LMFDB - Embedded newform 3380.2.a.s.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, 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("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.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:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 19x^{7} + 16x^{6} + 106x^{5} - 87x^{4} - 153x^{3} + 149x^{2} - 26x + 1 $ x^9 - x^8 - 19x^7 + 16x^6 + 106x^5 - 87x^4 - 153x^3 + 149x^2 - 26*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.161999$ of defining polynomial
Character	$\chi$	$=$	3380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.161999 q^{3} +1.00000 q^{5} +1.89771 q^{7} -2.97376 q^{9} +O(q^{10})$	$q-0.161999 q^{3} +1.00000 q^{5} +1.89771 q^{7} -2.97376 q^{9} +5.35728 q^{11} -0.161999 q^{15} +7.71387 q^{17} +0.553719 q^{19} -0.307427 q^{21} -3.14988 q^{23} +1.00000 q^{25} +0.967741 q^{27} -3.54250 q^{29} +6.01423 q^{31} -0.867873 q^{33} +1.89771 q^{35} -10.8657 q^{37} +8.74983 q^{41} -3.25198 q^{43} -2.97376 q^{45} +9.81442 q^{47} -3.39868 q^{49} -1.24964 q^{51} +8.87818 q^{53} +5.35728 q^{55} -0.0897018 q^{57} -13.4174 q^{59} -0.551418 q^{61} -5.64334 q^{63} +4.46051 q^{67} +0.510276 q^{69} -8.17861 q^{71} -12.6129 q^{73} -0.161999 q^{75} +10.1666 q^{77} +13.4130 q^{79} +8.76450 q^{81} +5.42318 q^{83} +7.71387 q^{85} +0.573881 q^{87} -1.46756 q^{89} -0.974297 q^{93} +0.553719 q^{95} -5.69045 q^{97} -15.9313 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} + 9 q^{5} + q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q - q^3 + 9 * q^5 + q^7 + 12 * q^9	$ 9 q - q^{3} + 9 q^{5} + q^{7} + 12 q^{9} - 7 q^{11} - q^{15} + 13 q^{17} - 4 q^{19} - 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} + 13 q^{31} + 34 q^{33} + q^{35} + q^{37} - 6 q^{41} + q^{43} + 12 q^{45} - 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} + 38 q^{57} + 15 q^{59} + 21 q^{61} - 17 q^{63} - 7 q^{67} + 15 q^{69} - 7 q^{71} - 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} + 45 q^{83} + 13 q^{85} + 28 q^{87} - 41 q^{89} - 11 q^{93} - 4 q^{95} + 8 q^{97} - 81 q^{99}+O(q^{100}) $ 9 * q - q^3 + 9 * q^5 + q^7 + 12 * q^9 - 7 * q^11 - q^15 + 13 * q^17 - 4 * q^19 - 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 + 13 * q^31 + 34 * q^33 + q^35 + q^37 - 6 * q^41 + q^43 + 12 * q^45 - 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 + 38 * q^57 + 15 * q^59 + 21 * q^61 - 17 * q^63 - 7 * q^67 + 15 * q^69 - 7 * q^71 - 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 + 45 * q^83 + 13 * q^85 + 28 * q^87 - 41 * q^89 - 11 * q^93 - 4 * q^95 + 8 * q^97 - 81 * 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
$3$	−0.161999	−0.0935300	−0.0467650	−	0.998906i	$-0.514891\pi$
$3$	−0.161999	−0.0935300	−0.0467650	+	0.998906i	$0.514891\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.89771	0.717268	0.358634	−	0.933478i	$-0.383243\pi$
$7$	1.89771	0.717268	0.358634	+	0.933478i	$0.383243\pi$
$8$	0	0
$9$	−2.97376	−0.991252
$10$	0	0
$11$	5.35728	1.61528	0.807641	−	0.589674i	$-0.200744\pi$
$11$	5.35728	1.61528	0.807641	+	0.589674i	$0.200744\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−0.161999	−0.0418279
$16$	0	0
$17$	7.71387	1.87089	0.935445	−	0.353473i	$-0.114999\pi$
$17$	7.71387	1.87089	0.935445	+	0.353473i	$0.114999\pi$
$18$	0	0
$19$	0.553719	0.127032	0.0635160	−	0.997981i	$-0.479769\pi$
$19$	0.553719	0.127032	0.0635160	+	0.997981i	$0.479769\pi$
$20$	0	0
$21$	−0.307427	−0.0670861
$22$	0	0
$23$	−3.14988	−0.656795	−0.328398	−	0.944540i	$-0.606509\pi$
$23$	−3.14988	−0.656795	−0.328398	+	0.944540i	$0.606509\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.967741	0.186242
$28$	0	0
$29$	−3.54250	−0.657826	−0.328913	−	0.944360i	$-0.606682\pi$
$29$	−3.54250	−0.657826	−0.328913	+	0.944360i	$0.606682\pi$
$30$	0	0
$31$	6.01423	1.08019	0.540094	−	0.841605i	$-0.318389\pi$
$31$	6.01423	1.08019	0.540094	+	0.841605i	$0.318389\pi$
$32$	0	0
$33$	−0.867873	−0.151077
$34$	0	0
$35$	1.89771	0.320772
$36$	0	0
$37$	−10.8657	−1.78631	−0.893157	−	0.449746i	$-0.851515\pi$
$37$	−10.8657	−1.78631	−0.893157	+	0.449746i	$0.851515\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.74983	1.36649	0.683247	−	0.730187i	$-0.260567\pi$
$41$	8.74983	1.36649	0.683247	+	0.730187i	$0.260567\pi$
$42$	0	0
$43$	−3.25198	−0.495923	−0.247961	−	0.968770i	$-0.579761\pi$
$43$	−3.25198	−0.495923	−0.247961	+	0.968770i	$0.579761\pi$
$44$	0	0
$45$	−2.97376	−0.443301
$46$	0	0
$47$	9.81442	1.43158	0.715790	−	0.698315i	$-0.246067\pi$
$47$	9.81442	1.43158	0.715790	+	0.698315i	$0.246067\pi$
$48$	0	0
$49$	−3.39868	−0.485526
$50$	0	0
$51$	−1.24964	−0.174984
$52$	0	0
$53$	8.87818	1.21951	0.609756	−	0.792590i	$-0.291268\pi$
$53$	8.87818	1.21951	0.609756	+	0.792590i	$0.291268\pi$
$54$	0	0
$55$	5.35728	0.722376
$56$	0	0
