LMFDB - Embedded newform 640.4.l.a.161.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [640,4,Mod(161,640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(640, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("640.161");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 640 = 2^{7} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	640.l (of order $4$, degree $2$, not minimal)

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:	no
Analytic conductor:	$37.7612224037$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			161.15
Character	$\chi$	$=$	640.161
Dual form			640.4.l.a.481.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.30312 + 1.30312i) q^{3} +(-3.53553 + 3.53553i) q^{5} +17.9744i q^{7} -23.6037i q^{9} +(9.66542 - 9.66542i) q^{11} +(-19.1969 - 19.1969i) q^{13} -9.21448 q^{15} +103.565 q^{17} +(16.2553 + 16.2553i) q^{19} +(-23.4229 + 23.4229i) q^{21} +49.6583i q^{23} -25.0000i q^{25} +(65.9429 - 65.9429i) q^{27} +(68.7899 + 68.7899i) q^{29} +249.904 q^{31} +25.1905 q^{33} +(-63.5492 - 63.5492i) q^{35} +(-210.860 + 210.860i) q^{37} -50.0318i q^{39} +129.785i q^{41} +(-344.071 + 344.071i) q^{43} +(83.4518 + 83.4518i) q^{45} +174.483 q^{47} +19.9202 q^{49} +(134.958 + 134.958i) q^{51} +(-322.832 + 322.832i) q^{53} +68.3448i q^{55} +42.3653i q^{57} +(-21.0376 + 21.0376i) q^{59} +(-489.653 - 489.653i) q^{61} +424.264 q^{63} +135.742 q^{65} +(261.470 + 261.470i) q^{67} +(-64.7110 + 64.7110i) q^{69} +218.795i q^{71} +1085.01i q^{73} +(32.5781 - 32.5781i) q^{75} +(173.730 + 173.730i) q^{77} +1155.60 q^{79} -465.437 q^{81} +(777.042 + 777.042i) q^{83} +(-366.157 + 366.157i) q^{85} +179.283i q^{87} -15.9331i q^{89} +(345.052 - 345.052i) q^{91} +(325.656 + 325.656i) q^{93} -114.942 q^{95} -923.624 q^{97} +(-228.140 - 228.140i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 40 q^{11} - 120 q^{15} + 24 q^{19} - 264 q^{27} - 400 q^{29} - 16 q^{37} + 808 q^{43} - 1880 q^{47} - 2352 q^{49} + 2144 q^{51} - 752 q^{53} - 2728 q^{59} + 912 q^{61} + 2520 q^{63} - 2040 q^{67}+ \cdots - 4456 q^{99}+O(q^{100}) $ 48 * q - 40 * q^11 - 120 * q^15 + 24 * q^19 - 264 * q^27 - 400 * q^29 - 16 * q^37 + 808 * q^43 - 1880 * q^47 - 2352 * q^49 + 2144 * q^51 - 752 * q^53 - 2728 * q^59 + 912 * q^61 + 2520 * q^63 - 2040 * q^67 + 528 * q^69 - 1904 * q^77 - 2832 * q^79 - 3888 * q^81 + 2440 * q^83 + 240 * q^85 - 4448 * q^91 - 4272 * q^93 + 3040 * q^95 - 4456 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/640\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$

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
$3$	1.30312	+	1.30312i	0.250786	+	0.250786i	0.821293	−	0.570507i	$-0.193253\pi$
$3$	1.30312	+	1.30312i	0.250786	+	0.250786i	−0.570507	+	0.821293i	$0.693253\pi$
$4$		0			0
$5$	−3.53553	+	3.53553i	−0.316228	+	0.316228i
$5$	−3.53553	+	3.53553i	−0.316228	+	0.316228i
$6$		0			0
$7$			17.9744i			0.970528i	0.874368	+	0.485264i	$0.161276\pi$
$7$			17.9744i			0.970528i	−0.874368	+	0.485264i	$0.838724\pi$
$8$		0			0
$9$		−	23.6037i		−	0.874213i
$10$		0			0
$11$	9.66542	−	9.66542i	0.264930	−	0.264930i	−0.562123	−	0.827054i	$-0.690015\pi$
$11$	9.66542	−	9.66542i	0.264930	−	0.264930i	0.827054	+	0.562123i	$0.190015\pi$
$12$		0			0
$13$	−19.1969	−	19.1969i	−0.409558	−	0.409558i	0.472027	−	0.881584i	$-0.343523\pi$
$13$	−19.1969	−	19.1969i	−0.409558	−	0.409558i	−0.881584	+	0.472027i	$0.843523\pi$
$14$		0			0
$15$	−9.21448			−0.158611
$16$		0			0
$17$	103.565			1.47754			0.738770	−	0.673957i	$-0.235407\pi$
$17$	103.565			1.47754			0.738770	+	0.673957i	$0.235407\pi$
$18$		0			0
$19$	16.2553	+	16.2553i	0.196275	+	0.196275i	0.798401	−	0.602126i	$-0.205680\pi$
$19$	16.2553	+	16.2553i	0.196275	+	0.196275i	−0.602126	+	0.798401i	$0.705680\pi$
$20$		0			0
$21$	−23.4229	+	23.4229i	−0.243395	+	0.243395i
$22$		0			0
$23$			49.6583i			0.450195i	0.974336	+	0.225097i	$0.0722700\pi$
$23$			49.6583i			0.450195i	−0.974336	+	0.225097i	$0.927730\pi$
$24$		0			0
$25$		−	25.0000i		−	0.200000i
$26$		0			0
$27$	65.9429	−	65.9429i	0.470027	−	0.470027i
$28$		0			0
$29$	68.7899	+	68.7899i	0.440481	+	0.440481i	0.892174	−	0.451692i	$-0.149180\pi$
$29$	68.7899	+	68.7899i	0.440481	+	0.440481i	−0.451692	+	0.892174i	$0.649180\pi$
$30$		0			0
$31$	249.904			1.44787			0.723937	−	0.689866i	$-0.242330\pi$
$31$	249.904			1.44787			0.723937	+	0.689866i	$0.242330\pi$
$32$		0			0
$33$	25.1905			0.132882
$34$		0			0
$35$	−63.5492	−	63.5492i	−0.306908	−	0.306908i
$36$		0			0
$37$	−210.860	+	210.860i	−0.936897	+	0.936897i	−0.998124	−	0.0612269i	$-0.980499\pi$
$37$	−210.860	+	210.860i	−0.936897	+	0.936897i	0.0612269	+	0.998124i	$0.480499\pi$
$38$		0			0
$39$		−	50.0318i		−	0.205423i
$40$		0			0
$41$			129.785i			0.494367i	0.968969	+	0.247183i	$0.0795050\pi$
$41$			129.785i			0.494367i	−0.968969	+	0.247183i	$0.920495\pi$
$42$		0			0
$43$	−344.071	+	344.071i	−1.22024	+	1.22024i	−0.252696	+	0.967546i	$0.581317\pi$
$43$	−344.071	+	344.071i	−1.22024	+	1.22024i	−0.967546	+	0.252696i	$0.918683\pi$
$44$		0			0
$45$	83.4518	+	83.4518i	0.276450	+	0.276450i
$46$		0			0
$47$	174.483			0.541511			0.270755	−	0.962648i	$-0.412727\pi$
$47$	174.483			0.541511			0.270755	+	0.962648i	$0.412727\pi$
$48$		0			0
$49$	19.9202			0.0580764
$50$		0			0
$51$	134.958	+	134.958i	0.370547	+	0.370547i
$52$		0			0
$53$	−322.832	+	322.832i	−0.836687	+	0.836687i	−0.988421	−	0.151735i	$-0.951514\pi$
$53$	−322.832	+	322.832i	−0.836687	+	0.836687i	0.151735	+	0.988421i	$0.451514\pi$
$54$		0			0
$55$			68.3448i			0.167557i
$56$		0			0
$57$			42.3653i			0.0984461i
$58$		0			0
$59$	−21.0376	+	21.0376i	−0.0464213	+	0.0464213i	−0.729936	−	0.683515i	$-0.760450\pi$
$59$	−21.0376	+	21.0376i	−0.0464213	+	0.0464213i	0.683515	+	0.729936i	$0.260450\pi$
$60$		0			0
$61$	−489.653	−	489.653i	−1.02777	−	1.02777i	−0.999603	−	0.0281617i	$-0.991035\pi$
$61$	−489.653	−	489.653i	−1.02777	−	1.02777i	−0.0281617	−	0.999603i	$-0.508965\pi$
$62$		0			0
$63$	424.264			0.848447
$64$		0			0
$65$	135.742			0.259027
$66$		0			0
$67$	261.470	+	261.470i	0.476771	+	0.476771i	0.904097	−	0.427327i	$-0.140545\pi$
$67$	261.470	+	261.470i	0.476771	+	0.476771i	−0.427327	+	0.904097i	$0.640545\pi$
