LMFDB - Embedded newform 1170.2.e.e.469.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1170,2,Mod(469,1170)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1170.469"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1170, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.34249703649$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			469.1
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	1170.469
Dual form			1170.2.e.e.469.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} -2.23607i q^{5} +1.23607i q^{7} +1.00000i q^{8} -2.23607 q^{10} +4.47214 q^{11} +1.00000i q^{13} +1.23607 q^{14} +1.00000 q^{16} +2.76393i q^{17} +7.23607 q^{19} +2.23607i q^{20} -4.47214i q^{22} -7.23607i q^{23} -5.00000 q^{25} +1.00000 q^{26} -1.23607i q^{28} +9.70820 q^{29} -4.00000 q^{31} -1.00000i q^{32} +2.76393 q^{34} +2.76393 q^{35} -6.94427i q^{37} -7.23607i q^{38} +2.23607 q^{40} -12.4721 q^{41} +6.47214i q^{43} -4.47214 q^{44} -7.23607 q^{46} -4.94427i q^{47} +5.47214 q^{49} +5.00000i q^{50} -1.00000i q^{52} -8.47214i q^{53} -10.0000i q^{55} -1.23607 q^{56} -9.70820i q^{58} +0.472136 q^{59} +6.94427 q^{61} +4.00000i q^{62} -1.00000 q^{64} +2.23607 q^{65} -2.76393i q^{68} -2.76393i q^{70} -6.47214 q^{71} +8.76393i q^{73} -6.94427 q^{74} -7.23607 q^{76} +5.52786i q^{77} +4.00000 q^{79} -2.23607i q^{80} +12.4721i q^{82} -12.9443i q^{83} +6.18034 q^{85} +6.47214 q^{86} +4.47214i q^{88} -8.47214 q^{89} -1.23607 q^{91} +7.23607i q^{92} -4.94427 q^{94} -16.1803i q^{95} -9.70820i q^{97} -5.47214i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{14} + 4 q^{16} + 20 q^{19} - 20 q^{25} + 4 q^{26} + 12 q^{29} - 16 q^{31} + 20 q^{34} + 20 q^{35} - 32 q^{41} - 20 q^{46} + 4 q^{49} + 4 q^{56} - 16 q^{59} - 8 q^{61} - 4 q^{64} - 8 q^{71}+ \cdots + 16 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^14 + 4 * q^16 + 20 * q^19 - 20 * q^25 + 4 * q^26 + 12 * q^29 - 16 * q^31 + 20 * q^34 + 20 * q^35 - 32 * q^41 - 20 * q^46 + 4 * q^49 + 4 * q^56 - 16 * q^59 - 8 * q^61 - 4 * q^64 - 8 * q^71 + 8 * q^74 - 20 * q^76 + 16 * q^79 - 20 * q^85 + 8 * q^86 - 16 * q^89 + 4 * q^91 + 16 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$.

$n$	$911$	$937$	$1081$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	− 2.23607i	− 1.00000i
$5$	− 2.23607i	− 1.00000i
$6$	0	0
$7$	1.23607i	0.467190i	0.972334	+	0.233595i	$0.0750489\pi$
$7$	1.23607i	0.467190i	−0.972334	+	0.233595i	$0.924951\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	−2.23607	−0.707107
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	1.23607	0.330353
$15$	0	0
$16$	1.00000	0.250000
$17$	2.76393i	0.670352i	0.942156	+	0.335176i	$0.108796\pi$
$17$	2.76393i	0.670352i	−0.942156	+	0.335176i	$0.891204\pi$
$18$	0	0
$19$	7.23607	1.66007	0.830034	−	0.557713i	$-0.188321\pi$
$19$	7.23607	1.66007	0.830034	+	0.557713i	$0.188321\pi$
$20$	2.23607i	0.500000i
$21$	0	0
$22$	− 4.47214i	− 0.953463i
$23$	− 7.23607i	− 1.50882i	−0.656401	−	0.754412i	$-0.727922\pi$
$23$	− 7.23607i	− 1.50882i	0.656401	−	0.754412i	$-0.272078\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	1.00000	0.196116
$27$	0	0
$28$	− 1.23607i	− 0.233595i
$29$	9.70820	1.80277	0.901384	−	0.433020i	$-0.142552\pi$
$29$	9.70820	1.80277	0.901384	+	0.433020i	$0.142552\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	2.76393	0.474010
$35$	2.76393	0.467190
$36$	0	0
$37$	− 6.94427i	− 1.14163i	−0.821078	−	0.570816i	$-0.806627\pi$
$37$	− 6.94427i	− 1.14163i	0.821078	−	0.570816i	$-0.193373\pi$
$38$	− 7.23607i	− 1.17385i
$39$	0	0
$40$	2.23607	0.353553
$41$	−12.4721	−1.94782	−0.973910	−	0.226934i	$-0.927130\pi$
$41$	−12.4721	−1.94782	−0.973910	+	0.226934i	$0.927130\pi$
$42$	0	0
$43$	6.47214i	0.986991i	0.869748	+	0.493496i	$0.164281\pi$
$43$	6.47214i	0.986991i	−0.869748	+	0.493496i	$0.835719\pi$
$44$	−4.47214	−0.674200
$45$	0	0
$46$	−7.23607	−1.06690
$47$	− 4.94427i	− 0.721196i	−0.932721	−	0.360598i	$-0.882573\pi$
$47$	− 4.94427i	− 0.721196i	0.932721	−	0.360598i	$-0.117427\pi$
$48$	0	0
$49$	5.47214	0.781734
$50$	5.00000i	0.707107i
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	− 8.47214i	− 1.16374i	−0.813283	−	0.581869i	$-0.802322\pi$
$53$	− 8.47214i	− 1.16374i	0.813283	−	0.581869i	$-0.197678\pi$
$54$	0	0
$55$	− 10.0000i	− 1.34840i
$56$	−1.23607	−0.165177
$57$	0	0
$58$	− 9.70820i	− 1.27475i
$59$	0.472136	0.0614669	0.0307334	−	0.999528i	$-0.490216\pi$
$59$	0.472136	0.0614669	0.0307334	+	0.999528i	$0.490216\pi$
$60$	0	0
$61$	6.94427	0.889123	0.444561	−	0.895748i	$-0.353360\pi$
$61$	6.94427	0.889123	0.444561	+	0.895748i	$0.353360\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	2.23607	0.277350
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	− 2.76393i	− 0.335176i
$69$	0	0
$70$	− 2.76393i	− 0.330353i
$71$	−6.47214	−0.768101	−0.384051	−	0.923312i	$-0.625471\pi$
$71$	−6.47214	−0.768101	−0.384051	+	0.923312i	$0.625471\pi$
$72$	0	0
$73$	8.76393i	1.02574i	0.858466	+	0.512870i	$0.171418\pi$
$73$	8.76393i	1.02574i	−0.858466	+	0.512870i	$0.828582\pi$
$74$	−6.94427	−0.807255
$75$	0	0
$76$	−7.23607	−0.830034