$57$	−0.0897018	−0.0118813
$58$	0	0
$59$	−13.4174	−1.74680	−0.873401	−	0.487002i	$-0.838091\pi$
$59$	−13.4174	−1.74680	−0.873401	+	0.487002i	$0.838091\pi$
$60$	0	0
$61$	−0.551418	−0.0706018	−0.0353009	−	0.999377i	$-0.511239\pi$
$61$	−0.551418	−0.0706018	−0.0353009	+	0.999377i	$0.511239\pi$
$62$	0	0
$63$	−5.64334	−0.710994
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.46051	0.544938	0.272469	−	0.962165i	$-0.412160\pi$
$67$	4.46051	0.544938	0.272469	+	0.962165i	$0.412160\pi$
$68$	0	0
$69$	0.510276	0.0614301
$70$	0	0
$71$	−8.17861	−0.970623	−0.485311	−	0.874341i	$-0.661294\pi$
$71$	−8.17861	−0.970623	−0.485311	+	0.874341i	$0.661294\pi$
$72$	0	0
$73$	−12.6129	−1.47623	−0.738113	−	0.674677i	$-0.764283\pi$
$73$	−12.6129	−1.47623	−0.738113	+	0.674677i	$0.764283\pi$
$74$	0	0
$75$	−0.161999	−0.0187060
$76$	0	0
$77$	10.1666	1.15859
$78$	0	0
$79$	13.4130	1.50908	0.754540	−	0.656255i	$-0.227860\pi$
$79$	13.4130	1.50908	0.754540	+	0.656255i	$0.227860\pi$
$80$	0	0
$81$	8.76450	0.973833
$82$	0	0
$83$	5.42318	0.595271	0.297636	−	0.954680i	$-0.403802\pi$
$83$	5.42318	0.595271	0.297636	+	0.954680i	$0.403802\pi$
$84$	0	0
$85$	7.71387	0.836687
$86$	0	0
$87$	0.573881	0.0615265
$88$	0	0
$89$	−1.46756	−0.155561	−0.0777804	−	0.996971i	$-0.524783\pi$
$89$	−1.46756	−0.155561	−0.0777804	+	0.996971i	$0.524783\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.974297	−0.101030
$94$	0	0
$95$	0.553719	0.0568104
$96$	0	0
$97$	−5.69045	−0.577777	−0.288889	−	0.957363i	$-0.593286\pi$
$97$	−5.69045	−0.577777	−0.288889	+	0.957363i	$0.593286\pi$
$98$	0	0
$99$	−15.9313	−1.60115
$100$	0	0
$101$	6.80029	0.676654	0.338327	−	0.941029i	$-0.390139\pi$
$101$	6.80029	0.676654	0.338327	+	0.941029i	$0.390139\pi$
$102$	0	0
$103$	12.9081	1.27187	0.635936	−	0.771742i	$-0.280614\pi$
$103$	12.9081	1.27187	0.635936	+	0.771742i	$0.280614\pi$
$104$	0	0
$105$	−0.307427	−0.0300018
$106$	0	0
$107$	−4.23968	−0.409865	−0.204933	−	0.978776i	$-0.565697\pi$
$107$	−4.23968	−0.409865	−0.204933	+	0.978776i	$0.565697\pi$
$108$	0	0
$109$	7.00088	0.670563	0.335281	−	0.942118i	$-0.391169\pi$
$109$	7.00088	0.670563	0.335281	+	0.942118i	$0.391169\pi$
$110$	0	0
$111$	1.76023	0.167074
$112$	0	0
$113$	2.29176	0.215590	0.107795	−	0.994173i	$-0.465621\pi$
$113$	2.29176	0.215590	0.107795	+	0.994173i	$0.465621\pi$
$114$	0	0
$115$	−3.14988	−0.293728
$116$	0	0
$117$	0	0
$118$	0	0
$119$	14.6387	1.34193
$120$	0	0
$121$	17.7005	1.60914
$122$	0	0
$123$	−1.41746	−0.127808
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−0.526702	−0.0467372	−0.0233686	−	0.999727i	$-0.507439\pi$
$127$	−0.526702	−0.0467372	−0.0233686	+	0.999727i	$0.507439\pi$
$128$	0	0
$129$	0.526817	0.0463837
$130$	0	0
$131$	17.8745	1.56171	0.780853	−	0.624715i	$-0.214785\pi$
$131$	17.8745	1.56171	0.780853	+	0.624715i	$0.214785\pi$
$132$	0	0
$133$	1.05080	0.0911159
$134$	0	0
$135$	0.967741	0.0832899
$136$	0	0
$137$	8.58940	0.733842	0.366921	−	0.930252i	$-0.380412\pi$
$137$	8.58940	0.733842	0.366921	+	0.930252i	$0.380412\pi$
$138$	0	0
$139$	2.68254	0.227530	0.113765	−	0.993508i	$-0.463709\pi$
$139$	2.68254	0.227530	0.113765	+	0.993508i	$0.463709\pi$
$140$	0	0
$141$	−1.58992	−0.133896
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−3.54250	−0.294189
$146$	0	0
$147$	0.550583	0.0454113
$148$	0	0
$149$	−4.22172	−0.345857	−0.172929	−	0.984934i	$-0.555323\pi$
$149$	−4.22172	−0.345857	−0.172929	+	0.984934i	$0.555323\pi$
$150$	0	0
$151$	−7.26015	−0.590822	−0.295411	−	0.955370i	$-0.595457\pi$
$151$	−7.26015	−0.590822	−0.295411	+	0.955370i	$0.595457\pi$
$152$	0	0
$153$	−22.9392	−1.85452
$154$	0	0
$155$	6.01423	0.483074
$156$	0	0
$157$	−10.7458	−0.857609	−0.428805	−	0.903397i	$-0.641065\pi$
$157$	−10.7458	−0.857609	−0.428805	+	0.903397i	$0.641065\pi$
$158$	0	0
$159$	−1.43825	−0.114061
$160$	0	0
$161$	−5.97757	−0.471098
$162$	0	0
$163$	−8.75744	−0.685935	−0.342968	−	0.939347i	$-0.611432\pi$
$163$	−8.75744	−0.685935	−0.342968	+	0.939347i	$0.611432\pi$
$164$	0	0
$165$	−0.867873	−0.0675639
$166$	0	0
$167$	−3.29670	−0.255106	−0.127553	−	0.991832i	$-0.540712\pi$
$167$	−3.29670	−0.255106	−0.127553	+	0.991832i	$0.540712\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−1.64663	−0.125921
$172$	0	0
$173$	20.6617	1.57088	0.785440	−	0.618938i	$-0.212437\pi$
$173$	20.6617	1.57088	0.785440	+	0.618938i	$0.212437\pi$
$174$	0	0
$175$	1.89771	0.143454
$176$	0	0
$177$	2.17361	0.163378
$178$	0	0
$179$	−19.5721	−1.46289	−0.731446	−	0.681900i	$-0.761154\pi$
$179$	−19.5721	−1.46289	−0.731446	+	0.681900i	$0.761154\pi$
$180$	0	0
$181$	10.7679	0.800369	0.400185	−	0.916435i	$-0.368946\pi$
$181$	10.7679	0.800369	0.400185	+	0.916435i	$0.368946\pi$
$182$	0	0
$183$	0.0893290	0.00660339
$184$	0	0
$185$	−10.8657	−0.798863
$186$	0	0
$187$	41.3254	3.02201
$188$	0	0
$189$	1.83649	0.133585
$190$	0	0