$68$		0			0
$69$	−64.7110	+	64.7110i	−0.112903	+	0.112903i
$70$		0			0
$71$			218.795i			0.365720i	0.983139	+	0.182860i	$0.0585356\pi$
$71$			218.795i			0.365720i	−0.983139	+	0.182860i	$0.941464\pi$
$72$		0			0
$73$			1085.01i			1.73960i	0.493401	+	0.869802i	$0.335754\pi$
$73$			1085.01i			1.73960i	−0.493401	+	0.869802i	$0.664246\pi$
$74$		0			0
$75$	32.5781	−	32.5781i	0.0501573	−	0.0501573i
$76$		0			0
$77$	173.730	+	173.730i	0.257122	+	0.257122i
$78$		0			0
$79$	1155.60			1.64576			0.822880	−	0.568215i	$-0.192366\pi$
$79$	1155.60			1.64576			0.822880	+	0.568215i	$0.192366\pi$
$80$		0			0
$81$	−465.437			−0.638460
$82$		0			0
$83$	777.042	+	777.042i	1.02761	+	1.02761i	0.999608	+	0.0280004i	$0.00891397\pi$
$83$	777.042	+	777.042i	1.02761	+	1.02761i	0.0280004	+	0.999608i	$0.491086\pi$
$84$		0			0
$85$	−366.157	+	366.157i	−0.467239	+	0.467239i
$86$		0			0
$87$			179.283i			0.220933i
$88$		0			0
$89$		−	15.9331i		−	0.0189765i	−0.999955	−	0.00948823i	$-0.996980\pi$
$89$		−	15.9331i		−	0.0189765i	0.999955	−	0.00948823i	$-0.00302024\pi$
$90$		0			0
$91$	345.052	−	345.052i	0.397487	−	0.397487i
$92$		0			0
$93$	325.656	+	325.656i	0.363107	+	0.363107i
$94$		0			0
$95$	−114.942			−0.124135
$96$		0			0
$97$	−923.624			−0.966802			−0.483401	−	0.875399i	$-0.660599\pi$
$97$	−923.624			−0.966802			−0.483401	+	0.875399i	$0.660599\pi$
$98$		0			0
$99$	−228.140	−	228.140i	−0.231605	−	0.231605i
$100$		0			0
$101$	385.353	−	385.353i	0.379644	−	0.379644i	−0.491330	−	0.870974i	$-0.663489\pi$
$101$	385.353	−	385.353i	0.379644	−	0.379644i	0.870974	+	0.491330i	$0.163489\pi$
$102$		0			0
$103$		−	1445.05i		−	1.38238i	−0.722672	−	0.691191i	$-0.757086\pi$
$103$		−	1445.05i		−	1.38238i	0.722672	−	0.691191i	$-0.242914\pi$
$104$		0			0
$105$		−	165.625i		−	0.153936i
$106$		0			0
$107$	488.994	−	488.994i	0.441802	−	0.441802i	−0.450815	−	0.892617i	$-0.648867\pi$
$107$	488.994	−	488.994i	0.441802	−	0.441802i	0.892617	+	0.450815i	$0.148867\pi$
$108$		0			0
$109$	640.953	+	640.953i	0.563231	+	0.563231i	0.930224	−	0.366993i	$-0.119613\pi$
$109$	640.953	+	640.953i	0.563231	+	0.563231i	−0.366993	+	0.930224i	$0.619613\pi$
$110$		0			0
$111$	−549.554			−0.469922
$112$		0			0
$113$	214.312			0.178414			0.0892069	−	0.996013i	$-0.471567\pi$
$113$	214.312			0.178414			0.0892069	+	0.996013i	$0.471567\pi$
$114$		0			0
$115$	−175.569	−	175.569i	−0.142364	−	0.142364i
$116$		0			0
$117$	−453.118	+	453.118i	−0.358040	+	0.358040i
$118$		0			0
$119$			1861.52i			1.43399i
$120$		0			0
$121$			1144.16i			0.859624i
$122$		0			0
$123$	−169.126	+	169.126i	−0.123980	+	0.123980i
$124$		0			0
$125$	88.3883	+	88.3883i	0.0632456	+	0.0632456i
$126$		0			0
$127$	−68.9099			−0.0481478			−0.0240739	−	0.999710i	$-0.507664\pi$
$127$	−68.9099			−0.0481478			−0.0240739	+	0.999710i	$0.507664\pi$
$128$		0			0
$129$	−896.735			−0.612040
$130$		0			0
$131$	488.197	+	488.197i	0.325603	+	0.325603i	0.850912	−	0.525309i	$-0.176050\pi$
$131$	488.197	+	488.197i	0.325603	+	0.325603i	−0.525309	+	0.850912i	$0.676050\pi$
$132$		0			0
$133$	−292.180	+	292.180i	−0.190490	+	0.190490i
$134$		0			0
$135$			466.287i			0.297271i
$136$		0			0
$137$			2605.52i			1.62485i	0.583066	+	0.812425i	$0.301853\pi$
$137$			2605.52i			1.62485i	−0.583066	+	0.812425i	$0.698147\pi$
$138$		0			0
$139$	2138.54	−	2138.54i	1.30495	−	1.30495i	0.379944	−	0.925009i	$-0.375943\pi$
$139$	2138.54	−	2138.54i	1.30495	−	1.30495i	0.925009	−	0.379944i	$-0.124057\pi$
$140$		0			0
$141$	227.373	+	227.373i	0.135803	+	0.135803i
$142$		0			0
$143$	−371.091			−0.217009
$144$		0			0
$145$	−486.418			−0.278585
$146$		0			0
$147$	25.9585	+	25.9585i	0.0145648	+	0.0145648i
$148$		0			0
$149$	−1394.57	+	1394.57i	−0.766764	+	0.766764i	−0.977535	−	0.210771i	$-0.932402\pi$
$149$	−1394.57	+	1394.57i	−0.766764	+	0.766764i	0.210771	+	0.977535i	$0.432402\pi$
$150$		0			0
$151$			1946.09i			1.04881i	0.851469	+	0.524405i	$0.175712\pi$
$151$			1946.09i			1.04881i	−0.851469	+	0.524405i	$0.824288\pi$
$152$		0			0
$153$		−	2444.52i		−	1.29168i
$154$		0			0
$155$	−883.545	+	883.545i	−0.457858	+	0.457858i
$156$		0			0
$157$	514.931	+	514.931i	0.261758	+	0.261758i	0.825768	−	0.564010i	$-0.190742\pi$
$157$	514.931	+	514.931i	0.261758	+	0.261758i	−0.564010	+	0.825768i	$0.690742\pi$
$158$		0			0
$159$	−841.380			−0.419659
$160$		0			0
$161$	−892.580			−0.436926
$162$		0			0
$163$	1228.72	+	1228.72i	0.590433	+	0.590433i	0.937748	−	0.347316i	$-0.112907\pi$
$163$	1228.72	+	1228.72i	0.590433	+	0.590433i	−0.347316	+	0.937748i	$0.612907\pi$
$164$		0			0
$165$	−89.0618	+	89.0618i	−0.0420209	+	0.0420209i
$166$		0			0
$167$		−	587.588i		−	0.272269i	−0.990690	−	0.136135i	$-0.956532\pi$
$167$		−	587.588i		−	0.272269i	0.990690	−	0.136135i	$-0.0434679\pi$
$168$		0			0
$169$		−	1459.96i		−	0.664525i
$170$		0			0
$171$	383.686	−	383.686i	0.171586	−	0.171586i
$172$		0			0
$173$	−1062.06	−	1062.06i	−0.466744	−	0.466744i	0.434114	−	0.900858i	$-0.357062\pi$
$173$	−1062.06	−	1062.06i	−0.466744	−	0.466744i	−0.900858	+	0.434114i	$0.857062\pi$
$174$		0			0
$175$	449.361			0.194106
$176$		0			0
$177$	−54.8291			−0.0232836
$178$		0			0
$179$	1954.76	+	1954.76i	0.816234	+	0.816234i	0.985560	−	0.169326i	$-0.0541590\pi$
$179$	1954.76	+	1954.76i	0.816234	+	0.816234i	−0.169326	+	0.985560i	$0.554159\pi$
$180$		0			0
$181$	2384.68	−	2384.68i	0.979293	−	0.979293i	−0.0204973	−	0.999790i	$-0.506525\pi$
$181$	2384.68	−	2384.68i	0.979293	−	0.979293i	0.999790	+	0.0204973i	$0.00652494\pi$
$182$		0			0
$183$		−	1276.16i		−	0.515499i
$184$		0			0
$185$		−	1491.01i		−	0.592546i
$186$		0			0
$187$	1001.00	−	1001.00i	0.391445	−	0.391445i
$188$		0			0
$189$	1185.29	+	1185.29i	0.456174	+	0.456174i
$190$		0			0
$191$	−3326.75			−1.26029			−0.630145	−	0.776478i	$-0.717004\pi$
$191$	−3326.75			−1.26029			−0.630145	+	0.776478i	$0.717004\pi$
$192$		0			0
$193$	2489.93			0.928648			0.464324	−	0.885665i	$-0.346297\pi$
$193$	2489.93			0.928648			0.464324	+	0.885665i	$0.346297\pi$
$194$		0			0
$195$	176.889	+	176.889i	0.0649604	+	0.0649604i
$196$		0			0
$197$	1788.50	−	1788.50i	0.646829	−	0.646829i	−0.305396	−	0.952225i	$-0.598789\pi$
$197$	1788.50	−	1788.50i	0.646829	−	0.646829i	0.952225	+	0.305396i	$0.0987890\pi$
