LMFDB - Embedded newform 1100.4.b.c.749.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,4,Mod(749,1100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1100.749");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1100.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:	$64.9021010063$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 44)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			749.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1100.749
Dual form			1100.4.b.c.749.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000i q^{3} +26.0000i q^{7} +2.00000 q^{9} +O(q^{10})$	$q-5.00000i q^{3} +26.0000i q^{7} +2.00000 q^{9} -11.0000 q^{11} +52.0000i q^{13} -46.0000i q^{17} +96.0000 q^{19} +130.000 q^{21} +27.0000i q^{23} -145.000i q^{27} -16.0000 q^{29} -293.000 q^{31} +55.0000i q^{33} +29.0000i q^{37} +260.000 q^{39} -472.000 q^{41} -110.000i q^{43} +224.000i q^{47} -333.000 q^{49} -230.000 q^{51} +754.000i q^{53} -480.000i q^{57} -825.000 q^{59} -548.000 q^{61} +52.0000i q^{63} +123.000i q^{67} +135.000 q^{69} +1001.00 q^{71} -1020.00i q^{73} -286.000i q^{77} -526.000 q^{79} -671.000 q^{81} -158.000i q^{83} +80.0000i q^{87} +1217.00 q^{89} -1352.00 q^{91} +1465.00i q^{93} +263.000i q^{97} -22.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9}+O(q^{10}) $ 2 * q + 4 * q^9	$ 2 q + 4 q^{9} - 22 q^{11} + 192 q^{19} + 260 q^{21} - 32 q^{29} - 586 q^{31} + 520 q^{39} - 944 q^{41} - 666 q^{49} - 460 q^{51} - 1650 q^{59} - 1096 q^{61} + 270 q^{69} + 2002 q^{71} - 1052 q^{79} - 1342 q^{81} + 2434 q^{89} - 2704 q^{91} - 44 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 - 22 * q^11 + 192 * q^19 + 260 * q^21 - 32 * q^29 - 586 * q^31 + 520 * q^39 - 944 * q^41 - 666 * q^49 - 460 * q^51 - 1650 * q^59 - 1096 * q^61 + 270 * q^69 + 2002 * q^71 - 1052 * q^79 - 1342 * q^81 + 2434 * q^89 - 2704 * q^91 - 44 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$551$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 5.00000i	− 0.962250i	−0.876652	−	0.481125i	$-0.840228\pi$
$3$	− 5.00000i	− 0.962250i	0.876652	−	0.481125i	$-0.159772\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	26.0000i	1.40387i	0.712242	+	0.701934i	$0.247680\pi$
$7$	26.0000i	1.40387i	−0.712242	+	0.701934i	$0.752320\pi$
$8$	0	0
$9$	2.00000	0.0740741
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	52.0000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	52.0000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 46.0000i	− 0.656273i	−0.944630	−	0.328136i	$-0.893579\pi$
$17$	− 46.0000i	− 0.656273i	0.944630	−	0.328136i	$-0.106421\pi$
$18$	0	0
$19$	96.0000	1.15915	0.579577	−	0.814918i	$-0.303218\pi$
$19$	96.0000	1.15915	0.579577	+	0.814918i	$0.303218\pi$
$20$	0	0
$21$	130.000	1.35087
$22$	0	0
$23$	27.0000i	0.244778i	0.992482	+	0.122389i	$0.0390555\pi$
$23$	27.0000i	0.244778i	−0.992482	+	0.122389i	$0.960944\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 145.000i	− 1.03353i
$28$	0	0
$29$	−16.0000	−0.102453	−0.0512263	−	0.998687i	$-0.516313\pi$
$29$	−16.0000	−0.102453	−0.0512263	+	0.998687i	$0.516313\pi$
$30$	0	0
$31$	−293.000	−1.69756	−0.848780	−	0.528746i	$-0.822662\pi$
$31$	−293.000	−1.69756	−0.848780	+	0.528746i	$0.822662\pi$
$32$	0	0
$33$	55.0000i	0.290129i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	29.0000i	0.128853i	0.997922	+	0.0644266i	$0.0205218\pi$
$37$	29.0000i	0.128853i	−0.997922	+	0.0644266i	$0.979478\pi$
$38$	0	0
$39$	260.000	1.06752
$40$	0	0
$41$	−472.000	−1.79790	−0.898951	−	0.438048i	$-0.855670\pi$
$41$	−472.000	−1.79790	−0.898951	+	0.438048i	$0.855670\pi$
$42$	0	0
$43$	− 110.000i	− 0.390113i	−0.980792	−	0.195056i	$-0.937511\pi$
$43$	− 110.000i	− 0.390113i	0.980792	−	0.195056i	$-0.0624890\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	224.000i	0.695186i	0.937645	+	0.347593i	$0.113001\pi$
$47$	224.000i	0.695186i	−0.937645	+	0.347593i	$0.886999\pi$
$48$	0	0
$49$	−333.000	−0.970845
$50$	0	0
$51$	−230.000	−0.631499
$52$	0	0
$53$	754.000i	1.95415i	0.212899	+	0.977074i	$0.431709\pi$
$53$	754.000i	1.95415i	−0.212899	+	0.977074i	$0.568291\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 480.000i	− 1.11540i
$58$	0	0
$59$	−825.000	−1.82044	−0.910219	−	0.414127i	$-0.864087\pi$
$59$	−825.000	−1.82044	−0.910219	+	0.414127i	$0.864087\pi$
$60$	0	0
$61$	−548.000	−1.15023	−0.575116	−	0.818072i	$-0.695043\pi$
$61$	−548.000	−1.15023	−0.575116	+	0.818072i	$0.695043\pi$
$62$	0	0
$63$	52.0000i	0.103990i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	123.000i	0.224281i	0.993692	+	0.112141i	$0.0357707\pi$
$67$	123.000i	0.224281i	−0.993692	+	0.112141i	$0.964229\pi$
$68$	0	0
$69$	135.000	0.235538
$70$	0	0
$71$	1001.00	1.67319	0.836597	−	0.547818i	$-0.184541\pi$
$71$	1001.00	1.67319	0.836597	+	0.547818i	$0.184541\pi$
$72$	0	0
