LMFDB - Embedded newform 3420.2.a.m.1.2

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			1.2
Root			$-2.08613$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} +0.648061 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +0.648061 q^{7} +3.52420 q^{11} +1.35194 q^{13} -6.87614 q^{17} -1.00000 q^{19} -5.46838 q^{23} +1.00000 q^{25} -5.52420 q^{29} -3.46838 q^{31} -0.648061 q^{35} +0.0558176 q^{37} -9.52420 q^{41} +7.69646 q^{43} +1.46838 q^{47} -6.58002 q^{49} +13.2207 q^{53} -3.52420 q^{55} +9.64064 q^{59} -0.172260 q^{61} -1.35194 q^{65} -5.40776 q^{67} -13.6406 q^{71} +6.34452 q^{73} +2.28390 q^{77} +4.34452 q^{79} +4.17226 q^{83} +6.87614 q^{85} -5.52420 q^{89} +0.876139 q^{91} +1.00000 q^{95} -2.64806 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	0.648061	0.244944	0.122472	−	0.992472i	$-0.460918\pi$
$7$	0.648061	0.244944	0.122472	+	0.992472i	$0.460918\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.52420	1.06259	0.531293	−	0.847188i	$-0.321706\pi$
$11$	3.52420	1.06259	0.531293	+	0.847188i	$0.321706\pi$
$12$	0	0
$13$	1.35194	0.374960	0.187480	−	0.982268i	$-0.439968\pi$
$13$	1.35194	0.374960	0.187480	+	0.982268i	$0.439968\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.87614	−1.66771	−0.833854	−	0.551985i	$-0.813871\pi$
$17$	−6.87614	−1.66771	−0.833854	+	0.551985i	$0.813871\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.46838	−1.14024	−0.570118	−	0.821563i	$-0.693103\pi$
$23$	−5.46838	−1.14024	−0.570118	+	0.821563i	$0.693103\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.52420	−1.02582	−0.512909	−	0.858443i	$-0.671432\pi$
$29$	−5.52420	−1.02582	−0.512909	+	0.858443i	$0.671432\pi$
$30$	0	0
$31$	−3.46838	−0.622940	−0.311470	−	0.950256i	$-0.600821\pi$
$31$	−3.46838	−0.622940	−0.311470	+	0.950256i	$0.600821\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.648061	−0.109542
$36$	0	0
$37$	0.0558176	0.00917636	0.00458818	−	0.999989i	$-0.498540\pi$
$37$	0.0558176	0.00917636	0.00458818	+	0.999989i	$0.498540\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.52420	−1.48743	−0.743715	−	0.668497i	$-0.766938\pi$
$41$	−9.52420	−1.48743	−0.743715	+	0.668497i	$0.766938\pi$
$42$	0	0
$43$	7.69646	1.17370	0.586850	−	0.809696i	$-0.300368\pi$
$43$	7.69646	1.17370	0.586850	+	0.809696i	$0.300368\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.46838	0.214186	0.107093	−	0.994249i	$-0.465846\pi$
$47$	1.46838	0.214186	0.107093	+	0.994249i	$0.465846\pi$
$48$	0	0
$49$	−6.58002	−0.940002
$50$	0	0
$51$	0	0
$52$	0	0
$53$	13.2207	1.81600	0.907999	−	0.418973i	$-0.137610\pi$
$53$	13.2207	1.81600	0.907999	+	0.418973i	$0.137610\pi$
$54$	0	0
$55$	−3.52420	−0.475203
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.64064	1.25510	0.627552	−	0.778574i	$-0.284057\pi$
$59$	9.64064	1.25510	0.627552	+	0.778574i	$0.284057\pi$
$60$	0	0
$61$	−0.172260	−0.0220557	−0.0110278	−	0.999939i	$-0.503510\pi$
$61$	−0.172260	−0.0220557	−0.0110278	+	0.999939i	$0.503510\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.35194	−0.167687
$66$	0	0
$67$	−5.40776	−0.660663	−0.330331	−	0.943865i	$-0.607160\pi$
$67$	−5.40776	−0.660663	−0.330331	+	0.943865i	$0.607160\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.6406	−1.61885	−0.809423	−	0.587226i	$-0.800220\pi$
$71$	−13.6406	−1.61885	−0.809423	+	0.587226i	$0.800220\pi$
$72$	0	0
$73$	6.34452	0.742570	0.371285	−	0.928519i	$-0.378917\pi$
$73$	6.34452	0.742570	0.371285	+	0.928519i	$0.378917\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.28390	0.260274
$78$	0	0
$79$	4.34452	0.488797	0.244398	−	0.969675i	$-0.421410\pi$
$79$	4.34452	0.488797	0.244398	+	0.969675i	$0.421410\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.17226	0.457965	0.228983	−	0.973431i	$-0.426460\pi$
$83$	4.17226	0.457965	0.228983	+	0.973431i	$0.426460\pi$
$84$	0	0
$85$	6.87614	0.745822
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.52420	−0.585564	−0.292782	−	0.956179i	$-0.594581\pi$
$89$	−5.52420	−0.585564	−0.292782	+	0.956179i	$0.594581\pi$
$90$	0	0
$91$	0.876139	0.0918443
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−2.64806	−0.268870	−0.134435	−	0.990922i	$-0.542922\pi$
