LMFDB - Embedded newform 1344.4.b.d.895.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(895,1344)]

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

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

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

chi := DirichletCharacter("1344.895");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.b (of order $2$, degree $1$, not 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:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1344.895
Dual form			1344.4.b.d.895.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +13.8564i q^{5} +(14.0000 - 12.1244i) q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +13.8564i q^{5} +(14.0000 - 12.1244i) q^{7} +9.00000 q^{9} -3.46410i q^{11} +13.8564i q^{13} +41.5692i q^{15} -76.2102i q^{17} -52.0000 q^{19} +(42.0000 - 36.3731i) q^{21} +114.315i q^{23} -67.0000 q^{25} +27.0000 q^{27} +246.000 q^{29} +116.000 q^{31} -10.3923i q^{33} +(168.000 + 193.990i) q^{35} +314.000 q^{37} +41.5692i q^{39} -270.200i q^{41} +377.587i q^{43} +124.708i q^{45} -192.000 q^{47} +(49.0000 - 339.482i) q^{49} -228.631i q^{51} +150.000 q^{53} +48.0000 q^{55} -156.000 q^{57} +204.000 q^{59} +581.969i q^{61} +(126.000 - 109.119i) q^{63} -192.000 q^{65} -509.223i q^{67} +342.946i q^{69} -814.064i q^{71} -124.708i q^{73} -201.000 q^{75} +(-42.0000 - 48.4974i) q^{77} +1375.25i q^{79} +81.0000 q^{81} +252.000 q^{83} +1056.00 q^{85} +738.000 q^{87} +214.774i q^{89} +(168.000 + 193.990i) q^{91} +348.000 q^{93} -720.533i q^{95} +1441.07i q^{97} -31.1769i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 28 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 + 28 * q^7 + 18 * q^9	$ 2 q + 6 q^{3} + 28 q^{7} + 18 q^{9} - 104 q^{19} + 84 q^{21} - 134 q^{25} + 54 q^{27} + 492 q^{29} + 232 q^{31} + 336 q^{35} + 628 q^{37} - 384 q^{47} + 98 q^{49} + 300 q^{53} + 96 q^{55} - 312 q^{57} + 408 q^{59} + 252 q^{63} - 384 q^{65} - 402 q^{75} - 84 q^{77} + 162 q^{81} + 504 q^{83} + 2112 q^{85} + 1476 q^{87} + 336 q^{91} + 696 q^{93}+O(q^{100}) $ 2 * q + 6 * q^3 + 28 * q^7 + 18 * q^9 - 104 * q^19 + 84 * q^21 - 134 * q^25 + 54 * q^27 + 492 * q^29 + 232 * q^31 + 336 * q^35 + 628 * q^37 - 384 * q^47 + 98 * q^49 + 300 * q^53 + 96 * q^55 - 312 * q^57 + 408 * q^59 + 252 * q^63 - 384 * q^65 - 402 * q^75 - 84 * q^77 + 162 * q^81 + 504 * q^83 + 2112 * q^85 + 1476 * q^87 + 336 * q^91 + 696 * q^93

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
$3$	3.00000			0.577350
$3$	3.00000			0.577350
$4$		0			0
$5$			13.8564i			1.23935i	0.784857	+	0.619677i	$0.212737\pi$
$5$			13.8564i			1.23935i	−0.784857	+	0.619677i	$0.787263\pi$
$6$		0			0
$7$	14.0000	−	12.1244i	0.755929	−	0.654654i
$7$	14.0000	−	12.1244i	0.755929	−	0.654654i
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$		−	3.46410i		−	0.0949514i	−0.998872	−	0.0474757i	$-0.984882\pi$
$11$		−	3.46410i		−	0.0949514i	0.998872	−	0.0474757i	$-0.0151177\pi$
$12$		0			0
$13$			13.8564i			0.295621i	0.989016	+	0.147811i	$0.0472226\pi$
$13$			13.8564i			0.295621i	−0.989016	+	0.147811i	$0.952777\pi$
$14$		0			0
$15$			41.5692i			0.715542i
$16$		0			0
$17$		−	76.2102i		−	1.08728i	−0.839320	−	0.543638i	$-0.817046\pi$
$17$		−	76.2102i		−	1.08728i	0.839320	−	0.543638i	$-0.182954\pi$
$18$		0			0
$19$	−52.0000			−0.627875			−0.313937	−	0.949444i	$-0.601648\pi$
$19$	−52.0000			−0.627875			−0.313937	+	0.949444i	$0.601648\pi$
$20$		0			0
$21$	42.0000	−	36.3731i	0.436436	−	0.377964i
$22$		0			0
$23$			114.315i			1.03637i	0.855270	+	0.518183i	$0.173391\pi$
$23$			114.315i			1.03637i	−0.855270	+	0.518183i	$0.826609\pi$
$24$		0			0
$25$	−67.0000			−0.536000
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	246.000			1.57521			0.787604	−	0.616181i	$-0.211321\pi$
$29$	246.000			1.57521			0.787604	+	0.616181i	$0.211321\pi$
$30$		0			0
$31$	116.000			0.672071			0.336036	−	0.941849i	$-0.390914\pi$
$31$	116.000			0.672071			0.336036	+	0.941849i	$0.390914\pi$
$32$		0			0
$33$		−	10.3923i		−	0.0548202i
$34$		0			0
$35$	168.000	+	193.990i	0.811348	+	0.936864i
$36$		0			0
$37$	314.000			1.39517			0.697585	−	0.716502i	$-0.254258\pi$
$37$	314.000			1.39517			0.697585	+	0.716502i	$0.254258\pi$
$38$		0			0
$39$			41.5692i			0.170677i
$40$		0			0
$41$		−	270.200i		−	1.02922i	−0.857423	−	0.514611i	$-0.827936\pi$
$41$		−	270.200i		−	1.02922i	0.857423	−	0.514611i	$-0.172064\pi$
$42$		0			0
$43$			377.587i			1.33910i	0.742765	+	0.669552i	$0.233514\pi$
$43$			377.587i			1.33910i	−0.742765	+	0.669552i	$0.766486\pi$
$44$		0			0
$45$			124.708i			0.413118i
$46$		0			0
$47$	−192.000			−0.595874			−0.297937	−	0.954586i	$-0.596299\pi$
$47$	−192.000			−0.595874			−0.297937	+	0.954586i	$0.596299\pi$
$48$		0			0
$49$	49.0000	−	339.482i	0.142857	−	0.989743i
$50$		0			0
$51$		−	228.631i		−	0.627739i
$52$		0			0
$53$	150.000			0.388756			0.194378	−	0.980927i	$-0.437731\pi$
$53$	150.000			0.388756			0.194378	+	0.980927i	$0.437731\pi$
$54$		0			0
$55$	48.0000			0.117679
$56$		0			0
$57$	−156.000			−0.362504
$58$		0			0
$59$	204.000			0.450145			0.225072	−	0.974342i	$-0.427738\pi$
$59$	204.000			0.450145			0.225072	+	0.974342i	$0.427738\pi$
$60$		0			0
$61$			581.969i			1.22153i	0.791811	+	0.610766i	$0.209139\pi$
$61$			581.969i			1.22153i	−0.791811	+	0.610766i	$0.790861\pi$
$62$		0			0
$63$	126.000	−	109.119i	0.251976	−	0.218218i
$64$		0			0
$65$	−192.000			−0.366380
$66$		0			0
$67$		−	509.223i		−	0.928530i	−0.885696	−	0.464265i	$-0.846319\pi$
$67$		−	509.223i		−	0.928530i	0.885696	−	0.464265i	$-0.153681\pi$
$68$		0			0
