LMFDB - Embedded newform 3640.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.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:	$29.0655463357$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.1194649.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} - 3x^{2} + 10x + 4 $ x^5 - 8x^3 - 3x^2 + 10*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.21592$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.21592 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.52153 q^{9} +O(q^{10})$	$q-1.21592 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.52153 q^{9} -5.76039 q^{11} -1.00000 q^{13} +1.21592 q^{15} +3.42408 q^{17} +3.23886 q^{19} -1.21592 q^{21} +9.23529 q^{23} +1.00000 q^{25} +5.49784 q^{27} +6.64610 q^{29} +1.84266 q^{31} +7.00420 q^{33} -1.00000 q^{35} +2.07375 q^{37} +1.21592 q^{39} -12.0423 q^{41} -4.52762 q^{43} +1.52153 q^{45} +1.33441 q^{47} +1.00000 q^{49} -4.16343 q^{51} -5.05858 q^{53} +5.76039 q^{55} -3.93821 q^{57} -5.78827 q^{59} -9.00420 q^{61} -1.52153 q^{63} +1.00000 q^{65} -7.28359 q^{67} -11.2294 q^{69} +6.33273 q^{71} -0.186363 q^{73} -1.21592 q^{75} -5.76039 q^{77} -1.75947 q^{79} -2.12038 q^{81} -10.1320 q^{83} -3.42408 q^{85} -8.08116 q^{87} -9.15566 q^{89} -1.00000 q^{91} -2.24053 q^{93} -3.23886 q^{95} -8.37746 q^{97} +8.76458 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} + 5 q^{7} + q^{9}+O(q^{10}) $ 5 * q - 5 * q^5 + 5 * q^7 + q^9	$ 5 q - 5 q^{5} + 5 q^{7} + q^{9} - 7 q^{11} - 5 q^{13} + q^{17} + 3 q^{19} - 5 q^{23} + 5 q^{25} - 9 q^{27} - 9 q^{29} + 6 q^{31} + q^{33} - 5 q^{35} - 10 q^{37} - 8 q^{41} + 6 q^{43} - q^{45} - 13 q^{47} + 5 q^{49} - 4 q^{51} - 16 q^{53} + 7 q^{55} - 10 q^{57} - q^{59} - 11 q^{61} + q^{63} + 5 q^{65} - 30 q^{67} - 15 q^{69} - 12 q^{71} + 23 q^{73} - 7 q^{77} + 2 q^{79} - 11 q^{81} - 2 q^{83} - q^{85} - 5 q^{87} - 25 q^{89} - 5 q^{91} - 22 q^{93} - 3 q^{95} - 5 q^{97} - 12 q^{99}+O(q^{100}) $ 5 * q - 5 * q^5 + 5 * q^7 + q^9 - 7 * q^11 - 5 * q^13 + q^17 + 3 * q^19 - 5 * q^23 + 5 * q^25 - 9 * q^27 - 9 * q^29 + 6 * q^31 + q^33 - 5 * q^35 - 10 * q^37 - 8 * q^41 + 6 * q^43 - q^45 - 13 * q^47 + 5 * q^49 - 4 * q^51 - 16 * q^53 + 7 * q^55 - 10 * q^57 - q^59 - 11 * q^61 + q^63 + 5 * q^65 - 30 * q^67 - 15 * q^69 - 12 * q^71 + 23 * q^73 - 7 * q^77 + 2 * q^79 - 11 * q^81 - 2 * q^83 - q^85 - 5 * q^87 - 25 * q^89 - 5 * q^91 - 22 * q^93 - 3 * q^95 - 5 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.21592	−0.702015	−0.351007	−	0.936373i	$-0.614161\pi$
$3$	−1.21592	−0.702015	−0.351007	+	0.936373i	$0.614161\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.52153	−0.507176
$10$	0	0
$11$	−5.76039	−1.73682	−0.868411	−	0.495846i	$-0.834858\pi$
$11$	−5.76039	−1.73682	−0.868411	+	0.495846i	$0.834858\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.21592	0.313950
$16$	0	0
$17$	3.42408	0.830462	0.415231	−	0.909716i	$-0.363701\pi$
$17$	3.42408	0.830462	0.415231	+	0.909716i	$0.363701\pi$
$18$	0	0
$19$	3.23886	0.743045	0.371523	−	0.928424i	$-0.378836\pi$
$19$	3.23886	0.743045	0.371523	+	0.928424i	$0.378836\pi$
$20$	0	0
$21$	−1.21592	−0.265337
$22$	0	0
$23$	9.23529	1.92569	0.962845	−	0.270053i	$-0.0870413\pi$
$23$	9.23529	1.92569	0.962845	+	0.270053i	$0.0870413\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.49784	1.05806
$28$	0	0
$29$	6.64610	1.23415	0.617075	−	0.786905i	$-0.288318\pi$
$29$	6.64610	1.23415	0.617075	+	0.786905i	$0.288318\pi$
$30$	0	0
$31$	1.84266	0.330951	0.165476	−	0.986214i	$-0.447084\pi$
$31$	1.84266	0.330951	0.165476	+	0.986214i	$0.447084\pi$
$32$	0	0
$33$	7.00420	1.21927
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	2.07375	0.340923	0.170461	−	0.985364i	$-0.445474\pi$
$37$	2.07375	0.340923	0.170461	+	0.985364i	$0.445474\pi$
$38$	0	0
$39$	1.21592	0.194704
$40$	0	0
$41$	−12.0423	−1.88069	−0.940345	−	0.340221i	$-0.889498\pi$
$41$	−12.0423	−1.88069	−0.940345	+	0.340221i	$0.889498\pi$
$42$	0	0
$43$	−4.52762	−0.690455	−0.345227	−	0.938519i	$-0.612198\pi$
$43$	−4.52762	−0.690455	−0.345227	+	0.938519i	$0.612198\pi$
$44$	0	0
$45$	1.52153	0.226816
$46$	0	0
$47$	1.33441	0.194643	0.0973217	−	0.995253i	$-0.468972\pi$
$47$	1.33441	0.194643	0.0973217	+	0.995253i	$0.468972\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.16343	−0.582997
$52$	0	0
$53$	−5.05858	−0.694850	−0.347425	−	0.937708i	$-0.612944\pi$
$53$	−5.05858	−0.694850	−0.347425	+	0.937708i	$0.612944\pi$
$54$	0	0
$55$	5.76039	0.776730
$56$	0	0
$57$	−3.93821	−0.521628
$58$	0	0
$59$	−5.78827	−0.753569	−0.376784	−	0.926301i	$-0.622970\pi$
$59$	−5.78827	−0.753569	−0.376784	+	0.926301i	$0.622970\pi$
$60$	0	0
$61$	−9.00420	−1.15287	−0.576435	−	0.817143i	$-0.695556\pi$
$61$	−9.00420	−1.15287	−0.576435	+	0.817143i	$0.695556\pi$
