LMFDB - Embedded newform 3267.2.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3267,2,Mod(1,3267)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3267, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3267.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3267 = 3^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3267.a (trivial)

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:	yes
Analytic conductor:	$26.0871263404$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} + 9 $ x^4 - 8*x^2 + 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.16372$ of defining polynomial
Character	$\chi$	$=$	3267.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.16372 q^{2} -0.645751 q^{4} -1.64575 q^{5} +1.16372 q^{7} -3.07892 q^{8} +O(q^{10})$	$q+1.16372 q^{2} -0.645751 q^{4} -1.64575 q^{5} +1.16372 q^{7} -3.07892 q^{8} -1.91520 q^{10} +4.24264 q^{13} +1.35425 q^{14} -2.29150 q^{16} +3.07892 q^{17} -6.57008 q^{19} +1.06275 q^{20} +1.35425 q^{23} -2.29150 q^{25} +4.93725 q^{26} -0.751475 q^{28} -1.16372 q^{29} -3.64575 q^{31} +3.49117 q^{32} +3.58301 q^{34} -1.91520 q^{35} -3.35425 q^{37} -7.64575 q^{38} +5.06713 q^{40} +9.64900 q^{41} +10.0613 q^{43} +1.57597 q^{46} +12.2915 q^{47} -5.64575 q^{49} -2.66667 q^{50} -2.73969 q^{52} +6.00000 q^{53} -3.58301 q^{56} -1.35425 q^{58} +7.93725 q^{59} +1.91520 q^{61} -4.24264 q^{62} +8.64575 q^{64} -6.98233 q^{65} -3.64575 q^{67} -1.98822 q^{68} -2.22876 q^{70} -9.29150 q^{71} +10.8127 q^{73} -3.90341 q^{74} +4.24264 q^{76} -0.751475 q^{79} +3.77124 q^{80} +11.2288 q^{82} -8.89753 q^{83} -5.06713 q^{85} +11.7085 q^{86} +12.5830 q^{89} +4.93725 q^{91} -0.874508 q^{92} +14.3039 q^{94} +10.8127 q^{95} -5.93725 q^{97} -6.57008 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} + 4 q^{5}+O(q^{10}) $ 4 * q + 8 * q^4 + 4 * q^5	$ 4 q + 8 q^{4} + 4 q^{5} + 16 q^{14} + 12 q^{16} + 36 q^{20} + 16 q^{23} + 12 q^{25} - 12 q^{26} - 4 q^{31} - 28 q^{34} - 24 q^{37} - 20 q^{38} + 28 q^{47} - 12 q^{49} + 24 q^{53} + 28 q^{56} - 16 q^{58} + 24 q^{64} - 4 q^{67} + 44 q^{70} - 16 q^{71} + 68 q^{80} - 8 q^{82} + 68 q^{86} + 8 q^{89} - 12 q^{91} + 60 q^{92} + 8 q^{97}+O(q^{100}) $ 4 * q + 8 * q^4 + 4 * q^5 + 16 * q^14 + 12 * q^16 + 36 * q^20 + 16 * q^23 + 12 * q^25 - 12 * q^26 - 4 * q^31 - 28 * q^34 - 24 * q^37 - 20 * q^38 + 28 * q^47 - 12 * q^49 + 24 * q^53 + 28 * q^56 - 16 * q^58 + 24 * q^64 - 4 * q^67 + 44 * q^70 - 16 * q^71 + 68 * q^80 - 8 * q^82 + 68 * q^86 + 8 * q^89 - 12 * q^91 + 60 * q^92 + 8 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	1.16372	0.822876	0.411438	−	0.911438i	$-0.365027\pi$
$2$	1.16372	0.822876	0.411438	+	0.911438i	$0.365027\pi$
$3$	0	0
$3$	0	0
$4$	−0.645751	−0.322876
$5$	−1.64575	−0.736002	−0.368001	−	0.929825i	$-0.619958\pi$
$5$	−1.64575	−0.736002	−0.368001	+	0.929825i	$0.619958\pi$
$6$	0	0
$7$	1.16372	0.439846	0.219923	−	0.975517i	$-0.429419\pi$
$7$	1.16372	0.439846	0.219923	+	0.975517i	$0.429419\pi$
$8$	−3.07892	−1.08856
$9$	0	0
$10$	−1.91520	−0.605638
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.24264	1.17670	0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264	1.17670	0.588348	+	0.808608i	$0.299778\pi$
$14$	1.35425	0.361938
$15$	0	0
$16$	−2.29150	−0.572876
$17$	3.07892	0.746747	0.373374	−	0.927681i	$-0.378201\pi$
$17$	3.07892	0.746747	0.373374	+	0.927681i	$0.378201\pi$
$18$	0	0
$19$	−6.57008	−1.50728	−0.753640	−	0.657287i	$-0.771704\pi$
$19$	−6.57008	−1.50728	−0.753640	+	0.657287i	$0.771704\pi$
$20$	1.06275	0.237637
$21$	0	0
$22$	0	0
$23$	1.35425	0.282380	0.141190	−	0.989982i	$-0.454907\pi$
$23$	1.35425	0.282380	0.141190	+	0.989982i	$0.454907\pi$
$24$	0	0
$25$	−2.29150	−0.458301
$26$	4.93725	0.968275
$27$	0	0
$28$	−0.751475	−0.142015
$29$	−1.16372	−0.216098	−0.108049	−	0.994146i	$-0.534460\pi$
$29$	−1.16372	−0.216098	−0.108049	+	0.994146i	$0.534460\pi$
$30$	0	0
$31$	−3.64575	−0.654796	−0.327398	−	0.944886i	$-0.606172\pi$
$31$	−3.64575	−0.654796	−0.327398	+	0.944886i	$0.606172\pi$
$32$	3.49117	0.617157
$33$	0	0
$34$	3.58301	0.614480
$35$	−1.91520	−0.323727
$36$	0	0
$37$	−3.35425	−0.551435	−0.275718	−	0.961239i	$-0.588915\pi$
$37$	−3.35425	−0.551435	−0.275718	+	0.961239i	$0.588915\pi$
$38$	−7.64575	−1.24030
$39$	0	0
$40$	5.06713	0.801184
$41$	9.64900	1.50692	0.753461	−	0.657493i	$-0.228383\pi$
$41$	9.64900	1.50692	0.753461	+	0.657493i	$0.228383\pi$
$42$	0	0
$43$	10.0613	1.53433	0.767163	−	0.641452i	$-0.221668\pi$
$43$	10.0613	1.53433	0.767163	+	0.641452i	$0.221668\pi$
$44$	0	0
$45$	0	0
$46$	1.57597	0.232364
$47$	12.2915	1.79290	0.896450	−	0.443145i	$-0.146137\pi$
$47$	12.2915	1.79290	0.896450	+	0.443145i	$0.146137\pi$
$48$	0	0
$49$	−5.64575	−0.806536
$50$	−2.66667	−0.377124
$51$	0	0
$52$	−2.73969	−0.379927
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	−3.58301	−0.478799
$57$	0	0
$58$	−1.35425	−0.177822
$59$	7.93725	1.03334	0.516671	−	0.856184i	$-0.327171\pi$