$69$			342.946i			0.598346i
$70$		0			0
$71$		−	814.064i		−	1.36073i	−0.732875	−	0.680363i	$-0.761822\pi$
$71$		−	814.064i		−	1.36073i	0.732875	−	0.680363i	$-0.238178\pi$
$72$		0			0
$73$		−	124.708i		−	0.199944i	−0.994990	−	0.0999721i	$-0.968125\pi$
$73$		−	124.708i		−	0.199944i	0.994990	−	0.0999721i	$-0.0318754\pi$
$74$		0			0
$75$	−201.000			−0.309460
$76$		0			0
$77$	−42.0000	−	48.4974i	−0.0621603	−	0.0717765i
$78$		0			0
$79$			1375.25i			1.95858i	0.202472	+	0.979288i	$0.435103\pi$
$79$			1375.25i			1.95858i	−0.202472	+	0.979288i	$0.564897\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	252.000			0.333260			0.166630	−	0.986019i	$-0.446711\pi$
$83$	252.000			0.333260			0.166630	+	0.986019i	$0.446711\pi$
$84$		0			0
$85$	1056.00			1.34752
$86$		0			0
$87$	738.000			0.909447
$88$		0			0
$89$			214.774i			0.255798i	0.991787	+	0.127899i	$0.0408234\pi$
$89$			214.774i			0.255798i	−0.991787	+	0.127899i	$0.959177\pi$
$90$		0			0
$91$	168.000	+	193.990i	0.193530	+	0.223469i
$92$		0			0
$93$	348.000			0.388021
$94$		0			0
$95$		−	720.533i		−	0.778159i
$96$		0			0
$97$			1441.07i			1.50843i	0.656625	+	0.754217i	$0.271983\pi$
$97$			1441.07i			1.50843i	−0.656625	+	0.754217i	$0.728017\pi$
$98$		0			0
$99$		−	31.1769i		−	0.0316505i
$100$		0			0
$101$		−	221.703i		−	0.218418i	−0.994019	−	0.109209i	$-0.965168\pi$
$101$		−	221.703i		−	0.218418i	0.994019	−	0.109209i	$-0.0348318\pi$
$102$		0			0
$103$	−508.000			−0.485968			−0.242984	−	0.970030i	$-0.578126\pi$
$103$	−508.000			−0.485968			−0.242984	+	0.970030i	$0.578126\pi$
$104$		0			0
$105$	504.000	+	581.969i	0.468432	+	0.540899i
$106$		0			0
$107$			287.520i			0.259772i	0.991529	+	0.129886i	$0.0414612\pi$
$107$			287.520i			0.259772i	−0.991529	+	0.129886i	$0.958539\pi$
$108$		0			0
$109$	−974.000			−0.855892			−0.427946	−	0.903804i	$-0.640763\pi$
$109$	−974.000			−0.855892			−0.427946	+	0.903804i	$0.640763\pi$
$110$		0			0
$111$	942.000			0.805502
$112$		0			0
$113$	−246.000			−0.204794			−0.102397	−	0.994744i	$-0.532651\pi$
$113$	−246.000			−0.204794			−0.102397	+	0.994744i	$0.532651\pi$
$114$		0			0
$115$	−1584.00			−1.28442
$116$		0			0
$117$			124.708i			0.0985404i
$118$		0			0
$119$	−924.000	−	1066.94i	−0.711789	−	0.821904i
$120$		0			0
$121$	1319.00			0.990984
$122$		0			0
$123$		−	810.600i		−	0.594222i
$124$		0			0
$125$			803.672i			0.575061i
$126$		0			0
$127$			1846.37i			1.29007i	0.764154	+	0.645033i	$0.223157\pi$
$127$			1846.37i			1.29007i	−0.764154	+	0.645033i	$0.776843\pi$
$128$		0			0
$129$			1132.76i			0.773132i
$130$		0			0
$131$	2772.00			1.84878			0.924392	−	0.381443i	$-0.124573\pi$
$131$	2772.00			1.84878			0.924392	+	0.381443i	$0.124573\pi$
$132$		0			0
$133$	−728.000	+	630.466i	−0.474629	+	0.411040i
$134$		0			0
$135$			374.123i			0.238514i
$136$		0			0
$137$	2418.00			1.50791			0.753955	−	0.656926i	$-0.228144\pi$
$137$	2418.00			1.50791			0.753955	+	0.656926i	$0.228144\pi$
$138$		0			0
$139$	−1676.00			−1.02271			−0.511354	−	0.859370i	$-0.670856\pi$
$139$	−1676.00			−1.02271			−0.511354	+	0.859370i	$0.670856\pi$
$140$		0			0
$141$	−576.000			−0.344028
$142$		0			0
$143$	48.0000			0.0280697
$144$		0			0
$145$			3408.68i			1.95224i
$146$		0			0
$147$	147.000	−	1018.45i	0.0824786	−	0.571429i
$148$		0			0
$149$	−594.000			−0.326593			−0.163297	−	0.986577i	$-0.552213\pi$
$149$	−594.000			−0.326593			−0.163297	+	0.986577i	$0.552213\pi$
$150$		0			0
$151$		−	966.484i		−	0.520870i	−0.965491	−	0.260435i	$-0.916134\pi$
$151$		−	966.484i		−	0.520870i	0.965491	−	0.260435i	$-0.0838660\pi$
$152$		0			0
$153$		−	685.892i		−	0.362425i
$154$		0			0
$155$			1607.34i			0.832935i
$156$		0			0
$157$			2549.58i			1.29604i	0.761623	+	0.648021i	$0.224403\pi$
$157$			2549.58i			1.29604i	−0.761623	+	0.648021i	$0.775597\pi$
$158$		0			0
$159$	450.000			0.224449
$160$		0			0
$161$	1386.00	+	1600.41i	0.678460	+	0.783418i
$162$		0			0
$163$			2095.78i			1.00708i	0.863971	+	0.503541i	$0.167970\pi$
$163$			2095.78i			1.00708i	−0.863971	+	0.503541i	$0.832030\pi$
$164$		0			0
$165$	144.000			0.0679417
$166$		0			0
$167$	−2760.00			−1.27889			−0.639447	−	0.768835i	$-0.720837\pi$
$167$	−2760.00			−1.27889			−0.639447	+	0.768835i	$0.720837\pi$
$168$		0			0
$169$	2005.00			0.912608
$170$		0			0
$171$	−468.000			−0.209292
$172$		0			0
$173$			332.554i			0.146148i	0.997327	+	0.0730740i	$0.0232809\pi$
$173$			332.554i			0.146148i	−0.997327	+	0.0730740i	$0.976719\pi$
$174$		0			0
$175$	−938.000	+	812.332i	−0.405178	+	0.350894i
$176$		0			0
$177$	612.000			0.259891
$178$		0			0
$179$			606.218i			0.253133i	0.991958	+	0.126567i	$0.0403957\pi$
$179$			606.218i			0.253133i	−0.991958	+	0.126567i	$0.959604\pi$
$180$		0			0
$181$		−	3796.66i		−	1.55913i	−0.626319	−	0.779567i	$-0.715439\pi$
$181$		−	3796.66i		−	1.55913i	0.626319	−	0.779567i	$-0.284561\pi$
$182$		0			0
$183$			1745.91i			0.705252i
$184$		0			0
$185$			4350.91i			1.72911i
$186$		0			0
$187$	−264.000			−0.103238
$188$		0			0
$189$	378.000	−	327.358i	0.145479	−	0.125988i
$190$		0			0
$191$		−	4887.85i		−	1.85169i	−0.377907	−	0.925844i	$-0.623356\pi$
$191$		−	4887.85i		−	1.85169i	0.377907	−	0.925844i	$-0.376644\pi$
$192$		0			0
$193$	274.000			0.102191			0.0510957	−	0.998694i	$-0.483729\pi$
$193$	274.000			0.102191			0.0510957	+	0.998694i	$0.483729\pi$
$194$		0			0
$195$	−576.000			−0.211529
$196$		0			0
$197$	1998.00			0.722597			0.361298	−	0.932450i	$-0.382334\pi$
$197$	1998.00			0.722597			0.361298	+	0.932450i	$0.382334\pi$
$198$		0			0
$199$	4948.00			1.76259			0.881293	−	0.472571i	$-0.156674\pi$