$191$	−13.3596	−0.966667	−0.483334	−	0.875436i	$-0.660574\pi$
$191$	−13.3596	−0.966667	−0.483334	+	0.875436i	$0.660574\pi$
$192$	0	0
$193$	3.69764	0.266162	0.133081	−	0.991105i	$-0.457513\pi$
$193$	3.69764	0.266162	0.133081	+	0.991105i	$0.457513\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.5498	1.25037	0.625187	−	0.780475i	$-0.285023\pi$
$197$	17.5498	1.25037	0.625187	+	0.780475i	$0.285023\pi$
$198$	0	0
$199$	19.2570	1.36510	0.682548	−	0.730841i	$-0.260872\pi$
$199$	19.2570	1.36510	0.682548	+	0.730841i	$0.260872\pi$
$200$	0	0
$201$	−0.722597	−0.0509681
$202$	0	0
$203$	−6.72265	−0.471838
$204$	0	0
$205$	8.74983	0.611115
$206$	0	0
$207$	9.36697	0.651050
$208$	0	0
$209$	2.96643	0.205192
$210$	0	0
$211$	−26.0925	−1.79628	−0.898142	−	0.439705i	$-0.855083\pi$
$211$	−26.0925	−1.79628	−0.898142	+	0.439705i	$0.855083\pi$
$212$	0	0
$213$	1.32492	0.0907824
$214$	0	0
$215$	−3.25198	−0.221783
$216$	0	0
$217$	11.4133	0.774784
$218$	0	0
$219$	2.04327	0.138071
$220$	0	0
$221$	0	0
$222$	0	0
$223$	9.30627	0.623194	0.311597	−	0.950214i	$-0.399136\pi$
$223$	9.30627	0.623194	0.311597	+	0.950214i	$0.399136\pi$
$224$	0	0
$225$	−2.97376	−0.198250
$226$	0	0
$227$	−1.77639	−0.117903	−0.0589515	−	0.998261i	$-0.518776\pi$
$227$	−1.77639	−0.117903	−0.0589515	+	0.998261i	$0.518776\pi$
$228$	0	0
$229$	12.8216	0.847272	0.423636	−	0.905833i	$-0.360754\pi$
$229$	12.8216	0.847272	0.423636	+	0.905833i	$0.360754\pi$
$230$	0	0
$231$	−1.64697	−0.108363
$232$	0	0
$233$	−16.8823	−1.10599	−0.552997	−	0.833183i	$-0.686516\pi$
$233$	−16.8823	−1.10599	−0.552997	+	0.833183i	$0.686516\pi$
$234$	0	0
$235$	9.81442	0.640222
$236$	0	0
$237$	−2.17289	−0.141144
$238$	0	0
$239$	−20.5027	−1.32621	−0.663106	−	0.748526i	$-0.730762\pi$
$239$	−20.5027	−1.32621	−0.663106	+	0.748526i	$0.730762\pi$
$240$	0	0
$241$	−11.0766	−0.713504	−0.356752	−	0.934199i	$-0.616116\pi$
$241$	−11.0766	−0.713504	−0.356752	+	0.934199i	$0.616116\pi$
$242$	0	0
$243$	−4.32306	−0.277324
$244$	0	0
$245$	−3.39868	−0.217134
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.878548	−0.0556757
$250$	0	0
$251$	−1.61833	−0.102148	−0.0510740	−	0.998695i	$-0.516264\pi$
$251$	−1.61833	−0.102148	−0.0510740	+	0.998695i	$0.516264\pi$
$252$	0	0
$253$	−16.8748	−1.06091
$254$	0	0
$255$	−1.24964	−0.0782554
$256$	0	0
$257$	13.6190	0.849532	0.424766	−	0.905303i	$-0.360356\pi$
$257$	13.6190	0.849532	0.424766	+	0.905303i	$0.360356\pi$
$258$	0	0
$259$	−20.6200	−1.28127
$260$	0	0
$261$	10.5345	0.652071
$262$	0	0
$263$	11.1385	0.686828	0.343414	−	0.939184i	$-0.388417\pi$
$263$	11.1385	0.686828	0.343414	+	0.939184i	$0.388417\pi$
$264$	0	0
$265$	8.87818	0.545382
$266$	0	0
$267$	0.237743	0.0145496
$268$	0	0
$269$	18.2455	1.11245	0.556223	−	0.831033i	$-0.312250\pi$
$269$	18.2455	1.11245	0.556223	+	0.831033i	$0.312250\pi$
$270$	0	0
$271$	−17.2410	−1.04732	−0.523658	−	0.851929i	$-0.675433\pi$
$271$	−17.2410	−1.04732	−0.523658	+	0.851929i	$0.675433\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.35728	0.323056
$276$	0	0
$277$	−16.7759	−1.00797	−0.503983	−	0.863713i	$-0.668133\pi$
$277$	−16.7759	−1.00797	−0.503983	+	0.863713i	$0.668133\pi$
$278$	0	0
$279$	−17.8849	−1.07074
$280$	0	0
$281$	−3.71383	−0.221549	−0.110774	−	0.993846i	$-0.535333\pi$
$281$	−3.71383	−0.221549	−0.110774	+	0.993846i	$0.535333\pi$
$282$	0	0
$283$	29.2742	1.74017	0.870086	−	0.492900i	$-0.164063\pi$
$283$	29.2742	1.74017	0.870086	+	0.492900i	$0.164063\pi$
$284$	0	0
$285$	−0.0897018	−0.00531348
$286$	0	0
$287$	16.6047	0.980143
$288$	0	0
$289$	42.5039	2.50023
$290$	0	0
$291$	0.921845	0.0540395
$292$	0	0
$293$	7.58006	0.442832	0.221416	−	0.975179i	$-0.428932\pi$
$293$	7.58006	0.442832	0.221416	+	0.975179i	$0.428932\pi$
$294$	0	0
$295$	−13.4174	−0.781194
$296$	0	0
$297$	5.18446	0.300833
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−6.17133	−0.355710
$302$	0	0
$303$	−1.10164	−0.0632875
$304$	0	0
$305$	−0.551418	−0.0315741
$306$	0	0
$307$	−2.94812	−0.168258	−0.0841292	−	0.996455i	$-0.526811\pi$
$307$	−2.94812	−0.168258	−0.0841292	+	0.996455i	$0.526811\pi$
$308$	0	0
$309$	−2.09110	−0.118958
$310$	0	0
$311$	−5.64858	−0.320302	−0.160151	−	0.987093i	$-0.551198\pi$
$311$	−5.64858	−0.320302	−0.160151	+	0.987093i	$0.551198\pi$
$312$	0	0
$313$	16.4132	0.927726	0.463863	−	0.885907i	$-0.346463\pi$
$313$	16.4132	0.927726	0.463863	+	0.885907i	$0.346463\pi$
$314$	0	0
$315$	−5.64334	−0.317966
$316$	0	0
$317$	25.3498	1.42378	0.711892	−	0.702289i	$-0.247838\pi$
$317$	25.3498	1.42378	0.711892	+	0.702289i	$0.247838\pi$
$318$	0	0
$319$	−18.9782	−1.06257
$320$	0	0
$321$	0.686822	0.0383347
$322$	0	0
$323$	4.27132	0.237663
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.13413	−0.0627178
$328$	0	0
$329$	18.6250	1.02683
$330$	0	0
$331$	16.8750	0.927534	0.463767	−	0.885957i	$-0.346497\pi$