$59$	7.93725	1.03334	0.516671	+	0.856184i	$0.327171\pi$
$60$	0	0
$61$	1.91520	0.245216	0.122608	−	0.992455i	$-0.460874\pi$
$61$	1.91520	0.245216	0.122608	+	0.992455i	$0.460874\pi$
$62$	−4.24264	−0.538816
$63$	0	0
$64$	8.64575	1.08072
$65$	−6.98233	−0.866052
$66$	0	0
$67$	−3.64575	−0.445399	−0.222700	−	0.974887i	$-0.571487\pi$
$67$	−3.64575	−0.445399	−0.222700	+	0.974887i	$0.571487\pi$
$68$	−1.98822	−0.241107
$69$	0	0
$70$	−2.22876	−0.266387
$71$	−9.29150	−1.10270	−0.551349	−	0.834275i	$-0.685887\pi$
$71$	−9.29150	−1.10270	−0.551349	+	0.834275i	$0.685887\pi$
$72$	0	0
$73$	10.8127	1.26553	0.632767	−	0.774342i	$-0.281919\pi$
$73$	10.8127	1.26553	0.632767	+	0.774342i	$0.281919\pi$
$74$	−3.90341	−0.453763
$75$	0	0
$76$	4.24264	0.486664
$77$	0	0
$78$	0	0
$79$	−0.751475	−0.0845475	−0.0422738	−	0.999106i	$-0.513460\pi$
$79$	−0.751475	−0.0845475	−0.0422738	+	0.999106i	$0.513460\pi$
$80$	3.77124	0.421638
$81$	0	0
$82$	11.2288	1.24001
$83$	−8.89753	−0.976631	−0.488315	−	0.872667i	$-0.662388\pi$
$83$	−8.89753	−0.976631	−0.488315	+	0.872667i	$0.662388\pi$
$84$	0	0
$85$	−5.06713	−0.549608
$86$	11.7085	1.26256
$87$	0	0
$88$	0	0
$89$	12.5830	1.33380	0.666898	−	0.745149i	$-0.267622\pi$
$89$	12.5830	1.33380	0.666898	+	0.745149i	$0.267622\pi$
$90$	0	0
$91$	4.93725	0.517565
$92$	−0.874508	−0.0911737
$93$	0	0
$94$	14.3039	1.47533
$95$	10.8127	1.10936
$96$	0	0
$97$	−5.93725	−0.602837	−0.301418	−	0.953492i	$-0.597460\pi$
$97$	−5.93725	−0.602837	−0.301418	+	0.953492i	$0.597460\pi$
$98$	−6.57008	−0.663679
$99$	0	0
$100$	1.47974	0.147974
$101$	10.0613	1.00113	0.500566	−	0.865698i	$-0.333125\pi$
$101$	10.0613	1.00113	0.500566	+	0.865698i	$0.333125\pi$
$102$	0	0
$103$	16.2288	1.59907	0.799533	−	0.600622i	$-0.205080\pi$
$103$	16.2288	1.59907	0.799533	+	0.600622i	$0.205080\pi$
$104$	−13.0627	−1.28091
$105$	0	0
$106$	6.98233	0.678184
$107$	−6.57008	−0.635154	−0.317577	−	0.948232i	$-0.602869\pi$
$107$	−6.57008	−0.635154	−0.317577	+	0.948232i	$0.602869\pi$
$108$	0	0
$109$	15.4676	1.48153	0.740764	−	0.671765i	$-0.234464\pi$
$109$	15.4676	1.48153	0.740764	+	0.671765i	$0.234464\pi$
$110$	0	0
$111$	0	0
$112$	−2.66667	−0.251977
$113$	16.9373	1.59332	0.796661	−	0.604426i	$-0.206597\pi$
$113$	16.9373	1.59332	0.796661	+	0.604426i	$0.206597\pi$
$114$	0	0
$115$	−2.22876	−0.207833
$116$	0.751475	0.0697727
$117$	0	0
$118$	9.23676	0.850312
$119$	3.58301	0.328454
$120$	0	0
$121$	0	0
$122$	2.22876	0.201782
$123$	0	0
$124$	2.35425	0.211418
$125$	12.0000	1.07331
$126$	0	0
$127$	−2.73969	−0.243108	−0.121554	−	0.992585i	$-0.538788\pi$
$127$	−2.73969	−0.243108	−0.121554	+	0.992585i	$0.538788\pi$
$128$	3.07892	0.272141
$129$	0	0
$130$	−8.12549	−0.712653
$131$	20.1225	1.75811	0.879056	−	0.476719i	$-0.158174\pi$
$131$	20.1225	1.75811	0.879056	+	0.476719i	$0.158174\pi$
$132$	0	0
$133$	−7.64575	−0.662971
$134$	−4.24264	−0.366508
$135$	0	0
$136$	−9.47974	−0.812881
$137$	−9.29150	−0.793827	−0.396913	−	0.917856i	$-0.629919\pi$
$137$	−9.29150	−0.793827	−0.396913	+	0.917856i	$0.629919\pi$
$138$	0	0
$139$	0.751475	0.0637393	0.0318696	−	0.999492i	$-0.489854\pi$
$139$	0.751475	0.0637393	0.0318696	+	0.999492i	$0.489854\pi$
$140$	1.23674	0.104524
$141$	0	0
$142$	−10.8127	−0.907384
$143$	0	0
$144$	0	0
$145$	1.91520	0.159048
$146$	12.5830	1.04138
$147$	0	0
$148$	2.16601	0.178045
$149$	15.0554	1.23338	0.616692	−	0.787205i	$-0.288472\pi$
$149$	15.0554	1.23338	0.616692	+	0.787205i	$0.288472\pi$
$150$	0	0
$151$	−12.7279	−1.03578	−0.517892	−	0.855446i	$-0.673283\pi$
$151$	−12.7279	−1.03578	−0.517892	+	0.855446i	$0.673283\pi$
$152$	20.2288	1.64077
$153$	0	0
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	−2.29150	−0.182882	−0.0914409	−	0.995811i	$-0.529147\pi$
$157$	−2.29150	−0.182882	−0.0914409	+	0.995811i	$0.529147\pi$
$158$	−0.874508	−0.0695721
$159$	0	0
$160$	−5.74559	−0.454229
$161$	1.57597	0.124204
$162$	0	0
$163$	17.2915	1.35438	0.677188	−	0.735810i	$-0.263199\pi$
$163$	17.2915	1.35438	0.677188	+	0.735810i	$0.263199\pi$
$164$	−6.23086	−0.486548
$165$	0	0
$166$	−10.3542	−0.803646
$167$	−22.0377	−1.70533	−0.852664	−	0.522459i	$-0.825015\pi$
$167$	−22.0377	−1.70533	−0.852664	+	0.522459i	$0.825015\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	−5.89674	−0.452259
$171$	0	0
$172$	−6.49707	−0.495397
$173$	−12.7279	−0.967686	−0.483843	−	0.875155i	$-0.660759\pi$
$173$	−12.7279	−0.967686	−0.483843	+	0.875155i	$0.660759\pi$
$174$	0	0
$175$	−2.66667	−0.201581
$176$	0	0
$177$	0	0
$178$	14.6431	1.09755
$179$	18.2915	1.36717	0.683586	−	0.729870i	$-0.260420\pi$
$179$	18.2915	1.36717	0.683586	+	0.729870i	$0.260420\pi$
$180$	0	0
$181$	20.8745	1.55159	0.775795	−	0.630985i	$-0.217349\pi$
$181$	20.8745	1.55159	0.775795	+	0.630985i	$0.217349\pi$
$182$	5.74559	0.425892
$183$	0	0
$184$	−4.16962	−0.307389
$185$	5.52026	0.405858
$186$	0	0
$187$	0	0
$188$	−7.93725	−0.578884
$189$	0	0
$190$	12.5830	0.912867