$62$	0	0
$63$	−1.52153	−0.191694
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−7.28359	−0.889832	−0.444916	−	0.895572i	$-0.646766\pi$
$67$	−7.28359	−0.889832	−0.444916	+	0.895572i	$0.646766\pi$
$68$	0	0
$69$	−11.2294	−1.35186
$70$	0	0
$71$	6.33273	0.751557	0.375778	−	0.926710i	$-0.377375\pi$
$71$	6.33273	0.751557	0.375778	+	0.926710i	$0.377375\pi$
$72$	0	0
$73$	−0.186363	−0.0218121	−0.0109061	−	0.999941i	$-0.503472\pi$
$73$	−0.186363	−0.0218121	−0.0109061	+	0.999941i	$0.503472\pi$
$74$	0	0
$75$	−1.21592	−0.140403
$76$	0	0
$77$	−5.76039	−0.656457
$78$	0	0
$79$	−1.75947	−0.197955	−0.0989776	−	0.995090i	$-0.531557\pi$
$79$	−1.75947	−0.197955	−0.0989776	+	0.995090i	$0.531557\pi$
$80$	0	0
$81$	−2.12038	−0.235597
$82$	0	0
$83$	−10.1320	−1.11213	−0.556064	−	0.831139i	$-0.687689\pi$
$83$	−10.1320	−1.11213	−0.556064	+	0.831139i	$0.687689\pi$
$84$	0	0
$85$	−3.42408	−0.371394
$86$	0	0
$87$	−8.08116	−0.866391
$88$	0	0
$89$	−9.15566	−0.970498	−0.485249	−	0.874376i	$-0.661271\pi$
$89$	−9.15566	−0.970498	−0.485249	+	0.874376i	$0.661271\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−2.24053	−0.232333
$94$	0	0
$95$	−3.23886	−0.332300
$96$	0	0
$97$	−8.37746	−0.850602	−0.425301	−	0.905052i	$-0.639832\pi$
$97$	−8.37746	−0.850602	−0.425301	+	0.905052i	$0.639832\pi$
$98$	0	0
$99$	8.76458	0.880873
$100$	0	0
$101$	3.61540	0.359746	0.179873	−	0.983690i	$-0.442431\pi$
$101$	3.61540	0.359746	0.179873	+	0.983690i	$0.442431\pi$
$102$	0	0
$103$	15.4421	1.52156	0.760779	−	0.649011i	$-0.224817\pi$
$103$	15.4421	1.52156	0.760779	+	0.649011i	$0.224817\pi$
$104$	0	0
$105$	1.21592	0.118662
$106$	0	0
$107$	1.91641	0.185266	0.0926332	−	0.995700i	$-0.470472\pi$
$107$	1.91641	0.185266	0.0926332	+	0.995700i	$0.470472\pi$
$108$	0	0
$109$	−8.08611	−0.774509	−0.387254	−	0.921973i	$-0.626576\pi$
$109$	−8.08611	−0.774509	−0.387254	+	0.921973i	$0.626576\pi$
$110$	0	0
$111$	−2.52153	−0.239333
$112$	0	0
$113$	−11.4360	−1.07581	−0.537906	−	0.843005i	$-0.680785\pi$
$113$	−11.4360	−1.07581	−0.537906	+	0.843005i	$0.680785\pi$
$114$	0	0
$115$	−9.23529	−0.861195
$116$	0	0
$117$	1.52153	0.140665
$118$	0	0
$119$	3.42408	0.313885
$120$	0	0
$121$	22.1820	2.01655
$122$	0	0
$123$	14.6425	1.32027
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−21.1576	−1.87743	−0.938715	−	0.344693i	$-0.887983\pi$
$127$	−21.1576	−1.87743	−0.938715	+	0.344693i	$0.887983\pi$
$128$	0	0
$129$	5.50524	0.484709
$130$	0	0
$131$	2.24075	0.195775	0.0978877	−	0.995197i	$-0.468791\pi$
$131$	2.24075	0.195775	0.0978877	+	0.995197i	$0.468791\pi$
$132$	0	0
$133$	3.23886	0.280845
$134$	0	0
$135$	−5.49784	−0.473178
$136$	0	0
$137$	0.158477	0.0135396	0.00676982	−	0.999977i	$-0.497845\pi$
$137$	0.158477	0.0135396	0.00676982	+	0.999977i	$0.497845\pi$
$138$	0	0
$139$	20.7671	1.76144	0.880721	−	0.473635i	$-0.157058\pi$
$139$	20.7671	1.76144	0.880721	+	0.473635i	$0.157058\pi$
$140$	0	0
$141$	−1.62254	−0.136642
$142$	0	0
$143$	5.76039	0.481708
$144$	0	0
$145$	−6.64610	−0.551928
$146$	0	0
$147$	−1.21592	−0.100288
$148$	0	0
$149$	−21.4664	−1.75859	−0.879297	−	0.476273i	$-0.841987\pi$
$149$	−21.4664	−1.75859	−0.879297	+	0.476273i	$0.841987\pi$
$150$	0	0
$151$	6.42219	0.522631	0.261315	−	0.965254i	$-0.415844\pi$
$151$	6.42219	0.522631	0.261315	+	0.965254i	$0.415844\pi$
$152$	0	0
$153$	−5.20984	−0.421190
$154$	0	0
$155$	−1.84266	−0.148006
$156$	0	0
$157$	13.8755	1.10739	0.553693	−	0.832721i	$-0.313218\pi$
$157$	13.8755	1.10739	0.553693	+	0.832721i	$0.313218\pi$
$158$	0	0
$159$	6.15086	0.487795
$160$	0	0
$161$	9.23529	0.727843
$162$	0	0
$163$	−5.79793	−0.454129	−0.227064	−	0.973880i	$-0.572913\pi$
$163$	−5.79793	−0.454129	−0.227064	+	0.973880i	$0.572913\pi$
$164$	0	0
$165$	−7.00420	−0.545276
$166$	0	0
$167$	−10.3424	−0.800318	−0.400159	−	0.916446i	$-0.631045\pi$
$167$	−10.3424	−0.800318	−0.400159	+	0.916446i	$0.631045\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.92801	−0.376854
$172$	0	0
$173$	11.1083	0.844547	0.422274	−	0.906468i	$-0.361232\pi$
$173$	11.1083	0.844547	0.422274	+	0.906468i	$0.361232\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	7.03810	0.529016
$178$	0	0
$179$	−22.4812	−1.68032	−0.840160	−	0.542338i	$-0.817539\pi$
$179$	−22.4812	−1.68032	−0.840160	+	0.542338i	$0.817539\pi$
$180$	0	0
$181$	−25.1517	−1.86951	−0.934756	−	0.355291i	$-0.884382\pi$
$181$	−25.1517	−1.86951	−0.934756	+	0.355291i	$0.884382\pi$
$182$	0	0
$183$	10.9484	0.809331
$184$	0	0
$185$	−2.07375	−0.152465
$186$	0	0
$187$	−19.7240	−1.44237
$188$	0	0
$189$	5.49784	0.399909
$190$	0	0
$191$	8.78143	0.635402	0.317701	−	0.948191i	$-0.397089\pi$
$191$	8.78143	0.635402	0.317701	+	0.948191i	$0.397089\pi$
$192$	0	0
$193$	−6.56039	−0.472227	−0.236113	−	0.971726i	$-0.575874\pi$
