LMFDB - Embedded newform 3420.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,2,Mod(1,3420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3420, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3420.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3420.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:	$27.3088374913$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1140)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.51414$ of defining polynomial
Character	$\chi$	$=$	3420.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -1.32088 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -1.32088 q^{7} -3.70739 q^{11} +3.32088 q^{13} -1.61350 q^{17} -1.00000 q^{19} +7.67004 q^{23} +1.00000 q^{25} +1.70739 q^{29} +9.67004 q^{31} +1.32088 q^{35} +5.96265 q^{37} -2.29261 q^{41} -8.73566 q^{43} -11.6700 q^{47} -5.25526 q^{49} -10.4431 q^{53} +3.70739 q^{55} -12.6983 q^{59} +9.02827 q^{61} -3.32088 q^{65} -13.2835 q^{67} +8.69832 q^{71} -12.0565 q^{73} +4.89703 q^{77} -14.0565 q^{79} -5.02827 q^{83} +1.61350 q^{85} +1.70739 q^{89} -4.38650 q^{91} +1.00000 q^{95} -0.679116 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 4 q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 + 4 * q^7	$ 3 q - 3 q^{5} + 4 q^{7} - 6 q^{11} + 2 q^{13} - 2 q^{17} - 3 q^{19} - 6 q^{23} + 3 q^{25} - 4 q^{35} - 6 q^{37} - 12 q^{41} - 8 q^{43} - 6 q^{47} + 3 q^{49} - 8 q^{53} + 6 q^{55} + 4 q^{59} + 14 q^{61} - 2 q^{65} - 8 q^{67} - 16 q^{71} - 10 q^{73} - 20 q^{77} - 16 q^{79} - 2 q^{83} + 2 q^{85} - 16 q^{91} + 3 q^{95} - 10 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + 4 * q^7 - 6 * q^11 + 2 * q^13 - 2 * q^17 - 3 * q^19 - 6 * q^23 + 3 * q^25 - 4 * q^35 - 6 * q^37 - 12 * q^41 - 8 * q^43 - 6 * q^47 + 3 * q^49 - 8 * q^53 + 6 * q^55 + 4 * q^59 + 14 * q^61 - 2 * q^65 - 8 * q^67 - 16 * q^71 - 10 * q^73 - 20 * q^77 - 16 * q^79 - 2 * q^83 + 2 * q^85 - 16 * q^91 + 3 * q^95 - 10 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.32088	−0.499247	−0.249624	−	0.968343i	$-0.580307\pi$
$7$	−1.32088	−0.499247	−0.249624	+	0.968343i	$0.580307\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.70739	−1.11782	−0.558910	−	0.829228i	$-0.688780\pi$
$11$	−3.70739	−1.11782	−0.558910	+	0.829228i	$0.688780\pi$
$12$	0	0
$13$	3.32088	0.921048	0.460524	−	0.887647i	$-0.347662\pi$
$13$	3.32088	0.921048	0.460524	+	0.887647i	$0.347662\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.61350	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$17$	−1.61350	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.67004	1.59931	0.799657	−	0.600457i	$-0.205015\pi$
$23$	7.67004	1.59931	0.799657	+	0.600457i	$0.205015\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.70739	0.317054	0.158527	−	0.987355i	$-0.449325\pi$
$29$	1.70739	0.317054	0.158527	+	0.987355i	$0.449325\pi$
$30$	0	0
$31$	9.67004	1.73679	0.868395	−	0.495872i	$-0.165152\pi$
$31$	9.67004	1.73679	0.868395	+	0.495872i	$0.165152\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.32088	0.223270
$36$	0	0
$37$	5.96265	0.980254	0.490127	−	0.871651i	$-0.336950\pi$
$37$	5.96265	0.980254	0.490127	+	0.871651i	$0.336950\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.29261	−0.358046	−0.179023	−	0.983845i	$-0.557294\pi$
$41$	−2.29261	−0.358046	−0.179023	+	0.983845i	$0.557294\pi$
$42$	0	0
$43$	−8.73566	−1.33218	−0.666088	−	0.745873i	$-0.732033\pi$
$43$	−8.73566	−1.33218	−0.666088	+	0.745873i	$0.732033\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.6700	−1.70225	−0.851125	−	0.524963i	$-0.824079\pi$
$47$	−11.6700	−1.70225	−0.851125	+	0.524963i	$0.824079\pi$
$48$	0	0
$49$	−5.25526	−0.750752
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.4431	−1.43446	−0.717232	−	0.696835i	$-0.754591\pi$
$53$	−10.4431	−1.43446	−0.717232	+	0.696835i	$0.754591\pi$
$54$	0	0
$55$	3.70739	0.499904
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.6983	−1.65318	−0.826590	−	0.562805i	$-0.809722\pi$
$59$	−12.6983	−1.65318	−0.826590	+	0.562805i	$0.809722\pi$
$60$	0	0
$61$	9.02827	1.15595	0.577976	−	0.816054i	$-0.303843\pi$
$61$	9.02827	1.15595	0.577976	+	0.816054i	$0.303843\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.32088	−0.411905
$66$	0	0
$67$	−13.2835	−1.62284	−0.811421	−	0.584462i	$-0.801306\pi$
$67$	−13.2835	−1.62284	−0.811421	+	0.584462i	$0.801306\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.69832	1.03230	0.516150	−	0.856498i	$-0.327365\pi$
