LMFDB - Embedded newform 816.4.c.b.577.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [816,4,Mod(577,816)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("816.577"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(816, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 816 = 2^{4} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	816.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-36,0,0,0,74] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$48.1455585647$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{569})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 285x^{2} + 20164 $ x^4 + 285*x^2 + 20164
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 102)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.3
Root			$-12.4269i$ of defining polynomial
Character	$\chi$	$=$	816.577
Dual form			816.4.c.b.577.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -16.4269i q^{5} -6.00000i q^{7} -9.00000 q^{9} -1.57314i q^{11} -17.2806 q^{13} +49.2806 q^{15} +(-28.2806 - 64.1343i) q^{17} +145.842 q^{19} +18.0000 q^{21} +30.9880i q^{23} -144.842 q^{25} -27.0000i q^{27} -59.7074i q^{29} -30.0000i q^{31} +4.71942 q^{33} -98.5612 q^{35} +47.1223i q^{37} -51.8417i q^{39} -228.988i q^{41} -376.086 q^{43} +147.842i q^{45} -456.806 q^{47} +307.000 q^{49} +(192.403 - 84.8417i) q^{51} -563.122 q^{53} -25.8417 q^{55} +437.525i q^{57} +120.000 q^{59} +408.878i q^{61} +54.0000i q^{63} +283.866i q^{65} -205.122 q^{67} -92.9641 q^{69} +895.391i q^{71} -1067.05i q^{73} -434.525i q^{75} -9.43884 q^{77} +1038.81i q^{79} +81.0000 q^{81} -1099.68 q^{83} +(-1053.53 + 464.561i) q^{85} +179.122 q^{87} -794.489 q^{89} +103.683i q^{91} +90.0000 q^{93} -2395.72i q^{95} -1006.24i q^{97} +14.1583i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{9} + 74 q^{13} + 54 q^{15} + 30 q^{17} + 154 q^{19} + 72 q^{21} - 150 q^{25} + 162 q^{33} - 108 q^{35} + 70 q^{43} - 396 q^{47} + 1228 q^{49} + 54 q^{51} - 1680 q^{53} + 326 q^{55} + 480 q^{59}+ \cdots + 360 q^{93}+O(q^{100}) $ 4 * q - 36 * q^9 + 74 * q^13 + 54 * q^15 + 30 * q^17 + 154 * q^19 + 72 * q^21 - 150 * q^25 + 162 * q^33 - 108 * q^35 + 70 * q^43 - 396 * q^47 + 1228 * q^49 + 54 * q^51 - 1680 * q^53 + 326 * q^55 + 480 * q^59 - 248 * q^67 + 630 * q^69 - 324 * q^77 + 324 * q^81 - 3540 * q^83 - 2926 * q^85 + 144 * q^87 - 888 * q^89 + 360 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/816\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$545$	$613$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$			3.00000i			0.577350i
$3$			3.00000i			0.577350i
$4$		0			0
$5$		−	16.4269i		−	1.46926i	−0.678466	−	0.734632i	$-0.737355\pi$
$5$		−	16.4269i		−	1.46926i	0.678466	−	0.734632i	$-0.262645\pi$
$6$		0			0
$7$		−	6.00000i		−	0.323970i	−0.986793	−	0.161985i	$-0.948210\pi$
$7$		−	6.00000i		−	0.323970i	0.986793	−	0.161985i	$-0.0517895\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$		−	1.57314i		−	0.0431199i	−0.999768	−	0.0215600i	$-0.993137\pi$
$11$		−	1.57314i		−	0.0431199i	0.999768	−	0.0215600i	$-0.00686328\pi$
$12$		0			0
$13$	−17.2806			−0.368675			−0.184337	−	0.982863i	$-0.559014\pi$
$13$	−17.2806			−0.368675			−0.184337	+	0.982863i	$0.559014\pi$
$14$		0			0
$15$	49.2806			0.848279
$16$		0			0
$17$	−28.2806	−	64.1343i	−0.403473	−	0.914991i
$17$	−28.2806	−	64.1343i	−0.403473	−	0.914991i
$18$		0			0
$19$	145.842			1.76097			0.880484	−	0.474076i	$-0.157218\pi$
$19$	145.842			1.76097			0.880484	+	0.474076i	$0.157218\pi$
$20$		0			0
$21$	18.0000			0.187044
$22$		0			0
$23$			30.9880i			0.280933i	0.990085	+	0.140466i	$0.0448602\pi$
$23$			30.9880i			0.280933i	−0.990085	+	0.140466i	$0.955140\pi$
$24$		0			0
$25$	−144.842			−1.15873
$26$		0			0
$27$		−	27.0000i		−	0.192450i
$28$		0			0
$29$		−	59.7074i		−	0.382324i	−0.981559	−	0.191162i	$-0.938774\pi$
$29$		−	59.7074i		−	0.382324i	0.981559	−	0.191162i	$-0.0612256\pi$
$30$		0			0
$31$		−	30.0000i		−	0.173812i	−0.996217	−	0.0869058i	$-0.972302\pi$
$31$		−	30.0000i		−	0.173812i	0.996217	−	0.0869058i	$-0.0276979\pi$
$32$		0			0
$33$	4.71942			0.0248953
$34$		0			0
$35$	−98.5612			−0.475996
$36$		0			0
$37$			47.1223i			0.209375i	0.994505	+	0.104687i	$0.0333842\pi$
$37$			47.1223i			0.209375i	−0.994505	+	0.104687i	$0.966616\pi$
$38$		0			0
$39$		−	51.8417i		−	0.212854i
$40$		0			0
$41$		−	228.988i		−	0.872242i	−0.899888	−	0.436121i	$-0.856352\pi$
$41$		−	228.988i		−	0.872242i	0.899888	−	0.436121i	$-0.143648\pi$
$42$		0			0
$43$	−376.086			−1.33378			−0.666891	−	0.745155i	$-0.732375\pi$
$43$	−376.086			−1.33378			−0.666891	+	0.745155i	$0.732375\pi$
$44$		0			0
$45$			147.842i			0.489754i
$46$		0			0
$47$	−456.806			−1.41770			−0.708851	−	0.705358i	$-0.750786\pi$
$47$	−456.806			−1.41770			−0.708851	+	0.705358i	$0.750786\pi$
$48$		0			0
$49$	307.000			0.895044
$50$		0			0
$51$	192.403	−	84.8417i	0.528271	−	0.232945i
$52$		0			0
$53$	−563.122			−1.45945			−0.729725	−	0.683741i	$-0.760352\pi$
$53$	−563.122			−1.45945			−0.729725	+	0.683741i	$0.760352\pi$
$54$		0			0
$55$	−25.8417			−0.0633545
$56$		0			0
$57$			437.525i			1.01670i
$58$		0			0
$59$	120.000			0.264791			0.132396	−	0.991197i	$-0.457733\pi$
$59$	120.000			0.264791			0.132396	+	0.991197i	$0.457733\pi$
$60$		0			0
$61$			408.878i			0.858220i	0.903252	+	0.429110i	$0.141173\pi$
$61$			408.878i			0.858220i	−0.903252	+	0.429110i	$0.858827\pi$
$62$		0			0
$63$			54.0000i			0.107990i
