LMFDB - Embedded newform 84.6.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [84,6,Mod(1,84)] 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("84.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(84, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	84.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$13.4722408643$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{5569}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1392 $ x^2 - x - 1392
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-36.8129$ of defining polynomial
Character	$\chi$	$=$	84.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} +71.6257 q^{5} -49.0000 q^{7} +81.0000 q^{9} -567.380 q^{11} +831.754 q^{13} -644.632 q^{15} +888.374 q^{17} +2915.02 q^{19} +441.000 q^{21} +3102.37 q^{23} +2005.25 q^{25} -729.000 q^{27} +8271.03 q^{29} -7029.05 q^{31} +5106.42 q^{33} -3509.66 q^{35} -10141.9 q^{37} -7485.79 q^{39} +3095.65 q^{41} +15026.2 q^{43} +5801.68 q^{45} +19895.4 q^{47} +2401.00 q^{49} -7995.37 q^{51} -9206.42 q^{53} -40639.0 q^{55} -26235.2 q^{57} -10301.3 q^{59} -22599.2 q^{61} -3969.00 q^{63} +59575.0 q^{65} +6419.09 q^{67} -27921.4 q^{69} -61279.0 q^{71} -29707.1 q^{73} -18047.2 q^{75} +27801.6 q^{77} -15630.8 q^{79} +6561.00 q^{81} +1668.23 q^{83} +63630.5 q^{85} -74439.3 q^{87} +75833.3 q^{89} -40756.0 q^{91} +63261.5 q^{93} +208790. q^{95} -98013.9 q^{97} -45957.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{3} - 6 q^{5} - 98 q^{7} + 162 q^{9} - 90 q^{11} + 768 q^{13} + 54 q^{15} + 1926 q^{17} + 2248 q^{19} + 882 q^{21} + 6354 q^{23} + 4906 q^{25} - 1458 q^{27} + 10572 q^{29} - 3312 q^{31} + 810 q^{33}+ \cdots - 7290 q^{99}+O(q^{100}) $ 2 * q - 18 * q^3 - 6 * q^5 - 98 * q^7 + 162 * q^9 - 90 * q^11 + 768 * q^13 + 54 * q^15 + 1926 * q^17 + 2248 * q^19 + 882 * q^21 + 6354 * q^23 + 4906 * q^25 - 1458 * q^27 + 10572 * q^29 - 3312 * q^31 + 810 * q^33 + 294 * q^35 + 2104 * q^37 - 6912 * q^39 + 1266 * q^41 - 5768 * q^43 - 486 * q^45 + 15612 * q^47 + 4802 * q^49 - 17334 * q^51 + 16512 * q^53 - 77696 * q^55 - 20232 * q^57 - 13140 * q^59 - 5796 * q^61 - 7938 * q^63 + 64524 * q^65 - 56116 * q^67 - 57186 * q^69 + 11022 * q^71 - 85384 * q^73 - 44154 * q^75 + 4410 * q^77 - 19620 * q^79 + 13122 * q^81 - 44424 * q^83 - 16916 * q^85 - 95148 * q^87 + 211218 * q^89 - 37632 * q^91 + 29808 * q^93 + 260568 * q^95 + 44864 * q^97 - 7290 * q^99

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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	71.6257	1.28128	0.640640	−	0.767841i	$-0.278669\pi$
$5$	71.6257	1.28128	0.640640	+	0.767841i	$0.278669\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−567.380	−1.41381	−0.706907	−	0.707306i	$-0.749910\pi$
$11$	−567.380	−1.41381	−0.706907	+	0.707306i	$0.749910\pi$
$12$	0	0
$13$	831.754	1.36501	0.682506	−	0.730880i	$-0.260890\pi$
$13$	831.754	1.36501	0.682506	+	0.730880i	$0.260890\pi$
$14$	0	0
$15$	−644.632	−0.739747
$16$	0	0
$17$	888.374	0.745545	0.372772	−	0.927923i	$-0.378407\pi$
$17$	888.374	0.745545	0.372772	+	0.927923i	$0.378407\pi$
$18$	0	0
$19$	2915.02	1.85250	0.926248	−	0.376915i	$-0.123015\pi$
$19$	2915.02	1.85250	0.926248	+	0.376915i	$0.123015\pi$
$20$	0	0
$21$	441.000	0.218218
$22$	0	0
$23$	3102.37	1.22285	0.611427	−	0.791301i	$-0.290596\pi$
$23$	3102.37	1.22285	0.611427	+	0.791301i	$0.290596\pi$
$24$	0	0
$25$	2005.25	0.641679
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	8271.03	1.82627	0.913134	−	0.407659i	$-0.133655\pi$
$29$	8271.03	1.82627	0.913134	+	0.407659i	$0.133655\pi$
$30$	0	0
$31$	−7029.05	−1.31369	−0.656845	−	0.754026i	$-0.728109\pi$