$73$	− 1020.00i	− 1.63537i	−0.575666	−	0.817685i	$-0.695257\pi$
$73$	− 1020.00i	− 1.63537i	0.575666	−	0.817685i	$-0.304743\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 286.000i	− 0.423282i
$78$	0	0
$79$	−526.000	−0.749109	−0.374555	−	0.927205i	$-0.622204\pi$
$79$	−526.000	−0.749109	−0.374555	+	0.927205i	$0.622204\pi$
$80$	0	0
$81$	−671.000	−0.920439
$82$	0	0
$83$	− 158.000i	− 0.208949i	−0.994528	−	0.104474i	$-0.966684\pi$
$83$	− 158.000i	− 0.208949i	0.994528	−	0.104474i	$-0.0333160\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	80.0000i	0.0985851i
$88$	0	0
$89$	1217.00	1.44946	0.724729	−	0.689034i	$-0.241965\pi$
$89$	1217.00	1.44946	0.724729	+	0.689034i	$0.241965\pi$
$90$	0	0
$91$	−1352.00	−1.55745
$92$	0	0
$93$	1465.00i	1.63348i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	263.000i	0.275295i	0.990481	+	0.137647i	$0.0439541\pi$
$97$	263.000i	0.275295i	−0.990481	+	0.137647i	$0.956046\pi$
$98$	0	0
$99$	−22.0000	−0.0223342
$100$	0	0
$101$	−814.000	−0.801941	−0.400970	−	0.916091i	$-0.631327\pi$
$101$	−814.000	−0.801941	−0.400970	+	0.916091i	$0.631327\pi$
$102$	0	0
$103$	376.000i	0.359693i	0.983695	+	0.179847i	$0.0575601\pi$
$103$	376.000i	0.359693i	−0.983695	+	0.179847i	$0.942440\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1494.00i	1.34982i	0.737901	+	0.674909i	$0.235817\pi$
$107$	1494.00i	1.34982i	−0.737901	+	0.674909i	$0.764183\pi$
$108$	0	0
$109$	−842.000	−0.739899	−0.369949	−	0.929052i	$-0.620625\pi$
$109$	−842.000	−0.739899	−0.369949	+	0.929052i	$0.620625\pi$
$110$	0	0
$111$	145.000	0.123989
$112$	0	0
$113$	1281.00i	1.06643i	0.845980	+	0.533214i	$0.179016\pi$
$113$	1281.00i	1.06643i	−0.845980	+	0.533214i	$0.820984\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	104.000i	0.0821778i
$118$	0	0
$119$	1196.00	0.921321
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	2360.00i	1.73003i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1736.00i	− 1.21295i	−0.795101	−	0.606477i	$-0.792582\pi$
$127$	− 1736.00i	− 1.21295i	0.795101	−	0.606477i	$-0.207418\pi$
$128$	0	0
$129$	−550.000	−0.375386
$130$	0	0
$131$	6.00000	0.00400170	0.00200085	−	0.999998i	$-0.499363\pi$
$131$	6.00000	0.00400170	0.00200085	+	0.999998i	$0.499363\pi$
$132$	0	0
$133$	2496.00i	1.62730i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1439.00i	0.897387i	0.893686	+	0.448694i	$0.148111\pi$
$137$	1439.00i	0.897387i	−0.893686	+	0.448694i	$0.851889\pi$
$138$	0	0
$139$	318.000	0.194046	0.0970231	−	0.995282i	$-0.469068\pi$
$139$	318.000	0.194046	0.0970231	+	0.995282i	$0.469068\pi$
$140$	0	0
$141$	1120.00	0.668943
$142$	0	0
$143$	− 572.000i	− 0.334497i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1665.00i	0.934196i
$148$	0	0
$149$	922.000	0.506934	0.253467	−	0.967344i	$-0.418429\pi$
$149$	922.000	0.506934	0.253467	+	0.967344i	$0.418429\pi$
$150$	0	0
$151$	−1030.00	−0.555101	−0.277550	−	0.960711i	$-0.589523\pi$
$151$	−1030.00	−0.555101	−0.277550	+	0.960711i	$0.589523\pi$
$152$	0	0
$153$	− 92.0000i	− 0.0486128i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1017.00i	− 0.516977i	−0.966014	−	0.258489i	$-0.916776\pi$
$157$	− 1017.00i	− 0.516977i	0.966014	−	0.258489i	$-0.0832245\pi$
$158$	0	0
$159$	3770.00	1.88038
$160$	0	0
$161$	−702.000	−0.343636
$162$	0	0
$163$	− 2444.00i	− 1.17441i	−0.809438	−	0.587205i	$-0.800228\pi$
$163$	− 2444.00i	− 1.17441i	0.809438	−	0.587205i	$-0.199772\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1452.00i	0.672809i	0.941718	+	0.336405i	$0.109211\pi$
$167$	1452.00i	0.672809i	−0.941718	+	0.336405i	$0.890789\pi$
$168$	0	0
$169$	−507.000	−0.230769
$170$	0	0
$171$	192.000	0.0858632
$172$	0	0
$173$	1914.00i	0.841149i	0.907258	+	0.420574i	$0.138171\pi$
$173$	1914.00i	0.841149i	−0.907258	+	0.420574i	$0.861829\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4125.00i	1.75172i
$178$	0	0
$179$	−1293.00	−0.539907	−0.269954	−	0.962873i	$-0.587008\pi$
$179$	−1293.00	−0.539907	−0.269954	+	0.962873i	$0.587008\pi$
$180$	0	0
$181$	455.000	0.186850	0.0934251	−	0.995626i	$-0.470218\pi$
$181$	455.000	0.186850	0.0934251	+	0.995626i	$0.470218\pi$
$182$	0	0
$183$	2740.00i	1.10681i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	506.000i	0.197874i
$188$	0	0
$189$	3770.00	1.45094
$190$	0	0
$191$	−1115.00	−0.422401	−0.211200	−	0.977443i	$-0.567737\pi$
$191$	−1115.00	−0.422401	−0.211200	+	0.977443i	$0.567737\pi$
$192$	0	0
$193$	5012.00i	1.86928i	0.355591	+	0.934642i	$0.384280\pi$
$193$	5012.00i	1.86928i	−0.355591	+	0.934642i	$0.615720\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4146.00i	1.49944i	0.661753	+	0.749721i	$0.269813\pi$
$197$	4146.00i	1.49944i	−0.661753	+	0.749721i	$0.730187\pi$
$198$	0	0
$199$	1240.00	0.441715	0.220857	−	0.975306i	$-0.429114\pi$
$199$	1240.00	0.441715	0.220857	+	0.975306i	$0.429114\pi$
$200$	0	0