$331$	16.8750	0.927534	0.463767	+	0.885957i	$0.346497\pi$
$332$	0	0
$333$	32.3120	1.77069
$334$	0	0
$335$	4.46051	0.243704
$336$	0	0
$337$	33.2141	1.80929	0.904643	−	0.426170i	$-0.140138\pi$
$337$	33.2141	1.80929	0.904643	+	0.426170i	$0.140138\pi$
$338$	0	0
$339$	−0.371261	−0.0201642
$340$	0	0
$341$	32.2199	1.74481
$342$	0	0
$343$	−19.7337	−1.06552
$344$	0	0
$345$	0.510276	0.0274724
$346$	0	0
$347$	−32.1558	−1.72621	−0.863107	−	0.505021i	$-0.831485\pi$
$347$	−32.1558	−1.72621	−0.863107	+	0.505021i	$0.831485\pi$
$348$	0	0
$349$	−2.18315	−0.116861	−0.0584307	−	0.998291i	$-0.518610\pi$
$349$	−2.18315	−0.116861	−0.0584307	+	0.998291i	$0.518610\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.8132	0.575529	0.287764	−	0.957701i	$-0.407088\pi$
$353$	10.8132	0.575529	0.287764	+	0.957701i	$0.407088\pi$
$354$	0	0
$355$	−8.17861	−0.434076
$356$	0	0
$357$	−2.37145	−0.125511
$358$	0	0
$359$	−20.4897	−1.08140	−0.540702	−	0.841214i	$-0.681841\pi$
$359$	−20.4897	−1.08140	−0.540702	+	0.841214i	$0.681841\pi$
$360$	0	0
$361$	−18.6934	−0.983863
$362$	0	0
$363$	−2.86746	−0.150503
$364$	0	0
$365$	−12.6129	−0.660188
$366$	0	0
$367$	−32.5527	−1.69924	−0.849619	−	0.527398i	$-0.823168\pi$
$367$	−32.5527	−1.69924	−0.849619	+	0.527398i	$0.823168\pi$
$368$	0	0
$369$	−26.0199	−1.35454
$370$	0	0
$371$	16.8482	0.874717
$372$	0	0
$373$	18.8558	0.976317	0.488159	−	0.872755i	$-0.337669\pi$
$373$	18.8558	0.976317	0.488159	+	0.872755i	$0.337669\pi$
$374$	0	0
$375$	−0.161999	−0.00836558
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−13.3031	−0.683332	−0.341666	−	0.939821i	$-0.610991\pi$
$379$	−13.3031	−0.683332	−0.341666	+	0.939821i	$0.610991\pi$
$380$	0	0
$381$	0.0853250	0.00437133
$382$	0	0
$383$	28.3902	1.45067	0.725336	−	0.688395i	$-0.241685\pi$
$383$	28.3902	1.45067	0.725336	+	0.688395i	$0.241685\pi$
$384$	0	0
$385$	10.1666	0.518137
$386$	0	0
$387$	9.67061	0.491585
$388$	0	0
$389$	−7.38713	−0.374543	−0.187271	−	0.982308i	$-0.559964\pi$
$389$	−7.38713	−0.374543	−0.187271	+	0.982308i	$0.559964\pi$
$390$	0	0
$391$	−24.2978	−1.22879
$392$	0	0
$393$	−2.89565	−0.146066
$394$	0	0
$395$	13.4130	0.674881
$396$	0	0
$397$	16.1613	0.811114	0.405557	−	0.914070i	$-0.367078\pi$
$397$	16.1613	0.811114	0.405557	+	0.914070i	$0.367078\pi$
$398$	0	0
$399$	−0.170228	−0.00852208
$400$	0	0
$401$	−8.05887	−0.402441	−0.201220	−	0.979546i	$-0.564491\pi$
$401$	−8.05887	−0.402441	−0.201220	+	0.979546i	$0.564491\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	8.76450	0.435511
$406$	0	0
$407$	−58.2107	−2.88540
$408$	0	0
$409$	24.9429	1.23335	0.616674	−	0.787219i	$-0.288480\pi$
$409$	24.9429	1.23335	0.616674	+	0.787219i	$0.288480\pi$
$410$	0	0
$411$	−1.39147	−0.0686363
$412$	0	0
$413$	−25.4625	−1.25293
$414$	0	0
$415$	5.42318	0.266213
$416$	0	0
$417$	−0.434568	−0.0212809
$418$	0	0
$419$	−0.758173	−0.0370392	−0.0185196	−	0.999828i	$-0.505895\pi$
$419$	−0.758173	−0.0370392	−0.0185196	+	0.999828i	$0.505895\pi$
$420$	0	0
$421$	1.20278	0.0586199	0.0293099	−	0.999570i	$-0.490669\pi$
$421$	1.20278	0.0586199	0.0293099	+	0.999570i	$0.490669\pi$
$422$	0	0
$423$	−29.1857	−1.41906
$424$	0	0
$425$	7.71387	0.374178
$426$	0	0
$427$	−1.04643	−0.0506404
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3.55978	−0.171469	−0.0857344	−	0.996318i	$-0.527324\pi$
$431$	−3.55978	−0.171469	−0.0857344	+	0.996318i	$0.527324\pi$
$432$	0	0
$433$	37.2920	1.79214	0.896070	−	0.443913i	$-0.146410\pi$
$433$	37.2920	1.79214	0.896070	+	0.443913i	$0.146410\pi$
$434$	0	0
$435$	0.573881	0.0275155
$436$	0	0
$437$	−1.74415	−0.0834339
$438$	0	0
$439$	−13.0813	−0.624336	−0.312168	−	0.950027i	$-0.601055\pi$
$439$	−13.0813	−0.624336	−0.312168	+	0.950027i	$0.601055\pi$
$440$	0	0
$441$	10.1069	0.481279
$442$	0	0
$443$	−22.1758	−1.05360	−0.526802	−	0.849988i	$-0.676609\pi$
$443$	−22.1758	−1.05360	−0.526802	+	0.849988i	$0.676609\pi$
$444$	0	0
$445$	−1.46756	−0.0695689
$446$	0	0
$447$	0.683914	0.0323480
$448$	0	0
$449$	−3.92762	−0.185356	−0.0926778	−	0.995696i	$-0.529543\pi$
$449$	−3.92762	−0.185356	−0.0926778	+	0.995696i	$0.529543\pi$
$450$	0	0
$451$	46.8753	2.20727
$452$	0	0
$453$	1.17613	0.0552596
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−29.1786	−1.36492	−0.682458	−	0.730925i	$-0.739089\pi$
$457$	−29.1786	−1.36492	−0.682458	+	0.730925i	$0.739089\pi$
$458$	0	0
$459$	7.46503	0.348438
$460$	0	0
$461$	−13.8570	−0.645384	−0.322692	−	0.946504i	$-0.604588\pi$
$461$	−13.8570	−0.645384	−0.322692	+	0.946504i	$0.604588\pi$
$462$	0	0
$463$	−28.0020	−1.30136	−0.650682	−	0.759351i	$-0.725517\pi$
$463$	−28.0020	−1.30136	−0.650682	+	0.759351i	$0.725517\pi$
$464$	0	0
$465$	−0.974297	−0.0451820
$466$	0	0
$467$	7.17194	0.331878	0.165939	−	0.986136i	$-0.446935\pi$
$467$	7.17194	0.331878	0.165939	+	0.986136i	$0.446935\pi$
$468$	0	0
$469$	8.46477	0.390867