$199$	4948.00			1.76259			0.881293	+	0.472571i	$0.156674\pi$
$200$		0			0
$201$		−	1527.67i		−	0.536087i
$202$		0			0
$203$	3444.00	−	2982.59i	1.19075	−	1.03122i
$204$		0			0
$205$	3744.00			1.27557
$206$		0			0
$207$			1028.84i			0.345455i
$208$		0			0
$209$			180.133i			0.0596176i
$210$		0			0
$211$			58.8897i			0.0192139i	0.999954	+	0.00960696i	$0.00305804\pi$
$211$			58.8897i			0.0192139i	−0.999954	+	0.00960696i	$0.996942\pi$
$212$		0			0
$213$		−	2442.19i		−	0.785616i
$214$		0			0
$215$	−5232.00			−1.65963
$216$		0			0
$217$	1624.00	−	1406.43i	0.508038	−	0.439974i
$218$		0			0
$219$		−	374.123i		−	0.115438i
$220$		0			0
$221$	1056.00			0.321422
$222$		0			0
$223$	2572.00			0.772349			0.386175	−	0.922426i	$-0.373796\pi$
$223$	2572.00			0.772349			0.386175	+	0.922426i	$0.373796\pi$
$224$		0			0
$225$	−603.000			−0.178667
$226$		0			0
$227$	−6348.00			−1.85609			−0.928043	−	0.372473i	$-0.878510\pi$
$227$	−6348.00			−1.85609			−0.928043	+	0.372473i	$0.878510\pi$
$228$		0			0
$229$			1801.33i			0.519805i	0.965635	+	0.259903i	$0.0836905\pi$
$229$			1801.33i			0.519805i	−0.965635	+	0.259903i	$0.916310\pi$
$230$		0			0
$231$	−126.000	−	145.492i	−0.0358883	−	0.0414402i
$232$		0			0
$233$	4434.00			1.24670			0.623350	−	0.781943i	$-0.285771\pi$
$233$	4434.00			1.24670			0.623350	+	0.781943i	$0.285771\pi$
$234$		0			0
$235$		−	2660.43i		−	0.738499i
$236$		0			0
$237$			4125.75i			1.13078i
$238$		0			0
$239$			4742.36i			1.28350i	0.766912	+	0.641752i	$0.221792\pi$
$239$			4742.36i			1.28350i	−0.766912	+	0.641752i	$0.778208\pi$
$240$		0			0
$241$		−	5002.16i		−	1.33700i	−0.743711	−	0.668501i	$-0.766936\pi$
$241$		−	5002.16i		−	1.33700i	0.743711	−	0.668501i	$-0.233064\pi$
$242$		0			0
$243$	243.000			0.0641500
$244$		0			0
$245$	4704.00	+	678.964i	1.22664	+	0.177051i
$246$		0			0
$247$		−	720.533i		−	0.185613i
$248$		0			0
$249$	756.000			0.192408
$250$		0			0
$251$	−2772.00			−0.697080			−0.348540	−	0.937294i	$-0.613322\pi$
$251$	−2772.00			−0.697080			−0.348540	+	0.937294i	$0.613322\pi$
$252$		0			0
$253$	396.000			0.0984044
$254$		0			0
$255$	3168.00			0.777992
$256$		0			0
$257$		−	3471.03i		−	0.842478i	−0.906950	−	0.421239i	$-0.861595\pi$
$257$		−	3471.03i		−	0.842478i	0.906950	−	0.421239i	$-0.138405\pi$
$258$		0			0
$259$	4396.00	−	3807.05i	1.05465	−	0.913353i
$260$		0			0
$261$	2214.00			0.525070
$262$		0			0
$263$		−	5525.24i		−	1.29544i	−0.761878	−	0.647721i	$-0.775722\pi$
$263$		−	5525.24i		−	1.29544i	0.761878	−	0.647721i	$-0.224278\pi$
$264$		0			0
$265$			2078.46i			0.481807i
$266$		0			0
$267$			644.323i			0.147685i
$268$		0			0
$269$			3824.37i			0.866825i	0.901196	+	0.433412i	$0.142691\pi$
$269$			3824.37i			0.866825i	−0.901196	+	0.433412i	$0.857309\pi$
$270$		0			0
$271$	−6388.00			−1.43189			−0.715947	−	0.698154i	$-0.754005\pi$
$271$	−6388.00			−1.43189			−0.715947	+	0.698154i	$0.754005\pi$
$272$		0			0
$273$	504.000	+	581.969i	0.111734	+	0.129020i
$274$		0			0
$275$			232.095i			0.0508940i
$276$		0			0
$277$	3602.00			0.781311			0.390656	−	0.920537i	$-0.372248\pi$
$277$	3602.00			0.781311			0.390656	+	0.920537i	$0.372248\pi$
$278$		0			0
$279$	1044.00			0.224024
$280$		0			0
$281$	258.000			0.0547722			0.0273861	−	0.999625i	$-0.491282\pi$
$281$	258.000			0.0547722			0.0273861	+	0.999625i	$0.491282\pi$
$282$		0			0
$283$	−4228.00			−0.888087			−0.444043	−	0.896005i	$-0.646456\pi$
$283$	−4228.00			−0.888087			−0.444043	+	0.896005i	$0.646456\pi$
$284$		0			0
$285$		−	2161.60i		−	0.449271i
$286$		0			0
$287$	−3276.00	−	3782.80i	−0.673785	−	0.778019i
$288$		0			0
$289$	−895.000			−0.182170
$290$		0			0
$291$			4323.20i			0.870895i
$292$		0			0
$293$			7939.72i			1.58308i	0.611115	+	0.791542i	$0.290721\pi$
$293$			7939.72i			1.58308i	−0.611115	+	0.791542i	$0.709279\pi$
$294$		0			0
$295$			2826.71i			0.557889i
$296$		0			0
$297$		−	93.5307i		−	0.0182734i
$298$		0			0
$299$	−1584.00			−0.306372
$300$		0			0
$301$	4578.00	+	5286.22i	0.876650	+	1.01227i
$302$		0			0
$303$		−	665.108i		−	0.126104i
$304$		0			0
$305$	−8064.00			−1.51391
$306$		0			0
$307$	5108.00			0.949606			0.474803	−	0.880092i	$-0.342519\pi$
$307$	5108.00			0.949606			0.474803	+	0.880092i	$0.342519\pi$
$308$		0			0
$309$	−1524.00			−0.280574
$310$		0			0
$311$	2928.00			0.533864			0.266932	−	0.963715i	$-0.413990\pi$
$311$	2928.00			0.533864			0.266932	+	0.963715i	$0.413990\pi$
$312$		0			0
$313$			1898.33i			0.342811i	0.985201	+	0.171405i	$0.0548308\pi$
$313$			1898.33i			0.342811i	−0.985201	+	0.171405i	$0.945169\pi$
$314$		0			0
$315$	1512.00	+	1745.91i	0.270449	+	0.312288i
$316$		0			0
$317$	3438.00			0.609140			0.304570	−	0.952490i	$-0.401487\pi$
$317$	3438.00			0.609140			0.304570	+	0.952490i	$0.401487\pi$
$318$		0			0
$319$		−	852.169i		−	0.149568i
$320$		0			0
$321$			862.561i			0.149980i
$322$		0			0
$323$			3962.93i			0.682673i
$324$		0			0
$325$		−	928.379i		−	0.158453i
$326$		0			0
$327$	−2922.00			−0.494150
$328$		0			0
$329$	−2688.00	+	2327.88i	−0.450438	+	0.390091i
$330$		0			0
$331$		−	1146.62i		−	0.190404i	−0.995458	−	0.0952021i	$-0.969650\pi$
$331$		−	1146.62i		−	0.190404i	0.995458	−	0.0952021i	$-0.0303497\pi$
$332$		0			0
$333$	2826.00			0.465057
$334$		0			0
$335$	7056.00			1.15078
$336$		0			0
$337$	−574.000			−0.0927827			−0.0463914	−	0.998923i	$-0.514772\pi$
$337$	−574.000			−0.0927827			−0.0463914	+	0.998923i	$0.514772\pi$
$338$		0			0
$339$	−738.000			−0.118238
$340$		0			0
$341$		−	401.836i		−	0.0638142i
$342$		0			0
$343$	−3430.00	−	5346.84i	−0.539949	−	0.841698i
$344$		0			0
$345$	−4752.00			−0.741563
