LMFDB - Embedded newform 2646.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2646,2,Mod(1,2646)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2646, 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("2646.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2646 = 2 \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2646.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:	$21.1284163748$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2646.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.58579 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.58579 q^{5} -1.00000 q^{8} +1.58579 q^{10} -1.00000 q^{11} +4.24264 q^{13} +1.00000 q^{16} -2.82843 q^{17} +0.171573 q^{19} -1.58579 q^{20} +1.00000 q^{22} -3.24264 q^{23} -2.48528 q^{25} -4.24264 q^{26} +8.24264 q^{29} -1.24264 q^{31} -1.00000 q^{32} +2.82843 q^{34} -3.24264 q^{37} -0.171573 q^{38} +1.58579 q^{40} -11.8284 q^{41} +10.4853 q^{43} -1.00000 q^{44} +3.24264 q^{46} -0.343146 q^{47} +2.48528 q^{50} +4.24264 q^{52} +8.00000 q^{53} +1.58579 q^{55} -8.24264 q^{58} -0.343146 q^{59} -9.17157 q^{61} +1.24264 q^{62} +1.00000 q^{64} -6.72792 q^{65} -10.2426 q^{67} -2.82843 q^{68} +7.24264 q^{71} +8.82843 q^{73} +3.24264 q^{74} +0.171573 q^{76} +10.4853 q^{79} -1.58579 q^{80} +11.8284 q^{82} -15.8995 q^{83} +4.48528 q^{85} -10.4853 q^{86} +1.00000 q^{88} -14.6569 q^{89} -3.24264 q^{92} +0.343146 q^{94} -0.272078 q^{95} -11.3137 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 6 q^{5} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 6 * q^5 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 6 q^{5} - 2 q^{8} + 6 q^{10} - 2 q^{11} + 2 q^{16} + 6 q^{19} - 6 q^{20} + 2 q^{22} + 2 q^{23} + 12 q^{25} + 8 q^{29} + 6 q^{31} - 2 q^{32} + 2 q^{37} - 6 q^{38} + 6 q^{40} - 18 q^{41} + 4 q^{43} - 2 q^{44} - 2 q^{46} - 12 q^{47} - 12 q^{50} + 16 q^{53} + 6 q^{55} - 8 q^{58} - 12 q^{59} - 24 q^{61} - 6 q^{62} + 2 q^{64} + 12 q^{65} - 12 q^{67} + 6 q^{71} + 12 q^{73} - 2 q^{74} + 6 q^{76} + 4 q^{79} - 6 q^{80} + 18 q^{82} - 12 q^{83} - 8 q^{85} - 4 q^{86} + 2 q^{88} - 18 q^{89} + 2 q^{92} + 12 q^{94} - 26 q^{95}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 6 * q^5 - 2 * q^8 + 6 * q^10 - 2 * q^11 + 2 * q^16 + 6 * q^19 - 6 * q^20 + 2 * q^22 + 2 * q^23 + 12 * q^25 + 8 * q^29 + 6 * q^31 - 2 * q^32 + 2 * q^37 - 6 * q^38 + 6 * q^40 - 18 * q^41 + 4 * q^43 - 2 * q^44 - 2 * q^46 - 12 * q^47 - 12 * q^50 + 16 * q^53 + 6 * q^55 - 8 * q^58 - 12 * q^59 - 24 * q^61 - 6 * q^62 + 2 * q^64 + 12 * q^65 - 12 * q^67 + 6 * q^71 + 12 * q^73 - 2 * q^74 + 6 * q^76 + 4 * q^79 - 6 * q^80 + 18 * q^82 - 12 * q^83 - 8 * q^85 - 4 * q^86 + 2 * q^88 - 18 * q^89 + 2 * q^92 + 12 * q^94 - 26 * q^95

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.58579	−0.709185	−0.354593	−	0.935021i	$-0.615380\pi$
$5$	−1.58579	−0.709185	−0.354593	+	0.935021i	$0.615380\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.58579	0.501470
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	4.24264	1.17670	0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264	1.17670	0.588348	+	0.808608i	$0.299778\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	0.171573	0.0393615	0.0196808	−	0.999806i	$-0.493735\pi$
$19$	0.171573	0.0393615	0.0196808	+	0.999806i	$0.493735\pi$
$20$	−1.58579	−0.354593
$21$	0	0
$22$	1.00000	0.213201
$23$	−3.24264	−0.676137	−0.338069	−	0.941121i	$-0.609774\pi$
$23$	−3.24264	−0.676137	−0.338069	+	0.941121i	$0.609774\pi$
$24$	0	0
$25$	−2.48528	−0.497056
$26$	−4.24264	−0.832050
$27$	0	0
$28$	0	0
$29$	8.24264	1.53062	0.765310	−	0.643662i	$-0.222586\pi$
$29$	8.24264	1.53062	0.765310	+	0.643662i	$0.222586\pi$
$30$	0	0
$31$	−1.24264	−0.223185	−0.111592	−	0.993754i	$-0.535595\pi$
$31$	−1.24264	−0.223185	−0.111592	+	0.993754i	$0.535595\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.82843	0.485071
$35$	0	0
$36$	0	0
$37$	−3.24264	−0.533087	−0.266543	−	0.963823i	$-0.585882\pi$
$37$	−3.24264	−0.533087	−0.266543	+	0.963823i	$0.585882\pi$
$38$	−0.171573	−0.0278328
$39$	0	0
$40$	1.58579	0.250735
$41$	−11.8284	−1.84729	−0.923645	−	0.383249i	$-0.874805\pi$
$41$	−11.8284	−1.84729	−0.923645	+	0.383249i	$0.874805\pi$
$42$	0	0
$43$	10.4853	1.59899	0.799495	−	0.600672i	$-0.205100\pi$
$43$	10.4853	1.59899	0.799495	+	0.600672i	$0.205100\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	3.24264	0.478101
$47$	−0.343146	−0.0500530	−0.0250265	−	0.999687i	$-0.507967\pi$
$47$	−0.343146	−0.0500530	−0.0250265	+	0.999687i	$0.507967\pi$
$48$	0	0
$49$	0	0
$50$	2.48528	0.351472
$51$	0	0
$52$	4.24264	0.588348
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	0	0
$55$	1.58579	0.213827
$56$	0	0
$57$	0	0
$58$	−8.24264	−1.08231
$59$	−0.343146	−0.0446738	−0.0223369	−	0.999751i	$-0.507111\pi$
$59$	−0.343146	−0.0446738	−0.0223369	+	0.999751i	$0.507111\pi$
$60$	0	0
$61$	−9.17157	−1.17430	−0.587150	−	0.809478i	$-0.699750\pi$