$71$	8.69832	1.03230	0.516150	+	0.856498i	$0.327365\pi$
$72$	0	0
$73$	−12.0565	−1.41111	−0.705556	−	0.708654i	$-0.749303\pi$
$73$	−12.0565	−1.41111	−0.705556	+	0.708654i	$0.749303\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.89703	0.558069
$78$	0	0
$79$	−14.0565	−1.58149	−0.790743	−	0.612149i	$-0.790305\pi$
$79$	−14.0565	−1.58149	−0.790743	+	0.612149i	$0.790305\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.02827	−0.551925	−0.275962	−	0.961168i	$-0.588997\pi$
$83$	−5.02827	−0.551925	−0.275962	+	0.961168i	$0.588997\pi$
$84$	0	0
$85$	1.61350	0.175008
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.70739	0.180983	0.0904915	−	0.995897i	$-0.471156\pi$
$89$	1.70739	0.180983	0.0904915	+	0.995897i	$0.471156\pi$
$90$	0	0
$91$	−4.38650	−0.459831
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−0.679116	−0.0689537	−0.0344769	−	0.999405i	$-0.510977\pi$
$97$	−0.679116	−0.0689537	−0.0344769	+	0.999405i	$0.510977\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.6418	−1.25790	−0.628952	−	0.777445i	$-0.716516\pi$
$101$	−12.6418	−1.25790	−0.628952	+	0.777445i	$0.716516\pi$
$102$	0	0
$103$	6.05655	0.596769	0.298385	−	0.954446i	$-0.403552\pi$
$103$	6.05655	0.596769	0.298385	+	0.954446i	$0.403552\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	10.6983	1.02471	0.512356	−	0.858773i	$-0.328773\pi$
$109$	10.6983	1.02471	0.512356	+	0.858773i	$0.328773\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.3118	1.53449	0.767243	−	0.641356i	$-0.221628\pi$
$113$	16.3118	1.53449	0.767243	+	0.641356i	$0.221628\pi$
$114$	0	0
$115$	−7.67004	−0.715235
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.13124	0.195371
$120$	0	0
$121$	2.74474	0.249521
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.29261	−0.375047	−0.187524	−	0.982260i	$-0.560046\pi$
$131$	−4.29261	−0.375047	−0.187524	+	0.982260i	$0.560046\pi$
$132$	0	0
$133$	1.32088	0.114535
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	−8.58522	−0.728189	−0.364094	−	0.931362i	$-0.618621\pi$
$139$	−8.58522	−0.728189	−0.364094	+	0.931362i	$0.618621\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−12.3118	−1.02957
$144$	0	0
$145$	−1.70739	−0.141791
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.0565	−0.987711	−0.493855	−	0.869544i	$-0.664413\pi$
$149$	−12.0565	−0.987711	−0.493855	+	0.869544i	$0.664413\pi$
$150$	0	0
$151$	−22.9536	−1.86794	−0.933968	−	0.357357i	$-0.883678\pi$
$151$	−22.9536	−1.86794	−0.933968	+	0.357357i	$0.883678\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.67004	−0.776717
$156$	0	0
$157$	−11.8688	−0.947230	−0.473615	−	0.880732i	$-0.657051\pi$
$157$	−11.8688	−0.947230	−0.473615	+	0.880732i	$0.657051\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−10.1312	−0.798454
$162$	0	0
$163$	−0.0373465	−0.00292520	−0.00146260	−	0.999999i	$-0.500466\pi$
$163$	−0.0373465	−0.00292520	−0.00146260	+	0.999999i	$0.500466\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.0283	0.853393	0.426697	−	0.904395i	$-0.359677\pi$
$167$	11.0283	0.853393	0.426697	+	0.904395i	$0.359677\pi$
$168$	0	0
$169$	−1.97173	−0.151671
$170$	0	0
$171$	0	0
$172$	0	0
$173$	17.8578	1.35771	0.678853	−	0.734274i	$-0.262477\pi$
$173$	17.8578	1.35771	0.678853	+	0.734274i	$0.262477\pi$
$174$	0	0
$175$	−1.32088	−0.0998495
$176$	0	0
$177$	0	0
$178$	0	0
$179$	19.9253	1.48929	0.744644	−	0.667462i	$-0.232619\pi$
$179$	19.9253	1.48929	0.744644	+	0.667462i	$0.232619\pi$
$180$	0	0
$181$	−15.4713	−1.14997	−0.574987	−	0.818162i	$-0.694993\pi$
$181$	−15.4713	−1.14997	−0.574987	+	0.818162i	$0.694993\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.96265	−0.438383
$186$	0	0
$187$	5.98185	0.437437
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.3492	−1.32770	−0.663849	−	0.747866i	$-0.731078\pi$
$191$	−18.3492	−1.32770	−0.663849	+	0.747866i	$0.731078\pi$
$192$	0	0
$193$	12.0192	0.865161	0.432581	−	0.901595i	$-0.357603\pi$
$193$	12.0192	0.865161	0.432581	+	0.901595i	$0.357603\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.84049	−0.344870	−0.172435	−	0.985021i	$-0.555164\pi$
$197$	−4.84049	−0.344870	−0.172435	+	0.985021i	$0.555164\pi$
$198$	0	0
$199$	−20.6983	−1.46726	−0.733632	−	0.679547i	$-0.762177\pi$
$199$	−20.6983	−1.46726	−0.733632	+	0.679547i	$0.762177\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.25526	−0.158289
$204$	0	0
$205$	2.29261	0.160123
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.70739	0.256445