$346$		0			0
$347$		−	10742.2i		−	1.66187i	−0.556366	−	0.830937i	$-0.687805\pi$
$347$		−	10742.2i		−	1.66187i	0.556366	−	0.830937i	$-0.312195\pi$
$348$		0			0
$349$		−	1510.35i		−	0.231654i	−0.993269	−	0.115827i	$-0.963048\pi$
$349$		−	1510.35i		−	0.231654i	0.993269	−	0.115827i	$-0.0369517\pi$
$350$		0			0
$351$			374.123i			0.0568923i
$352$		0			0
$353$		−	2542.65i		−	0.383376i	−0.981456	−	0.191688i	$-0.938604\pi$
$353$		−	2542.65i		−	0.383376i	0.981456	−	0.191688i	$-0.0613961\pi$
$354$		0			0
$355$	11280.0			1.68642
$356$		0			0
$357$	−2772.00	−	3200.83i	−0.410952	−	0.474526i
$358$		0			0
$359$			11199.4i			1.64647i	0.567698	+	0.823237i	$0.307834\pi$
$359$			11199.4i			1.64647i	−0.567698	+	0.823237i	$0.692166\pi$
$360$		0			0
$361$	−4155.00			−0.605773
$362$		0			0
$363$	3957.00			0.572145
$364$		0			0
$365$	1728.00			0.247802
$366$		0			0
$367$	11956.0			1.70054			0.850270	−	0.526347i	$-0.176439\pi$
$367$	11956.0			1.70054			0.850270	+	0.526347i	$0.176439\pi$
$368$		0			0
$369$		−	2431.80i		−	0.343074i
$370$		0			0
$371$	2100.00	−	1818.65i	0.293872	−	0.254501i
$372$		0			0
$373$	−1646.00			−0.228490			−0.114245	−	0.993453i	$-0.536445\pi$
$373$	−1646.00			−0.228490			−0.114245	+	0.993453i	$0.536445\pi$
$374$		0			0
$375$			2411.01i			0.332011i
$376$		0			0
$377$			3408.68i			0.465665i
$378$		0			0
$379$			11906.1i			1.61366i	0.590785	+	0.806829i	$0.298818\pi$
$379$			11906.1i			1.61366i	−0.590785	+	0.806829i	$0.701182\pi$
$380$		0			0
$381$			5539.10i			0.744821i
$382$		0			0
$383$	−9768.00			−1.30319			−0.651595	−	0.758567i	$-0.725900\pi$
$383$	−9768.00			−1.30319			−0.651595	+	0.758567i	$0.725900\pi$
$384$		0			0
$385$	672.000	−	581.969i	0.0889566	−	0.0770387i
$386$		0			0
$387$			3398.28i			0.446368i
$388$		0			0
$389$	−11850.0			−1.54452			−0.772261	−	0.635306i	$-0.780874\pi$
$389$	−11850.0			−1.54452			−0.772261	+	0.635306i	$0.780874\pi$
$390$		0			0
$391$	8712.00			1.12682
$392$		0			0
$393$	8316.00			1.06740
$394$		0			0
$395$	−19056.0			−2.42737
$396$		0			0
$397$			7136.05i			0.902136i	0.892490	+	0.451068i	$0.148957\pi$
$397$			7136.05i			0.902136i	−0.892490	+	0.451068i	$0.851043\pi$
$398$		0			0
$399$	−2184.00	+	1891.40i	−0.274027	+	0.237314i
$400$		0			0
$401$	11178.0			1.39203			0.696013	−	0.718029i	$-0.254955\pi$
$401$	11178.0			1.39203			0.696013	+	0.718029i	$0.254955\pi$
$402$		0			0
$403$			1607.34i			0.198679i
$404$		0			0
$405$			1122.37i			0.137706i
$406$		0			0
$407$		−	1087.73i		−	0.132473i
$408$		0			0
$409$			4891.31i			0.591344i	0.955289	+	0.295672i	$0.0955436\pi$
$409$			4891.31i			0.591344i	−0.955289	+	0.295672i	$0.904456\pi$
$410$		0			0
$411$	7254.00			0.870592
$412$		0			0
$413$	2856.00	−	2473.37i	0.340277	−	0.294689i
$414$		0			0
$415$			3491.81i			0.413028i
$416$		0			0
$417$	−5028.00			−0.590461
$418$		0			0
$419$	−7524.00			−0.877259			−0.438629	−	0.898668i	$-0.644536\pi$
$419$	−7524.00			−0.877259			−0.438629	+	0.898668i	$0.644536\pi$
$420$		0			0
$421$	−15406.0			−1.78347			−0.891737	−	0.452554i	$-0.850513\pi$
$421$	−15406.0			−1.78347			−0.891737	+	0.452554i	$0.850513\pi$
$422$		0			0
$423$	−1728.00			−0.198625
$424$		0			0
$425$			5106.09i			0.582780i
$426$		0			0
$427$	7056.00	+	8147.57i	0.799681	+	0.923392i
$428$		0			0
$429$	144.000			0.0162060
$430$		0			0
$431$			5407.46i			0.604335i	0.953255	+	0.302167i	$0.0977102\pi$
$431$			5407.46i			0.604335i	−0.953255	+	0.302167i	$0.902290\pi$
$432$		0			0
$433$		−	3755.09i		−	0.416762i	−0.978048	−	0.208381i	$-0.933181\pi$
$433$		−	3755.09i		−	0.416762i	0.978048	−	0.208381i	$-0.0668194\pi$
$434$		0			0
$435$			10226.0i			1.12713i
$436$		0			0
$437$		−	5944.40i		−	0.650707i
$438$		0			0
$439$	436.000			0.0474012			0.0237006	−	0.999719i	$-0.492455\pi$
$439$	436.000			0.0474012			0.0237006	+	0.999719i	$0.492455\pi$
$440$		0			0
$441$	441.000	−	3055.34i	0.0476190	−	0.329914i
$442$		0			0
$443$		−	12280.2i		−	1.31705i	−0.752560	−	0.658523i	$-0.771181\pi$
$443$		−	12280.2i		−	1.31705i	0.752560	−	0.658523i	$-0.228819\pi$
$444$		0			0
$445$	−2976.00			−0.317025
$446$		0			0
$447$	−1782.00			−0.188559
$448$		0			0
$449$	−3774.00			−0.396673			−0.198336	−	0.980134i	$-0.563554\pi$
$449$	−3774.00			−0.396673			−0.198336	+	0.980134i	$0.563554\pi$
$450$		0			0
$451$	−936.000			−0.0977262
$452$		0			0
$453$		−	2899.45i		−	0.300724i
$454$		0			0
$455$	−2688.00	+	2327.88i	−0.276957	+	0.239852i
$456$		0			0
$457$	−2710.00			−0.277393			−0.138696	−	0.990335i	$-0.544291\pi$
$457$	−2710.00			−0.277393			−0.138696	+	0.990335i	$0.544291\pi$
$458$		0			0
$459$		−	2057.68i		−	0.209246i
$460$		0			0
$461$		−	13329.9i		−	1.34671i	−0.739319	−	0.673355i	$-0.764852\pi$
$461$		−	13329.9i		−	1.34671i	0.739319	−	0.673355i	$-0.235148\pi$
$462$		0			0
$463$		−	1063.48i		−	0.106747i	−0.998575	−	0.0533737i	$-0.983003\pi$
$463$		−	1063.48i		−	0.106747i	0.998575	−	0.0533737i	$-0.0169975\pi$
$464$		0			0
$465$			4822.03i			0.480895i
$466$		0			0
$467$	−9708.00			−0.961954			−0.480977	−	0.876733i	$-0.659718\pi$
$467$	−9708.00			−0.961954			−0.480977	+	0.876733i	$0.659718\pi$
$468$		0			0
$469$	−6174.00	−	7129.12i	−0.607865	−	0.701902i
$470$		0			0
$471$			7648.74i			0.748270i
$472$		0			0
$473$	1308.00			0.127150
$474$		0			0
$475$	3484.00			0.336541
$476$		0			0
$477$	1350.00			0.129585
$478$		0			0
$479$	12000.0			1.14466			0.572332	−	0.820022i	$-0.306039\pi$
$479$	12000.0			1.14466			0.572332	+	0.820022i	$0.306039\pi$
$480$		0			0
$481$			4350.91i			0.412442i
$482$		0			0
$483$	4158.00	+	4801.24i	0.391709	+	0.452307i
$484$		0			0