$61$	−9.17157	−1.17430	−0.587150	+	0.809478i	$0.699750\pi$
$62$	1.24264	0.157816
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.72792	−0.834496
$66$	0	0
$67$	−10.2426	−1.25134	−0.625669	−	0.780089i	$-0.715174\pi$
$67$	−10.2426	−1.25134	−0.625669	+	0.780089i	$0.715174\pi$
$68$	−2.82843	−0.342997
$69$	0	0
$70$	0	0
$71$	7.24264	0.859543	0.429772	−	0.902938i	$-0.358594\pi$
$71$	7.24264	0.859543	0.429772	+	0.902938i	$0.358594\pi$
$72$	0	0
$73$	8.82843	1.03329	0.516645	−	0.856200i	$-0.327181\pi$
$73$	8.82843	1.03329	0.516645	+	0.856200i	$0.327181\pi$
$74$	3.24264	0.376949
$75$	0	0
$76$	0.171573	0.0196808
$77$	0	0
$78$	0	0
$79$	10.4853	1.17969	0.589843	−	0.807518i	$-0.299190\pi$
$79$	10.4853	1.17969	0.589843	+	0.807518i	$0.299190\pi$
$80$	−1.58579	−0.177296
$81$	0	0
$82$	11.8284	1.30623
$83$	−15.8995	−1.74520	−0.872598	−	0.488439i	$-0.837567\pi$
$83$	−15.8995	−1.74520	−0.872598	+	0.488439i	$0.837567\pi$
$84$	0	0
$85$	4.48528	0.486497
$86$	−10.4853	−1.13066
$87$	0	0
$88$	1.00000	0.106600
$89$	−14.6569	−1.55362	−0.776812	−	0.629733i	$-0.783164\pi$
$89$	−14.6569	−1.55362	−0.776812	+	0.629733i	$0.783164\pi$
$90$	0	0
$91$	0	0
$92$	−3.24264	−0.338069
$93$	0	0
$94$	0.343146	0.0353928
$95$	−0.272078	−0.0279146
$96$	0	0
$97$	−11.3137	−1.14873	−0.574367	−	0.818598i	$-0.694752\pi$
$97$	−11.3137	−1.14873	−0.574367	+	0.818598i	$0.694752\pi$
$98$	0	0
$99$	0	0
$100$	−2.48528	−0.248528
$101$	−2.48528	−0.247295	−0.123647	−	0.992326i	$-0.539459\pi$
$101$	−2.48528	−0.247295	−0.123647	+	0.992326i	$0.539459\pi$
$102$	0	0
$103$	4.07107	0.401134	0.200567	−	0.979680i	$-0.435722\pi$
$103$	4.07107	0.401134	0.200567	+	0.979680i	$0.435722\pi$
$104$	−4.24264	−0.416025
$105$	0	0
$106$	−8.00000	−0.777029
$107$	−18.9706	−1.83395	−0.916977	−	0.398941i	$-0.869378\pi$
$107$	−18.9706	−1.83395	−0.916977	+	0.398941i	$0.869378\pi$
$108$	0	0
$109$	−6.75736	−0.647238	−0.323619	−	0.946188i	$-0.604900\pi$
$109$	−6.75736	−0.647238	−0.323619	+	0.946188i	$0.604900\pi$
$110$	−1.58579	−0.151199
$111$	0	0
$112$	0	0
$113$	−4.24264	−0.399114	−0.199557	−	0.979886i	$-0.563950\pi$
$113$	−4.24264	−0.399114	−0.199557	+	0.979886i	$0.563950\pi$
$114$	0	0
$115$	5.14214	0.479507
$116$	8.24264	0.765310
$117$	0	0
$118$	0.343146	0.0315891
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	9.17157	0.830355
$123$	0	0
$124$	−1.24264	−0.111592
$125$	11.8701	1.06169
$126$	0	0
$127$	−3.75736	−0.333412	−0.166706	−	0.986007i	$-0.553313\pi$
$127$	−3.75736	−0.333412	−0.166706	+	0.986007i	$0.553313\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	6.72792	0.590078
$131$	2.48528	0.217140	0.108570	−	0.994089i	$-0.465373\pi$
$131$	2.48528	0.217140	0.108570	+	0.994089i	$0.465373\pi$
$132$	0	0
$133$	0	0
$134$	10.2426	0.884829
$135$	0	0
$136$	2.82843	0.242536
$137$	10.2426	0.875088	0.437544	−	0.899197i	$-0.355848\pi$
$137$	10.2426	0.875088	0.437544	+	0.899197i	$0.355848\pi$
$138$	0	0
$139$	3.17157	0.269009	0.134505	−	0.990913i	$-0.457056\pi$
$139$	3.17157	0.269009	0.134505	+	0.990913i	$0.457056\pi$
$140$	0	0
$141$	0	0
$142$	−7.24264	−0.607789
$143$	−4.24264	−0.354787
$144$	0	0
$145$	−13.0711	−1.08549
$146$	−8.82843	−0.730646
$147$	0	0
$148$	−3.24264	−0.266543
$149$	−14.2426	−1.16680	−0.583401	−	0.812184i	$-0.698278\pi$
$149$	−14.2426	−1.16680	−0.583401	+	0.812184i	$0.698278\pi$
$150$	0	0
$151$	−10.7279	−0.873026	−0.436513	−	0.899698i	$-0.643787\pi$
$151$	−10.7279	−0.873026	−0.436513	+	0.899698i	$0.643787\pi$
$152$	−0.171573	−0.0139164
$153$	0	0
$154$	0	0
$155$	1.97056	0.158279
$156$	0	0
$157$	−19.0711	−1.52204	−0.761018	−	0.648730i	$-0.775300\pi$
$157$	−19.0711	−1.52204	−0.761018	+	0.648730i	$0.775300\pi$
$158$	−10.4853	−0.834164
$159$	0	0
$160$	1.58579	0.125367
$161$	0	0
$162$	0	0
$163$	10.7279	0.840276	0.420138	−	0.907460i	$-0.361982\pi$
$163$	10.7279	0.840276	0.420138	+	0.907460i	$0.361982\pi$
$164$	−11.8284	−0.923645
$165$	0	0
$166$	15.8995	1.23404
$167$	−24.3848	−1.88695	−0.943475	−	0.331443i	$-0.892465\pi$
$167$	−24.3848	−1.88695	−0.943475	+	0.331443i	$0.892465\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	−4.48528	−0.344005
$171$	0	0
$172$	10.4853	0.799495
$173$	4.75736	0.361695	0.180848	−	0.983511i	$-0.442116\pi$
$173$	4.75736	0.361695	0.180848	+	0.983511i	$0.442116\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	14.6569	1.09858
$179$	22.9706	1.71690	0.858450	−	0.512897i	$-0.171428\pi$
$179$	22.9706	1.71690	0.858450	+	0.512897i	$0.171428\pi$
$180$	0	0
$181$	−1.75736	−0.130623	−0.0653117	−	0.997865i	$-0.520804\pi$
$181$	−1.75736	−0.130623	−0.0653117	+	0.997865i	$0.520804\pi$
$182$	0	0
$183$	0	0
$184$	3.24264	0.239051
$185$	5.14214	0.378057
$186$	0	0
$187$	2.82843	0.206835
$188$	−0.343146	−0.0250265
$189$	0	0
$190$	0.272078	0.0197386
$191$	−9.24264	−0.668774	−0.334387	−	0.942436i	$-0.608529\pi$
$191$	−9.24264	−0.668774	−0.334387	+	0.942436i	$0.608529\pi$
$192$	0	0