$210$	0	0
$211$	2.05655	0.141579	0.0707893	−	0.997491i	$-0.477448\pi$
$211$	2.05655	0.141579	0.0707893	+	0.997491i	$0.477448\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.73566	0.595767
$216$	0	0
$217$	−12.7730	−0.867088
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.35823	−0.360434
$222$	0	0
$223$	6.05655	0.405576	0.202788	−	0.979223i	$-0.435000\pi$
$223$	6.05655	0.405576	0.202788	+	0.979223i	$0.435000\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	28.3118	1.87912	0.939560	−	0.342383i	$-0.111234\pi$
$227$	28.3118	1.87912	0.939560	+	0.342383i	$0.111234\pi$
$228$	0	0
$229$	−1.80128	−0.119032	−0.0595161	−	0.998227i	$-0.518956\pi$
$229$	−1.80128	−0.119032	−0.0595161	+	0.998227i	$0.518956\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.94345	0.258344	0.129172	−	0.991622i	$-0.458768\pi$
$233$	3.94345	0.258344	0.129172	+	0.991622i	$0.458768\pi$
$234$	0	0
$235$	11.6700	0.761270
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.8031	1.08690	0.543452	−	0.839440i	$-0.317117\pi$
$239$	16.8031	1.08690	0.543452	+	0.839440i	$0.317117\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5.25526	0.335747
$246$	0	0
$247$	−3.32088	−0.211303
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.9909	−1.57741	−0.788707	−	0.614770i	$-0.789249\pi$
$251$	−24.9909	−1.57741	−0.788707	+	0.614770i	$0.789249\pi$
$252$	0	0
$253$	−28.4358	−1.78775
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.91518	0.181844	0.0909219	−	0.995858i	$-0.471019\pi$
$257$	2.91518	0.181844	0.0909219	+	0.995858i	$0.471019\pi$
$258$	0	0
$259$	−7.87598	−0.489389
$260$	0	0
$261$	0	0
$262$	0	0
$263$	22.3118	1.37581	0.687903	−	0.725803i	$-0.258532\pi$
$263$	22.3118	1.37581	0.687903	+	0.725803i	$0.258532\pi$
$264$	0	0
$265$	10.4431	0.641512
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−15.0656	−0.918567	−0.459284	−	0.888290i	$-0.651894\pi$
$269$	−15.0656	−0.918567	−0.459284	+	0.888290i	$0.651894\pi$
$270$	0	0
$271$	5.47133	0.332359	0.166180	−	0.986095i	$-0.446857\pi$
$271$	5.47133	0.332359	0.166180	+	0.986095i	$0.446857\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.70739	−0.223564
$276$	0	0
$277$	−10.7730	−0.647287	−0.323644	−	0.946179i	$-0.604908\pi$
$277$	−10.7730	−0.647287	−0.323644	+	0.946179i	$0.604908\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−17.8205	−1.06308	−0.531540	−	0.847033i	$-0.678387\pi$
$281$	−17.8205	−1.06308	−0.531540	+	0.847033i	$0.678387\pi$
$282$	0	0
$283$	−30.2070	−1.79562	−0.897810	−	0.440384i	$-0.854842\pi$
$283$	−30.2070	−1.79562	−0.897810	+	0.440384i	$0.854842\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.02827	0.178753
$288$	0	0
$289$	−14.3966	−0.846861
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.2553	1.06648	0.533242	−	0.845963i	$-0.320974\pi$
$293$	18.2553	1.06648	0.533242	+	0.845963i	$0.320974\pi$
$294$	0	0
$295$	12.6983	0.739325
$296$	0	0
$297$	0	0
$298$	0	0
$299$	25.4713	1.47304
$300$	0	0
$301$	11.5388	0.665085
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.02827	−0.516957
$306$	0	0
$307$	−16.5852	−0.946569	−0.473284	−	0.880910i	$-0.656932\pi$
$307$	−16.5852	−0.946569	−0.473284	+	0.880910i	$0.656932\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−11.5196	−0.653217	−0.326608	−	0.945160i	$-0.605906\pi$
$311$	−11.5196	−0.653217	−0.326608	+	0.945160i	$0.605906\pi$
$312$	0	0
$313$	32.7549	1.85141	0.925707	−	0.378241i	$-0.123471\pi$
$313$	32.7549	1.85141	0.925707	+	0.378241i	$0.123471\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.2161	−0.854619	−0.427310	−	0.904105i	$-0.640539\pi$
$317$	−15.2161	−0.854619	−0.427310	+	0.904105i	$0.640539\pi$
$318$	0	0
$319$	−6.32996	−0.354410
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.61350	0.0897773
$324$	0	0
$325$	3.32088	0.184210
$326$	0	0
$327$	0	0
$328$	0	0
$329$	15.4148	0.849844
$330$	0	0
$331$	−26.9536	−1.48150	−0.740751	−	0.671779i	$-0.765530\pi$
$331$	−26.9536	−1.48150	−0.740751	+	0.671779i	$0.765530\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	13.2835	0.725757
$336$	0	0
$337$	−17.5652	−0.956839	−0.478419	−	0.878132i	$-0.658790\pi$
$337$	−17.5652	−0.956839	−0.478419	+	0.878132i	$0.658790\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−35.8506	−1.94142
$342$	0	0
$343$	16.1878	0.874058
$344$	0	0
$345$	0	0
$346$	0	0
$347$	34.3118	1.84195	0.920977	−	0.389616i	$-0.127392\pi$
$347$	34.3118	1.84195	0.920977	+	0.389616i	$0.127392\pi$
$348$	0	0