$198$		0			0
$199$			1891.36i			0.673743i	0.941551	+	0.336872i	$0.109369\pi$
$199$			1891.36i			0.673743i	−0.941551	+	0.336872i	$0.890631\pi$
$200$		0			0
$201$			681.455i			0.239135i
$202$		0			0
$203$	−1236.46	+	1236.46i	−0.427499	+	0.427499i
$204$		0			0
$205$	−458.860	−	458.860i	−0.156333	−	0.156333i
$206$		0			0
$207$	1172.12			0.393566
$208$		0			0
$209$	314.229			0.103998
$210$		0			0
$211$	−3007.32	−	3007.32i	−0.981197	−	0.981197i	0.0186292	−	0.999826i	$-0.494070\pi$
$211$	−3007.32	−	3007.32i	−0.981197	−	0.981197i	−0.999826	+	0.0186292i	$0.994070\pi$
$212$		0			0
$213$	−285.116	+	285.116i	−0.0917176	+	0.0917176i
$214$		0			0
$215$		−	2432.95i		−	0.771749i
$216$		0			0
$217$			4491.88i			1.40520i
$218$		0			0
$219$	−1413.91	+	1413.91i	−0.436269	+	0.436269i
$220$		0			0
$221$	−1988.12	−	1988.12i	−0.605138	−	0.605138i
$222$		0			0
$223$	5155.90			1.54827			0.774136	−	0.633019i	$-0.218185\pi$
$223$	5155.90			1.54827			0.774136	+	0.633019i	$0.218185\pi$
$224$		0			0
$225$	−590.093			−0.174843
$226$		0			0
$227$	−363.535	−	363.535i	−0.106294	−	0.106294i	0.651960	−	0.758254i	$-0.273947\pi$
$227$	−363.535	−	363.535i	−0.106294	−	0.106294i	−0.758254	+	0.651960i	$0.773947\pi$
$228$		0			0
$229$	2227.85	−	2227.85i	0.642884	−	0.642884i	−0.308380	−	0.951263i	$-0.599787\pi$
$229$	2227.85	−	2227.85i	0.642884	−	0.642884i	0.951263	+	0.308380i	$0.0997867\pi$
$230$		0			0
$231$			452.784i			0.128965i
$232$		0			0
$233$		−	3753.04i		−	1.05524i	−0.849482	−	0.527618i	$-0.823085\pi$
$233$		−	3753.04i		−	1.05524i	0.849482	−	0.527618i	$-0.176915\pi$
$234$		0			0
$235$	−616.892	+	616.892i	−0.171241	+	0.171241i
$236$		0			0
$237$	1505.89	+	1505.89i	0.412734	+	0.412734i
$238$		0			0
$239$	1005.32			0.272088			0.136044	−	0.990703i	$-0.456561\pi$
$239$	1005.32			0.272088			0.136044	+	0.990703i	$0.456561\pi$
$240$		0			0
$241$	−1393.93			−0.372576			−0.186288	−	0.982495i	$-0.559646\pi$
$241$	−1393.93			−0.372576			−0.186288	+	0.982495i	$0.559646\pi$
$242$		0			0
$243$	−2386.98	−	2386.98i	−0.630144	−	0.630144i
$244$		0			0
$245$	−70.4285	+	70.4285i	−0.0183654	+	0.0183654i
$246$		0			0
$247$		−	624.101i		−	0.160772i
$248$		0			0
$249$			2025.16i			0.515420i
$250$		0			0
$251$	−4032.99	+	4032.99i	−1.01418	+	1.01418i	−0.0142864	+	0.999898i	$0.504548\pi$
$251$	−4032.99	+	4032.99i	−1.01418	+	1.01418i	−0.999898	+	0.0142864i	$0.995452\pi$
$252$		0			0
$253$	479.969	+	479.969i	0.119270	+	0.119270i
$254$		0			0
$255$	−954.297			−0.234354
$256$		0			0
$257$	2814.94			0.683234			0.341617	−	0.939839i	$-0.389025\pi$
$257$	2814.94			0.683234			0.341617	+	0.939839i	$0.389025\pi$
$258$		0			0
$259$	−3790.09	−	3790.09i	−0.909284	−	0.909284i
$260$		0			0
$261$	1623.70	−	1623.70i	0.385074	−	0.385074i
$262$		0			0
$263$		−	5568.12i		−	1.30549i	−0.757576	−	0.652747i	$-0.773616\pi$
$263$		−	5568.12i		−	1.30549i	0.757576	−	0.652747i	$-0.226384\pi$
$264$		0			0
$265$		−	2282.77i		−	0.529167i
$266$		0			0
$267$	20.7628	−	20.7628i	0.00475904	−	0.00475904i
$268$		0			0
$269$	−3381.13	−	3381.13i	−0.766362	−	0.766362i	0.211102	−	0.977464i	$-0.432295\pi$
$269$	−3381.13	−	3381.13i	−0.766362	−	0.766362i	−0.977464	+	0.211102i	$0.932295\pi$
$270$		0			0
$271$	−8497.20			−1.90468			−0.952339	−	0.305040i	$-0.901330\pi$
$271$	−8497.20			−1.90468			−0.952339	+	0.305040i	$0.901330\pi$
$272$		0			0
$273$	899.292			0.199369
$274$		0			0
$275$	−241.635	−	241.635i	−0.0529861	−	0.0529861i
$276$		0			0
$277$	−776.937	+	776.937i	−0.168526	+	0.168526i	−0.786331	−	0.617805i	$-0.788022\pi$
$277$	−776.937	+	776.937i	−0.168526	+	0.168526i	0.617805	+	0.786331i	$0.288022\pi$
$278$		0			0
$279$		−	5898.67i		−	1.26575i
$280$		0			0
$281$		−	8079.78i		−	1.71530i	−0.514234	−	0.857650i	$-0.671924\pi$
$281$		−	8079.78i		−	1.71530i	0.514234	−	0.857650i	$-0.328076\pi$
$282$		0			0
$283$	−2189.72	+	2189.72i	−0.459949	+	0.459949i	−0.898639	−	0.438690i	$-0.855443\pi$
$283$	−2189.72	+	2189.72i	−0.459949	+	0.459949i	0.438690	+	0.898639i	$0.355443\pi$
$284$		0			0
$285$	−149.784	−	149.784i	−0.0311314	−	0.0311314i
$286$		0			0
$287$	−2332.81			−0.479797
$288$		0			0
$289$	5812.70			1.18313
$290$		0			0
$291$	−1203.60	−	1203.60i	−0.242461	−	0.242461i
$292$		0			0
$293$	−5552.61	+	5552.61i	−1.10712	+	1.10712i	−0.113596	+	0.993527i	$0.536237\pi$
$293$	−5552.61	+	5552.61i	−1.10712	+	1.10712i	−0.993527	+	0.113596i	$0.963763\pi$
$294$		0			0
$295$		−	148.758i		−	0.0293594i
$296$		0			0
$297$		−	1274.73i		−	0.249049i
$298$		0			0
$299$	953.284	−	953.284i	0.184381	−	0.184381i
$300$		0			0
$301$	−6184.48	−	6184.48i	−1.18428	−	1.18428i
$302$		0			0
$303$	1004.32			0.190419
$304$		0			0
$305$	3462.37			0.650016
$306$		0			0
$307$	−4301.90	−	4301.90i	−0.799747	−	0.799747i	0.183309	−	0.983055i	$-0.441319\pi$
$307$	−4301.90	−	4301.90i	−0.799747	−	0.799747i	−0.983055	+	0.183309i	$0.941319\pi$
$308$		0			0
$309$	1883.08	−	1883.08i	0.346682	−	0.346682i
$310$		0			0
$311$			4756.88i			0.867325i	0.901075	+	0.433663i	$0.142779\pi$
$311$			4756.88i			0.867325i	−0.901075	+	0.433663i	$0.857221\pi$
$312$		0			0
$313$		−	5433.84i		−	0.981274i	−0.871364	−	0.490637i	$-0.836764\pi$
$313$		−	5433.84i		−	0.981274i	0.871364	−	0.490637i	$-0.163236\pi$
$314$		0			0
$315$	−1500.00	+	1500.00i	−0.268303	+	0.268303i
$316$		0			0
$317$	728.663	+	728.663i	0.129103	+	0.129103i	0.768706	−	0.639602i	$-0.220901\pi$
$317$	728.663	+	728.663i	0.129103	+	0.129103i	−0.639602	+	0.768706i	$0.720901\pi$
$318$		0			0
$319$	1329.77			0.233394
$320$		0			0
$321$	1274.44			0.221596
$322$		0			0
$323$	1683.48	+	1683.48i	0.290004	+	0.290004i
$324$		0			0
$325$	−479.921	+	479.921i	−0.0819115	+	0.0819115i
$326$		0			0
$327$			1670.48i			0.282501i
$328$		0			0
$329$			3136.24i			0.525551i
$330$		0			0
$331$	2514.44	−	2514.44i	0.417541	−	0.417541i	−0.466814	−	0.884355i	$-0.654598\pi$
$331$	2514.44	−	2514.44i	0.417541	−	0.417541i	0.884355	+	0.466814i	$0.154598\pi$
$332$		0			0
$333$	4977.09	+	4977.09i	0.819047	+	0.819047i
$334$		0			0
$335$	−1848.87			−0.301536
$336$		0			0
$337$	82.5425			0.0133424			0.00667118	−	0.999978i	$-0.497876\pi$
$337$	82.5425			0.0133424			0.00667118	+	0.999978i	$0.497876\pi$
