LMFDB - Embedded newform 2704.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2704,2,Mod(1,2704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2704, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2704.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2704 = 2^{4} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2704.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:	$21.5915487066$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1352)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	2704.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.44504 q^{3} -1.44504 q^{5} +0.445042 q^{7} -0.911854 q^{9} +O(q^{10})$	$q+1.44504 q^{3} -1.44504 q^{5} +0.445042 q^{7} -0.911854 q^{9} -2.33513 q^{11} -2.08815 q^{15} +6.89977 q^{17} -6.74094 q^{19} +0.643104 q^{21} +5.71379 q^{23} -2.91185 q^{25} -5.65279 q^{27} -8.14675 q^{29} +4.54288 q^{31} -3.37435 q^{33} -0.643104 q^{35} -8.07069 q^{37} +1.71379 q^{41} +3.86294 q^{43} +1.31767 q^{45} +0.829085 q^{47} -6.80194 q^{49} +9.97046 q^{51} -7.02715 q^{53} +3.37435 q^{55} -9.74094 q^{57} +3.59179 q^{59} -1.65519 q^{61} -0.405813 q^{63} -13.3448 q^{67} +8.25667 q^{69} -14.4916 q^{71} -6.35690 q^{73} -4.20775 q^{75} -1.03923 q^{77} +6.47219 q^{79} -5.43296 q^{81} -2.41789 q^{83} -9.97046 q^{85} -11.7724 q^{87} -13.2295 q^{89} +6.56465 q^{93} +9.74094 q^{95} -11.0368 q^{97} +2.12929 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{3} - 4 q^{5} + q^{7} + q^{9}+O(q^{10}) $ 3 * q + 4 * q^3 - 4 * q^5 + q^7 + q^9	$ 3 q + 4 q^{3} - 4 q^{5} + q^{7} + q^{9} - 6 q^{11} - 10 q^{15} - 2 q^{17} - 6 q^{19} + 6 q^{21} + 9 q^{23} - 5 q^{25} + q^{27} + 3 q^{29} - 5 q^{31} - 22 q^{33} - 6 q^{35} - 12 q^{37} - 3 q^{41} + 17 q^{43} - 13 q^{45} - 8 q^{47} - 16 q^{49} - 5 q^{51} - 15 q^{53} + 22 q^{55} - 15 q^{57} - 17 q^{59} - 28 q^{61} + 12 q^{63} - 17 q^{67} - 2 q^{69} + 7 q^{71} - 15 q^{73} + 5 q^{75} - 16 q^{77} + 13 q^{79} + 3 q^{81} - 13 q^{83} + 5 q^{85} + 4 q^{87} - 19 q^{89} - 2 q^{93} + 15 q^{95} - 5 q^{97} - 37 q^{99}+O(q^{100}) $ 3 * q + 4 * q^3 - 4 * q^5 + q^7 + q^9 - 6 * q^11 - 10 * q^15 - 2 * q^17 - 6 * q^19 + 6 * q^21 + 9 * q^23 - 5 * q^25 + q^27 + 3 * q^29 - 5 * q^31 - 22 * q^33 - 6 * q^35 - 12 * q^37 - 3 * q^41 + 17 * q^43 - 13 * q^45 - 8 * q^47 - 16 * q^49 - 5 * q^51 - 15 * q^53 + 22 * q^55 - 15 * q^57 - 17 * q^59 - 28 * q^61 + 12 * q^63 - 17 * q^67 - 2 * q^69 + 7 * q^71 - 15 * q^73 + 5 * q^75 - 16 * q^77 + 13 * q^79 + 3 * q^81 - 13 * q^83 + 5 * q^85 + 4 * q^87 - 19 * q^89 - 2 * q^93 + 15 * q^95 - 5 * q^97 - 37 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.44504	0.834295	0.417148	−	0.908839i	$-0.363030\pi$
$3$	1.44504	0.834295	0.417148	+	0.908839i	$0.363030\pi$
$4$	0	0
$5$	−1.44504	−0.646242	−0.323121	−	0.946358i	$-0.604732\pi$
$5$	−1.44504	−0.646242	−0.323121	+	0.946358i	$0.604732\pi$
$6$	0	0
$7$	0.445042	0.168210	0.0841050	−	0.996457i	$-0.473197\pi$
$7$	0.445042	0.168210	0.0841050	+	0.996457i	$0.473197\pi$
$8$	0	0
$9$	−0.911854	−0.303951
$10$	0	0
$11$	−2.33513	−0.704067	−0.352033	−	0.935987i	$-0.614510\pi$
$11$	−2.33513	−0.704067	−0.352033	+	0.935987i	$0.614510\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−2.08815	−0.539157
$16$	0	0
$17$	6.89977	1.67344	0.836720	−	0.547630i	$-0.184470\pi$
$17$	6.89977	1.67344	0.836720	+	0.547630i	$0.184470\pi$
$18$	0	0
$19$	−6.74094	−1.54648	−0.773239	−	0.634115i	$-0.781365\pi$
$19$	−6.74094	−1.54648	−0.773239	+	0.634115i	$0.781365\pi$
$20$	0	0
$21$	0.643104	0.140337
$22$	0	0
$23$	5.71379	1.19141	0.595704	−	0.803204i	$-0.296873\pi$
$23$	5.71379	1.19141	0.595704	+	0.803204i	$0.296873\pi$
$24$	0	0
$25$	−2.91185	−0.582371
$26$	0	0
$27$	−5.65279	−1.08788
$28$	0	0
$29$	−8.14675	−1.51281	−0.756407	−	0.654101i	$-0.773047\pi$
$29$	−8.14675	−1.51281	−0.756407	+	0.654101i	$0.773047\pi$
$30$	0	0
$31$	4.54288	0.815925	0.407962	−	0.912999i	$-0.366239\pi$
$31$	4.54288	0.815925	0.407962	+	0.912999i	$0.366239\pi$
$32$	0	0
$33$	−3.37435	−0.587400
$34$	0	0
$35$	−0.643104	−0.108704
$36$	0	0
$37$	−8.07069	−1.32681	−0.663406	−	0.748259i	$-0.730890\pi$
$37$	−8.07069	−1.32681	−0.663406	+	0.748259i	$0.730890\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.71379	0.267649	0.133825	−	0.991005i	$-0.457274\pi$
$41$	1.71379	0.267649	0.133825	+	0.991005i	$0.457274\pi$
$42$	0	0
$43$	3.86294	0.589092	0.294546	−	0.955637i	$-0.404832\pi$
$43$	3.86294	0.589092	0.294546	+	0.955637i	$0.404832\pi$
$44$	0	0
$45$	1.31767	0.196426
$46$	0	0
$47$	0.829085	0.120934	0.0604672	−	0.998170i	$-0.480741\pi$
$47$	0.829085	0.120934	0.0604672	+	0.998170i	$0.480741\pi$
$48$	0	0
$49$	−6.80194	−0.971705
$50$	0	0
$51$	9.97046	1.39614
$52$	0	0
$53$	−7.02715	−0.965253	−0.482626	−	0.875826i	$-0.660317\pi$
$53$	−7.02715	−0.965253	−0.482626	+	0.875826i	$0.660317\pi$
$54$	0	0
$55$	3.37435	0.454998
$56$	0	0
$57$	−9.74094	−1.29022
$58$	0	0