$470$	0	0
$471$	1.74081	0.0802122
$472$	0	0
$473$	−17.4218	−0.801056
$474$	0	0
$475$	0.553719	0.0254064
$476$	0	0
$477$	−26.4015	−1.20884
$478$	0	0
$479$	15.4839	0.707478	0.353739	−	0.935344i	$-0.384910\pi$
$479$	15.4839	0.707478	0.353739	+	0.935344i	$0.384910\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0.968358	0.0440618
$484$	0	0
$485$	−5.69045	−0.258390
$486$	0	0
$487$	34.4727	1.56211	0.781054	−	0.624464i	$-0.214683\pi$
$487$	34.4727	1.56211	0.781054	+	0.624464i	$0.214683\pi$
$488$	0	0
$489$	1.41869	0.0641555
$490$	0	0
$491$	−31.8837	−1.43889	−0.719446	−	0.694548i	$-0.755604\pi$
$491$	−31.8837	−1.43889	−0.719446	+	0.694548i	$0.755604\pi$
$492$	0	0
$493$	−27.3264	−1.23072
$494$	0	0
$495$	−15.9313	−0.716057
$496$	0	0
$497$	−15.5207	−0.696197
$498$	0	0
$499$	−4.94343	−0.221298	−0.110649	−	0.993860i	$-0.535293\pi$
$499$	−4.94343	−0.221298	−0.110649	+	0.993860i	$0.535293\pi$
$500$	0	0
$501$	0.534061	0.0238601
$502$	0	0
$503$	−30.1426	−1.34399	−0.671996	−	0.740555i	$-0.734563\pi$
$503$	−30.1426	−1.34399	−0.671996	+	0.740555i	$0.734563\pi$
$504$	0	0
$505$	6.80029	0.302609
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−28.3768	−1.25778	−0.628889	−	0.777495i	$-0.716490\pi$
$509$	−28.3768	−1.25778	−0.628889	+	0.777495i	$0.716490\pi$
$510$	0	0
$511$	−23.9356	−1.05885
$512$	0	0
$513$	0.535857	0.0236587
$514$	0	0
$515$	12.9081	0.568799
$516$	0	0
$517$	52.5787	2.31241
$518$	0	0
$519$	−3.34717	−0.146924
$520$	0	0
$521$	−24.8552	−1.08893	−0.544463	−	0.838785i	$-0.683267\pi$
$521$	−24.8552	−1.08893	−0.544463	+	0.838785i	$0.683267\pi$
$522$	0	0
$523$	3.95417	0.172904	0.0864520	−	0.996256i	$-0.472447\pi$
$523$	3.95417	0.172904	0.0864520	+	0.996256i	$0.472447\pi$
$524$	0	0
$525$	−0.307427	−0.0134172
$526$	0	0
$527$	46.3930	2.02091
$528$	0	0
$529$	−13.0783	−0.568620
$530$	0	0
$531$	39.9002	1.73152
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−4.23968	−0.183297
$536$	0	0
$537$	3.17066	0.136824
$538$	0	0
$539$	−18.2077	−0.784262
$540$	0	0
$541$	11.8032	0.507461	0.253731	−	0.967275i	$-0.418342\pi$
$541$	11.8032	0.507461	0.253731	+	0.967275i	$0.418342\pi$
$542$	0	0
$543$	−1.74438	−0.0748585
$544$	0	0
$545$	7.00088	0.299885
$546$	0	0
$547$	−22.2805	−0.952644	−0.476322	−	0.879271i	$-0.658030\pi$
$547$	−22.2805	−0.952644	−0.476322	+	0.879271i	$0.658030\pi$
$548$	0	0
$549$	1.63978	0.0699842
$550$	0	0
$551$	−1.96155	−0.0835649
$552$	0	0
$553$	25.4540	1.08241
$554$	0	0
$555$	1.76023	0.0747177
$556$	0	0
$557$	−34.7144	−1.47090	−0.735448	−	0.677581i	$-0.763028\pi$
$557$	−34.7144	−1.47090	−0.735448	+	0.677581i	$0.763028\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−6.69467	−0.282649
$562$	0	0
$563$	−20.0781	−0.846191	−0.423095	−	0.906085i	$-0.639056\pi$
$563$	−20.0781	−0.846191	−0.423095	+	0.906085i	$0.639056\pi$
$564$	0	0
$565$	2.29176	0.0964149
$566$	0	0
$567$	16.6325	0.698499
$568$	0	0
$569$	−11.2353	−0.471010	−0.235505	−	0.971873i	$-0.575674\pi$
$569$	−11.2353	−0.471010	−0.235505	+	0.971873i	$0.575674\pi$
$570$	0	0
$571$	−25.7673	−1.07833	−0.539164	−	0.842201i	$-0.681260\pi$
$571$	−25.7673	−1.07833	−0.539164	+	0.842201i	$0.681260\pi$
$572$	0	0
$573$	2.16424	0.0904124
$574$	0	0
$575$	−3.14988	−0.131359
$576$	0	0
$577$	1.07927	0.0449307	0.0224654	−	0.999748i	$-0.492848\pi$
$577$	1.07927	0.0449307	0.0224654	+	0.999748i	$0.492848\pi$
$578$	0	0
$579$	−0.599014	−0.0248942
$580$	0	0
$581$	10.2916	0.426969
$582$	0	0
$583$	47.5629	1.96985
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−29.9803	−1.23742	−0.618710	−	0.785620i	$-0.712344\pi$
$587$	−29.9803	−1.23742	−0.618710	+	0.785620i	$0.712344\pi$
$588$	0	0
$589$	3.33019	0.137218
$590$	0	0
$591$	−2.84305	−0.116947
$592$	0	0
$593$	7.87304	0.323307	0.161653	−	0.986848i	$-0.448317\pi$
$593$	7.87304	0.323307	0.161653	+	0.986848i	$0.448317\pi$
$594$	0	0
$595$	14.6387	0.600129
$596$	0	0
$597$	−3.11962	−0.127677
$598$	0	0
$599$	19.7409	0.806590	0.403295	−	0.915070i	$-0.367865\pi$
$599$	19.7409	0.806590	0.403295	+	0.915070i	$0.367865\pi$
$600$	0	0
$601$	−3.33925	−0.136211	−0.0681054	−	0.997678i	$-0.521695\pi$
$601$	−3.33925	−0.136211	−0.0681054	+	0.997678i	$0.521695\pi$
$602$	0	0
$603$	−13.2645	−0.540171
$604$	0	0
$605$	17.7005	0.719628
$606$	0	0
$607$	22.2968	0.904998	0.452499	−	0.891765i	$-0.350533\pi$
$607$	22.2968	0.904998	0.452499	+	0.891765i	$0.350533\pi$
$608$	0	0
$609$	1.08906	0.0441310
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−21.6710	−0.875285	−0.437643	−	0.899149i	$-0.644186\pi$
$613$	−21.6710	−0.875285	−0.437643	+	0.899149i	$0.644186\pi$
$614$	0	0
$615$	−1.41746	−0.0571576
$616$	0	0
$617$	−29.1369	−1.17301	−0.586504	−	0.809946i	$-0.699496\pi$
$617$	−29.1369	−1.17301	−0.586504	+	0.809946i	$0.699496\pi$
$618$	0	0
$619$	−33.1986	−1.33437	−0.667183	−	0.744894i	$-0.732500\pi$