$77$	5.52786i	0.629959i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	− 2.23607i	− 0.250000i
$81$	0	0
$82$	12.4721i	1.37732i
$83$	− 12.9443i	− 1.42082i	−0.703789	−	0.710409i	$-0.748510\pi$
$83$	− 12.9443i	− 1.42082i	0.703789	−	0.710409i	$-0.251490\pi$
$84$	0	0
$85$	6.18034	0.670352
$86$	6.47214	0.697908
$87$	0	0
$88$	4.47214i	0.476731i
$89$	−8.47214	−0.898045	−0.449022	−	0.893521i	$-0.648228\pi$
$89$	−8.47214	−0.898045	−0.449022	+	0.893521i	$0.648228\pi$
$90$	0	0
$91$	−1.23607	−0.129575
$92$	7.23607i	0.754412i
$93$	0	0
$94$	−4.94427	−0.509963
$95$	− 16.1803i	− 1.66007i
$96$	0	0
$97$	− 9.70820i	− 0.985719i	−0.870109	−	0.492859i	$-0.835952\pi$
$97$	− 9.70820i	− 0.985719i	0.870109	−	0.492859i	$-0.164048\pi$
$98$	− 5.47214i	− 0.552769i
$99$	0	0
$100$	5.00000	0.500000
$101$	4.76393	0.474029	0.237014	−	0.971506i	$-0.423831\pi$
$101$	4.76393	0.474029	0.237014	+	0.971506i	$0.423831\pi$
$102$	0	0
$103$	− 0.472136i	− 0.0465209i	−0.999729	−	0.0232605i	$-0.992595\pi$
$103$	− 0.472136i	− 0.0465209i	0.999729	−	0.0232605i	$-0.00740471\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−8.47214	−0.822887
$107$	− 10.4721i	− 1.01238i	−0.862422	−	0.506190i	$-0.831054\pi$
$107$	− 10.4721i	− 1.01238i	0.862422	−	0.506190i	$-0.168946\pi$
$108$	0	0
$109$	15.7082	1.50457	0.752287	−	0.658836i	$-0.228951\pi$
$109$	15.7082	1.50457	0.752287	+	0.658836i	$0.228951\pi$
$110$	−10.0000	−0.953463
$111$	0	0
$112$	1.23607i	0.116797i
$113$	12.6525i	1.19024i	0.803635	+	0.595122i	$0.202896\pi$
$113$	12.6525i	1.19024i	−0.803635	+	0.595122i	$0.797104\pi$
$114$	0	0
$115$	−16.1803	−1.50882
$116$	−9.70820	−0.901384
$117$	0	0
$118$	− 0.472136i	− 0.0434636i
$119$	−3.41641	−0.313182
$120$	0	0
$121$	9.00000	0.818182
$122$	− 6.94427i	− 0.628705i
$123$	0	0
$124$	4.00000	0.359211
$125$	11.1803i	1.00000i
$126$	0	0
$127$	5.41641i	0.480628i	0.970695	+	0.240314i	$0.0772505\pi$
$127$	5.41641i	0.480628i	−0.970695	+	0.240314i	$0.922750\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	− 2.23607i	− 0.196116i
$131$	3.70820	0.323987	0.161994	−	0.986792i	$-0.448208\pi$
$131$	3.70820	0.323987	0.161994	+	0.986792i	$0.448208\pi$
$132$	0	0
$133$	8.94427i	0.775567i
$134$	0	0
$135$	0	0
$136$	−2.76393	−0.237005
$137$	15.8885i	1.35745i	0.734393	+	0.678725i	$0.237467\pi$
$137$	15.8885i	1.35745i	−0.734393	+	0.678725i	$0.762533\pi$
$138$	0	0
$139$	−2.47214	−0.209684	−0.104842	−	0.994489i	$-0.533434\pi$
$139$	−2.47214	−0.209684	−0.104842	+	0.994489i	$0.533434\pi$
$140$	−2.76393	−0.233595
$141$	0	0
$142$	6.47214i	0.543130i
$143$	4.47214i	0.373979i
$144$	0	0
$145$	− 21.7082i	− 1.80277i
$146$	8.76393	0.725308
$147$	0	0
$148$	6.94427i	0.570816i
$149$	22.4721	1.84099	0.920495	−	0.390755i	$-0.127786\pi$
$149$	22.4721	1.84099	0.920495	+	0.390755i	$0.127786\pi$
$150$	0	0
$151$	7.41641	0.603539	0.301769	−	0.953381i	$-0.402423\pi$
$151$	7.41641	0.603539	0.301769	+	0.953381i	$0.402423\pi$
$152$	7.23607i	0.586923i
$153$	0	0
$154$	5.52786	0.445448
$155$	8.94427i	0.718421i
$156$	0	0
$157$	1.05573i	0.0842563i	0.999112	+	0.0421281i	$0.0134138\pi$
$157$	1.05573i	0.0842563i	−0.999112	+	0.0421281i	$0.986586\pi$
$158$	− 4.00000i	− 0.318223i
$159$	0	0
$160$	−2.23607	−0.176777
$161$	8.94427	0.704907
$162$	0	0
$163$	− 4.94427i	− 0.387265i	−0.981074	−	0.193633i	$-0.937973\pi$
$163$	− 4.94427i	− 0.387265i	0.981074	−	0.193633i	$-0.0620270\pi$
$164$	12.4721	0.973910
$165$	0	0
$166$	−12.9443	−1.00467
$167$	7.41641i	0.573899i	0.957946	+	0.286949i	$0.0926412\pi$
$167$	7.41641i	0.573899i	−0.957946	+	0.286949i	$0.907359\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	− 6.18034i	− 0.474010i
$171$	0	0
$172$	− 6.47214i	− 0.493496i
$173$	5.05573i	0.384380i	0.981358	+	0.192190i	$0.0615590\pi$
$173$	5.05573i	0.384380i	−0.981358	+	0.192190i	$0.938441\pi$
$174$	0	0
$175$	− 6.18034i	− 0.467190i
$176$	4.47214	0.337100
$177$	0	0
$178$	8.47214i	0.635013i
$179$	−14.1803	−1.05989	−0.529944	−	0.848033i	$-0.677787\pi$
$179$	−14.1803	−1.05989	−0.529944	+	0.848033i	$0.677787\pi$
$180$	0	0
$181$	−9.41641	−0.699916	−0.349958	−	0.936765i	$-0.613804\pi$
$181$	−9.41641	−0.699916	−0.349958	+	0.936765i	$0.613804\pi$
$182$	1.23607i	0.0916235i
$183$	0	0
$184$	7.23607	0.533450
$185$	−15.5279	−1.14163
$186$	0	0
$187$	12.3607i	0.903902i
$188$	4.94427i	0.360598i
$189$	0	0
$190$	−16.1803	−1.17385
$191$	−13.5279	−0.978842	−0.489421	−	0.872048i	$-0.662792\pi$
$191$	−13.5279	−0.978842	−0.489421	+	0.872048i	$0.662792\pi$
$192$	0	0
$193$	− 21.7082i	− 1.56259i	−0.624161	−	0.781295i	$-0.714559\pi$
$193$	− 21.7082i	− 1.56259i	0.624161	−	0.781295i	$-0.285441\pi$
$194$	−9.70820	−0.697008
$195$	0	0
$196$	−5.47214	−0.390867
$197$	2.94427i	0.209771i	0.994484	+	0.104885i	$0.0334476\pi$
$197$	2.94427i	0.209771i	−0.994484	+	0.104885i	$0.966552\pi$
$198$	0	0
$199$	−16.9443	−1.20115	−0.600574	−	0.799569i	$-0.705061\pi$
$199$	−16.9443	−1.20115	−0.600574	+	0.799569i	$0.705061\pi$
$200$	− 5.00000i	− 0.353553i
$201$	0	0
$202$	− 4.76393i	− 0.335189i
$203$	12.0000i	0.842235i
$204$	0	0
$205$	27.8885i	1.94782i
$206$	−0.472136	−0.0328953