$59$	3.59179	0.467612	0.233806	−	0.972283i	$-0.424882\pi$
$59$	3.59179	0.467612	0.233806	+	0.972283i	$0.424882\pi$
$60$	0	0
$61$	−1.65519	−0.211925	−0.105962	−	0.994370i	$-0.533792\pi$
$61$	−1.65519	−0.211925	−0.105962	+	0.994370i	$0.533792\pi$
$62$	0	0
$63$	−0.405813	−0.0511277
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−13.3448	−1.63033	−0.815164	−	0.579230i	$-0.803353\pi$
$67$	−13.3448	−1.63033	−0.815164	+	0.579230i	$0.803353\pi$
$68$	0	0
$69$	8.25667	0.993986
$70$	0	0
$71$	−14.4916	−1.71983	−0.859916	−	0.510435i	$-0.829484\pi$
$71$	−14.4916	−1.71983	−0.859916	+	0.510435i	$0.829484\pi$
$72$	0	0
$73$	−6.35690	−0.744018	−0.372009	−	0.928229i	$-0.621331\pi$
$73$	−6.35690	−0.744018	−0.372009	+	0.928229i	$0.621331\pi$
$74$	0	0
$75$	−4.20775	−0.485869
$76$	0	0
$77$	−1.03923	−0.118431
$78$	0	0
$79$	6.47219	0.728178	0.364089	−	0.931364i	$-0.381380\pi$
$79$	6.47219	0.728178	0.364089	+	0.931364i	$0.381380\pi$
$80$	0	0
$81$	−5.43296	−0.603662
$82$	0	0
$83$	−2.41789	−0.265398	−0.132699	−	0.991156i	$-0.542364\pi$
$83$	−2.41789	−0.265398	−0.132699	+	0.991156i	$0.542364\pi$
$84$	0	0
$85$	−9.97046	−1.08145
$86$	0	0
$87$	−11.7724	−1.26213
$88$	0	0
$89$	−13.2295	−1.40233	−0.701163	−	0.713001i	$-0.747336\pi$
$89$	−13.2295	−1.40233	−0.701163	+	0.713001i	$0.747336\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.56465	0.680722
$94$	0	0
$95$	9.74094	0.999399
$96$	0	0
$97$	−11.0368	−1.12062	−0.560310	−	0.828283i	$-0.689318\pi$
$97$	−11.0368	−1.12062	−0.560310	+	0.828283i	$0.689318\pi$
$98$	0	0
$99$	2.12929	0.214002
$100$	0	0
$101$	−5.89977	−0.587049	−0.293525	−	0.955952i	$-0.594828\pi$
$101$	−5.89977	−0.587049	−0.293525	+	0.955952i	$0.594828\pi$
$102$	0	0
$103$	19.4426	1.91574	0.957871	−	0.287200i	$-0.0927245\pi$
$103$	19.4426	1.91574	0.957871	+	0.287200i	$0.0927245\pi$
$104$	0	0
$105$	−0.929312	−0.0906916
$106$	0	0
$107$	8.18598	0.791369	0.395684	−	0.918387i	$-0.370507\pi$
$107$	8.18598	0.791369	0.395684	+	0.918387i	$0.370507\pi$
$108$	0	0
$109$	15.0858	1.44495	0.722477	−	0.691395i	$-0.243004\pi$
$109$	15.0858	1.44495	0.722477	+	0.691395i	$0.243004\pi$
$110$	0	0
$111$	−11.6625	−1.10695
$112$	0	0
$113$	−10.9119	−1.02650	−0.513250	−	0.858239i	$-0.671559\pi$
$113$	−10.9119	−1.02650	−0.513250	+	0.858239i	$0.671559\pi$
$114$	0	0
$115$	−8.25667	−0.769938
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.07069	0.281489
$120$	0	0
$121$	−5.54719	−0.504290
$122$	0	0
$123$	2.47650	0.223299
$124$	0	0
$125$	11.4330	1.02260
$126$	0	0
$127$	−4.53079	−0.402043	−0.201022	−	0.979587i	$-0.564426\pi$
$127$	−4.53079	−0.402043	−0.201022	+	0.979587i	$0.564426\pi$
$128$	0	0
$129$	5.58211	0.491477
$130$	0	0
$131$	−10.1032	−0.882722	−0.441361	−	0.897330i	$-0.645504\pi$
$131$	−10.1032	−0.882722	−0.441361	+	0.897330i	$0.645504\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	8.16852	0.703034
$136$	0	0
$137$	−6.21983	−0.531396	−0.265698	−	0.964056i	$-0.585602\pi$
$137$	−6.21983	−0.531396	−0.265698	+	0.964056i	$0.585602\pi$
$138$	0	0
$139$	10.9541	0.929112	0.464556	−	0.885544i	$-0.346214\pi$
$139$	10.9541	0.929112	0.464556	+	0.885544i	$0.346214\pi$
$140$	0	0
$141$	1.19806	0.100895
$142$	0	0
$143$	0	0
$144$	0	0
$145$	11.7724	0.977644
$146$	0	0
$147$	−9.82908	−0.810689
$148$	0	0
$149$	4.40581	0.360938	0.180469	−	0.983581i	$-0.442238\pi$
$149$	4.40581	0.360938	0.180469	+	0.983581i	$0.442238\pi$
$150$	0	0
$151$	16.3056	1.32693	0.663465	−	0.748207i	$-0.269085\pi$
$151$	16.3056	1.32693	0.663465	+	0.748207i	$0.269085\pi$
$152$	0	0
$153$	−6.29159	−0.508645
$154$	0	0
$155$	−6.56465	−0.527285
$156$	0	0
$157$	−11.2131	−0.894905	−0.447453	−	0.894308i	$-0.647669\pi$
$157$	−11.2131	−0.894905	−0.447453	+	0.894308i	$0.647669\pi$
$158$	0	0
$159$	−10.1545	−0.805306
$160$	0	0
$161$	2.54288	0.200407
$162$	0	0
$163$	−3.41789	−0.267710	−0.133855	−	0.991001i	$-0.542736\pi$
$163$	−3.41789	−0.267710	−0.133855	+	0.991001i	$0.542736\pi$
$164$	0	0
$165$	4.87608	0.379603
$166$	0	0
$167$	−5.32975	−0.412428	−0.206214	−	0.978507i	$-0.566114\pi$
$167$	−5.32975	−0.412428	−0.206214	+	0.978507i	$0.566114\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	6.14675	0.470054
$172$	0	0
$173$	8.63533	0.656532	0.328266	−	0.944585i	$-0.393536\pi$
$173$	8.63533	0.656532	0.328266	+	0.944585i	$0.393536\pi$
$174$	0	0
$175$	−1.29590	−0.0979606
$176$	0	0
$177$	5.19029	0.390126
$178$	0	0
$179$	−18.6920	−1.39711	−0.698554	−	0.715558i	$-0.746173\pi$
$179$	−18.6920	−1.39711	−0.698554	+	0.715558i	$0.746173\pi$
$180$	0	0
$181$	19.8092	1.47241	0.736204	−	0.676759i	$-0.236616\pi$
$181$	19.8092	1.47241	0.736204	+	0.676759i	$0.236616\pi$
$182$	0	0
$183$	−2.39181	−0.176808
$184$	0	0
$185$	11.6625	0.857443
$186$	0	0
$187$	−16.1118	−1.17821
$188$	0	0
$189$	−2.51573	−0.182992
$190$	0	0
$191$	−10.4547	−0.756478	−0.378239	−	0.925708i	$-0.623470\pi$
$191$	−10.4547	−0.756478	−0.378239	+	0.925708i	$0.623470\pi$