$485$	−19968.0			−1.86948
$486$		0			0
$487$		−	8310.38i		−	0.773263i	−0.922234	−	0.386632i	$-0.873638\pi$
$487$		−	8310.38i		−	0.773263i	0.922234	−	0.386632i	$-0.126362\pi$
$488$		0			0
$489$			6287.34i			0.581439i
$490$		0			0
$491$		−	12252.5i		−	1.12617i	−0.826399	−	0.563084i	$-0.809615\pi$
$491$		−	12252.5i		−	1.12617i	0.826399	−	0.563084i	$-0.190385\pi$
$492$		0			0
$493$		−	18747.7i		−	1.71269i
$494$		0			0
$495$	432.000			0.0392262
$496$		0			0
$497$	−9870.00	−	11396.9i	−0.890805	−	1.02861i
$498$		0			0
$499$		−	717.069i		−	0.0643295i	−0.999483	−	0.0321647i	$-0.989760\pi$
$499$		−	717.069i		−	0.0643295i	0.999483	−	0.0321647i	$-0.0102401\pi$
$500$		0			0
$501$	−8280.00			−0.738369
$502$		0			0
$503$	1272.00			0.112755			0.0563774	−	0.998410i	$-0.482045\pi$
$503$	1272.00			0.112755			0.0563774	+	0.998410i	$0.482045\pi$
$504$		0			0
$505$	3072.00			0.270697
$506$		0			0
$507$	6015.00			0.526895
$508$		0			0
$509$		−	13288.3i		−	1.15716i	−0.815626	−	0.578579i	$-0.803608\pi$
$509$		−	13288.3i		−	1.15716i	0.815626	−	0.578579i	$-0.196392\pi$
$510$		0			0
$511$	−1512.00	−	1745.91i	−0.130894	−	0.151144i
$512$		0			0
$513$	−1404.00			−0.120835
$514$		0			0
$515$		−	7039.05i		−	0.602287i
$516$		0			0
$517$			665.108i			0.0565791i
$518$		0			0
$519$			997.661i			0.0843786i
$520$		0			0
$521$			3277.04i			0.275566i	0.990462	+	0.137783i	$0.0439976\pi$
$521$			3277.04i			0.275566i	−0.990462	+	0.137783i	$0.956002\pi$
$522$		0			0
$523$	8332.00			0.696621			0.348311	−	0.937379i	$-0.386755\pi$
$523$	8332.00			0.696621			0.348311	+	0.937379i	$0.386755\pi$
$524$		0			0
$525$	−2814.00	+	2437.00i	−0.233930	+	0.202589i
$526$		0			0
$527$		−	8840.39i		−	0.730727i
$528$		0			0
$529$	−901.000			−0.0740528
$530$		0			0
$531$	1836.00			0.150048
$532$		0			0
$533$	3744.00			0.304260
$534$		0			0
$535$	−3984.00			−0.321950
$536$		0			0
$537$			1818.65i			0.146147i
$538$		0			0
$539$	−1176.00	−	169.741i	−0.0939776	−	0.0135645i
$540$		0			0
$541$	12890.0			1.02437			0.512185	−	0.858875i	$-0.328836\pi$
$541$	12890.0			1.02437			0.512185	+	0.858875i	$0.328836\pi$
$542$		0			0
$543$		−	11390.0i		−	0.900166i
$544$		0			0
$545$		−	13496.1i		−	1.06075i
$546$		0			0
$547$		−	7409.71i		−	0.579189i	−0.957149	−	0.289595i	$-0.906480\pi$
$547$		−	7409.71i		−	0.579189i	0.957149	−	0.289595i	$-0.0935205\pi$
$548$		0			0
$549$			5237.72i			0.407178i
$550$		0			0
$551$	−12792.0			−0.989034
$552$		0			0
$553$	16674.0	+	19253.5i	1.28219	+	1.48054i
$554$		0			0
$555$			13052.7i			0.998302i
$556$		0			0
$557$	−6954.00			−0.528995			−0.264498	−	0.964386i	$-0.585206\pi$
$557$	−6954.00			−0.528995			−0.264498	+	0.964386i	$0.585206\pi$
$558$		0			0
$559$	−5232.00			−0.395868
$560$		0			0
$561$	−792.000			−0.0596048
$562$		0			0
$563$	−16788.0			−1.25671			−0.628357	−	0.777925i	$-0.716272\pi$
$563$	−16788.0			−1.25671			−0.628357	+	0.777925i	$0.716272\pi$
$564$		0			0
$565$		−	3408.68i		−	0.253813i
$566$		0			0
$567$	1134.00	−	982.073i	0.0839921	−	0.0727393i
$568$		0			0
$569$	17418.0			1.28331			0.641653	−	0.766995i	$-0.278249\pi$
$569$	17418.0			1.28331			0.641653	+	0.766995i	$0.278249\pi$
$570$		0			0
$571$			6405.12i			0.469433i	0.972064	+	0.234716i	$0.0754161\pi$
$571$			6405.12i			0.469433i	−0.972064	+	0.234716i	$0.924584\pi$
$572$		0			0
$573$		−	14663.5i		−	1.06907i
$574$		0			0
$575$		−	7659.13i		−	0.555492i
$576$		0			0
$577$		−	23569.7i		−	1.70056i	−0.526333	−	0.850279i	$-0.676433\pi$
$577$		−	23569.7i		−	1.70056i	0.526333	−	0.850279i	$-0.323567\pi$
$578$		0			0
$579$	822.000			0.0590003
$580$		0			0
$581$	3528.00	−	3055.34i	0.251921	−	0.218170i
$582$		0			0
$583$		−	519.615i		−	0.0369130i
$584$		0			0
$585$	−1728.00			−0.122127
$586$		0			0
$587$	−252.000			−0.0177192			−0.00885959	−	0.999961i	$-0.502820\pi$
$587$	−252.000			−0.0177192			−0.00885959	+	0.999961i	$0.502820\pi$
$588$		0			0
$589$	−6032.00			−0.421977
$590$		0			0
$591$	5994.00			0.417192
$592$		0			0
$593$		−	21415.1i		−	1.48299i	−0.670960	−	0.741494i	$-0.734118\pi$
$593$		−	21415.1i		−	1.48299i	0.670960	−	0.741494i	$-0.265882\pi$
$594$		0			0
$595$	14784.0	−	12803.3i	1.01863	−	0.882160i
$596$		0			0
$597$	14844.0			1.01763
$598$		0			0
$599$		−	16000.7i		−	1.09144i	−0.837969	−	0.545718i	$-0.816257\pi$
$599$		−	16000.7i		−	1.09144i	0.837969	−	0.545718i	$-0.183743\pi$
$600$		0			0
$601$		−	23334.2i		−	1.58373i	−0.610697	−	0.791865i	$-0.709110\pi$
$601$		−	23334.2i		−	1.58373i	0.610697	−	0.791865i	$-0.290890\pi$
$602$		0			0
$603$		−	4583.01i		−	0.309510i
$604$		0			0
$605$			18276.6i			1.22818i
$606$		0			0
$607$	−16436.0			−1.09904			−0.549519	−	0.835481i	$-0.685189\pi$
$607$	−16436.0			−1.09904			−0.549519	+	0.835481i	$0.685189\pi$
$608$		0			0
$609$	10332.0	−	8947.77i	0.687477	−	0.595373i
$610$		0			0
$611$		−	2660.43i		−	0.176153i
$612$		0			0
$613$	−11822.0			−0.778933			−0.389467	−	0.921041i	$-0.627341\pi$
$613$	−11822.0			−0.778933			−0.389467	+	0.921041i	$0.627341\pi$
$614$		0			0
$615$	11232.0			0.736452
$616$		0			0
$617$	−25494.0			−1.66345			−0.831726	−	0.555186i	$-0.812647\pi$
$617$	−25494.0			−1.66345			−0.831726	+	0.555186i	$0.812647\pi$
$618$		0			0
$619$	−11660.0			−0.757116			−0.378558	−	0.925578i	$-0.623580\pi$
$619$	−11660.0			−0.757116			−0.378558	+	0.925578i	$0.623580\pi$
$620$		0			0
$621$			3086.51i			0.199449i
$622$		0			0
$623$	2604.00	+	3006.84i	0.167459	+	0.193365i
$624$		0			0
$625$	−19511.0			−1.24870
$626$		0			0
$627$			540.400i			0.0344202i
$628$		0			0
$629$		−	23930.0i		−	1.51694i
$630$		0			0