$31$	−7029.05	−1.31369	−0.656845	+	0.754026i	$0.728109\pi$
$32$	0	0
$33$	5106.42	0.816266
$34$	0	0
$35$	−3509.66	−0.484278
$36$	0	0
$37$	−10141.9	−1.21790	−0.608952	−	0.793207i	$-0.708410\pi$
$37$	−10141.9	−1.21790	−0.608952	+	0.793207i	$0.708410\pi$
$38$	0	0
$39$	−7485.79	−0.788091
$40$	0	0
$41$	3095.65	0.287602	0.143801	−	0.989607i	$-0.454067\pi$
$41$	3095.65	0.287602	0.143801	+	0.989607i	$0.454067\pi$
$42$	0	0
$43$	15026.2	1.23930	0.619651	−	0.784877i	$-0.287274\pi$
$43$	15026.2	1.23930	0.619651	+	0.784877i	$0.287274\pi$
$44$	0	0
$45$	5801.68	0.427093
$46$	0	0
$47$	19895.4	1.31373	0.656867	−	0.754007i	$-0.271881\pi$
$47$	19895.4	1.31373	0.656867	+	0.754007i	$0.271881\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−7995.37	−0.430440
$52$	0	0
$53$	−9206.42	−0.450196	−0.225098	−	0.974336i	$-0.572270\pi$
$53$	−9206.42	−0.450196	−0.225098	+	0.974336i	$0.572270\pi$
$54$	0	0
$55$	−40639.0	−1.81149
$56$	0	0
$57$	−26235.2	−1.06954
$58$	0	0
$59$	−10301.3	−0.385267	−0.192633	−	0.981271i	$-0.561703\pi$
$59$	−10301.3	−0.385267	−0.192633	+	0.981271i	$0.561703\pi$
$60$	0	0
$61$	−22599.2	−0.777622	−0.388811	−	0.921318i	$-0.627114\pi$
$61$	−22599.2	−0.777622	−0.388811	+	0.921318i	$0.627114\pi$
$62$	0	0
$63$	−3969.00	−0.125988
$64$	0	0
$65$	59575.0	1.74896
$66$	0	0
$67$	6419.09	0.174697	0.0873487	−	0.996178i	$-0.472161\pi$
$67$	6419.09	0.174697	0.0873487	+	0.996178i	$0.472161\pi$
$68$	0	0
$69$	−27921.4	−0.706015
$70$	0	0
$71$	−61279.0	−1.44267	−0.721333	−	0.692588i	$-0.756470\pi$
$71$	−61279.0	−1.44267	−0.721333	+	0.692588i	$0.756470\pi$
$72$	0	0
$73$	−29707.1	−0.652459	−0.326230	−	0.945291i	$-0.605778\pi$
$73$	−29707.1	−0.652459	−0.326230	+	0.945291i	$0.605778\pi$
$74$	0	0
$75$	−18047.2	−0.370473
$76$	0	0
$77$	27801.6	0.534372
$78$	0	0
$79$	−15630.8	−0.281782	−0.140891	−	0.990025i	$-0.544997\pi$
$79$	−15630.8	−0.281782	−0.140891	+	0.990025i	$0.544997\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	1668.23	0.0265804	0.0132902	−	0.999912i	$-0.495769\pi$
$83$	1668.23	0.0265804	0.0132902	+	0.999912i	$0.495769\pi$
$84$	0	0
$85$	63630.5	0.955252
$86$	0	0
$87$	−74439.3	−1.05440
$88$	0	0
$89$	75833.3	1.01481	0.507405	−	0.861707i	$-0.330605\pi$
$89$	75833.3	1.01481	0.507405	+	0.861707i	$0.330605\pi$
$90$	0	0
$91$	−40756.0	−0.515926
$92$	0	0
$93$	63261.5	0.758459
$94$	0	0
$95$	208790.	2.37357
$96$	0	0
$97$	−98013.9	−1.05769	−0.528845	−	0.848718i	$-0.677375\pi$
$97$	−98013.9	−1.05769	−0.528845	+	0.848718i	$0.677375\pi$
$98$	0	0
$99$	−45957.8	−0.471271
$100$	0	0
$101$	54466.9	0.531287	0.265644	−	0.964071i	$-0.414416\pi$
$101$	54466.9	0.531287	0.265644	+	0.964071i	$0.414416\pi$
$102$	0	0
$103$	−178867.	−1.66126	−0.830631	−	0.556823i	$-0.812020\pi$
$103$	−178867.	−1.66126	−0.830631	+	0.556823i	$0.812020\pi$
$104$	0	0
$105$	31586.9	0.279598
$106$	0	0
$107$	−6591.90	−0.0556610	−0.0278305	−	0.999613i	$-0.508860\pi$
$107$	−6591.90	−0.0556610	−0.0278305	+	0.999613i	$0.508860\pi$
$108$	0	0
$109$	177946.	1.43457	0.717285	−	0.696780i	$-0.245385\pi$
$109$	177946.	1.43457	0.717285	+	0.696780i	$0.245385\pi$
$110$	0	0
$111$	91276.7	0.703158
$112$	0	0
$113$	101861.	0.750432	0.375216	−	0.926937i	$-0.377569\pi$
$113$	101861.	0.750432	0.375216	+	0.926937i	$0.377569\pi$
$114$	0	0
$115$	222210.	1.56682
$116$	0	0
$117$	67372.1	0.455004
$118$	0	0
$119$	−43530.3	−0.281789
$120$	0	0
$121$	160869.	0.998871
$122$	0	0
$123$	−27860.8	−0.166047
$124$	0	0
$125$	−80203.2	−0.459110
$126$	0	0
$127$	−153984.	−0.847163	−0.423581	−	0.905858i	$-0.639227\pi$
$127$	−153984.	−0.847163	−0.423581	+	0.905858i	$0.639227\pi$
$128$	0	0
$129$	−135236.	−0.715512