$349$	35.2835	1.88868	0.944342	−	0.328965i	$-0.106700\pi$
$349$	35.2835	1.88868	0.944342	+	0.328965i	$0.106700\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.71646	0.463930	0.231965	−	0.972724i	$-0.425484\pi$
$353$	8.71646	0.463930	0.231965	+	0.972724i	$0.425484\pi$
$354$	0	0
$355$	−8.69832	−0.461659
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.4623	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$359$	−30.4623	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.0565	0.631069
$366$	0	0
$367$	−27.5652	−1.43889	−0.719446	−	0.694548i	$-0.755604\pi$
$367$	−27.5652	−1.43889	−0.719446	+	0.694548i	$0.755604\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	13.7941	0.716152
$372$	0	0
$373$	16.0939	0.833310	0.416655	−	0.909065i	$-0.363202\pi$
$373$	16.0939	0.833310	0.416655	+	0.909065i	$0.363202\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.67004	0.292022
$378$	0	0
$379$	9.01013	0.462819	0.231410	−	0.972856i	$-0.425666\pi$
$379$	9.01013	0.462819	0.231410	+	0.972856i	$0.425666\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	23.3401	1.19262	0.596311	−	0.802753i	$-0.296632\pi$
$383$	23.3401	1.19262	0.596311	+	0.802753i	$0.296632\pi$
$384$	0	0
$385$	−4.89703	−0.249576
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−20.0565	−1.01691	−0.508454	−	0.861089i	$-0.669783\pi$
$389$	−20.0565	−1.01691	−0.508454	+	0.861089i	$0.669783\pi$
$390$	0	0
$391$	−12.3756	−0.625860
$392$	0	0
$393$	0	0
$394$	0	0
$395$	14.0565	0.707262
$396$	0	0
$397$	−21.7375	−1.09097	−0.545487	−	0.838119i	$-0.683655\pi$
$397$	−21.7375	−1.09097	−0.545487	+	0.838119i	$0.683655\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.17872	0.358488	0.179244	−	0.983805i	$-0.442635\pi$
$401$	7.17872	0.358488	0.179244	+	0.983805i	$0.442635\pi$
$402$	0	0
$403$	32.1131	1.59967
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−22.1059	−1.09575
$408$	0	0
$409$	16.2443	0.803231	0.401615	−	0.915808i	$-0.368449\pi$
$409$	16.2443	0.803231	0.401615	+	0.915808i	$0.368449\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	16.7730	0.825346
$414$	0	0
$415$	5.02827	0.246828
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−16.9909	−0.830061	−0.415031	−	0.909807i	$-0.636229\pi$
$419$	−16.9909	−0.830061	−0.415031	+	0.909807i	$0.636229\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.61350	−0.0782660
$426$	0	0
$427$	−11.9253	−0.577106
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.6418	−1.09062	−0.545308	−	0.838236i	$-0.683587\pi$
$431$	−22.6418	−1.09062	−0.545308	+	0.838236i	$0.683587\pi$
$432$	0	0
$433$	−12.2070	−0.586630	−0.293315	−	0.956016i	$-0.594759\pi$
$433$	−12.2070	−0.586630	−0.293315	+	0.956016i	$0.594759\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.67004	−0.366908
$438$	0	0
$439$	−31.8506	−1.52015	−0.760073	−	0.649837i	$-0.774837\pi$
$439$	−31.8506	−1.52015	−0.760073	+	0.649837i	$0.774837\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.95358	−0.425397	−0.212699	−	0.977118i	$-0.568225\pi$
$443$	−8.95358	−0.425397	−0.212699	+	0.977118i	$0.568225\pi$
$444$	0	0
$445$	−1.70739	−0.0809380
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.51960	−0.260486	−0.130243	−	0.991482i	$-0.541576\pi$
$449$	−5.51960	−0.260486	−0.130243	+	0.991482i	$0.541576\pi$
$450$	0	0
$451$	8.49960	0.400231
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.38650	0.205643
$456$	0	0
$457$	0.131241	0.00613918	0.00306959	−	0.999995i	$-0.499023\pi$
$457$	0.131241	0.00613918	0.00306959	+	0.999995i	$0.499023\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14.1131	0.657312	0.328656	−	0.944450i	$-0.393404\pi$
$461$	14.1131	0.657312	0.328656	+	0.944450i	$0.393404\pi$
$462$	0	0
$463$	5.98080	0.277951	0.138976	−	0.990296i	$-0.455619\pi$
$463$	5.98080	0.277951	0.138976	+	0.990296i	$0.455619\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.38650	0.295532	0.147766	−	0.989022i	$-0.452792\pi$
$467$	6.38650	0.295532	0.147766	+	0.989022i	$0.452792\pi$
$468$	0	0
$469$	17.5460	0.810200
$470$	0	0
$471$	0	0
$472$	0	0
$473$	32.3865	1.48913
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−18.8597	−0.861721	−0.430861	−	0.902419i	$-0.641790\pi$
$479$	−18.8597	−0.861721	−0.430861	+	0.902419i	$0.641790\pi$
$480$	0	0
$481$	19.8013	0.902861
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.679116	0.0308370
$486$	0	0
$487$	10.1312	0.459090	0.229545	−	0.973298i	$-0.426276\pi$