$64$		0			0
$65$			283.866i			0.541680i
$66$		0			0
$67$	−205.122			−0.374025			−0.187013	−	0.982358i	$-0.559881\pi$
$67$	−205.122			−0.374025			−0.187013	+	0.982358i	$0.559881\pi$
$68$		0			0
$69$	−92.9641			−0.162197
$70$		0			0
$71$			895.391i			1.49667i	0.663323	+	0.748333i	$0.269146\pi$
$71$			895.391i			1.49667i	−0.663323	+	0.748333i	$0.730854\pi$
$72$		0			0
$73$		−	1067.05i		−	1.71081i	−0.517963	−	0.855403i	$-0.673310\pi$
$73$		−	1067.05i		−	1.71081i	0.517963	−	0.855403i	$-0.326690\pi$
$74$		0			0
$75$		−	434.525i		−	0.668995i
$76$		0			0
$77$	−9.43884			−0.0139695
$78$		0			0
$79$			1038.81i			1.47943i	0.672922	+	0.739714i	$0.265039\pi$
$79$			1038.81i			1.47943i	−0.672922	+	0.739714i	$0.734961\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	−1099.68			−1.45429			−0.727144	−	0.686485i	$-0.759153\pi$
$83$	−1099.68			−1.45429			−0.727144	+	0.686485i	$0.759153\pi$
$84$		0			0
$85$	−1053.53	+	464.561i	−1.34436	+	0.592809i
$86$		0			0
$87$	179.122			0.220735
$88$		0			0
$89$	−794.489			−0.946244			−0.473122	−	0.880997i	$-0.656873\pi$
$89$	−794.489			−0.946244			−0.473122	+	0.880997i	$0.656873\pi$
$90$		0			0
$91$			103.683i			0.119439i
$92$		0			0
$93$	90.0000			0.100350
$94$		0			0
$95$		−	2395.72i		−	2.58733i
$96$		0			0
$97$		−	1006.24i		−	1.05329i	−0.850087	−	0.526643i	$-0.823451\pi$
$97$		−	1006.24i		−	1.05329i	0.850087	−	0.526643i	$-0.176549\pi$
$98$		0			0
$99$			14.1583i			0.0143733i
$100$		0			0
$101$	−394.173			−0.388333			−0.194167	−	0.980969i	$-0.562200\pi$
$101$	−394.173			−0.388333			−0.194167	+	0.980969i	$0.562200\pi$
$102$		0			0
$103$	−748.892			−0.716413			−0.358207	−	0.933642i	$-0.616612\pi$
$103$	−748.892			−0.716413			−0.358207	+	0.933642i	$0.616612\pi$
$104$		0			0
$105$		−	295.683i		−	0.274817i
$106$		0			0
$107$		−	1200.48i		−	1.08462i	−0.840177	−	0.542312i	$-0.817549\pi$
$107$		−	1200.48i		−	1.08462i	0.840177	−	0.542312i	$-0.182451\pi$
$108$		0			0
$109$			1083.44i			0.952061i	0.879429	+	0.476030i	$0.157925\pi$
$109$			1083.44i			0.952061i	−0.879429	+	0.476030i	$0.842075\pi$
$110$		0			0
$111$	−141.367			−0.120883
$112$		0			0
$113$		−	1027.06i		−	0.855024i	−0.904010	−	0.427512i	$-0.859390\pi$
$113$		−	1027.06i		−	0.855024i	0.904010	−	0.427512i	$-0.140610\pi$
$114$		0			0
$115$	509.036			0.412764
$116$		0			0
$117$	155.525			0.122892
$118$		0			0
$119$	−384.806	+	169.683i	−0.296429	+	0.130713i
$120$		0			0
$121$	1328.53			0.998141
$122$		0			0
$123$	686.964			0.503589
$124$		0			0
$125$			325.938i			0.233222i
$126$		0			0
$127$	−2291.63			−1.60117			−0.800586	−	0.599217i	$-0.795479\pi$
$127$	−2291.63			−1.60117			−0.800586	+	0.599217i	$0.795479\pi$
$128$		0			0
$129$		−	1128.26i		−	0.770060i
$130$		0			0
$131$		−	1935.04i		−	1.29057i	−0.763940	−	0.645287i	$-0.776738\pi$
$131$		−	1935.04i		−	1.29057i	0.763940	−	0.645287i	$-0.223262\pi$
$132$		0			0
$133$		−	875.050i		−	0.570500i
$134$		0			0
$135$	−443.525			−0.282760
$136$		0			0
$137$	973.540			0.607118			0.303559	−	0.952813i	$-0.401825\pi$
$137$	973.540			0.607118			0.303559	+	0.952813i	$0.401825\pi$
$138$		0			0
$139$			310.388i			0.189401i	0.995506	+	0.0947007i	$0.0301894\pi$
$139$			310.388i			0.189401i	−0.995506	+	0.0947007i	$0.969811\pi$
$140$		0			0
$141$		−	1370.42i		−	0.818510i
$142$		0			0
$143$			27.1848i			0.0158972i
$144$		0			0
$145$	−980.806			−0.561734
$146$		0			0
$147$			921.000i			0.516754i
$148$		0			0
$149$	−2680.76			−1.47394			−0.736969	−	0.675927i	$-0.763743\pi$
$149$	−2680.76			−1.47394			−0.736969	+	0.675927i	$0.763743\pi$
$150$		0			0
$151$	226.878			0.122272			0.0611359	−	0.998129i	$-0.480528\pi$
$151$	226.878			0.122272			0.0611359	+	0.998129i	$0.480528\pi$
$152$		0			0
$153$	254.525	+	577.209i	0.134491	+	0.304997i
$154$		0			0
$155$	−492.806			−0.255375
$156$		0			0
$157$	353.108			0.179497			0.0897486	−	0.995964i	$-0.471394\pi$
$157$	353.108			0.179497			0.0897486	+	0.995964i	$0.471394\pi$
$158$		0			0
$159$		−	1689.37i		−	0.842613i
$160$		0			0
$161$	185.928			0.0910136
$162$		0			0
$163$		−	1070.70i		−	0.514504i	−0.966344	−	0.257252i	$-0.917183\pi$
$163$		−	1070.70i		−	0.514504i	0.966344	−	0.257252i	$-0.0828170\pi$
$164$		0			0
$165$		−	77.5252i		−	0.0365778i
$166$		0			0
$167$		−	744.408i		−	0.344934i	−0.985015	−	0.172467i	$-0.944826\pi$
$167$		−	744.408i		−	0.344934i	0.985015	−	0.172467i	$-0.0551739\pi$
$168$		0			0
$169$	−1898.38			−0.864079
$170$		0			0
$171$	−1312.58			−0.586989
$172$		0			0
$173$		−	2870.53i		−	1.26152i	−0.775980	−	0.630758i	$-0.782744\pi$
$173$		−	2870.53i		−	1.26152i	0.775980	−	0.630758i	$-0.217256\pi$
$174$		0			0
$175$			869.050i			0.375395i
$176$		0			0
$177$			360.000i			0.152877i
$178$		0			0
$179$	1144.03			0.477702			0.238851	−	0.971056i	$-0.423229\pi$
$179$	1144.03			0.477702			0.238851	+	0.971056i	$0.423229\pi$
$180$		0			0
$181$		−	3516.07i		−	1.44391i	−0.691940	−	0.721955i	$-0.743244\pi$
$181$		−	3516.07i		−	1.44391i	0.691940	−	0.721955i	$-0.256756\pi$
$182$		0			0
$183$	−1226.63			−0.495494
$184$		0			0
$185$	774.072			0.307626
$186$		0			0
$187$	−100.892	+	44.4893i	−0.0394544	+	0.0173978i
$188$		0			0
$189$	−162.000			−0.0623480
$190$		0			0
$191$	4153.61			1.57353			0.786766	−	0.617251i	$-0.211754\pi$
$191$	4153.61			1.57353			0.786766	+	0.617251i	$0.211754\pi$
$192$		0			0
$193$			4090.91i			1.52575i	0.646546	+	0.762875i	$0.276213\pi$