$130$	0	0
$131$	245130.	1.24801	0.624005	−	0.781420i	$-0.285504\pi$
$131$	245130.	1.24801	0.624005	+	0.781420i	$0.285504\pi$
$132$	0	0
$133$	−142836.	−0.700178
$134$	0	0
$135$	−52215.2	−0.246582
$136$	0	0
$137$	85541.3	0.389381	0.194690	−	0.980865i	$-0.437630\pi$
$137$	85541.3	0.389381	0.194690	+	0.980865i	$0.437630\pi$
$138$	0	0
$139$	−195746.	−0.859320	−0.429660	−	0.902991i	$-0.641367\pi$
$139$	−195746.	−0.859320	−0.429660	+	0.902991i	$0.641367\pi$
$140$	0	0
$141$	−179058.	−0.758485
$142$	0	0
$143$	−471921.	−1.92987
$144$	0	0
$145$	592419.	2.33996
$146$	0	0
$147$	−21609.0	−0.0824786
$148$	0	0
$149$	−276266.	−1.01944	−0.509720	−	0.860340i	$-0.670251\pi$
$149$	−276266.	−1.01944	−0.509720	+	0.860340i	$0.670251\pi$
$150$	0	0
$151$	281649.	1.00523	0.502616	−	0.864510i	$-0.332371\pi$
$151$	281649.	1.00523	0.502616	+	0.864510i	$0.332371\pi$
$152$	0	0
$153$	71958.3	0.248515
$154$	0	0
$155$	−503461.	−1.68320
$156$	0	0
$157$	−455344.	−1.47432	−0.737158	−	0.675720i	$-0.763833\pi$
$157$	−455344.	−1.47432	−0.737158	+	0.675720i	$0.763833\pi$
$158$	0	0
$159$	82857.8	0.259921
$160$	0	0
$161$	−152016.	−0.462195
$162$	0	0
$163$	−375179.	−1.10604	−0.553018	−	0.833169i	$-0.686524\pi$
$163$	−375179.	−1.10604	−0.553018	+	0.833169i	$0.686524\pi$
$164$	0	0
$165$	365751.	1.04587
$166$	0	0
$167$	279742.	0.776186	0.388093	−	0.921620i	$-0.373134\pi$
$167$	279742.	0.776186	0.388093	+	0.921620i	$0.373134\pi$
$168$	0	0
$169$	320522.	0.863260
$170$	0	0
$171$	236116.	0.617499
$172$	0	0
$173$	−162065.	−0.411693	−0.205847	−	0.978584i	$-0.565995\pi$
$173$	−162065.	−0.411693	−0.205847	+	0.978584i	$0.565995\pi$
$174$	0	0
$175$	−98257.0	−0.242532
$176$	0	0
$177$	92711.6	0.222434
$178$	0	0
$179$	−153376.	−0.357787	−0.178894	−	0.983868i	$-0.557252\pi$
$179$	−153376.	−0.357787	−0.178894	+	0.983868i	$0.557252\pi$
$180$	0	0
$181$	−323034.	−0.732911	−0.366456	−	0.930435i	$-0.619429\pi$
$181$	−323034.	−0.732911	−0.366456	+	0.930435i	$0.619429\pi$
$182$	0	0
$183$	203393.	0.448960
$184$	0	0
$185$	−726418.	−1.56048
$186$	0	0
$187$	−504046.	−1.05406
$188$	0	0
$189$	35721.0	0.0727393
$190$	0	0
$191$	654819.	1.29879	0.649394	−	0.760453i	$-0.275023\pi$
$191$	654819.	1.29879	0.649394	+	0.760453i	$0.275023\pi$
$192$	0	0
$193$	776059.	1.49969	0.749845	−	0.661613i	$-0.230128\pi$
$193$	776059.	1.49969	0.749845	+	0.661613i	$0.230128\pi$
$194$	0	0
$195$	−536175.	−1.00976
$196$	0	0
$197$	−399284.	−0.733020	−0.366510	−	0.930414i	$-0.619447\pi$
$197$	−399284.	−0.733020	−0.366510	+	0.930414i	$0.619447\pi$
$198$	0	0
$199$	9524.77	0.0170499	0.00852495	−	0.999964i	$-0.497286\pi$
$199$	9524.77	0.0170499	0.00852495	+	0.999964i	$0.497286\pi$
$200$	0	0
$201$	−57771.8	−0.100862
$202$	0	0
$203$	−405280.	−0.690265
$204$	0	0
$205$	221728.	0.368499
$206$	0	0
$207$	251292.	0.407618
$208$	0	0
$209$	−1.65392e6	−2.61908
$210$	0	0
$211$	−481254.	−0.744164	−0.372082	−	0.928200i	$-0.621356\pi$
$211$	−481254.	−0.744164	−0.372082	+	0.928200i	$0.621356\pi$
$212$	0	0
$213$	551511.	0.832924
$214$	0	0
$215$	1.07626e6	1.58789
$216$	0	0
$217$	344424.	0.496528
$218$	0	0
$219$	267364.	0.376698
$220$	0	0
$221$	738909.	1.01768
$222$	0	0
$223$	765559.	1.03090	0.515450	−	0.856919i	$-0.327625\pi$
$223$	765559.	1.03090	0.515450	+	0.856919i	$0.327625\pi$
$224$	0	0
$225$	162425.	0.213893
$226$	0	0
$227$	657810.	0.847297	0.423649	−	0.905827i	$-0.360749\pi$
$227$	657810.	0.847297	0.423649	+	0.905827i	$0.360749\pi$
$228$	0	0
$229$	499749.	0.629743	0.314871	−	0.949134i	$-0.398039\pi$
$229$	499749.	0.629743	0.314871	+	0.949134i	$0.398039\pi$
$230$	0	0
$231$	−250215.	−0.308520
$232$	0	0