$192$	0	0
$193$	3.72587	0.268194	0.134097	−	0.990968i	$-0.457187\pi$
$193$	3.72587	0.268194	0.134097	+	0.990968i	$0.457187\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.8310	−1.55539	−0.777697	−	0.628639i	$-0.783612\pi$
$197$	−21.8310	−1.55539	−0.777697	+	0.628639i	$0.783612\pi$
$198$	0	0
$199$	14.3013	1.01379	0.506895	−	0.862008i	$-0.330793\pi$
$199$	14.3013	1.01379	0.506895	+	0.862008i	$0.330793\pi$
$200$	0	0
$201$	−19.2838	−1.36018
$202$	0	0
$203$	−3.62565	−0.254470
$204$	0	0
$205$	−2.47650	−0.172966
$206$	0	0
$207$	−5.21014	−0.362130
$208$	0	0
$209$	15.7409	1.08882
$210$	0	0
$211$	2.18300	0.150284	0.0751418	−	0.997173i	$-0.476059\pi$
$211$	2.18300	0.150284	0.0751418	+	0.997173i	$0.476059\pi$
$212$	0	0
$213$	−20.9409	−1.43485
$214$	0	0
$215$	−5.58211	−0.380696
$216$	0	0
$217$	2.02177	0.137247
$218$	0	0
$219$	−9.18598	−0.620731
$220$	0	0
$221$	0	0
$222$	0	0
$223$	28.0489	1.87829	0.939147	−	0.343515i	$-0.111618\pi$
$223$	28.0489	1.87829	0.939147	+	0.343515i	$0.111618\pi$
$224$	0	0
$225$	2.65519	0.177012
$226$	0	0
$227$	−5.76510	−0.382643	−0.191322	−	0.981527i	$-0.561277\pi$
$227$	−5.76510	−0.382643	−0.191322	+	0.981527i	$0.561277\pi$
$228$	0	0
$229$	16.1323	1.06605	0.533025	−	0.846099i	$-0.321055\pi$
$229$	16.1323	1.06605	0.533025	+	0.846099i	$0.321055\pi$
$230$	0	0
$231$	−1.50173	−0.0988065
$232$	0	0
$233$	9.13169	0.598237	0.299118	−	0.954216i	$-0.403307\pi$
$233$	9.13169	0.598237	0.299118	+	0.954216i	$0.403307\pi$
$234$	0	0
$235$	−1.19806	−0.0781530
$236$	0	0
$237$	9.35258	0.607516
$238$	0	0
$239$	11.4179	0.738562	0.369281	−	0.929318i	$-0.379604\pi$
$239$	11.4179	0.738562	0.369281	+	0.929318i	$0.379604\pi$
$240$	0	0
$241$	11.3666	0.732186	0.366093	−	0.930578i	$-0.380695\pi$
$241$	11.3666	0.732186	0.366093	+	0.930578i	$0.380695\pi$
$242$	0	0
$243$	9.10752	0.584248
$244$	0	0
$245$	9.82908	0.627957
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−3.49396	−0.221421
$250$	0	0
$251$	−4.61596	−0.291357	−0.145678	−	0.989332i	$-0.546536\pi$
$251$	−4.61596	−0.291357	−0.145678	+	0.989332i	$0.546536\pi$
$252$	0	0
$253$	−13.3424	−0.838831
$254$	0	0
$255$	−14.4077	−0.902247
$256$	0	0
$257$	−4.24459	−0.264770	−0.132385	−	0.991198i	$-0.542264\pi$
$257$	−4.24459	−0.264770	−0.132385	+	0.991198i	$0.542264\pi$
$258$	0	0
$259$	−3.59179	−0.223183
$260$	0	0
$261$	7.42865	0.459822
$262$	0	0
$263$	−31.0780	−1.91635	−0.958175	−	0.286182i	$-0.907614\pi$
$263$	−31.0780	−1.91635	−0.958175	+	0.286182i	$0.907614\pi$
$264$	0	0
$265$	10.1545	0.623787
$266$	0	0
$267$	−19.1172	−1.16995
$268$	0	0
$269$	5.88471	0.358797	0.179398	−	0.983776i	$-0.442585\pi$
$269$	5.88471	0.358797	0.179398	+	0.983776i	$0.442585\pi$
$270$	0	0
$271$	32.5827	1.97926	0.989629	−	0.143647i	$-0.0458831\pi$
$271$	32.5827	1.97926	0.989629	+	0.143647i	$0.0458831\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.79954	0.410028
$276$	0	0
$277$	−17.0465	−1.02423	−0.512113	−	0.858918i	$-0.671137\pi$
$277$	−17.0465	−1.02423	−0.512113	+	0.858918i	$0.671137\pi$
$278$	0	0
$279$	−4.14244	−0.248001
$280$	0	0
$281$	−12.6920	−0.757143	−0.378571	−	0.925572i	$-0.623585\pi$
$281$	−12.6920	−0.757143	−0.378571	+	0.925572i	$0.623585\pi$
$282$	0	0
$283$	−5.15346	−0.306341	−0.153171	−	0.988200i	$-0.548948\pi$
$283$	−5.15346	−0.306341	−0.153171	+	0.988200i	$0.548948\pi$
$284$	0	0
$285$	14.0761	0.833794
$286$	0	0
$287$	0.762709	0.0450213
$288$	0	0
$289$	30.6069	1.80040
$290$	0	0
$291$	−15.9487	−0.934929
$292$	0	0
$293$	15.3696	0.897900	0.448950	−	0.893557i	$-0.351798\pi$
$293$	15.3696	0.897900	0.448950	+	0.893557i	$0.351798\pi$
$294$	0	0
$295$	−5.19029	−0.302191
$296$	0	0
$297$	13.2000	0.765941
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.71917	0.0990912
$302$	0	0
$303$	−8.52542	−0.489772
$304$	0	0
$305$	2.39181	0.136955
$306$	0	0
$307$	17.9651	1.02532	0.512661	−	0.858591i	$-0.328660\pi$
$307$	17.9651	1.02532	0.512661	+	0.858591i	$0.328660\pi$
$308$	0	0
$309$	28.0954	1.59829
$310$	0	0
$311$	−24.3230	−1.37923	−0.689617	−	0.724175i	$-0.742221\pi$
$311$	−24.3230	−1.37923	−0.689617	+	0.724175i	$0.742221\pi$
$312$	0	0
$313$	−24.6625	−1.39401	−0.697003	−	0.717068i	$-0.745484\pi$
$313$	−24.6625	−1.39401	−0.697003	+	0.717068i	$0.745484\pi$
$314$	0	0
$315$	0.586417	0.0330409
$316$	0	0
$317$	12.6896	0.712721	0.356360	−	0.934349i	$-0.384018\pi$
$317$	12.6896	0.712721	0.356360	+	0.934349i	$0.384018\pi$
$318$	0	0
$319$	19.0237	1.06512
$320$	0	0
$321$	11.8291	0.660235
$322$	0	0
$323$	−46.5109	−2.58794
$324$	0	0
$325$	0	0
$326$	0	0
$327$	21.7995	1.20552
$328$	0	0
$329$	0.368977	0.0203424
$330$	0	0
$331$	28.8374	1.58505	0.792525	−	0.609840i	$-0.208766\pi$
$331$	28.8374	1.58505	0.792525	+	0.609840i	$0.208766\pi$
$332$	0	0
$333$	7.35929	0.403287
$334$	0	0
$335$	19.2838	1.05359
$336$	0	0
$337$	−9.80194	−0.533946	−0.266973	−	0.963704i	$-0.586023\pi$
$337$	−9.80194	−0.533946	−0.266973	+	0.963704i	$0.586023\pi$
