.343i q^{43} +188.807i q^{45} +446.379 q^{47} +49.0000 q^{49} +57.4353i q^{51} +384.874i q^{53} -218.325 q^{55} +73.1863 q^{57} +94.7027i q^{59} +376.701i q^{61} -63.0000 q^{63} -1068.21 q^{65} -338.575i q^{67} -304.795i q^{69} +268.469 q^{71} -634.633 q^{73} -945.300i q^{75} -72.8493i q^{77} -1134.80 q^{79} +81.0000 q^{81} +589.208i q^{83} -401.637i q^{85} +537.724 q^{87} -459.183 q^{89} -356.433i q^{91} +792.488i q^{93} -511.781 q^{95} -1365.75 q^{97} +93.6634i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 84 q^{7} - 108 q^{9}+O(q^{10}) \) 12 * q + 84 * q^7 - 108 * q^9	\( 12 q + 84 q^{7} - 108 q^{9} + 24 q^{15} + 24 q^{17} - 80 q^{23} - 564 q^{25} + 640 q^{31} - 408 q^{33} - 120 q^{39} + 1416 q^{41} + 1536 q^{47} + 588 q^{49} - 1392 q^{55} - 336 q^{57} - 756 q^{63} - 2880 q^{65} - 1392 q^{71} + 2472 q^{73} + 544 q^{79} + 972 q^{81} - 720 q^{87} + 888 q^{89} - 2368 q^{95} - 2712 q^{97}+O(q^{100}) \) 12 * q + 84 * q^7 - 108 * q^9 + 24 * q^15 + 24 * q^17 - 80 * q^23 - 564 * q^25 + 640 * q^31 - 408 * q^33 - 120 * q^39 + 1416 * q^41 + 1536 * q^47 + 588 * q^49 - 1392 * q^55 - 336 * q^57 - 756 * q^63 - 2880 * q^65 - 1392 * q^71 + 2472 * q^73 + 544 * q^79 + 972 * q^81 - 720 * q^87 + 888 * q^89 - 2368 * q^95 - 2712 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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