$193$	−6.56039	−0.472227	−0.236113	+	0.971726i	$0.575874\pi$
$194$	0	0
$195$	−1.21592	−0.0870742
$196$	0	0
$197$	−21.5945	−1.53855	−0.769273	−	0.638920i	$-0.779381\pi$
$197$	−21.5945	−1.53855	−0.769273	+	0.638920i	$0.779381\pi$
$198$	0	0
$199$	26.2336	1.85965	0.929826	−	0.368000i	$-0.119957\pi$
$199$	26.2336	1.85965	0.929826	+	0.368000i	$0.119957\pi$
$200$	0	0
$201$	8.85630	0.624675
$202$	0	0
$203$	6.64610	0.466465
$204$	0	0
$205$	12.0423	0.841071
$206$	0	0
$207$	−14.0517	−0.976663
$208$	0	0
$209$	−18.6571	−1.29054
$210$	0	0
$211$	−8.09290	−0.557138	−0.278569	−	0.960416i	$-0.589860\pi$
$211$	−8.09290	−0.557138	−0.278569	+	0.960416i	$0.589860\pi$
$212$	0	0
$213$	−7.70012	−0.527604
$214$	0	0
$215$	4.52762	0.308781
$216$	0	0
$217$	1.84266	0.125088
$218$	0	0
$219$	0.226603	0.0153124
$220$	0	0
$221$	−3.42408	−0.230329
$222$	0	0
$223$	−0.245487	−0.0164390	−0.00821951	−	0.999966i	$-0.502616\pi$
$223$	−0.245487	−0.0164390	−0.00821951	+	0.999966i	$0.502616\pi$
$224$	0	0
$225$	−1.52153	−0.101435
$226$	0	0
$227$	−15.3972	−1.02195	−0.510974	−	0.859596i	$-0.670715\pi$
$227$	−15.3972	−1.02195	−0.510974	+	0.859596i	$0.670715\pi$
$228$	0	0
$229$	0.287784	0.0190173	0.00950865	−	0.999955i	$-0.496973\pi$
$229$	0.287784	0.0190173	0.00950865	+	0.999955i	$0.496973\pi$
$230$	0	0
$231$	7.00420	0.460842
$232$	0	0
$233$	−10.3619	−0.678833	−0.339416	−	0.940636i	$-0.610230\pi$
$233$	−10.3619	−0.678833	−0.339416	+	0.940636i	$0.610230\pi$
$234$	0	0
$235$	−1.33441	−0.0870471
$236$	0	0
$237$	2.13938	0.138967
$238$	0	0
$239$	2.14163	0.138531	0.0692653	−	0.997598i	$-0.477934\pi$
$239$	2.14163	0.138531	0.0692653	+	0.997598i	$0.477934\pi$
$240$	0	0
$241$	20.8399	1.34241	0.671207	−	0.741270i	$-0.265776\pi$
$241$	20.8399	1.34241	0.671207	+	0.741270i	$0.265776\pi$
$242$	0	0
$243$	−13.9153	−0.892666
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−3.23886	−0.206084
$248$	0	0
$249$	12.3197	0.780730
$250$	0	0
$251$	−1.03453	−0.0652991	−0.0326495	−	0.999467i	$-0.510395\pi$
$251$	−1.03453	−0.0652991	−0.0326495	+	0.999467i	$0.510395\pi$
$252$	0	0
$253$	−53.1988	−3.34458
$254$	0	0
$255$	4.16343	0.260724
$256$	0	0
$257$	−27.0499	−1.68733	−0.843664	−	0.536871i	$-0.819606\pi$
$257$	−27.0499	−1.68733	−0.843664	+	0.536871i	$0.819606\pi$
$258$	0	0
$259$	2.07375	0.128857
$260$	0	0
$261$	−10.1122	−0.625930
$262$	0	0
$263$	26.1363	1.61164	0.805818	−	0.592164i	$-0.201726\pi$
$263$	26.1363	1.61164	0.805818	+	0.592164i	$0.201726\pi$
$264$	0	0
$265$	5.05858	0.310746
$266$	0	0
$267$	11.1326	0.681304
$268$	0	0
$269$	−15.6154	−0.952088	−0.476044	−	0.879422i	$-0.657930\pi$
$269$	−15.6154	−0.952088	−0.476044	+	0.879422i	$0.657930\pi$
$270$	0	0
$271$	11.9670	0.726944	0.363472	−	0.931605i	$-0.381591\pi$
$271$	11.9670	0.726944	0.363472	+	0.931605i	$0.381591\pi$
$272$	0	0
$273$	1.21592	0.0735911
$274$	0	0
$275$	−5.76039	−0.347364
$276$	0	0
$277$	22.4524	1.34903	0.674517	−	0.738259i	$-0.264352\pi$
$277$	22.4524	1.34903	0.674517	+	0.738259i	$0.264352\pi$
$278$	0	0
$279$	−2.80366	−0.167850
$280$	0	0
$281$	−18.5756	−1.10813	−0.554063	−	0.832475i	$-0.686923\pi$
$281$	−18.5756	−1.10813	−0.554063	+	0.832475i	$0.686923\pi$
$282$	0	0
$283$	0.784470	0.0466319	0.0233159	−	0.999728i	$-0.492578\pi$
$283$	0.784470	0.0466319	0.0233159	+	0.999728i	$0.492578\pi$
$284$	0	0
$285$	3.93821	0.233279
$286$	0	0
$287$	−12.0423	−0.710834
$288$	0	0
$289$	−5.27565	−0.310332
$290$	0	0
$291$	10.1864	0.597135
$292$	0	0
$293$	2.48611	0.145240	0.0726200	−	0.997360i	$-0.476864\pi$
$293$	2.48611	0.145240	0.0726200	+	0.997360i	$0.476864\pi$
$294$	0	0
$295$	5.78827	0.337006
$296$	0	0
$297$	−31.6697	−1.83766
$298$	0	0
$299$	−9.23529	−0.534091
$300$	0	0
$301$	−4.52762	−0.260967
$302$	0	0
$303$	−4.39605	−0.252547
$304$	0	0
$305$	9.00420	0.515579
$306$	0	0
$307$	29.0041	1.65535	0.827675	−	0.561208i	$-0.189663\pi$
$307$	29.0041	1.65535	0.827675	+	0.561208i	$0.189663\pi$
$308$	0	0
$309$	−18.7765	−1.06816
$310$	0	0
$311$	8.36055	0.474083	0.237042	−	0.971499i	$-0.423822\pi$
$311$	8.36055	0.474083	0.237042	+	0.971499i	$0.423822\pi$
$312$	0	0
$313$	−9.79414	−0.553598	−0.276799	−	0.960928i	$-0.589274\pi$
$313$	−9.79414	−0.553598	−0.276799	+	0.960928i	$0.589274\pi$
$314$	0	0
$315$	1.52153	0.0857283
$316$	0	0
$317$	−27.1043	−1.52233	−0.761165	−	0.648558i	$-0.775372\pi$
$317$	−27.1043	−1.52233	−0.761165	+	0.648558i	$0.775372\pi$
$318$	0	0
$319$	−38.2841	−2.14350
$320$	0	0
$321$	−2.33021	−0.130060
$322$	0	0
$323$	11.0901	0.617071
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	9.83210	0.543716
$328$	0	0
$329$	1.33441	0.0735683
$330$	0	0
$331$	34.5153	1.89713	0.948566	−	0.316578i	$-0.102534\pi$
$331$	34.5153	1.89713	0.948566	+	0.316578i	$0.102534\pi$
$332$	0	0
$333$	−3.15527	−0.172908