$631$		−	855.633i		−	0.0539813i	−0.999636	−	0.0269907i	$-0.991408\pi$
$631$		−	855.633i		−	0.0539813i	0.999636	−	0.0269907i	$-0.00859244\pi$
$632$		0			0
$633$			176.669i			0.0110932i
$634$		0			0
$635$	−25584.0			−1.59885
$636$		0			0
$637$	4704.00	+	678.964i	0.292589	+	0.0422316i
$638$		0			0
$639$		−	7326.57i		−	0.453576i
$640$		0			0
$641$	−8214.00			−0.506136			−0.253068	−	0.967448i	$-0.581440\pi$
$641$	−8214.00			−0.506136			−0.253068	+	0.967448i	$0.581440\pi$
$642$		0			0
$643$	7868.00			0.482556			0.241278	−	0.970456i	$-0.422433\pi$
$643$	7868.00			0.482556			0.241278	+	0.970456i	$0.422433\pi$
$644$		0			0
$645$	−15696.0			−0.958185
$646$		0			0
$647$	−17232.0			−1.04708			−0.523539	−	0.852002i	$-0.675389\pi$
$647$	−17232.0			−1.04708			−0.523539	+	0.852002i	$0.675389\pi$
$648$		0			0
$649$		−	706.677i		−	0.0427419i
$650$		0			0
$651$	4872.00	−	4219.28i	0.293316	−	0.254019i
$652$		0			0
$653$	−7338.00			−0.439752			−0.219876	−	0.975528i	$-0.570565\pi$
$653$	−7338.00			−0.439752			−0.219876	+	0.975528i	$0.570565\pi$
$654$		0			0
$655$			38410.0i			2.29130i
$656$		0			0
$657$		−	1122.37i		−	0.0666481i
$658$		0			0
$659$		−	26136.6i		−	1.54498i	−0.635029	−	0.772488i	$-0.719012\pi$
$659$		−	26136.6i		−	1.54498i	0.635029	−	0.772488i	$-0.280988\pi$
$660$		0			0
$661$		−	23320.3i		−	1.37225i	−0.727485	−	0.686124i	$-0.759311\pi$
$661$		−	23320.3i		−	1.37225i	0.727485	−	0.686124i	$-0.240689\pi$
$662$		0			0
$663$	3168.00			0.185573
$664$		0			0
$665$	−8736.00	−	10087.5i	−0.509425	−	0.588233i
$666$		0			0
$667$			28121.6i			1.63249i
$668$		0			0
$669$	7716.00			0.445916
$670$		0			0
$671$	2016.00			0.115986
$672$		0			0
$673$	−24158.0			−1.38369			−0.691844	−	0.722047i	$-0.743202\pi$
$673$	−24158.0			−1.38369			−0.691844	+	0.722047i	$0.743202\pi$
$674$		0			0
$675$	−1809.00			−0.103153
$676$		0			0
$677$		−	3131.55i		−	0.177777i	−0.996042	−	0.0888886i	$-0.971668\pi$
$677$		−	3131.55i		−	0.177777i	0.996042	−	0.0888886i	$-0.0283315\pi$
$678$		0			0
$679$	17472.0	+	20174.9i	0.987502	+	1.14027i
$680$		0			0
$681$	−19044.0			−1.07161
$682$		0			0
$683$		−	3356.71i		−	0.188054i	−0.995570	−	0.0940272i	$-0.970026\pi$
$683$		−	3356.71i		−	0.188054i	0.995570	−	0.0940272i	$-0.0299741\pi$
$684$		0			0
$685$			33504.8i			1.86884i
$686$		0			0
$687$			5404.00i			0.300110i
$688$		0			0
$689$			2078.46i			0.114925i
$690$		0			0
$691$	−36188.0			−1.99227			−0.996133	−	0.0878532i	$-0.971999\pi$
$691$	−36188.0			−1.99227			−0.996133	+	0.0878532i	$0.971999\pi$
$692$		0			0
$693$	−378.000	−	436.477i	−0.0207201	−	0.0239255i
$694$		0			0
$695$		−	23223.3i		−	1.26750i
$696$		0			0
$697$	−20592.0			−1.11905
$698$		0			0
$699$	13302.0			0.719782
$700$		0			0
$701$	33006.0			1.77834			0.889172	−	0.457573i	$-0.151281\pi$
$701$	33006.0			1.77834			0.889172	+	0.457573i	$0.151281\pi$
$702$		0			0
$703$	−16328.0			−0.875992
$704$		0			0
$705$		−	7981.29i		−	0.426373i
$706$		0			0
$707$	−2688.00	−	3103.84i	−0.142988	−	0.165109i
$708$		0			0
$709$	−1478.00			−0.0782898			−0.0391449	−	0.999234i	$-0.512463\pi$
$709$	−1478.00			−0.0782898			−0.0391449	+	0.999234i	$0.512463\pi$
$710$		0			0
$711$			12377.2i			0.652859i
$712$		0			0
$713$			13260.6i			0.696511i
$714$		0			0
$715$			665.108i			0.0347883i
$716$		0			0
$717$			14227.1i			0.741031i
$718$		0			0
$719$	−22800.0			−1.18261			−0.591305	−	0.806448i	$-0.701387\pi$
$719$	−22800.0			−1.18261			−0.591305	+	0.806448i	$0.701387\pi$
$720$		0			0
$721$	−7112.00	+	6159.17i	−0.367357	+	0.318141i
$722$		0			0
$723$		−	15006.5i		−	0.771919i
$724$		0			0
$725$	−16482.0			−0.844312
$726$		0			0
$727$	31532.0			1.60861			0.804303	−	0.594219i	$-0.202539\pi$
$727$	31532.0			1.60861			0.804303	+	0.594219i	$0.202539\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	28776.0			1.45598
$732$		0			0
$733$			15061.9i			0.758969i	0.925198	+	0.379485i	$0.123899\pi$
$733$			15061.9i			0.758969i	−0.925198	+	0.379485i	$0.876101\pi$
$734$		0			0
$735$	14112.0	+	2036.89i	0.708203	+	0.102220i
$736$		0			0
$737$	−1764.00			−0.0881652
$738$		0			0
$739$			38953.8i			1.93903i	0.245042	+	0.969513i	$0.421198\pi$
$739$			38953.8i			1.93903i	−0.245042	+	0.969513i	$0.578802\pi$
$740$		0			0
$741$		−	2161.60i		−	0.107164i
$742$		0			0
$743$		−	23774.1i		−	1.17387i	−0.809633	−	0.586937i	$-0.800334\pi$
$743$		−	23774.1i		−	1.17387i	0.809633	−	0.586937i	$-0.199666\pi$
$744$		0			0
$745$		−	8230.71i		−	0.404765i
$746$		0			0
$747$	2268.00			0.111087
$748$		0			0
$749$	3486.00	+	4025.29i	0.170061	+	0.196369i
$750$		0			0
$751$		−	8850.78i		−	0.430053i	−0.976608	−	0.215026i	$-0.931016\pi$
$751$		−	8850.78i		−	0.430053i	0.976608	−	0.215026i	$-0.0689837\pi$
$752$		0			0
$753$	−8316.00			−0.402459
$754$		0			0
$755$	13392.0			0.645543
$756$		0			0
$757$	6394.00			0.306993			0.153497	−	0.988149i	$-0.450947\pi$
$757$	6394.00			0.306993			0.153497	+	0.988149i	$0.450947\pi$
$758$		0			0
$759$	1188.00			0.0568138
$760$		0			0
$761$			9803.41i			0.466982i	0.972359	+	0.233491i	$0.0750149\pi$
$761$			9803.41i			0.466982i	−0.972359	+	0.233491i	$0.924985\pi$
$762$		0			0
$763$	−13636.0	+	11809.1i	−0.646994	+	0.560313i
$764$		0			0
$765$	9504.00			0.449174
$766$		0			0
$767$			2826.71i			0.133072i
$768$		0			0
$769$		−	9865.76i		−	0.462638i	−0.972878	−	0.231319i	$-0.925696\pi$
$769$		−	9865.76i		−	0.462638i	0.972878	−	0.231319i	$-0.0743041\pi$
$770$		0			0
$771$		−	10413.1i		−	0.486405i
$772$		0			0
$773$			12179.8i			0.566722i	0.959013	+	0.283361i	$0.0914495\pi$
$773$			12179.8i			0.566722i	−0.959013	+	0.283361i	$0.908551\pi$
$774$		0			0