$233$	−480588.	−0.579940	−0.289970	−	0.957036i	$-0.593645\pi$
$233$	−480588.	−0.579940	−0.289970	+	0.957036i	$0.593645\pi$
$234$	0	0
$235$	1.42502e6	1.68326
$236$	0	0
$237$	140677.	0.162687
$238$	0	0
$239$	869603.	0.984751	0.492376	−	0.870383i	$-0.336129\pi$
$239$	869603.	0.984751	0.492376	+	0.870383i	$0.336129\pi$
$240$	0	0
$241$	−671527.	−0.744768	−0.372384	−	0.928079i	$-0.621460\pi$
$241$	−671527.	−0.744768	−0.372384	+	0.928079i	$0.621460\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	171973.	0.183040
$246$	0	0
$247$	2.42458e6	2.52868
$248$	0	0
$249$	−15014.1	−0.0153462
$250$	0	0
$251$	−405312.	−0.406074	−0.203037	−	0.979171i	$-0.565081\pi$
$251$	−405312.	−0.406074	−0.203037	+	0.979171i	$0.565081\pi$
$252$	0	0
$253$	−1.76023e6	−1.72889
$254$	0	0
$255$	−572674.	−0.551515
$256$	0	0
$257$	−837649.	−0.791096	−0.395548	−	0.918445i	$-0.629445\pi$
$257$	−837649.	−0.791096	−0.395548	+	0.918445i	$0.629445\pi$
$258$	0	0
$259$	496951.	0.460325
$260$	0	0
$261$	669953.	0.608756
$262$	0	0
$263$	629562.	0.561241	0.280620	−	0.959819i	$-0.409460\pi$
$263$	629562.	0.561241	0.280620	+	0.959819i	$0.409460\pi$
$264$	0	0
$265$	−659417.	−0.576827
$266$	0	0
$267$	−682500.	−0.585901
$268$	0	0
$269$	−319041.	−0.268822	−0.134411	−	0.990926i	$-0.542914\pi$
$269$	−319041.	−0.268822	−0.134411	+	0.990926i	$0.542914\pi$
$270$	0	0
$271$	715999.	0.592229	0.296114	−	0.955153i	$-0.404309\pi$
$271$	715999.	0.592229	0.296114	+	0.955153i	$0.404309\pi$
$272$	0	0
$273$	366804.	0.297870
$274$	0	0
$275$	−1.13774e6	−0.907214
$276$	0	0
$277$	−869625.	−0.680977	−0.340489	−	0.940249i	$-0.610592\pi$
$277$	−869625.	−0.680977	−0.340489	+	0.940249i	$0.610592\pi$
$278$	0	0
$279$	−569353.	−0.437896
$280$	0	0
$281$	1.58664e6	1.19870	0.599351	−	0.800486i	$-0.295425\pi$
$281$	1.58664e6	1.19870	0.599351	+	0.800486i	$0.295425\pi$
$282$	0	0
$283$	−1.60231e6	−1.18927	−0.594636	−	0.803995i	$-0.702704\pi$
$283$	−1.60231e6	−1.18927	−0.594636	+	0.803995i	$0.702704\pi$
$284$	0	0
$285$	−1.87911e6	−1.37038
$286$	0	0
$287$	−151687.	−0.108703
$288$	0	0
$289$	−630648.	−0.444163
$290$	0	0
$291$	882125.	0.610658
$292$	0	0
$293$	1.18717e6	0.807874	0.403937	−	0.914787i	$-0.367641\pi$
$293$	1.18717e6	0.807874	0.403937	+	0.914787i	$0.367641\pi$
$294$	0	0
$295$	−737837.	−0.493635
$296$	0	0
$297$	413620.	0.272089
$298$	0	0
$299$	2.58041e6	1.66921
$300$	0	0
$301$	−736283.	−0.468412
$302$	0	0
$303$	−490202.	−0.306739
$304$	0	0
$305$	−1.61868e6	−0.996351
$306$	0	0
$307$	−2.24511e6	−1.35954	−0.679769	−	0.733426i	$-0.737920\pi$
$307$	−2.24511e6	−1.35954	−0.679769	+	0.733426i	$0.737920\pi$
$308$	0	0
$309$	1.60981e6	0.959130
$310$	0	0
$311$	562022.	0.329498	0.164749	−	0.986336i	$-0.447319\pi$
$311$	562022.	0.329498	0.164749	+	0.986336i	$0.447319\pi$
$312$	0	0
$313$	320869.	0.185126	0.0925629	−	0.995707i	$-0.470494\pi$
$313$	320869.	0.185126	0.0925629	+	0.995707i	$0.470494\pi$
$314$	0	0
$315$	−284283.	−0.161426
$316$	0	0
$317$	−2.18659e6	−1.22213	−0.611067	−	0.791579i	$-0.709259\pi$
$317$	−2.18659e6	−1.22213	−0.611067	+	0.791579i	$0.709259\pi$
$318$	0	0
$319$	−4.69282e6	−2.58200
$320$	0	0
$321$	59327.1	0.0321359
$322$	0	0
$323$	2.58963e6	1.38112
$324$	0	0
$325$	1.66787e6	0.875899
$326$	0	0
$327$	−1.60151e6	−0.828249
$328$	0	0
$329$	−974873.	−0.496545
$330$	0	0
$331$	211399.	0.106056	0.0530278	−	0.998593i	$-0.483113\pi$
$331$	211399.	0.106056	0.0530278	+	0.998593i	$0.483113\pi$
$332$	0	0
$333$	−821491.	−0.405968
$334$	0	0
$335$	459772.	0.223836
$336$	0	0
$337$	−1.37707e6	−0.660511	−0.330256	−	0.943892i	$-0.607135\pi$
$337$	−1.37707e6	−0.660511	−0.330256	+	0.943892i	$0.607135\pi$
$338$	0	0