$338$		0			0
$339$	279.275	+	279.275i	0.0447438	+	0.0447438i
$340$		0			0
$341$	2415.43	−	2415.43i	0.383586	−	0.383586i
$342$		0			0
$343$			6523.28i			1.02689i
$344$		0			0
$345$		−	457.576i		−	0.0714059i
$346$		0			0
$347$	7202.76	−	7202.76i	1.11431	−	1.11431i	0.121745	−	0.992561i	$-0.461151\pi$
$347$	7202.76	−	7202.76i	1.11431	−	1.11431i	0.992561	−	0.121745i	$-0.0388490\pi$
$348$		0			0
$349$	2134.41	+	2134.41i	0.327370	+	0.327370i	0.851586	−	0.524215i	$-0.175641\pi$
$349$	2134.41	+	2134.41i	0.327370	+	0.327370i	−0.524215	+	0.851586i	$0.675641\pi$
$350$		0			0
$351$	−2531.79			−0.385006
$352$		0			0
$353$	705.357			0.106352			0.0531761	−	0.998585i	$-0.483066\pi$
$353$	705.357			0.106352			0.0531761	+	0.998585i	$0.483066\pi$
$354$		0			0
$355$	−773.556	−	773.556i	−0.115651	−	0.115651i
$356$		0			0
$357$	−2425.79	+	2425.79i	−0.359626	+	0.359626i
$358$		0			0
$359$		−	11015.7i		−	1.61946i	−0.586803	−	0.809730i	$-0.699614\pi$
$359$		−	11015.7i		−	1.61946i	0.586803	−	0.809730i	$-0.300386\pi$
$360$		0			0
$361$		−	6330.53i		−	0.922952i
$362$		0			0
$363$	−1490.98	+	1490.98i	−0.215582	+	0.215582i
$364$		0			0
$365$	−3836.10	−	3836.10i	−0.550111	−	0.550111i
$366$		0			0
$367$	12269.4			1.74512			0.872560	−	0.488506i	$-0.162458\pi$
$367$	12269.4			1.74512			0.872560	+	0.488506i	$0.162458\pi$
$368$		0			0
$369$	3063.42			0.432182
$370$		0			0
$371$	−5802.72	−	5802.72i	−0.812028	−	0.812028i
$372$		0			0
$373$	1651.92	−	1651.92i	0.229312	−	0.229312i	−0.583093	−	0.812405i	$-0.698158\pi$
$373$	1651.92	−	1651.92i	0.229312	−	0.229312i	0.812405	+	0.583093i	$0.198158\pi$
$374$		0			0
$375$			230.362i			0.0317222i
$376$		0			0
$377$		−	2641.10i		−	0.360805i
$378$		0			0
$379$	−2757.81	+	2757.81i	−0.373771	+	0.373771i	−0.868849	−	0.495078i	$-0.835140\pi$
$379$	−2757.81	+	2757.81i	−0.373771	+	0.373771i	0.495078	+	0.868849i	$0.335140\pi$
$380$		0			0
$381$	−89.7982	−	89.7982i	−0.0120748	−	0.0120748i
$382$		0			0
$383$	−10375.4			−1.38423			−0.692116	−	0.721787i	$-0.743321\pi$
$383$	−10375.4			−1.38423			−0.692116	+	0.721787i	$0.743321\pi$
$384$		0			0
$385$	−1228.46			−0.162618
$386$		0			0
$387$	8121.37	+	8121.37i	1.06675	+	1.06675i
$388$		0			0
$389$	2933.34	−	2933.34i	0.382330	−	0.382330i	−0.489611	−	0.871941i	$-0.662861\pi$
$389$	2933.34	−	2933.34i	0.382330	−	0.382330i	0.871941	+	0.489611i	$0.162861\pi$
$390$		0			0
$391$			5142.87i			0.665181i
$392$		0			0
$393$			1272.36i			0.163313i
$394$		0			0
$395$	−4085.66	+	4085.66i	−0.520435	+	0.520435i
$396$		0			0
$397$	2310.28	+	2310.28i	0.292065	+	0.292065i	0.837895	−	0.545831i	$-0.183786\pi$
$397$	2310.28	+	2310.28i	0.292065	+	0.292065i	−0.545831	+	0.837895i	$0.683786\pi$
$398$		0			0
$399$	−761.492			−0.0955446
$400$		0			0
$401$	−15396.7			−1.91740			−0.958698	−	0.284427i	$-0.908197\pi$
$401$	−15396.7			−1.91740			−0.958698	+	0.284427i	$0.908197\pi$
$402$		0			0
$403$	−4797.37	−	4797.37i	−0.592988	−	0.592988i
$404$		0			0
$405$	1645.57	−	1645.57i	0.201899	−	0.201899i
$406$		0			0
$407$			4076.10i			0.496425i
$408$		0			0
$409$			12575.6i			1.52036i	0.649715	+	0.760178i	$0.274888\pi$
$409$			12575.6i			1.52036i	−0.649715	+	0.760178i	$0.725112\pi$
$410$		0			0
$411$	−3395.31	+	3395.31i	−0.407490	+	0.407490i
$412$		0			0
$413$	−378.138	−	378.138i	−0.0450531	−	0.0450531i
$414$		0			0
$415$	−5494.52			−0.649917
$416$		0			0
$417$	5573.56			0.654529
$418$		0			0
$419$	8665.37	+	8665.37i	1.01034	+	1.01034i	0.999946	+	0.0103906i	$0.00330748\pi$
$419$	8665.37	+	8665.37i	1.01034	+	1.01034i	0.0103906	+	0.999946i	$0.496693\pi$
$420$		0			0
$421$	7662.55	−	7662.55i	0.887054	−	0.887054i	−0.107185	−	0.994239i	$-0.534184\pi$
$421$	7662.55	−	7662.55i	0.887054	−	0.887054i	0.994239	+	0.107185i	$0.0341837\pi$
$422$		0			0
$423$		−	4118.46i		−	0.473395i
$424$		0			0
$425$		−	2589.12i		−	0.295508i
$426$		0			0
$427$	8801.23	−	8801.23i	0.997474	−	0.997474i
$428$		0			0
$429$	−483.578	−	483.578i	−0.0544228	−	0.0544228i
$430$		0			0
$431$	1895.81			0.211874			0.105937	−	0.994373i	$-0.466216\pi$
$431$	1895.81			0.211874			0.105937	+	0.994373i	$0.466216\pi$
$432$		0			0
$433$	−2200.34			−0.244207			−0.122104	−	0.992517i	$-0.538964\pi$
$433$	−2200.34			−0.244207			−0.122104	+	0.992517i	$0.538964\pi$
$434$		0			0
$435$	−633.863	−	633.863i	−0.0698653	−	0.0698653i
$436$		0			0
$437$	−807.211	+	807.211i	−0.0883619	+	0.0883619i
$438$		0			0
$439$			6043.58i			0.657049i	0.944495	+	0.328524i	$0.106551\pi$
$439$			6043.58i			0.657049i	−0.944495	+	0.328524i	$0.893449\pi$
$440$		0			0
$441$		−	470.191i		−	0.0507711i
$442$		0			0
$443$	−2766.30	+	2766.30i	−0.296684	+	0.296684i	−0.839714	−	0.543030i	$-0.817277\pi$
$443$	−2766.30	+	2766.30i	−0.296684	+	0.296684i	0.543030	+	0.839714i	$0.317277\pi$
$444$		0			0
$445$	56.3320	+	56.3320i	0.00600088	+	0.00600088i
$446$		0			0
$447$	−3634.60			−0.384588
$448$		0			0
$449$	−17830.6			−1.87411			−0.937056	−	0.349179i	$-0.886461\pi$
$449$	−17830.6			−1.87411			−0.937056	+	0.349179i	$0.886461\pi$
$450$		0			0
$451$	1254.43	+	1254.43i	0.130973	+	0.130973i
$452$		0			0
$453$	−2535.99	+	2535.99i	−0.263027	+	0.263027i
$454$		0			0
$455$			2439.89i			0.251393i
$456$		0			0
$457$			14018.4i			1.43491i	0.696606	+	0.717454i	$0.254693\pi$
$457$			14018.4i			1.43491i	−0.696606	+	0.717454i	$0.745307\pi$
$458$		0			0
$459$	6829.38	−	6829.38i	0.694484	−	0.694484i
$460$		0			0
$461$	−2477.06	−	2477.06i	−0.250256	−	0.250256i	0.570819	−	0.821076i	$-0.306626\pi$
$461$	−2477.06	−	2477.06i	−0.250256	−	0.250256i	−0.821076	+	0.570819i	$0.806626\pi$
$462$		0			0
$463$	6279.68			0.630327			0.315163	−	0.949037i	$-0.397941\pi$
$463$	6279.68			0.630327			0.315163	+	0.949037i	$0.397941\pi$
$464$		0			0
$465$	−2302.74			−0.229649
$466$		0			0
$467$	5797.97	+	5797.97i	0.574514	+	0.574514i	0.933386	−	0.358873i	$-0.116839\pi$
$467$	5797.97	+	5797.97i	0.574514	+	0.574514i	−0.358873	+	0.933386i	$0.616839\pi$
$468$		0			0
$469$	−4699.77	+	4699.77i	−0.462719	+	0.462719i
$470$		0			0
$471$			1342.04i			0.131290i
$472$		0			0
$473$			6651.19i			0.646558i
$474$		0			0
$475$	406.383	−	406.383i	0.0392550	−	0.0392550i
$476$		0			0
$477$	7620.04	+	7620.04i	0.731442	+	0.731442i
$478$		0			0