$487$	10.1312	0.459090	0.229545	+	0.973298i	$0.426276\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.06562	−0.0480908	−0.0240454	−	0.999711i	$-0.507655\pi$
$491$	−1.06562	−0.0480908	−0.0240454	+	0.999711i	$0.507655\pi$
$492$	0	0
$493$	−2.75486	−0.124073
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−11.4895	−0.515373
$498$	0	0
$499$	−14.2443	−0.637664	−0.318832	−	0.947811i	$-0.603291\pi$
$499$	−14.2443	−0.637664	−0.318832	+	0.947811i	$0.603291\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−30.9717	−1.38096	−0.690481	−	0.723351i	$-0.742601\pi$
$503$	−30.9717	−1.38096	−0.690481	+	0.723351i	$0.742601\pi$
$504$	0	0
$505$	12.6418	0.562551
$506$	0	0
$507$	0	0
$508$	0	0
$509$	30.4057	1.34771	0.673855	−	0.738864i	$-0.264637\pi$
$509$	30.4057	1.34771	0.673855	+	0.738864i	$0.264637\pi$
$510$	0	0
$511$	15.9253	0.704494
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.05655	−0.266883
$516$	0	0
$517$	43.2654	1.90281
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.34916	0.190540	0.0952700	−	0.995451i	$-0.469629\pi$
$521$	4.34916	0.190540	0.0952700	+	0.995451i	$0.469629\pi$
$522$	0	0
$523$	−8.69832	−0.380351	−0.190175	−	0.981750i	$-0.560906\pi$
$523$	−8.69832	−0.380351	−0.190175	+	0.981750i	$0.560906\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.6026	−0.679659
$528$	0	0
$529$	35.8296	1.55781
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7.61350	−0.329777
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	19.4833	0.839206
$540$	0	0
$541$	8.45398	0.363465	0.181733	−	0.983348i	$-0.441830\pi$
$541$	8.45398	0.363465	0.181733	+	0.983348i	$0.441830\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10.6983	−0.458266
$546$	0	0
$547$	11.5279	0.492896	0.246448	−	0.969156i	$-0.420736\pi$
$547$	11.5279	0.492896	0.246448	+	0.969156i	$0.420736\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.70739	−0.0727372
$552$	0	0
$553$	18.5671	0.789552
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.22699	−0.0519892	−0.0259946	−	0.999662i	$-0.508275\pi$
$557$	−1.22699	−0.0519892	−0.0259946	+	0.999662i	$0.508275\pi$
$558$	0	0
$559$	−29.0101	−1.22700
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−19.9144	−0.839291	−0.419646	−	0.907688i	$-0.637846\pi$
$563$	−19.9144	−0.839291	−0.419646	+	0.907688i	$0.637846\pi$
$564$	0	0
$565$	−16.3118	−0.686243
$566$	0	0
$567$	0	0
$568$	0	0
$569$	27.5015	1.15292	0.576460	−	0.817125i	$-0.304433\pi$
$569$	27.5015	1.15292	0.576460	+	0.817125i	$0.304433\pi$
$570$	0	0
$571$	−9.47133	−0.396363	−0.198181	−	0.980165i	$-0.563503\pi$
$571$	−9.47133	−0.396363	−0.198181	+	0.980165i	$0.563503\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.67004	0.319863
$576$	0	0
$577$	36.6418	1.52542	0.762708	−	0.646743i	$-0.223869\pi$
$577$	36.6418	1.52542	0.762708	+	0.646743i	$0.223869\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.64177	0.275547
$582$	0	0
$583$	38.7165	1.60347
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.67004	0.151479	0.0757394	−	0.997128i	$-0.475868\pi$
$587$	3.67004	0.151479	0.0757394	+	0.997128i	$0.475868\pi$
$588$	0	0
$589$	−9.67004	−0.398447
$590$	0	0
$591$	0	0
$592$	0	0
$593$	17.2270	0.707428	0.353714	−	0.935354i	$-0.384919\pi$
$593$	17.2270	0.707428	0.353714	+	0.935354i	$0.384919\pi$
$594$	0	0
$595$	−2.13124	−0.0873724
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.11310	0.331492	0.165746	−	0.986168i	$-0.446997\pi$
$599$	8.11310	0.331492	0.165746	+	0.986168i	$0.446997\pi$
$600$	0	0
$601$	7.35823	0.300149	0.150074	−	0.988675i	$-0.452049\pi$
$601$	7.35823	0.300149	0.150074	+	0.988675i	$0.452049\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.74474	−0.111589
$606$	0	0
$607$	19.9253	0.808743	0.404372	−	0.914595i	$-0.367490\pi$
$607$	19.9253	0.808743	0.404372	+	0.914595i	$0.367490\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−38.7549	−1.56785
$612$	0	0
$613$	39.3219	1.58820	0.794099	−	0.607788i	$-0.207943\pi$
$613$	39.3219	1.58820	0.794099	+	0.607788i	$0.207943\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−0.0674757	−0.00271647	−0.00135823	−	0.999999i	$-0.500432\pi$
$617$	−0.0674757	−0.00271647	−0.00135823	+	0.999999i	$0.500432\pi$
$618$	0	0
$619$	38.8680	1.56224	0.781118	−	0.624384i	$-0.214650\pi$
$619$	38.8680	1.56224	0.781118	+	0.624384i	$0.214650\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.25526	−0.0903552
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−9.62071	−0.383603
$630$	0	0