$334$	0	0
$335$	7.28359	0.397945
$336$	0	0
$337$	−17.1854	−0.936148	−0.468074	−	0.883689i	$-0.655052\pi$
$337$	−17.1854	−0.936148	−0.468074	+	0.883689i	$0.655052\pi$
$338$	0	0
$339$	13.9054	0.755236
$340$	0	0
$341$	−10.6144	−0.574803
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	11.2294	0.604571
$346$	0	0
$347$	−30.6376	−1.64471	−0.822357	−	0.568971i	$-0.807342\pi$
$347$	−30.6376	−1.64471	−0.822357	+	0.568971i	$0.807342\pi$
$348$	0	0
$349$	14.9218	0.798745	0.399373	−	0.916789i	$-0.369228\pi$
$349$	14.9218	0.798745	0.399373	+	0.916789i	$0.369228\pi$
$350$	0	0
$351$	−5.49784	−0.293453
$352$	0	0
$353$	−21.8509	−1.16300	−0.581502	−	0.813545i	$-0.697535\pi$
$353$	−21.8509	−1.16300	−0.581502	+	0.813545i	$0.697535\pi$
$354$	0	0
$355$	−6.33273	−0.336106
$356$	0	0
$357$	−4.16343	−0.220352
$358$	0	0
$359$	−19.4095	−1.02439	−0.512196	−	0.858868i	$-0.671168\pi$
$359$	−19.4095	−1.02439	−0.512196	+	0.858868i	$0.671168\pi$
$360$	0	0
$361$	−8.50980	−0.447884
$362$	0	0
$363$	−26.9717	−1.41565
$364$	0	0
$365$	0.186363	0.00975467
$366$	0	0
$367$	−21.9817	−1.14743	−0.573717	−	0.819053i	$-0.694499\pi$
$367$	−21.9817	−1.14743	−0.573717	+	0.819053i	$0.694499\pi$
$368$	0	0
$369$	18.3227	0.953840
$370$	0	0
$371$	−5.05858	−0.262629
$372$	0	0
$373$	−6.25628	−0.323938	−0.161969	−	0.986796i	$-0.551784\pi$
$373$	−6.25628	−0.323938	−0.161969	+	0.986796i	$0.551784\pi$
$374$	0	0
$375$	1.21592	0.0627901
$376$	0	0
$377$	−6.64610	−0.342291
$378$	0	0
$379$	−8.34826	−0.428821	−0.214411	−	0.976744i	$-0.568783\pi$
$379$	−8.34826	−0.428821	−0.214411	+	0.976744i	$0.568783\pi$
$380$	0	0
$381$	25.7260	1.31798
$382$	0	0
$383$	21.5426	1.10077	0.550387	−	0.834909i	$-0.314480\pi$
$383$	21.5426	1.10077	0.550387	+	0.834909i	$0.314480\pi$
$384$	0	0
$385$	5.76039	0.293576
$386$	0	0
$387$	6.88889	0.350182
$388$	0	0
$389$	−24.8498	−1.25993	−0.629967	−	0.776622i	$-0.716931\pi$
$389$	−24.8498	−1.25993	−0.629967	+	0.776622i	$0.716931\pi$
$390$	0	0
$391$	31.6224	1.59921
$392$	0	0
$393$	−2.72459	−0.137437
$394$	0	0
$395$	1.75947	0.0885283
$396$	0	0
$397$	10.0823	0.506017	0.253009	−	0.967464i	$-0.418580\pi$
$397$	10.0823	0.506017	0.253009	+	0.967464i	$0.418580\pi$
$398$	0	0
$399$	−3.93821	−0.197157
$400$	0	0
$401$	10.0239	0.500571	0.250285	−	0.968172i	$-0.419476\pi$
$401$	10.0239	0.500571	0.250285	+	0.968172i	$0.419476\pi$
$402$	0	0
$403$	−1.84266	−0.0917894
$404$	0	0
$405$	2.12038	0.105362
$406$	0	0
$407$	−11.9456	−0.592122
$408$	0	0
$409$	18.2033	0.900094	0.450047	−	0.893005i	$-0.351407\pi$
$409$	18.2033	0.900094	0.450047	+	0.893005i	$0.351407\pi$
$410$	0	0
$411$	−0.192697	−0.00950503
$412$	0	0
$413$	−5.78827	−0.284822
$414$	0	0
$415$	10.1320	0.497359
$416$	0	0
$417$	−25.2512	−1.23656
$418$	0	0
$419$	4.78539	0.233782	0.116891	−	0.993145i	$-0.462707\pi$
$419$	4.78539	0.233782	0.116891	+	0.993145i	$0.462707\pi$
$420$	0	0
$421$	23.5835	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$421$	23.5835	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$422$	0	0
$423$	−2.03034	−0.0987183
$424$	0	0
$425$	3.42408	0.166092
$426$	0	0
$427$	−9.00420	−0.435744
$428$	0	0
$429$	−7.00420	−0.338166
$430$	0	0
$431$	10.4874	0.505159	0.252580	−	0.967576i	$-0.418721\pi$
$431$	10.4874	0.505159	0.252580	+	0.967576i	$0.418721\pi$
$432$	0	0
$433$	−1.79832	−0.0864219	−0.0432110	−	0.999066i	$-0.513759\pi$
$433$	−1.79832	−0.0864219	−0.0432110	+	0.999066i	$0.513759\pi$
$434$	0	0
$435$	8.08116	0.387462
$436$	0	0
$437$	29.9118	1.43087
$438$	0	0
$439$	27.2308	1.29966	0.649828	−	0.760082i	$-0.274841\pi$
$439$	27.2308	1.29966	0.649828	+	0.760082i	$0.274841\pi$
$440$	0	0
$441$	−1.52153	−0.0724537
$442$	0	0
$443$	31.5791	1.50037	0.750183	−	0.661230i	$-0.229965\pi$
$443$	31.5791	1.50037	0.750183	+	0.661230i	$0.229965\pi$
$444$	0	0
$445$	9.15566	0.434020
$446$	0	0
$447$	26.1015	1.23456
$448$	0	0
$449$	−20.7151	−0.977607	−0.488803	−	0.872394i	$-0.662566\pi$
$449$	−20.7151	−0.977607	−0.488803	+	0.872394i	$0.662566\pi$
$450$	0	0
$451$	69.3683	3.26642
$452$	0	0
$453$	−7.80890	−0.366894
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	31.3789	1.46784	0.733922	−	0.679233i	$-0.237688\pi$
$457$	31.3789	1.46784	0.733922	+	0.679233i	$0.237688\pi$
$458$	0	0
$459$	18.8251	0.878678
$460$	0	0
$461$	29.3419	1.36659	0.683294	−	0.730143i	$-0.260547\pi$
$461$	29.3419	1.36659	0.683294	+	0.730143i	$0.260547\pi$
$462$	0	0
$463$	−24.0641	−1.11835	−0.559177	−	0.829048i	$-0.688883\pi$
$463$	−24.0641	−1.11835	−0.559177	+	0.829048i	$0.688883\pi$
$464$	0	0
$465$	2.24053	0.103902
$466$	0	0
$467$	8.94737	0.414035	0.207018	−	0.978337i	$-0.433624\pi$
$467$	8.94737	0.414035	0.207018	+	0.978337i	$0.433624\pi$
$468$	0	0
$469$	−7.28359	−0.336325
$470$	0	0
$471$	−16.8716	−0.777401
$472$	0	0