$479$	−965.448			−0.0920927			−0.0460464	−	0.998939i	$-0.514662\pi$
$479$	−965.448			−0.0920927			−0.0460464	+	0.998939i	$0.514662\pi$
$480$		0			0
$481$	8095.70			0.767427
$482$		0			0
$483$	−1163.14	−	1163.14i	−0.109575	−	0.109575i
$484$		0			0
$485$	3265.50	−	3265.50i	0.305730	−	0.305730i
$486$		0			0
$487$		−	13452.2i		−	1.25170i	−0.779942	−	0.625852i	$-0.784752\pi$
$487$		−	13452.2i		−	1.25170i	0.779942	−	0.625852i	$-0.215248\pi$
$488$		0			0
$489$			3202.34i			0.296145i
$490$		0			0
$491$	−7716.82	+	7716.82i	−0.709277	+	0.709277i	−0.966383	−	0.257106i	$-0.917231\pi$
$491$	−7716.82	+	7716.82i	−0.709277	+	0.709277i	0.257106	+	0.966383i	$0.417231\pi$
$492$		0			0
$493$	7124.22	+	7124.22i	0.650829	+	0.650829i
$494$		0			0
$495$	1613.19			0.146480
$496$		0			0
$497$	−3932.71			−0.354942
$498$		0			0
$499$	3025.88	+	3025.88i	0.271457	+	0.271457i	0.829686	−	0.558230i	$-0.188519\pi$
$499$	3025.88	+	3025.88i	0.271457	+	0.271457i	−0.558230	+	0.829686i	$0.688519\pi$
$500$		0			0
$501$	765.700	−	765.700i	0.0682813	−	0.0682813i
$502$		0			0
$503$			7264.11i			0.643918i	0.946754	+	0.321959i	$0.104341\pi$
$503$			7264.11i			0.643918i	−0.946754	+	0.321959i	$0.895659\pi$
$504$		0			0
$505$			2724.86i			0.240108i
$506$		0			0
$507$	1902.51	−	1902.51i	0.166654	−	0.166654i
$508$		0			0
$509$	6106.12	+	6106.12i	0.531727	+	0.531727i	0.921086	−	0.389359i	$-0.127303\pi$
$509$	6106.12	+	6106.12i	0.531727	+	0.531727i	−0.389359	+	0.921086i	$0.627303\pi$
$510$		0			0
$511$	−19502.5			−1.68833
$512$		0			0
$513$	2143.84			0.184509
$514$		0			0
$515$	5109.04	+	5109.04i	0.437148	+	0.437148i
$516$		0			0
$517$	1686.45	−	1686.45i	0.143463	−	0.143463i
$518$		0			0
$519$		−	2767.98i		−	0.234106i
$520$		0			0
$521$		−	15120.8i		−	1.27151i	−0.771893	−	0.635753i	$-0.780690\pi$
$521$		−	15120.8i		−	1.27151i	0.771893	−	0.635753i	$-0.219310\pi$
$522$		0			0
$523$	15970.5	−	15970.5i	1.33526	−	1.33526i	0.434671	−	0.900589i	$-0.356865\pi$
$523$	15970.5	−	15970.5i	1.33526	−	1.33526i	0.900589	−	0.434671i	$-0.143135\pi$
$524$		0			0
$525$	585.572	+	585.572i	0.0486790	+	0.0486790i
$526$		0			0
$527$	25881.3			2.13929
$528$		0			0
$529$	9701.05			0.797325
$530$		0			0
$531$	496.565	+	496.565i	0.0405821	+	0.0405821i
$532$		0			0
$533$	2491.47	−	2491.47i	0.202472	−	0.202472i
$534$		0			0
$535$			3457.71i			0.279420i
$536$		0			0
$537$			5094.60i			0.409401i
$538$		0			0
$539$	192.537	−	192.537i	0.0153862	−	0.0153862i
$540$		0			0
$541$	−16156.7	−	16156.7i	−1.28398	−	1.28398i	−0.938385	−	0.345592i	$-0.887678\pi$
$541$	−16156.7	−	16156.7i	−1.28398	−	1.28398i	−0.345592	−	0.938385i	$-0.612322\pi$
$542$		0			0
$543$	6215.07			0.491186
$544$		0			0
$545$	−4532.22			−0.356218
$546$		0			0
$547$	6368.79	+	6368.79i	0.497824	+	0.497824i	0.910760	−	0.412936i	$-0.135497\pi$
$547$	6368.79	+	6368.79i	0.497824	+	0.497824i	−0.412936	+	0.910760i	$0.635497\pi$
$548$		0			0
$549$	−11557.6	+	11557.6i	−0.898485	+	0.898485i
$550$		0			0
$551$			2236.40i			0.172911i
$552$		0			0
$553$			20771.2i			1.59726i
$554$		0			0
$555$	1942.97	−	1942.97i	0.148602	−	0.148602i
$556$		0			0
$557$	9746.29	+	9746.29i	0.741407	+	0.741407i	0.972849	−	0.231442i	$-0.0743443\pi$
$557$	9746.29	+	9746.29i	0.741407	+	0.741407i	−0.231442	+	0.972849i	$0.574344\pi$
$558$		0			0
$559$	13210.2			0.999519
$560$		0			0
$561$	2608.85			0.196338
$562$		0			0
$563$	7196.60	+	7196.60i	0.538722	+	0.538722i	0.923154	−	0.384431i	$-0.125602\pi$
$563$	7196.60	+	7196.60i	0.538722	+	0.538722i	−0.384431	+	0.923154i	$0.625602\pi$
$564$		0			0
$565$	−757.707	+	757.707i	−0.0564194	+	0.0564194i
$566$		0			0
$567$		−	8365.97i		−	0.619643i
$568$		0			0
$569$		−	5236.83i		−	0.385834i	−0.981215	−	0.192917i	$-0.938205\pi$
$569$		−	5236.83i		−	0.385834i	0.981215	−	0.192917i	$-0.0617948\pi$
$570$		0			0
$571$	−7407.25	+	7407.25i	−0.542879	+	0.542879i	−0.924372	−	0.381493i	$-0.875410\pi$
$571$	−7407.25	+	7407.25i	−0.542879	+	0.542879i	0.381493	+	0.924372i	$0.375410\pi$
$572$		0			0
$573$	−4335.17	−	4335.17i	−0.316063	−	0.316063i
$574$		0			0
$575$	1241.46			0.0900390
$576$		0			0
$577$	−7726.80			−0.557488			−0.278744	−	0.960365i	$-0.589918\pi$
$577$	−7726.80			−0.557488			−0.278744	+	0.960365i	$0.589918\pi$
$578$		0			0
$579$	3244.69	+	3244.69i	0.232892	+	0.232892i
$580$		0			0
$581$	−13966.9	+	13966.9i	−0.997322	+	0.997322i
$582$		0			0
$583$			6240.62i			0.443327i
$584$		0			0
$585$		−	3204.03i		−	0.226445i
$586$		0			0
$587$	−8359.85	+	8359.85i	−0.587816	+	0.587816i	−0.937039	−	0.349224i	$-0.886445\pi$
$587$	−8359.85	+	8359.85i	−0.587816	+	0.587816i	0.349224	+	0.937039i	$0.386445\pi$
$588$		0			0
$589$	4062.27	+	4062.27i	0.284181	+	0.284181i
$590$		0			0
$591$	4661.27			0.324432
$592$		0			0
$593$	−8292.91			−0.574281			−0.287141	−	0.957888i	$-0.592705\pi$
$593$	−8292.91			−0.574281			−0.287141	+	0.957888i	$0.592705\pi$
$594$		0			0
$595$	−6581.47	−	6581.47i	−0.453469	−	0.453469i
$596$		0			0
$597$	−2464.68	+	2464.68i	−0.168966	+	0.168966i
$598$		0			0
$599$			27431.8i			1.87117i	0.353097	+	0.935587i	$0.385129\pi$
$599$			27431.8i			1.87117i	−0.353097	+	0.935587i	$0.614871\pi$
$600$		0			0
$601$		−	2790.89i		−	0.189422i	−0.995505	−	0.0947112i	$-0.969807\pi$
$601$		−	2790.89i		−	0.189422i	0.995505	−	0.0947112i	$-0.0301928\pi$
$602$		0			0
$603$	6171.67	−	6171.67i	0.416799	−	0.416799i
$604$		0			0
$605$	−4045.21	−	4045.21i	−0.271837	−	0.271837i
$606$		0			0
$607$	10317.1			0.689882			0.344941	−	0.938624i	$-0.387899\pi$
$607$	10317.1			0.689882			0.344941	+	0.938624i	$0.387899\pi$
$608$		0			0
$609$	−3222.52			−0.214422
$610$		0			0
$611$	−3349.53	−	3349.53i	−0.221780	−	0.221780i
$612$		0			0
$613$	2563.99	−	2563.99i	0.168938	−	0.168938i	−0.617575	−	0.786512i	$-0.711885\pi$
$613$	2563.99	−	2563.99i	0.168938	−	0.168938i	0.786512	+	0.617575i	$0.211885\pi$
$614$		0			0
$615$		−	1195.90i		−	0.0784121i
$616$		0			0
$617$			9519.59i			0.621141i	0.950550	+	0.310571i	$0.100520\pi$
$617$			9519.59i			0.621141i	−0.950550	+	0.310571i	$0.899480\pi$
$618$		0			0
$619$	5737.81	−	5737.81i	0.372572	−	0.372572i	−0.495841	−	0.868413i	$-0.665140\pi$
$619$	5737.81	−	5737.81i	0.372572	−	0.372572i	0.868413	+	0.495841i	$0.165140\pi$