$207$	0	0
$208$	1.00000i	0.0693375i
$209$	32.3607	2.23844
$210$	0	0
$211$	−21.8885	−1.50687	−0.753435	−	0.657523i	$-0.771604\pi$
$211$	−21.8885	−1.50687	−0.753435	+	0.657523i	$0.771604\pi$
$212$	8.47214i	0.581869i
$213$	0	0
$214$	−10.4721	−0.715860
$215$	14.4721	0.986991
$216$	0	0
$217$	− 4.94427i	− 0.335639i
$218$	− 15.7082i	− 1.06389i
$219$	0	0
$220$	10.0000i	0.674200i
$221$	−2.76393	−0.185922
$222$	0	0
$223$	21.2361i	1.42207i	0.703155	+	0.711036i	$0.251774\pi$
$223$	21.2361i	1.42207i	−0.703155	+	0.711036i	$0.748226\pi$
$224$	1.23607	0.0825883
$225$	0	0
$226$	12.6525	0.841630
$227$	24.9443i	1.65561i	0.561016	+	0.827805i	$0.310410\pi$
$227$	24.9443i	1.65561i	−0.561016	+	0.827805i	$0.689590\pi$
$228$	0	0
$229$	3.70820	0.245045	0.122523	−	0.992466i	$-0.460902\pi$
$229$	3.70820	0.245045	0.122523	+	0.992466i	$0.460902\pi$
$230$	16.1803i	1.06690i
$231$	0	0
$232$	9.70820i	0.637375i
$233$	2.18034i	0.142839i	0.997446	+	0.0714194i	$0.0227529\pi$
$233$	2.18034i	0.142839i	−0.997446	+	0.0714194i	$0.977247\pi$
$234$	0	0
$235$	−11.0557	−0.721196
$236$	−0.472136	−0.0307334
$237$	0	0
$238$	3.41641i	0.221453i
$239$	13.8885	0.898375	0.449188	−	0.893437i	$-0.351713\pi$
$239$	13.8885	0.898375	0.449188	+	0.893437i	$0.351713\pi$
$240$	0	0
$241$	12.4721	0.803401	0.401700	−	0.915771i	$-0.368419\pi$
$241$	12.4721	0.803401	0.401700	+	0.915771i	$0.368419\pi$
$242$	− 9.00000i	− 0.578542i
$243$	0	0
$244$	−6.94427	−0.444561
$245$	− 12.2361i	− 0.781734i
$246$	0	0
$247$	7.23607i	0.460420i
$248$	− 4.00000i	− 0.254000i
$249$	0	0
$250$	11.1803	0.707107
$251$	−15.7082	−0.991493	−0.495747	−	0.868467i	$-0.665105\pi$
$251$	−15.7082	−0.991493	−0.495747	+	0.868467i	$0.665105\pi$
$252$	0	0
$253$	− 32.3607i	− 2.03450i
$254$	5.41641	0.339856
$255$	0	0
$256$	1.00000	0.0625000
$257$	5.23607i	0.326617i	0.986575	+	0.163308i	$0.0522166\pi$
$257$	5.23607i	0.326617i	−0.986575	+	0.163308i	$0.947783\pi$
$258$	0	0
$259$	8.58359	0.533358
$260$	−2.23607	−0.138675
$261$	0	0
$262$	− 3.70820i	− 0.229094i
$263$	12.1803i	0.751072i	0.926808	+	0.375536i	$0.122541\pi$
$263$	12.1803i	0.751072i	−0.926808	+	0.375536i	$0.877459\pi$
$264$	0	0
$265$	−18.9443	−1.16374
$266$	8.94427	0.548408
$267$	0	0
$268$	0	0
$269$	11.2361	0.685075	0.342538	−	0.939504i	$-0.388714\pi$
$269$	11.2361	0.685075	0.342538	+	0.939504i	$0.388714\pi$
$270$	0	0
$271$	15.4164	0.936480	0.468240	−	0.883601i	$-0.344888\pi$
$271$	15.4164	0.936480	0.468240	+	0.883601i	$0.344888\pi$
$272$	2.76393i	0.167588i
$273$	0	0
$274$	15.8885	0.959862
$275$	−22.3607	−1.34840
$276$	0	0
$277$	32.8328i	1.97273i	0.164564	+	0.986366i	$0.447378\pi$
$277$	32.8328i	1.97273i	−0.164564	+	0.986366i	$0.552622\pi$
$278$	2.47214i	0.148269i
$279$	0	0
$280$	2.76393i	0.165177i
$281$	−15.5279	−0.926315	−0.463157	−	0.886276i	$-0.653284\pi$
$281$	−15.5279	−0.926315	−0.463157	+	0.886276i	$0.653284\pi$
$282$	0	0
$283$	15.4164i	0.916410i	0.888846	+	0.458205i	$0.151508\pi$
$283$	15.4164i	0.916410i	−0.888846	+	0.458205i	$0.848492\pi$
$284$	6.47214	0.384051
$285$	0	0
$286$	4.47214	0.264443
$287$	− 15.4164i	− 0.910002i
$288$	0	0
$289$	9.36068	0.550628
$290$	−21.7082	−1.27475
$291$	0	0
$292$	− 8.76393i	− 0.512870i
$293$	14.9443i	0.873054i	0.899691	+	0.436527i	$0.143792\pi$
$293$	14.9443i	0.873054i	−0.899691	+	0.436527i	$0.856208\pi$
$294$	0	0
$295$	− 1.05573i	− 0.0614669i
$296$	6.94427	0.403628
$297$	0	0
$298$	− 22.4721i	− 1.30178i
$299$	7.23607	0.418473
$300$	0	0
$301$	−8.00000	−0.461112
$302$	− 7.41641i	− 0.426766i
$303$	0	0
$304$	7.23607	0.415017
$305$	− 15.5279i	− 0.889123i
$306$	0	0
$307$	− 20.3607i	− 1.16205i	−0.813887	−	0.581023i	$-0.802653\pi$
$307$	− 20.3607i	− 1.16205i	0.813887	−	0.581023i	$-0.197347\pi$
$308$	− 5.52786i	− 0.314979i
$309$	0	0
$310$	8.94427	0.508001
$311$	13.5279	0.767095	0.383547	−	0.923521i	$-0.374702\pi$
$311$	13.5279	0.767095	0.383547	+	0.923521i	$0.374702\pi$
$312$	0	0
$313$	7.41641i	0.419200i	0.977787	+	0.209600i	$0.0672162\pi$
$313$	7.41641i	0.419200i	−0.977787	+	0.209600i	$0.932784\pi$
$314$	1.05573	0.0595782
$315$	0	0
$316$	−4.00000	−0.225018
$317$	− 13.4164i	− 0.753541i	−0.926307	−	0.376770i	$-0.877035\pi$
$317$	− 13.4164i	− 0.753541i	0.926307	−	0.376770i	$-0.122965\pi$
$318$	0	0
$319$	43.4164	2.43085
$320$	2.23607i	0.125000i
$321$	0	0
$322$	− 8.94427i	− 0.498445i
$323$	20.0000i	1.11283i
$324$	0	0
$325$	− 5.00000i	− 0.277350i
$326$	−4.94427	−0.273838
$327$	0	0
$328$	− 12.4721i	− 0.688659i
$329$	6.11146	0.336935
$330$	0	0
$331$	−7.23607	−0.397730	−0.198865	−	0.980027i	$-0.563726\pi$
$331$	−7.23607	−0.397730	−0.198865	+	0.980027i	$0.563726\pi$
$332$	12.9443i	0.710409i
$333$	0	0
$334$	7.41641	0.405808
$335$	0	0
$336$	0	0
$337$	− 2.47214i	− 0.134666i	−0.997731	−	0.0673329i	$-0.978551\pi$
$337$	− 2.47214i	− 0.134666i	0.997731	−	0.0673329i	$-0.0214490\pi$
$338$	1.00000i	0.0543928i
$339$	0	0
$340$	−6.18034	−0.335176
$341$	−17.8885	−0.968719
$342$	0	0
$343$	15.4164i	0.832408i
$344$	−6.47214	−0.348954
$345$	0	0
$346$	5.05573	0.271798
$347$	10.4721i	0.562174i	0.959682	+	0.281087i	$0.0906949\pi$