$193$			4090.91i			1.52575i	−0.646546	+	0.762875i	$0.723787\pi$
$194$		0			0
$195$	−851.597			−0.312739
$196$		0			0
$197$			4823.75i			1.74456i	0.489010	+	0.872278i	$0.337358\pi$
$197$			4823.75i			1.74456i	−0.489010	+	0.872278i	$0.662642\pi$
$198$		0			0
$199$			2985.74i			1.06359i	0.846874	+	0.531793i	$0.178482\pi$
$199$			2985.74i			1.06359i	−0.846874	+	0.531793i	$0.821518\pi$
$200$		0			0
$201$		−	615.367i		−	0.215943i
$202$		0			0
$203$	−358.245			−0.123861
$204$		0			0
$205$	−3761.55			−1.28155
$206$		0			0
$207$		−	278.892i		−	0.0936442i
$208$		0			0
$209$		−	229.429i		−	0.0759328i
$210$		0			0
$211$		−	4387.40i		−	1.43147i	−0.698370	−	0.715736i	$-0.746091\pi$
$211$		−	4387.40i		−	1.43147i	0.698370	−	0.715736i	$-0.253909\pi$
$212$		0			0
$213$	−2686.17			−0.864101
$214$		0			0
$215$			6177.92i			1.95968i
$216$		0			0
$217$	−180.000			−0.0563097
$218$		0			0
$219$	3201.15			0.987734
$220$		0			0
$221$	488.705	+	1108.28i	0.148750	+	0.337334i
$222$		0			0
$223$	−918.791			−0.275905			−0.137952	−	0.990439i	$-0.544052\pi$
$223$	−918.791			−0.275905			−0.137952	+	0.990439i	$0.544052\pi$
$224$		0			0
$225$	1303.58			0.386245
$226$		0			0
$227$			1754.31i			0.512942i	0.966552	+	0.256471i	$0.0825598\pi$
$227$			1754.31i			0.512942i	−0.966552	+	0.256471i	$0.917440\pi$
$228$		0			0
$229$	−1859.18			−0.536498			−0.268249	−	0.963350i	$-0.586445\pi$
$229$	−1859.18			−0.536498			−0.268249	+	0.963350i	$0.586445\pi$
$230$		0			0
$231$		−	28.3165i		−	0.00806532i
$232$		0			0
$233$			1537.43i			0.432276i	0.976363	+	0.216138i	$0.0693461\pi$
$233$			1537.43i			0.432276i	−0.976363	+	0.216138i	$0.930654\pi$
$234$		0			0
$235$			7503.89i			2.08298i
$236$		0			0
$237$	−3116.42			−0.854148
$238$		0			0
$239$	3904.69			1.05679			0.528396	−	0.848998i	$-0.322794\pi$
$239$	3904.69			1.05679			0.528396	+	0.848998i	$0.322794\pi$
$240$		0			0
$241$			2369.64i			0.633369i	0.948531	+	0.316685i	$0.102570\pi$
$241$			2369.64i			0.633369i	−0.948531	+	0.316685i	$0.897430\pi$
$242$		0			0
$243$			243.000i			0.0641500i
$244$		0			0
$245$		−	5043.05i		−	1.31505i
$246$		0			0
$247$	−2520.23			−0.649224
$248$		0			0
$249$		−	3299.05i		−	0.839634i
$250$		0			0
$251$	−7356.23			−1.84989			−0.924943	−	0.380107i	$-0.875887\pi$
$251$	−7356.23			−1.84989			−0.924943	+	0.380107i	$0.875887\pi$
$252$		0			0
$253$	48.7485			0.0121138
$254$		0			0
$255$	−1393.68	−	3160.58i	−0.342258	−	0.776168i
$256$		0			0
$257$	3415.32			0.828957			0.414479	−	0.910059i	$-0.363964\pi$
$257$	3415.32			0.828957			0.414479	+	0.910059i	$0.363964\pi$
$258$		0			0
$259$	282.734			0.0678310
$260$		0			0
$261$			537.367i			0.127441i
$262$		0			0
$263$	3030.22			0.710460			0.355230	−	0.934779i	$-0.384402\pi$
$263$	3030.22			0.710460			0.355230	+	0.934779i	$0.384402\pi$
$264$		0			0
$265$			9250.33i			2.14431i
$266$		0			0
$267$		−	2383.47i		−	0.546314i
$268$		0			0
$269$		−	5449.58i		−	1.23519i	−0.786496	−	0.617596i	$-0.788107\pi$
$269$		−	5449.58i		−	1.23519i	0.786496	−	0.617596i	$-0.211893\pi$
$270$		0			0
$271$	−2628.14			−0.589109			−0.294554	−	0.955635i	$-0.595171\pi$
$271$	−2628.14			−0.589109			−0.294554	+	0.955635i	$0.595171\pi$
$272$		0			0
$273$	−311.050			−0.0689584
$274$		0			0
$275$			227.856i			0.0499645i
$276$		0			0
$277$			7133.14i			1.54725i	0.633643	+	0.773626i	$0.281559\pi$
$277$			7133.14i			1.54725i	−0.633643	+	0.773626i	$0.718441\pi$
$278$		0			0
$279$			270.000i			0.0579372i
$280$		0			0
$281$	6010.26			1.27595			0.637975	−	0.770057i	$-0.279772\pi$
$281$	6010.26			1.27595			0.637975	+	0.770057i	$0.279772\pi$
$282$		0			0
$283$			1226.19i			0.257559i	0.991673	+	0.128780i	$0.0411060\pi$
$283$			1226.19i			0.257559i	−0.991673	+	0.128780i	$0.958894\pi$
$284$		0			0
$285$	7187.17			1.49379
$286$		0			0
$287$	−1373.93			−0.282580
$288$		0			0
$289$	−3313.42	+	3627.51i	−0.674418	+	0.738349i
$290$		0			0
$291$	3018.73			0.608114
$292$		0			0
$293$	8459.73			1.68677			0.843383	−	0.537312i	$-0.180560\pi$
$293$	8459.73			1.68677			0.843383	+	0.537312i	$0.180560\pi$
$294$		0			0
$295$		−	1971.22i		−	0.389048i
$296$		0			0
$297$	−42.4748			−0.00829844
$298$		0			0
$299$		−	535.491i		−	0.103573i
$300$		0			0
$301$			2256.52i			0.432105i
$302$		0			0
$303$		−	1182.52i		−	0.224204i
$304$		0			0
$305$	6716.58			1.26095
$306$		0			0
$307$	3384.78			0.629249			0.314624	−	0.949216i	$-0.398121\pi$
$307$	3384.78			0.629249			0.314624	+	0.949216i	$0.398121\pi$
$308$		0			0
$309$		−	2246.68i		−	0.413621i
$310$		0			0
$311$			861.003i			0.156987i	0.996915	+	0.0784935i	$0.0250110\pi$
$311$			861.003i			0.156987i	−0.996915	+	0.0784935i	$0.974989\pi$
$312$		0			0
$313$			950.648i			0.171674i	0.996309	+	0.0858368i	$0.0273564\pi$
$313$			950.648i			0.171674i	−0.996309	+	0.0858368i	$0.972644\pi$
$314$		0			0
$315$	887.050			0.158665
$316$		0			0
$317$		−	8401.18i		−	1.48851i	−0.667897	−	0.744254i	$-0.732805\pi$
$317$		−	8401.18i		−	1.48851i	0.667897	−	0.744254i	$-0.267195\pi$
$318$		0			0
$319$	−93.9281			−0.0164858
$320$		0			0
$321$	3601.44			0.626208
$322$		0			0
$323$	−4124.49	−	9353.46i	−0.710504	−	1.61127i
$324$		0			0
$325$	2502.95			0.427196
$326$		0			0
$327$	−3250.32			−0.549673
$328$		0			0
$329$			2740.83i			0.459292i
$330$		0			0
$331$	−4128.20			−0.685518			−0.342759	−	0.939423i	$-0.611361\pi$
$331$	−4128.20			−0.685518			−0.342759	+	0.939423i	$0.611361\pi$
$332$		0			0
$333$		−	424.101i		−	0.0697916i