$620$		0			0
$621$	3274.62	+	3274.62i	0.211604	+	0.211604i
$622$		0			0
$623$	286.388			0.0184172
$624$		0			0
$625$	−625.000			−0.0400000
$626$		0			0
$627$	409.479	+	409.479i	0.0260814	+	0.0260814i
$628$		0			0
$629$	−21837.7	+	21837.7i	−1.38430	+	1.38430i
$630$		0			0
$631$		−	2962.42i		−	0.186897i	−0.995624	−	0.0934485i	$-0.970211\pi$
$631$		−	2962.42i		−	0.186897i	0.995624	−	0.0934485i	$-0.0297890\pi$
$632$		0			0
$633$		−	7837.83i		−	0.492142i
$634$		0			0
$635$	243.633	−	243.633i	0.0152257	−	0.0152257i
$636$		0			0
$637$	−382.405	−	382.405i	−0.0237856	−	0.0237856i
$638$		0			0
$639$	5164.37			0.319717
$640$		0			0
$641$	−14378.2			−0.885964			−0.442982	−	0.896530i	$-0.646079\pi$
$641$	−14378.2			−0.885964			−0.442982	+	0.896530i	$0.646079\pi$
$642$		0			0
$643$	6022.70	+	6022.70i	0.369381	+	0.369381i	0.867251	−	0.497870i	$-0.165884\pi$
$643$	6022.70	+	6022.70i	0.369381	+	0.369381i	−0.497870	+	0.867251i	$0.665884\pi$
$644$		0			0
$645$	3170.44	−	3170.44i	0.193544	−	0.193544i
$646$		0			0
$647$			14764.5i			0.897147i	0.893746	+	0.448573i	$0.148068\pi$
$647$			14764.5i			0.897147i	−0.893746	+	0.448573i	$0.851932\pi$
$648$		0			0
$649$			406.674i			0.0245968i
$650$		0			0
$651$	−5853.48	+	5853.48i	−0.352405	+	0.352405i
$652$		0			0
$653$	−9815.04	−	9815.04i	−0.588197	−	0.588197i	0.348946	−	0.937143i	$-0.386540\pi$
$653$	−9815.04	−	9815.04i	−0.588197	−	0.588197i	−0.937143	+	0.348946i	$0.886540\pi$
$654$		0			0
$655$	−3452.08			−0.205929
$656$		0			0
$657$	25610.4			1.52078
$658$		0			0
$659$	−19820.3	−	19820.3i	−1.17161	−	1.17161i	−0.981826	−	0.189783i	$-0.939221\pi$
$659$	−19820.3	−	19820.3i	−1.17161	−	1.17161i	−0.189783	−	0.981826i	$-0.560779\pi$
$660$		0			0
$661$	−12939.4	+	12939.4i	−0.761397	+	0.761397i	−0.976575	−	0.215178i	$-0.930967\pi$
$661$	−12939.4	+	12939.4i	−0.761397	+	0.761397i	0.215178	+	0.976575i	$0.430967\pi$
$662$		0			0
$663$		−	5181.54i		−	0.303521i
$664$		0			0
$665$		−	2066.02i		−	0.120477i
$666$		0			0
$667$	−3415.99	+	3415.99i	−0.198302	+	0.198302i
$668$		0			0
$669$	6718.78	+	6718.78i	0.388285	+	0.388285i
$670$		0			0
$671$	−9465.41			−0.544572
$672$		0			0
$673$	−3822.26			−0.218926			−0.109463	−	0.993991i	$-0.534913\pi$
$673$	−3822.26			−0.218926			−0.109463	+	0.993991i	$0.534913\pi$
$674$		0			0
$675$	−1648.57	−	1648.57i	−0.0940053	−	0.0940053i
$676$		0			0
$677$	−20574.4	+	20574.4i	−1.16800	+	1.16800i	−0.185324	+	0.982677i	$0.559334\pi$
$677$	−20574.4	+	20574.4i	−1.16800	+	1.16800i	−0.982677	+	0.185324i	$0.940666\pi$
$678$		0			0
$679$		−	16601.6i		−	0.938308i
$680$		0			0
$681$		−	947.463i		−	0.0533140i
$682$		0			0
$683$	5442.92	−	5442.92i	0.304931	−	0.304931i	−0.538009	−	0.842939i	$-0.680823\pi$
$683$	5442.92	−	5442.92i	0.304931	−	0.304931i	0.842939	+	0.538009i	$0.180823\pi$
$684$		0			0
$685$	−9211.90	−	9211.90i	−0.513823	−	0.513823i
$686$		0			0
$687$	5806.33			0.322453
$688$		0			0
$689$	12394.7			0.685343
$690$		0			0
$691$	7826.48	+	7826.48i	0.430873	+	0.430873i	0.888925	−	0.458052i	$-0.151453\pi$
$691$	7826.48	+	7826.48i	0.430873	+	0.430873i	−0.458052	+	0.888925i	$0.651453\pi$
$692$		0			0
$693$	4100.69	−	4100.69i	0.224779	−	0.224779i
$694$		0			0
$695$			15121.8i			0.825325i
$696$		0			0
$697$			13441.2i			0.730447i
$698$		0			0
$699$	4890.68	−	4890.68i	0.264639	−	0.264639i
$700$		0			0
$701$	5762.25	+	5762.25i	0.310467	+	0.310467i	0.845090	−	0.534624i	$-0.179547\pi$
$701$	5762.25	+	5762.25i	0.310467	+	0.310467i	−0.534624	+	0.845090i	$0.679547\pi$
$702$		0			0
$703$	−6855.19			−0.367779
$704$		0			0
$705$	−1607.77			−0.0858896
$706$		0			0
$707$	6926.49	+	6926.49i	0.368455	+	0.368455i
$708$		0			0
$709$	16981.3	−	16981.3i	0.899502	−	0.899502i	−0.0958899	−	0.995392i	$-0.530570\pi$
$709$	16981.3	−	16981.3i	0.899502	−	0.899502i	0.995392	+	0.0958899i	$0.0305697\pi$
$710$		0			0
$711$		−	27276.5i		−	1.43874i
$712$		0			0
$713$			12409.8i			0.651826i
$714$		0			0
$715$	1312.01	−	1312.01i	0.0686241	−	0.0686241i
$716$		0			0
$717$	1310.06	+	1310.06i	0.0682359	+	0.0682359i
$718$		0			0
$719$	1795.42			0.0931262			0.0465631	−	0.998915i	$-0.485173\pi$
$719$	1795.42			0.0931262			0.0465631	+	0.998915i	$0.485173\pi$
$720$		0			0
$721$	25974.0			1.34164
$722$		0			0
$723$	−1816.46	−	1816.46i	−0.0934368	−	0.0934368i
$724$		0			0
$725$	1719.75	−	1719.75i	0.0880963	−	0.0880963i
$726$		0			0
$727$		−	15599.8i		−	0.795825i	−0.917423	−	0.397912i	$-0.869735\pi$
$727$		−	15599.8i		−	0.795825i	0.917423	−	0.397912i	$-0.130265\pi$
$728$		0			0
$729$			6345.74i			0.322397i
$730$		0			0
$731$	−35633.7	+	35633.7i	−1.80296	+	1.80296i
$732$		0			0
$733$	−12450.9	−	12450.9i	−0.627401	−	0.627401i	0.320013	−	0.947413i	$-0.396313\pi$
$733$	−12450.9	−	12450.9i	−0.627401	−	0.627401i	−0.947413	+	0.320013i	$0.896313\pi$
$734$		0			0
$735$	−183.554			−0.00921156
$736$		0			0
$737$	5054.43			0.252622
$738$		0			0
$739$	637.719	+	637.719i	0.0317441	+	0.0317441i	0.722801	−	0.691057i	$-0.242854\pi$
$739$	637.719	+	637.719i	0.0317441	+	0.0317441i	−0.691057	+	0.722801i	$0.742854\pi$
$740$		0			0
$741$	813.281	−	813.281i	0.0403193	−	0.0403193i
$742$		0			0
$743$		−	1952.47i		−	0.0964053i	−0.998838	−	0.0482026i	$-0.984651\pi$
$743$		−	1952.47i		−	0.0964053i	0.998838	−	0.0482026i	$-0.0153493\pi$
$744$		0			0
$745$		−	9861.12i		−	0.484944i
$746$		0			0
$747$	18341.1	−	18341.1i	0.898348	−	0.898348i
$748$		0			0
$749$	8789.38	+	8789.38i	0.428781	+	0.428781i
$750$		0			0
$751$	13710.1			0.666161			0.333081	−	0.942898i	$-0.391912\pi$
$751$	13710.1			0.666161			0.333081	+	0.942898i	$0.391912\pi$
$752$		0			0
$753$	−10511.0			−0.508687
$754$		0			0
$755$	−6880.46	−	6880.46i	−0.331663	−	0.331663i
$756$		0			0
$757$	−5425.87	+	5425.87i	−0.260510	+	0.260510i	−0.825261	−	0.564751i	$-0.808972\pi$
$757$	−5425.87	+	5425.87i	−0.260510	+	0.260510i	0.564751	+	0.825261i	$0.308972\pi$
$758$		0			0
$759$			1250.92i			0.0598227i
$760$		0			0
$761$		−	18568.0i		−	0.884481i	−0.896896	−	0.442241i	$-0.854184\pi$
$761$		−	18568.0i		−	0.884481i	0.896896	−	0.442241i	$-0.145816\pi$
$762$		0			0
$763$	−11520.8	+	11520.8i	−0.546631	+	0.546631i
$764$		0			0
$765$	8642.68	+	8642.68i	0.408467	+	0.408467i
$766$		0			0