$193$	−12.4853	−0.898710	−0.449355	−	0.893353i	$-0.648346\pi$
$193$	−12.4853	−0.898710	−0.449355	+	0.893353i	$0.648346\pi$
$194$	11.3137	0.812277
$195$	0	0
$196$	0	0
$197$	22.7279	1.61930	0.809649	−	0.586915i	$-0.199658\pi$
$197$	22.7279	1.61930	0.809649	+	0.586915i	$0.199658\pi$
$198$	0	0
$199$	−13.2426	−0.938746	−0.469373	−	0.883000i	$-0.655520\pi$
$199$	−13.2426	−0.938746	−0.469373	+	0.883000i	$0.655520\pi$
$200$	2.48528	0.175736
$201$	0	0
$202$	2.48528	0.174864
$203$	0	0
$204$	0	0
$205$	18.7574	1.31007
$206$	−4.07107	−0.283645
$207$	0	0
$208$	4.24264	0.294174
$209$	−0.171573	−0.0118679
$210$	0	0
$211$	−12.7279	−0.876226	−0.438113	−	0.898920i	$-0.644353\pi$
$211$	−12.7279	−0.876226	−0.438113	+	0.898920i	$0.644353\pi$
$212$	8.00000	0.549442
$213$	0	0
$214$	18.9706	1.29680
$215$	−16.6274	−1.13398
$216$	0	0
$217$	0	0
$218$	6.75736	0.457666
$219$	0	0
$220$	1.58579	0.106914
$221$	−12.0000	−0.807207
$222$	0	0
$223$	1.24264	0.0832134	0.0416067	−	0.999134i	$-0.486752\pi$
$223$	1.24264	0.0832134	0.0416067	+	0.999134i	$0.486752\pi$
$224$	0	0
$225$	0	0
$226$	4.24264	0.282216
$227$	12.7279	0.844782	0.422391	−	0.906414i	$-0.361191\pi$
$227$	12.7279	0.844782	0.422391	+	0.906414i	$0.361191\pi$
$228$	0	0
$229$	−12.0000	−0.792982	−0.396491	−	0.918039i	$-0.629772\pi$
$229$	−12.0000	−0.792982	−0.396491	+	0.918039i	$0.629772\pi$
$230$	−5.14214	−0.339062
$231$	0	0
$232$	−8.24264	−0.541156
$233$	7.75736	0.508202	0.254101	−	0.967178i	$-0.418221\pi$
$233$	7.75736	0.508202	0.254101	+	0.967178i	$0.418221\pi$
$234$	0	0
$235$	0.544156	0.0354968
$236$	−0.343146	−0.0223369
$237$	0	0
$238$	0	0
$239$	−18.0000	−1.16432	−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000	−1.16432	−0.582162	+	0.813073i	$0.697793\pi$
$240$	0	0
$241$	−14.4853	−0.933079	−0.466539	−	0.884500i	$-0.654499\pi$
$241$	−14.4853	−0.933079	−0.466539	+	0.884500i	$0.654499\pi$
$242$	10.0000	0.642824
$243$	0	0
$244$	−9.17157	−0.587150
$245$	0	0
$246$	0	0
$247$	0.727922	0.0463166
$248$	1.24264	0.0789078
$249$	0	0
$250$	−11.8701	−0.750728
$251$	11.3137	0.714115	0.357057	−	0.934082i	$-0.383780\pi$
$251$	11.3137	0.714115	0.357057	+	0.934082i	$0.383780\pi$
$252$	0	0
$253$	3.24264	0.203863
$254$	3.75736	0.235758
$255$	0	0
$256$	1.00000	0.0625000
$257$	−5.48528	−0.342162	−0.171081	−	0.985257i	$-0.554726\pi$
$257$	−5.48528	−0.342162	−0.171081	+	0.985257i	$0.554726\pi$
$258$	0	0
$259$	0	0
$260$	−6.72792	−0.417248
$261$	0	0
$262$	−2.48528	−0.153541
$263$	−21.7279	−1.33980	−0.669901	−	0.742451i	$-0.733663\pi$
$263$	−21.7279	−1.33980	−0.669901	+	0.742451i	$0.733663\pi$
$264$	0	0
$265$	−12.6863	−0.779313
$266$	0	0
$267$	0	0
$268$	−10.2426	−0.625669
$269$	−21.3848	−1.30385	−0.651926	−	0.758282i	$-0.726039\pi$
$269$	−21.3848	−1.30385	−0.651926	+	0.758282i	$0.726039\pi$
$270$	0	0
$271$	−29.3137	−1.78068	−0.890340	−	0.455295i	$-0.849534\pi$
$271$	−29.3137	−1.78068	−0.890340	+	0.455295i	$0.849534\pi$
$272$	−2.82843	−0.171499
$273$	0	0
$274$	−10.2426	−0.618781
$275$	2.48528	0.149868
$276$	0	0
$277$	28.2132	1.69517	0.847584	−	0.530662i	$-0.178057\pi$
$277$	28.2132	1.69517	0.847584	+	0.530662i	$0.178057\pi$
$278$	−3.17157	−0.190218
$279$	0	0
$280$	0	0
$281$	−6.48528	−0.386879	−0.193440	−	0.981112i	$-0.561964\pi$
$281$	−6.48528	−0.386879	−0.193440	+	0.981112i	$0.561964\pi$
$282$	0	0
$283$	−20.1421	−1.19733	−0.598663	−	0.801001i	$-0.704301\pi$
$283$	−20.1421	−1.19733	−0.598663	+	0.801001i	$0.704301\pi$
$284$	7.24264	0.429772
$285$	0	0
$286$	4.24264	0.250873
$287$	0	0
$288$	0	0
$289$	−9.00000	−0.529412
$290$	13.0711	0.767560
$291$	0	0
$292$	8.82843	0.516645
$293$	−11.6569	−0.681001	−0.340500	−	0.940244i	$-0.610596\pi$
$293$	−11.6569	−0.681001	−0.340500	+	0.940244i	$0.610596\pi$
$294$	0	0
$295$	0.544156	0.0316820
$296$	3.24264	0.188475
$297$	0	0
$298$	14.2426	0.825054
$299$	−13.7574	−0.795609
$300$	0	0
$301$	0	0
$302$	10.7279	0.617323
$303$	0	0
$304$	0.171573	0.00984038
$305$	14.5442	0.832796
$306$	0	0
$307$	16.7990	0.958769	0.479384	−	0.877605i	$-0.340860\pi$
$307$	16.7990	0.958769	0.479384	+	0.877605i	$0.340860\pi$
$308$	0	0
$309$	0	0
$310$	−1.97056	−0.111920
$311$	15.5563	0.882120	0.441060	−	0.897478i	$-0.354603\pi$
$311$	15.5563	0.882120	0.441060	+	0.897478i	$0.354603\pi$
$312$	0	0
$313$	10.2426	0.578948	0.289474	−	0.957186i	$-0.406520\pi$
$313$	10.2426	0.578948	0.289474	+	0.957186i	$0.406520\pi$
$314$	19.0711	1.07624
$315$	0	0
$316$	10.4853	0.589843
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	−8.24264	−0.461499
$320$	−1.58579	−0.0886482
$321$	0	0
$322$	0	0
$323$	−0.485281	−0.0270018
$324$	0	0
$325$	−10.5442	−0.584885
$326$	−10.7279	−0.594165
$327$	0	0
$328$	11.8284	0.653116
$329$	0	0
$330$	0	0
$331$	24.4853	1.34583	0.672916	−	0.739719i	$-0.265041\pi$
$331$	24.4853	1.34583	0.672916	+	0.739719i	$0.265041\pi$
$332$	−15.8995	−0.872598
$333$	0	0
$334$	24.3848	1.33428
$335$	16.2426	0.887430
$336$	0	0