$334$		0			0
$335$			3369.52i			0.549541i
$336$		0			0
$337$		−	40.2293i		−	0.00650276i	−0.999995	−	0.00325138i	$-0.998965\pi$
$337$		−	40.2293i		−	0.00650276i	0.999995	−	0.00325138i	$-0.00103495\pi$
$338$		0			0
$339$	3081.18			0.493648
$340$		0			0
$341$	−47.1942			−0.00749475
$342$		0			0
$343$		−	3900.00i		−	0.613936i
$344$		0			0
$345$			1527.11i			0.238309i
$346$		0			0
$347$			1137.51i			0.175978i	0.996121	+	0.0879892i	$0.0280441\pi$
$347$			1137.51i			0.175978i	−0.996121	+	0.0879892i	$0.971956\pi$
$348$		0			0
$349$	625.741			0.0959746			0.0479873	−	0.998848i	$-0.484719\pi$
$349$	625.741			0.0959746			0.0479873	+	0.998848i	$0.484719\pi$
$350$		0			0
$351$			466.576i			0.0709515i
$352$		0			0
$353$	−6354.81			−0.958165			−0.479082	−	0.877770i	$-0.659031\pi$
$353$	−6354.81			−0.958165			−0.479082	+	0.877770i	$0.659031\pi$
$354$		0			0
$355$	14708.5			2.19900
$356$		0			0
$357$	−509.050	−	1154.42i	−0.0754672	−	0.171144i
$358$		0			0
$359$	10466.9			1.53878			0.769388	−	0.638781i	$-0.220561\pi$
$359$	10466.9			1.53878			0.769388	+	0.638781i	$0.220561\pi$
$360$		0			0
$361$	14410.8			2.10101
$362$		0			0
$363$			3985.58i			0.576277i
$364$		0			0
$365$	−17528.3			−2.51362
$366$		0			0
$367$		−	8320.39i		−	1.18344i	−0.806145	−	0.591718i	$-0.798450\pi$
$367$		−	8320.39i		−	1.18344i	0.806145	−	0.591718i	$-0.201550\pi$
$368$		0			0
$369$			2060.89i			0.290747i
$370$		0			0
$371$			3378.73i			0.472817i
$372$		0			0
$373$	265.251			0.0368208			0.0184104	−	0.999831i	$-0.494139\pi$
$373$	265.251			0.0368208			0.0184104	+	0.999831i	$0.494139\pi$
$374$		0			0
$375$	−977.813			−0.134651
$376$		0			0
$377$			1031.78i			0.140953i
$378$		0			0
$379$			10168.5i			1.37815i	0.724688	+	0.689077i	$0.241984\pi$
$379$			10168.5i			1.37815i	−0.724688	+	0.689077i	$0.758016\pi$
$380$		0			0
$381$		−	6874.88i		−	0.924438i
$382$		0			0
$383$	1646.79			0.219705			0.109853	−	0.993948i	$-0.464962\pi$
$383$	1646.79			0.219705			0.109853	+	0.993948i	$0.464962\pi$
$384$		0			0
$385$			155.050i			0.0205249i
$386$		0			0
$387$	3384.78			0.444594
$388$		0			0
$389$	6268.68			0.817055			0.408528	−	0.912746i	$-0.366042\pi$
$389$	6268.68			0.817055			0.408528	+	0.912746i	$0.366042\pi$
$390$		0			0
$391$	1987.40	−	876.359i	0.257051	−	0.113349i
$392$		0			0
$393$	5805.12			0.745114
$394$		0			0
$395$	17064.3			2.17367
$396$		0			0
$397$		−	13537.7i		−	1.71143i	−0.517446	−	0.855716i	$-0.673117\pi$
$397$		−	13537.7i		−	1.71143i	0.517446	−	0.855716i	$-0.326883\pi$
$398$		0			0
$399$	2625.15			0.329378
$400$		0			0
$401$		−	6490.51i		−	0.808281i	−0.914697	−	0.404141i	$-0.867571\pi$
$401$		−	6490.51i		−	0.808281i	0.914697	−	0.404141i	$-0.132429\pi$
$402$		0			0
$403$			518.417i			0.0640799i
$404$		0			0
$405$		−	1330.58i		−	0.163251i
$406$		0			0
$407$	74.1300			0.00902822
$408$		0			0
$409$	−7733.80			−0.934992			−0.467496	−	0.883995i	$-0.654844\pi$
$409$	−7733.80			−0.934992			−0.467496	+	0.883995i	$0.654844\pi$
$410$		0			0
$411$			2920.62i			0.350520i
$412$		0			0
$413$		−	720.000i		−	0.0857842i
$414$		0			0
$415$			18064.3i			2.13673i
$416$		0			0
$417$	−931.165			−0.109351
$418$		0			0
$419$		−	8170.70i		−	0.952660i	−0.879267	−	0.476330i	$-0.841967\pi$
$419$		−	8170.70i		−	0.952660i	0.879267	−	0.476330i	$-0.158033\pi$
$420$		0			0
$421$	−13732.6			−1.58975			−0.794874	−	0.606774i	$-0.792463\pi$
$421$	−13732.6			−1.58975			−0.794874	+	0.606774i	$0.792463\pi$
$422$		0			0
$423$	4111.25			0.472567
$424$		0			0
$425$	4096.21	+	9289.32i	0.467518	+	1.06023i
$426$		0			0
$427$	2453.27			0.278037
$428$		0			0
$429$	−81.5543			−0.00917827
$430$		0			0
$431$		−	13075.7i		−	1.46133i	−0.682734	−	0.730667i	$-0.739209\pi$
$431$		−	13075.7i		−	1.46133i	0.682734	−	0.730667i	$-0.260791\pi$
$432$		0			0
$433$	16873.0			1.87266			0.936331	−	0.351118i	$-0.114198\pi$
$433$	16873.0			1.87266			0.936331	+	0.351118i	$0.114198\pi$
$434$		0			0
$435$		−	2942.42i		−	0.324318i
$436$		0			0
$437$			4519.35i			0.494713i
$438$		0			0
$439$			16207.3i			1.76203i	0.473089	+	0.881014i	$0.343139\pi$
$439$			16207.3i			1.76203i	−0.473089	+	0.881014i	$0.656861\pi$
$440$		0			0
$441$	−2763.00			−0.298348
$442$		0			0
$443$	−8022.65			−0.860423			−0.430212	−	0.902728i	$-0.641561\pi$
$443$	−8022.65			−0.860423			−0.430212	+	0.902728i	$0.641561\pi$
$444$		0			0
$445$			13051.0i			1.39028i
$446$		0			0
$447$		−	8042.29i		−	0.850978i
$448$		0			0
$449$			4730.44i			0.497201i	0.968606	+	0.248600i	$0.0799706\pi$
$449$			4730.44i			0.497201i	−0.968606	+	0.248600i	$0.920029\pi$
$450$		0			0
$451$	−360.230			−0.0376110
$452$		0			0
$453$			680.633i			0.0705937i
$454$		0			0
$455$	1703.19			0.175488
$456$		0			0
$457$	840.447			0.0860273			0.0430136	−	0.999074i	$-0.486304\pi$
$457$	840.447			0.0860273			0.0430136	+	0.999074i	$0.486304\pi$
$458$		0			0
$459$	−1731.63	+	763.576i	−0.176090	+	0.0776485i
$460$		0			0
$461$	6788.88			0.685878			0.342939	−	0.939358i	$-0.388578\pi$
$461$	6788.88			0.685878			0.342939	+	0.939358i	$0.388578\pi$
$462$		0			0
$463$	11545.7			1.15890			0.579452	−	0.815006i	$-0.303267\pi$
$463$	11545.7			1.15890			0.579452	+	0.815006i	$0.303267\pi$
$464$		0			0
$465$		−	1478.42i		−	0.147441i
$466$		0			0
$467$	−174.303			−0.0172715			−0.00863573	−	0.999963i	$-0.502749\pi$
$467$	−174.303			−0.0172715			−0.00863573	+	0.999963i	$0.502749\pi$
$468$		0			0
$469$			1230.73i			0.121173i
$470$		0			0