$339$	−916748.	−0.433262
$340$	0	0
$341$	3.98814e6	1.85731
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−1.99989e6	−0.904603
$346$	0	0
$347$	−1.62901e6	−0.726273	−0.363136	−	0.931736i	$-0.618294\pi$
$347$	−1.62901e6	−0.726273	−0.363136	+	0.931736i	$0.618294\pi$
$348$	0	0
$349$	3.30453e6	1.45227	0.726133	−	0.687554i	$-0.241316\pi$
$349$	3.30453e6	1.45227	0.726133	+	0.687554i	$0.241316\pi$
$350$	0	0
$351$	−606349.	−0.262697
$352$	0	0
$353$	−2.46031e6	−1.05088	−0.525440	−	0.850831i	$-0.676099\pi$
$353$	−2.46031e6	−1.05088	−0.525440	+	0.850831i	$0.676099\pi$
$354$	0	0
$355$	−4.38916e6	−1.84846
$356$	0	0
$357$	391773.	0.162691
$358$	0	0
$359$	4.00216e6	1.63892	0.819462	−	0.573134i	$-0.194272\pi$
$359$	4.00216e6	1.63892	0.819462	+	0.573134i	$0.194272\pi$
$360$	0	0
$361$	6.02123e6	2.43174
$362$	0	0
$363$	−1.44782e6	−0.576699
$364$	0	0
$365$	−2.12779e6	−0.835983
$366$	0	0
$367$	4.74112e6	1.83745	0.918725	−	0.394897i	$-0.129220\pi$
$367$	4.74112e6	1.83745	0.918725	+	0.394897i	$0.129220\pi$
$368$	0	0
$369$	250748.	0.0958674
$370$	0	0
$371$	451115.	0.170158
$372$	0	0
$373$	−3.18391e6	−1.18492	−0.592460	−	0.805600i	$-0.701843\pi$
$373$	−3.18391e6	−1.18492	−0.592460	+	0.805600i	$0.701843\pi$
$374$	0	0
$375$	721829.	0.265067
$376$	0	0
$377$	6.87946e6	2.49288
$378$	0	0
$379$	980638.	0.350680	0.175340	−	0.984508i	$-0.443898\pi$
$379$	980638.	0.350680	0.175340	+	0.984508i	$0.443898\pi$
$380$	0	0
$381$	1.38586e6	0.489110
$382$	0	0
$383$	42711.2	0.0148780	0.00743901	−	0.999972i	$-0.497632\pi$
$383$	42711.2	0.0148780	0.00743901	+	0.999972i	$0.497632\pi$
$384$	0	0
$385$	1.99131e6	0.684680
$386$	0	0
$387$	1.21712e6	0.413101
$388$	0	0
$389$	−4.96471e6	−1.66349	−0.831744	−	0.555159i	$-0.812657\pi$
$389$	−4.96471e6	−1.66349	−0.831744	+	0.555159i	$0.812657\pi$
$390$	0	0
$391$	2.75607e6	0.911692
$392$	0	0
$393$	−2.20617e6	−0.720539
$394$	0	0
$395$	−1.11957e6	−0.361042
$396$	0	0
$397$	−2.38304e6	−0.758848	−0.379424	−	0.925223i	$-0.623878\pi$
$397$	−2.38304e6	−0.758848	−0.379424	+	0.925223i	$0.623878\pi$
$398$	0	0
$399$	1.28552e6	0.404248
$400$	0	0
$401$	−4.53716e6	−1.40904	−0.704520	−	0.709684i	$-0.748838\pi$
$401$	−4.53716e6	−1.40904	−0.704520	+	0.709684i	$0.748838\pi$
$402$	0	0
$403$	−5.84645e6	−1.79320
$404$	0	0
$405$	469936.	0.142364
$406$	0	0
$407$	5.75429e6	1.72189
$408$	0	0
$409$	1.20350e6	0.355746	0.177873	−	0.984053i	$-0.443078\pi$
$409$	1.20350e6	0.355746	0.177873	+	0.984053i	$0.443078\pi$
$410$	0	0
$411$	−769872.	−0.224809
$412$	0	0
$413$	504763.	0.145617
$414$	0	0
$415$	119489.	0.0340570
$416$	0	0
$417$	1.76171e6	0.496129
$418$	0	0
$419$	4.44364e6	1.23653	0.618264	−	0.785970i	$-0.287836\pi$
$419$	4.44364e6	1.23653	0.618264	+	0.785970i	$0.287836\pi$
$420$	0	0
$421$	4.10480e6	1.12872	0.564361	−	0.825528i	$-0.309123\pi$
$421$	4.10480e6	1.12872	0.564361	+	0.825528i	$0.309123\pi$
$422$	0	0
$423$	1.61152e6	0.437911
$424$	0	0
$425$	1.78141e6	0.478400
$426$	0	0
$427$	1.10736e6	0.293913
$428$	0	0
$429$	4.24729e6	1.11421
$430$	0	0
$431$	−1.92814e6	−0.499971	−0.249986	−	0.968250i	$-0.580426\pi$
$431$	−1.92814e6	−0.499971	−0.249986	+	0.968250i	$0.580426\pi$
$432$	0	0
$433$	−4.07929e6	−1.04560	−0.522799	−	0.852456i	$-0.675112\pi$
$433$	−4.07929e6	−1.04560	−0.522799	+	0.852456i	$0.675112\pi$
$434$	0	0
$435$	−5.33177e6	−1.35098
$436$	0	0
$437$	9.04348e6	2.26533
$438$	0	0
$439$	−5.79028e6	−1.43396	−0.716982	−	0.697091i	$-0.754477\pi$
$439$	−5.79028e6	−1.43396	−0.716982	+	0.697091i	$0.754477\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	−1.95039e6	−0.472184	−0.236092	−	0.971731i	$-0.575867\pi$
$443$	−1.95039e6	−0.472184	−0.236092	+	0.971731i	$0.575867\pi$