$347$	10.4721i	0.562174i	−0.959682	+	0.281087i	$0.909305\pi$
$348$	0	0
$349$	10.7639	0.576180	0.288090	−	0.957603i	$-0.406980\pi$
$349$	10.7639	0.576180	0.288090	+	0.957603i	$0.406980\pi$
$350$	−6.18034	−0.330353
$351$	0	0
$352$	− 4.47214i	− 0.238366i
$353$	9.05573i	0.481988i	0.970527	+	0.240994i	$0.0774734\pi$
$353$	9.05573i	0.481988i	−0.970527	+	0.240994i	$0.922527\pi$
$354$	0	0
$355$	14.4721i	0.768101i
$356$	8.47214	0.449022
$357$	0	0
$358$	14.1803i	0.749454i
$359$	−2.47214	−0.130474	−0.0652372	−	0.997870i	$-0.520780\pi$
$359$	−2.47214	−0.130474	−0.0652372	+	0.997870i	$0.520780\pi$
$360$	0	0
$361$	33.3607	1.75583
$362$	9.41641i	0.494915i
$363$	0	0
$364$	1.23607	0.0647876
$365$	19.5967	1.02574
$366$	0	0
$367$	8.47214i	0.442242i	0.975246	+	0.221121i	$0.0709716\pi$
$367$	8.47214i	0.442242i	−0.975246	+	0.221121i	$0.929028\pi$
$368$	− 7.23607i	− 0.377206i
$369$	0	0
$370$	15.5279i	0.807255i
$371$	10.4721	0.543686
$372$	0	0
$373$	2.00000i	0.103556i	0.998659	+	0.0517780i	$0.0164888\pi$
$373$	2.00000i	0.103556i	−0.998659	+	0.0517780i	$0.983511\pi$
$374$	12.3607	0.639156
$375$	0	0
$376$	4.94427	0.254981
$377$	9.70820i	0.499998i
$378$	0	0
$379$	−5.12461	−0.263234	−0.131617	−	0.991301i	$-0.542017\pi$
$379$	−5.12461	−0.263234	−0.131617	+	0.991301i	$0.542017\pi$
$380$	16.1803i	0.830034i
$381$	0	0
$382$	13.5279i	0.692146i
$383$	− 14.4721i	− 0.739492i	−0.929133	−	0.369746i	$-0.879445\pi$
$383$	− 14.4721i	− 0.739492i	0.929133	−	0.369746i	$-0.120555\pi$
$384$	0	0
$385$	12.3607	0.629959
$386$	−21.7082	−1.10492
$387$	0	0
$388$	9.70820i	0.492859i
$389$	−18.6525	−0.945718	−0.472859	−	0.881138i	$-0.656778\pi$
$389$	−18.6525	−0.945718	−0.472859	+	0.881138i	$0.656778\pi$
$390$	0	0
$391$	20.0000	1.01144
$392$	5.47214i	0.276385i
$393$	0	0
$394$	2.94427	0.148330
$395$	− 8.94427i	− 0.450035i
$396$	0	0
$397$	9.41641i	0.472596i	0.971681	+	0.236298i	$0.0759341\pi$
$397$	9.41641i	0.472596i	−0.971681	+	0.236298i	$0.924066\pi$
$398$	16.9443i	0.849340i
$399$	0	0
$400$	−5.00000	−0.250000
$401$	15.5279	0.775425	0.387712	−	0.921780i	$-0.373265\pi$
$401$	15.5279	0.775425	0.387712	+	0.921780i	$0.373265\pi$
$402$	0	0
$403$	− 4.00000i	− 0.199254i
$404$	−4.76393	−0.237014
$405$	0	0
$406$	12.0000	0.595550
$407$	− 31.0557i	− 1.53938i
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	27.8885	1.37732
$411$	0	0
$412$	0.472136i	0.0232605i
$413$	0.583592i	0.0287167i
$414$	0	0
$415$	−28.9443	−1.42082
$416$	1.00000	0.0490290
$417$	0	0
$418$	− 32.3607i	− 1.58281i
$419$	−21.5967	−1.05507	−0.527535	−	0.849533i	$-0.676884\pi$
$419$	−21.5967	−1.05507	−0.527535	+	0.849533i	$0.676884\pi$
$420$	0	0
$421$	−0.291796	−0.0142213	−0.00711064	−	0.999975i	$-0.502263\pi$
$421$	−0.291796	−0.0142213	−0.00711064	+	0.999975i	$0.502263\pi$
$422$	21.8885i	1.06552i
$423$	0	0
$424$	8.47214	0.411443
$425$	− 13.8197i	− 0.670352i
$426$	0	0
$427$	8.58359i	0.415389i
$428$	10.4721i	0.506190i
$429$	0	0
$430$	− 14.4721i	− 0.697908i
$431$	−38.8328	−1.87051	−0.935255	−	0.353973i	$-0.884830\pi$
$431$	−38.8328	−1.87051	−0.935255	+	0.353973i	$0.884830\pi$
$432$	0	0
$433$	− 18.4721i	− 0.887714i	−0.896098	−	0.443857i	$-0.853610\pi$
$433$	− 18.4721i	− 0.887714i	0.896098	−	0.443857i	$-0.146390\pi$
$434$	−4.94427	−0.237333
$435$	0	0
$436$	−15.7082	−0.752287
$437$	− 52.3607i	− 2.50475i
$438$	0	0
$439$	−12.9443	−0.617796	−0.308898	−	0.951095i	$-0.599960\pi$
$439$	−12.9443	−0.617796	−0.308898	+	0.951095i	$0.599960\pi$
$440$	10.0000	0.476731
$441$	0	0
$442$	2.76393i	0.131467i
$443$	13.5279i	0.642728i	0.946956	+	0.321364i	$0.104141\pi$
$443$	13.5279i	0.642728i	−0.946956	+	0.321364i	$0.895859\pi$
$444$	0	0
$445$	18.9443i	0.898045i
$446$	21.2361	1.00556
$447$	0	0
$448$	− 1.23607i	− 0.0583987i
$449$	−8.47214	−0.399825	−0.199912	−	0.979814i	$-0.564066\pi$
$449$	−8.47214	−0.399825	−0.199912	+	0.979814i	$0.564066\pi$
$450$	0	0
$451$	−55.7771	−2.62644
$452$	− 12.6525i	− 0.595122i
$453$	0	0
$454$	24.9443	1.17069
$455$	2.76393i	0.129575i
$456$	0	0
$457$	− 9.70820i	− 0.454131i	−0.973880	−	0.227065i	$-0.927087\pi$
$457$	− 9.70820i	− 0.454131i	0.973880	−	0.227065i	$-0.0729131\pi$
$458$	− 3.70820i	− 0.173273i
$459$	0	0
$460$	16.1803	0.754412
$461$	−6.11146	−0.284639	−0.142319	−	0.989821i	$-0.545456\pi$
$461$	−6.11146	−0.284639	−0.142319	+	0.989821i	$0.545456\pi$
$462$	0	0
$463$	− 35.7082i	− 1.65950i	−0.558135	−	0.829750i	$-0.688483\pi$
$463$	− 35.7082i	− 1.65950i	0.558135	−	0.829750i	$-0.311517\pi$
$464$	9.70820	0.450692
$465$	0	0
$466$	2.18034	0.101002
$467$	− 2.47214i	− 0.114397i	−0.998363	−	0.0571984i	$-0.981783\pi$
$467$	− 2.47214i	− 0.114397i	0.998363	−	0.0571984i	$-0.0182168\pi$
$468$	0	0
$469$	0	0
$470$	11.0557i	0.509963i
$471$	0	0
$472$	0.472136i	0.0217318i
$473$	28.9443i	1.33086i
$474$	0	0
$475$	−36.1803	−1.66007
$476$	3.41641	0.156591
$477$	0	0
$478$	− 13.8885i	− 0.635247i
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	6.94427	0.316632
$482$	− 12.4721i	− 0.568090i
$483$	0	0
$484$	−9.00000	−0.409091
$485$	−21.7082	−0.985719
$486$	0	0
$487$	− 25.5967i	− 1.15990i	−0.814652	−	0.579950i	$-0.803072\pi$