$338$	0	0
$339$	−15.7681	−0.856405
$340$	0	0
$341$	−10.6082	−0.574466
$342$	0	0
$343$	−6.14244	−0.331661
$344$	0	0
$345$	−11.9312	−0.642356
$346$	0	0
$347$	17.1879	0.922695	0.461347	−	0.887220i	$-0.347366\pi$
$347$	17.1879	0.922695	0.461347	+	0.887220i	$0.347366\pi$
$348$	0	0
$349$	−12.3556	−0.661378	−0.330689	−	0.943740i	$-0.607281\pi$
$349$	−12.3556	−0.661378	−0.330689	+	0.943740i	$0.607281\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−24.4209	−1.29979	−0.649896	−	0.760023i	$-0.725188\pi$
$353$	−24.4209	−1.29979	−0.649896	+	0.760023i	$0.725188\pi$
$354$	0	0
$355$	20.9409	1.11143
$356$	0	0
$357$	4.43727	0.234845
$358$	0	0
$359$	−12.1293	−0.640160	−0.320080	−	0.947391i	$-0.603710\pi$
$359$	−12.1293	−0.640160	−0.320080	+	0.947391i	$0.603710\pi$
$360$	0	0
$361$	26.4403	1.39159
$362$	0	0
$363$	−8.01592	−0.420727
$364$	0	0
$365$	9.18598	0.480816
$366$	0	0
$367$	−5.96615	−0.311430	−0.155715	−	0.987802i	$-0.549768\pi$
$367$	−5.96615	−0.311430	−0.155715	+	0.987802i	$0.549768\pi$
$368$	0	0
$369$	−1.56273	−0.0813524
$370$	0	0
$371$	−3.12737	−0.162365
$372$	0	0
$373$	−6.09352	−0.315511	−0.157755	−	0.987478i	$-0.550426\pi$
$373$	−6.09352	−0.315511	−0.157755	+	0.987478i	$0.550426\pi$
$374$	0	0
$375$	16.5211	0.853146
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−32.9627	−1.69318	−0.846590	−	0.532246i	$-0.821348\pi$
$379$	−32.9627	−1.69318	−0.846590	+	0.532246i	$0.821348\pi$
$380$	0	0
$381$	−6.54719	−0.335423
$382$	0	0
$383$	24.4155	1.24757	0.623787	−	0.781594i	$-0.285593\pi$
$383$	24.4155	1.24757	0.623787	+	0.781594i	$0.285593\pi$
$384$	0	0
$385$	1.50173	0.0765352
$386$	0	0
$387$	−3.52243	−0.179055
$388$	0	0
$389$	−8.78746	−0.445542	−0.222771	−	0.974871i	$-0.571510\pi$
$389$	−8.78746	−0.445542	−0.222771	+	0.974871i	$0.571510\pi$
$390$	0	0
$391$	39.4239	1.99375
$392$	0	0
$393$	−14.5996	−0.736451
$394$	0	0
$395$	−9.35258	−0.470580
$396$	0	0
$397$	14.4964	0.727551	0.363776	−	0.931487i	$-0.381488\pi$
$397$	14.4964	0.727551	0.363776	+	0.931487i	$0.381488\pi$
$398$	0	0
$399$	−4.33513	−0.217028
$400$	0	0
$401$	−2.08038	−0.103889	−0.0519445	−	0.998650i	$-0.516542\pi$
$401$	−2.08038	−0.103889	−0.0519445	+	0.998650i	$0.516542\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	7.85086	0.390112
$406$	0	0
$407$	18.8461	0.934165
$408$	0	0
$409$	0.951083	0.0470280	0.0235140	−	0.999724i	$-0.492515\pi$
$409$	0.951083	0.0470280	0.0235140	+	0.999724i	$0.492515\pi$
$410$	0	0
$411$	−8.98792	−0.443341
$412$	0	0
$413$	1.59850	0.0786570
$414$	0	0
$415$	3.49396	0.171512
$416$	0	0
$417$	15.8291	0.775154
$418$	0	0
$419$	8.30665	0.405806	0.202903	−	0.979199i	$-0.434962\pi$
$419$	8.30665	0.405806	0.202903	+	0.979199i	$0.434962\pi$
$420$	0	0
$421$	27.4077	1.33577	0.667886	−	0.744264i	$-0.267200\pi$
$421$	27.4077	1.33577	0.667886	+	0.744264i	$0.267200\pi$
$422$	0	0
$423$	−0.756004	−0.0367582
$424$	0	0
$425$	−20.0911	−0.974563
$426$	0	0
$427$	−0.736627	−0.0356479
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.0930	−1.01602	−0.508008	−	0.861352i	$-0.669618\pi$
$431$	−21.0930	−1.01602	−0.508008	+	0.861352i	$0.669618\pi$
$432$	0	0
$433$	−14.0175	−0.673636	−0.336818	−	0.941570i	$-0.609351\pi$
$433$	−14.0175	−0.673636	−0.336818	+	0.941570i	$0.609351\pi$
$434$	0	0
$435$	17.0116	0.815644
$436$	0	0
$437$	−38.5163	−1.84249
$438$	0	0
$439$	−26.1377	−1.24748	−0.623741	−	0.781631i	$-0.714388\pi$
$439$	−26.1377	−1.24748	−0.623741	+	0.781631i	$0.714388\pi$
$440$	0	0
$441$	6.20237	0.295351
$442$	0	0
$443$	11.1884	0.531576	0.265788	−	0.964032i	$-0.414368\pi$
$443$	11.1884	0.531576	0.265788	+	0.964032i	$0.414368\pi$
$444$	0	0
$445$	19.1172	0.906243
$446$	0	0
$447$	6.36658	0.301129
$448$	0	0
$449$	21.7060	1.02437	0.512185	−	0.858875i	$-0.328836\pi$
$449$	21.7060	1.02437	0.512185	+	0.858875i	$0.328836\pi$
$450$	0	0
$451$	−4.00192	−0.188443
$452$	0	0
$453$	23.5623	1.10705
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12.7657	0.597154	0.298577	−	0.954386i	$-0.403488\pi$
$457$	12.7657	0.597154	0.298577	+	0.954386i	$0.403488\pi$
$458$	0	0
$459$	−39.0030	−1.82050
$460$	0	0
$461$	−1.80864	−0.0842369	−0.0421184	−	0.999113i	$-0.513411\pi$
$461$	−1.80864	−0.0842369	−0.0421184	+	0.999113i	$0.513411\pi$
$462$	0	0
$463$	−6.39911	−0.297392	−0.148696	−	0.988883i	$-0.547508\pi$
$463$	−6.39911	−0.297392	−0.148696	+	0.988883i	$0.547508\pi$
$464$	0	0
$465$	−9.48619	−0.439912
$466$	0	0
$467$	−20.3817	−0.943150	−0.471575	−	0.881826i	$-0.656314\pi$
$467$	−20.3817	−0.943150	−0.471575	+	0.881826i	$0.656314\pi$
$468$	0	0
$469$	−5.93900	−0.274238
$470$	0	0
$471$	−16.2034	−0.746615
$472$	0	0
$473$	−9.02044	−0.414760
$474$	0	0
$475$	19.6286	0.900623
$476$	0	0
$477$	6.40773	0.293390
$478$	0	0
$479$	33.8092	1.54478	0.772392	−	0.635147i	$-0.219060\pi$
$479$	33.8092	1.54478	0.772392	+	0.635147i	$0.219060\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	3.67456	0.167198