$337$	13.4853	0.734590	0.367295	−	0.930104i	$-0.380284\pi$
$337$	13.4853	0.734590	0.367295	+	0.930104i	$0.380284\pi$
$338$	−5.00000	−0.271964
$339$	0	0
$340$	4.48528	0.243249
$341$	1.24264	0.0672928
$342$	0	0
$343$	0	0
$344$	−10.4853	−0.565328
$345$	0	0
$346$	−4.75736	−0.255757
$347$	−35.4853	−1.90495	−0.952475	−	0.304617i	$-0.901471\pi$
$347$	−35.4853	−1.90495	−0.952475	+	0.304617i	$0.901471\pi$
$348$	0	0
$349$	−22.2426	−1.19062	−0.595311	−	0.803496i	$-0.702971\pi$
$349$	−22.2426	−1.19062	−0.595311	+	0.803496i	$0.702971\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	0.0533002
$353$	8.65685	0.460758	0.230379	−	0.973101i	$-0.426003\pi$
$353$	8.65685	0.460758	0.230379	+	0.973101i	$0.426003\pi$
$354$	0	0
$355$	−11.4853	−0.609575
$356$	−14.6569	−0.776812
$357$	0	0
$358$	−22.9706	−1.21403
$359$	33.4558	1.76573	0.882866	−	0.469625i	$-0.155611\pi$
$359$	33.4558	1.76573	0.882866	+	0.469625i	$0.155611\pi$
$360$	0	0
$361$	−18.9706	−0.998451
$362$	1.75736	0.0923648
$363$	0	0
$364$	0	0
$365$	−14.0000	−0.732793
$366$	0	0
$367$	11.1005	0.579442	0.289721	−	0.957111i	$-0.406438\pi$
$367$	11.1005	0.579442	0.289721	+	0.957111i	$0.406438\pi$
$368$	−3.24264	−0.169034
$369$	0	0
$370$	−5.14214	−0.267327
$371$	0	0
$372$	0	0
$373$	−6.75736	−0.349883	−0.174941	−	0.984579i	$-0.555974\pi$
$373$	−6.75736	−0.349883	−0.174941	+	0.984579i	$0.555974\pi$
$374$	−2.82843	−0.146254
$375$	0	0
$376$	0.343146	0.0176964
$377$	34.9706	1.80108
$378$	0	0
$379$	−30.9706	−1.59085	−0.795425	−	0.606051i	$-0.792753\pi$
$379$	−30.9706	−1.59085	−0.795425	+	0.606051i	$0.792753\pi$
$380$	−0.272078	−0.0139573
$381$	0	0
$382$	9.24264	0.472895
$383$	30.0416	1.53506	0.767528	−	0.641016i	$-0.221487\pi$
$383$	30.0416	1.53506	0.767528	+	0.641016i	$0.221487\pi$
$384$	0	0
$385$	0	0
$386$	12.4853	0.635484
$387$	0	0
$388$	−11.3137	−0.574367
$389$	−3.02944	−0.153599	−0.0767993	−	0.997047i	$-0.524470\pi$
$389$	−3.02944	−0.153599	−0.0767993	+	0.997047i	$0.524470\pi$
$390$	0	0
$391$	9.17157	0.463826
$392$	0	0
$393$	0	0
$394$	−22.7279	−1.14502
$395$	−16.6274	−0.836616
$396$	0	0
$397$	−16.5858	−0.832417	−0.416208	−	0.909269i	$-0.636641\pi$
$397$	−16.5858	−0.832417	−0.416208	+	0.909269i	$0.636641\pi$
$398$	13.2426	0.663794
$399$	0	0
$400$	−2.48528	−0.124264
$401$	10.2426	0.511493	0.255747	−	0.966744i	$-0.417679\pi$
$401$	10.2426	0.511493	0.255747	+	0.966744i	$0.417679\pi$
$402$	0	0
$403$	−5.27208	−0.262621
$404$	−2.48528	−0.123647
$405$	0	0
$406$	0	0
$407$	3.24264	0.160732
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	−18.7574	−0.926360
$411$	0	0
$412$	4.07107	0.200567
$413$	0	0
$414$	0	0
$415$	25.2132	1.23767
$416$	−4.24264	−0.208013
$417$	0	0
$418$	0.171573	0.00839190
$419$	−12.0416	−0.588272	−0.294136	−	0.955764i	$-0.595032\pi$
$419$	−12.0416	−0.588272	−0.294136	+	0.955764i	$0.595032\pi$
$420$	0	0
$421$	18.2132	0.887657	0.443829	−	0.896112i	$-0.353620\pi$
$421$	18.2132	0.887657	0.443829	+	0.896112i	$0.353620\pi$
$422$	12.7279	0.619586
$423$	0	0
$424$	−8.00000	−0.388514
$425$	7.02944	0.340978
$426$	0	0
$427$	0	0
$428$	−18.9706	−0.916977
$429$	0	0
$430$	16.6274	0.801845
$431$	19.2426	0.926885	0.463443	−	0.886127i	$-0.346614\pi$
$431$	19.2426	0.926885	0.463443	+	0.886127i	$0.346614\pi$
$432$	0	0
$433$	−30.7279	−1.47669	−0.738345	−	0.674423i	$-0.764392\pi$
$433$	−30.7279	−1.47669	−0.738345	+	0.674423i	$0.764392\pi$
$434$	0	0
$435$	0	0
$436$	−6.75736	−0.323619
$437$	−0.556349	−0.0266138
$438$	0	0
$439$	28.9706	1.38269	0.691345	−	0.722525i	$-0.257019\pi$
$439$	28.9706	1.38269	0.691345	+	0.722525i	$0.257019\pi$
$440$	−1.58579	−0.0755994
$441$	0	0
$442$	12.0000	0.570782
$443$	28.9411	1.37503	0.687517	−	0.726168i	$-0.258701\pi$
$443$	28.9411	1.37503	0.687517	+	0.726168i	$0.258701\pi$
$444$	0	0
$445$	23.2426	1.10181
$446$	−1.24264	−0.0588407
$447$	0	0
$448$	0	0
$449$	11.2721	0.531962	0.265981	−	0.963978i	$-0.414304\pi$
$449$	11.2721	0.531962	0.265981	+	0.963978i	$0.414304\pi$
$450$	0	0
$451$	11.8284	0.556979
$452$	−4.24264	−0.199557
$453$	0	0
$454$	−12.7279	−0.597351
$455$	0	0
$456$	0	0
$457$	−13.9706	−0.653515	−0.326758	−	0.945108i	$-0.605956\pi$
$457$	−13.9706	−0.653515	−0.326758	+	0.945108i	$0.605956\pi$
$458$	12.0000	0.560723
$459$	0	0
$460$	5.14214	0.239753
$461$	−6.89949	−0.321342	−0.160671	−	0.987008i	$-0.551366\pi$
$461$	−6.89949	−0.321342	−0.160671	+	0.987008i	$0.551366\pi$
$462$	0	0
$463$	−13.2721	−0.616806	−0.308403	−	0.951256i	$-0.599794\pi$
$463$	−13.2721	−0.616806	−0.308403	+	0.951256i	$0.599794\pi$
$464$	8.24264	0.382655
$465$	0	0
$466$	−7.75736	−0.359353
$467$	23.6569	1.09471	0.547354	−	0.836901i	$-0.315635\pi$
$467$	23.6569	1.09471	0.547354	+	0.836901i	$0.315635\pi$
$468$	0	0
$469$	0	0
$470$	−0.544156	−0.0251000
$471$	0	0
$472$	0.343146	0.0157946
$473$	−10.4853	−0.482114
$474$	0	0
$475$	−0.426407	−0.0195649
$476$	0	0
$477$	0	0
$478$	18.0000	0.823301
$479$	−18.3431	−0.838120	−0.419060	−	0.907959i	$-0.637640\pi$