$473$	26.0808	1.19920
$474$	0	0
$475$	3.23886	0.148609
$476$	0	0
$477$	7.69677	0.352411
$478$	0	0
$479$	4.46238	0.203892	0.101946	−	0.994790i	$-0.467493\pi$
$479$	4.46238	0.203892	0.101946	+	0.994790i	$0.467493\pi$
$480$	0	0
$481$	−2.07375	−0.0945550
$482$	0	0
$483$	−11.2294	−0.510956
$484$	0	0
$485$	8.37746	0.380401
$486$	0	0
$487$	−35.3995	−1.60410	−0.802052	−	0.597254i	$-0.796259\pi$
$487$	−35.3995	−1.60410	−0.802052	+	0.597254i	$0.796259\pi$
$488$	0	0
$489$	7.04985	0.318805
$490$	0	0
$491$	−34.4846	−1.55627	−0.778134	−	0.628098i	$-0.783834\pi$
$491$	−34.4846	−1.55627	−0.778134	+	0.628098i	$0.783834\pi$
$492$	0	0
$493$	22.7568	1.02491
$494$	0	0
$495$	−8.76458	−0.393939
$496$	0	0
$497$	6.33273	0.284062
$498$	0	0
$499$	−32.0327	−1.43398	−0.716991	−	0.697083i	$-0.754481\pi$
$499$	−32.0327	−1.43398	−0.716991	+	0.697083i	$0.754481\pi$
$500$	0	0
$501$	12.5756	0.561835
$502$	0	0
$503$	−12.2736	−0.547253	−0.273627	−	0.961836i	$-0.588223\pi$
$503$	−12.2736	−0.547253	−0.273627	+	0.961836i	$0.588223\pi$
$504$	0	0
$505$	−3.61540	−0.160883
$506$	0	0
$507$	−1.21592	−0.0540011
$508$	0	0
$509$	28.5051	1.26347	0.631733	−	0.775186i	$-0.282344\pi$
$509$	28.5051	1.26347	0.631733	+	0.775186i	$0.282344\pi$
$510$	0	0
$511$	−0.186363	−0.00824420
$512$	0	0
$513$	17.8067	0.786186
$514$	0	0
$515$	−15.4421	−0.680462
$516$	0	0
$517$	−7.68670	−0.338061
$518$	0	0
$519$	−13.5068	−0.592884
$520$	0	0
$521$	−21.3801	−0.936681	−0.468341	−	0.883548i	$-0.655148\pi$
$521$	−21.3801	−0.936681	−0.468341	+	0.883548i	$0.655148\pi$
$522$	0	0
$523$	−7.11733	−0.311219	−0.155610	−	0.987819i	$-0.549734\pi$
$523$	−7.11733	−0.311219	−0.155610	+	0.987819i	$0.549734\pi$
$524$	0	0
$525$	−1.21592	−0.0530673
$526$	0	0
$527$	6.30942	0.274843
$528$	0	0
$529$	62.2906	2.70828
$530$	0	0
$531$	8.80701	0.382192
$532$	0	0
$533$	12.0423	0.521610
$534$	0	0
$535$	−1.91641	−0.0828537
$536$	0	0
$537$	27.3354	1.17961
$538$	0	0
$539$	−5.76039	−0.248117
$540$	0	0
$541$	−22.8395	−0.981947	−0.490974	−	0.871174i	$-0.663359\pi$
$541$	−22.8395	−0.981947	−0.490974	+	0.871174i	$0.663359\pi$
$542$	0	0
$543$	30.5826	1.31242
$544$	0	0
$545$	8.08611	0.346371
$546$	0	0
$547$	−29.3842	−1.25638	−0.628188	−	0.778061i	$-0.716203\pi$
$547$	−29.3842	−1.25638	−0.628188	+	0.778061i	$0.716203\pi$
$548$	0	0
$549$	13.7001	0.584707
$550$	0	0
$551$	21.5258	0.917029
$552$	0	0
$553$	−1.75947	−0.0748201
$554$	0	0
$555$	2.52153	0.107033
$556$	0	0
$557$	−34.7093	−1.47068	−0.735340	−	0.677699i	$-0.762977\pi$
$557$	−34.7093	−1.47068	−0.735340	+	0.677699i	$0.762977\pi$
$558$	0	0
$559$	4.52762	0.191498
$560$	0	0
$561$	23.9830	1.01256
$562$	0	0
$563$	−7.28611	−0.307073	−0.153536	−	0.988143i	$-0.549066\pi$
$563$	−7.28611	−0.307073	−0.153536	+	0.988143i	$0.549066\pi$
$564$	0	0
$565$	11.4360	0.481118
$566$	0	0
$567$	−2.12038	−0.0890474
$568$	0	0
$569$	9.58278	0.401731	0.200865	−	0.979619i	$-0.435625\pi$
$569$	9.58278	0.401731	0.200865	+	0.979619i	$0.435625\pi$
$570$	0	0
$571$	29.6721	1.24174	0.620870	−	0.783914i	$-0.286780\pi$
$571$	29.6721	1.24174	0.620870	+	0.783914i	$0.286780\pi$
$572$	0	0
$573$	−10.6776	−0.446061
$574$	0	0
$575$	9.23529	0.385138
$576$	0	0
$577$	−8.34423	−0.347375	−0.173687	−	0.984801i	$-0.555568\pi$
$577$	−8.34423	−0.347375	−0.173687	+	0.984801i	$0.555568\pi$
$578$	0	0
$579$	7.97694	0.331510
$580$	0	0
$581$	−10.1320	−0.420345
$582$	0	0
$583$	29.1394	1.20683
$584$	0	0
$585$	−1.52153	−0.0629074
$586$	0	0
$587$	21.3263	0.880231	0.440116	−	0.897941i	$-0.354937\pi$
$587$	21.3263	0.880231	0.440116	+	0.897941i	$0.354937\pi$
$588$	0	0
$589$	5.96811	0.245912
$590$	0	0
$591$	26.2573	1.08008
$592$	0	0
$593$	40.8869	1.67902	0.839512	−	0.543341i	$-0.182841\pi$
$593$	40.8869	1.67902	0.839512	+	0.543341i	$0.182841\pi$
$594$	0	0
$595$	−3.42408	−0.140374
$596$	0	0
$597$	−31.8981	−1.30550
$598$	0	0
$599$	−9.70407	−0.396498	−0.198249	−	0.980152i	$-0.563525\pi$
$599$	−9.70407	−0.396498	−0.198249	+	0.980152i	$0.563525\pi$
$600$	0	0
$601$	−30.9164	−1.26111	−0.630554	−	0.776145i	$-0.717172\pi$
$601$	−30.9164	−1.26111	−0.630554	+	0.776145i	$0.717172\pi$
$602$	0	0
$603$	11.0822	0.451301
$604$	0	0
$605$	−22.1820	−0.901828
$606$	0	0
$607$	−33.0668	−1.34214	−0.671070	−	0.741394i	$-0.734165\pi$
$607$	−33.0668	−1.34214	−0.671070	+	0.741394i	$0.734165\pi$
$608$	0	0
$609$	−8.08116	−0.327465
$610$	0	0
$611$	−1.33441	−0.0539843
$612$	0	0
$613$	−34.1268	−1.37837	−0.689184	−	0.724586i	$-0.742031\pi$
$613$	−34.1268	−1.37837	−0.689184	+	0.724586i	$0.742031\pi$
$614$	0	0
$615$	−14.6425	−0.590444
$616$	0	0
$617$	35.2205	1.41793	0.708963	−	0.705246i	$-0.249163\pi$
$617$	35.2205	1.41793	0.708963	+	0.705246i	$0.249163\pi$
$618$	0	0
$619$	25.1615	1.01133	0.505664	−	0.862731i	$-0.331248\pi$