$471$			1059.32i			0.103633i
$472$		0			0
$473$			591.636i			0.0575126i
$474$		0			0
$475$	−21124.0			−2.04049
$476$		0			0
$477$	5068.10			0.486483
$478$		0			0
$479$		−	16182.7i		−	1.54365i	−0.635837	−	0.771824i	$-0.719345\pi$
$479$		−	16182.7i		−	1.54365i	0.635837	−	0.771824i	$-0.280655\pi$
$480$		0			0
$481$		−	814.301i		−	0.0771911i
$482$		0			0
$483$			557.784i			0.0525467i
$484$		0			0
$485$	−16529.4			−1.54755
$486$		0			0
$487$			17815.3i			1.65767i	0.559491	+	0.828837i	$0.310997\pi$
$487$			17815.3i			1.65767i	−0.559491	+	0.828837i	$0.689003\pi$
$488$		0			0
$489$	3212.11			0.297049
$490$		0			0
$491$	−19445.2			−1.78727			−0.893635	−	0.448794i	$-0.851854\pi$
$491$	−19445.2			−1.78727			−0.893635	+	0.448794i	$0.851854\pi$
$492$		0			0
$493$	−3829.30	+	1688.56i	−0.349823	+	0.154258i
$494$		0			0
$495$	232.576			0.0211182
$496$		0			0
$497$	5372.35			0.484875
$498$		0			0
$499$			9524.72i			0.854479i	0.904138	+	0.427240i	$0.140514\pi$
$499$			9524.72i			0.854479i	−0.904138	+	0.427240i	$0.859486\pi$
$500$		0			0
$501$	2233.22			0.199148
$502$		0			0
$503$			1592.37i			0.141154i	0.997506	+	0.0705768i	$0.0224840\pi$
$503$			1592.37i			0.141154i	−0.997506	+	0.0705768i	$0.977516\pi$
$504$		0			0
$505$			6475.02i			0.570564i
$506$		0			0
$507$		−	5695.14i		−	0.498876i
$508$		0			0
$509$	−14893.7			−1.29696			−0.648480	−	0.761232i	$-0.724595\pi$
$509$	−14893.7			−1.29696			−0.648480	+	0.761232i	$0.724595\pi$
$510$		0			0
$511$	−6402.30			−0.554249
$512$		0			0
$513$		−	3937.73i		−	0.338898i
$514$		0			0
$515$			12301.9i			1.05260i
$516$		0			0
$517$			718.619i			0.0611312i
$518$		0			0
$519$	8611.58			0.728336
$520$		0			0
$521$		−	4403.38i		−	0.370279i	−0.982712	−	0.185140i	$-0.940726\pi$
$521$		−	4403.38i		−	0.370279i	0.982712	−	0.185140i	$-0.0592737\pi$
$522$		0			0
$523$	6728.23			0.562533			0.281267	−	0.959630i	$-0.409245\pi$
$523$	6728.23			0.562533			0.281267	+	0.959630i	$0.409245\pi$
$524$		0			0
$525$	−2607.15			−0.216734
$526$		0			0
$527$	−1924.03	+	848.417i	−0.159036	+	0.0701284i
$528$		0			0
$529$	11206.7			0.921077
$530$		0			0
$531$	−1080.00			−0.0882637
$532$		0			0
$533$			3957.05i			0.321574i
$534$		0			0
$535$	−19720.1			−1.59360
$536$		0			0
$537$			3432.09i			0.275802i
$538$		0			0
$539$		−	482.954i		−	0.0385942i
$540$		0			0
$541$			13624.4i			1.08273i	0.840787	+	0.541366i	$0.182093\pi$
$541$			13624.4i			1.08273i	−0.840787	+	0.541366i	$0.817907\pi$
$542$		0			0
$543$	10548.2			0.833641
$544$		0			0
$545$	17797.5			1.39883
$546$		0			0
$547$		−	3158.10i		−	0.246857i	−0.992353	−	0.123428i	$-0.960611\pi$
$547$		−	3158.10i		−	0.246857i	0.992353	−	0.123428i	$-0.0393889\pi$
$548$		0			0
$549$		−	3679.90i		−	0.286073i
$550$		0			0
$551$		−	8707.84i		−	0.673260i
$552$		0			0
$553$	6232.83			0.479290
$554$		0			0
$555$			2322.22i			0.177608i
$556$		0			0
$557$	−5831.35			−0.443595			−0.221797	−	0.975093i	$-0.571192\pi$
$557$	−5831.35			−0.443595			−0.221797	+	0.975093i	$0.571192\pi$
$558$		0			0
$559$	6498.99			0.491732
$560$		0			0
$561$	−133.468	−	302.677i	−0.0100446	−	0.0227790i
$562$		0			0
$563$	12511.5			0.936587			0.468294	−	0.883573i	$-0.344869\pi$
$563$	12511.5			0.936587			0.468294	+	0.883573i	$0.344869\pi$
$564$		0			0
$565$	−16871.4			−1.25625
$566$		0			0
$567$		−	486.000i		−	0.0359966i
$568$		0			0
$569$	−8469.14			−0.623980			−0.311990	−	0.950085i	$-0.600996\pi$
$569$	−8469.14			−0.623980			−0.311990	+	0.950085i	$0.600996\pi$
$570$		0			0
$571$		−	5857.64i		−	0.429308i	−0.976690	−	0.214654i	$-0.931138\pi$
$571$		−	5857.64i		−	0.429308i	0.976690	−	0.214654i	$-0.0688623\pi$
$572$		0			0
$573$			12460.8i			0.908480i
$574$		0			0
$575$		−	4488.36i		−	0.325526i
$576$		0			0
$577$	15537.9			1.12106			0.560528	−	0.828135i	$-0.310598\pi$
$577$	15537.9			1.12106			0.560528	+	0.828135i	$0.310598\pi$
$578$		0			0
$579$	−12272.7			−0.880893
$580$		0			0
$581$			6598.10i			0.471145i
$582$		0			0
$583$			885.870i			0.0629314i
$584$		0			0
$585$		−	2554.79i		−	0.180560i
$586$		0			0
$587$	−1491.54			−0.104876			−0.0524382	−	0.998624i	$-0.516699\pi$
$587$	−1491.54			−0.104876			−0.0524382	+	0.998624i	$0.516699\pi$
$588$		0			0
$589$		−	4375.25i		−	0.306077i
$590$		0			0
$591$	−14471.2			−1.00722
$592$		0			0
$593$	14418.0			0.998442			0.499221	−	0.866475i	$-0.333620\pi$
$593$	14418.0			0.998442			0.499221	+	0.866475i	$0.333620\pi$
$594$		0			0
$595$	2787.37	+	6321.15i	0.192052	+	0.435533i
$596$		0			0
$597$	−8957.22			−0.614062
$598$		0			0
$599$	−3644.49			−0.248597			−0.124299	−	0.992245i	$-0.539668\pi$
$599$	−3644.49			−0.248597			−0.124299	+	0.992245i	$0.539668\pi$
$600$		0			0
$601$		−	6984.34i		−	0.474039i	−0.971505	−	0.237019i	$-0.923829\pi$
$601$		−	6984.34i		−	0.474039i	0.971505	−	0.237019i	$-0.0761705\pi$
$602$		0			0
$603$	1846.10			0.124675
$604$		0			0
$605$		−	21823.5i		−	1.46653i
$606$		0			0
$607$		−	14362.8i		−	0.960407i	−0.877157	−	0.480204i	$-0.840563\pi$
$607$		−	14362.8i		−	0.960407i	0.877157	−	0.480204i	$-0.159437\pi$
$608$		0			0
$609$		−	1074.73i		−	0.0715114i
$610$		0			0
$611$	7893.87			0.522671
$612$		0			0
$613$	−8703.81			−0.573481			−0.286740	−	0.958008i	$-0.592572\pi$
$613$	−8703.81			−0.573481			−0.286740	+	0.958008i	$0.592572\pi$
$614$		0			0
$615$		−	11284.7i		−	0.739905i
$616$		0			0
$617$		−	26903.7i		−	1.75543i	−0.479182	−	0.877715i	$-0.659067\pi$