$444$	0	0
$445$	5.43162e6	1.30026
$446$	0	0
$447$	2.48639e6	0.588574
$448$	0	0
$449$	−422170.	−0.0988260	−0.0494130	−	0.998778i	$-0.515735\pi$
$449$	−422170.	−0.0988260	−0.0494130	+	0.998778i	$0.515735\pi$
$450$	0	0
$451$	−1.75641e6	−0.406616
$452$	0	0
$453$	−2.53484e6	−0.580371
$454$	0	0
$455$	−2.91918e6	−0.661046
$456$	0	0
$457$	4.28191e6	0.959063	0.479532	−	0.877525i	$-0.340807\pi$
$457$	4.28191e6	0.959063	0.479532	+	0.877525i	$0.340807\pi$
$458$	0	0
$459$	−647625.	−0.143480
$460$	0	0
$461$	−5.33830e6	−1.16990	−0.584952	−	0.811068i	$-0.698887\pi$
$461$	−5.33830e6	−1.16990	−0.584952	+	0.811068i	$0.698887\pi$
$462$	0	0
$463$	3.36278e6	0.729030	0.364515	−	0.931198i	$-0.381235\pi$
$463$	3.36278e6	0.729030	0.364515	+	0.931198i	$0.381235\pi$
$464$	0	0
$465$	4.53115e6	0.971798
$466$	0	0
$467$	−5.26556e6	−1.11725	−0.558627	−	0.829419i	$-0.688672\pi$
$467$	−5.26556e6	−1.11725	−0.558627	+	0.829419i	$0.688672\pi$
$468$	0	0
$469$	−314535.	−0.0660294
$470$	0	0
$471$	4.09810e6	0.851197
$472$	0	0
$473$	−8.52555e6	−1.75214
$474$	0	0
$475$	5.84533e6	1.18871
$476$	0	0
$477$	−745720.	−0.150065
$478$	0	0
$479$	−5.27654e6	−1.05078	−0.525388	−	0.850863i	$-0.676080\pi$
$479$	−5.27654e6	−1.05078	−0.525388	+	0.850863i	$0.676080\pi$
$480$	0	0
$481$	−8.43554e6	−1.66246
$482$	0	0
$483$	1.36815e6	0.266849
$484$	0	0
$485$	−7.02032e6	−1.35520
$486$	0	0
$487$	−3.72209e6	−0.711155	−0.355578	−	0.934647i	$-0.615716\pi$
$487$	−3.72209e6	−0.711155	−0.355578	+	0.934647i	$0.615716\pi$
$488$	0	0
$489$	3.37661e6	0.638571
$490$	0	0
$491$	−5.19917e6	−0.973263	−0.486631	−	0.873607i	$-0.661775\pi$
$491$	−5.19917e6	−0.973263	−0.486631	+	0.873607i	$0.661775\pi$
$492$	0	0
$493$	7.34777e6	1.36156
$494$	0	0
$495$	−3.29176e6	−0.603831
$496$	0	0
$497$	3.00267e6	0.545277
$498$	0	0
$499$	−5.32512e6	−0.957366	−0.478683	−	0.877988i	$-0.658886\pi$
$499$	−5.32512e6	−0.957366	−0.478683	+	0.877988i	$0.658886\pi$
$500$	0	0
$501$	−2.51767e6	−0.448131
$502$	0	0
$503$	−4.26899e6	−0.752324	−0.376162	−	0.926554i	$-0.622756\pi$
$503$	−4.26899e6	−0.752324	−0.376162	+	0.926554i	$0.622756\pi$
$504$	0	0
$505$	3.90123e6	0.680728
$506$	0	0
$507$	−2.88470e6	−0.498403
$508$	0	0
$509$	1.07761e7	1.84361	0.921805	−	0.387655i	$-0.126715\pi$
$509$	1.07761e7	1.84361	0.921805	+	0.387655i	$0.126715\pi$
$510$	0	0
$511$	1.45565e6	0.246606
$512$	0	0
$513$	−2.12505e6	−0.356513
$514$	0	0
$515$	−1.28115e7	−2.12854
$516$	0	0
$517$	−1.12882e7	−1.85738
$518$	0	0
$519$	1.45859e6	0.237691
$520$	0	0
$521$	−6.57993e6	−1.06201	−0.531003	−	0.847370i	$-0.678185\pi$
$521$	−6.57993e6	−1.06201	−0.531003	+	0.847370i	$0.678185\pi$
$522$	0	0
$523$	−4.93301e6	−0.788602	−0.394301	−	0.918981i	$-0.629013\pi$
$523$	−4.93301e6	−0.788602	−0.394301	+	0.918981i	$0.629013\pi$
$524$	0	0
$525$	884313.	0.140026
$526$	0	0
$527$	−6.24443e6	−0.979414
$528$	0	0
$529$	3.18838e6	0.495372
$530$	0	0
$531$	−834404.	−0.128422
$532$	0	0
$533$	2.57482e6	0.392581
$534$	0	0
$535$	−472149.	−0.0713173
$536$	0	0
$537$	1.38038e6	0.206569
$538$	0	0
$539$	−1.36228e6	−0.201973
$540$	0	0
$541$	8.39789e6	1.23361	0.616804	−	0.787117i	$-0.288427\pi$
$541$	8.39789e6	1.23361	0.616804	+	0.787117i	$0.288427\pi$
$542$	0	0
$543$	2.90730e6	0.423147
$544$	0	0
$545$	1.27455e7	1.83808
$546$	0	0
$547$	7.34611e6	1.04976	0.524879	−	0.851177i	$-0.324110\pi$
$547$	7.34611e6	1.04976	0.524879	+	0.851177i	$0.324110\pi$
$548$	0	0
$549$	−1.83053e6	−0.259207
$550$	0	0
$551$	2.41102e7	3.38315
$552$	0	0
$553$	765910.	0.106504
$554$	0	0
$555$	6.53776e6	0.900942
$556$	0	0
$557$	−1.36793e7	−1.86821	−0.934107	−	0.356994i	$-0.883802\pi$