$619$	−33.1986	−1.33437	−0.667183	+	0.744894i	$0.732500\pi$
$620$	0	0
$621$	−3.04827	−0.122323
$622$	0	0
$623$	−2.78500	−0.111579
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.480558	−0.0191916
$628$	0	0
$629$	−83.8168	−3.34199
$630$	0	0
$631$	−46.5904	−1.85473	−0.927366	−	0.374155i	$-0.877933\pi$
$631$	−46.5904	−1.85473	−0.927366	+	0.374155i	$0.877933\pi$
$632$	0	0
$633$	4.22696	0.168006
$634$	0	0
$635$	−0.526702	−0.0209015
$636$	0	0
$637$	0	0
$638$	0	0
$639$	24.3212	0.962132
$640$	0	0
$641$	−30.9608	−1.22288	−0.611438	−	0.791292i	$-0.709409\pi$
$641$	−30.9608	−1.22288	−0.611438	+	0.791292i	$0.709409\pi$
$642$	0	0
$643$	−24.0391	−0.948010	−0.474005	−	0.880522i	$-0.657192\pi$
$643$	−24.0391	−0.948010	−0.474005	+	0.880522i	$0.657192\pi$
$644$	0	0
$645$	0.526817	0.0207434
$646$	0	0
$647$	41.9606	1.64964	0.824822	−	0.565393i	$-0.191275\pi$
$647$	41.9606	1.64964	0.824822	+	0.565393i	$0.191275\pi$
$648$	0	0
$649$	−71.8811	−2.82158
$650$	0	0
$651$	−1.84894	−0.0724656
$652$	0	0
$653$	−1.92478	−0.0753223	−0.0376612	−	0.999291i	$-0.511991\pi$
$653$	−1.92478	−0.0753223	−0.0376612	+	0.999291i	$0.511991\pi$
$654$	0	0
$655$	17.8745	0.698416
$656$	0	0
$657$	37.5076	1.46331
$658$	0	0
$659$	2.78444	0.108466	0.0542332	−	0.998528i	$-0.482729\pi$
$659$	2.78444	0.108466	0.0542332	+	0.998528i	$0.482729\pi$
$660$	0	0
$661$	−12.5553	−0.488346	−0.244173	−	0.969732i	$-0.578516\pi$
$661$	−12.5553	−0.488346	−0.244173	+	0.969732i	$0.578516\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.05080	0.0407483
$666$	0	0
$667$	11.1584	0.432057
$668$	0	0
$669$	−1.50760	−0.0582874
$670$	0	0
$671$	−2.95410	−0.114042
$672$	0	0
$673$	46.3999	1.78859	0.894293	−	0.447481i	$-0.147679\pi$
$673$	46.3999	1.78859	0.894293	+	0.447481i	$0.147679\pi$
$674$	0	0
$675$	0.967741	0.0372484
$676$	0	0
$677$	−45.7333	−1.75767	−0.878837	−	0.477123i	$-0.841680\pi$
$677$	−45.7333	−1.75767	−0.878837	+	0.477123i	$0.841680\pi$
$678$	0	0
$679$	−10.7988	−0.414421
$680$	0	0
$681$	0.287773	0.0110275
$682$	0	0
$683$	−40.5600	−1.55198	−0.775992	−	0.630743i	$-0.782750\pi$
$683$	−40.5600	−1.55198	−0.775992	+	0.630743i	$0.782750\pi$
$684$	0	0
$685$	8.58940	0.328184
$686$	0	0
$687$	−2.07708	−0.0792454
$688$	0	0
$689$	0	0
$690$	0	0
$691$	17.9242	0.681868	0.340934	−	0.940087i	$-0.389257\pi$
$691$	17.9242	0.681868	0.340934	+	0.940087i	$0.389257\pi$
$692$	0	0
$693$	−30.2330	−1.14846
$694$	0	0
$695$	2.68254	0.101755
$696$	0	0
$697$	67.4951	2.55656
$698$	0	0
$699$	2.73491	0.103444
$700$	0	0
$701$	−19.5457	−0.738230	−0.369115	−	0.929384i	$-0.620339\pi$
$701$	−19.5457	−0.738230	−0.369115	+	0.929384i	$0.620339\pi$
$702$	0	0
$703$	−6.01656	−0.226919
$704$	0	0
$705$	−1.58992	−0.0598800
$706$	0	0
$707$	12.9050	0.485343
$708$	0	0
$709$	−46.6490	−1.75194	−0.875970	−	0.482366i	$-0.839778\pi$
$709$	−46.6490	−1.75194	−0.875970	+	0.482366i	$0.839778\pi$
$710$	0	0
$711$	−39.8870	−1.49588
$712$	0	0
$713$	−18.9441	−0.709462
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.32142	0.124041
$718$	0	0
$719$	−40.4882	−1.50995	−0.754977	−	0.655751i	$-0.772352\pi$
$719$	−40.4882	−1.50995	−0.754977	+	0.655751i	$0.772352\pi$
$720$	0	0
$721$	24.4959	0.912274
$722$	0	0
$723$	1.79439	0.0667341
$724$	0	0
$725$	−3.54250	−0.131565
$726$	0	0
$727$	22.9582	0.851471	0.425736	−	0.904848i	$-0.360015\pi$
$727$	22.9582	0.851471	0.425736	+	0.904848i	$0.360015\pi$
$728$	0	0
$729$	−25.5932	−0.947895
$730$	0	0
$731$	−25.0854	−0.927817
$732$	0	0
$733$	−21.3753	−0.789513	−0.394757	−	0.918786i	$-0.629171\pi$
$733$	−21.3753	−0.789513	−0.394757	+	0.918786i	$0.629171\pi$
$734$	0	0
$735$	0.550583	0.0203085
$736$	0	0
$737$	23.8962	0.880229
$738$	0	0
$739$	35.1671	1.29364	0.646821	−	0.762642i	$-0.276098\pi$
$739$	35.1671	1.29364	0.646821	+	0.762642i	$0.276098\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−19.9324	−0.731249	−0.365625	−	0.930762i	$-0.619145\pi$
$743$	−19.9324	−0.731249	−0.365625	+	0.930762i	$0.619145\pi$
$744$	0	0
$745$	−4.22172	−0.154672
$746$	0	0
$747$	−16.1272	−0.590064
$748$	0	0
$749$	−8.04569	−0.293983
$750$	0	0
$751$	33.7277	1.23074	0.615370	−	0.788238i	$-0.289006\pi$
$751$	33.7277	1.23074	0.615370	+	0.788238i	$0.289006\pi$
$752$	0	0
$753$	0.262167	0.00955390
$754$	0	0
$755$	−7.26015	−0.264224
$756$	0	0
$757$	23.3007	0.846877	0.423439	−	0.905925i	$-0.360823\pi$
$757$	23.3007	0.846877	0.423439	+	0.905925i	$0.360823\pi$
$758$	0	0
$759$	2.73370	0.0992269
$760$	0	0
$761$	−7.72440	−0.280009	−0.140005	−	0.990151i	$-0.544712\pi$
$761$	−7.72440	−0.280009	−0.140005	+	0.990151i	$0.544712\pi$
$762$	0	0
$763$	13.2857	0.480973
$764$	0	0
$765$	−22.9392	−0.829368
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−6.44221	−0.232312	−0.116156	−	0.993231i	$-0.537057\pi$
$769$	−6.44221	−0.232312	−0.116156	+	0.993231i	$0.537057\pi$
$770$	0	0
$771$	−2.20627	−0.0794568