$487$	− 25.5967i	− 1.15990i	0.814652	−	0.579950i	$-0.196928\pi$
$488$	6.94427i	0.314352i
$489$	0	0
$490$	−12.2361	−0.552769
$491$	−6.18034	−0.278915	−0.139457	−	0.990228i	$-0.544536\pi$
$491$	−6.18034	−0.278915	−0.139457	+	0.990228i	$0.544536\pi$
$492$	0	0
$493$	26.8328i	1.20849i
$494$	7.23607	0.325566
$495$	0	0
$496$	−4.00000	−0.179605
$497$	− 8.00000i	− 0.358849i
$498$	0	0
$499$	−1.12461	−0.0503445	−0.0251723	−	0.999683i	$-0.508013\pi$
$499$	−1.12461	−0.0503445	−0.0251723	+	0.999683i	$0.508013\pi$
$500$	− 11.1803i	− 0.500000i
$501$	0	0
$502$	15.7082i	0.701091i
$503$	35.2361i	1.57110i	0.618799	+	0.785549i	$0.287620\pi$
$503$	35.2361i	1.57110i	−0.618799	+	0.785549i	$0.712380\pi$
$504$	0	0
$505$	− 10.6525i	− 0.474029i
$506$	−32.3607	−1.43861
$507$	0	0
$508$	− 5.41641i	− 0.240314i
$509$	−32.9443	−1.46023	−0.730115	−	0.683325i	$-0.760533\pi$
$509$	−32.9443	−1.46023	−0.730115	+	0.683325i	$0.760533\pi$
$510$	0	0
$511$	−10.8328	−0.479216
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	5.23607	0.230953
$515$	−1.05573	−0.0465209
$516$	0	0
$517$	− 22.1115i	− 0.972461i
$518$	− 8.58359i	− 0.377141i
$519$	0	0
$520$	2.23607i	0.0980581i
$521$	32.8328	1.43843	0.719216	−	0.694787i	$-0.244501\pi$
$521$	32.8328	1.43843	0.719216	+	0.694787i	$0.244501\pi$
$522$	0	0
$523$	8.00000i	0.349816i	0.984585	+	0.174908i	$0.0559627\pi$
$523$	8.00000i	0.349816i	−0.984585	+	0.174908i	$0.944037\pi$
$524$	−3.70820	−0.161994
$525$	0	0
$526$	12.1803	0.531088
$527$	− 11.0557i	− 0.481595i
$528$	0	0
$529$	−29.3607	−1.27655
$530$	18.9443i	0.822887i
$531$	0	0
$532$	− 8.94427i	− 0.387783i
$533$	− 12.4721i	− 0.540228i
$534$	0	0
$535$	−23.4164	−1.01238
$536$	0	0
$537$	0	0
$538$	− 11.2361i	− 0.484421i
$539$	24.4721	1.05409
$540$	0	0
$541$	−9.23607	−0.397090	−0.198545	−	0.980092i	$-0.563622\pi$
$541$	−9.23607	−0.397090	−0.198545	+	0.980092i	$0.563622\pi$
$542$	− 15.4164i	− 0.662191i
$543$	0	0
$544$	2.76393	0.118503
$545$	− 35.1246i	− 1.50457i
$546$	0	0
$547$	− 12.0000i	− 0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$	− 12.0000i	− 0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$	− 15.8885i	− 0.678725i
$549$	0	0
$550$	22.3607i	0.953463i
$551$	70.2492	2.99272
$552$	0	0
$553$	4.94427i	0.210252i
$554$	32.8328	1.39493
$555$	0	0
$556$	2.47214	0.104842
$557$	− 43.3050i	− 1.83489i	−0.397863	−	0.917445i	$-0.630248\pi$
$557$	− 43.3050i	− 1.83489i	0.397863	−	0.917445i	$-0.369752\pi$
$558$	0	0
$559$	−6.47214	−0.273742
$560$	2.76393	0.116797
$561$	0	0
$562$	15.5279i	0.655003i
$563$	− 7.41641i	− 0.312564i	−0.987712	−	0.156282i	$-0.950049\pi$
$563$	− 7.41641i	− 0.312564i	0.987712	−	0.156282i	$-0.0499509\pi$
$564$	0	0
$565$	28.2918	1.19024
$566$	15.4164	0.648000
$567$	0	0
$568$	− 6.47214i	− 0.271565i
$569$	17.0557	0.715013	0.357507	−	0.933911i	$-0.383627\pi$
$569$	17.0557	0.715013	0.357507	+	0.933911i	$0.383627\pi$
$570$	0	0
$571$	−35.4164	−1.48213	−0.741065	−	0.671433i	$-0.765679\pi$
$571$	−35.4164	−1.48213	−0.741065	+	0.671433i	$0.765679\pi$
$572$	− 4.47214i	− 0.186989i
$573$	0	0
$574$	−15.4164	−0.643468
$575$	36.1803i	1.50882i
$576$	0	0
$577$	2.87539i	0.119704i	0.998207	+	0.0598520i	$0.0190629\pi$
$577$	2.87539i	0.119704i	−0.998207	+	0.0598520i	$0.980937\pi$
$578$	− 9.36068i	− 0.389353i
$579$	0	0
$580$	21.7082i	0.901384i
$581$	16.0000	0.663792
$582$	0	0
$583$	− 37.8885i	− 1.56918i
$584$	−8.76393	−0.362654
$585$	0	0
$586$	14.9443	0.617342
$587$	− 37.8885i	− 1.56383i	−0.623387	−	0.781914i	$-0.714244\pi$
$587$	− 37.8885i	− 1.56383i	0.623387	−	0.781914i	$-0.285756\pi$
$588$	0	0
$589$	−28.9443	−1.19263
$590$	−1.05573	−0.0434636
$591$	0	0
$592$	− 6.94427i	− 0.285408i
$593$	− 12.4721i	− 0.512169i	−0.966654	−	0.256085i	$-0.917567\pi$
$593$	− 12.4721i	− 0.512169i	0.966654	−	0.256085i	$-0.0824326\pi$
$594$	0	0
$595$	7.63932i	0.313182i
$596$	−22.4721	−0.920495
$597$	0	0
$598$	− 7.23607i	− 0.295905i
$599$	−41.8885	−1.71152	−0.855760	−	0.517373i	$-0.826910\pi$
$599$	−41.8885	−1.71152	−0.855760	+	0.517373i	$0.826910\pi$
$600$	0	0
$601$	−13.4164	−0.547267	−0.273633	−	0.961834i	$-0.588225\pi$
$601$	−13.4164	−0.547267	−0.273633	+	0.961834i	$0.588225\pi$
$602$	8.00000i	0.326056i
$603$	0	0
$604$	−7.41641	−0.301769
$605$	− 20.1246i	− 0.818182i
$606$	0	0
$607$	25.4164i	1.03162i	0.856703	+	0.515810i	$0.172509\pi$
$607$	25.4164i	1.03162i	−0.856703	+	0.515810i	$0.827491\pi$
$608$	− 7.23607i	− 0.293461i
$609$	0	0
$610$	−15.5279	−0.628705
$611$	4.94427	0.200024
$612$	0	0
$613$	18.0000i	0.727013i	0.931592	+	0.363507i	$0.118421\pi$
$613$	18.0000i	0.727013i	−0.931592	+	0.363507i	$0.881579\pi$
$614$	−20.3607	−0.821690
$615$	0	0
$616$	−5.52786	−0.222724
$617$	0.111456i	0.00448706i	0.999997	+	0.00224353i	$0.000714138\pi$
$617$	0.111456i	0.00448706i	−0.999997	+	0.00224353i	$0.999286\pi$
$618$	0	0
$619$	20.7639	0.834573	0.417286	−	0.908775i	$-0.362981\pi$
$619$	20.7639	0.834573	0.417286	+	0.908775i	$0.362981\pi$
$620$	− 8.94427i	− 0.359211i
$621$	0	0
$622$	− 13.5279i	− 0.542418i
$623$	− 10.4721i	− 0.419557i
$624$	0	0
$625$	25.0000	1.00000
$626$	7.41641	0.296419