$767$	807.710			0.0380244
$768$		0			0
$769$	13501.1			0.633110			0.316555	−	0.948574i	$-0.397474\pi$
$769$	13501.1			0.633110			0.316555	+	0.948574i	$0.397474\pi$
$770$		0			0
$771$	3668.22	+	3668.22i	0.171346	+	0.171346i
$772$		0			0
$773$	19002.7	−	19002.7i	0.884191	−	0.884191i	−0.109767	−	0.993957i	$-0.535010\pi$
$773$	19002.7	−	19002.7i	0.884191	−	0.884191i	0.993957	+	0.109767i	$0.0350103\pi$
$774$		0			0
$775$		−	6247.60i		−	0.289575i
$776$		0			0
$777$		−	9877.91i		−	0.456072i
$778$		0			0
$779$	−2109.70	+	2109.70i	−0.0970318	+	0.0970318i
$780$		0			0
$781$	2114.74	+	2114.74i	0.0968904	+	0.0968904i
$782$		0			0
$783$	9072.41			0.414076
$784$		0			0
$785$	−3641.11			−0.165550
$786$		0			0
$787$	−18237.9	−	18237.9i	−0.826061	−	0.826061i	0.160908	−	0.986969i	$-0.448558\pi$
$787$	−18237.9	−	18237.9i	−0.826061	−	0.826061i	−0.986969	+	0.160908i	$0.948558\pi$
$788$		0			0
$789$	7255.95	−	7255.95i	0.327400	−	0.327400i
$790$		0			0
$791$			3852.13i			0.173156i
$792$		0			0
$793$			18799.6i			0.841858i
$794$		0			0
$795$	2974.73	−	2974.73i	0.132708	−	0.132708i
$796$		0			0
$797$	−14529.0	−	14529.0i	−0.645727	−	0.645727i	0.306230	−	0.951958i	$-0.400932\pi$
$797$	−14529.0	−	14529.0i	−0.645727	−	0.645727i	−0.951958	+	0.306230i	$0.900932\pi$
$798$		0			0
$799$	18070.4			0.800104
$800$		0			0
$801$	−376.081			−0.0165895
$802$		0			0
$803$	10487.1	+	10487.1i	0.460874	+	0.460874i
$804$		0			0
$805$	3155.75	−	3155.75i	0.138168	−	0.138168i
$806$		0			0
$807$		−	8812.07i		−	0.384386i
$808$		0			0
$809$		−	35390.4i		−	1.53802i	−0.639234	−	0.769012i	$-0.720748\pi$
$809$		−	35390.4i		−	1.53802i	0.639234	−	0.769012i	$-0.279252\pi$
$810$		0			0
$811$	3671.23	−	3671.23i	0.158957	−	0.158957i	−0.623147	−	0.782104i	$-0.714146\pi$
$811$	3671.23	−	3671.23i	0.158957	−	0.158957i	0.782104	+	0.623147i	$0.214146\pi$
$812$		0			0
$813$	−11072.9	−	11072.9i	−0.477667	−	0.477667i
$814$		0			0
$815$	−8688.34			−0.373422
$816$		0			0
$817$	−11186.0			−0.479006
$818$		0			0
$819$	−8144.53	−	8144.53i	−0.347488	−	0.347488i
$820$		0			0
$821$	32787.6	−	32787.6i	1.39378	−	1.39378i	0.577131	−	0.816652i	$-0.304172\pi$
$821$	32787.6	−	32787.6i	1.39378	−	1.39378i	0.816652	−	0.577131i	$-0.195828\pi$
$822$		0			0
$823$			24158.6i			1.02323i	0.859216	+	0.511613i	$0.170952\pi$
$823$			24158.6i			1.02323i	−0.859216	+	0.511613i	$0.829048\pi$
$824$		0			0
$825$		−	629.762i		−	0.0265764i
$826$		0			0
$827$	−24624.9	+	24624.9i	−1.03542	+	1.03542i	−0.0360706	+	0.999349i	$0.511484\pi$
$827$	−24624.9	+	24624.9i	−1.03542	+	1.03542i	−0.999349	+	0.0360706i	$0.988516\pi$
$828$		0			0
$829$	−2211.20	−	2211.20i	−0.0926394	−	0.0926394i	0.659268	−	0.751908i	$-0.270866\pi$
$829$	−2211.20	−	2211.20i	−0.0926394	−	0.0926394i	−0.751908	+	0.659268i	$0.770866\pi$
$830$		0			0
$831$	−2024.89			−0.0845279
$832$		0			0
$833$	2063.03			0.0858102
$834$		0			0
$835$	2077.44	+	2077.44i	0.0860990	+	0.0860990i
$836$		0			0
$837$	16479.4	−	16479.4i	0.680540	−	0.680540i
$838$		0			0
$839$		−	14313.9i		−	0.589001i	−0.955651	−	0.294501i	$-0.904847\pi$
$839$		−	14313.9i		−	0.589001i	0.955651	−	0.294501i	$-0.0951533\pi$
$840$		0			0
$841$		−	14924.9i		−	0.611952i
$842$		0			0
$843$	10529.0	−	10529.0i	0.430174	−	0.430174i
$844$		0			0
$845$	5161.74	+	5161.74i	0.210141	+	0.210141i
$846$		0			0
$847$	−20565.6			−0.834289
$848$		0			0
$849$	−5706.96			−0.230698
$850$		0			0
$851$	−10471.0	−	10471.0i	−0.421786	−	0.421786i
$852$		0			0
$853$	21680.6	−	21680.6i	0.870257	−	0.870257i	−0.122243	−	0.992500i	$-0.539009\pi$
$853$	21680.6	−	21680.6i	0.870257	−	0.870257i	0.992500	+	0.122243i	$0.0390088\pi$
$854$		0			0
$855$			2713.07i			0.108520i
$856$		0			0
$857$		−	19741.2i		−	0.786870i	−0.919352	−	0.393435i	$-0.871287\pi$
$857$		−	19741.2i		−	0.786870i	0.919352	−	0.393435i	$-0.128713\pi$
$858$		0			0
$859$	−4986.65	+	4986.65i	−0.198070	+	0.198070i	−0.799172	−	0.601102i	$-0.794729\pi$
$859$	−4986.65	+	4986.65i	−0.198070	+	0.198070i	0.601102	+	0.799172i	$0.294729\pi$
$860$		0			0
$861$	−3039.94	−	3039.94i	−0.120326	−	0.120326i
$862$		0			0
$863$	−38207.3			−1.50706			−0.753529	−	0.657415i	$-0.771650\pi$
$863$	−38207.3			−1.50706			−0.753529	+	0.657415i	$0.771650\pi$
$864$		0			0
$865$	7509.88			0.295195
$866$		0			0
$867$	7574.67	+	7574.67i	0.296712	+	0.296712i
$868$		0			0
$869$	11169.4	−	11169.4i	0.436012	−	0.436012i
$870$		0			0
$871$		−	10038.8i		−	0.390530i
$872$		0			0
$873$			21801.0i			0.845190i
$874$		0			0
$875$	−1588.73	+	1588.73i	−0.0613815	+	0.0613815i
$876$		0			0
$877$	24917.7	+	24917.7i	0.959421	+	0.959421i	0.999208	−	0.0397868i	$-0.0126679\pi$
$877$	24917.7	+	24917.7i	0.959421	+	0.959421i	−0.0397868	+	0.999208i	$0.512668\pi$
$878$		0			0
$879$	−14471.5			−0.555302
$880$		0			0
$881$	−28327.0			−1.08327			−0.541635	−	0.840614i	$-0.682195\pi$
$881$	−28327.0			−1.08327			−0.541635	+	0.840614i	$0.682195\pi$
$882$		0			0
$883$	−18768.9	−	18768.9i	−0.715314	−	0.715314i	0.252328	−	0.967642i	$-0.418804\pi$
$883$	−18768.9	−	18768.9i	−0.715314	−	0.715314i	−0.967642	+	0.252328i	$0.918804\pi$
$884$		0			0
$885$	193.850	−	193.850i	0.00736293	−	0.00736293i
$886$		0			0
$887$			26017.3i			0.984865i	0.870351	+	0.492433i	$0.163892\pi$
$887$			26017.3i			0.984865i	−0.870351	+	0.492433i	$0.836108\pi$
$888$		0			0
$889$		−	1238.62i		−	0.0467288i
$890$		0			0
$891$	−4498.65	+	4498.65i	−0.169147	+	0.169147i
$892$		0			0
$893$	2836.28	+	2836.28i	0.106285	+	0.106285i
$894$		0			0
$895$	−13822.3			−0.516232
$896$		0			0
$897$	2484.49			0.0924803
$898$		0			0
$899$	17190.9	+	17190.9i	0.637762	+	0.637762i
$900$		0			0
$901$	−33434.1	+	33434.1i	−1.23624	+	1.23624i
$902$		0			0
$903$		−	16118.3i		−	0.594001i
$904$		0			0
$905$			16862.2i			0.619359i
$906$		0			0
$907$	26216.3	−	26216.3i	0.959757	−	0.959757i	−0.0394644	−	0.999221i	$-0.512565\pi$
$907$	26216.3	−	26216.3i	0.959757	−	0.959757i	0.999221	+	0.0394644i	$0.0125652\pi$
$908$		0			0
$909$	−9095.77	−	9095.77i	−0.331889	−	0.331889i
$910$		0			0
$911$	−1507.54			−0.0548265			−0.0274132	−	0.999624i	$-0.508727\pi$
$911$	−1507.54			−0.0548265			−0.0274132	+	0.999624i	$0.508727\pi$
$912$		0			0
$913$	15020.9			0.544489
$914$		0			0