$479$	−18.3431	−0.838120	−0.419060	+	0.907959i	$0.637640\pi$
$480$	0	0
$481$	−13.7574	−0.627282
$482$	14.4853	0.659786
$483$	0	0
$484$	−10.0000	−0.454545
$485$	17.9411	0.814665
$486$	0	0
$487$	−21.5147	−0.974925	−0.487462	−	0.873144i	$-0.662077\pi$
$487$	−21.5147	−0.974925	−0.487462	+	0.873144i	$0.662077\pi$
$488$	9.17157	0.415178
$489$	0	0
$490$	0	0
$491$	0.514719	0.0232289	0.0116145	−	0.999933i	$-0.496303\pi$
$491$	0.514719	0.0232289	0.0116145	+	0.999933i	$0.496303\pi$
$492$	0	0
$493$	−23.3137	−1.05000
$494$	−0.727922	−0.0327508
$495$	0	0
$496$	−1.24264	−0.0557962
$497$	0	0
$498$	0	0
$499$	18.9706	0.849239	0.424620	−	0.905372i	$-0.360408\pi$
$499$	18.9706	0.849239	0.424620	+	0.905372i	$0.360408\pi$
$500$	11.8701	0.530845
$501$	0	0
$502$	−11.3137	−0.504956
$503$	39.5563	1.76373	0.881865	−	0.471502i	$-0.156288\pi$
$503$	39.5563	1.76373	0.881865	+	0.471502i	$0.156288\pi$
$504$	0	0
$505$	3.94113	0.175378
$506$	−3.24264	−0.144153
$507$	0	0
$508$	−3.75736	−0.166706
$509$	22.2843	0.987733	0.493866	−	0.869538i	$-0.335583\pi$
$509$	22.2843	0.987733	0.493866	+	0.869538i	$0.335583\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	5.48528	0.241945
$515$	−6.45584	−0.284478
$516$	0	0
$517$	0.343146	0.0150915
$518$	0	0
$519$	0	0
$520$	6.72792	0.295039
$521$	5.14214	0.225281	0.112641	−	0.993636i	$-0.464069\pi$
$521$	5.14214	0.225281	0.112641	+	0.993636i	$0.464069\pi$
$522$	0	0
$523$	0.514719	0.0225071	0.0112535	−	0.999937i	$-0.496418\pi$
$523$	0.514719	0.0225071	0.0112535	+	0.999937i	$0.496418\pi$
$524$	2.48528	0.108570
$525$	0	0
$526$	21.7279	0.947382
$527$	3.51472	0.153104
$528$	0	0
$529$	−12.4853	−0.542838
$530$	12.6863	0.551057
$531$	0	0
$532$	0	0
$533$	−50.1838	−2.17370
$534$	0	0
$535$	30.0833	1.30061
$536$	10.2426	0.442415
$537$	0	0
$538$	21.3848	0.921963
$539$	0	0
$540$	0	0
$541$	−33.2426	−1.42921	−0.714606	−	0.699527i	$-0.753394\pi$
$541$	−33.2426	−1.42921	−0.714606	+	0.699527i	$0.753394\pi$
$542$	29.3137	1.25913
$543$	0	0
$544$	2.82843	0.121268
$545$	10.7157	0.459011
$546$	0	0
$547$	7.51472	0.321306	0.160653	−	0.987011i	$-0.448640\pi$
$547$	7.51472	0.321306	0.160653	+	0.987011i	$0.448640\pi$
$548$	10.2426	0.437544
$549$	0	0
$550$	−2.48528	−0.105973
$551$	1.41421	0.0602475
$552$	0	0
$553$	0	0
$554$	−28.2132	−1.19866
$555$	0	0
$556$	3.17157	0.134505
$557$	−6.72792	−0.285071	−0.142536	−	0.989790i	$-0.545526\pi$
$557$	−6.72792	−0.285071	−0.142536	+	0.989790i	$0.545526\pi$
$558$	0	0
$559$	44.4853	1.88153
$560$	0	0
$561$	0	0
$562$	6.48528	0.273565
$563$	19.4142	0.818212	0.409106	−	0.912487i	$-0.365841\pi$
$563$	19.4142	0.818212	0.409106	+	0.912487i	$0.365841\pi$
$564$	0	0
$565$	6.72792	0.283046
$566$	20.1421	0.846637
$567$	0	0
$568$	−7.24264	−0.303894
$569$	36.2426	1.51937	0.759685	−	0.650291i	$-0.225353\pi$
$569$	36.2426	1.51937	0.759685	+	0.650291i	$0.225353\pi$
$570$	0	0
$571$	−41.2132	−1.72472	−0.862359	−	0.506297i	$-0.831014\pi$
$571$	−41.2132	−1.72472	−0.862359	+	0.506297i	$0.831014\pi$
$572$	−4.24264	−0.177394
$573$	0	0
$574$	0	0
$575$	8.05887	0.336078
$576$	0	0
$577$	28.2426	1.17576	0.587878	−	0.808949i	$-0.299963\pi$
$577$	28.2426	1.17576	0.587878	+	0.808949i	$0.299963\pi$
$578$	9.00000	0.374351
$579$	0	0
$580$	−13.0711	−0.542747
$581$	0	0
$582$	0	0
$583$	−8.00000	−0.331326
$584$	−8.82843	−0.365323
$585$	0	0
$586$	11.6569	0.481540
$587$	16.2426	0.670406	0.335203	−	0.942146i	$-0.391195\pi$
$587$	16.2426	0.670406	0.335203	+	0.942146i	$0.391195\pi$
$588$	0	0
$589$	−0.213203	−0.00878489
$590$	−0.544156	−0.0224025
$591$	0	0
$592$	−3.24264	−0.133272
$593$	−8.65685	−0.355494	−0.177747	−	0.984076i	$-0.556881\pi$
$593$	−8.65685	−0.355494	−0.177747	+	0.984076i	$0.556881\pi$
$594$	0	0
$595$	0	0
$596$	−14.2426	−0.583401
$597$	0	0
$598$	13.7574	0.562580
$599$	−5.72792	−0.234037	−0.117018	−	0.993130i	$-0.537334\pi$
$599$	−5.72792	−0.234037	−0.117018	+	0.993130i	$0.537334\pi$
$600$	0	0
$601$	−5.27208	−0.215053	−0.107526	−	0.994202i	$-0.534293\pi$
$601$	−5.27208	−0.215053	−0.107526	+	0.994202i	$0.534293\pi$
$602$	0	0
$603$	0	0
$604$	−10.7279	−0.436513
$605$	15.8579	0.644714
$606$	0	0
$607$	30.0000	1.21766	0.608831	−	0.793300i	$-0.291639\pi$
$607$	30.0000	1.21766	0.608831	+	0.793300i	$0.291639\pi$
$608$	−0.171573	−0.00695820
$609$	0	0
$610$	−14.5442	−0.588876
$611$	−1.45584	−0.0588971
$612$	0	0
$613$	3.72792	0.150569	0.0752847	−	0.997162i	$-0.476013\pi$
$613$	3.72792	0.150569	0.0752847	+	0.997162i	$0.476013\pi$
$614$	−16.7990	−0.677952
$615$	0	0
$616$	0	0
$617$	6.97056	0.280624	0.140312	−	0.990107i	$-0.455189\pi$
$617$	6.97056	0.280624	0.140312	+	0.990107i	$0.455189\pi$
$618$	0	0
$619$	−22.7990	−0.916369	−0.458184	−	0.888857i	$-0.651500\pi$
$619$	−22.7990	−0.916369	−0.458184	+	0.888857i	$0.651500\pi$
$620$	1.97056	0.0791397
$621$	0	0
$622$	−15.5563	−0.623753
$623$	0	0
$624$	0	0
$625$	−6.39697	−0.255879
$626$	−10.2426	−0.409378
$627$	0	0