$484$	0	0
$485$	15.9487	0.724193
$486$	0	0
$487$	−7.47650	−0.338793	−0.169396	−	0.985548i	$-0.554182\pi$
$487$	−7.47650	−0.338793	−0.169396	+	0.985548i	$0.554182\pi$
$488$	0	0
$489$	−4.93900	−0.223349
$490$	0	0
$491$	33.7885	1.52485	0.762427	−	0.647074i	$-0.224007\pi$
$491$	33.7885	1.52485	0.762427	+	0.647074i	$0.224007\pi$
$492$	0	0
$493$	−56.2107	−2.53160
$494$	0	0
$495$	−3.07692	−0.138297
$496$	0	0
$497$	−6.44935	−0.289293
$498$	0	0
$499$	−22.9855	−1.02897	−0.514487	−	0.857498i	$-0.672018\pi$
$499$	−22.9855	−1.02897	−0.514487	+	0.857498i	$0.672018\pi$
$500$	0	0
$501$	−7.70171	−0.344087
$502$	0	0
$503$	−22.1726	−0.988626	−0.494313	−	0.869284i	$-0.664580\pi$
$503$	−22.1726	−0.988626	−0.494313	+	0.869284i	$0.664580\pi$
$504$	0	0
$505$	8.52542	0.379376
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−10.0258	−0.444387	−0.222193	−	0.975003i	$-0.571322\pi$
$509$	−10.0258	−0.444387	−0.222193	+	0.975003i	$0.571322\pi$
$510$	0	0
$511$	−2.82908	−0.125151
$512$	0	0
$513$	38.1051	1.68238
$514$	0	0
$515$	−28.0954	−1.23803
$516$	0	0
$517$	−1.93602	−0.0851459
$518$	0	0
$519$	12.4784	0.547742
$520$	0	0
$521$	6.02284	0.263865	0.131933	−	0.991259i	$-0.457882\pi$
$521$	6.02284	0.263865	0.131933	+	0.991259i	$0.457882\pi$
$522$	0	0
$523$	45.5056	1.98982	0.994910	−	0.100770i	$-0.0321306\pi$
$523$	45.5056	1.98982	0.994910	+	0.100770i	$0.0321306\pi$
$524$	0	0
$525$	−1.87263	−0.0817281
$526$	0	0
$527$	31.3448	1.36540
$528$	0	0
$529$	9.64742	0.419453
$530$	0	0
$531$	−3.27519	−0.142131
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−11.8291	−0.511416
$536$	0	0
$537$	−27.0108	−1.16560
$538$	0	0
$539$	15.8834	0.684146
$540$	0	0
$541$	−22.1051	−0.950374	−0.475187	−	0.879885i	$-0.657620\pi$
$541$	−22.1051	−0.950374	−0.475187	+	0.879885i	$0.657620\pi$
$542$	0	0
$543$	28.6252	1.22842
$544$	0	0
$545$	−21.7995	−0.933790
$546$	0	0
$547$	7.77240	0.332324	0.166162	−	0.986098i	$-0.446863\pi$
$547$	7.77240	0.332324	0.166162	+	0.986098i	$0.446863\pi$
$548$	0	0
$549$	1.50929	0.0644148
$550$	0	0
$551$	54.9168	2.33953
$552$	0	0
$553$	2.88040	0.122487
$554$	0	0
$555$	16.8528	0.715360
$556$	0	0
$557$	−21.2959	−0.902336	−0.451168	−	0.892439i	$-0.648992\pi$
$557$	−21.2959	−0.902336	−0.451168	+	0.892439i	$0.648992\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−23.2823	−0.982978
$562$	0	0
$563$	−11.7144	−0.493702	−0.246851	−	0.969053i	$-0.579396\pi$
$563$	−11.7144	−0.493702	−0.246851	+	0.969053i	$0.579396\pi$
$564$	0	0
$565$	15.7681	0.663368
$566$	0	0
$567$	−2.41789	−0.101542
$568$	0	0
$569$	29.9474	1.25546	0.627729	−	0.778432i	$-0.283984\pi$
$569$	29.9474	1.25546	0.627729	+	0.778432i	$0.283984\pi$
$570$	0	0
$571$	−11.1967	−0.468569	−0.234284	−	0.972168i	$-0.575275\pi$
$571$	−11.1967	−0.468569	−0.234284	+	0.972168i	$0.575275\pi$
$572$	0	0
$573$	−15.1075	−0.631126
$574$	0	0
$575$	−16.6377	−0.693841
$576$	0	0
$577$	6.89785	0.287161	0.143581	−	0.989639i	$-0.454138\pi$
$577$	6.89785	0.287161	0.143581	+	0.989639i	$0.454138\pi$
$578$	0	0
$579$	5.38404	0.223753
$580$	0	0
$581$	−1.07606	−0.0446427
$582$	0	0
$583$	16.4093	0.679603
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.4765	−1.01025	−0.505127	−	0.863045i	$-0.668554\pi$
$587$	−24.4765	−1.01025	−0.505127	+	0.863045i	$0.668554\pi$
$588$	0	0
$589$	−30.6233	−1.26181
$590$	0	0
$591$	−31.5467	−1.29766
$592$	0	0
$593$	15.5496	0.638545	0.319272	−	0.947663i	$-0.396562\pi$
$593$	15.5496	0.638545	0.319272	+	0.947663i	$0.396562\pi$
$594$	0	0
$595$	−4.43727	−0.181910
$596$	0	0
$597$	20.6659	0.845801
$598$	0	0
$599$	−41.2978	−1.68738	−0.843692	−	0.536828i	$-0.819622\pi$
$599$	−41.2978	−1.68738	−0.843692	+	0.536828i	$0.819622\pi$
$600$	0	0
$601$	−47.1909	−1.92496	−0.962478	−	0.271359i	$-0.912527\pi$
$601$	−47.1909	−1.92496	−0.962478	+	0.271359i	$0.912527\pi$
$602$	0	0
$603$	12.1685	0.495541
$604$	0	0
$605$	8.01592	0.325893
$606$	0	0
$607$	3.51706	0.142753	0.0713765	−	0.997449i	$-0.477261\pi$
$607$	3.51706	0.142753	0.0713765	+	0.997449i	$0.477261\pi$
$608$	0	0
$609$	−5.23921	−0.212303
$610$	0	0
$611$	0	0
$612$	0	0
$613$	19.6926	0.795377	0.397689	−	0.917520i	$-0.369812\pi$
$613$	19.6926	0.795377	0.397689	+	0.917520i	$0.369812\pi$
$614$	0	0
$615$	−3.57865	−0.144305
$616$	0	0
$617$	37.7827	1.52107	0.760537	−	0.649295i	$-0.224936\pi$
$617$	37.7827	1.52107	0.760537	+	0.649295i	$0.224936\pi$
$618$	0	0
$619$	12.4440	0.500166	0.250083	−	0.968224i	$-0.419542\pi$
$619$	12.4440	0.500166	0.250083	+	0.968224i	$0.419542\pi$
$620$	0	0
$621$	−32.2989	−1.29611
$622$	0	0
$623$	−5.88769	−0.235885
$624$	0	0
$625$	−1.96184	−0.0784735
$626$	0	0
$627$	22.7463	0.908400
$628$	0	0
$629$	−55.6859	−2.22034
$630$	0	0
$631$	−22.3696	−0.890518	−0.445259	−	0.895402i	$-0.646888\pi$
$631$	−22.3696	−0.890518	−0.445259	+	0.895402i	$0.646888\pi$
$632$	0	0
$633$	3.15452	0.125381
$634$	0	0