$97$	−2.64806	−0.268870	−0.134435	+	0.990922i	$0.542922\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−8.70388	−0.866068	−0.433034	−	0.901378i	$-0.642557\pi$
$101$	−8.70388	−0.866068	−0.433034	+	0.901378i	$0.642557\pi$
$102$	0	0
$103$	−12.3445	−1.21634	−0.608171	−	0.793806i	$-0.708096\pi$
$103$	−12.3445	−1.21634	−0.608171	+	0.793806i	$0.708096\pi$
$104$	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
$109$	−11.6406	−1.11497	−0.557486	−	0.830187i	$-0.688234\pi$
$109$	−11.6406	−1.11497	−0.557486	+	0.830187i	$0.688234\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.764504	−0.0719184	−0.0359592	−	0.999353i	$-0.511449\pi$
$113$	−0.764504	−0.0719184	−0.0359592	+	0.999353i	$0.511449\pi$
$114$	0	0
$115$	5.46838	0.509929
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.45616	−0.408495
$120$	0	0
$121$	1.41998	0.129089
$122$	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
$131$	−11.5242	−1.00687	−0.503437	−	0.864032i	$-0.667931\pi$
$131$	−11.5242	−1.00687	−0.503437	+	0.864032i	$0.667931\pi$
$132$	0	0
$133$	−0.648061	−0.0561940
$134$	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
$139$	−23.0484	−1.95494	−0.977470	−	0.211075i	$-0.932304\pi$
$139$	−23.0484	−1.95494	−0.977470	+	0.211075i	$0.932304\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.76450	0.398428
$144$	0	0
$145$	5.52420	0.458760
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.34452	0.519763	0.259882	−	0.965640i	$-0.416316\pi$
$149$	6.34452	0.519763	0.259882	+	0.965640i	$0.416316\pi$
$150$	0	0
$151$	−1.93937	−0.157824	−0.0789120	−	0.996882i	$-0.525145\pi$
$151$	−1.93937	−0.157824	−0.0789120	+	0.996882i	$0.525145\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.46838	0.278587
$156$	0	0
$157$	−18.4562	−1.47296	−0.736481	−	0.676458i	$-0.763514\pi$
$157$	−18.4562	−1.47296	−0.736481	+	0.676458i	$0.763514\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.54384	−0.279294
$162$	0	0
$163$	−5.94418	−0.465584	−0.232792	−	0.972527i	$-0.574786\pi$
$163$	−5.94418	−0.465584	−0.232792	+	0.972527i	$0.574786\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.82774	0.141435	0.0707174	−	0.997496i	$-0.477471\pi$
$167$	1.82774	0.141435	0.0707174	+	0.997496i	$0.477471\pi$
$168$	0	0
$169$	−11.1723	−0.859405
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.2691	−1.54103	−0.770514	−	0.637423i	$-0.780000\pi$
$173$	−20.2691	−1.54103	−0.770514	+	0.637423i	$0.780000\pi$
$174$	0	0
$175$	0.648061	0.0489888
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.11164	0.606292	0.303146	−	0.952944i	$-0.401963\pi$
$179$	8.11164	0.606292	0.303146	+	0.952944i	$0.401963\pi$
$180$	0	0
$181$	17.3929	1.29281	0.646403	−	0.762996i	$-0.276273\pi$
$181$	17.3929	1.29281	0.646403	+	0.762996i	$0.276273\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.0558176	−0.00410379
$186$	0	0
$187$	−24.2329	−1.77208
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.17968	−0.519503	−0.259752	−	0.965675i	$-0.583641\pi$
$191$	−7.17968	−0.519503	−0.259752	+	0.965675i	$0.583641\pi$
$192$	0	0
$193$	−12.2887	−0.884560	−0.442280	−	0.896877i	$-0.645830\pi$
$193$	−12.2887	−0.884560	−0.442280	+	0.896877i	$0.645830\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.6284	−1.46971	−0.734857	−	0.678222i	$-0.762751\pi$
$197$	−20.6284	−1.46971	−0.734857	+	0.678222i	$0.762751\pi$
$198$	0	0
$199$	1.64064	0.116302	0.0581510	−	0.998308i	$-0.481480\pi$
$199$	1.64064	0.116302	0.0581510	+	0.998308i	$0.481480\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.58002	−0.251268
$204$	0	0
$205$	9.52420	0.665199
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.52420	−0.243774
$210$	0	0
$211$	−16.3445	−1.12520	−0.562602	−	0.826728i	$-0.690199\pi$
$211$	−16.3445	−1.12520	−0.562602	+	0.826728i	$0.690199\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7.69646	−0.524894
$216$	0	0
$217$	−2.24772	−0.152585
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9.29612	−0.625325
$222$	0	0
$223$	−12.3445	−0.826650	−0.413325	−	0.910584i	$-0.635633\pi$
$223$	−12.3445	−0.826650	−0.413325	+	0.910584i	$0.635633\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.2355	0.745726	0.372863	−	0.927886i	$-0.378376\pi$
$227$	11.2355	0.745726	0.372863	+	0.927886i	$0.378376\pi$
$228$	0	0