$628$	−19.0711	−0.761018
$629$	9.17157	0.365695
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	−10.4853	−0.417082
$633$	0	0
$634$	10.0000	0.397151
$635$	5.95837	0.236451
$636$	0	0
$637$	0	0
$638$	8.24264	0.326329
$639$	0	0
$640$	1.58579	0.0626837
$641$	1.21320	0.0479187	0.0239593	−	0.999713i	$-0.492373\pi$
$641$	1.21320	0.0479187	0.0239593	+	0.999713i	$0.492373\pi$
$642$	0	0
$643$	0.171573	0.00676617	0.00338309	−	0.999994i	$-0.498923\pi$
$643$	0.171573	0.00676617	0.00338309	+	0.999994i	$0.498923\pi$
$644$	0	0
$645$	0	0
$646$	0.485281	0.0190931
$647$	21.5147	0.845831	0.422915	−	0.906169i	$-0.361007\pi$
$647$	21.5147	0.845831	0.422915	+	0.906169i	$0.361007\pi$
$648$	0	0
$649$	0.343146	0.0134696
$650$	10.5442	0.413576
$651$	0	0
$652$	10.7279	0.420138
$653$	27.9411	1.09342	0.546710	−	0.837322i	$-0.315880\pi$
$653$	27.9411	1.09342	0.546710	+	0.837322i	$0.315880\pi$
$654$	0	0
$655$	−3.94113	−0.153993
$656$	−11.8284	−0.461822
$657$	0	0
$658$	0	0
$659$	−24.9411	−0.971568	−0.485784	−	0.874079i	$-0.661466\pi$
$659$	−24.9411	−0.971568	−0.485784	+	0.874079i	$0.661466\pi$
$660$	0	0
$661$	−0.727922	−0.0283129	−0.0141564	−	0.999900i	$-0.504506\pi$
$661$	−0.727922	−0.0283129	−0.0141564	+	0.999900i	$0.504506\pi$
$662$	−24.4853	−0.951647
$663$	0	0
$664$	15.8995	0.617020
$665$	0	0
$666$	0	0
$667$	−26.7279	−1.03491
$668$	−24.3848	−0.943475
$669$	0	0
$670$	−16.2426	−0.627508
$671$	9.17157	0.354065
$672$	0	0
$673$	49.4558	1.90638	0.953191	−	0.302368i	$-0.0977771\pi$
$673$	49.4558	1.90638	0.953191	+	0.302368i	$0.0977771\pi$
$674$	−13.4853	−0.519434
$675$	0	0
$676$	5.00000	0.192308
$677$	38.6985	1.48730	0.743652	−	0.668567i	$-0.233092\pi$
$677$	38.6985	1.48730	0.743652	+	0.668567i	$0.233092\pi$
$678$	0	0
$679$	0	0
$680$	−4.48528	−0.172003
$681$	0	0
$682$	−1.24264	−0.0475832
$683$	13.9706	0.534569	0.267284	−	0.963618i	$-0.413874\pi$
$683$	13.9706	0.534569	0.267284	+	0.963618i	$0.413874\pi$
$684$	0	0
$685$	−16.2426	−0.620599
$686$	0	0
$687$	0	0
$688$	10.4853	0.399748
$689$	33.9411	1.29305
$690$	0	0
$691$	21.9411	0.834680	0.417340	−	0.908750i	$-0.362962\pi$
$691$	21.9411	0.834680	0.417340	+	0.908750i	$0.362962\pi$
$692$	4.75736	0.180848
$693$	0	0
$694$	35.4853	1.34700
$695$	−5.02944	−0.190777
$696$	0	0
$697$	33.4558	1.26723
$698$	22.2426	0.841896
$699$	0	0
$700$	0	0
$701$	−19.7574	−0.746225	−0.373113	−	0.927786i	$-0.621709\pi$
$701$	−19.7574	−0.746225	−0.373113	+	0.927786i	$0.621709\pi$
$702$	0	0
$703$	−0.556349	−0.0209831
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−8.65685	−0.325805
$707$	0	0
$708$	0	0
$709$	−27.2426	−1.02312	−0.511559	−	0.859248i	$-0.670932\pi$
$709$	−27.2426	−1.02312	−0.511559	+	0.859248i	$0.670932\pi$
$710$	11.4853	0.431035
$711$	0	0
$712$	14.6569	0.549289
$713$	4.02944	0.150904
$714$	0	0
$715$	6.72792	0.251610
$716$	22.9706	0.858450
$717$	0	0
$718$	−33.4558	−1.24856
$719$	23.6569	0.882252	0.441126	−	0.897445i	$-0.354579\pi$
$719$	23.6569	0.882252	0.441126	+	0.897445i	$0.354579\pi$
$720$	0	0
$721$	0	0
$722$	18.9706	0.706011
$723$	0	0
$724$	−1.75736	−0.0653117
$725$	−20.4853	−0.760804
$726$	0	0
$727$	14.1421	0.524503	0.262251	−	0.965000i	$-0.415535\pi$
$727$	14.1421	0.524503	0.262251	+	0.965000i	$0.415535\pi$
$728$	0	0
$729$	0	0
$730$	14.0000	0.518163
$731$	−29.6569	−1.09690
$732$	0	0
$733$	−27.9411	−1.03203	−0.516015	−	0.856580i	$-0.672585\pi$
$733$	−27.9411	−1.03203	−0.516015	+	0.856580i	$0.672585\pi$
$734$	−11.1005	−0.409727
$735$	0	0
$736$	3.24264	0.119525
$737$	10.2426	0.377293
$738$	0	0
$739$	48.7279	1.79249	0.896243	−	0.443564i	$-0.146286\pi$
$739$	48.7279	1.79249	0.896243	+	0.443564i	$0.146286\pi$
$740$	5.14214	0.189029
$741$	0	0
$742$	0	0
$743$	−14.7574	−0.541395	−0.270698	−	0.962664i	$-0.587254\pi$
$743$	−14.7574	−0.541395	−0.270698	+	0.962664i	$0.587254\pi$
$744$	0	0
$745$	22.5858	0.827479
$746$	6.75736	0.247405
$747$	0	0
$748$	2.82843	0.103418
$749$	0	0
$750$	0	0
$751$	39.4558	1.43976	0.719882	−	0.694096i	$-0.244196\pi$
$751$	39.4558	1.43976	0.719882	+	0.694096i	$0.244196\pi$
$752$	−0.343146	−0.0125132
$753$	0	0
$754$	−34.9706	−1.27355
$755$	17.0122	0.619137
$756$	0	0
$757$	−47.4558	−1.72481	−0.862406	−	0.506217i	$-0.831043\pi$
$757$	−47.4558	−1.72481	−0.862406	+	0.506217i	$0.831043\pi$
$758$	30.9706	1.12490
$759$	0	0
$760$	0.272078	0.00986930
$761$	29.3137	1.06262	0.531311	−	0.847177i	$-0.321700\pi$
$761$	29.3137	1.06262	0.531311	+	0.847177i	$0.321700\pi$
$762$	0	0
$763$	0	0
$764$	−9.24264	−0.334387
$765$	0	0
$766$	−30.0416	−1.08545
$767$	−1.45584	−0.0525675
$768$	0	0
$769$	44.1838	1.59331	0.796654	−	0.604436i	$-0.206601\pi$
$769$	44.1838	1.59331	0.796654	+	0.604436i	$0.206601\pi$
$770$	0	0
$771$	0	0
$772$	−12.4853	−0.449355
$773$	−39.3848	−1.41657	−0.708286	−	0.705926i	$-0.750531\pi$
$773$	−39.3848	−1.41657	−0.708286	+	0.705926i	$0.750531\pi$
$774$	0	0
$775$	3.08831	0.110935
$776$	11.3137	0.406138
$777$	0	0