$772$	0	0
$773$	39.1922	1.40964	0.704822	−	0.709384i	$-0.251027\pi$
$773$	39.1922	1.40964	0.704822	+	0.709384i	$0.251027\pi$
$774$	0	0
$775$	6.01423	0.216037
$776$	0	0
$777$	3.34042	0.119837
$778$	0	0
$779$	4.84495	0.173588
$780$	0	0
$781$	−43.8152	−1.56783
$782$	0	0
$783$	−3.42822	−0.122515
$784$	0	0
$785$	−10.7458	−0.383535
$786$	0	0
$787$	27.0865	0.965529	0.482765	−	0.875750i	$-0.339633\pi$
$787$	27.0865	0.965529	0.482765	+	0.875750i	$0.339633\pi$
$788$	0	0
$789$	−1.80442	−0.0642390
$790$	0	0
$791$	4.34909	0.154636
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−1.43825	−0.0510096
$796$	0	0
$797$	3.49750	0.123888	0.0619440	−	0.998080i	$-0.480270\pi$
$797$	3.49750	0.123888	0.0619440	+	0.998080i	$0.480270\pi$
$798$	0	0
$799$	75.7072	2.67833
$800$	0	0
$801$	4.36416	0.154200
$802$	0	0
$803$	−67.5708	−2.38452
$804$	0	0
$805$	−5.97757	−0.210682
$806$	0	0
$807$	−2.95574	−0.104047
$808$	0	0
$809$	12.6258	0.443898	0.221949	−	0.975058i	$-0.428758\pi$
$809$	12.6258	0.443898	0.221949	+	0.975058i	$0.428758\pi$
$810$	0	0
$811$	−40.7350	−1.43040	−0.715200	−	0.698920i	$-0.753664\pi$
$811$	−40.7350	−1.43040	−0.715200	+	0.698920i	$0.753664\pi$
$812$	0	0
$813$	2.79302	0.0979554
$814$	0	0
$815$	−8.75744	−0.306760
$816$	0	0
$817$	−1.80069	−0.0629980
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26.0805	−0.910214	−0.455107	−	0.890437i	$-0.650399\pi$
$821$	−26.0805	−0.910214	−0.455107	+	0.890437i	$0.650399\pi$
$822$	0	0
$823$	−4.24240	−0.147881	−0.0739404	−	0.997263i	$-0.523557\pi$
$823$	−4.24240	−0.147881	−0.0739404	+	0.997263i	$0.523557\pi$
$824$	0	0
$825$	−0.867873	−0.0302155
$826$	0	0
$827$	41.1905	1.43234	0.716168	−	0.697928i	$-0.245895\pi$
$827$	41.1905	1.43234	0.716168	+	0.697928i	$0.245895\pi$
$828$	0	0
$829$	−26.2757	−0.912591	−0.456296	−	0.889828i	$-0.650824\pi$
$829$	−26.2757	−0.912591	−0.456296	+	0.889828i	$0.650824\pi$
$830$	0	0
$831$	2.71768	0.0942752
$832$	0	0
$833$	−26.2170	−0.908366
$834$	0	0
$835$	−3.29670	−0.114087
$836$	0	0
$837$	5.82022	0.201176
$838$	0	0
$839$	−29.6372	−1.02319	−0.511594	−	0.859227i	$-0.670945\pi$
$839$	−29.6372	−1.02319	−0.511594	+	0.859227i	$0.670945\pi$
$840$	0	0
$841$	−16.4507	−0.567265
$842$	0	0
$843$	0.601637	0.0207215
$844$	0	0
$845$	0	0
$846$	0	0
$847$	33.5905	1.15418
$848$	0	0
$849$	−4.74239	−0.162758
$850$	0	0
$851$	34.2257	1.17324
$852$	0	0
$853$	−37.5743	−1.28652	−0.643259	−	0.765648i	$-0.722418\pi$
$853$	−37.5743	−1.28652	−0.643259	+	0.765648i	$0.722418\pi$
$854$	0	0
$855$	−1.64663	−0.0563134
$856$	0	0
$857$	41.4208	1.41491	0.707453	−	0.706760i	$-0.249844\pi$
$857$	41.4208	1.41491	0.707453	+	0.706760i	$0.249844\pi$
$858$	0	0
$859$	−6.17985	−0.210854	−0.105427	−	0.994427i	$-0.533621\pi$
$859$	−6.17985	−0.210854	−0.105427	+	0.994427i	$0.533621\pi$
$860$	0	0
$861$	−2.68994	−0.0916728
$862$	0	0
$863$	−18.0410	−0.614124	−0.307062	−	0.951690i	$-0.599346\pi$
$863$	−18.0410	−0.614124	−0.307062	+	0.951690i	$0.599346\pi$
$864$	0	0
$865$	20.6617	0.702519
$866$	0	0
$867$	−6.88557	−0.233846
$868$	0	0
$869$	71.8572	2.43759
$870$	0	0
$871$	0	0
$872$	0	0
$873$	16.9220	0.572723
$874$	0	0
$875$	1.89771	0.0641544
$876$	0	0
$877$	15.3180	0.517251	0.258625	−	0.965978i	$-0.416730\pi$
$877$	15.3180	0.517251	0.258625	+	0.965978i	$0.416730\pi$
$878$	0	0
$879$	−1.22796	−0.0414181
$880$	0	0
$881$	38.9448	1.31208	0.656042	−	0.754725i	$-0.272230\pi$
$881$	38.9448	1.31208	0.656042	+	0.754725i	$0.272230\pi$
$882$	0	0
$883$	−53.4157	−1.79758	−0.898791	−	0.438376i	$-0.855554\pi$
$883$	−53.4157	−1.79758	−0.898791	+	0.438376i	$0.855554\pi$
$884$	0	0
$885$	2.17361	0.0730651
$886$	0	0
$887$	15.9682	0.536161	0.268080	−	0.963397i	$-0.413611\pi$
$887$	15.9682	0.536161	0.268080	+	0.963397i	$0.413611\pi$
$888$	0	0
$889$	−0.999529	−0.0335231
$890$	0	0
$891$	46.9539	1.57302
$892$	0	0
$893$	5.43444	0.181856
$894$	0	0
$895$	−19.5721	−0.654225
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−21.3054	−0.710575
$900$	0	0
$901$	68.4851	2.28157
$902$	0	0
$903$	0.999748	0.0332695
$904$	0	0
$905$	10.7679	0.357936
$906$	0	0
$907$	9.85604	0.327265	0.163632	−	0.986521i	$-0.447679\pi$
$907$	9.85604	0.327265	0.163632	+	0.986521i	$0.447679\pi$
$908$	0	0
$909$	−20.2224	−0.670735
$910$	0	0
$911$	46.1345	1.52850	0.764252	−	0.644918i	$-0.223109\pi$
$911$	46.1345	1.52850	0.764252	+	0.644918i	$0.223109\pi$
$912$	0	0
$913$	29.0535	0.961531
$914$	0	0
$915$	0.0893290	0.00295312
$916$	0	0
$917$	33.9208	1.12016
$918$	0	0
$919$	15.6260	0.515455	0.257727	−	0.966218i	$-0.417026\pi$
$919$	15.6260	0.515455	0.257727	+	0.966218i	$0.417026\pi$
$920$	0	0
$921$	0.477592	0.0157372
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−10.8657	−0.357263
$926$	0	0
$927$	−38.3855	−1.26075
$928$	0	0
$929$	−18.2052	−0.597294	−0.298647	−	0.954364i	$-0.596535\pi$