$619$	25.1615	1.01133	0.505664	+	0.862731i	$0.331248\pi$
$620$	0	0
$621$	50.7741	2.03749
$622$	0	0
$623$	−9.15566	−0.366814
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	22.6856	0.905975
$628$	0	0
$629$	7.10070	0.283124
$630$	0	0
$631$	−37.9251	−1.50978	−0.754888	−	0.655854i	$-0.772309\pi$
$631$	−37.9251	−1.50978	−0.754888	+	0.655854i	$0.772309\pi$
$632$	0	0
$633$	9.84036	0.391119
$634$	0	0
$635$	21.1576	0.839613
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−9.63542	−0.381171
$640$	0	0
$641$	−1.72801	−0.0682523	−0.0341261	−	0.999418i	$-0.510865\pi$
$641$	−1.72801	−0.0682523	−0.0341261	+	0.999418i	$0.510865\pi$
$642$	0	0
$643$	0.848893	0.0334771	0.0167385	−	0.999860i	$-0.494672\pi$
$643$	0.848893	0.0334771	0.0167385	+	0.999860i	$0.494672\pi$
$644$	0	0
$645$	−5.50524	−0.216769
$646$	0	0
$647$	−24.2319	−0.952653	−0.476326	−	0.879269i	$-0.658032\pi$
$647$	−24.2319	−0.952653	−0.476326	+	0.879269i	$0.658032\pi$
$648$	0	0
$649$	33.3427	1.30881
$650$	0	0
$651$	−2.24053	−0.0878135
$652$	0	0
$653$	−21.1121	−0.826181	−0.413090	−	0.910690i	$-0.635551\pi$
$653$	−21.1121	−0.826181	−0.413090	+	0.910690i	$0.635551\pi$
$654$	0	0
$655$	−2.24075	−0.0875534
$656$	0	0
$657$	0.283556	0.0110626
$658$	0	0
$659$	−20.3932	−0.794407	−0.397203	−	0.917731i	$-0.630019\pi$
$659$	−20.3932	−0.794407	−0.397203	+	0.917731i	$0.630019\pi$
$660$	0	0
$661$	−38.0862	−1.48138	−0.740691	−	0.671846i	$-0.765502\pi$
$661$	−38.0862	−1.48138	−0.740691	+	0.671846i	$0.765502\pi$
$662$	0	0
$663$	4.16343	0.161694
$664$	0	0
$665$	−3.23886	−0.125598
$666$	0	0
$667$	61.3786	2.37659
$668$	0	0
$669$	0.298494	0.0115404
$670$	0	0
$671$	51.8676	2.00233
$672$	0	0
$673$	−5.13952	−0.198114	−0.0990570	−	0.995082i	$-0.531583\pi$
$673$	−5.13952	−0.198114	−0.0990570	+	0.995082i	$0.531583\pi$
$674$	0	0
$675$	5.49784	0.211612
$676$	0	0
$677$	−20.2373	−0.777781	−0.388891	−	0.921284i	$-0.627142\pi$
$677$	−20.2373	−0.777781	−0.388891	+	0.921284i	$0.627142\pi$
$678$	0	0
$679$	−8.37746	−0.321497
$680$	0	0
$681$	18.7218	0.717422
$682$	0	0
$683$	−14.0301	−0.536848	−0.268424	−	0.963301i	$-0.586503\pi$
$683$	−14.0301	−0.536848	−0.268424	+	0.963301i	$0.586503\pi$
$684$	0	0
$685$	−0.158477	−0.00605511
$686$	0	0
$687$	−0.349923	−0.0133504
$688$	0	0
$689$	5.05858	0.192717
$690$	0	0
$691$	−11.9198	−0.453451	−0.226725	−	0.973959i	$-0.572802\pi$
$691$	−11.9198	−0.453451	−0.226725	+	0.973959i	$0.572802\pi$
$692$	0	0
$693$	8.76458	0.332939
$694$	0	0
$695$	−20.7671	−0.787741
$696$	0	0
$697$	−41.2338	−1.56184
$698$	0	0
$699$	12.5993	0.476550
$700$	0	0
$701$	9.91398	0.374446	0.187223	−	0.982317i	$-0.440051\pi$
$701$	9.91398	0.374446	0.187223	+	0.982317i	$0.440051\pi$
$702$	0	0
$703$	6.71659	0.253321
$704$	0	0
$705$	1.62254	0.0611084
$706$	0	0
$707$	3.61540	0.135971
$708$	0	0
$709$	−17.5122	−0.657686	−0.328843	−	0.944385i	$-0.606659\pi$
$709$	−17.5122	−0.657686	−0.328843	+	0.944385i	$0.606659\pi$
$710$	0	0
$711$	2.67707	0.100398
$712$	0	0
$713$	17.0175	0.637310
$714$	0	0
$715$	−5.76039	−0.215426
$716$	0	0
$717$	−2.60406	−0.0972506
$718$	0	0
$719$	−6.49703	−0.242298	−0.121149	−	0.992634i	$-0.538658\pi$
$719$	−6.49703	−0.242298	−0.121149	+	0.992634i	$0.538658\pi$
$720$	0	0
$721$	15.4421	0.575095
$722$	0	0
$723$	−25.3397	−0.942394
$724$	0	0
$725$	6.64610	0.246830
$726$	0	0
$727$	−39.9801	−1.48278	−0.741390	−	0.671074i	$-0.765833\pi$
$727$	−39.9801	−1.48278	−0.741390	+	0.671074i	$0.765833\pi$
$728$	0	0
$729$	23.2811	0.862262
$730$	0	0
$731$	−15.5029	−0.573397
$732$	0	0
$733$	−36.3708	−1.34339	−0.671693	−	0.740830i	$-0.734432\pi$
$733$	−36.3708	−1.34339	−0.671693	+	0.740830i	$0.734432\pi$
$734$	0	0
$735$	1.21592	0.0448501
$736$	0	0
$737$	41.9563	1.54548
$738$	0	0
$739$	−1.74696	−0.0642630	−0.0321315	−	0.999484i	$-0.510230\pi$
$739$	−1.74696	−0.0642630	−0.0321315	+	0.999484i	$0.510230\pi$
$740$	0	0
$741$	3.93821	0.144674
$742$	0	0
$743$	−11.6190	−0.426259	−0.213129	−	0.977024i	$-0.568366\pi$
$743$	−11.6190	−0.426259	−0.213129	+	0.977024i	$0.568366\pi$
$744$	0	0
$745$	21.4664	0.786467
$746$	0	0
$747$	15.4161	0.564045
$748$	0	0
$749$	1.91641	0.0700241
$750$	0	0
$751$	24.7968	0.904849	0.452424	−	0.891803i	$-0.350559\pi$
$751$	24.7968	0.904849	0.452424	+	0.891803i	$0.350559\pi$
$752$	0	0
$753$	1.25791	0.0458409
$754$	0	0
$755$	−6.42219	−0.233727
$756$	0	0
$757$	6.75492	0.245512	0.122756	−	0.992437i	$-0.460827\pi$
$757$	6.75492	0.245512	0.122756	+	0.992437i	$0.460827\pi$
$758$	0	0
$759$	64.6858	2.34794
$760$	0	0
$761$	8.66104	0.313962	0.156981	−	0.987602i	$-0.449824\pi$
$761$	8.66104	0.313962	0.156981	+	0.987602i	$0.449824\pi$
$762$	0	0
$763$	−8.08611	−0.292737
$764$	0	0
$765$	5.20984	0.188362
$766$	0	0
$767$	5.78827	0.209002
$768$	0	0
$769$	−43.1871	−1.55737	−0.778683	−	0.627418i	$-0.784112\pi$