$631$	29.2835	1.16576	0.582880	−	0.812559i	$-0.301926\pi$
$631$	29.2835	1.16576	0.582880	+	0.812559i	$0.301926\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.00000	0.158735
$636$	0	0
$637$	−17.4521	−0.691478
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−10.4057	−0.411001	−0.205500	−	0.978657i	$-0.565882\pi$
$641$	−10.4057	−0.411001	−0.205500	+	0.978657i	$0.565882\pi$
$642$	0	0
$643$	−11.3774	−0.448682	−0.224341	−	0.974511i	$-0.572023\pi$
$643$	−11.3774	−0.448682	−0.224341	+	0.974511i	$0.572023\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−45.5388	−1.79032	−0.895158	−	0.445750i	$-0.852937\pi$
$647$	−45.5388	−1.79032	−0.895158	+	0.445750i	$0.852937\pi$
$648$	0	0
$649$	47.0776	1.84796
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.27341	0.0889654	0.0444827	−	0.999010i	$-0.485836\pi$
$653$	2.27341	0.0889654	0.0444827	+	0.999010i	$0.485836\pi$
$654$	0	0
$655$	4.29261	0.167726
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8.77301	0.341748	0.170874	−	0.985293i	$-0.445341\pi$
$659$	8.77301	0.341748	0.170874	+	0.985293i	$0.445341\pi$
$660$	0	0
$661$	−44.4924	−1.73055	−0.865277	−	0.501295i	$-0.832857\pi$
$661$	−44.4924	−1.73055	−0.865277	+	0.501295i	$0.832857\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.32088	−0.0512217
$666$	0	0
$667$	13.0957	0.507069
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−33.4713	−1.29215
$672$	0	0
$673$	−5.96265	−0.229843	−0.114922	−	0.993375i	$-0.536662\pi$
$673$	−5.96265	−0.229843	−0.114922	+	0.993375i	$0.536662\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.32996	−0.243280	−0.121640	−	0.992574i	$-0.538815\pi$
$677$	−6.32996	−0.243280	−0.121640	+	0.992574i	$0.538815\pi$
$678$	0	0
$679$	0.897033	0.0344250
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−39.8506	−1.52484	−0.762421	−	0.647082i	$-0.775989\pi$
$683$	−39.8506	−1.52484	−0.762421	+	0.647082i	$0.775989\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−34.6802	−1.32121
$690$	0	0
$691$	8.07469	0.307176	0.153588	−	0.988135i	$-0.450917\pi$
$691$	8.07469	0.307176	0.153588	+	0.988135i	$0.450917\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.58522	0.325656
$696$	0	0
$697$	3.69912	0.140114
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.77301	0.104735	0.0523676	−	0.998628i	$-0.483323\pi$
$701$	2.77301	0.104735	0.0523676	+	0.998628i	$0.483323\pi$
$702$	0	0
$703$	−5.96265	−0.224886
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16.6983	0.628005
$708$	0	0
$709$	10.1987	0.383021	0.191510	−	0.981491i	$-0.438661\pi$
$709$	10.1987	0.383021	0.191510	+	0.981491i	$0.438661\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	74.1696	2.77767
$714$	0	0
$715$	12.3118	0.460436
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−34.2745	−1.27822	−0.639111	−	0.769115i	$-0.720698\pi$
$719$	−34.2745	−1.27822	−0.639111	+	0.769115i	$0.720698\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.70739	0.0634108
$726$	0	0
$727$	22.0939	0.819417	0.409709	−	0.912216i	$-0.365630\pi$
$727$	22.0939	0.819417	0.409709	+	0.912216i	$0.365630\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	14.0950	0.521321
$732$	0	0
$733$	−13.3401	−0.492727	−0.246364	−	0.969177i	$-0.579236\pi$
$733$	−13.3401	−0.492727	−0.246364	+	0.969177i	$0.579236\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	49.2472	1.81405
$738$	0	0
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−32.8861	−1.20647	−0.603237	−	0.797562i	$-0.706123\pi$
$743$	−32.8861	−1.20647	−0.603237	+	0.797562i	$0.706123\pi$
$744$	0	0
$745$	12.0565	0.441718
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	16.4996	0.602079	0.301039	−	0.953612i	$-0.402666\pi$
$751$	16.4996	0.602079	0.301039	+	0.953612i	$0.402666\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	22.9536	0.835366
$756$	0	0
$757$	−42.5489	−1.54647	−0.773234	−	0.634121i	$-0.781362\pi$
$757$	−42.5489	−1.54647	−0.773234	+	0.634121i	$0.781362\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5.34009	−0.193578	−0.0967890	−	0.995305i	$-0.530857\pi$
$761$	−5.34009	−0.193578	−0.0967890	+	0.995305i	$0.530857\pi$
$762$	0	0
$763$	−14.1312	−0.511585
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−42.1696	−1.52266
$768$	0	0
$769$	−7.28354	−0.262651	−0.131326	−	0.991339i	$-0.541923\pi$
$769$	−7.28354	−0.262651	−0.131326	+	0.991339i	$0.541923\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18.1806	0.653910	0.326955	−	0.945040i	$-0.393977\pi$