$778$	3.02944	0.108611
$779$	−2.02944	−0.0727121
$780$	0	0
$781$	−7.24264	−0.259162
$782$	−9.17157	−0.327975
$783$	0	0
$784$	0	0
$785$	30.2426	1.07941
$786$	0	0
$787$	53.3137	1.90043	0.950214	−	0.311597i	$-0.100864\pi$
$787$	53.3137	1.90043	0.950214	+	0.311597i	$0.100864\pi$
$788$	22.7279	0.809649
$789$	0	0
$790$	16.6274	0.591577
$791$	0	0
$792$	0	0
$793$	−38.9117	−1.38179
$794$	16.5858	0.588608
$795$	0	0
$796$	−13.2426	−0.469373
$797$	25.2426	0.894140	0.447070	−	0.894499i	$-0.352467\pi$
$797$	25.2426	0.894140	0.447070	+	0.894499i	$0.352467\pi$
$798$	0	0
$799$	0.970563	0.0343360
$800$	2.48528	0.0878680
$801$	0	0
$802$	−10.2426	−0.361680
$803$	−8.82843	−0.311548
$804$	0	0
$805$	0	0
$806$	5.27208	0.185701
$807$	0	0
$808$	2.48528	0.0874319
$809$	−18.2426	−0.641377	−0.320689	−	0.947185i	$-0.603914\pi$
$809$	−18.2426	−0.641377	−0.320689	+	0.947185i	$0.603914\pi$
$810$	0	0
$811$	45.4264	1.59514	0.797568	−	0.603228i	$-0.206119\pi$
$811$	45.4264	1.59514	0.797568	+	0.603228i	$0.206119\pi$
$812$	0	0
$813$	0	0
$814$	−3.24264	−0.113654
$815$	−17.0122	−0.595911
$816$	0	0
$817$	1.79899	0.0629387
$818$	−6.00000	−0.209785
$819$	0	0
$820$	18.7574	0.655035
$821$	−13.0294	−0.454730	−0.227365	−	0.973810i	$-0.573011\pi$
$821$	−13.0294	−0.454730	−0.227365	+	0.973810i	$0.573011\pi$
$822$	0	0
$823$	29.7574	1.03728	0.518638	−	0.854994i	$-0.326439\pi$
$823$	29.7574	1.03728	0.518638	+	0.854994i	$0.326439\pi$
$824$	−4.07107	−0.141822
$825$	0	0
$826$	0	0
$827$	12.9411	0.450007	0.225004	−	0.974358i	$-0.427761\pi$
$827$	12.9411	0.450007	0.225004	+	0.974358i	$0.427761\pi$
$828$	0	0
$829$	15.8579	0.550766	0.275383	−	0.961335i	$-0.411195\pi$
$829$	15.8579	0.550766	0.275383	+	0.961335i	$0.411195\pi$
$830$	−25.2132	−0.875163
$831$	0	0
$832$	4.24264	0.147087
$833$	0	0
$834$	0	0
$835$	38.6690	1.33820
$836$	−0.171573	−0.00593397
$837$	0	0
$838$	12.0416	0.415971
$839$	−17.6152	−0.608145	−0.304073	−	0.952649i	$-0.598347\pi$
$839$	−17.6152	−0.608145	−0.304073	+	0.952649i	$0.598347\pi$
$840$	0	0
$841$	38.9411	1.34280
$842$	−18.2132	−0.627668
$843$	0	0
$844$	−12.7279	−0.438113
$845$	−7.92893	−0.272764
$846$	0	0
$847$	0	0
$848$	8.00000	0.274721
$849$	0	0
$850$	−7.02944	−0.241108
$851$	10.5147	0.360440
$852$	0	0
$853$	−24.0000	−0.821744	−0.410872	−	0.911693i	$-0.634776\pi$
$853$	−24.0000	−0.821744	−0.410872	+	0.911693i	$0.634776\pi$
$854$	0	0
$855$	0	0
$856$	18.9706	0.648400
$857$	33.7696	1.15355	0.576773	−	0.816904i	$-0.304312\pi$
$857$	33.7696	1.15355	0.576773	+	0.816904i	$0.304312\pi$
$858$	0	0
$859$	29.8284	1.01773	0.508866	−	0.860846i	$-0.330065\pi$
$859$	29.8284	1.01773	0.508866	+	0.860846i	$0.330065\pi$
$860$	−16.6274	−0.566990
$861$	0	0
$862$	−19.2426	−0.655407
$863$	2.97056	0.101119	0.0505596	−	0.998721i	$-0.483900\pi$
$863$	2.97056	0.101119	0.0505596	+	0.998721i	$0.483900\pi$
$864$	0	0
$865$	−7.54416	−0.256509
$866$	30.7279	1.04418
$867$	0	0
$868$	0	0
$869$	−10.4853	−0.355689
$870$	0	0
$871$	−43.4558	−1.47245
$872$	6.75736	0.228833
$873$	0	0
$874$	0.556349	0.0188188
$875$	0	0
$876$	0	0
$877$	−14.9706	−0.505520	−0.252760	−	0.967529i	$-0.581338\pi$
$877$	−14.9706	−0.505520	−0.252760	+	0.967529i	$0.581338\pi$
$878$	−28.9706	−0.977709
$879$	0	0
$880$	1.58579	0.0534568
$881$	−54.9411	−1.85101	−0.925507	−	0.378731i	$-0.876361\pi$
$881$	−54.9411	−1.85101	−0.925507	+	0.378731i	$0.876361\pi$
$882$	0	0
$883$	11.5147	0.387501	0.193751	−	0.981051i	$-0.437935\pi$
$883$	11.5147	0.387501	0.193751	+	0.981051i	$0.437935\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−28.9411	−0.972296
$887$	−47.3553	−1.59004	−0.795018	−	0.606585i	$-0.792539\pi$
$887$	−47.3553	−1.59004	−0.795018	+	0.606585i	$0.792539\pi$
$888$	0	0
$889$	0	0
$890$	−23.2426	−0.779095
$891$	0	0
$892$	1.24264	0.0416067
$893$	−0.0588745	−0.00197016
$894$	0	0
$895$	−36.4264	−1.21760
$896$	0	0
$897$	0	0
$898$	−11.2721	−0.376154
$899$	−10.2426	−0.341611
$900$	0	0
$901$	−22.6274	−0.753829
$902$	−11.8284	−0.393844
$903$	0	0
$904$	4.24264	0.141108
$905$	2.78680	0.0926363
$906$	0	0
$907$	−8.72792	−0.289806	−0.144903	−	0.989446i	$-0.546287\pi$
$907$	−8.72792	−0.289806	−0.144903	+	0.989446i	$0.546287\pi$
$908$	12.7279	0.422391
$909$	0	0
$910$	0	0
$911$	12.4853	0.413656	0.206828	−	0.978377i	$-0.433686\pi$
$911$	12.4853	0.413656	0.206828	+	0.978377i	$0.433686\pi$
$912$	0	0
$913$	15.8995	0.526196
$914$	13.9706	0.462105
$915$	0	0
$916$	−12.0000	−0.396491
$917$	0	0
$918$	0	0
$919$	12.4853	0.411851	0.205926	−	0.978568i	$-0.433979\pi$
$919$	12.4853	0.411851	0.205926	+	0.978568i	$0.433979\pi$
$920$	−5.14214	−0.169531
$921$	0	0
$922$	6.89949	0.227223
$923$	30.7279	1.01142
$924$	0	0
$925$	8.05887	0.264974
$926$	13.2721	0.436148
$927$	0	0
$928$	−8.24264	−0.270578
$929$	−16.2843	−0.534270	−0.267135	−	0.963659i	$-0.586077\pi$
$929$	−16.2843	−0.534270	−0.267135	+	0.963659i	$0.586077\pi$
$930$	0	0
$931$	0	0
$932$	7.75736	0.254101
$933$	0	0