$617$		−	26903.7i		−	1.75543i	0.479182	−	0.877715i	$-0.340933\pi$
$618$		0			0
$619$		−	8641.76i		−	0.561133i	−0.959835	−	0.280567i	$-0.909478\pi$
$619$		−	8641.76i		−	0.561133i	0.959835	−	0.280567i	$-0.0905224\pi$
$620$		0			0
$621$	836.677			0.0540655
$622$		0			0
$623$			4766.94i			0.306554i
$624$		0			0
$625$	−12751.1			−0.816070
$626$		0			0
$627$	688.288			0.0438398
$628$		0			0
$629$	3022.16	−	1332.65i	0.191576	−	0.0844771i
$630$		0			0
$631$	10472.0			0.660672			0.330336	−	0.943863i	$-0.392838\pi$
$631$	10472.0			0.660672			0.330336	+	0.943863i	$0.392838\pi$
$632$		0			0
$633$	13162.2			0.826461
$634$		0			0
$635$			37644.2i			2.35254i
$636$		0			0
$637$	−5305.14			−0.329980
$638$		0			0
$639$		−	8058.52i		−	0.498889i
$640$		0			0
$641$			9806.52i			0.604266i	0.953266	+	0.302133i	$0.0976986\pi$
$641$			9806.52i			0.604266i	−0.953266	+	0.302133i	$0.902301\pi$
$642$		0			0
$643$		−	13782.6i		−	0.845307i	−0.906291	−	0.422653i	$-0.861099\pi$
$643$		−	13782.6i		−	0.845307i	0.906291	−	0.422653i	$-0.138901\pi$
$644$		0			0
$645$	−18533.8			−1.13142
$646$		0			0
$647$	−13972.3			−0.849005			−0.424502	−	0.905427i	$-0.639551\pi$
$647$	−13972.3			−0.849005			−0.424502	+	0.905427i	$0.639551\pi$
$648$		0			0
$649$		−	188.777i		−	0.0114178i
$650$		0			0
$651$		−	540.000i		−	0.0325104i
$652$		0			0
$653$		−	469.753i		−	0.0281514i	−0.999901	−	0.0140757i	$-0.995519\pi$
$653$		−	469.753i		−	0.0281514i	0.999901	−	0.0140757i	$-0.00448058\pi$
$654$		0			0
$655$	−31786.6			−1.89619
$656$		0			0
$657$			9603.45i			0.570269i
$658$		0			0
$659$	−6126.73			−0.362160			−0.181080	−	0.983468i	$-0.557959\pi$
$659$	−6126.73			−0.362160			−0.181080	+	0.983468i	$0.557959\pi$
$660$		0			0
$661$	−19862.7			−1.16879			−0.584393	−	0.811471i	$-0.698667\pi$
$661$	−19862.7			−1.16879			−0.584393	+	0.811471i	$0.698667\pi$
$662$		0			0
$663$	−3324.83	+	1466.11i	−0.194760	+	0.0858811i
$664$		0			0
$665$	−14374.3			−0.838215
$666$		0			0
$667$	1850.22			0.107407
$668$		0			0
$669$		−	2756.37i		−	0.159294i
$670$		0			0
$671$	643.222			0.0370064
$672$		0			0
$673$		−	29895.9i		−	1.71234i	−0.516697	−	0.856169i	$-0.672838\pi$
$673$		−	29895.9i		−	1.71234i	0.516697	−	0.856169i	$-0.327162\pi$
$674$		0			0
$675$			3910.73i			0.222998i
$676$		0			0
$677$			3689.27i			0.209439i	0.994502	+	0.104719i	$0.0333945\pi$
$677$			3689.27i			0.209439i	−0.994502	+	0.104719i	$0.966606\pi$
$678$		0			0
$679$	−6037.47			−0.341232
$680$		0			0
$681$	−5262.94			−0.296147
$682$		0			0
$683$		−	14206.2i		−	0.795879i	−0.917412	−	0.397940i	$-0.869725\pi$
$683$		−	14206.2i		−	0.795879i	0.917412	−	0.397940i	$-0.130275\pi$
$684$		0			0
$685$		−	15992.2i		−	0.892016i
$686$		0			0
$687$		−	5577.54i		−	0.309747i
$688$		0			0
$689$	9731.08			0.538062
$690$		0			0
$691$		−	36094.2i		−	1.98710i	−0.113391	−	0.993550i	$-0.536171\pi$
$691$		−	36094.2i		−	1.98710i	0.113391	−	0.993550i	$-0.463829\pi$
$692$		0			0
$693$	84.9495			0.00465652
$694$		0			0
$695$	5098.71			0.278281
$696$		0			0
$697$	−14686.0	+	6475.91i	−0.798094	+	0.351926i
$698$		0			0
$699$	−4612.29			−0.249575
$700$		0			0
$701$	17089.4			0.920767			0.460383	−	0.887720i	$-0.347712\pi$
$701$	17089.4			0.920767			0.460383	+	0.887720i	$0.347712\pi$
$702$		0			0
$703$			6872.40i			0.368702i
$704$		0			0
$705$	−22511.7			−1.20261
$706$		0			0
$707$			2365.04i			0.125808i
$708$		0			0
$709$		−	17325.6i		−	0.917737i	−0.888504	−	0.458868i	$-0.848255\pi$
$709$		−	17325.6i		−	0.917737i	0.888504	−	0.458868i	$-0.151745\pi$
$710$		0			0
$711$		−	9349.25i		−	0.493143i
$712$		0			0
$713$	929.641			0.0488293
$714$		0			0
$715$	446.560			0.0233572
$716$		0			0
$717$			11714.1i			0.610140i
$718$		0			0
$719$		−	3861.73i		−	0.200304i	−0.994972	−	0.100152i	$-0.968067\pi$
$719$		−	3861.73i		−	0.200304i	0.994972	−	0.100152i	$-0.0319329\pi$
$720$		0			0
$721$			4493.35i			0.232096i
$722$		0			0
$723$	−7108.92			−0.365676
$724$		0			0
$725$			8648.13i			0.443012i
$726$		0			0
$727$	−3016.23			−0.153873			−0.0769366	−	0.997036i	$-0.524514\pi$
$727$	−3016.23			−0.153873			−0.0769366	+	0.997036i	$0.524514\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$	10635.9	+	24120.0i	0.538146	+	1.22040i
$732$		0			0
$733$	−6778.92			−0.341590			−0.170795	−	0.985307i	$-0.554634\pi$
$733$	−6778.92			−0.341590			−0.170795	+	0.985307i	$0.554634\pi$
$734$		0			0
$735$	15129.1			0.759247
$736$		0			0
$737$			322.686i			0.0161279i
$738$		0			0
$739$	−25629.9			−1.27579			−0.637896	−	0.770123i	$-0.720195\pi$
$739$	−25629.9			−1.27579			−0.637896	+	0.770123i	$0.720195\pi$
$740$		0			0
$741$		−	7560.69i		−	0.374830i
$742$		0			0
$743$		−	8981.07i		−	0.443450i	−0.975109	−	0.221725i	$-0.928831\pi$
$743$		−	8981.07i		−	0.443450i	0.975109	−	0.221725i	$-0.0711688\pi$
$744$		0			0
$745$			44036.5i			2.16560i
$746$		0			0
$747$	9897.15			0.484763
$748$		0			0
$749$	−7202.88			−0.351385
$750$		0			0
$751$		−	4198.37i		−	0.203996i	−0.994785	−	0.101998i	$-0.967477\pi$
$751$		−	4198.37i		−	0.203996i	0.994785	−	0.101998i	$-0.0325235\pi$
$752$		0			0
$753$		−	22068.7i		−	1.06803i
$754$		0			0
$755$		−	3726.89i		−	0.179649i
$756$		0			0
$757$	11693.1			0.561418			0.280709	−	0.959793i	$-0.409430\pi$
$757$	11693.1			0.561418			0.280709	+	0.959793i	$0.409430\pi$
$758$		0			0
$759$			146.245i			0.00699390i
$760$		0			0
$761$	13505.6			0.643333			0.321667	−	0.946853i	$-0.395757\pi$
$761$	13505.6			0.643333			0.321667	+	0.946853i	$0.395757\pi$