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	0	0
$193$	−23.5406	−1.69449	−0.847246	−	0.531200i	$-0.821741\pi$
$193$	−23.5406	−1.69449	−0.847246	+	0.531200i	$0.821741\pi$
$194$	−6.90931	−0.496060
$195$	0	0
$196$	3.64575	0.260411
$197$	1.16372	0.0829118	0.0414559	−	0.999140i	$-0.486800\pi$
$197$	1.16372	0.0829118	0.0414559	+	0.999140i	$0.486800\pi$
$198$	0	0
$199$	8.35425	0.592217	0.296108	−	0.955154i	$-0.404311\pi$
$199$	8.35425	0.592217	0.296108	+	0.955154i	$0.404311\pi$
$200$	7.05535	0.498889
$201$	0	0
$202$	11.7085	0.823807
$203$	−1.35425	−0.0950496
$204$	0	0
$205$	−15.8799	−1.10910
$206$	18.8858	1.31583
$207$	0	0
$208$	−9.72202	−0.674101
$209$	0	0
$210$	0	0
$211$	−5.81861	−0.400570	−0.200285	−	0.979738i	$-0.564187\pi$
$211$	−5.81861	−0.400570	−0.200285	+	0.979738i	$0.564187\pi$
$212$	−3.87451	−0.266102
$213$	0	0
$214$	−7.64575	−0.522653
$215$	−16.5583	−1.12927
$216$	0	0
$217$	−4.24264	−0.288009
$218$	18.0000	1.21911
$219$	0	0
$220$	0	0
$221$	13.0627	0.878695
$222$	0	0
$223$	−5.06275	−0.339027	−0.169513	−	0.985528i	$-0.554220\pi$
$223$	−5.06275	−0.339027	−0.169513	+	0.985528i	$0.554220\pi$
$224$	4.06275	0.271454
$225$	0	0
$226$	19.7103	1.31111
$227$	−22.0377	−1.46269	−0.731347	−	0.682006i	$-0.761108\pi$
$227$	−22.0377	−1.46269	−0.731347	+	0.682006i	$0.761108\pi$
$228$	0	0
$229$	−19.8118	−1.30920	−0.654599	−	0.755976i	$-0.727163\pi$
$229$	−19.8118	−1.30920	−0.654599	+	0.755976i	$0.727163\pi$
$230$	−2.59365	−0.171020
$231$	0	0
$232$	3.58301	0.235236
$233$	25.8681	1.69467	0.847337	−	0.531055i	$-0.178204\pi$
$233$	25.8681	1.69467	0.847337	+	0.531055i	$0.178204\pi$
$234$	0	0
$235$	−20.2288	−1.31958
$236$	−5.12549	−0.333641
$237$	0	0
$238$	4.16962	0.270276
$239$	7.39458	0.478316	0.239158	−	0.970981i	$-0.423129\pi$
$239$	7.39458	0.478316	0.239158	+	0.970981i	$0.423129\pi$
$240$	0	0
$241$	−12.3157	−0.793322	−0.396661	−	0.917965i	$-0.629831\pi$
$241$	−12.3157	−0.793322	−0.396661	+	0.917965i	$0.629831\pi$
$242$	0	0
$243$	0	0
$244$	−1.23674	−0.0791742
$245$	9.29150	0.593612
$246$	0	0
$247$	−27.8745	−1.77361
$248$	11.2250	0.712786
$249$	0	0
$250$	13.9647	0.883203
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	0	0
$253$	0	0
$254$	−3.18824	−0.200048
$255$	0	0
$256$	−13.7085	−0.856781
$257$	10.3542	0.645880	0.322940	−	0.946419i	$-0.395329\pi$
$257$	10.3542	0.645880	0.322940	+	0.946419i	$0.395329\pi$
$258$	0	0
$259$	−3.90341	−0.242546
$260$	4.50885	0.279627
$261$	0	0
$262$	23.4170	1.44671
$263$	−26.2803	−1.62052	−0.810258	−	0.586074i	$-0.800673\pi$
$263$	−26.2803	−1.62052	−0.810258	+	0.586074i	$0.800673\pi$
$264$	0	0
$265$	−9.87451	−0.606586
$266$	−8.89753	−0.545542
$267$	0	0
$268$	2.35425	0.143809
$269$	−5.52026	−0.336576	−0.168288	−	0.985738i	$-0.553824\pi$
$269$	−5.52026	−0.336576	−0.168288	+	0.985738i	$0.553824\pi$
$270$	0	0
$271$	−23.2014	−1.40939	−0.704693	−	0.709512i	$-0.748915\pi$
$271$	−23.2014	−1.40939	−0.704693	+	0.709512i	$0.748915\pi$
$272$	−7.05535	−0.427793
$273$	0	0
$274$	−10.8127	−0.653221
$275$	0	0
$276$	0	0
$277$	−8.48528	−0.509831	−0.254916	−	0.966963i	$-0.582048\pi$
$277$	−8.48528	−0.509831	−0.254916	+	0.966963i	$0.582048\pi$
$278$	0.874508	0.0524495
$279$	0	0
$280$	5.89674	0.352397
$281$	−13.8916	−0.828706	−0.414353	−	0.910116i	$-0.635992\pi$
$281$	−13.8916	−0.828706	−0.414353	+	0.910116i	$0.635992\pi$
$282$	0	0
$283$	21.2132	1.26099	0.630497	−	0.776192i	$-0.282851\pi$
$283$	21.2132	1.26099	0.630497	+	0.776192i	$0.282851\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	11.2288	0.662813
$288$	0	0
$289$	−7.52026	−0.442368
$290$	2.22876	0.130877
$291$	0	0
$292$	−6.98233	−0.408610
$293$	−10.4005	−0.607602	−0.303801	−	0.952735i	$-0.598256\pi$
$293$	−10.4005	−0.607602	−0.303801	+	0.952735i	$0.598256\pi$
$294$	0	0
$295$	−13.0627	−0.760542
$296$	10.3275	0.600271
$297$	0	0
$298$	17.5203	1.01492
$299$	5.74559	0.332276
$300$	0	0
$301$	11.7085	0.674867
$302$	−14.8118	−0.852321
$303$	0	0
$304$	15.0554	0.863484
$305$	−3.15194	−0.180479
$306$	0	0
$307$	−14.3039	−0.816366	−0.408183	−	0.912900i	$-0.633838\pi$
$307$	−14.3039	−0.816366	−0.408183	+	0.912900i	$0.633838\pi$
$308$	0	0
$309$	0	0
$310$	6.98233	0.396570
$311$	19.9373	1.13054	0.565269	−	0.824907i	$-0.308772\pi$
$311$	19.9373	1.13054	0.565269	+	0.824907i	$0.308772\pi$
$312$	0	0
$313$	4.58301	0.259047	0.129523	−	0.991576i	$-0.458655\pi$
$313$	4.58301	0.259047	0.129523	+	0.991576i	$0.458655\pi$
$314$	−2.66667	−0.150489
$315$	0	0
$316$	0.485266	0.0272983
$317$	−9.87451	−0.554608	−0.277304	−	0.960782i	$-0.589441\pi$
$317$	−9.87451	−0.554608	−0.277304	+	0.960782i	$0.589441\pi$
$318$	0	0
$319$	0	0
$320$	−14.2288	−0.795412
$321$	0	0
$322$	1.83399	0.102204
$323$	−20.2288	−1.12556
$324$	0	0
$325$	−9.72202	−0.539281
$326$	20.1225	1.11448
$327$	0	0
$328$	−29.7085	−1.64038
$329$	14.3039	0.788599
$330$	0	0
$331$	10.2288	0.562223	0.281112	−	0.959675i	$-0.409297\pi$
$331$	10.2288	0.562223	0.281112	+	0.959675i	$0.409297\pi$