$229$	17.9245	1.18449	0.592243	−	0.805759i	$-0.298242\pi$
$229$	17.9245	1.18449	0.592243	+	0.805759i	$0.298242\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.3445	1.46384	0.731919	−	0.681392i	$-0.238625\pi$
$233$	22.3445	1.46384	0.731919	+	0.681392i	$0.238625\pi$
$234$	0	0
$235$	−1.46838	−0.0957867
$236$	0	0
$237$	0	0
$238$	0	0
$239$	26.6842	1.72606	0.863030	−	0.505153i	$-0.168564\pi$
$239$	26.6842	1.72606	0.863030	+	0.505153i	$0.168564\pi$
$240$	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
$245$	6.58002	0.420382
$246$	0	0
$247$	−1.35194	−0.0860218
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.88356	−0.623845	−0.311922	−	0.950108i	$-0.600973\pi$
$251$	−9.88356	−0.623845	−0.311922	+	0.950108i	$0.600973\pi$
$252$	0	0
$253$	−19.2717	−1.21160
$254$	0	0
$255$	0	0
$256$	0	0
$257$	30.5168	1.90358	0.951792	−	0.306743i	$-0.0992393\pi$
$257$	30.5168	1.90358	0.951792	+	0.306743i	$0.0992393\pi$
$258$	0	0
$259$	0.0361732	0.00224769
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.23550	0.322835	0.161417	−	0.986886i	$-0.448394\pi$
$263$	5.23550	0.322835	0.161417	+	0.986886i	$0.448394\pi$
$264$	0	0
$265$	−13.2207	−0.812139
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−11.7719	−0.717747	−0.358873	−	0.933386i	$-0.616839\pi$
$269$	−11.7719	−0.717747	−0.358873	+	0.933386i	$0.616839\pi$
$270$	0	0
$271$	−27.3929	−1.66400	−0.832001	−	0.554775i	$-0.812804\pi$
$271$	−27.3929	−1.66400	−0.832001	+	0.554775i	$0.812804\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.52420	0.212517
$276$	0	0
$277$	−0.247722	−0.0148842	−0.00744210	−	0.999972i	$-0.502369\pi$
$277$	−0.247722	−0.0148842	−0.00744210	+	0.999972i	$0.502369\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.2132	1.56375	0.781875	−	0.623435i	$-0.214263\pi$
$281$	26.2132	1.56375	0.781875	+	0.623435i	$0.214263\pi$
$282$	0	0
$283$	19.0894	1.13475	0.567373	−	0.823461i	$-0.307960\pi$
$283$	19.0894	1.13475	0.567373	+	0.823461i	$0.307960\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.17226	−0.364337
$288$	0	0
$289$	30.2813	1.78125
$290$	0	0
$291$	0	0
$292$	0	0
$293$	19.5800	1.14388	0.571938	−	0.820297i	$-0.306192\pi$
$293$	19.5800	1.14388	0.571938	+	0.820297i	$0.306192\pi$
$294$	0	0
$295$	−9.64064	−0.561300
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.39292	−0.427544
$300$	0	0
$301$	4.98777	0.287491
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.172260	0.00986360
$306$	0	0
$307$	−31.0484	−1.77203	−0.886013	−	0.463661i	$-0.846536\pi$
$307$	−31.0484	−1.77203	−0.886013	+	0.463661i	$0.846536\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−29.2765	−1.66012	−0.830058	−	0.557677i	$-0.811693\pi$
$311$	−29.2765	−1.66012	−0.830058	+	0.557677i	$0.811693\pi$
$312$	0	0
$313$	−7.98516	−0.451348	−0.225674	−	0.974203i	$-0.572458\pi$
$313$	−7.98516	−0.451348	−0.225674	+	0.974203i	$0.572458\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.9729	1.06563	0.532813	−	0.846233i	$-0.321135\pi$
$317$	18.9729	1.06563	0.532813	+	0.846233i	$0.321135\pi$
$318$	0	0
$319$	−19.4684	−1.09002
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.87614	0.382599
$324$	0	0
$325$	1.35194	0.0749921
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.951601	0.0524635
$330$	0	0
$331$	−5.93937	−0.326458	−0.163229	−	0.986588i	$-0.552191\pi$
$331$	−5.93937	−0.326458	−0.163229	+	0.986588i	$0.552191\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.40776	0.295457
$336$	0	0
$337$	27.7933	1.51400	0.756998	−	0.653418i	$-0.226665\pi$
$337$	27.7933	1.51400	0.756998	+	0.653418i	$0.226665\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−12.2233	−0.661927
$342$	0	0
$343$	−8.80068	−0.475192
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.2355	0.925250	0.462625	−	0.886554i	$-0.346908\pi$
$347$	17.2355	0.925250	0.462625	+	0.886554i	$0.346908\pi$
$348$	0	0
$349$	27.4078	1.46710	0.733552	−	0.679634i	$-0.237861\pi$
$349$	27.4078	1.46710	0.733552	+	0.679634i	$0.237861\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	16.5922	0.883116	0.441558	−	0.897233i	$-0.354426\pi$
$353$	16.5922	0.883116	0.441558	+	0.897233i	$0.354426\pi$
$354$	0	0
$355$	13.6406	0.723970
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.5094	0.924109	0.462054	−	0.886852i	$-0.347112\pi$