$762$		0			0
$763$	6500.63			0.308439
$764$		0			0
$765$	9481.73	−	4181.05i	0.448121	−	0.197603i
$766$		0			0
$767$	−2073.67			−0.0976217
$768$		0			0
$769$	15256.1			0.715407			0.357704	−	0.933835i	$-0.383560\pi$
$769$	15256.1			0.715407			0.357704	+	0.933835i	$0.383560\pi$
$770$		0			0
$771$			10246.0i			0.478599i
$772$		0			0
$773$	−16038.3			−0.746256			−0.373128	−	0.927780i	$-0.621715\pi$
$773$	−16038.3			−0.746256			−0.373128	+	0.927780i	$0.621715\pi$
$774$		0			0
$775$			4345.25i			0.201401i
$776$		0			0
$777$			848.202i			0.0391623i
$778$		0			0
$779$		−	33396.0i		−	1.53599i
$780$		0			0
$781$	1408.57			0.0645362
$782$		0			0
$783$	−1612.10			−0.0735783
$784$		0			0
$785$		−	5800.45i		−	0.263729i
$786$		0			0
$787$			26105.8i			1.18243i	0.806514	+	0.591215i	$0.201352\pi$
$787$			26105.8i			1.18243i	−0.806514	+	0.591215i	$0.798648\pi$
$788$		0			0
$789$			9090.65i			0.410185i
$790$		0			0
$791$	−6162.36			−0.277002
$792$		0			0
$793$		−	7065.64i		−	0.316404i
$794$		0			0
$795$	−27751.0			−1.23802
$796$		0			0
$797$	5988.53			0.266154			0.133077	−	0.991106i	$-0.457514\pi$
$797$	5988.53			0.266154			0.133077	+	0.991106i	$0.457514\pi$
$798$		0			0
$799$	12918.7	+	29296.9i	0.572005	+	1.29718i
$800$		0			0
$801$	7150.40			0.315415
$802$		0			0
$803$	−1678.62			−0.0737698
$804$		0			0
$805$		−	3054.22i		−	0.133723i
$806$		0			0
$807$	16348.7			0.713138
$808$		0			0
$809$			44380.7i			1.92873i	0.264577	+	0.964365i	$0.414768\pi$
$809$			44380.7i			1.92873i	−0.264577	+	0.964365i	$0.585232\pi$
$810$		0			0
$811$		−	33546.5i		−	1.45250i	−0.687432	−	0.726249i	$-0.741262\pi$
$811$		−	33546.5i		−	1.45250i	0.687432	−	0.726249i	$-0.258738\pi$
$812$		0			0
$813$		−	7884.43i		−	0.340122i
$814$		0			0
$815$	−17588.3			−0.755941
$816$		0			0
$817$	−54849.1			−2.34875
$818$		0			0
$819$		−	933.151i		−	0.0398131i
$820$		0			0
$821$		−	13701.7i		−	0.582451i	−0.956654	−	0.291226i	$-0.905937\pi$
$821$		−	13701.7i		−	0.582451i	0.956654	−	0.291226i	$-0.0940630\pi$
$822$		0			0
$823$			4440.76i			0.188087i	0.995568	+	0.0940433i	$0.0299792\pi$
$823$			4440.76i			0.188087i	−0.995568	+	0.0940433i	$0.970021\pi$
$824$		0			0
$825$	−683.569			−0.0288470
$826$		0			0
$827$			8705.42i			0.366043i	0.983109	+	0.183021i	$0.0585877\pi$
$827$			8705.42i			0.366043i	−0.983109	+	0.183021i	$0.941412\pi$
$828$		0			0
$829$	5755.30			0.241121			0.120561	−	0.992706i	$-0.461531\pi$
$829$	5755.30			0.241121			0.120561	+	0.992706i	$0.461531\pi$
$830$		0			0
$831$	−21399.4			−0.893306
$832$		0			0
$833$	−8682.14	−	19689.2i	−0.361126	−	0.818957i
$834$		0			0
$835$	−12228.3			−0.506799
$836$		0			0
$837$	−810.000			−0.0334501
$838$		0			0
$839$			29272.2i			1.20451i	0.798302	+	0.602257i	$0.205732\pi$
$839$			29272.2i			1.20451i	−0.798302	+	0.602257i	$0.794268\pi$
$840$		0			0
$841$	20824.0			0.853828
$842$		0			0
$843$			18030.8i			0.736670i
$844$		0			0
$845$			31184.4i			1.26956i
$846$		0			0
$847$		−	7971.15i		−	0.323367i
$848$		0			0
$849$	−3678.56			−0.148702
$850$		0			0
$851$	−1460.23			−0.0588202
$852$		0			0
$853$		−	45569.0i		−	1.82913i	−0.404434	−	0.914567i	$-0.632531\pi$
$853$		−	45569.0i		−	1.82913i	0.404434	−	0.914567i	$-0.367469\pi$
$854$		0			0
$855$			21561.5i			0.862442i
$856$		0			0
$857$		−	33462.5i		−	1.33379i	−0.745151	−	0.666895i	$-0.767623\pi$
$857$		−	33462.5i		−	1.33379i	0.745151	−	0.666895i	$-0.232377\pi$
$858$		0			0
$859$	−7859.19			−0.312168			−0.156084	−	0.987744i	$-0.549887\pi$
$859$	−7859.19			−0.312168			−0.156084	+	0.987744i	$0.549887\pi$
$860$		0			0
$861$		−	4121.78i		−	0.163148i
$862$		0			0
$863$	17613.4			0.694746			0.347373	−	0.937727i	$-0.387074\pi$
$863$	17613.4			0.694746			0.347373	+	0.937727i	$0.387074\pi$
$864$		0			0
$865$	−47153.8			−1.85350
$866$		0			0
$867$	−10882.5	−	9940.25i	−0.426286	−	0.389376i
$868$		0			0
$869$	1634.19			0.0637928
$870$		0			0
$871$	3544.63			0.137894
$872$		0			0
$873$			9056.20i			0.351095i
$874$		0			0
$875$	1955.63			0.0755568
$876$		0			0
$877$		−	38660.0i		−	1.48855i	−0.667874	−	0.744274i	$-0.732796\pi$
$877$		−	38660.0i		−	1.48855i	0.667874	−	0.744274i	$-0.267204\pi$
$878$		0			0
$879$			25379.2i			0.973855i
$880$		0			0
$881$			15647.5i			0.598384i	0.954193	+	0.299192i	$0.0967172\pi$
$881$			15647.5i			0.598384i	−0.954193	+	0.299192i	$0.903283\pi$
$882$		0			0
$883$	12759.6			0.486291			0.243145	−	0.969990i	$-0.421821\pi$
$883$	12759.6			0.486291			0.243145	+	0.969990i	$0.421821\pi$
$884$		0			0
$885$	5913.67			0.224617
$886$		0			0
$887$		−	18022.4i		−	0.682225i	−0.940022	−	0.341113i	$-0.889196\pi$
$887$		−	18022.4i		−	0.682225i	0.940022	−	0.341113i	$-0.110804\pi$
$888$		0			0
$889$			13749.8i			0.518731i
$890$		0			0
$891$		−	127.424i		−	0.00479111i
$892$		0			0
$893$	−66621.4			−2.49653
$894$		0			0
$895$		−	18792.8i		−	0.701871i
$896$		0			0
$897$	1606.47			0.0597977
$898$		0			0
$899$	−1791.22			−0.0664523
$900$		0			0
$901$	15925.4	+	36115.5i	0.588849	+	1.33538i
$902$		0			0
$903$	−6769.56			−0.249476
$904$		0			0
$905$	−57758.0			−2.12148
$906$		0			0
$907$		−	51434.0i		−	1.88295i	−0.337078	−	0.941477i	$-0.609439\pi$
$907$		−	51434.0i		−	1.88295i	0.337078	−	0.941477i	$-0.390561\pi$
$908$		0			0
$909$	3547.56			0.129444
$910$		0			0
$911$		−	7890.19i		−	0.286953i	−0.989654	−	0.143476i	$-0.954172\pi$
$911$		−	7890.19i		−	0.286953i	0.989654	−	0.143476i	$-0.0458281\pi$