$201$	615.000	0.215815
$202$	0	0
$203$	− 416.000i	− 0.143830i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	54.0000i	0.0181317i
$208$	0	0
$209$	−1056.00	−0.349498
$210$	0	0
$211$	−2820.00	−0.920080	−0.460040	−	0.887898i	$-0.652165\pi$
$211$	−2820.00	−0.920080	−0.460040	+	0.887898i	$0.652165\pi$
$212$	0	0
$213$	− 5005.00i	− 1.61003i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 7618.00i	− 2.38315i
$218$	0	0
$219$	−5100.00	−1.57363
$220$	0	0
$221$	2392.00	0.728069
$222$	0	0
$223$	3695.00i	1.10958i	0.831992	+	0.554788i	$0.187201\pi$
$223$	3695.00i	1.10958i	−0.831992	+	0.554788i	$0.812799\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	486.000i	0.142101i	0.997473	+	0.0710506i	$0.0226352\pi$
$227$	486.000i	0.142101i	−0.997473	+	0.0710506i	$0.977365\pi$
$228$	0	0
$229$	−4231.00	−1.22093	−0.610464	−	0.792044i	$-0.709017\pi$
$229$	−4231.00	−1.22093	−0.610464	+	0.792044i	$0.709017\pi$
$230$	0	0
$231$	−1430.00	−0.407303
$232$	0	0
$233$	− 3336.00i	− 0.937977i	−0.883204	−	0.468988i	$-0.844619\pi$
$233$	− 3336.00i	− 0.937977i	0.883204	−	0.468988i	$-0.155381\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2630.00i	0.720831i
$238$	0	0
$239$	−3610.00	−0.977036	−0.488518	−	0.872554i	$-0.662462\pi$
$239$	−3610.00	−0.977036	−0.488518	+	0.872554i	$0.662462\pi$
$240$	0	0
$241$	−6408.00	−1.71276	−0.856381	−	0.516345i	$-0.827292\pi$
$241$	−6408.00	−1.71276	−0.856381	+	0.516345i	$0.827292\pi$
$242$	0	0
$243$	− 560.000i	− 0.147835i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4992.00i	1.28596i
$248$	0	0
$249$	−790.000	−0.201061
$250$	0	0
$251$	−1787.00	−0.449380	−0.224690	−	0.974430i	$-0.572137\pi$
$251$	−1787.00	−0.449380	−0.224690	+	0.974430i	$0.572137\pi$
$252$	0	0
$253$	− 297.000i	− 0.0738033i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	354.000i	0.0859218i	0.999077	+	0.0429609i	$0.0136791\pi$
$257$	354.000i	0.0859218i	−0.999077	+	0.0429609i	$0.986321\pi$
$258$	0	0
$259$	−754.000	−0.180893
$260$	0	0
$261$	−32.0000	−0.00758908
$262$	0	0
$263$	− 2026.00i	− 0.475013i	−0.971386	−	0.237507i	$-0.923670\pi$
$263$	− 2026.00i	− 0.475013i	0.971386	−	0.237507i	$-0.0763302\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 6085.00i	− 1.39474i
$268$	0	0
$269$	8750.00	1.98326	0.991630	−	0.129112i	$-0.0412128\pi$
$269$	8750.00	1.98326	0.991630	+	0.129112i	$0.0412128\pi$
$270$	0	0
$271$	−5036.00	−1.12884	−0.564419	−	0.825488i	$-0.690900\pi$
$271$	−5036.00	−1.12884	−0.564419	+	0.825488i	$0.690900\pi$
$272$	0	0
$273$	6760.00i	1.49866i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4306.00i	0.934016i	0.884253	+	0.467008i	$0.154668\pi$
$277$	4306.00i	0.934016i	−0.884253	+	0.467008i	$0.845332\pi$
$278$	0	0
$279$	−586.000	−0.125745
$280$	0	0
$281$	−1202.00	−0.255179	−0.127590	−	0.991827i	$-0.540724\pi$
$281$	−1202.00	−0.255179	−0.127590	+	0.991827i	$0.540724\pi$
$282$	0	0
$283$	− 6620.00i	− 1.39052i	−0.718757	−	0.695262i	$-0.755288\pi$
$283$	− 6620.00i	− 1.39052i	0.718757	−	0.695262i	$-0.244712\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 12272.0i	− 2.52402i
$288$	0	0
$289$	2797.00	0.569306
$290$	0	0
$291$	1315.00	0.264903
$292$	0	0
$293$	6968.00i	1.38933i	0.719331	+	0.694667i	$0.244448\pi$
$293$	6968.00i	1.38933i	−0.719331	+	0.694667i	$0.755552\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1595.00i	0.311620i
$298$	0	0
$299$	−1404.00	−0.271557
$300$	0	0
$301$	2860.00	0.547667
$302$	0	0
$303$	4070.00i	0.771668i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 7640.00i	− 1.42032i	−0.704041	−	0.710159i	$-0.748623\pi$
$307$	− 7640.00i	− 1.42032i	0.704041	−	0.710159i	$-0.251377\pi$
$308$	0	0
$309$	1880.00	0.346115
$310$	0	0
$311$	−652.000	−0.118880	−0.0594398	−	0.998232i	$-0.518931\pi$
$311$	−652.000	−0.118880	−0.0594398	+	0.998232i	$0.518931\pi$
$312$	0	0
$313$	8055.00i	1.45462i	0.686310	+	0.727309i	$0.259229\pi$
$313$	8055.00i	1.45462i	−0.686310	+	0.727309i	$0.740771\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5675.00i	1.00549i	0.864435	+	0.502744i	$0.167676\pi$
$317$	5675.00i	1.00549i	−0.864435	+	0.502744i	$0.832324\pi$
$318$	0	0
$319$	176.000	0.0308906
$320$	0	0
$321$	7470.00	1.29886
$322$	0	0
$323$	− 4416.00i	− 0.760721i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4210.00i	0.711968i
$328$	0	0
$329$	−5824.00	−0.975950
$330$	0	0
$331$	−2245.00	−0.372799	−0.186399	−	0.982474i	$-0.559682\pi$
$331$	−2245.00	−0.372799	−0.186399	+	0.982474i	$0.559682\pi$
$332$	0	0
$333$	58.0000i	0.00954469i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	694.000i	0.112180i	0.998426	+	0.0560899i	$0.0178633\pi$
$337$	694.000i	0.112180i	−0.998426	+	0.0560899i	$0.982137\pi$
$338$	0	0
$339$	6405.00	1.02617
$340$	0	0
$341$	3223.00	0.511834
$342$	0	0
$343$	260.000i	0.0409291i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4092.00i	− 0.633055i	−0.948583	−	0.316527i	$-0.897483\pi$