$359$	17.5094	0.924109	0.462054	+	0.886852i	$0.347112\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.34452	−0.332087
$366$	0	0
$367$	17.7933	0.928801	0.464400	−	0.885625i	$-0.346270\pi$
$367$	17.7933	0.928801	0.464400	+	0.885625i	$0.346270\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.56779	0.444818
$372$	0	0
$373$	3.59966	0.186383	0.0931917	−	0.995648i	$-0.470293\pi$
$373$	3.59966	0.186383	0.0931917	+	0.995648i	$0.470293\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.46838	−0.384641
$378$	0	0
$379$	−30.4051	−1.56181	−0.780904	−	0.624651i	$-0.785241\pi$
$379$	−30.4051	−1.56181	−0.780904	+	0.624651i	$0.785241\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.93676	−0.150062	−0.0750308	−	0.997181i	$-0.523906\pi$
$383$	−2.93676	−0.150062	−0.0750308	+	0.997181i	$0.523906\pi$
$384$	0	0
$385$	−2.28390	−0.116398
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.65548	−0.0839361	−0.0419681	−	0.999119i	$-0.513363\pi$
$389$	−1.65548	−0.0839361	−0.0419681	+	0.999119i	$0.513363\pi$
$390$	0	0
$391$	37.6014	1.90158
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.34452	−0.218597
$396$	0	0
$397$	−34.9123	−1.75220	−0.876099	−	0.482131i	$-0.839863\pi$
$397$	−34.9123	−1.75220	−0.876099	+	0.482131i	$0.839863\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−32.9171	−1.64380	−0.821901	−	0.569630i	$-0.807087\pi$
$401$	−32.9171	−1.64380	−0.821901	+	0.569630i	$0.807087\pi$
$402$	0	0
$403$	−4.68904	−0.233578
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.196712	0.00975067
$408$	0	0
$409$	−27.1452	−1.34224	−0.671122	−	0.741347i	$-0.734187\pi$
$409$	−27.1452	−1.34224	−0.671122	+	0.741347i	$0.734187\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.24772	0.307430
$414$	0	0
$415$	−4.17226	−0.204808
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.88356	−0.0920178	−0.0460089	−	0.998941i	$-0.514650\pi$
$419$	−1.88356	−0.0920178	−0.0460089	+	0.998941i	$0.514650\pi$
$420$	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
$425$	−6.87614	−0.333542
$426$	0	0
$427$	−0.111635	−0.00540241
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−18.7039	−0.900934	−0.450467	−	0.892793i	$-0.648743\pi$
$431$	−18.7039	−0.900934	−0.450467	+	0.892793i	$0.648743\pi$
$432$	0	0
$433$	37.0894	1.78240	0.891201	−	0.453609i	$-0.149864\pi$
$433$	37.0894	1.78240	0.891201	+	0.453609i	$0.149864\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.46838	0.261588
$438$	0	0
$439$	−8.22327	−0.392475	−0.196238	−	0.980556i	$-0.562872\pi$
$439$	−8.22327	−0.392475	−0.196238	+	0.980556i	$0.562872\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.0606	0.573018	0.286509	−	0.958078i	$-0.407505\pi$
$443$	12.0606	0.573018	0.286509	+	0.958078i	$0.407505\pi$
$444$	0	0
$445$	5.52420	0.261872
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−23.2765	−1.09848	−0.549242	−	0.835663i	$-0.685084\pi$
$449$	−23.2765	−1.09848	−0.549242	+	0.835663i	$0.685084\pi$
$450$	0	0
$451$	−33.5652	−1.58052
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.876139	−0.0410740
$456$	0	0
$457$	−6.45616	−0.302006	−0.151003	−	0.988533i	$-0.548250\pi$
$457$	−6.45616	−0.302006	−0.151003	+	0.988533i	$0.548250\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−22.6890	−1.05673	−0.528367	−	0.849016i	$-0.677195\pi$
$461$	−22.6890	−1.05673	−0.528367	+	0.849016i	$0.677195\pi$
$462$	0	0
$463$	30.2887	1.40764	0.703818	−	0.710381i	$-0.251477\pi$
$463$	30.2887	1.40764	0.703818	+	0.710381i	$0.251477\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.12386	0.0520061	0.0260030	−	0.999662i	$-0.491722\pi$
$467$	1.12386	0.0520061	0.0260030	+	0.999662i	$0.491722\pi$
$468$	0	0
$469$	−3.50456	−0.161825
$470$	0	0
$471$	0	0
$472$	0	0
$473$	27.1239	1.24716
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10.3397	−0.472434	−0.236217	−	0.971700i	$-0.575908\pi$
$479$	−10.3397	−0.472434	−0.236217	+	0.971700i	$0.575908\pi$
$480$	0	0
$481$	0.0754620	0.00344077
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.64806	0.120242
$486$	0	0
$487$	3.54384	0.160587	0.0802935	−	0.996771i	$-0.474414\pi$
$487$	3.54384	0.160587	0.0802935	+	0.996771i	$0.474414\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.22808	0.100552	0.0502759	−	0.998735i	$-0.483990\pi$