$627$	0	0
$628$	− 1.05573i	− 0.0421281i
$629$	19.1935	0.765295
$630$	0	0
$631$	−39.4164	−1.56914	−0.784571	−	0.620039i	$-0.787117\pi$
$631$	−39.4164	−1.56914	−0.784571	+	0.620039i	$0.787117\pi$
$632$	4.00000i	0.159111i
$633$	0	0
$634$	−13.4164	−0.532834
$635$	12.1115	0.480628
$636$	0	0
$637$	5.47214i	0.216814i
$638$	− 43.4164i	− 1.71887i
$639$	0	0
$640$	2.23607	0.0883883
$641$	−39.3050	−1.55245	−0.776226	−	0.630455i	$-0.782869\pi$
$641$	−39.3050	−1.55245	−0.776226	+	0.630455i	$0.782869\pi$
$642$	0	0
$643$	36.9443i	1.45694i	0.685078	+	0.728470i	$0.259768\pi$
$643$	36.9443i	1.45694i	−0.685078	+	0.728470i	$0.740232\pi$
$644$	−8.94427	−0.352454
$645$	0	0
$646$	20.0000	0.786889
$647$	9.70820i	0.381669i	0.981622	+	0.190834i	$0.0611194\pi$
$647$	9.70820i	0.381669i	−0.981622	+	0.190834i	$0.938881\pi$
$648$	0	0
$649$	2.11146	0.0828819
$650$	−5.00000	−0.196116
$651$	0	0
$652$	4.94427i	0.193633i
$653$	− 14.3607i	− 0.561977i	−0.959711	−	0.280988i	$-0.909338\pi$
$653$	− 14.3607i	− 0.561977i	0.959711	−	0.280988i	$-0.0906623\pi$
$654$	0	0
$655$	− 8.29180i	− 0.323987i
$656$	−12.4721	−0.486955
$657$	0	0
$658$	− 6.11146i	− 0.238249i
$659$	48.0689	1.87250	0.936249	−	0.351337i	$-0.114273\pi$
$659$	48.0689	1.87250	0.936249	+	0.351337i	$0.114273\pi$
$660$	0	0
$661$	−28.2918	−1.10042	−0.550212	−	0.835025i	$-0.685453\pi$
$661$	−28.2918	−1.10042	−0.550212	+	0.835025i	$0.685453\pi$
$662$	7.23607i	0.281238i
$663$	0	0
$664$	12.9443	0.502335
$665$	20.0000	0.775567
$666$	0	0
$667$	− 70.2492i	− 2.72006i
$668$	− 7.41641i	− 0.286949i
$669$	0	0
$670$	0	0
$671$	31.0557	1.19889
$672$	0	0
$673$	44.0000i	1.69608i	0.529936	+	0.848038i	$0.322216\pi$
$673$	44.0000i	1.69608i	−0.529936	+	0.848038i	$0.677784\pi$
$674$	−2.47214	−0.0952231
$675$	0	0
$676$	1.00000	0.0384615
$677$	36.4721i	1.40174i	0.713290	+	0.700869i	$0.247204\pi$
$677$	36.4721i	1.40174i	−0.713290	+	0.700869i	$0.752796\pi$
$678$	0	0
$679$	12.0000	0.460518
$680$	6.18034i	0.237005i
$681$	0	0
$682$	17.8885i	0.684988i
$683$	6.11146i	0.233848i	0.993141	+	0.116924i	$0.0373035\pi$
$683$	6.11146i	0.233848i	−0.993141	+	0.116924i	$0.962697\pi$
$684$	0	0
$685$	35.5279	1.35745
$686$	15.4164	0.588601
$687$	0	0
$688$	6.47214i	0.246748i
$689$	8.47214	0.322763
$690$	0	0
$691$	13.7082	0.521485	0.260742	−	0.965408i	$-0.416033\pi$
$691$	13.7082	0.521485	0.260742	+	0.965408i	$0.416033\pi$
$692$	− 5.05573i	− 0.192190i
$693$	0	0
$694$	10.4721	0.397517
$695$	5.52786i	0.209684i
$696$	0	0
$697$	− 34.4721i	− 1.30573i
$698$	− 10.7639i	− 0.407421i
$699$	0	0
$700$	6.18034i	0.233595i
$701$	−34.0689	−1.28676	−0.643382	−	0.765545i	$-0.722469\pi$
$701$	−34.0689	−1.28676	−0.643382	+	0.765545i	$0.722469\pi$
$702$	0	0
$703$	− 50.2492i	− 1.89519i
$704$	−4.47214	−0.168550
$705$	0	0
$706$	9.05573	0.340817
$707$	5.88854i	0.221461i
$708$	0	0
$709$	34.5410	1.29722	0.648608	−	0.761123i	$-0.275352\pi$
$709$	34.5410	1.29722	0.648608	+	0.761123i	$0.275352\pi$
$710$	14.4721	0.543130
$711$	0	0
$712$	− 8.47214i	− 0.317507i
$713$	28.9443i	1.08397i
$714$	0	0
$715$	10.0000	0.373979
$716$	14.1803	0.529944
$717$	0	0
$718$	2.47214i	0.0922593i
$719$	3.05573	0.113959	0.0569797	−	0.998375i	$-0.481853\pi$
$719$	3.05573	0.113959	0.0569797	+	0.998375i	$0.481853\pi$
$720$	0	0
$721$	0.583592	0.0217341
$722$	− 33.3607i	− 1.24156i
$723$	0	0
$724$	9.41641	0.349958
$725$	−48.5410	−1.80277
$726$	0	0
$727$	− 18.0000i	− 0.667583i	−0.942647	−	0.333792i	$-0.891672\pi$
$727$	− 18.0000i	− 0.667583i	0.942647	−	0.333792i	$-0.108328\pi$
$728$	− 1.23607i	− 0.0458117i
$729$	0	0
$730$	− 19.5967i	− 0.725308i
$731$	−17.8885	−0.661632
$732$	0	0
$733$	6.58359i	0.243171i	0.992581	+	0.121585i	$0.0387978\pi$
$733$	6.58359i	0.243171i	−0.992581	+	0.121585i	$0.961202\pi$
$734$	8.47214	0.312712
$735$	0	0
$736$	−7.23607	−0.266725
$737$	0	0
$738$	0	0
$739$	13.7082	0.504264	0.252132	−	0.967693i	$-0.418868\pi$
$739$	13.7082	0.504264	0.252132	+	0.967693i	$0.418868\pi$
$740$	15.5279	0.570816
$741$	0	0
$742$	− 10.4721i	− 0.384444i
$743$	16.3607i	0.600215i	0.953905	+	0.300108i	$0.0970226\pi$
$743$	16.3607i	0.600215i	−0.953905	+	0.300108i	$0.902977\pi$
$744$	0	0
$745$	− 50.2492i	− 1.84099i
$746$	2.00000	0.0732252
$747$	0	0
$748$	− 12.3607i	− 0.451951i
$749$	12.9443	0.472973
$750$	0	0
$751$	6.83282	0.249333	0.124666	−	0.992199i	$-0.460214\pi$
$751$	6.83282	0.249333	0.124666	+	0.992199i	$0.460214\pi$
$752$	− 4.94427i	− 0.180299i
$753$	0	0
$754$	9.70820	0.353552
$755$	− 16.5836i	− 0.603539i
$756$	0	0
$757$	38.0000i	1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$	38.0000i	1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$	5.12461i	0.186134i
$759$	0	0
$760$	16.1803	0.586923
$761$	−16.4721	−0.597114	−0.298557	−	0.954392i	$-0.596505\pi$
$761$	−16.4721	−0.597114	−0.298557	+	0.954392i	$0.596505\pi$
$762$	0	0
$763$	19.4164i	0.702921i
$764$	13.5279	0.489421
$765$	0	0
$766$	−14.4721	−0.522900
$767$	0.472136i	0.0170478i
$768$	0	0
$769$	−48.8328	−1.76096	−0.880478	−	0.474087i	$-0.842778\pi$
$769$	−48.8328	−1.76096	−0.880478	+	0.474087i	$0.842778\pi$
$770$	− 12.3607i	− 0.445448i