$332$	5.74559	0.315330
$333$	0	0
$334$	−25.6458	−1.40327
$335$	6.00000	0.327815
$336$	0	0
$337$	−29.0200	−1.58082	−0.790411	−	0.612577i	$-0.790133\pi$
$337$	−29.0200	−1.58082	−0.790411	+	0.612577i	$0.790133\pi$
$338$	5.81861	0.316491
$339$	0	0
$340$	3.27211	0.177455
$341$	0	0
$342$	0	0
$343$	−14.7161	−0.794597
$344$	−30.9778	−1.67021
$345$	0	0
$346$	−14.8118	−0.796285
$347$	11.6372	0.624719	0.312359	−	0.949964i	$-0.398881\pi$
$347$	11.6372	0.624719	0.312359	+	0.949964i	$0.398881\pi$
$348$	0	0
$349$	28.1955	1.50927	0.754636	−	0.656143i	$-0.227813\pi$
$349$	28.1955	1.50927	0.754636	+	0.656143i	$0.227813\pi$
$350$	−3.10326	−0.165876
$351$	0	0
$352$	0	0
$353$	20.7085	1.10220	0.551101	−	0.834439i	$-0.314208\pi$
$353$	20.7085	1.10220	0.551101	+	0.834439i	$0.314208\pi$
$354$	0	0
$355$	15.2915	0.811589
$356$	−8.12549	−0.430650
$357$	0	0
$358$	21.2862	1.12501
$359$	19.7103	1.04027	0.520134	−	0.854085i	$-0.325882\pi$
$359$	19.7103	1.04027	0.520134	+	0.854085i	$0.325882\pi$
$360$	0	0
$361$	24.1660	1.27190
$362$	24.2921	1.27677
$363$	0	0
$364$	−3.18824	−0.167109
$365$	−17.7951	−0.931436
$366$	0	0
$367$	−0.125492	−0.00655064	−0.00327532	−	0.999995i	$-0.501043\pi$
$367$	−0.125492	−0.00655064	−0.00327532	+	0.999995i	$0.501043\pi$
$368$	−3.10326	−0.161769
$369$	0	0
$370$	6.42405	0.333970
$371$	6.98233	0.362505
$372$	0	0
$373$	21.2132	1.09838	0.549189	−	0.835698i	$-0.314937\pi$
$373$	21.2132	1.09838	0.549189	+	0.835698i	$0.314937\pi$
$374$	0	0
$375$	0	0
$376$	−37.8445	−1.95168
$377$	−4.93725	−0.254282
$378$	0	0
$379$	12.3542	0.634595	0.317298	−	0.948326i	$-0.397225\pi$
$379$	12.3542	0.634595	0.317298	+	0.948326i	$0.397225\pi$
$380$	−6.98233	−0.358186
$381$	0	0
$382$	3.49117	0.178624
$383$	−15.8745	−0.811149	−0.405575	−	0.914062i	$-0.632929\pi$
$383$	−15.8745	−0.811149	−0.405575	+	0.914062i	$0.632929\pi$
$384$	0	0
$385$	0	0
$386$	−27.3948	−1.39436
$387$	0	0
$388$	3.83399	0.194641
$389$	34.4575	1.74707	0.873533	−	0.486766i	$-0.161823\pi$
$389$	34.4575	1.74707	0.873533	+	0.486766i	$0.161823\pi$
$390$	0	0
$391$	4.16962	0.210867
$392$	17.3828	0.877964
$393$	0	0
$394$	1.35425	0.0682261
$395$	1.23674	0.0622272
$396$	0	0
$397$	27.1660	1.36342	0.681711	−	0.731621i	$-0.261236\pi$
$397$	27.1660	1.36342	0.681711	+	0.731621i	$0.261236\pi$
$398$	9.72202	0.487321
$399$	0	0
$400$	5.25098	0.262549
$401$	31.7490	1.58547	0.792735	−	0.609566i	$-0.208656\pi$
$401$	31.7490	1.58547	0.792735	+	0.609566i	$0.208656\pi$
$402$	0	0
$403$	−15.4676	−0.770497
$404$	−6.49707	−0.323241
$405$	0	0
$406$	−1.57597	−0.0782140
$407$	0	0
$408$	0	0
$409$	−22.4499	−1.11008	−0.555039	−	0.831824i	$-0.687297\pi$
$409$	−22.4499	−1.11008	−0.555039	+	0.831824i	$0.687297\pi$
$410$	−18.4797	−0.912649
$411$	0	0
$412$	−10.4797	−0.516300
$413$	9.23676	0.454511
$414$	0	0
$415$	14.6431	0.718802
$416$	14.8118	0.726206
$417$	0	0
$418$	0	0
$419$	−9.29150	−0.453920	−0.226960	−	0.973904i	$-0.572879\pi$
$419$	−9.29150	−0.453920	−0.226960	+	0.973904i	$0.572879\pi$
$420$	0	0
$421$	−17.0000	−0.828529	−0.414265	−	0.910156i	$-0.635961\pi$
$421$	−17.0000	−0.828529	−0.414265	+	0.910156i	$0.635961\pi$
$422$	−6.77124	−0.329619
$423$	0	0
$424$	−18.4735	−0.897153
$425$	−7.05535	−0.342235
$426$	0	0
$427$	2.22876	0.107857
$428$	4.24264	0.205076
$429$	0	0
$430$	−19.2693	−0.929247
$431$	−17.3828	−0.837300	−0.418650	−	0.908148i	$-0.637497\pi$
$431$	−17.3828	−0.837300	−0.418650	+	0.908148i	$0.637497\pi$
$432$	0	0
$433$	−37.1033	−1.78307	−0.891535	−	0.452953i	$-0.850371\pi$
$433$	−37.1033	−1.78307	−0.891535	+	0.452953i	$0.850371\pi$
$434$	−4.93725	−0.236996
$435$	0	0
$436$	−9.98823	−0.478350
$437$	−8.89753	−0.425627
$438$	0	0
$439$	16.2191	0.774095	0.387047	−	0.922060i	$-0.373495\pi$
$439$	16.2191	0.774095	0.387047	+	0.922060i	$0.373495\pi$
$440$	0	0
$441$	0	0
$442$	15.2014	0.723057
$443$	2.12549	0.100985	0.0504926	−	0.998724i	$-0.483921\pi$
$443$	2.12549	0.100985	0.0504926	+	0.998724i	$0.483921\pi$
$444$	0	0
$445$	−20.7085	−0.981677
$446$	−5.89163	−0.278977
$447$	0	0
$448$	10.0613	0.475349
$449$	−28.9373	−1.36563	−0.682817	−	0.730590i	$-0.739245\pi$
$449$	−28.9373	−1.36563	−0.682817	+	0.730590i	$0.739245\pi$
$450$	0	0
$451$	0	0
$452$	−10.9373	−0.514445
$453$	0	0
$454$	−25.6458	−1.20361
$455$	−8.12549	−0.380929
$456$	0	0
$457$	−2.73969	−0.128157	−0.0640787	−	0.997945i	$-0.520411\pi$
$457$	−2.73969	−0.128157	−0.0640787	+	0.997945i	$0.520411\pi$
$458$	−23.0554	−1.07731
$459$	0	0
$460$	1.43922	0.0671041
$461$	−26.6926	−1.24320	−0.621599	−	0.783336i	$-0.713517\pi$
$461$	−26.6926	−1.24320	−0.621599	+	0.783336i	$0.713517\pi$
$462$	0	0
$463$	−9.41699	−0.437645	−0.218822	−	0.975765i	$-0.570222\pi$
$463$	−9.41699	−0.437645	−0.218822	+	0.975765i	$0.570222\pi$
$464$	2.66667	0.123797
$465$	0	0
$466$	30.1033	1.39451
$467$	12.7712	0.590983	0.295491	−	0.955345i	$-0.404517\pi$
$467$	12.7712	0.590983	0.295491	+	0.955345i	$0.404517\pi$
$468$	0	0
$469$	−4.24264	−0.195907