$773$	18.1806	0.653910	0.326955	+	0.945040i	$0.393977\pi$
$774$	0	0
$775$	9.67004	0.347358
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.29261	0.0821413
$780$	0	0
$781$	−32.2480	−1.15393
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11.8688	0.423614
$786$	0	0
$787$	20.8114	0.741847	0.370923	−	0.928663i	$-0.379041\pi$
$787$	20.8114	0.741847	0.370923	+	0.928663i	$0.379041\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−21.5460	−0.766088
$792$	0	0
$793$	29.9819	1.06469
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−40.9354	−1.45001	−0.725004	−	0.688745i	$-0.758162\pi$
$797$	−40.9354	−1.45001	−0.725004	+	0.688745i	$0.758162\pi$
$798$	0	0
$799$	18.8296	0.666142
$800$	0	0
$801$	0	0
$802$	0	0
$803$	44.6983	1.57737
$804$	0	0
$805$	10.1312	0.357079
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.48947	0.0523670	0.0261835	−	0.999657i	$-0.491665\pi$
$809$	1.48947	0.0523670	0.0261835	+	0.999657i	$0.491665\pi$
$810$	0	0
$811$	21.5460	0.756583	0.378292	−	0.925687i	$-0.376512\pi$
$811$	21.5460	0.756583	0.378292	+	0.925687i	$0.376512\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0.0373465	0.00130819
$816$	0	0
$817$	8.73566	0.305622
$818$	0	0
$819$	0	0
$820$	0	0
$821$	35.9819	1.25578	0.627888	−	0.778304i	$-0.283920\pi$
$821$	35.9819	1.25578	0.627888	+	0.778304i	$0.283920\pi$
$822$	0	0
$823$	15.8880	0.553819	0.276910	−	0.960896i	$-0.410690\pi$
$823$	15.8880	0.553819	0.276910	+	0.960896i	$0.410690\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−26.8789	−0.934671	−0.467335	−	0.884080i	$-0.654786\pi$
$827$	−26.8789	−0.934671	−0.467335	+	0.884080i	$0.654786\pi$
$828$	0	0
$829$	17.9253	0.622572	0.311286	−	0.950316i	$-0.399240\pi$
$829$	17.9253	0.622572	0.311286	+	0.950316i	$0.399240\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	8.47934	0.293792
$834$	0	0
$835$	−11.0283	−0.381649
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.30168	−0.252082	−0.126041	−	0.992025i	$-0.540227\pi$
$839$	−7.30168	−0.252082	−0.126041	+	0.992025i	$0.540227\pi$
$840$	0	0
$841$	−26.0848	−0.899477
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.97173	0.0678294
$846$	0	0
$847$	−3.62548	−0.124573
$848$	0	0
$849$	0	0
$850$	0	0
$851$	45.7338	1.56773
$852$	0	0
$853$	−37.4148	−1.28106	−0.640529	−	0.767934i	$-0.721285\pi$
$853$	−37.4148	−1.28106	−0.640529	+	0.767934i	$0.721285\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.1806	0.347762	0.173881	−	0.984767i	$-0.444369\pi$
$857$	10.1806	0.347762	0.173881	+	0.984767i	$0.444369\pi$
$858$	0	0
$859$	−40.7367	−1.38992	−0.694959	−	0.719049i	$-0.744578\pi$
$859$	−40.7367	−1.38992	−0.694959	+	0.719049i	$0.744578\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8.11310	0.276173	0.138086	−	0.990420i	$-0.455905\pi$
$863$	8.11310	0.276173	0.138086	+	0.990420i	$0.455905\pi$
$864$	0	0
$865$	−17.8578	−0.607184
$866$	0	0
$867$	0	0
$868$	0	0
$869$	52.1131	1.76782
$870$	0	0
$871$	−44.1131	−1.49472
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.32088	0.0446540
$876$	0	0
$877$	29.8880	1.00924	0.504622	−	0.863340i	$-0.331632\pi$
$877$	29.8880	1.00924	0.504622	+	0.863340i	$0.331632\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−34.8114	−1.17283	−0.586413	−	0.810012i	$-0.699460\pi$
$881$	−34.8114	−1.17283	−0.586413	+	0.810012i	$0.699460\pi$
$882$	0	0
$883$	27.8496	0.937212	0.468606	−	0.883407i	$-0.344756\pi$
$883$	27.8496	0.937212	0.468606	+	0.883407i	$0.344756\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	50.7933	1.70547	0.852735	−	0.522343i	$-0.174942\pi$
$887$	50.7933	1.70547	0.852735	+	0.522343i	$0.174942\pi$
$888$	0	0
$889$	5.28354	0.177204
$890$	0	0
$891$	0	0
$892$	0	0
$893$	11.6700	0.390523
$894$	0	0
$895$	−19.9253	−0.666030
$896$	0	0
$897$	0	0
$898$	0	0
$899$	16.5105	0.550657
$900$	0	0
$901$	16.8498	0.561349
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15.4713	0.514284
$906$	0	0
$907$	−32.3009	−1.07253	−0.536267	−	0.844049i	$-0.680166\pi$
$907$	−32.3009	−1.07253	−0.536267	+	0.844049i	$0.680166\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	52.7367	1.74725	0.873623	−	0.486604i	$-0.161764\pi$
$911$	52.7367	1.74725	0.873623	+	0.486604i	$0.161764\pi$
$912$	0	0
$913$	18.6418	0.616953
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.67004	0.187241
$918$	0	0
$919$	52.5105	1.73216	0.866081	−	0.499903i	$-0.166631\pi$
$919$	52.5105	1.73216	0.866081	+	0.499903i	$0.166631\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	28.8861	0.950798