$557$	−1.36793e7	−1.86821	−0.934107	+	0.356994i	$0.883802\pi$
$558$	0	0
$559$	1.24981e7	1.69166
$560$	0	0
$561$	4.53641e6	0.608563
$562$	0	0
$563$	−1.35081e7	−1.79607	−0.898037	−	0.439920i	$-0.855007\pi$
$563$	−1.35081e7	−1.79607	−0.898037	+	0.439920i	$0.855007\pi$
$564$	0	0
$565$	7.29586e6	0.961513
$566$	0	0
$567$	−321489.	−0.0419961
$568$	0	0
$569$	−1.31087e7	−1.69738	−0.848692	−	0.528888i	$-0.822609\pi$
$569$	−1.31087e7	−1.69738	−0.848692	+	0.528888i	$0.822609\pi$
$570$	0	0
$571$	−2.25105e6	−0.288931	−0.144466	−	0.989510i	$-0.546146\pi$
$571$	−2.25105e6	−0.288931	−0.144466	+	0.989510i	$0.546146\pi$
$572$	0	0
$573$	−5.89337e6	−0.749855
$574$	0	0
$575$	6.22102e6	0.784679
$576$	0	0
$577$	3.93495e6	0.492039	0.246020	−	0.969265i	$-0.420877\pi$
$577$	3.93495e6	0.492039	0.246020	+	0.969265i	$0.420877\pi$
$578$	0	0
$579$	−6.98453e6	−0.865847
$580$	0	0
$581$	−81743.5	−0.0100465
$582$	0	0
$583$	5.22354e6	0.636493
$584$	0	0
$585$	4.82558e6	0.582988
$586$	0	0
$587$	2.57017e6	0.307869	0.153935	−	0.988081i	$-0.450806\pi$
$587$	2.57017e6	0.307869	0.153935	+	0.988081i	$0.450806\pi$
$588$	0	0
$589$	−2.04898e7	−2.43360
$590$	0	0
$591$	3.59355e6	0.423209
$592$	0	0
$593$	1.17677e6	0.137422	0.0687111	−	0.997637i	$-0.478111\pi$
$593$	1.17677e6	0.137422	0.0687111	+	0.997637i	$0.478111\pi$
$594$	0	0
$595$	−3.11789e6	−0.361051
$596$	0	0
$597$	−85722.9	−0.00984376
$598$	0	0
$599$	9.31387e6	1.06063	0.530314	−	0.847801i	$-0.322074\pi$
$599$	9.31387e6	1.06063	0.530314	+	0.847801i	$0.322074\pi$
$600$	0	0
$601$	−9.65552e6	−1.09041	−0.545205	−	0.838303i	$-0.683548\pi$
$601$	−9.65552e6	−1.09041	−0.545205	+	0.838303i	$0.683548\pi$
$602$	0	0
$603$	519946.	0.0582325
$604$	0	0
$605$	1.15224e7	1.27983
$606$	0	0
$607$	1.42813e6	0.157325	0.0786625	−	0.996901i	$-0.474935\pi$
$607$	1.42813e6	0.157325	0.0786625	+	0.996901i	$0.474935\pi$
$608$	0	0
$609$	3.64752e6	0.398524
$610$	0	0
$611$	1.65481e7	1.79326
$612$	0	0
$613$	−9.41130e6	−1.01158	−0.505788	−	0.862658i	$-0.668798\pi$
$613$	−9.41130e6	−1.01158	−0.505788	+	0.862658i	$0.668798\pi$
$614$	0	0
$615$	−1.99555e6	−0.212753
$616$	0	0
$617$	1.68556e7	1.78251	0.891256	−	0.453501i	$-0.149825\pi$
$617$	1.68556e7	1.78251	0.891256	+	0.453501i	$0.149825\pi$
$618$	0	0
$619$	1.07193e7	1.12444	0.562222	−	0.826986i	$-0.309947\pi$
$619$	1.07193e7	1.12444	0.562222	+	0.826986i	$0.309947\pi$
$620$	0	0
$621$	−2.26163e6	−0.235338
$622$	0	0
$623$	−3.71583e6	−0.383562
$624$	0	0
$625$	−1.20110e7	−1.22993
$626$	0	0
$627$	1.48853e7	1.51213
$628$	0	0
$629$	−9.00977e6	−0.908002
$630$	0	0
$631$	368207.	0.0368145	0.0184073	−	0.999831i	$-0.494140\pi$
$631$	368207.	0.0368145	0.0184073	+	0.999831i	$0.494140\pi$
$632$	0	0
$633$	4.33129e6	0.429643
$634$	0	0
$635$	−1.10292e7	−1.08545
$636$	0	0
$637$	1.99704e6	0.195002
$638$	0	0
$639$	−4.96360e6	−0.480889
$640$	0	0
$641$	1.43233e7	1.37689	0.688445	−	0.725289i	$-0.258294\pi$
$641$	1.43233e7	1.37689	0.688445	+	0.725289i	$0.258294\pi$
$642$	0	0
$643$	−7.65392e6	−0.730056	−0.365028	−	0.930996i	$-0.618941\pi$
$643$	−7.65392e6	−0.730056	−0.365028	+	0.930996i	$0.618941\pi$
$644$	0	0
$645$	−9.68635e6	−0.916771
$646$	0	0
$647$	−8.84497e6	−0.830684	−0.415342	−	0.909665i	$-0.636338\pi$
$647$	−8.84497e6	−0.830684	−0.415342	+	0.909665i	$0.636338\pi$
$648$	0	0
$649$	5.84475e6	0.544696
$650$	0	0
$651$	−3.09981e6	−0.286670
$652$	0	0
$653$	5.22481e6	0.479499	0.239749	−	0.970835i	$-0.422935\pi$
$653$	5.22481e6	0.479499	0.239749	+	0.970835i	$0.422935\pi$
$654$	0	0
$655$	1.75576e7	1.59905
$656$	0	0
$657$	−2.40628e6	−0.217486
$658$	0	0
$659$	7.14374e6	0.640785	0.320392	−	0.947285i	$-0.396185\pi$