$470$	−23.5406	−1.08585
$471$	0	0
$472$	−24.4382	−1.12486
$473$	0	0
$474$	0	0
$475$	15.0554	0.690788
$476$	−2.31373	−0.106050
$477$	0	0
$478$	8.60523	0.393594
$479$	20.8010	0.950420	0.475210	−	0.879872i	$-0.342372\pi$
$479$	20.8010	0.950420	0.475210	+	0.879872i	$0.342372\pi$
$480$	0	0
$481$	−14.2309	−0.648872
$482$	−14.3320	−0.652806
$483$	0	0
$484$	0	0
$485$	9.77124	0.443689
$486$	0	0
$487$	20.4575	0.927018	0.463509	−	0.886092i	$-0.346590\pi$
$487$	20.4575	0.927018	0.463509	+	0.886092i	$0.346590\pi$
$488$	−5.89674	−0.266933
$489$	0	0
$490$	10.8127	0.488469
$491$	6.98233	0.315108	0.157554	−	0.987510i	$-0.449639\pi$
$491$	6.98233	0.315108	0.157554	+	0.987510i	$0.449639\pi$
$492$	0	0
$493$	−3.58301	−0.161370
$494$	−32.4382	−1.45946
$495$	0	0
$496$	8.35425	0.375117
$497$	−10.8127	−0.485017
$498$	0	0
$499$	36.8118	1.64792	0.823960	−	0.566647i	$-0.191760\pi$
$499$	36.8118	1.64792	0.823960	+	0.566647i	$0.191760\pi$
$500$	−7.74902	−0.346547
$501$	0	0
$502$	−10.4735	−0.467455
$503$	−6.98233	−0.311327	−0.155663	−	0.987810i	$-0.549752\pi$
$503$	−6.98233	−0.311327	−0.155663	+	0.987810i	$0.549752\pi$
$504$	0	0
$505$	−16.5583	−0.736835
$506$	0	0
$507$	0	0
$508$	1.76916	0.0784937
$509$	−22.9373	−1.01668	−0.508338	−	0.861158i	$-0.669740\pi$
$509$	−22.9373	−1.01668	−0.508338	+	0.861158i	$0.669740\pi$
$510$	0	0
$511$	12.5830	0.556639
$512$	−22.1107	−0.977165
$513$	0	0
$514$	12.0495	0.531479
$515$	−26.7085	−1.17692
$516$	0	0
$517$	0	0
$518$	−4.54249	−0.199585
$519$	0	0
$520$	21.4980	0.942751
$521$	−10.9373	−0.479170	−0.239585	−	0.970875i	$-0.577011\pi$
$521$	−10.9373	−0.479170	−0.239585	+	0.970875i	$0.577011\pi$
$522$	0	0
$523$	−23.8799	−1.04419	−0.522097	−	0.852886i	$-0.674850\pi$
$523$	−23.8799	−1.04419	−0.522097	+	0.852886i	$0.674850\pi$
$524$	−12.9941	−0.567651
$525$	0	0
$526$	−30.5830	−1.33348
$527$	−11.2250	−0.488967
$528$	0	0
$529$	−21.1660	−0.920261
$530$	−11.4912	−0.499145
$531$	0	0
$532$	4.93725	0.214057
$533$	40.9373	1.77319
$534$	0	0
$535$	10.8127	0.467475
$536$	11.2250	0.484845
$537$	0	0
$538$	−6.42405	−0.276960
$539$	0	0
$540$	0	0
$541$	−1.50295	−0.0646169	−0.0323084	−	0.999478i	$-0.510286\pi$
$541$	−1.50295	−0.0646169	−0.0323084	+	0.999478i	$0.510286\pi$
$542$	−27.0000	−1.15975
$543$	0	0
$544$	10.7490	0.460860
$545$	−25.4558	−1.09041
$546$	0	0
$547$	−23.2014	−0.992021	−0.496011	−	0.868317i	$-0.665202\pi$
$547$	−23.2014	−0.992021	−0.496011	+	0.868317i	$0.665202\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	0	0
$551$	7.64575	0.325720
$552$	0	0
$553$	−0.874508	−0.0371879
$554$	−9.87451	−0.419528
$555$	0	0
$556$	−0.485266	−0.0205799
$557$	−9.23676	−0.391374	−0.195687	−	0.980666i	$-0.562694\pi$
$557$	−9.23676	−0.391374	−0.195687	+	0.980666i	$0.562694\pi$
$558$	0	0
$559$	42.6863	1.80544
$560$	4.38868	0.185456
$561$	0	0
$562$	−16.1660	−0.681922
$563$	33.2627	1.40185	0.700927	−	0.713233i	$-0.252770\pi$
$563$	33.2627	1.40185	0.700927	+	0.713233i	$0.252770\pi$
$564$	0	0
$565$	−27.8745	−1.17269
$566$	24.6863	1.03764
$567$	0	0
$568$	28.6078	1.20036
$569$	22.0377	0.923868	0.461934	−	0.886914i	$-0.347156\pi$
$569$	22.0377	0.923868	0.461934	+	0.886914i	$0.347156\pi$
$570$	0	0
$571$	−4.99412	−0.208997	−0.104499	−	0.994525i	$-0.533324\pi$
$571$	−4.99412	−0.208997	−0.104499	+	0.994525i	$0.533324\pi$
$572$	0	0
$573$	0	0
$574$	13.0672	0.545412
$575$	−3.10326	−0.129415
$576$	0	0
$577$	34.8745	1.45184	0.725922	−	0.687777i	$-0.241413\pi$
$577$	34.8745	1.45184	0.725922	+	0.687777i	$0.241413\pi$
$578$	−8.75149	−0.364014
$579$	0	0
$580$	−1.23674	−0.0513529
$581$	−10.3542	−0.429567
$582$	0	0
$583$	0	0
$584$	−33.2915	−1.37761
$585$	0	0
$586$	−12.1033	−0.499981
$587$	−33.1033	−1.36632	−0.683159	−	0.730270i	$-0.739394\pi$
$587$	−33.1033	−1.36632	−0.683159	+	0.730270i	$0.739394\pi$
$588$	0	0
$589$	23.9529	0.986962
$590$	−15.2014	−0.625832
$591$	0	0
$592$	7.68627	0.315904
$593$	11.5642	0.474885	0.237442	−	0.971402i	$-0.423691\pi$
$593$	11.5642	0.474885	0.237442	+	0.971402i	$0.423691\pi$
$594$	0	0
$595$	−5.89674	−0.241743
$596$	−9.72202	−0.398230
$597$	0	0
$598$	6.68627	0.273422
$599$	4.06275	0.165999	0.0829997	−	0.996550i	$-0.473550\pi$
$599$	4.06275	0.165999	0.0829997	+	0.996550i	$0.473550\pi$
$600$	0	0
$601$	4.24264	0.173061	0.0865305	−	0.996249i	$-0.472422\pi$
$601$	4.24264	0.173061	0.0865305	+	0.996249i	$0.472422\pi$
$602$	13.6254	0.555331
$603$	0	0
$604$	8.21907	0.334429
$605$	0	0
$606$	0	0
$607$	24.3651	0.988951	0.494475	−	0.869192i	$-0.335360\pi$
$607$	24.3651	0.988951	0.494475	+	0.869192i	$0.335360\pi$
$608$	−22.9373	−0.930228
$609$	0	0
$610$	−3.66798	−0.148512
$611$	52.1484	2.10970
$612$	0	0
$613$	−8.89753	−0.359368	−0.179684	−	0.983724i	$-0.557507\pi$
$613$	−8.89753	−0.359368	−0.179684	+	0.983724i	$0.557507\pi$
$614$	−16.6458	−0.671768
$615$	0	0
$616$	0	0
$617$	19.6458	0.790908	0.395454	−	0.918486i	$-0.370587\pi$
$617$	19.6458	0.790908	0.395454	+	0.918486i	$0.370587\pi$