$934$	−23.6569	−0.774076
$935$	−4.48528	−0.146684
$936$	0	0
$937$	0.686292	0.0224202	0.0112101	−	0.999937i	$-0.496432\pi$
$937$	0.686292	0.0224202	0.0112101	+	0.999937i	$0.496432\pi$
$938$	0	0
$939$	0	0
$940$	0.544156	0.0177484
$941$	−10.0711	−0.328307	−0.164154	−	0.986435i	$-0.552489\pi$
$941$	−10.0711	−0.328307	−0.164154	+	0.986435i	$0.552489\pi$
$942$	0	0
$943$	38.3553	1.24902
$944$	−0.343146	−0.0111684
$945$	0	0
$946$	10.4853	0.340906
$947$	16.0294	0.520887	0.260443	−	0.965489i	$-0.416131\pi$
$947$	16.0294	0.520887	0.260443	+	0.965489i	$0.416131\pi$
$948$	0	0
$949$	37.4558	1.21587
$950$	0.426407	0.0138345
$951$	0	0
$952$	0	0
$953$	−17.7574	−0.575217	−0.287609	−	0.957748i	$-0.592860\pi$
$953$	−17.7574	−0.575217	−0.287609	+	0.957748i	$0.592860\pi$
$954$	0	0
$955$	14.6569	0.474285
$956$	−18.0000	−0.582162
$957$	0	0
$958$	18.3431	0.592640
$959$	0	0
$960$	0	0
$961$	−29.4558	−0.950189
$962$	13.7574	0.443555
$963$	0	0
$964$	−14.4853	−0.466539
$965$	19.7990	0.637352
$966$	0	0
$967$	−52.6690	−1.69372	−0.846861	−	0.531814i	$-0.821511\pi$
$967$	−52.6690	−1.69372	−0.846861	+	0.531814i	$0.821511\pi$
$968$	10.0000	0.321412
$969$	0	0
$970$	−17.9411	−0.576055
$971$	−55.4975	−1.78100	−0.890499	−	0.454984i	$-0.849645\pi$
$971$	−55.4975	−1.78100	−0.890499	+	0.454984i	$0.849645\pi$
$972$	0	0
$973$	0	0
$974$	21.5147	0.689376
$975$	0	0
$976$	−9.17157	−0.293575
$977$	−16.9706	−0.542936	−0.271468	−	0.962447i	$-0.587509\pi$
$977$	−16.9706	−0.542936	−0.271468	+	0.962447i	$0.587509\pi$
$978$	0	0
$979$	14.6569	0.468435
$980$	0	0
$981$	0	0
$982$	−0.514719	−0.0164253
$983$	1.79899	0.0573789	0.0286894	−	0.999588i	$-0.490867\pi$
$983$	1.79899	0.0573789	0.0286894	+	0.999588i	$0.490867\pi$
$984$	0	0
$985$	−36.0416	−1.14838
$986$	23.3137	0.742460
$987$	0	0
$988$	0.727922	0.0231583
$989$	−34.0000	−1.08114
$990$	0	0
$991$	0.242641	0.00770774	0.00385387	−	0.999993i	$-0.498773\pi$
$991$	0.242641	0.00770774	0.00385387	+	0.999993i	$0.498773\pi$
$992$	1.24264	0.0394539
$993$	0	0
$994$	0	0
$995$	21.0000	0.665745
$996$	0	0
$997$	−39.8995	−1.26363	−0.631815	−	0.775119i	$-0.717690\pi$
$997$	−39.8995	−1.26363	−0.631815	+	0.775119i	$0.717690\pi$
$998$	−18.9706	−0.600503
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.a.be.1.2	✓	2
3.2	odd	2		2646.2.a.bp.1.1	yes	2
7.6	odd	2		2646.2.a.bj.1.1	yes	2
21.20	even	2		2646.2.a.bk.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2646.2.a.be.1.2	✓	2	1.1	even	1	trivial
2646.2.a.bj.1.1	yes	2	7.6	odd	2
2646.2.a.bk.1.2	yes	2	21.20	even	2
2646.2.a.bp.1.1	yes	2	3.2	odd	2

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:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.58579 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.58579 q^{5} -1.00000 q^{8} +1.58579 q^{10} -1.00000 q^{11} +4.24264 q^{13} +1.00000 q^{16} -2.82843 q^{17} +0.171573 q^{19} -1.58579 q^{20} +1.00000 q^{22} -3.24264 q^{23} -2.48528 q^{25} -4.24264 q^{26} +8.24264 q^{29} -1.24264 q^{31} -1.00000 q^{32} +2.82843 q^{34} -3.24264 q^{37} -0.171573 q^{38} +1.58579 q^{40} -11.8284 q^{41} +10.4853 q^{43} -1.00000 q^{44} +3.24264 q^{46} -0.343146 q^{47} +2.48528 q^{50} +4.24264 q^{52} +8.00000 q^{53} +1.58579 q^{55} -8.24264 q^{58} -0.343146 q^{59} -9.17157 q^{61} +1.24264 q^{62} +1.00000 q^{64} -6.72792 q^{65} -10.2426 q^{67} -2.82843 q^{68} +7.24264 q^{71} +8.82843 q^{73} +3.24264 q^{74} +0.171573 q^{76} +10.4853 q^{79} -1.58579 q^{80} +11.8284 q^{82} -15.8995 q^{83} +4.48528 q^{85} -10.4853 q^{86} +1.00000 q^{88} -14.6569 q^{89} -3.24264 q^{92} +0.343146 q^{94} -0.272078 q^{95} -11.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 6 q^{5} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 6 * q^5 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 6 q^{5} - 2 q^{8} + 6 q^{10} - 2 q^{11} + 2 q^{16} + 6 q^{19} - 6 q^{20} + 2 q^{22} + 2 q^{23} + 12 q^{25} + 8 q^{29} + 6 q^{31} - 2 q^{32} + 2 q^{37} - 6 q^{38} + 6 q^{40} - 18 q^{41} + 4 q^{43} - 2 q^{44} - 2 q^{46} - 12 q^{47} - 12 q^{50} + 16 q^{53} + 6 q^{55} - 8 q^{58} - 12 q^{59} - 24 q^{61} - 6 q^{62} + 2 q^{64} + 12 q^{65} - 12 q^{67} + 6 q^{71} + 12 q^{73} - 2 q^{74} + 6 q^{76} + 4 q^{79} - 6 q^{80} + 18 q^{82} - 12 q^{83} - 8 q^{85} - 4 q^{86} + 2 q^{88} - 18 q^{89} + 2 q^{92} + 12 q^{94} - 26 q^{95}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 6 * q^5 - 2 * q^8 + 6 * q^10 - 2 * q^11 + 2 * q^16 + 6 * q^19 - 6 * q^20 + 2 * q^22 + 2 * q^23 + 12 * q^25 + 8 * q^29 + 6 * q^31 - 2 * q^32 + 2 * q^37 - 6 * q^38 + 6 * q^40 - 18 * q^41 + 4 * q^43 - 2 * q^44 - 2 * q^46 - 12 * q^47 - 12 * q^50 + 16 * q^53 + 6 * q^55 - 8 * q^58 - 12 * q^59 - 24 * q^61 - 6 * q^62 + 2 * q^64 + 12 * q^65 - 12 * q^67 + 6 * q^71 + 12 * q^73 - 2 * q^74 + 6 * q^76 + 4 * q^79 - 6 * q^80 + 18 * q^82 - 12 * q^83 - 8 * q^85 - 4 * q^86 + 2 * q^88 - 18 * q^89 + 2 * q^92 + 12 * q^94 - 26 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more