$347$	− 4092.00i	− 0.633055i	0.948583	−	0.316527i	$-0.102517\pi$
$348$	0	0
$349$	−334.000	−0.0512281	−0.0256141	−	0.999672i	$-0.508154\pi$
$349$	−334.000	−0.0512281	−0.0256141	+	0.999672i	$0.508154\pi$
$350$	0	0
$351$	7540.00	1.14660
$352$	0	0
$353$	891.000i	0.134343i	0.997741	+	0.0671716i	$0.0213975\pi$
$353$	891.000i	0.134343i	−0.997741	+	0.0671716i	$0.978603\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 5980.00i	− 0.886541i
$358$	0	0
$359$	−2476.00	−0.364006	−0.182003	−	0.983298i	$-0.558258\pi$
$359$	−2476.00	−0.364006	−0.182003	+	0.983298i	$0.558258\pi$
$360$	0	0
$361$	2357.00	0.343636
$362$	0	0
$363$	− 605.000i	− 0.0874773i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1379.00i	− 0.196140i	−0.995180	−	0.0980698i	$-0.968733\pi$
$367$	− 1379.00i	− 0.196140i	0.995180	−	0.0980698i	$-0.0312668\pi$
$368$	0	0
$369$	−944.000	−0.133178
$370$	0	0
$371$	−19604.0	−2.74337
$372$	0	0
$373$	− 6266.00i	− 0.869816i	−0.900475	−	0.434908i	$-0.856781\pi$
$373$	− 6266.00i	− 0.869816i	0.900475	−	0.434908i	$-0.143219\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 832.000i	− 0.113661i
$378$	0	0
$379$	−151.000	−0.0204653	−0.0102327	−	0.999948i	$-0.503257\pi$
$379$	−151.000	−0.0204653	−0.0102327	+	0.999948i	$0.503257\pi$
$380$	0	0
$381$	−8680.00	−1.16717
$382$	0	0
$383$	− 1989.00i	− 0.265361i	−0.991159	−	0.132680i	$-0.957642\pi$
$383$	− 1989.00i	− 0.265361i	0.991159	−	0.132680i	$-0.0423584\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 220.000i	− 0.0288972i
$388$	0	0
$389$	−6817.00	−0.888523	−0.444262	−	0.895897i	$-0.646534\pi$
$389$	−6817.00	−0.888523	−0.444262	+	0.895897i	$0.646534\pi$
$390$	0	0
$391$	1242.00	0.160641
$392$	0	0
$393$	− 30.0000i	− 0.00385064i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4170.00i	0.527170i	0.964636	+	0.263585i	$0.0849049\pi$
$397$	4170.00i	0.527170i	−0.964636	+	0.263585i	$0.915095\pi$
$398$	0	0
$399$	12480.0	1.56587
$400$	0	0
$401$	10914.0	1.35915	0.679575	−	0.733606i	$-0.262164\pi$
$401$	10914.0	1.35915	0.679575	+	0.733606i	$0.262164\pi$
$402$	0	0
$403$	− 15236.0i	− 1.88327i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 319.000i	− 0.0388507i
$408$	0	0
$409$	1102.00	0.133228	0.0666142	−	0.997779i	$-0.478780\pi$
$409$	1102.00	0.133228	0.0666142	+	0.997779i	$0.478780\pi$
$410$	0	0
$411$	7195.00	0.863511
$412$	0	0
$413$	− 21450.0i	− 2.55565i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1590.00i	− 0.186721i
$418$	0	0
$419$	−11028.0	−1.28581	−0.642903	−	0.765947i	$-0.722270\pi$
$419$	−11028.0	−1.28581	−0.642903	+	0.765947i	$0.722270\pi$
$420$	0	0
$421$	2622.00	0.303536	0.151768	−	0.988416i	$-0.451503\pi$
$421$	2622.00	0.303536	0.151768	+	0.988416i	$0.451503\pi$
$422$	0	0
$423$	448.000i	0.0514953i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 14248.0i	− 1.61478i
$428$	0	0
$429$	−2860.00	−0.321870
$430$	0	0
$431$	16598.0	1.85498	0.927491	−	0.373845i	$-0.121961\pi$
$431$	16598.0	1.85498	0.927491	+	0.373845i	$0.121961\pi$
$432$	0	0
$433$	− 5763.00i	− 0.639612i	−0.947483	−	0.319806i	$-0.896382\pi$
$433$	− 5763.00i	− 0.639612i	0.947483	−	0.319806i	$-0.103618\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2592.00i	0.283735i
$438$	0	0
$439$	3128.00	0.340071	0.170036	−	0.985438i	$-0.445612\pi$
$439$	3128.00	0.340071	0.170036	+	0.985438i	$0.445612\pi$
$440$	0	0
$441$	−666.000	−0.0719145
$442$	0	0
$443$	6369.00i	0.683071i	0.939869	+	0.341535i	$0.110947\pi$
$443$	6369.00i	0.683071i	−0.939869	+	0.341535i	$0.889053\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 4610.00i	− 0.487798i
$448$	0	0
$449$	−8691.00	−0.913483	−0.456741	−	0.889600i	$-0.650983\pi$
$449$	−8691.00	−0.913483	−0.456741	+	0.889600i	$0.650983\pi$
$450$	0	0
$451$	5192.00	0.542088
$452$	0	0
$453$	5150.00i	0.534146i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2260.00i	− 0.231331i	−0.993288	−	0.115666i	$-0.963100\pi$
$457$	− 2260.00i	− 0.231331i	0.993288	−	0.115666i	$-0.0369001\pi$
$458$	0	0
$459$	−6670.00	−0.678277
$460$	0	0
$461$	−12756.0	−1.28873	−0.644367	−	0.764717i	$-0.722879\pi$
$461$	−12756.0	−1.28873	−0.644367	+	0.764717i	$0.722879\pi$
$462$	0	0
$463$	− 6887.00i	− 0.691287i	−0.938366	−	0.345644i	$-0.887661\pi$
$463$	− 6887.00i	− 0.691287i	0.938366	−	0.345644i	$-0.112339\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1535.00i	0.152101i	0.997104	+	0.0760507i	$0.0242311\pi$
$467$	1535.00i	0.152101i	−0.997104	+	0.0760507i	$0.975769\pi$
$468$	0	0
$469$	−3198.00	−0.314861
$470$	0	0
$471$	−5085.00	−0.497462
$472$	0	0
$473$	1210.00i	0.117623i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1508.00i	0.144752i
$478$	0	0
$479$	17564.0	1.67541	0.837703	−	0.546126i	$-0.183898\pi$
$479$	17564.0	1.67541	0.837703	+	0.546126i	$0.183898\pi$
$480$	0	0
$481$	−1508.00	−0.142950