$775$	−7772.00			−0.360230
$776$		0			0
$777$	13188.0	−	11421.1i	0.608902	−	0.527325i
$778$		0			0
$779$			14050.4i			0.646223i
$780$		0			0
$781$	−2820.00			−0.129203
$782$		0			0
$783$	6642.00			0.303149
$784$		0			0
$785$	−35328.0			−1.60626
$786$		0			0
$787$	2020.00			0.0914933			0.0457466	−	0.998953i	$-0.485433\pi$
$787$	2020.00			0.0914933			0.0457466	+	0.998953i	$0.485433\pi$
$788$		0			0
$789$		−	16575.7i		−	0.747923i
$790$		0			0
$791$	−3444.00	+	2982.59i	−0.154810	+	0.134069i
$792$		0			0
$793$	−8064.00			−0.361111
$794$		0			0
$795$			6235.38i			0.278171i
$796$		0			0
$797$			34377.7i			1.52788i	0.645286	+	0.763941i	$0.276738\pi$
$797$			34377.7i			1.52788i	−0.645286	+	0.763941i	$0.723262\pi$
$798$		0			0
$799$			14632.4i			0.647880i
$800$		0			0
$801$			1932.97i			0.0852660i
$802$		0			0
$803$	−432.000			−0.0189850
$804$		0			0
$805$	−22176.0	+	19205.0i	−0.970933	+	0.840853i
$806$		0			0
$807$			11473.1i			0.500461i
$808$		0			0
$809$	27162.0			1.18043			0.590213	−	0.807247i	$-0.299044\pi$
$809$	27162.0			1.18043			0.590213	+	0.807247i	$0.299044\pi$
$810$		0			0
$811$	18124.0			0.784735			0.392367	−	0.919809i	$-0.371656\pi$
$811$	18124.0			0.784735			0.392367	+	0.919809i	$0.371656\pi$
$812$		0			0
$813$	−19164.0			−0.826705
$814$		0			0
$815$	−29040.0			−1.24813
$816$		0			0
$817$		−	19634.5i		−	0.840790i
$818$		0			0
$819$	1512.00	+	1745.91i	0.0645098	+	0.0744895i
$820$		0			0
$821$	−44010.0			−1.87084			−0.935420	−	0.353539i	$-0.884978\pi$
$821$	−44010.0			−1.87084			−0.935420	+	0.353539i	$0.884978\pi$
$822$		0			0
$823$		−	578.505i		−	0.0245023i	−0.999925	−	0.0122512i	$-0.996100\pi$
$823$		−	578.505i		−	0.0245023i	0.999925	−	0.0122512i	$-0.00389976\pi$
$824$		0			0
$825$			696.284i			0.0293837i
$826$		0			0
$827$		−	11241.0i		−	0.472658i	−0.971673	−	0.236329i	$-0.924056\pi$
$827$		−	11241.0i		−	0.472658i	0.971673	−	0.236329i	$-0.0759443\pi$
$828$		0			0
$829$		−	37772.6i		−	1.58250i	−0.611491	−	0.791252i	$-0.709430\pi$
$829$		−	37772.6i		−	1.58250i	0.611491	−	0.791252i	$-0.290570\pi$
$830$		0			0
$831$	10806.0			0.451090
$832$		0			0
$833$	−25872.0	−	3734.30i	−1.07612	−	0.155325i
$834$		0			0
$835$		−	38243.7i		−	1.58500i
$836$		0			0
$837$	3132.00			0.129340
$838$		0			0
$839$	−19392.0			−0.797957			−0.398979	−	0.916960i	$-0.630635\pi$
$839$	−19392.0			−0.797957			−0.398979	+	0.916960i	$0.630635\pi$
$840$		0			0
$841$	36127.0			1.48128
$842$		0			0
$843$	774.000			0.0316227
$844$		0			0
$845$			27782.1i			1.13105i
$846$		0			0
$847$	18466.0	−	15992.0i	0.749114	−	0.648751i
$848$		0			0
$849$	−12684.0			−0.512737
$850$		0			0
$851$			35895.0i			1.44591i
$852$		0			0
$853$		−	29015.3i		−	1.16467i	−0.812948	−	0.582336i	$-0.802139\pi$
$853$		−	29015.3i		−	1.16467i	0.812948	−	0.582336i	$-0.197861\pi$
$854$		0			0
$855$		−	6484.80i		−	0.259386i
$856$		0			0
$857$		−	3304.75i		−	0.131725i	−0.997829	−	0.0658624i	$-0.979020\pi$
$857$		−	3304.75i		−	0.131725i	0.997829	−	0.0658624i	$-0.0209799\pi$
$858$		0			0
$859$	−31516.0			−1.25182			−0.625909	−	0.779896i	$-0.715272\pi$
$859$	−31516.0			−1.25182			−0.625909	+	0.779896i	$0.715272\pi$
$860$		0			0
$861$	−9828.00	−	11348.4i	−0.389010	−	0.449190i
$862$		0			0
$863$		−	12730.6i		−	0.502148i	−0.967968	−	0.251074i	$-0.919216\pi$
$863$		−	12730.6i		−	0.502148i	0.967968	−	0.251074i	$-0.0807838\pi$
$864$		0			0
$865$	−4608.00			−0.181129
$866$		0			0
$867$	−2685.00			−0.105176
$868$		0			0
$869$	4764.00			0.185970
$870$		0			0
$871$	7056.00			0.274493
$872$		0			0
$873$			12969.6i			0.502811i
$874$		0			0
$875$	9744.00	+	11251.4i	0.376466	+	0.434705i
$876$		0			0
$877$	40714.0			1.56763			0.783816	−	0.620993i	$-0.213270\pi$
$877$	40714.0			1.56763			0.783816	+	0.620993i	$0.213270\pi$
$878$		0			0
$879$			23819.2i			0.913994i
$880$		0			0
$881$			26098.5i			0.998050i	0.866588	+	0.499025i	$0.166308\pi$
$881$			26098.5i			0.998050i	−0.866588	+	0.499025i	$0.833692\pi$
$882$		0			0
$883$		−	8753.78i		−	0.333622i	−0.985989	−	0.166811i	$-0.946653\pi$
$883$		−	8753.78i		−	0.333622i	0.985989	−	0.166811i	$-0.0533470\pi$
$884$		0			0
$885$			8480.12i			0.322097i
$886$		0			0
$887$	−9384.00			−0.355224			−0.177612	−	0.984101i	$-0.556837\pi$
$887$	−9384.00			−0.355224			−0.177612	+	0.984101i	$0.556837\pi$
$888$		0			0
$889$	22386.0	+	25849.1i	0.844547	+	0.975199i
$890$		0			0
$891$		−	280.592i		−	0.0105502i
$892$		0			0
$893$	9984.00			0.374134
$894$		0			0
$895$	−8400.00			−0.313722
$896$		0			0
$897$	−4752.00			−0.176884
$898$		0			0
$899$	28536.0			1.05865
$900$		0			0
$901$		−	11431.5i		−	0.422686i
$902$		0			0
$903$	13734.0	+	15858.7i	0.506134	+	0.584433i
$904$		0			0
$905$	52608.0			1.93232
$906$		0			0
$907$		−	28720.9i		−	1.05145i	−0.850656	−	0.525723i	$-0.823795\pi$
$907$		−	28720.9i		−	1.05145i	0.850656	−	0.525723i	$-0.176205\pi$
$908$		0			0
$909$		−	1995.32i		−	0.0728060i
$910$		0			0
$911$		−	35884.6i		−	1.30506i	−0.757762	−	0.652531i	$-0.773707\pi$
$911$		−	35884.6i		−	1.30506i	0.757762	−	0.652531i	$-0.226293\pi$
$912$		0			0
$913$		−	872.954i		−	0.0316435i
$914$		0			0
$915$	−24192.0			−0.874058
$916$		0			0
$917$	38808.0	−	33608.7i	1.39755	−	1.21031i
$918$		0			0
$919$			23559.4i			0.845649i	0.906212	+	0.422824i	$0.138961\pi$
$919$			23559.4i			0.845649i	−0.906212	+	0.422824i	$0.861039\pi$
$920$		0			0
$921$	15324.0			0.548255
$922$		0			0
$923$	11280.0			0.402260
$924$		0			0
$925$	−21038.0			−0.747811
$926$		0			0
$927$	−4572.00			−0.161989
$928$		0			0
$929$			24518.9i			0.865920i	0.901413	+	0.432960i	$0.142531\pi$