$635$	6.54719	0.259817
$636$	0	0
$637$	0	0
$638$	0	0
$639$	13.2142	0.522745
$640$	0	0
$641$	47.6564	1.88231	0.941157	−	0.337971i	$-0.109740\pi$
$641$	47.6564	1.88231	0.941157	+	0.337971i	$0.109740\pi$
$642$	0	0
$643$	23.7071	0.934916	0.467458	−	0.884015i	$-0.345170\pi$
$643$	23.7071	0.934916	0.467458	+	0.884015i	$0.345170\pi$
$644$	0	0
$645$	−8.06638	−0.317613
$646$	0	0
$647$	30.4155	1.19576	0.597878	−	0.801587i	$-0.296011\pi$
$647$	30.4155	1.19576	0.597878	+	0.801587i	$0.296011\pi$
$648$	0	0
$649$	−8.38729	−0.329230
$650$	0	0
$651$	2.92154	0.114504
$652$	0	0
$653$	6.87561	0.269063	0.134532	−	0.990909i	$-0.457047\pi$
$653$	6.87561	0.269063	0.134532	+	0.990909i	$0.457047\pi$
$654$	0	0
$655$	14.5996	0.570452
$656$	0	0
$657$	5.79656	0.226145
$658$	0	0
$659$	17.2218	0.670864	0.335432	−	0.942064i	$-0.391118\pi$
$659$	17.2218	0.670864	0.335432	+	0.942064i	$0.391118\pi$
$660$	0	0
$661$	29.3967	1.14340	0.571700	−	0.820463i	$-0.306284\pi$
$661$	29.3967	1.14340	0.571700	+	0.820463i	$0.306284\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.33513	0.168109
$666$	0	0
$667$	−46.5488	−1.80238
$668$	0	0
$669$	40.5319	1.56705
$670$	0	0
$671$	3.86507	0.149209
$672$	0	0
$673$	−26.8823	−1.03624	−0.518119	−	0.855309i	$-0.673367\pi$
$673$	−26.8823	−1.03624	−0.518119	+	0.855309i	$0.673367\pi$
$674$	0	0
$675$	16.4601	0.633550
$676$	0	0
$677$	42.4215	1.63039	0.815195	−	0.579187i	$-0.196630\pi$
$677$	42.4215	1.63039	0.815195	+	0.579187i	$0.196630\pi$
$678$	0	0
$679$	−4.91185	−0.188500
$680$	0	0
$681$	−8.33081	−0.319237
$682$	0	0
$683$	35.6176	1.36287	0.681435	−	0.731879i	$-0.261356\pi$
$683$	35.6176	1.36287	0.681435	+	0.731879i	$0.261356\pi$
$684$	0	0
$685$	8.98792	0.343411
$686$	0	0
$687$	23.3118	0.889401
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−9.21552	−0.350575	−0.175287	−	0.984517i	$-0.556085\pi$
$691$	−9.21552	−0.350575	−0.175287	+	0.984517i	$0.556085\pi$
$692$	0	0
$693$	0.947625	0.0359973
$694$	0	0
$695$	−15.8291	−0.600431
$696$	0	0
$697$	11.8248	0.447895
$698$	0	0
$699$	13.1957	0.499106
$700$	0	0
$701$	−39.1473	−1.47857	−0.739287	−	0.673390i	$-0.764837\pi$
$701$	−39.1473	−1.47857	−0.739287	+	0.673390i	$0.764837\pi$
$702$	0	0
$703$	54.4040	2.05189
$704$	0	0
$705$	−1.73125	−0.0652027
$706$	0	0
$707$	−2.62565	−0.0987476
$708$	0	0
$709$	−48.4389	−1.81916	−0.909581	−	0.415527i	$-0.863597\pi$
$709$	−48.4389	−1.81916	−0.909581	+	0.415527i	$0.863597\pi$
$710$	0	0
$711$	−5.90169	−0.221331
$712$	0	0
$713$	25.9571	0.972099
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.4993	0.616179
$718$	0	0
$719$	−18.8968	−0.704731	−0.352366	−	0.935862i	$-0.614623\pi$
$719$	−18.8968	−0.704731	−0.352366	+	0.935862i	$0.614623\pi$
$720$	0	0
$721$	8.65279	0.322247
$722$	0	0
$723$	16.4252	0.610859
$724$	0	0
$725$	23.7222	0.881019
$726$	0	0
$727$	20.7976	0.771341	0.385671	−	0.922637i	$-0.373970\pi$
$727$	20.7976	0.771341	0.385671	+	0.922637i	$0.373970\pi$
$728$	0	0
$729$	29.4596	1.09110
$730$	0	0
$731$	26.6534	0.985811
$732$	0	0
$733$	−6.44994	−0.238234	−0.119117	−	0.992880i	$-0.538006\pi$
$733$	−6.44994	−0.238234	−0.119117	+	0.992880i	$0.538006\pi$
$734$	0	0
$735$	14.2034	0.523902
$736$	0	0
$737$	31.1618	1.14786
$738$	0	0
$739$	18.9772	0.698086	0.349043	−	0.937107i	$-0.386507\pi$
$739$	18.9772	0.698086	0.349043	+	0.937107i	$0.386507\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.8933	0.473010	0.236505	−	0.971630i	$-0.423998\pi$
$743$	12.8933	0.473010	0.236505	+	0.971630i	$0.423998\pi$
$744$	0	0
$745$	−6.36658	−0.233254
$746$	0	0
$747$	2.20477	0.0806682
$748$	0	0
$749$	3.64310	0.133116
$750$	0	0
$751$	6.48321	0.236576	0.118288	−	0.992979i	$-0.462259\pi$
$751$	6.48321	0.236576	0.118288	+	0.992979i	$0.462259\pi$
$752$	0	0
$753$	−6.67025	−0.243077
$754$	0	0
$755$	−23.5623	−0.857518
$756$	0	0
$757$	−21.0543	−0.765231	−0.382616	−	0.923908i	$-0.624977\pi$
$757$	−21.0543	−0.765231	−0.382616	+	0.923908i	$0.624977\pi$
$758$	0	0
$759$	−19.2804	−0.699833
$760$	0	0
$761$	4.37973	0.158765	0.0793826	−	0.996844i	$-0.474705\pi$
$761$	4.37973	0.158765	0.0793826	+	0.996844i	$0.474705\pi$
$762$	0	0
$763$	6.71379	0.243056
$764$	0	0
$765$	9.09160	0.328708
$766$	0	0
$767$	0	0
$768$	0	0
$769$	51.1075	1.84298	0.921492	−	0.388397i	$-0.126971\pi$
$769$	51.1075	1.84298	0.921492	+	0.388397i	$0.126971\pi$
$770$	0	0
$771$	−6.13361	−0.220896
$772$	0	0
$773$	−18.9487	−0.681537	−0.340768	−	0.940147i	$-0.610687\pi$
$773$	−18.9487	−0.681537	−0.340768	+	0.940147i	$0.610687\pi$
$774$	0	0
$775$	−13.2282	−0.475171
$776$	0	0
$777$	−5.19029	−0.186201
$778$	0	0
$779$	−11.5526	−0.413914
$780$	0	0
$781$	33.8396	1.21088
$782$	0	0
$783$	46.0519	1.64576
$784$	0	0
$785$	16.2034	0.578326
$786$	0	0
$787$	−23.8877	−0.851504	−0.425752	−	0.904840i	$-0.639990\pi$
$787$	−23.8877	−0.851504	−0.425752	+	0.904840i	$0.639990\pi$
$788$	0	0
$789$	−44.9090	−1.59880
$790$	0	0
$791$	−4.85623	−0.172668