$491$	2.22808	0.100552	0.0502759	+	0.998735i	$0.483990\pi$
$492$	0	0
$493$	37.9852	1.71077
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.83997	−0.396527
$498$	0	0
$499$	29.1452	1.30472	0.652359	−	0.757910i	$-0.273779\pi$
$499$	29.1452	1.30472	0.652359	+	0.757910i	$0.273779\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−40.1723	−1.79119	−0.895596	−	0.444868i	$-0.853251\pi$
$503$	−40.1723	−1.79119	−0.895596	+	0.444868i	$0.853251\pi$
$504$	0	0
$505$	8.70388	0.387318
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0.835158	0.0370177	0.0185089	−	0.999829i	$-0.494108\pi$
$509$	0.835158	0.0370177	0.0185089	+	0.999829i	$0.494108\pi$
$510$	0	0
$511$	4.11164	0.181888
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12.3445	0.543965
$516$	0	0
$517$	5.17487	0.227591
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.82032	−0.298804	−0.149402	−	0.988777i	$-0.547735\pi$
$521$	−6.82032	−0.298804	−0.149402	+	0.988777i	$0.547735\pi$
$522$	0	0
$523$	13.6406	0.596464	0.298232	−	0.954493i	$-0.403603\pi$
$523$	13.6406	0.596464	0.298232	+	0.954493i	$0.403603\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	23.8491	1.03888
$528$	0	0
$529$	6.90320	0.300139
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12.8761	−0.557727
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−23.1893	−0.998834
$540$	0	0
$541$	29.5046	1.26850	0.634250	−	0.773128i	$-0.281309\pi$
$541$	29.5046	1.26850	0.634250	+	0.773128i	$0.281309\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	11.6406	0.498630
$546$	0	0
$547$	−39.7374	−1.69905	−0.849525	−	0.527548i	$-0.823111\pi$
$547$	−39.7374	−1.69905	−0.849525	+	0.527548i	$0.823111\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.52420	0.235339
$552$	0	0
$553$	2.81551	0.119728
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−11.7523	−0.497960	−0.248980	−	0.968509i	$-0.580095\pi$
$557$	−11.7523	−0.497960	−0.248980	+	0.968509i	$0.580095\pi$
$558$	0	0
$559$	10.4051	0.440091
$560$	0	0
$561$	0	0
$562$	0	0
$563$	36.6136	1.54308	0.771539	−	0.636182i	$-0.219487\pi$
$563$	36.6136	1.54308	0.771539	+	0.636182i	$0.219487\pi$
$564$	0	0
$565$	0.764504	0.0321629
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15.0436	0.630660	0.315330	−	0.948982i	$-0.397885\pi$
$569$	15.0436	0.630660	0.315330	+	0.948982i	$0.397885\pi$
$570$	0	0
$571$	23.3929	0.978963	0.489482	−	0.872014i	$-0.337186\pi$
$571$	23.3929	0.978963	0.489482	+	0.872014i	$0.337186\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.46838	−0.228047
$576$	0	0
$577$	32.7039	1.36148	0.680740	−	0.732525i	$-0.261658\pi$
$577$	32.7039	1.36148	0.680740	+	0.732525i	$0.261658\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.70388	0.112176
$582$	0	0
$583$	46.5922	1.92965
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9.46838	−0.390802	−0.195401	−	0.980723i	$-0.562601\pi$
$587$	−9.46838	−0.390802	−0.195401	+	0.980723i	$0.562601\pi$
$588$	0	0
$589$	3.46838	0.142912
$590$	0	0
$591$	0	0
$592$	0	0
$593$	27.7523	1.13965	0.569825	−	0.821766i	$-0.307011\pi$
$593$	27.7523	1.13965	0.569825	+	0.821766i	$0.307011\pi$
$594$	0	0
$595$	4.45616	0.182685
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−28.6890	−1.17220	−0.586101	−	0.810238i	$-0.699338\pi$
$599$	−28.6890	−1.17220	−0.586101	+	0.810238i	$0.699338\pi$
$600$	0	0
$601$	11.2961	0.460778	0.230389	−	0.973099i	$-0.426000\pi$
$601$	11.2961	0.460778	0.230389	+	0.973099i	$0.426000\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.41998	−0.0577305
$606$	0	0
$607$	8.11164	0.329241	0.164621	−	0.986357i	$-0.447360\pi$
$607$	8.11164	0.329241	0.164621	+	0.986357i	$0.447360\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.98516	0.0803111
$612$	0	0
$613$	−17.1696	−0.693476	−0.346738	−	0.937962i	$-0.612711\pi$
$613$	−17.1696	−0.693476	−0.346738	+	0.937962i	$0.612711\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−26.3807	−1.06205	−0.531023	−	0.847357i	$-0.678192\pi$
$617$	−26.3807	−1.06205	−0.531023	+	0.847357i	$0.678192\pi$
$618$	0	0
$619$	−38.6742	−1.55445	−0.777224	−	0.629224i	$-0.783373\pi$