$482$	0	0
$483$	3510.00i	0.330664i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7541.00i	0.701674i	0.936437	+	0.350837i	$0.114103\pi$
$487$	7541.00i	0.701674i	−0.936437	+	0.350837i	$0.885897\pi$
$488$	0	0
$489$	−12220.0	−1.13008
$490$	0	0
$491$	12552.0	1.15369	0.576847	−	0.816852i	$-0.304283\pi$
$491$	12552.0	1.15369	0.576847	+	0.816852i	$0.304283\pi$
$492$	0	0
$493$	736.000i	0.0672369i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	26026.0i	2.34894i
$498$	0	0
$499$	8396.00	0.753220	0.376610	−	0.926372i	$-0.377090\pi$
$499$	8396.00	0.753220	0.376610	+	0.926372i	$0.377090\pi$
$500$	0	0
$501$	7260.00	0.647411
$502$	0	0
$503$	− 12194.0i	− 1.08092i	−0.841369	−	0.540461i	$-0.818250\pi$
$503$	− 12194.0i	− 1.08092i	0.841369	−	0.540461i	$-0.181750\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2535.00i	0.222058i
$508$	0	0
$509$	−18295.0	−1.59315	−0.796573	−	0.604542i	$-0.793356\pi$
$509$	−18295.0	−1.59315	−0.796573	+	0.604542i	$0.793356\pi$
$510$	0	0
$511$	26520.0	2.29584
$512$	0	0
$513$	− 13920.0i	− 1.19802i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2464.00i	− 0.209607i
$518$	0	0
$519$	9570.00	0.809396
$520$	0	0
$521$	7101.00	0.597122	0.298561	−	0.954391i	$-0.403493\pi$
$521$	7101.00	0.597122	0.298561	+	0.954391i	$0.403493\pi$
$522$	0	0
$523$	− 4912.00i	− 0.410682i	−0.978690	−	0.205341i	$-0.934170\pi$
$523$	− 4912.00i	− 0.410682i	0.978690	−	0.205341i	$-0.0658304\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13478.0i	1.11406i
$528$	0	0
$529$	11438.0	0.940084
$530$	0	0
$531$	−1650.00	−0.134847
$532$	0	0
$533$	− 24544.0i	− 1.99459i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6465.00i	0.519526i
$538$	0	0
$539$	3663.00	0.292721
$540$	0	0
$541$	11496.0	0.913589	0.456794	−	0.889572i	$-0.348997\pi$
$541$	11496.0	0.913589	0.456794	+	0.889572i	$0.348997\pi$
$542$	0	0
$543$	− 2275.00i	− 0.179797i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 19048.0i	− 1.48891i	−0.667673	−	0.744455i	$-0.732709\pi$
$547$	− 19048.0i	− 1.48891i	0.667673	−	0.744455i	$-0.267291\pi$
$548$	0	0
$549$	−1096.00	−0.0852024
$550$	0	0
$551$	−1536.00	−0.118758
$552$	0	0
$553$	− 13676.0i	− 1.05165i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 10718.0i	− 0.815325i	−0.913133	−	0.407663i	$-0.866344\pi$
$557$	− 10718.0i	− 0.815325i	0.913133	−	0.407663i	$-0.133656\pi$
$558$	0	0
$559$	5720.00	0.432791
$560$	0	0
$561$	2530.00	0.190404
$562$	0	0
$563$	6660.00i	0.498553i	0.968432	+	0.249277i	$0.0801929\pi$
$563$	6660.00i	0.498553i	−0.968432	+	0.249277i	$0.919807\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 17446.0i	− 1.29217i
$568$	0	0
$569$	10960.0	0.807499	0.403750	−	0.914870i	$-0.367707\pi$
$569$	10960.0	0.807499	0.403750	+	0.914870i	$0.367707\pi$
$570$	0	0
$571$	18596.0	1.36290	0.681452	−	0.731863i	$-0.261349\pi$
$571$	18596.0	1.36290	0.681452	+	0.731863i	$0.261349\pi$
$572$	0	0
$573$	5575.00i	0.406455i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	21119.0i	1.52374i	0.647733	+	0.761868i	$0.275717\pi$
$577$	21119.0i	1.52374i	−0.647733	+	0.761868i	$0.724283\pi$
$578$	0	0
$579$	25060.0	1.79872
$580$	0	0
$581$	4108.00	0.293337
$582$	0	0
$583$	− 8294.00i	− 0.589198i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2836.00i	0.199411i	0.995017	+	0.0997055i	$0.0317901\pi$
$587$	2836.00i	0.199411i	−0.995017	+	0.0997055i	$0.968210\pi$
$588$	0	0
$589$	−28128.0	−1.96773
$590$	0	0
$591$	20730.0	1.44284
$592$	0	0
$593$	18044.0i	1.24954i	0.780808	+	0.624771i	$0.214808\pi$
$593$	18044.0i	1.24954i	−0.780808	+	0.624771i	$0.785192\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 6200.00i	− 0.425040i
$598$	0	0
$599$	9264.00	0.631914	0.315957	−	0.948773i	$-0.397674\pi$
$599$	9264.00	0.631914	0.315957	+	0.948773i	$0.397674\pi$
$600$	0	0
$601$	−19326.0	−1.31169	−0.655844	−	0.754897i	$-0.727687\pi$
$601$	−19326.0	−1.31169	−0.655844	+	0.754897i	$0.727687\pi$
$602$	0	0
$603$	246.000i	0.0166134i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 10082.0i	− 0.674161i	−0.941476	−	0.337081i	$-0.890561\pi$
$607$	− 10082.0i	− 0.674161i	0.941476	−	0.337081i	$-0.109439\pi$
$608$	0	0
$609$	−2080.00	−0.138400
$610$	0	0
$611$	−11648.0	−0.771240
$612$	0	0
$613$	13088.0i	0.862348i	0.902269	+	0.431174i	$0.141900\pi$
$613$	13088.0i	0.862348i	−0.902269	+	0.431174i	$0.858100\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 23426.0i	− 1.52852i	−0.644910	−	0.764259i	$-0.723105\pi$
$617$	− 23426.0i	− 1.52852i	0.644910	−	0.764259i	$-0.276895\pi$
$618$	0	0
$619$	−23587.0	−1.53157	−0.765785	−	0.643097i	$-0.777649\pi$
$619$	−23587.0	−1.53157	−0.765785	+	0.643097i	$0.777649\pi$
$620$	0	0
$621$	3915.00	0.252985
$622$	0	0
$623$	31642.0i	2.03485i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	5280.00i	0.336304i
$628$	0	0
$629$	1334.00	0.0845629
$630$	0	0