$792$	0	0
$793$	0	0
$794$	0	0
$795$	14.6737	0.520423
$796$	0	0
$797$	16.3338	0.578573	0.289286	−	0.957243i	$-0.406582\pi$
$797$	16.3338	0.578573	0.289286	+	0.957243i	$0.406582\pi$
$798$	0	0
$799$	5.72050	0.202377
$800$	0	0
$801$	12.0634	0.426239
$802$	0	0
$803$	14.8442	0.523839
$804$	0	0
$805$	−3.67456	−0.129511
$806$	0	0
$807$	8.50365	0.299343
$808$	0	0
$809$	1.66296	0.0584664	0.0292332	−	0.999573i	$-0.490693\pi$
$809$	1.66296	0.0584664	0.0292332	+	0.999573i	$0.490693\pi$
$810$	0	0
$811$	−5.99462	−0.210500	−0.105250	−	0.994446i	$-0.533564\pi$
$811$	−5.99462	−0.210500	−0.105250	+	0.994446i	$0.533564\pi$
$812$	0	0
$813$	47.0834	1.65129
$814$	0	0
$815$	4.93900	0.173006
$816$	0	0
$817$	−26.0398	−0.911018
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7.19136	0.250980	0.125490	−	0.992095i	$-0.459950\pi$
$821$	7.19136	0.250980	0.125490	+	0.992095i	$0.459950\pi$
$822$	0	0
$823$	1.54229	0.0537607	0.0268803	−	0.999639i	$-0.491443\pi$
$823$	1.54229	0.0537607	0.0268803	+	0.999639i	$0.491443\pi$
$824$	0	0
$825$	9.82563	0.342084
$826$	0	0
$827$	−20.3308	−0.706972	−0.353486	−	0.935440i	$-0.615004\pi$
$827$	−20.3308	−0.706972	−0.353486	+	0.935440i	$0.615004\pi$
$828$	0	0
$829$	−9.16123	−0.318183	−0.159091	−	0.987264i	$-0.550856\pi$
$829$	−9.16123	−0.318183	−0.159091	+	0.987264i	$0.550856\pi$
$830$	0	0
$831$	−24.6329	−0.854507
$832$	0	0
$833$	−46.9318	−1.62609
$834$	0	0
$835$	7.70171	0.266529
$836$	0	0
$837$	−25.6799	−0.887629
$838$	0	0
$839$	11.6582	0.402485	0.201242	−	0.979541i	$-0.435502\pi$
$839$	11.6582	0.402485	0.201242	+	0.979541i	$0.435502\pi$
$840$	0	0
$841$	37.3696	1.28861
$842$	0	0
$843$	−18.3405	−0.631680
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.46873	−0.0848266
$848$	0	0
$849$	−7.44696	−0.255579
$850$	0	0
$851$	−46.1142	−1.58078
$852$	0	0
$853$	−7.56033	−0.258861	−0.129430	−	0.991589i	$-0.541315\pi$
$853$	−7.56033	−0.258861	−0.129430	+	0.991589i	$0.541315\pi$
$854$	0	0
$855$	−8.88231	−0.303769
$856$	0	0
$857$	3.96077	0.135297	0.0676487	−	0.997709i	$-0.478450\pi$
$857$	3.96077	0.135297	0.0676487	+	0.997709i	$0.478450\pi$
$858$	0	0
$859$	−22.0388	−0.751953	−0.375976	−	0.926629i	$-0.622693\pi$
$859$	−22.0388	−0.751953	−0.375976	+	0.926629i	$0.622693\pi$
$860$	0	0
$861$	1.10215	0.0375611
$862$	0	0
$863$	12.6668	0.431183	0.215591	−	0.976484i	$-0.430832\pi$
$863$	12.6668	0.431183	0.215591	+	0.976484i	$0.430832\pi$
$864$	0	0
$865$	−12.4784	−0.424279
$866$	0	0
$867$	44.2282	1.50207
$868$	0	0
$869$	−15.1134	−0.512686
$870$	0	0
$871$	0	0
$872$	0	0
$873$	10.0640	0.340614
$874$	0	0
$875$	5.08815	0.172011
$876$	0	0
$877$	−26.5972	−0.898123	−0.449061	−	0.893501i	$-0.648242\pi$
$877$	−26.5972	−0.898123	−0.449061	+	0.893501i	$0.648242\pi$
$878$	0	0
$879$	22.2097	0.749114
$880$	0	0
$881$	5.47544	0.184472	0.0922361	−	0.995737i	$-0.470599\pi$
$881$	5.47544	0.184472	0.0922361	+	0.995737i	$0.470599\pi$
$882$	0	0
$883$	39.6015	1.33270	0.666348	−	0.745641i	$-0.267857\pi$
$883$	39.6015	1.33270	0.666348	+	0.745641i	$0.267857\pi$
$884$	0	0
$885$	−7.50019	−0.252116
$886$	0	0
$887$	19.0519	0.639700	0.319850	−	0.947468i	$-0.396367\pi$
$887$	19.0519	0.639700	0.319850	+	0.947468i	$0.396367\pi$
$888$	0	0
$889$	−2.01639	−0.0676277
$890$	0	0
$891$	12.6866	0.425019
$892$	0	0
$893$	−5.58881	−0.187022
$894$	0	0
$895$	27.0108	0.902870
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−37.0097	−1.23434
$900$	0	0
$901$	−48.4857	−1.61529
$902$	0	0
$903$	2.48427	0.0826713
$904$	0	0
$905$	−28.6252	−0.951533
$906$	0	0
$907$	−45.0127	−1.49462	−0.747311	−	0.664475i	$-0.768655\pi$
$907$	−45.0127	−1.49462	−0.747311	+	0.664475i	$0.768655\pi$
$908$	0	0
$909$	5.37973	0.178434
$910$	0	0
$911$	23.3884	0.774891	0.387445	−	0.921893i	$-0.373358\pi$
$911$	23.3884	0.774891	0.387445	+	0.921893i	$0.373358\pi$
$912$	0	0
$913$	5.64609	0.186858
$914$	0	0
$915$	3.45627	0.114261
$916$	0	0
$917$	−4.49635	−0.148483
$918$	0	0
$919$	13.6732	0.451038	0.225519	−	0.974239i	$-0.427592\pi$
$919$	13.6732	0.451038	0.225519	+	0.974239i	$0.427592\pi$
$920$	0	0
$921$	25.9603	0.855421
$922$	0	0
$923$	0	0
$924$	0	0
$925$	23.5007	0.772697
$926$	0	0
$927$	−17.7289	−0.582292
$928$	0	0
$929$	2.47086	0.0810663	0.0405332	−	0.999178i	$-0.487094\pi$
$929$	2.47086	0.0810663	0.0405332	+	0.999178i	$0.487094\pi$
$930$	0	0
$931$	45.8514	1.50272
$932$	0	0
$933$	−35.1478	−1.15069
$934$	0	0
$935$	23.2823	0.761412
$936$	0	0
$937$	−58.1390	−1.89932	−0.949659	−	0.313286i	$-0.898570\pi$
$937$	−58.1390	−1.89932	−0.949659	+	0.313286i	$0.898570\pi$
$938$	0	0
$939$	−35.6383	−1.16301
$940$	0	0
$941$	41.9154	1.36640	0.683202	−	0.730229i	$-0.260587\pi$
$941$	41.9154	1.36640	0.683202	+	0.730229i	$0.260587\pi$
$942$	0	0
$943$	9.79225	0.318880
$944$	0	0
$945$	3.63533	0.118257
$946$	0	0
$947$	−31.8745	−1.03578	−0.517892	−	0.855446i	$-0.673283\pi$