$618$	0	0
$619$	12.9373	0.519992	0.259996	−	0.965610i	$-0.416279\pi$
$619$	12.9373	0.519992	0.259996	+	0.965610i	$0.416279\pi$
$620$	−3.87451	−0.155604
$621$	0	0
$622$	23.2014	0.930292
$623$	14.6431	0.586664
$624$	0	0
$625$	−8.29150	−0.331660
$626$	5.33334	0.213163
$627$	0	0
$628$	1.47974	0.0590481
$629$	−10.3275	−0.411783
$630$	0	0
$631$	−22.2288	−0.884913	−0.442456	−	0.896790i	$-0.645893\pi$
$631$	−22.2288	−0.884913	−0.442456	+	0.896790i	$0.645893\pi$
$632$	2.31373	0.0920353
$633$	0	0
$634$	−11.4912	−0.456373
$635$	4.50885	0.178928
$636$	0	0
$637$	−23.9529	−0.949048
$638$	0	0
$639$	0	0
$640$	−5.06713	−0.200296
$641$	−40.9373	−1.61692	−0.808462	−	0.588548i	$-0.799700\pi$
$641$	−40.9373	−1.61692	−0.808462	+	0.588548i	$0.799700\pi$
$642$	0	0
$643$	−19.2915	−0.760783	−0.380391	−	0.924826i	$-0.624211\pi$
$643$	−19.2915	−0.760783	−0.380391	+	0.924826i	$0.624211\pi$
$644$	−1.01768	−0.0401024
$645$	0	0
$646$	−23.5406	−0.926194
$647$	−36.0000	−1.41531	−0.707653	−	0.706560i	$-0.750246\pi$
$647$	−36.0000	−1.41531	−0.707653	+	0.706560i	$0.750246\pi$
$648$	0	0
$649$	0	0
$650$	−11.3137	−0.443761
$651$	0	0
$652$	−11.1660	−0.437295
$653$	−1.16601	−0.0456295	−0.0228148	−	0.999740i	$-0.507263\pi$
$653$	−1.16601	−0.0456295	−0.0228148	+	0.999740i	$0.507263\pi$
$654$	0	0
$655$	−33.1166	−1.29397
$656$	−22.1107	−0.863278
$657$	0	0
$658$	16.6458	0.648919
$659$	33.5289	1.30610	0.653050	−	0.757315i	$-0.273489\pi$
$659$	33.5289	1.30610	0.653050	+	0.757315i	$0.273489\pi$
$660$	0	0
$661$	7.58301	0.294945	0.147472	−	0.989066i	$-0.452886\pi$
$661$	7.58301	0.294945	0.147472	+	0.989066i	$0.452886\pi$
$662$	11.9034	0.462640
$663$	0	0
$664$	27.3948	1.06312
$665$	12.5830	0.487948
$666$	0	0
$667$	−1.57597	−0.0610218
$668$	14.2309	0.550609
$669$	0	0
$670$	6.98233	0.269751
$671$	0	0
$672$	0	0
$673$	18.8858	0.727993	0.363996	−	0.931400i	$-0.381412\pi$
$673$	18.8858	0.727993	0.363996	+	0.931400i	$0.381412\pi$
$674$	−33.7712	−1.30082
$675$	0	0
$676$	−3.22876	−0.124183
$677$	−12.8009	−0.491980	−0.245990	−	0.969272i	$-0.579113\pi$
$677$	−12.8009	−0.491980	−0.245990	+	0.969272i	$0.579113\pi$
$678$	0	0
$679$	−6.90931	−0.265155
$680$	15.6013	0.598282
$681$	0	0
$682$	0	0
$683$	12.2915	0.470321	0.235161	−	0.971957i	$-0.424438\pi$
$683$	12.2915	0.470321	0.235161	+	0.971957i	$0.424438\pi$
$684$	0	0
$685$	15.2915	0.584258
$686$	−17.1255	−0.653854
$687$	0	0
$688$	−23.0554	−0.878978
$689$	25.4558	0.969790
$690$	0	0
$691$	−15.4170	−0.586490	−0.293245	−	0.956037i	$-0.594735\pi$
$691$	−15.4170	−0.586490	−0.293245	+	0.956037i	$0.594735\pi$
$692$	8.21907	0.312442
$693$	0	0
$694$	13.5425	0.514066
$695$	−1.23674	−0.0469123
$696$	0	0
$697$	29.7085	1.12529
$698$	32.8118	1.24194
$699$	0	0
$700$	1.72201	0.0650857
$701$	−36.3416	−1.37260	−0.686301	−	0.727317i	$-0.740767\pi$
$701$	−36.3416	−1.37260	−0.686301	+	0.727317i	$0.740767\pi$
$702$	0	0
$703$	22.0377	0.831167
$704$	0	0
$705$	0	0
$706$	24.0989	0.906975
$707$	11.7085	0.440343
$708$	0	0
$709$	31.1033	1.16811	0.584054	−	0.811715i	$-0.301466\pi$
$709$	31.1033	1.16811	0.584054	+	0.811715i	$0.301466\pi$
$710$	17.7951	0.667837
$711$	0	0
$712$	−38.7421	−1.45192
$713$	−4.93725	−0.184902
$714$	0	0
$715$	0	0
$716$	−11.8118	−0.441426
$717$	0	0
$718$	22.9373	0.856011
$719$	−3.87451	−0.144495	−0.0722474	−	0.997387i	$-0.523017\pi$
$719$	−3.87451	−0.144495	−0.0722474	+	0.997387i	$0.523017\pi$
$720$	0	0
$721$	18.8858	0.703342
$722$	28.1225	1.04661
$723$	0	0
$724$	−13.4797	−0.500971
$725$	2.66667	0.0990377
$726$	0	0
$727$	−4.81176	−0.178458	−0.0892292	−	0.996011i	$-0.528440\pi$
$727$	−4.81176	−0.178458	−0.0892292	+	0.996011i	$0.528440\pi$
$728$	−15.2014	−0.563402
$729$	0	0
$730$	−20.7085	−0.766456
$731$	30.9778	1.14575
$732$	0	0
$733$	38.3298	1.41574	0.707872	−	0.706341i	$-0.249655\pi$
$733$	38.3298	1.41574	0.707872	+	0.706341i	$0.249655\pi$
$734$	−0.146038	−0.00539036
$735$	0	0
$736$	4.72791	0.174273
$737$	0	0
$738$	0	0
$739$	−20.8740	−0.767862	−0.383931	−	0.923362i	$-0.625430\pi$
$739$	−20.8740	−0.767862	−0.383931	+	0.923362i	$0.625430\pi$
$740$	−3.56471	−0.131042
$741$	0	0
$742$	8.12549	0.298296
$743$	−13.9647	−0.512314	−0.256157	−	0.966635i	$-0.582456\pi$
$743$	−13.9647	−0.512314	−0.256157	+	0.966635i	$0.582456\pi$
$744$	0	0
$745$	−24.7774	−0.907773
$746$	24.6863	0.903829
$747$	0	0
$748$	0	0
$749$	−7.64575	−0.279370
$750$	0	0
$751$	11.6458	0.424960	0.212480	−	0.977165i	$-0.431846\pi$
$751$	11.6458	0.424960	0.212480	+	0.977165i	$0.431846\pi$
$752$	−28.1660	−1.02711
$753$	0	0
$754$	−5.74559	−0.209242
$755$	20.9470	0.762339
$756$	0	0
$757$	−15.4575	−0.561813	−0.280906	−	0.959735i	$-0.590635\pi$
$757$	−15.4575	−0.561813	−0.280906	+	0.959735i	$0.590635\pi$
$758$	14.3769	0.522193
$759$	0	0
$760$	−33.2915	−1.20761
$761$	31.3475	1.13634	0.568172	−	0.822909i	$-0.307651\pi$
$761$	31.3475	1.13634	0.568172	+	0.822909i	$0.307651\pi$
$762$	0	0
$763$	18.0000	0.651644