$769$	−43.1871	−1.55737	−0.778683	+	0.627418i	$0.784112\pi$
$770$	0	0
$771$	32.8907	1.18453
$772$	0	0
$773$	28.6635	1.03095	0.515477	−	0.856903i	$-0.327615\pi$
$773$	28.6635	1.03095	0.515477	+	0.856903i	$0.327615\pi$
$774$	0	0
$775$	1.84266	0.0661903
$776$	0	0
$777$	−2.52153	−0.0904593
$778$	0	0
$779$	−39.0033	−1.39744
$780$	0	0
$781$	−36.4790	−1.30532
$782$	0	0
$783$	36.5392	1.30580
$784$	0	0
$785$	−13.8755	−0.495238
$786$	0	0
$787$	−23.3225	−0.831358	−0.415679	−	0.909511i	$-0.636456\pi$
$787$	−23.3225	−0.831358	−0.415679	+	0.909511i	$0.636456\pi$
$788$	0	0
$789$	−31.7798	−1.13139
$790$	0	0
$791$	−11.4360	−0.406619
$792$	0	0
$793$	9.00420	0.319748
$794$	0	0
$795$	−6.15086	−0.218148
$796$	0	0
$797$	−20.6001	−0.729693	−0.364846	−	0.931068i	$-0.618879\pi$
$797$	−20.6001	−0.729693	−0.364846	+	0.931068i	$0.618879\pi$
$798$	0	0
$799$	4.56912	0.161644
$800$	0	0
$801$	13.9306	0.492213
$802$	0	0
$803$	1.07352	0.0378838
$804$	0	0
$805$	−9.23529	−0.325501
$806$	0	0
$807$	18.9872	0.668379
$808$	0	0
$809$	−9.06916	−0.318855	−0.159427	−	0.987210i	$-0.550965\pi$
$809$	−9.06916	−0.318855	−0.159427	+	0.987210i	$0.550965\pi$
$810$	0	0
$811$	−26.5630	−0.932753	−0.466376	−	0.884586i	$-0.654441\pi$
$811$	−26.5630	−0.932753	−0.466376	+	0.884586i	$0.654441\pi$
$812$	0	0
$813$	−14.5510	−0.510325
$814$	0	0
$815$	5.79793	0.203093
$816$	0	0
$817$	−14.6643	−0.513039
$818$	0	0
$819$	1.52153	0.0531664
$820$	0	0
$821$	37.1432	1.29631	0.648153	−	0.761510i	$-0.275542\pi$
$821$	37.1432	1.29631	0.648153	+	0.761510i	$0.275542\pi$
$822$	0	0
$823$	25.3832	0.884804	0.442402	−	0.896817i	$-0.354127\pi$
$823$	25.3832	0.884804	0.442402	+	0.896817i	$0.354127\pi$
$824$	0	0
$825$	7.00420	0.243855
$826$	0	0
$827$	−41.2224	−1.43344	−0.716722	−	0.697359i	$-0.754358\pi$
$827$	−41.2224	−1.43344	−0.716722	+	0.697359i	$0.754358\pi$
$828$	0	0
$829$	−22.3319	−0.775619	−0.387809	−	0.921740i	$-0.626768\pi$
$829$	−22.3319	−0.775619	−0.387809	+	0.921740i	$0.626768\pi$
$830$	0	0
$831$	−27.3005	−0.947042
$832$	0	0
$833$	3.42408	0.118637
$834$	0	0
$835$	10.3424	0.357913
$836$	0	0
$837$	10.1306	0.350166
$838$	0	0
$839$	−28.7826	−0.993686	−0.496843	−	0.867840i	$-0.665508\pi$
$839$	−28.7826	−0.993686	−0.496843	+	0.867840i	$0.665508\pi$
$840$	0	0
$841$	15.1706	0.523125
$842$	0	0
$843$	22.5865	0.777920
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	22.1820	0.762184
$848$	0	0
$849$	−0.953856	−0.0327363
$850$	0	0
$851$	19.1517	0.656512
$852$	0	0
$853$	4.98393	0.170646	0.0853232	−	0.996353i	$-0.472808\pi$
$853$	4.98393	0.170646	0.0853232	+	0.996353i	$0.472808\pi$
$854$	0	0
$855$	4.92801	0.168534
$856$	0	0
$857$	28.4706	0.972538	0.486269	−	0.873809i	$-0.338358\pi$
$857$	28.4706	0.972538	0.486269	+	0.873809i	$0.338358\pi$
$858$	0	0
$859$	−11.7352	−0.400400	−0.200200	−	0.979755i	$-0.564159\pi$
$859$	−11.7352	−0.400400	−0.200200	+	0.979755i	$0.564159\pi$
$860$	0	0
$861$	14.6425	0.499016
$862$	0	0
$863$	−27.8928	−0.949483	−0.474741	−	0.880125i	$-0.657458\pi$
$863$	−27.8928	−0.949483	−0.474741	+	0.880125i	$0.657458\pi$
$864$	0	0
$865$	−11.1083	−0.377693
$866$	0	0
$867$	6.41479	0.217858
$868$	0	0
$869$	10.1352	0.343813
$870$	0	0
$871$	7.28359	0.246795
$872$	0	0
$873$	12.7465	0.431405
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	42.0606	1.42029	0.710143	−	0.704058i	$-0.248630\pi$
$877$	42.0606	1.42029	0.710143	+	0.704058i	$0.248630\pi$
$878$	0	0
$879$	−3.02292	−0.101961
$880$	0	0
$881$	38.1111	1.28400	0.641999	−	0.766706i	$-0.278106\pi$
$881$	38.1111	1.28400	0.641999	+	0.766706i	$0.278106\pi$
$882$	0	0
$883$	30.5837	1.02922	0.514612	−	0.857423i	$-0.327936\pi$
$883$	30.5837	1.02922	0.514612	+	0.857423i	$0.327936\pi$
$884$	0	0
$885$	−7.03810	−0.236583
$886$	0	0
$887$	10.9330	0.367093	0.183546	−	0.983011i	$-0.441242\pi$
$887$	10.9330	0.367093	0.183546	+	0.983011i	$0.441242\pi$
$888$	0	0
$889$	−21.1576	−0.709602
$890$	0	0
$891$	12.2142	0.409190
$892$	0	0
$893$	4.32196	0.144629
$894$	0	0
$895$	22.4812	0.751462
$896$	0	0
$897$	11.2294	0.374939
$898$	0	0
$899$	12.2465	0.408443
$900$	0	0
$901$	−17.3210	−0.577047
$902$	0	0
$903$	5.50524	0.183203
$904$	0	0
$905$	25.1517	0.836071
$906$	0	0
$907$	−3.73143	−0.123900	−0.0619501	−	0.998079i	$-0.519732\pi$
$907$	−3.73143	−0.123900	−0.0619501	+	0.998079i	$0.519732\pi$
$908$	0	0
$909$	−5.50093	−0.182454
$910$	0	0
$911$	−31.5481	−1.04524	−0.522618	−	0.852567i	$-0.675045\pi$
$911$	−31.5481	−1.04524	−0.522618	+	0.852567i	$0.675045\pi$
$912$	0	0
$913$	58.3641	1.93157
$914$	0	0
$915$	−10.9484	−0.361944
$916$	0	0
$917$	2.24075	0.0739961
$918$	0	0
$919$	49.2113	1.62333	0.811666	−	0.584122i	$-0.198561\pi$
$919$	49.2113	1.62333	0.811666	+	0.584122i	$0.198561\pi$
$920$	0	0
$921$	−35.2668	−1.16208
$922$	0	0