$619$	−38.6742	−1.55445	−0.777224	+	0.629224i	$0.783373\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.58002	−0.143430
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−0.383810	−0.0153035
$630$	0	0
$631$	21.4078	0.852229	0.426115	−	0.904669i	$-0.359882\pi$
$631$	21.4078	0.852229	0.426115	+	0.904669i	$0.359882\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.00000	0.158735
$636$	0	0
$637$	−8.89578	−0.352464
$638$	0	0
$639$	0	0
$640$	0	0
$641$	19.1648	0.756966	0.378483	−	0.925608i	$-0.376446\pi$
$641$	19.1648	0.756966	0.378483	+	0.925608i	$0.376446\pi$
$642$	0	0
$643$	8.99258	0.354633	0.177316	−	0.984154i	$-0.443258\pi$
$643$	8.99258	0.354633	0.177316	+	0.984154i	$0.443258\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−38.9878	−1.53277	−0.766384	−	0.642383i	$-0.777946\pi$
$647$	−38.9878	−1.53277	−0.766384	+	0.642383i	$0.777946\pi$
$648$	0	0
$649$	33.9755	1.33366
$650$	0	0
$651$	0	0
$652$	0	0
$653$	33.8129	1.32320	0.661601	−	0.749856i	$-0.269877\pi$
$653$	33.8129	1.32320	0.661601	+	0.749856i	$0.269877\pi$
$654$	0	0
$655$	11.5242	0.450288
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.75228	−0.0682590	−0.0341295	−	0.999417i	$-0.510866\pi$
$659$	−1.75228	−0.0682590	−0.0341295	+	0.999417i	$0.510866\pi$
$660$	0	0
$661$	−16.9271	−0.658390	−0.329195	−	0.944262i	$-0.606777\pi$
$661$	−16.9271	−0.658390	−0.329195	+	0.944262i	$0.606777\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.648061	0.0251307
$666$	0	0
$667$	30.2084	1.16968
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.607080	−0.0234361
$672$	0	0
$673$	−0.0558176	−0.00215161	−0.00107581	−	0.999999i	$-0.500342\pi$
$673$	−0.0558176	−0.00215161	−0.00107581	+	0.999999i	$0.500342\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−19.4684	−0.748231	−0.374115	−	0.927382i	$-0.622054\pi$
$677$	−19.4684	−0.748231	−0.374115	+	0.927382i	$0.622054\pi$
$678$	0	0
$679$	−1.71610	−0.0658580
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−16.2233	−0.620766	−0.310383	−	0.950612i	$-0.600457\pi$
$683$	−16.2233	−0.620766	−0.310383	+	0.950612i	$0.600457\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	0	0
$688$	0	0
$689$	17.8735	0.680927
$690$	0	0
$691$	19.8884	0.756589	0.378295	−	0.925685i	$-0.376511\pi$
$691$	19.8884	0.756589	0.378295	+	0.925685i	$0.376511\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	23.0484	0.874276
$696$	0	0
$697$	65.4897	2.48060
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−7.75228	−0.292799	−0.146400	−	0.989226i	$-0.546769\pi$
$701$	−7.75228	−0.292799	−0.146400	+	0.989226i	$0.546769\pi$
$702$	0	0
$703$	−0.0558176	−0.00210520
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.64064	−0.212138
$708$	0	0
$709$	29.9245	1.12384	0.561920	−	0.827192i	$-0.310063\pi$
$709$	29.9245	1.12384	0.561920	+	0.827192i	$0.310063\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	18.9664	0.710299
$714$	0	0
$715$	−4.76450	−0.178182
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−11.2913	−0.421095	−0.210547	−	0.977584i	$-0.567525\pi$
$719$	−11.2913	−0.421095	−0.210547	+	0.977584i	$0.567525\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5.52420	−0.205164
$726$	0	0
$727$	9.59966	0.356032	0.178016	−	0.984028i	$-0.443032\pi$
$727$	9.59966	0.356032	0.178016	+	0.984028i	$0.443032\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−52.9219	−1.95739
$732$	0	0
$733$	12.9368	0.477830	0.238915	−	0.971040i	$-0.423208\pi$
$733$	12.9368	0.477830	0.238915	+	0.971040i	$0.423208\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−19.0580	−0.702011
$738$	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
$743$	14.4413	0.529801	0.264900	−	0.964276i	$-0.414661\pi$
$743$	14.4413	0.529801	0.264900	+	0.964276i	$0.414661\pi$
$744$	0	0
$745$	−6.34452	−0.232445
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−25.5652	−0.932887	−0.466443	−	0.884551i	$-0.654465\pi$
$751$	−25.5652	−0.932887	−0.466443	+	0.884551i	$0.654465\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.93937	0.0705811
$756$	0	0
$757$	3.41737	0.124206	0.0621032	−	0.998070i	$-0.480219\pi$
$757$	3.41737	0.124206	0.0621032	+	0.998070i	$0.480219\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	20.9368	0.758957	0.379479	−	0.925201i	$-0.376103\pi$