$764$	−1.93725	−0.0700874
$765$	0	0
$766$	−18.4735	−0.667475
$767$	33.6749	1.21593
$768$	0	0
$769$	27.1048	0.977425	0.488713	−	0.872445i	$-0.337467\pi$
$769$	27.1048	0.977425	0.488713	+	0.872445i	$0.337467\pi$
$770$	0	0
$771$	0	0
$772$	15.2014	0.547110
$773$	−7.64575	−0.274999	−0.137499	−	0.990502i	$-0.543906\pi$
$773$	−7.64575	−0.274999	−0.137499	+	0.990502i	$0.543906\pi$
$774$	0	0
$775$	8.35425	0.300093
$776$	18.2803	0.656225
$777$	0	0
$778$	40.0990	1.43762
$779$	−63.3948	−2.27135
$780$	0	0
$781$	0	0
$782$	4.85228	0.173517
$783$	0	0
$784$	12.9373	0.462045
$785$	3.77124	0.134601
$786$	0	0
$787$	5.74559	0.204808	0.102404	−	0.994743i	$-0.467347\pi$
$787$	5.74559	0.204808	0.102404	+	0.994743i	$0.467347\pi$
$788$	−0.751475	−0.0267702
$789$	0	0
$790$	1.43922	0.0512052
$791$	19.7103	0.700816
$792$	0	0
$793$	8.12549	0.288545
$794$	31.6137	1.12193
$795$	0	0
$796$	−5.39477	−0.191212
$797$	38.8118	1.37478	0.687392	−	0.726287i	$-0.258756\pi$
$797$	38.8118	1.37478	0.687392	+	0.726287i	$0.258756\pi$
$798$	0	0
$799$	37.8445	1.33884
$800$	−8.00002	−0.282843
$801$	0	0
$802$	36.9470	1.30464
$803$	0	0
$804$	0	0
$805$	−2.59365	−0.0914143
$806$	−18.0000	−0.634023
$807$	0	0
$808$	−30.9778	−1.08979
$809$	−28.9470	−1.01772	−0.508861	−	0.860849i	$-0.669933\pi$
$809$	−28.9470	−1.01772	−0.508861	+	0.860849i	$0.669933\pi$
$810$	0	0
$811$	2.66667	0.0936395	0.0468198	−	0.998903i	$-0.485091\pi$
$811$	2.66667	0.0936395	0.0468198	+	0.998903i	$0.485091\pi$
$812$	0.874508	0.0306892
$813$	0	0
$814$	0	0
$815$	−28.4575	−0.996823
$816$	0	0
$817$	−66.1033	−2.31266
$818$	−26.1255	−0.913456
$819$	0	0
$820$	10.2544	0.358101
$821$	−24.0259	−0.838510	−0.419255	−	0.907868i	$-0.637709\pi$
$821$	−24.0259	−0.838510	−0.419255	+	0.907868i	$0.637709\pi$
$822$	0	0
$823$	19.4170	0.676834	0.338417	−	0.940996i	$-0.390109\pi$
$823$	19.4170	0.676834	0.338417	+	0.940996i	$0.390109\pi$
$824$	−49.9670	−1.74068
$825$	0	0
$826$	10.7490	0.374006
$827$	36.0024	1.25192	0.625962	−	0.779853i	$-0.284706\pi$
$827$	36.0024	1.25192	0.625962	+	0.779853i	$0.284706\pi$
$828$	0	0
$829$	6.06275	0.210568	0.105284	−	0.994442i	$-0.466425\pi$
$829$	6.06275	0.210568	0.105284	+	0.994442i	$0.466425\pi$
$830$	17.0405	0.591485
$831$	0	0
$832$	36.6808	1.27168
$833$	−17.3828	−0.602279
$834$	0	0
$835$	36.2686	1.25513
$836$	0	0
$837$	0	0
$838$	−10.8127	−0.373519
$839$	26.5203	0.915581	0.457791	−	0.889060i	$-0.348641\pi$
$839$	26.5203	0.915581	0.457791	+	0.889060i	$0.348641\pi$
$840$	0	0
$841$	−27.6458	−0.953302
$842$	−19.7833	−0.681777
$843$	0	0
$844$	3.75737	0.129334
$845$	−8.22876	−0.283078
$846$	0	0
$847$	0	0
$848$	−13.7490	−0.472143
$849$	0	0
$850$	−8.21047	−0.281617
$851$	−4.54249	−0.155714
$852$	0	0
$853$	28.6078	0.979512	0.489756	−	0.871860i	$-0.337086\pi$
$853$	28.6078	0.979512	0.489756	+	0.871860i	$0.337086\pi$
$854$	2.59365	0.0887530
$855$	0	0
$856$	20.2288	0.691405
$857$	10.4735	0.357768	0.178884	−	0.983870i	$-0.442751\pi$
$857$	10.4735	0.357768	0.178884	+	0.983870i	$0.442751\pi$
$858$	0	0
$859$	14.4797	0.494042	0.247021	−	0.969010i	$-0.420548\pi$
$859$	14.4797	0.494042	0.247021	+	0.969010i	$0.420548\pi$
$860$	10.6926	0.364613
$861$	0	0
$862$	−20.2288	−0.688994
$863$	36.3948	1.23889	0.619446	−	0.785039i	$-0.287357\pi$
$863$	36.3948	1.23889	0.619446	+	0.785039i	$0.287357\pi$
$864$	0	0
$865$	20.9470	0.712219
$866$	−43.1779	−1.46724
$867$	0	0
$868$	2.73969	0.0929912
$869$	0	0
$870$	0	0
$871$	−15.4676	−0.524100
$872$	−47.6235	−1.61274
$873$	0	0
$874$	−10.3542	−0.350238
$875$	13.9647	0.472092
$876$	0	0
$877$	−37.0931	−1.25254	−0.626272	−	0.779605i	$-0.715420\pi$
$877$	−37.0931	−1.25254	−0.626272	+	0.779605i	$0.715420\pi$
$878$	18.8745	0.636984
$879$	0	0
$880$	0	0
$881$	34.9373	1.17707	0.588533	−	0.808473i	$-0.299706\pi$
$881$	34.9373	1.17707	0.588533	+	0.808473i	$0.299706\pi$
$882$	0	0
$883$	−35.2915	−1.18765	−0.593827	−	0.804593i	$-0.702384\pi$
$883$	−35.2915	−1.18765	−0.593827	+	0.804593i	$0.702384\pi$
$884$	−8.43529	−0.283709
$885$	0	0
$886$	2.47348	0.0830982
$887$	−52.1484	−1.75097	−0.875486	−	0.483243i	$-0.839459\pi$
$887$	−52.1484	−1.75097	−0.875486	+	0.483243i	$0.839459\pi$
$888$	0	0
$889$	−3.18824	−0.106930
$890$	−24.0989	−0.807798
$891$	0	0
$892$	3.26927	0.109463
$893$	−80.7562	−2.70240
$894$	0	0
$895$	−30.1033	−1.00624
$896$	3.58301	0.119700
$897$	0	0
$898$	−33.6749	−1.12375
$899$	4.24264	0.141500
$900$	0	0
$901$	18.4735	0.615442
$902$	0	0
$903$	0	0
$904$	−52.1484	−1.73443
$905$	−34.3542	−1.14197
$906$	0	0
$907$	−4.10326	−0.136247	−0.0681233	−	0.997677i	$-0.521701\pi$
$907$	−4.10326	−0.136247	−0.0681233	+	0.997677i	$0.521701\pi$
$908$	14.2309	0.472268
$909$	0	0
$910$	−9.45581	−0.313457
$911$	−23.1255	−0.766182	−0.383091	−	0.923711i	$-0.625140\pi$
$911$	−23.1255	−0.766182	−0.383091	+	0.923711i	$0.625140\pi$
$912$	0	0
$913$	0	0
$914$	−3.18824	−0.105458
$915$	0	0
$916$	12.7935	0.422708
$917$	23.4170	0.773297
$918$	0	0