$923$	−6.33273	−0.208444
$924$	0	0
$925$	2.07375	0.0681846
$926$	0	0
$927$	−23.4956	−0.771697
$928$	0	0
$929$	24.8680	0.815892	0.407946	−	0.913006i	$-0.366245\pi$
$929$	24.8680	0.815892	0.407946	+	0.913006i	$0.366245\pi$
$930$	0	0
$931$	3.23886	0.106149
$932$	0	0
$933$	−10.1658	−0.332813
$934$	0	0
$935$	19.7240	0.645045
$936$	0	0
$937$	−6.14670	−0.200804	−0.100402	−	0.994947i	$-0.532013\pi$
$937$	−6.14670	−0.200804	−0.100402	+	0.994947i	$0.532013\pi$
$938$	0	0
$939$	11.9089	0.388634
$940$	0	0
$941$	48.5207	1.58173	0.790864	−	0.611991i	$-0.209631\pi$
$941$	48.5207	1.58173	0.790864	+	0.611991i	$0.209631\pi$
$942$	0	0
$943$	−111.214	−3.62163
$944$	0	0
$945$	−5.49784	−0.178845
$946$	0	0
$947$	2.91715	0.0947946	0.0473973	−	0.998876i	$-0.484907\pi$
$947$	2.91715	0.0947946	0.0473973	+	0.998876i	$0.484907\pi$
$948$	0	0
$949$	0.186363	0.00604959
$950$	0	0
$951$	32.9568	1.06870
$952$	0	0
$953$	27.9625	0.905795	0.452897	−	0.891563i	$-0.350390\pi$
$953$	27.9625	0.905795	0.452897	+	0.891563i	$0.350390\pi$
$954$	0	0
$955$	−8.78143	−0.284160
$956$	0	0
$957$	46.5506	1.50477
$958$	0	0
$959$	0.158477	0.00511750
$960$	0	0
$961$	−27.6046	−0.890471
$962$	0	0
$963$	−2.91587	−0.0939626
$964$	0	0
$965$	6.56039	0.211186
$966$	0	0
$967$	45.5773	1.46567	0.732834	−	0.680407i	$-0.238197\pi$
$967$	45.5773	1.46567	0.732834	+	0.680407i	$0.238197\pi$
$968$	0	0
$969$	−13.4848	−0.433193
$970$	0	0
$971$	4.82863	0.154958	0.0774790	−	0.996994i	$-0.475313\pi$
$971$	4.82863	0.154958	0.0774790	+	0.996994i	$0.475313\pi$
$972$	0	0
$973$	20.7671	0.665763
$974$	0	0
$975$	1.21592	0.0389408
$976$	0	0
$977$	−13.4104	−0.429037	−0.214518	−	0.976720i	$-0.568818\pi$
$977$	−13.4104	−0.429037	−0.214518	+	0.976720i	$0.568818\pi$
$978$	0	0
$979$	52.7402	1.68558
$980$	0	0
$981$	12.3032	0.392812
$982$	0	0
$983$	12.2271	0.389983	0.194991	−	0.980805i	$-0.437532\pi$
$983$	12.2271	0.389983	0.194991	+	0.980805i	$0.437532\pi$
$984$	0	0
$985$	21.5945	0.688059
$986$	0	0
$987$	−1.62254	−0.0516460
$988$	0	0
$989$	−41.8138	−1.32960
$990$	0	0
$991$	−16.9075	−0.537085	−0.268542	−	0.963268i	$-0.586542\pi$
$991$	−16.9075	−0.537085	−0.268542	+	0.963268i	$0.586542\pi$
$992$	0	0
$993$	−41.9680	−1.33181
$994$	0	0
$995$	−26.2336	−0.831661
$996$	0	0
$997$	13.4648	0.426434	0.213217	−	0.977005i	$-0.431606\pi$
$997$	13.4648	0.426434	0.213217	+	0.977005i	$0.431606\pi$
$998$	0	0
$999$	11.4012	0.360716

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.y.1.2	✓	5
4.3	odd	2		7280.2.a.cc.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.y.1.2	✓	5	1.1	even	1	trivial
7280.2.a.cc.1.4		5	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q-1.21592 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.52153 q^{9} +O(q^{10})\)	\(q-1.21592 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.52153 q^{9} -5.76039 q^{11} -1.00000 q^{13} +1.21592 q^{15} +3.42408 q^{17} +3.23886 q^{19} -1.21592 q^{21} +9.23529 q^{23} +1.00000 q^{25} +5.49784 q^{27} +6.64610 q^{29} +1.84266 q^{31} +7.00420 q^{33} -1.00000 q^{35} +2.07375 q^{37} +1.21592 q^{39} -12.0423 q^{41} -4.52762 q^{43} +1.52153 q^{45} +1.33441 q^{47} +1.00000 q^{49} -4.16343 q^{51} -5.05858 q^{53} +5.76039 q^{55} -3.93821 q^{57} -5.78827 q^{59} -9.00420 q^{61} -1.52153 q^{63} +1.00000 q^{65} -7.28359 q^{67} -11.2294 q^{69} +6.33273 q^{71} -0.186363 q^{73} -1.21592 q^{75} -5.76039 q^{77} -1.75947 q^{79} -2.12038 q^{81} -10.1320 q^{83} -3.42408 q^{85} -8.08116 q^{87} -9.15566 q^{89} -1.00000 q^{91} -2.24053 q^{93} -3.23886 q^{95} -8.37746 q^{97} +8.76458 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} + 5 q^{7} + q^{9}+O(q^{10}) \) 5 * q - 5 * q^5 + 5 * q^7 + q^9	\( 5 q - 5 q^{5} + 5 q^{7} + q^{9} - 7 q^{11} - 5 q^{13} + q^{17} + 3 q^{19} - 5 q^{23} + 5 q^{25} - 9 q^{27} - 9 q^{29} + 6 q^{31} + q^{33} - 5 q^{35} - 10 q^{37} - 8 q^{41} + 6 q^{43} - q^{45} - 13 q^{47} + 5 q^{49} - 4 q^{51} - 16 q^{53} + 7 q^{55} - 10 q^{57} - q^{59} - 11 q^{61} + q^{63} + 5 q^{65} - 30 q^{67} - 15 q^{69} - 12 q^{71} + 23 q^{73} - 7 q^{77} + 2 q^{79} - 11 q^{81} - 2 q^{83} - q^{85} - 5 q^{87} - 25 q^{89} - 5 q^{91} - 22 q^{93} - 3 q^{95} - 5 q^{97} - 12 q^{99}+O(q^{100}) \) 5 * q - 5 * q^5 + 5 * q^7 + q^9 - 7 * q^11 - 5 * q^13 + q^17 + 3 * q^19 - 5 * q^23 + 5 * q^25 - 9 * q^27 - 9 * q^29 + 6 * q^31 + q^33 - 5 * q^35 - 10 * q^37 - 8 * q^41 + 6 * q^43 - q^45 - 13 * q^47 + 5 * q^49 - 4 * q^51 - 16 * q^53 + 7 * q^55 - 10 * q^57 - q^59 - 11 * q^61 + q^63 + 5 * q^65 - 30 * q^67 - 15 * q^69 - 12 * q^71 + 23 * q^73 - 7 * q^77 + 2 * q^79 - 11 * q^81 - 2 * q^83 - q^85 - 5 * q^87 - 25 * q^89 - 5 * q^91 - 22 * q^93 - 3 * q^95 - 5 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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