$631$	19683.0	1.24179	0.620894	−	0.783895i	$-0.286770\pi$
$631$	19683.0	1.24179	0.620894	+	0.783895i	$0.286770\pi$
$632$	0	0
$633$	14100.0i	0.885347i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 17316.0i	− 1.07706i
$638$	0	0
$639$	2002.00	0.123940
$640$	0	0
$641$	375.000	0.0231070	0.0115535	−	0.999933i	$-0.496322\pi$
$641$	375.000	0.0231070	0.0115535	+	0.999933i	$0.496322\pi$
$642$	0	0
$643$	− 21055.0i	− 1.29133i	−0.763619	−	0.645667i	$-0.776579\pi$
$643$	− 21055.0i	− 1.29133i	0.763619	−	0.645667i	$-0.223421\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6427.00i	0.390528i	0.980751	+	0.195264i	$0.0625563\pi$
$647$	6427.00i	0.390528i	−0.980751	+	0.195264i	$0.937444\pi$
$648$	0	0
$649$	9075.00	0.548883
$650$	0	0
$651$	−38090.0	−2.29319
$652$	0	0
$653$	− 7617.00i	− 0.456472i	−0.973606	−	0.228236i	$-0.926704\pi$
$653$	− 7617.00i	− 0.456472i	0.973606	−	0.228236i	$-0.0732958\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2040.00i	− 0.121138i
$658$	0	0
$659$	17630.0	1.04214	0.521068	−	0.853515i	$-0.325534\pi$
$659$	17630.0	1.04214	0.521068	+	0.853515i	$0.325534\pi$
$660$	0	0
$661$	4605.00	0.270974	0.135487	−	0.990779i	$-0.456740\pi$
$661$	4605.00	0.270974	0.135487	+	0.990779i	$0.456740\pi$
$662$	0	0
$663$	− 11960.0i	− 0.700585i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 432.000i	− 0.0250781i
$668$	0	0
$669$	18475.0	1.06769
$670$	0	0
$671$	6028.00	0.346808
$672$	0	0
$673$	− 2818.00i	− 0.161406i	−0.996738	−	0.0807028i	$-0.974284\pi$
$673$	− 2818.00i	− 0.161406i	0.996738	−	0.0807028i	$-0.0257165\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 8438.00i	− 0.479023i	−0.970894	−	0.239512i	$-0.923013\pi$
$677$	− 8438.00i	− 0.479023i	0.970894	−	0.239512i	$-0.0769873\pi$
$678$	0	0
$679$	−6838.00	−0.386478
$680$	0	0
$681$	2430.00	0.136737
$682$	0	0
$683$	− 17344.0i	− 0.971669i	−0.874051	−	0.485834i	$-0.838516\pi$
$683$	− 17344.0i	− 0.971669i	0.874051	−	0.485834i	$-0.161484\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	21155.0i	1.17484i
$688$	0	0
$689$	−39208.0	−2.16793
$690$	0	0
$691$	−3947.00	−0.217295	−0.108648	−	0.994080i	$-0.534652\pi$
$691$	−3947.00	−0.217295	−0.108648	+	0.994080i	$0.534652\pi$
$692$	0	0
$693$	− 572.000i	− 0.0313542i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	21712.0i	1.17991i
$698$	0	0
$699$	−16680.0	−0.902569
$700$	0	0
$701$	−7998.00	−0.430928	−0.215464	−	0.976512i	$-0.569126\pi$
$701$	−7998.00	−0.430928	−0.215464	+	0.976512i	$0.569126\pi$
$702$	0	0
$703$	2784.00i	0.149361i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 21164.0i	− 1.12582i
$708$	0	0
$709$	881.000	0.0466666	0.0233333	−	0.999728i	$-0.492572\pi$
$709$	881.000	0.0466666	0.0233333	+	0.999728i	$0.492572\pi$
$710$	0	0
$711$	−1052.00	−0.0554896
$712$	0	0
$713$	− 7911.00i	− 0.415525i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	18050.0i	0.940153i
$718$	0	0
$719$	26093.0	1.35341	0.676707	−	0.736252i	$-0.263406\pi$
$719$	26093.0	1.35341	0.676707	+	0.736252i	$0.263406\pi$
$720$	0	0
$721$	−9776.00	−0.504962
$722$	0	0
$723$	32040.0i	1.64811i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	7481.00i	0.381644i	0.981625	+	0.190822i	$0.0611153\pi$
$727$	7481.00i	0.381644i	−0.981625	+	0.190822i	$0.938885\pi$
$728$	0	0
$729$	−20917.0	−1.06269
$730$	0	0
$731$	−5060.00	−0.256020
$732$	0	0
$733$	17788.0i	0.896337i	0.893949	+	0.448168i	$0.147923\pi$
$733$	17788.0i	0.896337i	−0.893949	+	0.448168i	$0.852077\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1353.00i	− 0.0676233i
$738$	0	0
$739$	32182.0	1.60194	0.800970	−	0.598704i	$-0.204317\pi$
$739$	32182.0	1.60194	0.800970	+	0.598704i	$0.204317\pi$
$740$	0	0
$741$	24960.0	1.23742
$742$	0	0
$743$	− 32044.0i	− 1.58221i	−0.611682	−	0.791104i	$-0.709507\pi$
$743$	− 32044.0i	− 1.58221i	0.611682	−	0.791104i	$-0.290493\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 316.000i	− 0.0154777i
$748$	0	0
$749$	−38844.0	−1.89497
$750$	0	0
$751$	−33779.0	−1.64130	−0.820648	−	0.571434i	$-0.806387\pi$
$751$	−33779.0	−1.64130	−0.820648	+	0.571434i	$0.806387\pi$
$752$	0	0
$753$	8935.00i	0.432416i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	15630.0i	0.750439i	0.926936	+	0.375219i	$0.122433\pi$
$757$	15630.0i	0.750439i	−0.926936	+	0.375219i	$0.877567\pi$
$758$	0	0
$759$	−1485.00	−0.0710172
$760$	0	0
$761$	1948.00	0.0927923	0.0463962	−	0.998923i	$-0.485226\pi$
$761$	1948.00	0.0927923	0.0463962	+	0.998923i	$0.485226\pi$
$762$	0	0
$763$	− 21892.0i	− 1.03872i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 42900.0i	− 2.01959i
$768$	0	0
$769$	17420.0	0.816881	0.408440	−	0.912785i	$-0.366073\pi$
$769$	17420.0	0.816881	0.408440	+	0.912785i	$0.366073\pi$
$770$	0	0
$771$	1770.00	0.0826783
$772$	0	0
$773$	11122.0i	0.517504i	0.965944	+	0.258752i	$0.0833112\pi$