$912$		0			0
$913$			1729.96i			0.0627088i
$914$		0			0
$915$			20149.7i			0.728010i
$916$		0			0
$917$	−11610.2			−0.418107
$918$		0			0
$919$	46068.0			1.65358			0.826792	−	0.562507i	$-0.190163\pi$
$919$	46068.0			1.65358			0.826792	+	0.562507i	$0.190163\pi$
$920$		0			0
$921$			10154.3i			0.363297i
$922$		0			0
$923$		−	15472.9i		−	0.551783i
$924$		0			0
$925$		−	6825.28i		−	0.242610i
$926$		0			0
$927$	6740.03			0.238804
$928$		0			0
$929$			28007.0i			0.989106i	0.869147	+	0.494553i	$0.164668\pi$
$929$			28007.0i			0.989106i	−0.869147	+	0.494553i	$0.835332\pi$
$930$		0			0
$931$	44773.4			1.57614
$932$		0			0
$933$	−2583.01			−0.0906365
$934$		0			0
$935$	730.820	+	1657.34i	0.0255619	+	0.0579689i
$936$		0			0
$937$	42032.8			1.46548			0.732738	−	0.680511i	$-0.238242\pi$
$937$	42032.8			1.46548			0.732738	+	0.680511i	$0.238242\pi$
$938$		0			0
$939$	−2851.95			−0.0991158
$940$		0			0
$941$		−	31863.4i		−	1.10384i	−0.833896	−	0.551922i	$-0.813895\pi$
$941$		−	31863.4i		−	1.10384i	0.833896	−	0.551922i	$-0.186105\pi$
$942$		0			0
$943$	7095.89			0.245041
$944$		0			0
$945$			2661.15i			0.0916056i
$946$		0			0
$947$			42612.1i			1.46220i	0.682269	+	0.731102i	$0.260993\pi$
$947$			42612.1i			1.46220i	−0.682269	+	0.731102i	$0.739007\pi$
$948$		0			0
$949$			18439.3i			0.630731i
$950$		0			0
$951$	25203.5			0.859390
$952$		0			0
$953$	−6529.28			−0.221935			−0.110968	−	0.993824i	$-0.535395\pi$
$953$	−6529.28			−0.221935			−0.110968	+	0.993824i	$0.535395\pi$
$954$		0			0
$955$		−	68230.8i		−	2.31193i
$956$		0			0
$957$		−	281.784i		−	0.00951807i
$958$		0			0
$959$		−	5841.24i		−	0.196688i
$960$		0			0
$961$	28891.0			0.969790
$962$		0			0
$963$			10804.3i			0.361541i
$964$		0			0
$965$	67200.8			2.24173
$966$		0			0
$967$	−21784.1			−0.724436			−0.362218	−	0.932093i	$-0.617980\pi$
$967$	−21784.1			−0.724436			−0.362218	+	0.932093i	$0.617980\pi$
$968$		0			0
$969$	28060.4	−	12373.5i	0.930267	−	0.410210i
$970$		0			0
$971$	293.579			0.00970279			0.00485140	−	0.999988i	$-0.498456\pi$
$971$	293.579			0.00970279			0.00485140	+	0.999988i	$0.498456\pi$
$972$		0			0
$973$	1862.33			0.0613603
$974$		0			0
$975$			7508.85i			0.246642i
$976$		0			0
$977$	42000.5			1.37535			0.687673	−	0.726020i	$-0.258632\pi$
$977$	42000.5			1.37535			0.687673	+	0.726020i	$0.258632\pi$
$978$		0			0
$979$			1249.84i			0.0408020i
$980$		0			0
$981$		−	9750.95i		−	0.317354i
$982$		0			0
$983$		−	4754.22i		−	0.154258i	−0.997021	−	0.0771292i	$-0.975425\pi$
$983$		−	4754.22i		−	0.154258i	0.997021	−	0.0771292i	$-0.0245754\pi$
$984$		0			0
$985$	79239.0			2.56321
$986$		0			0
$987$	−8222.50			−0.265172
$988$		0			0
$989$		−	11654.2i		−	0.374703i
$990$		0			0
$991$			44080.5i			1.41298i	0.707722	+	0.706491i	$0.249723\pi$
$991$			44080.5i			1.41298i	−0.707722	+	0.706491i	$0.750277\pi$
$992$		0			0
$993$		−	12384.6i		−	0.395784i
$994$		0			0
$995$	49046.4			1.56269
$996$		0			0
$997$			6073.28i			0.192922i	0.995337	+	0.0964608i	$0.0307522\pi$
$997$			6073.28i			0.192922i	−0.995337	+	0.0964608i	$0.969248\pi$
$998$		0			0
$999$	1272.30			0.0402942

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	816.4.c.b.577.3		4
4.3	odd	2		102.4.b.a.67.1	✓	4
12.11	even	2		306.4.b.e.271.4		4
17.16	even	2	inner	816.4.c.b.577.2		4
68.47	odd	4		1734.4.a.s.1.2		2
68.55	odd	4		1734.4.a.m.1.1		2
68.67	odd	2		102.4.b.a.67.4	yes	4
204.203	even	2		306.4.b.e.271.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.4.b.a.67.1	✓	4	4.3	odd	2
102.4.b.a.67.4	yes	4	68.67	odd	2
306.4.b.e.271.1		4	204.203	even	2
306.4.b.e.271.4		4	12.11	even	2
816.4.c.b.577.2		4	17.16	even	2	inner
816.4.c.b.577.3		4	1.1	even	1	trivial
1734.4.a.m.1.1		2	68.55	odd	4
1734.4.a.s.1.2		2	68.47	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	816.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -16.4269i q^{5} -6.00000i q^{7} -9.00000 q^{9} -1.57314i q^{11} -17.2806 q^{13} +49.2806 q^{15} +(-28.2806 - 64.1343i) q^{17} +145.842 q^{19} +18.0000 q^{21} +30.9880i q^{23} -144.842 q^{25} -27.0000i q^{27} -59.7074i q^{29} -30.0000i q^{31} +4.71942 q^{33} -98.5612 q^{35} +47.1223i q^{37} -51.8417i q^{39} -228.988i q^{41} -376.086 q^{43} +147.842i q^{45} -456.806 q^{47} +307.000 q^{49} +(192.403 - 84.8417i) q^{51} -563.122 q^{53} -25.8417 q^{55} +437.525i q^{57} +120.000 q^{59} +408.878i q^{61} +54.0000i q^{63} +283.866i q^{65} -205.122 q^{67} -92.9641 q^{69} +895.391i q^{71} -1067.05i q^{73} -434.525i q^{75} -9.43884 q^{77} +1038.81i q^{79} +81.0000 q^{81} -1099.68 q^{83} +(-1053.53 + 464.561i) q^{85} +179.122 q^{87} -794.489 q^{89} +103.683i q^{91} +90.0000 q^{93} -2395.72i q^{95} -1006.24i q^{97} +14.1583i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{9} + 74 q^{13} + 54 q^{15} + 30 q^{17} + 154 q^{19} + 72 q^{21} - 150 q^{25} + 162 q^{33} - 108 q^{35} + 70 q^{43} - 396 q^{47} + 1228 q^{49} + 54 q^{51} - 1680 q^{53} + 326 q^{55} + 480 q^{59}+ \cdots + 360 q^{93}+O(q^{100}) \) 4 * q - 36 * q^9 + 74 * q^13 + 54 * q^15 + 30 * q^17 + 154 * q^19 + 72 * q^21 - 150 * q^25 + 162 * q^33 - 108 * q^35 + 70 * q^43 - 396 * q^47 + 1228 * q^49 + 54 * q^51 - 1680 * q^53 + 326 * q^55 + 480 * q^59 - 248 * q^67 + 630 * q^69 - 324 * q^77 + 324 * q^81 - 3540 * q^83 - 2926 * q^85 + 144 * q^87 - 888 * q^89 + 360 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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