$771$	0	0
$772$	21.7082i	0.781295i
$773$	− 14.9443i	− 0.537508i	−0.963209	−	0.268754i	$-0.913388\pi$
$773$	− 14.9443i	− 0.537508i	0.963209	−	0.268754i	$-0.0866119\pi$
$774$	0	0
$775$	20.0000	0.718421
$776$	9.70820	0.348504
$777$	0	0
$778$	18.6525i	0.668724i
$779$	−90.2492	−3.23351
$780$	0	0
$781$	−28.9443	−1.03571
$782$	− 20.0000i	− 0.715199i
$783$	0	0
$784$	5.47214	0.195433
$785$	2.36068	0.0842563
$786$	0	0
$787$	− 11.0557i	− 0.394094i	−0.980394	−	0.197047i	$-0.936865\pi$
$787$	− 11.0557i	− 0.394094i	0.980394	−	0.197047i	$-0.0631352\pi$
$788$	− 2.94427i	− 0.104885i
$789$	0	0
$790$	−8.94427	−0.318223
$791$	−15.6393	−0.556070
$792$	0	0
$793$	6.94427i	0.246598i
$794$	9.41641	0.334176
$795$	0	0
$796$	16.9443	0.600574
$797$	26.9443i	0.954415i	0.878791	+	0.477208i	$0.158351\pi$
$797$	26.9443i	0.954415i	−0.878791	+	0.477208i	$0.841649\pi$
$798$	0	0
$799$	13.6656	0.483455
$800$	5.00000i	0.176777i
$801$	0	0
$802$	− 15.5279i	− 0.548308i
$803$	39.1935i	1.38311i
$804$	0	0
$805$	− 20.0000i	− 0.704907i
$806$	−4.00000	−0.140894
$807$	0	0
$808$	4.76393i	0.167595i
$809$	40.4721	1.42292	0.711462	−	0.702724i	$-0.248033\pi$
$809$	40.4721	1.42292	0.711462	+	0.702724i	$0.248033\pi$
$810$	0	0
$811$	17.7082	0.621819	0.310910	−	0.950439i	$-0.399366\pi$
$811$	17.7082	0.621819	0.310910	+	0.950439i	$0.399366\pi$
$812$	− 12.0000i	− 0.421117i
$813$	0	0
$814$	−31.0557	−1.08850
$815$	−11.0557	−0.387265
$816$	0	0
$817$	46.8328i	1.63847i
$818$	6.00000i	0.209785i
$819$	0	0
$820$	− 27.8885i	− 0.973910i
$821$	21.5279	0.751328	0.375664	−	0.926756i	$-0.377415\pi$
$821$	21.5279	0.751328	0.375664	+	0.926756i	$0.377415\pi$
$822$	0	0
$823$	− 32.2492i	− 1.12414i	−0.827091	−	0.562069i	$-0.810006\pi$
$823$	− 32.2492i	− 1.12414i	0.827091	−	0.562069i	$-0.189994\pi$
$824$	0.472136	0.0164476
$825$	0	0
$826$	0.583592	0.0203058
$827$	− 6.11146i	− 0.212516i	−0.994339	−	0.106258i	$-0.966113\pi$
$827$	− 6.11146i	− 0.212516i	0.994339	−	0.106258i	$-0.0338870\pi$
$828$	0	0
$829$	46.7214	1.62270	0.811350	−	0.584561i	$-0.198733\pi$
$829$	46.7214	1.62270	0.811350	+	0.584561i	$0.198733\pi$
$830$	28.9443i	1.00467i
$831$	0	0
$832$	− 1.00000i	− 0.0346688i
$833$	15.1246i	0.524037i
$834$	0	0
$835$	16.5836	0.573899
$836$	−32.3607	−1.11922
$837$	0	0
$838$	21.5967i	0.746047i
$839$	19.7771	0.682781	0.341390	−	0.939922i	$-0.389102\pi$
$839$	19.7771	0.682781	0.341390	+	0.939922i	$0.389102\pi$
$840$	0	0
$841$	65.2492	2.24997
$842$	0.291796i	0.0100560i
$843$	0	0
$844$	21.8885	0.753435
$845$	2.23607i	0.0769231i
$846$	0	0
$847$	11.1246i	0.382246i
$848$	− 8.47214i	− 0.290934i
$849$	0	0
$850$	−13.8197	−0.474010
$851$	−50.2492	−1.72252
$852$	0	0
$853$	14.5836i	0.499333i	0.968332	+	0.249666i	$0.0803209\pi$
$853$	14.5836i	0.499333i	−0.968332	+	0.249666i	$0.919679\pi$
$854$	8.58359	0.293724
$855$	0	0
$856$	10.4721	0.357930
$857$	22.1803i	0.757666i	0.925465	+	0.378833i	$0.123675\pi$
$857$	22.1803i	0.757666i	−0.925465	+	0.378833i	$0.876325\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	−14.4721	−0.493496
$861$	0	0
$862$	38.8328i	1.32265i
$863$	− 13.5279i	− 0.460494i	−0.973132	−	0.230247i	$-0.926047\pi$
$863$	− 13.5279i	− 0.460494i	0.973132	−	0.230247i	$-0.0739534\pi$
$864$	0	0
$865$	11.3050	0.384380
$866$	−18.4721	−0.627709
$867$	0	0
$868$	4.94427i	0.167820i
$869$	17.8885	0.606827
$870$	0	0
$871$	0	0
$872$	15.7082i	0.531947i
$873$	0	0
$874$	−52.3607	−1.77113
$875$	−13.8197	−0.467190
$876$	0	0
$877$	− 15.5279i	− 0.524339i	−0.965022	−	0.262169i	$-0.915562\pi$
$877$	− 15.5279i	− 0.524339i	0.965022	−	0.262169i	$-0.0844379\pi$
$878$	12.9443i	0.436848i
$879$	0	0
$880$	− 10.0000i	− 0.337100i
$881$	10.3607	0.349060	0.174530	−	0.984652i	$-0.444159\pi$
$881$	10.3607	0.349060	0.174530	+	0.984652i	$0.444159\pi$
$882$	0	0
$883$	− 9.88854i	− 0.332776i	−0.986060	−	0.166388i	$-0.946790\pi$
$883$	− 9.88854i	− 0.332776i	0.986060	−	0.166388i	$-0.0532104\pi$
$884$	2.76393	0.0929611
$885$	0	0
$886$	13.5279	0.454477
$887$	12.1803i	0.408976i	0.978869	+	0.204488i	$0.0655529\pi$
$887$	12.1803i	0.408976i	−0.978869	+	0.204488i	$0.934447\pi$
$888$	0	0
$889$	−6.69505	−0.224545
$890$	18.9443	0.635013
$891$	0	0
$892$	− 21.2361i	− 0.711036i
$893$	− 35.7771i	− 1.19723i
$894$	0	0
$895$	31.7082i	1.05989i
$896$	−1.23607	−0.0412941
$897$	0	0
$898$	8.47214i	0.282719i
$899$	−38.8328	−1.29515
$900$	0	0
$901$	23.4164	0.780114
$902$	55.7771i	1.85717i
$903$	0	0
$904$	−12.6525	−0.420815
$905$	21.0557i	0.699916i
$906$	0	0
$907$	− 54.4721i	− 1.80872i	−0.426773	−	0.904359i	$-0.640350\pi$
$907$	− 54.4721i	− 1.80872i	0.426773	−	0.904359i	$-0.359650\pi$
$908$	− 24.9443i	− 0.827805i
$909$	0	0
$910$	2.76393	0.0916235
$911$	28.9443	0.958967	0.479483	−	0.877551i	$-0.340824\pi$
$911$	28.9443	0.958967	0.479483	+	0.877551i	$0.340824\pi$
$912$	0	0
$913$	− 57.8885i	− 1.91583i
$914$	−9.70820	−0.321119
$915$	0	0
$916$	−3.70820	−0.122523
$917$	4.58359i	0.151364i
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	− 16.1803i	− 0.533450i
$921$	0	0
$922$	6.11146i	0.201270i
$923$	− 6.47214i	− 0.213033i
$924$	0	0
$925$	34.7214i	1.14163i