$773$	11122.0i	0.517504i	−0.965944	+	0.258752i	$0.916689\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3770.00i	0.174064i
$778$	0	0
$779$	−45312.0	−2.08404
$780$	0	0
$781$	−11011.0	−0.504487
$782$	0	0
$783$	2320.00i	0.105888i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 29648.0i	− 1.34287i	−0.741064	−	0.671434i	$-0.765679\pi$
$787$	− 29648.0i	− 1.34287i	0.741064	−	0.671434i	$-0.234321\pi$
$788$	0	0
$789$	−10130.0	−0.457082
$790$	0	0
$791$	−33306.0	−1.49712
$792$	0	0
$793$	− 28496.0i	− 1.27607i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23219.0i	1.03194i	0.856606	+	0.515972i	$0.172569\pi$
$797$	23219.0i	1.03194i	−0.856606	+	0.515972i	$0.827431\pi$
$798$	0	0
$799$	10304.0	0.456232
$800$	0	0
$801$	2434.00	0.107367
$802$	0	0
$803$	11220.0i	0.493082i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 43750.0i	− 1.90839i
$808$	0	0
$809$	−9232.00	−0.401211	−0.200606	−	0.979672i	$-0.564291\pi$
$809$	−9232.00	−0.401211	−0.200606	+	0.979672i	$0.564291\pi$
$810$	0	0
$811$	−32286.0	−1.39792	−0.698961	−	0.715160i	$-0.746354\pi$
$811$	−32286.0	−1.39792	−0.698961	+	0.715160i	$0.746354\pi$
$812$	0	0
$813$	25180.0i	1.08623i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 10560.0i	− 0.452200i
$818$	0	0
$819$	−2704.00	−0.115367
$820$	0	0
$821$	−31706.0	−1.34780	−0.673902	−	0.738821i	$-0.735383\pi$
$821$	−31706.0	−1.34780	−0.673902	+	0.738821i	$0.735383\pi$
$822$	0	0
$823$	41139.0i	1.74242i	0.490906	+	0.871212i	$0.336666\pi$
$823$	41139.0i	1.74242i	−0.490906	+	0.871212i	$0.663334\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15252.0i	0.641311i	0.947196	+	0.320655i	$0.103903\pi$
$827$	15252.0i	0.641311i	−0.947196	+	0.320655i	$0.896097\pi$
$828$	0	0
$829$	−369.000	−0.0154595	−0.00772973	−	0.999970i	$-0.502460\pi$
$829$	−369.000	−0.0154595	−0.00772973	+	0.999970i	$0.502460\pi$
$830$	0	0
$831$	21530.0	0.898757
$832$	0	0
$833$	15318.0i	0.637140i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	42485.0i	1.75448i
$838$	0	0
$839$	10257.0	0.422063	0.211032	−	0.977479i	$-0.432318\pi$
$839$	10257.0	0.422063	0.211032	+	0.977479i	$0.432318\pi$
$840$	0	0
$841$	−24133.0	−0.989503
$842$	0	0
$843$	6010.00i	0.245546i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3146.00i	0.127624i
$848$	0	0
$849$	−33100.0	−1.33803
$850$	0	0
$851$	−783.000	−0.0315404
$852$	0	0
$853$	− 34386.0i	− 1.38025i	−0.723690	−	0.690126i	$-0.757555\pi$
$853$	− 34386.0i	− 1.38025i	0.723690	−	0.690126i	$-0.242445\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	23464.0i	0.935257i	0.883925	+	0.467628i	$0.154891\pi$
$857$	23464.0i	0.935257i	−0.883925	+	0.467628i	$0.845109\pi$
$858$	0	0
$859$	22475.0	0.892709	0.446355	−	0.894856i	$-0.352722\pi$
$859$	22475.0	0.892709	0.446355	+	0.894856i	$0.352722\pi$
$860$	0	0
$861$	−61360.0	−2.42874
$862$	0	0
$863$	2880.00i	0.113599i	0.998386	+	0.0567997i	$0.0180897\pi$
$863$	2880.00i	0.113599i	−0.998386	+	0.0567997i	$0.981910\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 13985.0i	− 0.547815i
$868$	0	0
$869$	5786.00	0.225865
$870$	0	0
$871$	−6396.00	−0.248818
$872$	0	0
$873$	526.000i	0.0203922i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11084.0i	0.426773i	0.976968	+	0.213387i	$0.0684494\pi$
$877$	11084.0i	0.426773i	−0.976968	+	0.213387i	$0.931551\pi$
$878$	0	0
$879$	34840.0	1.33689
$880$	0	0
$881$	41797.0	1.59838	0.799192	−	0.601076i	$-0.205261\pi$
$881$	41797.0	1.59838	0.799192	+	0.601076i	$0.205261\pi$
$882$	0	0
$883$	23780.0i	0.906298i	0.891435	+	0.453149i	$0.149699\pi$
$883$	23780.0i	0.906298i	−0.891435	+	0.453149i	$0.850301\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14334.0i	0.542603i	0.962494	+	0.271301i	$0.0874540\pi$
$887$	14334.0i	0.542603i	−0.962494	+	0.271301i	$0.912546\pi$
$888$	0	0
$889$	45136.0	1.70283
$890$	0	0
$891$	7381.00	0.277523
$892$	0	0
$893$	21504.0i	0.805827i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7020.00i	0.261305i
$898$	0	0
$899$	4688.00	0.173919
$900$	0	0
$901$	34684.0	1.28245
$902$	0	0
$903$	− 14300.0i	− 0.526992i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 25716.0i	− 0.941440i	−0.882283	−	0.470720i	$-0.843994\pi$
$907$	− 25716.0i	− 0.941440i	0.882283	−	0.470720i	$-0.156006\pi$
$908$	0	0
$909$	−1628.00	−0.0594030
$910$	0	0
$911$	−28300.0	−1.02922	−0.514611	−	0.857424i	$-0.672064\pi$
$911$	−28300.0	−1.02922	−0.514611	+	0.857424i	$0.672064\pi$
$912$	0	0
$913$	1738.00i	0.0630004i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	156.000i	0.00561786i
$918$	0	0
$919$	−21410.0	−0.768499	−0.384250	−	0.923229i	$-0.625540\pi$
$919$	−21410.0	−0.768499	−0.384250	+	0.923229i	$0.625540\pi$
$920$	0	0
$921$	−38200.0	−1.36670
$922$	0	0
$923$	52052.0i	1.85624i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	752.000i	0.0266439i
$928$	0	0
$929$	−19938.0	−0.704138	−0.352069	−	0.935974i	$-0.614522\pi$