$947$	−31.8745	−1.03578	−0.517892	+	0.855446i	$0.673283\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	18.3370	0.594619
$952$	0	0
$953$	−16.2360	−0.525934	−0.262967	−	0.964805i	$-0.584701\pi$
$953$	−16.2360	−0.525934	−0.262967	+	0.964805i	$0.584701\pi$
$954$	0	0
$955$	15.1075	0.488868
$956$	0	0
$957$	27.4900	0.888626
$958$	0	0
$959$	−2.76809	−0.0893862
$960$	0	0
$961$	−10.3623	−0.334267
$962$	0	0
$963$	−7.46442	−0.240538
$964$	0	0
$965$	−5.38404	−0.173318
$966$	0	0
$967$	1.35881	0.0436965	0.0218483	−	0.999761i	$-0.493045\pi$
$967$	1.35881	0.0436965	0.0218483	+	0.999761i	$0.493045\pi$
$968$	0	0
$969$	−67.2103	−2.15910
$970$	0	0
$971$	−33.3236	−1.06941	−0.534703	−	0.845040i	$-0.679577\pi$
$971$	−33.3236	−1.06941	−0.534703	+	0.845040i	$0.679577\pi$
$972$	0	0
$973$	4.87502	0.156286
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−34.3314	−1.09836	−0.549179	−	0.835704i	$-0.685060\pi$
$977$	−34.3314	−1.09836	−0.549179	+	0.835704i	$0.685060\pi$
$978$	0	0
$979$	30.8926	0.987332
$980$	0	0
$981$	−13.7560	−0.439195
$982$	0	0
$983$	18.3739	0.586036	0.293018	−	0.956107i	$-0.405340\pi$
$983$	18.3739	0.586036	0.293018	+	0.956107i	$0.405340\pi$
$984$	0	0
$985$	31.5467	1.00516
$986$	0	0
$987$	0.533188	0.0169716
$988$	0	0
$989$	22.0720	0.701849
$990$	0	0
$991$	−54.8133	−1.74120	−0.870601	−	0.491990i	$-0.836270\pi$
$991$	−54.8133	−1.74120	−0.870601	+	0.491990i	$0.836270\pi$
$992$	0	0
$993$	41.6713	1.32240
$994$	0	0
$995$	−20.6659	−0.655154
$996$	0	0
$997$	39.1976	1.24140	0.620700	−	0.784048i	$-0.286848\pi$
$997$	39.1976	1.24140	0.620700	+	0.784048i	$0.286848\pi$
$998$	0	0
$999$	45.6219	1.44341

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.2.a.bb.1.2		3
4.3	odd	2		1352.2.a.i.1.2	✓	3
13.5	odd	4		2704.2.f.p.337.4		6
13.8	odd	4		2704.2.f.p.337.3		6
13.12	even	2		2704.2.a.bc.1.2		3
52.3	odd	6		1352.2.i.i.529.2		6
52.7	even	12		1352.2.o.g.361.3		12
52.11	even	12		1352.2.o.g.1161.3		12
52.15	even	12		1352.2.o.g.1161.4		12
52.19	even	12		1352.2.o.g.361.4		12
52.23	odd	6		1352.2.i.j.529.2		6
52.31	even	4		1352.2.f.e.337.4		6
52.35	odd	6		1352.2.i.i.1329.2		6
52.43	odd	6		1352.2.i.j.1329.2		6
52.47	even	4		1352.2.f.e.337.3		6
52.51	odd	2		1352.2.a.j.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1352.2.a.i.1.2	✓	3	4.3	odd	2
1352.2.a.j.1.2	yes	3	52.51	odd	2
1352.2.f.e.337.3		6	52.47	even	4
1352.2.f.e.337.4		6	52.31	even	4
1352.2.i.i.529.2		6	52.3	odd	6
1352.2.i.i.1329.2		6	52.35	odd	6
1352.2.i.j.529.2		6	52.23	odd	6
1352.2.i.j.1329.2		6	52.43	odd	6
1352.2.o.g.361.3		12	52.7	even	12
1352.2.o.g.361.4		12	52.19	even	12
1352.2.o.g.1161.3		12	52.11	even	12
1352.2.o.g.1161.4		12	52.15	even	12
2704.2.a.bb.1.2		3	1.1	even	1	trivial
2704.2.a.bc.1.2		3	13.12	even	2
2704.2.f.p.337.3		6	13.8	odd	4
2704.2.f.p.337.4		6	13.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.a (trivial)

\(f(q)\)	\(=\)	\(q+1.44504 q^{3} -1.44504 q^{5} +0.445042 q^{7} -0.911854 q^{9} +O(q^{10})\)	\(q+1.44504 q^{3} -1.44504 q^{5} +0.445042 q^{7} -0.911854 q^{9} -2.33513 q^{11} -2.08815 q^{15} +6.89977 q^{17} -6.74094 q^{19} +0.643104 q^{21} +5.71379 q^{23} -2.91185 q^{25} -5.65279 q^{27} -8.14675 q^{29} +4.54288 q^{31} -3.37435 q^{33} -0.643104 q^{35} -8.07069 q^{37} +1.71379 q^{41} +3.86294 q^{43} +1.31767 q^{45} +0.829085 q^{47} -6.80194 q^{49} +9.97046 q^{51} -7.02715 q^{53} +3.37435 q^{55} -9.74094 q^{57} +3.59179 q^{59} -1.65519 q^{61} -0.405813 q^{63} -13.3448 q^{67} +8.25667 q^{69} -14.4916 q^{71} -6.35690 q^{73} -4.20775 q^{75} -1.03923 q^{77} +6.47219 q^{79} -5.43296 q^{81} -2.41789 q^{83} -9.97046 q^{85} -11.7724 q^{87} -13.2295 q^{89} +6.56465 q^{93} +9.74094 q^{95} -11.0368 q^{97} +2.12929 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{3} - 4 q^{5} + q^{7} + q^{9}+O(q^{10}) \) 3 * q + 4 * q^3 - 4 * q^5 + q^7 + q^9	\( 3 q + 4 q^{3} - 4 q^{5} + q^{7} + q^{9} - 6 q^{11} - 10 q^{15} - 2 q^{17} - 6 q^{19} + 6 q^{21} + 9 q^{23} - 5 q^{25} + q^{27} + 3 q^{29} - 5 q^{31} - 22 q^{33} - 6 q^{35} - 12 q^{37} - 3 q^{41} + 17 q^{43} - 13 q^{45} - 8 q^{47} - 16 q^{49} - 5 q^{51} - 15 q^{53} + 22 q^{55} - 15 q^{57} - 17 q^{59} - 28 q^{61} + 12 q^{63} - 17 q^{67} - 2 q^{69} + 7 q^{71} - 15 q^{73} + 5 q^{75} - 16 q^{77} + 13 q^{79} + 3 q^{81} - 13 q^{83} + 5 q^{85} + 4 q^{87} - 19 q^{89} - 2 q^{93} + 15 q^{95} - 5 q^{97} - 37 q^{99}+O(q^{100}) \) 3 * q + 4 * q^3 - 4 * q^5 + q^7 + q^9 - 6 * q^11 - 10 * q^15 - 2 * q^17 - 6 * q^19 + 6 * q^21 + 9 * q^23 - 5 * q^25 + q^27 + 3 * q^29 - 5 * q^31 - 22 * q^33 - 6 * q^35 - 12 * q^37 - 3 * q^41 + 17 * q^43 - 13 * q^45 - 8 * q^47 - 16 * q^49 - 5 * q^51 - 15 * q^53 + 22 * q^55 - 15 * q^57 - 17 * q^59 - 28 * q^61 + 12 * q^63 - 17 * q^67 - 2 * q^69 + 7 * q^71 - 15 * q^73 + 5 * q^75 - 16 * q^77 + 13 * q^79 + 3 * q^81 - 13 * q^83 + 5 * q^85 + 4 * q^87 - 19 * q^89 - 2 * q^93 + 15 * q^95 - 5 * q^97 - 37 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more