$926$	−35.7082	−1.17344
$927$	0	0
$928$	− 9.70820i	− 0.318687i
$929$	23.5279	0.771924	0.385962	−	0.922515i	$-0.373870\pi$
$929$	23.5279	0.771924	0.385962	+	0.922515i	$0.373870\pi$
$930$	0	0
$931$	39.5967	1.29773
$932$	− 2.18034i	− 0.0714194i
$933$	0	0
$934$	−2.47214	−0.0808908
$935$	27.6393	0.903902
$936$	0	0
$937$	51.7771i	1.69148i	0.533592	+	0.845742i	$0.320842\pi$
$937$	51.7771i	1.69148i	−0.533592	+	0.845742i	$0.679158\pi$
$938$	0	0
$939$	0	0
$940$	11.0557	0.360598
$941$	46.4721	1.51495	0.757474	−	0.652865i	$-0.226433\pi$
$941$	46.4721	1.51495	0.757474	+	0.652865i	$0.226433\pi$
$942$	0	0
$943$	90.2492i	2.93892i
$944$	0.472136	0.0153667
$945$	0	0
$946$	28.9443	0.941059
$947$	− 12.9443i	− 0.420632i	−0.977633	−	0.210316i	$-0.932551\pi$
$947$	− 12.9443i	− 0.420632i	0.977633	−	0.210316i	$-0.0674493\pi$
$948$	0	0
$949$	−8.76393	−0.284489
$950$	36.1803i	1.17385i
$951$	0	0
$952$	− 3.41641i	− 0.110726i
$953$	11.1246i	0.360362i	0.983634	+	0.180181i	$0.0576683\pi$
$953$	11.1246i	0.360362i	−0.983634	+	0.180181i	$0.942332\pi$
$954$	0	0
$955$	30.2492i	0.978842i
$956$	−13.8885	−0.449188
$957$	0	0
$958$	36.0000i	1.16311i
$959$	−19.6393	−0.634187
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 6.94427i	− 0.223892i
$963$	0	0
$964$	−12.4721	−0.401700
$965$	−48.5410	−1.56259
$966$	0	0
$967$	− 10.5410i	− 0.338976i	−0.985532	−	0.169488i	$-0.945789\pi$
$967$	− 10.5410i	− 0.338976i	0.985532	−	0.169488i	$-0.0542115\pi$
$968$	9.00000i	0.289271i
$969$	0	0
$970$	21.7082i	0.697008i
$971$	−18.7639	−0.602163	−0.301082	−	0.953598i	$-0.597348\pi$
$971$	−18.7639	−0.602163	−0.301082	+	0.953598i	$0.597348\pi$
$972$	0	0
$973$	− 3.05573i	− 0.0979621i
$974$	−25.5967	−0.820173
$975$	0	0
$976$	6.94427	0.222281
$977$	34.3607i	1.09930i	0.835397	+	0.549648i	$0.185238\pi$
$977$	34.3607i	1.09930i	−0.835397	+	0.549648i	$0.814762\pi$
$978$	0	0
$979$	−37.8885	−1.21092
$980$	12.2361i	0.390867i
$981$	0	0
$982$	6.18034i	0.197223i
$983$	− 19.4164i	− 0.619287i	−0.950853	−	0.309644i	$-0.899790\pi$
$983$	− 19.4164i	− 0.619287i	0.950853	−	0.309644i	$-0.100210\pi$
$984$	0	0
$985$	6.58359	0.209771
$986$	26.8328	0.854531
$987$	0	0
$988$	− 7.23607i	− 0.230210i
$989$	46.8328	1.48920
$990$	0	0
$991$	7.05573	0.224133	0.112066	−	0.993701i	$-0.464253\pi$
$991$	7.05573	0.224133	0.112066	+	0.993701i	$0.464253\pi$
$992$	4.00000i	0.127000i
$993$	0	0
$994$	−8.00000	−0.253745
$995$	37.8885i	1.20115i
$996$	0	0
$997$	− 8.11146i	− 0.256892i	−0.991716	−	0.128446i	$-0.959001\pi$
$997$	− 8.11146i	− 0.256892i	0.991716	−	0.128446i	$-0.0409990\pi$
$998$	1.12461i	0.0355990i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.e.e.469.1		4
3.2	odd	2		390.2.e.e.79.4	yes	4
5.2	odd	4		5850.2.a.cm.1.1		2
5.3	odd	4		5850.2.a.cf.1.2		2
5.4	even	2	inner	1170.2.e.e.469.4		4
12.11	even	2		3120.2.l.k.1249.2		4
15.2	even	4		1950.2.a.be.1.1		2
15.8	even	4		1950.2.a.bf.1.2		2
15.14	odd	2		390.2.e.e.79.1	✓	4
60.59	even	2		3120.2.l.k.1249.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.e.e.79.1	✓	4	15.14	odd	2
390.2.e.e.79.4	yes	4	3.2	odd	2
1170.2.e.e.469.1		4	1.1	even	1	trivial
1170.2.e.e.469.4		4	5.4	even	2	inner
1950.2.a.be.1.1		2	15.2	even	4
1950.2.a.bf.1.2		2	15.8	even	4
3120.2.l.k.1249.2		4	12.11	even	2
3120.2.l.k.1249.3		4	60.59	even	2
5850.2.a.cf.1.2		2	5.3	odd	4
5850.2.a.cm.1.1		2	5.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -2.23607i q^{5} +1.23607i q^{7} +1.00000i q^{8} -2.23607 q^{10} +4.47214 q^{11} +1.00000i q^{13} +1.23607 q^{14} +1.00000 q^{16} +2.76393i q^{17} +7.23607 q^{19} +2.23607i q^{20} -4.47214i q^{22} -7.23607i q^{23} -5.00000 q^{25} +1.00000 q^{26} -1.23607i q^{28} +9.70820 q^{29} -4.00000 q^{31} -1.00000i q^{32} +2.76393 q^{34} +2.76393 q^{35} -6.94427i q^{37} -7.23607i q^{38} +2.23607 q^{40} -12.4721 q^{41} +6.47214i q^{43} -4.47214 q^{44} -7.23607 q^{46} -4.94427i q^{47} +5.47214 q^{49} +5.00000i q^{50} -1.00000i q^{52} -8.47214i q^{53} -10.0000i q^{55} -1.23607 q^{56} -9.70820i q^{58} +0.472136 q^{59} +6.94427 q^{61} +4.00000i q^{62} -1.00000 q^{64} +2.23607 q^{65} -2.76393i q^{68} -2.76393i q^{70} -6.47214 q^{71} +8.76393i q^{73} -6.94427 q^{74} -7.23607 q^{76} +5.52786i q^{77} +4.00000 q^{79} -2.23607i q^{80} +12.4721i q^{82} -12.9443i q^{83} +6.18034 q^{85} +6.47214 q^{86} +4.47214i q^{88} -8.47214 q^{89} -1.23607 q^{91} +7.23607i q^{92} -4.94427 q^{94} -16.1803i q^{95} -9.70820i q^{97} -5.47214i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{14} + 4 q^{16} + 20 q^{19} - 20 q^{25} + 4 q^{26} + 12 q^{29} - 16 q^{31} + 20 q^{34} + 20 q^{35} - 32 q^{41} - 20 q^{46} + 4 q^{49} + 4 q^{56} - 16 q^{59} - 8 q^{61} - 4 q^{64} - 8 q^{71}+ \cdots + 16 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^14 + 4 * q^16 + 20 * q^19 - 20 * q^25 + 4 * q^26 + 12 * q^29 - 16 * q^31 + 20 * q^34 + 20 * q^35 - 32 * q^41 - 20 * q^46 + 4 * q^49 + 4 * q^56 - 16 * q^59 - 8 * q^61 - 4 * q^64 - 8 * q^71 + 8 * q^74 - 20 * q^76 + 16 * q^79 - 20 * q^85 + 8 * q^86 - 16 * q^89 + 4 * q^91 + 16 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more