$761$	20.9368	0.758957	0.379479	+	0.925201i	$0.376103\pi$
$762$	0	0
$763$	−7.54384	−0.273105
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.0336	0.470615
$768$	0	0
$769$	0.592243	0.0213568	0.0106784	−	0.999943i	$-0.496601\pi$
$769$	0.592243	0.0213568	0.0106784	+	0.999943i	$0.496601\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.69165	0.276650	0.138325	−	0.990387i	$-0.455828\pi$
$773$	7.69165	0.276650	0.138325	+	0.990387i	$0.455828\pi$
$774$	0	0
$775$	−3.46838	−0.124588
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.52420	0.341240
$780$	0	0
$781$	−48.0723	−1.72016
$782$	0	0
$783$	0	0
$784$	0	0
$785$	18.4562	0.658728
$786$	0	0
$787$	−38.3297	−1.36631	−0.683153	−	0.730275i	$-0.739392\pi$
$787$	−38.3297	−1.36631	−0.683153	+	0.730275i	$0.739392\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−0.495445	−0.0176160
$792$	0	0
$793$	−0.232886	−0.00827001
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10.2935	0.364615	0.182307	−	0.983242i	$-0.441643\pi$
$797$	10.2935	0.364615	0.182307	+	0.983242i	$0.441643\pi$
$798$	0	0
$799$	−10.0968	−0.357199
$800$	0	0
$801$	0	0
$802$	0	0
$803$	22.3594	0.789045
$804$	0	0
$805$	3.54384	0.124904
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.16003	−0.0407846	−0.0203923	−	0.999792i	$-0.506492\pi$
$809$	−1.16003	−0.0407846	−0.0203923	+	0.999792i	$0.506492\pi$
$810$	0	0
$811$	0.495445	0.0173974	0.00869871	−	0.999962i	$-0.497231\pi$
$811$	0.495445	0.0173974	0.00869871	+	0.999962i	$0.497231\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.94418	0.208216
$816$	0	0
$817$	−7.69646	−0.269265
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.76711	0.201274	0.100637	−	0.994923i	$-0.467912\pi$
$821$	5.76711	0.201274	0.100637	+	0.994923i	$0.467912\pi$
$822$	0	0
$823$	−1.83255	−0.0638786	−0.0319393	−	0.999490i	$-0.510168\pi$
$823$	−1.83255	−0.0638786	−0.0319393	+	0.999490i	$0.510168\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.94899	0.206867	0.103433	−	0.994636i	$-0.467017\pi$
$827$	5.94899	0.206867	0.103433	+	0.994636i	$0.467017\pi$
$828$	0	0
$829$	6.11164	0.212266	0.106133	−	0.994352i	$-0.466153\pi$
$829$	6.11164	0.212266	0.106133	+	0.994352i	$0.466153\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	45.2451	1.56765
$834$	0	0
$835$	−1.82774	−0.0632515
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−29.6406	−1.02331	−0.511654	−	0.859191i	$-0.670967\pi$
$839$	−29.6406	−1.02331	−0.511654	+	0.859191i	$0.670967\pi$
$840$	0	0
$841$	1.51678	0.0523028
$842$	0	0
$843$	0	0
$844$	0	0
$845$	11.1723	0.384337
$846$	0	0
$847$	0.920235	0.0316197
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−0.305232	−0.0104632
$852$	0	0
$853$	−22.9516	−0.785848	−0.392924	−	0.919571i	$-0.628536\pi$
$853$	−22.9516	−0.785848	−0.392924	+	0.919571i	$0.628536\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−0.308348	−0.0105330	−0.00526648	−	0.999986i	$-0.501676\pi$
$857$	−0.308348	−0.0105330	−0.00526648	+	0.999986i	$0.501676\pi$
$858$	0	0
$859$	30.2180	1.03103	0.515513	−	0.856882i	$-0.327601\pi$
$859$	30.2180	1.03103	0.515513	+	0.856882i	$0.327601\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−28.6890	−0.976586	−0.488293	−	0.872680i	$-0.662380\pi$
$863$	−28.6890	−0.976586	−0.488293	+	0.872680i	$0.662380\pi$
$864$	0	0
$865$	20.2691	0.689169
$866$	0	0
$867$	0	0
$868$	0	0
$869$	15.3110	0.519389
$870$	0	0
$871$	−7.31096	−0.247722
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.648061	−0.0219085
$876$	0	0
$877$	12.1675	0.410866	0.205433	−	0.978671i	$-0.434140\pi$
$877$	12.1675	0.410866	0.205433	+	0.978671i	$0.434140\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	24.3297	0.819688	0.409844	−	0.912156i	$-0.365583\pi$
$881$	24.3297	0.819688	0.409844	+	0.912156i	$0.365583\pi$
$882$	0	0
$883$	58.7449	1.97692	0.988461	−	0.151476i	$-0.0484026\pi$
$883$	58.7449	1.97692	0.988461	+	0.151476i	$0.0484026\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38.5626	−1.29480	−0.647402	−	0.762149i	$-0.724145\pi$
$887$	−38.5626	−1.29480	−0.647402	+	0.762149i	$0.724145\pi$
$888$	0	0
$889$	−2.59224	−0.0869410
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.46838	−0.0491375
$894$	0	0
$895$	−8.11164	−0.271142
$896$	0	0
$897$	0	0