$919$	46.0636	1.51950	0.759749	−	0.650216i	$-0.225322\pi$
$919$	46.0636	1.51950	0.759749	+	0.650216i	$0.225322\pi$
$920$	6.86216	0.226239
$921$	0	0
$922$	−31.0627	−1.02300
$923$	−39.4205	−1.29754
$924$	0	0
$925$	7.68627	0.252723
$926$	−10.9588	−0.360127
$927$	0	0
$928$	−4.06275	−0.133366
$929$	−25.7490	−0.844798	−0.422399	−	0.906410i	$-0.638812\pi$
$929$	−25.7490	−0.844798	−0.422399	+	0.906410i	$0.638812\pi$
$930$	0	0
$931$	37.0931	1.21568
$932$	−16.7044	−0.547169
$933$	0	0
$934$	14.8622	0.486305
$935$	0	0
$936$	0	0
$937$	57.6278	1.88262	0.941309	−	0.337545i	$-0.109597\pi$
$937$	57.6278	1.88262	0.941309	+	0.337545i	$0.109597\pi$
$938$	−4.93725	−0.161207
$939$	0	0
$940$	13.0627	0.426060
$941$	−4.31566	−0.140686	−0.0703432	−	0.997523i	$-0.522409\pi$
$941$	−4.31566	−0.140686	−0.0703432	+	0.997523i	$0.522409\pi$
$942$	0	0
$943$	13.0672	0.425525
$944$	−18.1882	−0.591977
$945$	0	0
$946$	0	0
$947$	−11.4170	−0.371002	−0.185501	−	0.982644i	$-0.559391\pi$
$947$	−11.4170	−0.371002	−0.185501	+	0.982644i	$0.559391\pi$
$948$	0	0
$949$	45.8745	1.48915
$950$	17.5203	0.568432
$951$	0	0
$952$	−11.0318	−0.357542
$953$	26.3534	0.853669	0.426834	−	0.904330i	$-0.359629\pi$
$953$	26.3534	0.853669	0.426834	+	0.904330i	$0.359629\pi$
$954$	0	0
$955$	−4.93725	−0.159766
$956$	−4.77506	−0.154436
$957$	0	0
$958$	24.2065	0.782077
$959$	−10.8127	−0.349161
$960$	0	0
$961$	−17.7085	−0.571242
$962$	−16.5608	−0.533941
$963$	0	0
$964$	7.95286	0.256144
$965$	38.7421	1.24715
$966$	0	0
$967$	−30.1838	−0.970644	−0.485322	−	0.874335i	$-0.661298\pi$
$967$	−30.1838	−0.970644	−0.485322	+	0.874335i	$0.661298\pi$
$968$	0	0
$969$	0	0
$970$	11.3710	0.365101
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0.874508	0.0280354
$974$	23.8069	0.762821
$975$	0	0
$976$	−4.38868	−0.140478
$977$	27.8745	0.891785	0.445892	−	0.895087i	$-0.352886\pi$
$977$	27.8745	0.891785	0.445892	+	0.895087i	$0.352886\pi$
$978$	0	0
$979$	0	0
$980$	−6.00000	−0.191663
$981$	0	0
$982$	8.12549	0.259295
$983$	−57.8745	−1.84591	−0.922955	−	0.384908i	$-0.874233\pi$
$983$	−57.8745	−1.84591	−0.922955	+	0.384908i	$0.874233\pi$
$984$	0	0
$985$	−1.91520	−0.0610232
$986$	−4.16962	−0.132788
$987$	0	0
$988$	18.0000	0.572656
$989$	13.6254	0.433264
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	−12.7279	−0.404112
$993$	0	0
$994$	−12.5830	−0.399109
$995$	−13.7490	−0.435873
$996$	0	0
$997$	32.4382	1.02733	0.513664	−	0.857992i	$-0.328288\pi$
$997$	32.4382	1.02733	0.513664	+	0.857992i	$0.328288\pi$
$998$	42.8387	1.35603
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3267.2.a.bb.1.3	yes	4
3.2	odd	2		3267.2.a.y.1.2	✓	4
11.10	odd	2	inner	3267.2.a.bb.1.2	yes	4
33.32	even	2		3267.2.a.y.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3267.2.a.y.1.2	✓	4	3.2	odd	2
3267.2.a.y.1.3	yes	4	33.32	even	2
3267.2.a.bb.1.2	yes	4	11.10	odd	2	inner
3267.2.a.bb.1.3	yes	4	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:	\( N \)	\(=\)	\( 3267 = 3^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3267.a (trivial)

\(f(q)\)	\(=\)	\(q+1.16372 q^{2} -0.645751 q^{4} -1.64575 q^{5} +1.16372 q^{7} -3.07892 q^{8} +O(q^{10})\)	\(q+1.16372 q^{2} -0.645751 q^{4} -1.64575 q^{5} +1.16372 q^{7} -3.07892 q^{8} -1.91520 q^{10} +4.24264 q^{13} +1.35425 q^{14} -2.29150 q^{16} +3.07892 q^{17} -6.57008 q^{19} +1.06275 q^{20} +1.35425 q^{23} -2.29150 q^{25} +4.93725 q^{26} -0.751475 q^{28} -1.16372 q^{29} -3.64575 q^{31} +3.49117 q^{32} +3.58301 q^{34} -1.91520 q^{35} -3.35425 q^{37} -7.64575 q^{38} +5.06713 q^{40} +9.64900 q^{41} +10.0613 q^{43} +1.57597 q^{46} +12.2915 q^{47} -5.64575 q^{49} -2.66667 q^{50} -2.73969 q^{52} +6.00000 q^{53} -3.58301 q^{56} -1.35425 q^{58} +7.93725 q^{59} +1.91520 q^{61} -4.24264 q^{62} +8.64575 q^{64} -6.98233 q^{65} -3.64575 q^{67} -1.98822 q^{68} -2.22876 q^{70} -9.29150 q^{71} +10.8127 q^{73} -3.90341 q^{74} +4.24264 q^{76} -0.751475 q^{79} +3.77124 q^{80} +11.2288 q^{82} -8.89753 q^{83} -5.06713 q^{85} +11.7085 q^{86} +12.5830 q^{89} +4.93725 q^{91} -0.874508 q^{92} +14.3039 q^{94} +10.8127 q^{95} -5.93725 q^{97} -6.57008 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} + 4 q^{5}+O(q^{10}) \) 4 * q + 8 * q^4 + 4 * q^5	\( 4 q + 8 q^{4} + 4 q^{5} + 16 q^{14} + 12 q^{16} + 36 q^{20} + 16 q^{23} + 12 q^{25} - 12 q^{26} - 4 q^{31} - 28 q^{34} - 24 q^{37} - 20 q^{38} + 28 q^{47} - 12 q^{49} + 24 q^{53} + 28 q^{56} - 16 q^{58} + 24 q^{64} - 4 q^{67} + 44 q^{70} - 16 q^{71} + 68 q^{80} - 8 q^{82} + 68 q^{86} + 8 q^{89} - 12 q^{91} + 60 q^{92} + 8 q^{97}+O(q^{100}) \) 4 * q + 8 * q^4 + 4 * q^5 + 16 * q^14 + 12 * q^16 + 36 * q^20 + 16 * q^23 + 12 * q^25 - 12 * q^26 - 4 * q^31 - 28 * q^34 - 24 * q^37 - 20 * q^38 + 28 * q^47 - 12 * q^49 + 24 * q^53 + 28 * q^56 - 16 * q^58 + 24 * q^64 - 4 * q^67 + 44 * q^70 - 16 * q^71 + 68 * q^80 - 8 * q^82 + 68 * q^86 + 8 * q^89 - 12 * q^91 + 60 * q^92 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more