$929$	−19938.0	−0.704138	−0.352069	+	0.935974i	$0.614522\pi$
$930$	0	0
$931$	−31968.0	−1.12536
$932$	0	0
$933$	3260.00i	0.114392i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 26728.0i	− 0.931874i	−0.884818	−	0.465937i	$-0.845717\pi$
$937$	− 26728.0i	− 0.931874i	0.884818	−	0.465937i	$-0.154283\pi$
$938$	0	0
$939$	40275.0	1.39971
$940$	0	0
$941$	8634.00	0.299108	0.149554	−	0.988754i	$-0.452216\pi$
$941$	8634.00	0.299108	0.149554	+	0.988754i	$0.452216\pi$
$942$	0	0
$943$	− 12744.0i	− 0.440087i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 24841.0i	− 0.852401i	−0.904629	−	0.426201i	$-0.859852\pi$
$947$	− 24841.0i	− 0.852401i	0.904629	−	0.426201i	$-0.140148\pi$
$948$	0	0
$949$	53040.0	1.81428
$950$	0	0
$951$	28375.0	0.967531
$952$	0	0
$953$	51234.0i	1.74148i	0.491742	+	0.870741i	$0.336360\pi$
$953$	51234.0i	1.74148i	−0.491742	+	0.870741i	$0.663640\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 880.000i	− 0.0297245i
$958$	0	0
$959$	−37414.0	−1.25981
$960$	0	0
$961$	56058.0	1.88171
$962$	0	0
$963$	2988.00i	0.0999865i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	26752.0i	0.889645i	0.895619	+	0.444822i	$0.146733\pi$
$967$	26752.0i	0.889645i	−0.895619	+	0.444822i	$0.853267\pi$
$968$	0	0
$969$	−22080.0	−0.732004
$970$	0	0
$971$	21155.0	0.699172	0.349586	−	0.936904i	$-0.386322\pi$
$971$	21155.0	0.699172	0.349586	+	0.936904i	$0.386322\pi$
$972$	0	0
$973$	8268.00i	0.272415i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	56323.0i	1.84435i	0.386770	+	0.922176i	$0.373591\pi$
$977$	56323.0i	1.84435i	−0.386770	+	0.922176i	$0.626409\pi$
$978$	0	0
$979$	−13387.0	−0.437028
$980$	0	0
$981$	−1684.00	−0.0548073
$982$	0	0
$983$	24683.0i	0.800880i	0.916323	+	0.400440i	$0.131143\pi$
$983$	24683.0i	0.800880i	−0.916323	+	0.400440i	$0.868857\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	29120.0i	0.939108i
$988$	0	0
$989$	2970.00	0.0954909
$990$	0	0
$991$	−26816.0	−0.859574	−0.429787	−	0.902930i	$-0.641411\pi$
$991$	−26816.0	−0.859574	−0.429787	+	0.902930i	$0.641411\pi$
$992$	0	0
$993$	11225.0i	0.358726i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 18614.0i	− 0.591285i	−0.955299	−	0.295643i	$-0.904466\pi$
$997$	− 18614.0i	− 0.591285i	0.955299	−	0.295643i	$-0.0955337\pi$
$998$	0	0
$999$	4205.00	0.133173

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.4.b.c.749.1		2
5.2	odd	4		44.4.a.a.1.1	✓	1
5.3	odd	4		1100.4.a.d.1.1		1
5.4	even	2	inner	1100.4.b.c.749.2		2
15.2	even	4		396.4.a.e.1.1		1
20.7	even	4		176.4.a.e.1.1		1
35.27	even	4		2156.4.a.b.1.1		1
40.27	even	4		704.4.a.c.1.1		1
40.37	odd	4		704.4.a.j.1.1		1
55.32	even	4		484.4.a.a.1.1		1
60.47	odd	4		1584.4.a.p.1.1		1
220.87	odd	4		1936.4.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.4.a.a.1.1	✓	1	5.2	odd	4
176.4.a.e.1.1		1	20.7	even	4
396.4.a.e.1.1		1	15.2	even	4
484.4.a.a.1.1		1	55.32	even	4
704.4.a.c.1.1		1	40.27	even	4
704.4.a.j.1.1		1	40.37	odd	4
1100.4.a.d.1.1		1	5.3	odd	4
1100.4.b.c.749.1		2	1.1	even	1	trivial
1100.4.b.c.749.2		2	5.4	even	2	inner
1584.4.a.p.1.1		1	60.47	odd	4
1936.4.a.m.1.1		1	220.87	odd	4
2156.4.a.b.1.1		1	35.27	even	4

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:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1100.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.00000i q^{3} +26.0000i q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q-5.00000i q^{3} +26.0000i q^{7} +2.00000 q^{9} -11.0000 q^{11} +52.0000i q^{13} -46.0000i q^{17} +96.0000 q^{19} +130.000 q^{21} +27.0000i q^{23} -145.000i q^{27} -16.0000 q^{29} -293.000 q^{31} +55.0000i q^{33} +29.0000i q^{37} +260.000 q^{39} -472.000 q^{41} -110.000i q^{43} +224.000i q^{47} -333.000 q^{49} -230.000 q^{51} +754.000i q^{53} -480.000i q^{57} -825.000 q^{59} -548.000 q^{61} +52.0000i q^{63} +123.000i q^{67} +135.000 q^{69} +1001.00 q^{71} -1020.00i q^{73} -286.000i q^{77} -526.000 q^{79} -671.000 q^{81} -158.000i q^{83} +80.0000i q^{87} +1217.00 q^{89} -1352.00 q^{91} +1465.00i q^{93} +263.000i q^{97} -22.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9}+O(q^{10}) \) 2 * q + 4 * q^9	\( 2 q + 4 q^{9} - 22 q^{11} + 192 q^{19} + 260 q^{21} - 32 q^{29} - 586 q^{31} + 520 q^{39} - 944 q^{41} - 666 q^{49} - 460 q^{51} - 1650 q^{59} - 1096 q^{61} + 270 q^{69} + 2002 q^{71} - 1052 q^{79} - 1342 q^{81} + 2434 q^{89} - 2704 q^{91} - 44 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 - 22 * q^11 + 192 * q^19 + 260 * q^21 - 32 * q^29 - 586 * q^31 + 520 * q^39 - 944 * q^41 - 666 * q^49 - 460 * q^51 - 1650 * q^59 - 1096 * q^61 + 270 * q^69 + 2002 * q^71 - 1052 * q^79 - 1342 * q^81 + 2434 * q^89 - 2704 * q^91 - 44 * q^99

Introduction

L-functions

Modular forms

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Database

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