$929$	−18.2052	−0.597294	−0.298647	+	0.954364i	$0.596535\pi$
$930$	0	0
$931$	−1.88192	−0.0616773
$932$	0	0
$933$	0.915063	0.0299578
$934$	0	0
$935$	41.3254	1.35149
$936$	0	0
$937$	−41.4735	−1.35488	−0.677440	−	0.735578i	$-0.736911\pi$
$937$	−41.4735	−1.35488	−0.677440	+	0.735578i	$0.736911\pi$
$938$	0	0
$939$	−2.65891	−0.0867702
$940$	0	0
$941$	−25.1745	−0.820665	−0.410333	−	0.911936i	$-0.634587\pi$
$941$	−25.1745	−0.820665	−0.410333	+	0.911936i	$0.634587\pi$
$942$	0	0
$943$	−27.5609	−0.897507
$944$	0	0
$945$	1.83649	0.0597412
$946$	0	0
$947$	27.6232	0.897634	0.448817	−	0.893624i	$-0.351845\pi$
$947$	27.6232	0.897634	0.448817	+	0.893624i	$0.351845\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−4.10663	−0.133167
$952$	0	0
$953$	−35.7517	−1.15811	−0.579055	−	0.815289i	$-0.696578\pi$
$953$	−35.7517	−1.15811	−0.579055	+	0.815289i	$0.696578\pi$
$954$	0	0
$955$	−13.3596	−0.432307
$956$	0	0
$957$	3.07444	0.0993826
$958$	0	0
$959$	16.3002	0.526362
$960$	0	0
$961$	5.17095	0.166805
$962$	0	0
$963$	12.6078	0.406280
$964$	0	0
$965$	3.69764	0.119031
$966$	0	0
$967$	47.4240	1.52505	0.762527	−	0.646957i	$-0.223959\pi$
$967$	47.4240	1.52505	0.762527	+	0.646957i	$0.223959\pi$
$968$	0	0
$969$	−0.691949	−0.0222286
$970$	0	0
$971$	−19.7029	−0.632295	−0.316147	−	0.948710i	$-0.602389\pi$
$971$	−19.7029	−0.632295	−0.316147	+	0.948710i	$0.602389\pi$
$972$	0	0
$973$	5.09069	0.163200
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−0.0105702	−0.000338172 0	−0.000169086	−	1.00000i	$-0.500054\pi$
$977$	−0.0105702	−0.000338172 0	−0.000169086		1.00000i	$0.500054\pi$
$978$	0	0
$979$	−7.86213	−0.251275
$980$	0	0
$981$	−20.8189	−0.664697
$982$	0	0
$983$	−17.7240	−0.565306	−0.282653	−	0.959222i	$-0.591215\pi$
$983$	−17.7240	−0.565306	−0.282653	+	0.959222i	$0.591215\pi$
$984$	0	0
$985$	17.5498	0.559184
$986$	0	0
$987$	−3.01722	−0.0960392
$988$	0	0
$989$	10.2434	0.325720
$990$	0	0
$991$	6.89901	0.219154	0.109577	−	0.993978i	$-0.465050\pi$
$991$	6.89901	0.219154	0.109577	+	0.993978i	$0.465050\pi$
$992$	0	0
$993$	−2.73373	−0.0867523
$994$	0	0
$995$	19.2570	0.610489
$996$	0	0
$997$	−4.36107	−0.138117	−0.0690583	−	0.997613i	$-0.521999\pi$
$997$	−4.36107	−0.138117	−0.0690583	+	0.997613i	$0.521999\pi$
$998$	0	0
$999$	−10.5152	−0.332686

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.s.1.5	yes	9
13.5	odd	4		3380.2.f.j.3041.9		18
13.8	odd	4		3380.2.f.j.3041.10		18
13.12	even	2		3380.2.a.r.1.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.2.a.r.1.5	✓	9	13.12	even	2
3380.2.a.s.1.5	yes	9	1.1	even	1	trivial
3380.2.f.j.3041.9		18	13.5	odd	4
3380.2.f.j.3041.10		18	13.8	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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.a (trivial)

\(f(q)\)	\(=\)	\(q-0.161999 q^{3} +1.00000 q^{5} +1.89771 q^{7} -2.97376 q^{9} +O(q^{10})\)	\(q-0.161999 q^{3} +1.00000 q^{5} +1.89771 q^{7} -2.97376 q^{9} +5.35728 q^{11} -0.161999 q^{15} +7.71387 q^{17} +0.553719 q^{19} -0.307427 q^{21} -3.14988 q^{23} +1.00000 q^{25} +0.967741 q^{27} -3.54250 q^{29} +6.01423 q^{31} -0.867873 q^{33} +1.89771 q^{35} -10.8657 q^{37} +8.74983 q^{41} -3.25198 q^{43} -2.97376 q^{45} +9.81442 q^{47} -3.39868 q^{49} -1.24964 q^{51} +8.87818 q^{53} +5.35728 q^{55} -0.0897018 q^{57} -13.4174 q^{59} -0.551418 q^{61} -5.64334 q^{63} +4.46051 q^{67} +0.510276 q^{69} -8.17861 q^{71} -12.6129 q^{73} -0.161999 q^{75} +10.1666 q^{77} +13.4130 q^{79} +8.76450 q^{81} +5.42318 q^{83} +7.71387 q^{85} +0.573881 q^{87} -1.46756 q^{89} -0.974297 q^{93} +0.553719 q^{95} -5.69045 q^{97} -15.9313 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} + 9 q^{5} + q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q - q^3 + 9 * q^5 + q^7 + 12 * q^9	\( 9 q - q^{3} + 9 q^{5} + q^{7} + 12 q^{9} - 7 q^{11} - q^{15} + 13 q^{17} - 4 q^{19} - 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} + 13 q^{31} + 34 q^{33} + q^{35} + q^{37} - 6 q^{41} + q^{43} + 12 q^{45} - 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} + 38 q^{57} + 15 q^{59} + 21 q^{61} - 17 q^{63} - 7 q^{67} + 15 q^{69} - 7 q^{71} - 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} + 45 q^{83} + 13 q^{85} + 28 q^{87} - 41 q^{89} - 11 q^{93} - 4 q^{95} + 8 q^{97} - 81 q^{99}+O(q^{100}) \) 9 * q - q^3 + 9 * q^5 + q^7 + 12 * q^9 - 7 * q^11 - q^15 + 13 * q^17 - 4 * q^19 - 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 + 13 * q^31 + 34 * q^33 + q^35 + q^37 - 6 * q^41 + q^43 + 12 * q^45 - 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 + 38 * q^57 + 15 * q^59 + 21 * q^61 - 17 * q^63 - 7 * q^67 + 15 * q^69 - 7 * q^71 - 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 + 45 * q^83 + 13 * q^85 + 28 * q^87 - 41 * q^89 - 11 * q^93 - 4 * q^95 + 8 * q^97 - 81 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more