$929$			24518.9i			0.865920i	−0.901413	+	0.432960i	$0.857469\pi$
$930$		0			0
$931$	−2548.00	+	17653.1i	−0.0896964	+	0.621435i
$932$		0			0
$933$	8784.00			0.308226
$934$		0			0
$935$		−	3658.09i		−	0.127949i
$936$		0			0
$937$			24040.9i			0.838187i	0.907943	+	0.419093i	$0.137652\pi$
$937$			24040.9i			0.838187i	−0.907943	+	0.419093i	$0.862348\pi$
$938$		0			0
$939$			5694.98i			0.197922i
$940$		0			0
$941$			235.559i			0.00816047i	0.999992	+	0.00408023i	$0.00129878\pi$
$941$			235.559i			0.00816047i	−0.999992	+	0.00408023i	$0.998701\pi$
$942$		0			0
$943$	30888.0			1.06665
$944$		0			0
$945$	4536.00	+	5237.72i	0.156144	+	0.180300i
$946$		0			0
$947$		−	41240.1i		−	1.41513i	−0.706650	−	0.707563i	$-0.749795\pi$
$947$		−	41240.1i		−	1.41513i	0.706650	−	0.707563i	$-0.250205\pi$
$948$		0			0
$949$	1728.00			0.0591077
$950$		0			0
$951$	10314.0			0.351687
$952$		0			0
$953$	28962.0			0.984440			0.492220	−	0.870471i	$-0.336186\pi$
$953$	28962.0			0.984440			0.492220	+	0.870471i	$0.336186\pi$
$954$		0			0
$955$	67728.0			2.29490
$956$		0			0
$957$		−	2556.51i		−	0.0863533i
$958$		0			0
$959$	33852.0	−	29316.7i	1.13987	−	0.987159i
$960$		0			0
$961$	−16335.0			−0.548320
$962$		0			0
$963$			2587.68i			0.0865908i
$964$		0			0
$965$			3796.66i			0.126651i
$966$		0			0
$967$		−	12855.3i		−	0.427506i	−0.976888	−	0.213753i	$-0.931431\pi$
$967$		−	12855.3i		−	0.427506i	0.976888	−	0.213753i	$-0.0685687\pi$
$968$		0			0
$969$			11888.8i			0.394142i
$970$		0			0
$971$	−47556.0			−1.57172			−0.785862	−	0.618401i	$-0.787781\pi$
$971$	−47556.0			−1.57172			−0.785862	+	0.618401i	$0.787781\pi$
$972$		0			0
$973$	−23464.0	+	20320.4i	−0.773095	+	0.669520i
$974$		0			0
$975$		−	2785.14i		−	0.0914829i
$976$		0			0
$977$	4434.00			0.145196			0.0725979	−	0.997361i	$-0.476871\pi$
$977$	4434.00			0.145196			0.0725979	+	0.997361i	$0.476871\pi$
$978$		0			0
$979$	744.000			0.0242884
$980$		0			0
$981$	−8766.00			−0.285297
$982$		0			0
$983$	−45816.0			−1.48658			−0.743288	−	0.668972i	$-0.766735\pi$
$983$	−45816.0			−1.48658			−0.743288	+	0.668972i	$0.766735\pi$
$984$		0			0
$985$			27685.1i			0.895554i
$986$		0			0
$987$	−8064.00	+	6983.63i	−0.260061	+	0.225219i
$988$		0			0
$989$	−43164.0			−1.38780
$990$		0			0
$991$			10381.9i			0.332787i	0.986059	+	0.166394i	$0.0532123\pi$
$991$			10381.9i			0.332787i	−0.986059	+	0.166394i	$0.946788\pi$
$992$		0			0
$993$		−	3439.85i		−	0.109930i
$994$		0			0
$995$			68561.5i			2.18447i
$996$		0			0
$997$			5819.69i			0.184866i	0.995719	+	0.0924330i	$0.0294644\pi$
$997$			5819.69i			0.184866i	−0.995719	+	0.0924330i	$0.970536\pi$
$998$		0			0
$999$	8478.00			0.268501

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.b.d.895.2		2
4.3	odd	2		1344.4.b.a.895.2		2
7.6	odd	2		1344.4.b.a.895.1		2
8.3	odd	2		336.4.b.c.223.1	yes	2
8.5	even	2		336.4.b.b.223.1	✓	2
24.5	odd	2		1008.4.b.e.559.2		2
24.11	even	2		1008.4.b.b.559.2		2
28.27	even	2	inner	1344.4.b.d.895.1		2
56.13	odd	2		336.4.b.c.223.2	yes	2
56.27	even	2		336.4.b.b.223.2	yes	2
168.83	odd	2		1008.4.b.e.559.1		2
168.125	even	2		1008.4.b.b.559.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.4.b.b.223.1	✓	2	8.5	even	2
336.4.b.b.223.2	yes	2	56.27	even	2
336.4.b.c.223.1	yes	2	8.3	odd	2
336.4.b.c.223.2	yes	2	56.13	odd	2
1008.4.b.b.559.1		2	168.125	even	2
1008.4.b.b.559.2		2	24.11	even	2
1008.4.b.e.559.1		2	168.83	odd	2
1008.4.b.e.559.2		2	24.5	odd	2
1344.4.b.a.895.1		2	7.6	odd	2
1344.4.b.a.895.2		2	4.3	odd	2
1344.4.b.d.895.1		2	28.27	even	2	inner
1344.4.b.d.895.2		2	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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +13.8564i q^{5} +(14.0000 - 12.1244i) q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +13.8564i q^{5} +(14.0000 - 12.1244i) q^{7} +9.00000 q^{9} -3.46410i q^{11} +13.8564i q^{13} +41.5692i q^{15} -76.2102i q^{17} -52.0000 q^{19} +(42.0000 - 36.3731i) q^{21} +114.315i q^{23} -67.0000 q^{25} +27.0000 q^{27} +246.000 q^{29} +116.000 q^{31} -10.3923i q^{33} +(168.000 + 193.990i) q^{35} +314.000 q^{37} +41.5692i q^{39} -270.200i q^{41} +377.587i q^{43} +124.708i q^{45} -192.000 q^{47} +(49.0000 - 339.482i) q^{49} -228.631i q^{51} +150.000 q^{53} +48.0000 q^{55} -156.000 q^{57} +204.000 q^{59} +581.969i q^{61} +(126.000 - 109.119i) q^{63} -192.000 q^{65} -509.223i q^{67} +342.946i q^{69} -814.064i q^{71} -124.708i q^{73} -201.000 q^{75} +(-42.0000 - 48.4974i) q^{77} +1375.25i q^{79} +81.0000 q^{81} +252.000 q^{83} +1056.00 q^{85} +738.000 q^{87} +214.774i q^{89} +(168.000 + 193.990i) q^{91} +348.000 q^{93} -720.533i q^{95} +1441.07i q^{97} -31.1769i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 28 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 + 28 * q^7 + 18 * q^9	\( 2 q + 6 q^{3} + 28 q^{7} + 18 q^{9} - 104 q^{19} + 84 q^{21} - 134 q^{25} + 54 q^{27} + 492 q^{29} + 232 q^{31} + 336 q^{35} + 628 q^{37} - 384 q^{47} + 98 q^{49} + 300 q^{53} + 96 q^{55} - 312 q^{57} + 408 q^{59} + 252 q^{63} - 384 q^{65} - 402 q^{75} - 84 q^{77} + 162 q^{81} + 504 q^{83} + 2112 q^{85} + 1476 q^{87} + 336 q^{91} + 696 q^{93}+O(q^{100}) \) 2 * q + 6 * q^3 + 28 * q^7 + 18 * q^9 - 104 * q^19 + 84 * q^21 - 134 * q^25 + 54 * q^27 + 492 * q^29 + 232 * q^31 + 336 * q^35 + 628 * q^37 - 384 * q^47 + 98 * q^49 + 300 * q^53 + 96 * q^55 - 312 * q^57 + 408 * q^59 + 252 * q^63 - 384 * q^65 - 402 * q^75 - 84 * q^77 + 162 * q^81 + 504 * q^83 + 2112 * q^85 + 1476 * q^87 + 336 * q^91 + 696 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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