$915$	4511.90	+	4511.90i	0.163015	+	0.163015i
$916$		0			0
$917$	−8775.06	+	8775.06i	−0.316007	+	0.316007i
$918$		0			0
$919$			2424.43i			0.0870236i	0.999053	+	0.0435118i	$0.0138546\pi$
$919$			2424.43i			0.0870236i	−0.999053	+	0.0435118i	$0.986145\pi$
$920$		0			0
$921$		−	11211.8i		−	0.401131i
$922$		0			0
$923$	4200.17	−	4200.17i	0.149784	−	0.149784i
$924$		0			0
$925$	5271.50	+	5271.50i	0.187379	+	0.187379i
$926$		0			0
$927$	−34108.7			−1.20850
$928$		0			0
$929$	−31265.1			−1.10417			−0.552086	−	0.833787i	$-0.686168\pi$
$929$	−31265.1			−1.10417			−0.552086	+	0.833787i	$0.686168\pi$
$930$		0			0
$931$	323.809	+	323.809i	0.0113989	+	0.0113989i
$932$		0			0
$933$	−6198.81	+	6198.81i	−0.217513	+	0.217513i
$934$		0			0
$935$			7078.13i			0.247572i
$936$		0			0
$937$		−	24317.7i		−	0.847839i	−0.905700	−	0.423919i	$-0.860654\pi$
$937$		−	24317.7i		−	0.847839i	0.905700	−	0.423919i	$-0.139346\pi$
$938$		0			0
$939$	7080.97	−	7080.97i	0.246090	−	0.246090i
$940$		0			0
$941$	−26684.6	−	26684.6i	−0.924433	−	0.924433i	0.0729055	−	0.997339i	$-0.476773\pi$
$941$	−26684.6	−	26684.6i	−0.924433	−	0.924433i	−0.997339	+	0.0729055i	$0.976773\pi$
$942$		0			0
$943$	−6444.92			−0.222561
$944$		0			0
$945$	−8381.24			−0.288510
$946$		0			0
$947$	−12759.1	−	12759.1i	−0.437819	−	0.437819i	0.453458	−	0.891278i	$-0.350190\pi$
$947$	−12759.1	−	12759.1i	−0.437819	−	0.437819i	−0.891278	+	0.453458i	$0.850190\pi$
$948$		0			0
$949$	20828.8	−	20828.8i	0.712468	−	0.712468i
$950$		0			0
$951$			1899.08i			0.0647547i
$952$		0			0
$953$		−	4436.11i		−	0.150787i	−0.997154	−	0.0753933i	$-0.975979\pi$
$953$		−	4436.11i		−	0.150787i	0.997154	−	0.0753933i	$-0.0240212\pi$
$954$		0			0
$955$	11761.8	−	11761.8i	0.398539	−	0.398539i
$956$		0			0
$957$	1732.85	+	1732.85i	0.0585320	+	0.0585320i
$958$		0			0
$959$	−46832.7			−1.57696
$960$		0			0
$961$	32661.1			1.09634
$962$		0			0
$963$	−11542.1	−	11542.1i	−0.386229	−	0.386229i
$964$		0			0
$965$	−8803.23	+	8803.23i	−0.293664	+	0.293664i
$966$		0			0
$967$		−	37242.9i		−	1.23852i	−0.785185	−	0.619261i	$-0.787432\pi$
$967$		−	37242.9i		−	1.23852i	0.785185	−	0.619261i	$-0.212568\pi$
$968$		0			0
$969$			4387.57i			0.145458i
$970$		0			0
$971$	28061.0	−	28061.0i	0.927414	−	0.927414i	−0.0701242	−	0.997538i	$-0.522340\pi$
$971$	28061.0	−	28061.0i	0.927414	−	0.927414i	0.997538	+	0.0701242i	$0.0223396\pi$
$972$		0			0
$973$	38439.0	+	38439.0i	1.26649	+	1.26649i
$974$		0			0
$975$	−1250.79			−0.0410846
$976$		0			0
$977$	11806.4			0.386611			0.193306	−	0.981139i	$-0.438079\pi$
$977$	11806.4			0.386611			0.193306	+	0.981139i	$0.438079\pi$
$978$		0			0
$979$	−154.000	−	154.000i	−0.00502744	−	0.00502744i
$980$		0			0
$981$	15128.9	−	15128.9i	0.492383	−	0.492383i
$982$		0			0
$983$			28865.2i			0.936579i	0.883575	+	0.468290i	$0.155130\pi$
$983$			28865.2i			0.936579i	−0.883575	+	0.468290i	$0.844870\pi$
$984$		0			0
$985$			12646.6i			0.409091i
$986$		0			0
$987$	−4086.90	+	4086.90i	−0.131801	+	0.131801i
$988$		0			0
$989$	−17086.0	−	17086.0i	−0.549347	−	0.549347i
$990$		0			0
$991$	−37756.7			−1.21027			−0.605137	−	0.796121i	$-0.706882\pi$
$991$	−37756.7			−1.21027			−0.605137	+	0.796121i	$0.706882\pi$
$992$		0			0
$993$	6553.25			0.209427
$994$		0			0
$995$	−6686.97	−	6686.97i	−0.213056	−	0.213056i
$996$		0			0
$997$	11348.0	−	11348.0i	0.360478	−	0.360478i	−0.503511	−	0.863989i	$-0.667959\pi$
$997$	11348.0	−	11348.0i	0.360478	−	0.360478i	0.863989	+	0.503511i	$0.167959\pi$
$998$		0			0
$999$			27809.5i			0.880733i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	640.4.l.a.161.15		48
4.3	odd	2		640.4.l.b.161.10		48
8.3	odd	2		80.4.l.a.61.7	yes	48
8.5	even	2		320.4.l.a.81.10		48
16.3	odd	4		80.4.l.a.21.7	✓	48
16.5	even	4	inner	640.4.l.a.481.15		48
16.11	odd	4		640.4.l.b.481.10		48
16.13	even	4		320.4.l.a.241.10		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.4.l.a.21.7	✓	48	16.3	odd	4
80.4.l.a.61.7	yes	48	8.3	odd	2
320.4.l.a.81.10		48	8.5	even	2
320.4.l.a.241.10		48	16.13	even	4
640.4.l.a.161.15		48	1.1	even	1	trivial
640.4.l.a.481.15		48	16.5	even	4	inner
640.4.l.b.161.10		48	4.3	odd	2
640.4.l.b.481.10		48	16.11	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 640 = 2^{7} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	640.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.30312 + 1.30312i) q^{3} +(-3.53553 + 3.53553i) q^{5} +17.9744i q^{7} -23.6037i q^{9} +(9.66542 - 9.66542i) q^{11} +(-19.1969 - 19.1969i) q^{13} -9.21448 q^{15} +103.565 q^{17} +(16.2553 + 16.2553i) q^{19} +(-23.4229 + 23.4229i) q^{21} +49.6583i q^{23} -25.0000i q^{25} +(65.9429 - 65.9429i) q^{27} +(68.7899 + 68.7899i) q^{29} +249.904 q^{31} +25.1905 q^{33} +(-63.5492 - 63.5492i) q^{35} +(-210.860 + 210.860i) q^{37} -50.0318i q^{39} +129.785i q^{41} +(-344.071 + 344.071i) q^{43} +(83.4518 + 83.4518i) q^{45} +174.483 q^{47} +19.9202 q^{49} +(134.958 + 134.958i) q^{51} +(-322.832 + 322.832i) q^{53} +68.3448i q^{55} +42.3653i q^{57} +(-21.0376 + 21.0376i) q^{59} +(-489.653 - 489.653i) q^{61} +424.264 q^{63} +135.742 q^{65} +(261.470 + 261.470i) q^{67} +(-64.7110 + 64.7110i) q^{69} +218.795i q^{71} +1085.01i q^{73} +(32.5781 - 32.5781i) q^{75} +(173.730 + 173.730i) q^{77} +1155.60 q^{79} -465.437 q^{81} +(777.042 + 777.042i) q^{83} +(-366.157 + 366.157i) q^{85} +179.283i q^{87} -15.9331i q^{89} +(345.052 - 345.052i) q^{91} +(325.656 + 325.656i) q^{93} -114.942 q^{95} -923.624 q^{97} +(-228.140 - 228.140i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 40 q^{11} - 120 q^{15} + 24 q^{19} - 264 q^{27} - 400 q^{29} - 16 q^{37} + 808 q^{43} - 1880 q^{47} - 2352 q^{49} + 2144 q^{51} - 752 q^{53} - 2728 q^{59} + 912 q^{61} + 2520 q^{63} - 2040 q^{67}+ \cdots - 4456 q^{99}+O(q^{100}) \) 48 * q - 40 * q^11 - 120 * q^15 + 24 * q^19 - 264 * q^27 - 400 * q^29 - 16 * q^37 + 808 * q^43 - 1880 * q^47 - 2352 * q^49 + 2144 * q^51 - 752 * q^53 - 2728 * q^59 + 912 * q^61 + 2520 * q^63 - 2040 * q^67 + 528 * q^69 - 1904 * q^77 - 2832 * q^79 - 3888 * q^81 + 2440 * q^83 + 240 * q^85 - 4448 * q^91 - 4272 * q^93 + 3040 * q^95 - 4456 * q^99

Introduction

L-functions

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