$898$	0	0
$899$	19.1600	0.639023
$900$	0	0
$901$	−90.9071	−3.02855
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17.3929	−0.578160
$906$	0	0
$907$	29.4897	0.979190	0.489595	−	0.871950i	$-0.337145\pi$
$907$	29.4897	0.979190	0.489595	+	0.871950i	$0.337145\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−18.2180	−0.603591	−0.301795	−	0.953373i	$-0.597586\pi$
$911$	−18.2180	−0.603591	−0.301795	+	0.953373i	$0.597586\pi$
$912$	0	0
$913$	14.7039	0.486627
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.46838	−0.246628
$918$	0	0
$919$	55.1600	1.81956	0.909781	−	0.415089i	$-0.136250\pi$
$919$	55.1600	1.81956	0.909781	+	0.415089i	$0.136250\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−18.4413	−0.607003
$924$	0	0
$925$	0.0558176	0.00183527
$926$	0	0
$927$	0	0
$928$	0	0
$929$	42.4562	1.39294	0.696471	−	0.717585i	$-0.254753\pi$
$929$	42.4562	1.39294	0.696471	+	0.717585i	$0.254753\pi$
$930$	0	0
$931$	6.58002	0.215651
$932$	0	0
$933$	0	0
$934$	0	0
$935$	24.2329	0.792500
$936$	0	0
$937$	−11.6406	−0.380283	−0.190142	−	0.981757i	$-0.560895\pi$
$937$	−11.6406	−0.380283	−0.190142	+	0.981757i	$0.560895\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	41.4126	1.35001	0.675006	−	0.737813i	$-0.264141\pi$
$941$	41.4126	1.35001	0.675006	+	0.737813i	$0.264141\pi$
$942$	0	0
$943$	52.0820	1.69602
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.3955	−0.662766	−0.331383	−	0.943496i	$-0.607515\pi$
$947$	−20.3955	−0.662766	−0.331383	+	0.943496i	$0.607515\pi$
$948$	0	0
$949$	8.57741	0.278434
$950$	0	0
$951$	0	0
$952$	0	0
$953$	41.3323	1.33888	0.669442	−	0.742864i	$-0.266533\pi$
$953$	41.3323	1.33888	0.669442	+	0.742864i	$0.266533\pi$
$954$	0	0
$955$	7.17968	0.232329
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.29612	−0.0418539
$960$	0	0
$961$	−18.9703	−0.611946
$962$	0	0
$963$	0	0
$964$	0	0
$965$	12.2887	0.395587
$966$	0	0
$967$	−4.15262	−0.133539	−0.0667696	−	0.997768i	$-0.521269\pi$
$967$	−4.15262	−0.133539	−0.0667696	+	0.997768i	$0.521269\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−41.7523	−1.33989	−0.669947	−	0.742409i	$-0.733683\pi$
$971$	−41.7523	−1.33989	−0.669947	+	0.742409i	$0.733683\pi$
$972$	0	0
$973$	−14.9368	−0.478851
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17.2207	0.550938	0.275469	−	0.961310i	$-0.411167\pi$
$977$	17.2207	0.550938	0.275469	+	0.961310i	$0.411167\pi$
$978$	0	0
$979$	−19.4684	−0.622212
$980$	0	0
$981$	0	0
$982$	0	0
$983$	25.6768	0.818963	0.409482	−	0.912318i	$-0.365710\pi$
$983$	25.6768	0.818963	0.409482	+	0.912318i	$0.365710\pi$
$984$	0	0
$985$	20.6284	0.657276
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−42.0872	−1.33829
$990$	0	0
$991$	−9.90320	−0.314586	−0.157293	−	0.987552i	$-0.550277\pi$
$991$	−9.90320	−0.314586	−0.157293	+	0.987552i	$0.550277\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.64064	−0.0520119
$996$	0	0
$997$	−41.0187	−1.29908	−0.649538	−	0.760329i	$-0.725038\pi$
$997$	−41.0187	−1.29908	−0.649538	+	0.760329i	$0.725038\pi$
$998$	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.1.2		3
3.2	odd	2		1140.2.a.g.1.2	✓	3
12.11	even	2		4560.2.a.br.1.2		3
15.2	even	4		5700.2.f.q.3649.2		6
15.8	even	4		5700.2.f.q.3649.5		6
15.14	odd	2		5700.2.a.w.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.a.g.1.2	✓	3	3.2	odd	2
3420.2.a.m.1.2		3	1.1	even	1	trivial
4560.2.a.br.1.2		3	12.11	even	2
5700.2.a.w.1.2		3	15.14	odd	2
5700.2.f.q.3649.2		6	15.2	even	4
5700.2.f.q.3649.5		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +0.648061 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +0.648061 q^{7} +3.52420 q^{11} +1.35194 q^{13} -6.87614 q^{17} -1.00000 q^{19} -5.46838 q^{23} +1.00000 q^{25} -5.52420 q^{29} -3.46838 q^{31} -0.648061 q^{35} +0.0558176 q^{37} -9.52420 q^{41} +7.69646 q^{43} +1.46838 q^{47} -6.58002 q^{49} +13.2207 q^{53} -3.52420 q^{55} +9.64064 q^{59} -0.172260 q^{61} -1.35194 q^{65} -5.40776 q^{67} -13.6406 q^{71} +6.34452 q^{73} +2.28390 q^{77} +4.34452 q^{79} +4.17226 q^{83} +6.87614 q^{85} -5.52420 q^{89} +0.876139 q^{91} +1.00000 q^{95} -2.64806 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 4 q^{7}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more