$924$	0	0
$925$	5.96265	0.196051
$926$	0	0
$927$	0	0
$928$	0	0
$929$	35.8688	1.17682	0.588408	−	0.808564i	$-0.299755\pi$
$929$	35.8688	1.17682	0.588408	+	0.808564i	$0.299755\pi$
$930$	0	0
$931$	5.25526	0.172234
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.98185	−0.195628
$936$	0	0
$937$	10.6983	0.349499	0.174749	−	0.984613i	$-0.444088\pi$
$937$	10.6983	0.349499	0.174749	+	0.984613i	$0.444088\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	22.3673	0.729153	0.364577	−	0.931173i	$-0.381214\pi$
$941$	22.3673	0.729153	0.364577	+	0.931173i	$0.381214\pi$
$942$	0	0
$943$	−17.5844	−0.572628
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−34.8223	−1.13157	−0.565787	−	0.824551i	$-0.691428\pi$
$947$	−34.8223	−1.13157	−0.565787	+	0.824551i	$0.691428\pi$
$948$	0	0
$949$	−40.0384	−1.29970
$950$	0	0
$951$	0	0
$952$	0	0
$953$	29.4823	0.955024	0.477512	−	0.878625i	$-0.341539\pi$
$953$	29.4823	0.955024	0.477512	+	0.878625i	$0.341539\pi$
$954$	0	0
$955$	18.3492	0.593765
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.64177	0.0853072
$960$	0	0
$961$	62.5097	2.01644
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−12.0192	−0.386912
$966$	0	0
$967$	18.8669	0.606719	0.303359	−	0.952876i	$-0.401892\pi$
$967$	18.8669	0.606719	0.303359	+	0.952876i	$0.401892\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−31.2270	−1.00212	−0.501061	−	0.865412i	$-0.667057\pi$
$971$	−31.2270	−1.00212	−0.501061	+	0.865412i	$0.667057\pi$
$972$	0	0
$973$	11.3401	0.363546
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6.44305	−0.206132	−0.103066	−	0.994675i	$-0.532865\pi$
$977$	−6.44305	−0.206132	−0.103066	+	0.994675i	$0.532865\pi$
$978$	0	0
$979$	−6.32996	−0.202306
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.57429	−0.145897	−0.0729486	−	0.997336i	$-0.523241\pi$
$983$	−4.57429	−0.145897	−0.0729486	+	0.997336i	$0.523241\pi$
$984$	0	0
$985$	4.84049	0.154231
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−67.0029	−2.13057
$990$	0	0
$991$	−38.8296	−1.23346	−0.616731	−	0.787174i	$-0.711543\pi$
$991$	−38.8296	−1.23346	−0.616731	+	0.787174i	$0.711543\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	20.6983	0.656181
$996$	0	0
$997$	54.9245	1.73948	0.869738	−	0.493513i	$-0.164288\pi$
$997$	54.9245	1.73948	0.869738	+	0.493513i	$0.164288\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.a.m.1.1		3
3.2	odd	2		1140.2.a.g.1.1	✓	3
12.11	even	2		4560.2.a.br.1.3		3
15.2	even	4		5700.2.f.q.3649.1		6
15.8	even	4		5700.2.f.q.3649.6		6
15.14	odd	2		5700.2.a.w.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.a.g.1.1	✓	3	3.2	odd	2
3420.2.a.m.1.1		3	1.1	even	1	trivial
4560.2.a.br.1.3		3	12.11	even	2
5700.2.a.w.1.3		3	15.14	odd	2
5700.2.f.q.3649.1		6	15.2	even	4
5700.2.f.q.3649.6		6	15.8	even	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:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3420.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -1.32088 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -1.32088 q^{7} -3.70739 q^{11} +3.32088 q^{13} -1.61350 q^{17} -1.00000 q^{19} +7.67004 q^{23} +1.00000 q^{25} +1.70739 q^{29} +9.67004 q^{31} +1.32088 q^{35} +5.96265 q^{37} -2.29261 q^{41} -8.73566 q^{43} -11.6700 q^{47} -5.25526 q^{49} -10.4431 q^{53} +3.70739 q^{55} -12.6983 q^{59} +9.02827 q^{61} -3.32088 q^{65} -13.2835 q^{67} +8.69832 q^{71} -12.0565 q^{73} +4.89703 q^{77} -14.0565 q^{79} -5.02827 q^{83} +1.61350 q^{85} +1.70739 q^{89} -4.38650 q^{91} +1.00000 q^{95} -0.679116 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 4 q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 + 4 * q^7	\( 3 q - 3 q^{5} + 4 q^{7} - 6 q^{11} + 2 q^{13} - 2 q^{17} - 3 q^{19} - 6 q^{23} + 3 q^{25} - 4 q^{35} - 6 q^{37} - 12 q^{41} - 8 q^{43} - 6 q^{47} + 3 q^{49} - 8 q^{53} + 6 q^{55} + 4 q^{59} + 14 q^{61} - 2 q^{65} - 8 q^{67} - 16 q^{71} - 10 q^{73} - 20 q^{77} - 16 q^{79} - 2 q^{83} + 2 q^{85} - 16 q^{91} + 3 q^{95} - 10 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + 4 * q^7 - 6 * q^11 + 2 * q^13 - 2 * q^17 - 3 * q^19 - 6 * q^23 + 3 * q^25 - 4 * q^35 - 6 * q^37 - 12 * q^41 - 8 * q^43 - 6 * q^47 + 3 * q^49 - 8 * q^53 + 6 * q^55 + 4 * q^59 + 14 * q^61 - 2 * q^65 - 8 * q^67 - 16 * q^71 - 10 * q^73 - 20 * q^77 - 16 * q^79 - 2 * q^83 + 2 * q^85 - 16 * q^91 + 3 * q^95 - 10 * q^97

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