$659$	7.14374e6	0.640785	0.320392	+	0.947285i	$0.396185\pi$
$660$	0	0
$661$	−1.05360e6	−0.0937934	−0.0468967	−	0.998900i	$-0.514933\pi$
$661$	−1.05360e6	−0.0937934	−0.0468967	+	0.998900i	$0.514933\pi$
$662$	0	0
$663$	−6.65018e6	−0.587557
$664$	0	0
$665$	−1.02307e7	−0.897123
$666$	0	0
$667$	2.56598e7	2.23326
$668$	0	0
$669$	−6.89004e6	−0.595191
$670$	0	0
$671$	1.28223e7	1.09941
$672$	0	0
$673$	−1.01967e7	−0.867802	−0.433901	−	0.900960i	$-0.642863\pi$
$673$	−1.01967e7	−0.867802	−0.433901	+	0.900960i	$0.642863\pi$
$674$	0	0
$675$	−1.46182e6	−0.123491
$676$	0	0
$677$	1.65785e7	1.39019	0.695094	−	0.718919i	$-0.255363\pi$
$677$	1.65785e7	1.39019	0.695094	+	0.718919i	$0.255363\pi$
$678$	0	0
$679$	4.80268e6	0.399769
$680$	0	0
$681$	−5.92029e6	−0.489187
$682$	0	0
$683$	−1.43664e6	−0.117841	−0.0589206	−	0.998263i	$-0.518766\pi$
$683$	−1.43664e6	−0.117841	−0.0589206	+	0.998263i	$0.518766\pi$
$684$	0	0
$685$	6.12696e6	0.498906
$686$	0	0
$687$	−4.49774e6	−0.363582
$688$	0	0
$689$	−7.65748e6	−0.614523
$690$	0	0
$691$	−1.15592e7	−0.920939	−0.460469	−	0.887676i	$-0.652319\pi$
$691$	−1.15592e7	−0.920939	−0.460469	+	0.887676i	$0.652319\pi$
$692$	0	0
$693$	2.25193e6	0.178124
$694$	0	0
$695$	−1.40204e7	−1.10103
$696$	0	0
$697$	2.75010e6	0.214420
$698$	0	0
$699$	4.32529e6	0.334829
$700$	0	0
$701$	8.28743e6	0.636979	0.318489	−	0.947926i	$-0.396824\pi$
$701$	8.28743e6	0.636979	0.318489	+	0.947926i	$0.396824\pi$
$702$	0	0
$703$	−2.95637e7	−2.25616
$704$	0	0
$705$	−1.28252e7	−0.971831
$706$	0	0
$707$	−2.66888e6	−0.200808
$708$	0	0
$709$	−3.40586e6	−0.254455	−0.127228	−	0.991874i	$-0.540608\pi$
$709$	−3.40586e6	−0.254455	−0.127228	+	0.991874i	$0.540608\pi$
$710$	0	0
$711$	−1.26610e6	−0.0939274
$712$	0	0
$713$	−2.18068e7	−1.60645
$714$	0	0
$715$	−3.38017e7	−2.47271
$716$	0	0
$717$	−7.82643e6	−0.568546
$718$	0	0
$719$	−1.81891e6	−0.131217	−0.0656083	−	0.997845i	$-0.520899\pi$
$719$	−1.81891e6	−0.131217	−0.0656083	+	0.997845i	$0.520899\pi$
$720$	0	0
$721$	8.76450e6	0.627898
$722$	0	0
$723$	6.04374e6	0.429992
$724$	0	0
$725$	1.65854e7	1.17188
$726$	0	0
$727$	6.14132e6	0.430949	0.215474	−	0.976510i	$-0.430870\pi$
$727$	6.14132e6	0.430949	0.215474	+	0.976510i	$0.430870\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.33489e7	0.923955
$732$	0	0
$733$	−7.62786e6	−0.524376	−0.262188	−	0.965017i	$-0.584444\pi$
$733$	−7.62786e6	−0.524376	−0.262188	+	0.965017i	$0.584444\pi$
$734$	0	0
$735$	−1.54776e6	−0.105678
$736$	0	0
$737$	−3.64206e6	−0.246990
$738$	0	0
$739$	2.76967e7	1.86560	0.932798	−	0.360400i	$-0.117360\pi$
$739$	2.76967e7	1.86560	0.932798	+	0.360400i	$0.117360\pi$
$740$	0	0
$741$	−2.18212e7	−1.45993
$742$	0	0
$743$	−1.10656e7	−0.735366	−0.367683	−	0.929951i	$-0.619849\pi$
$743$	−1.10656e7	−0.735366	−0.367683	+	0.929951i	$0.619849\pi$
$744$	0	0
$745$	−1.97878e7	−1.30619
$746$	0	0
$747$	135127.	0.00886014
$748$	0	0
$749$	323003.	0.0210379
$750$	0	0
$751$	2.77646e6	0.179635	0.0898176	−	0.995958i	$-0.471372\pi$
$751$	2.77646e6	0.179635	0.0898176	+	0.995958i	$0.471372\pi$
$752$	0	0
$753$	3.64781e6	0.234447
$754$	0	0
$755$	2.01733e7	1.28798
$756$	0	0
$757$	−2.67284e7	−1.69525	−0.847624	−	0.530597i	$-0.821968\pi$
$757$	−2.67284e7	−1.69525	−0.847624	+	0.530597i	$0.821968\pi$
$758$	0	0
$759$	1.58420e7	0.998174
$760$	0	0
$761$	2.43818e7	1.52618	0.763089	−	0.646294i	$-0.223682\pi$
$761$	2.43818e7	1.52618	0.763089	+	0.646294i	$0.223682\pi$
$762$	0	0
$763$	−8.71934e6	−0.542216
$764$	0	0
$765$	5.15407e6	0.318417
$766$	0	0
$767$	−8.56814e6	−0.525894
$768$	0	0
$769$	−1.88874e7	−1.15174	−0.575872	−	0.817540i	$-0.695337\pi$
$769$	−1.88874e7	−1.15174	−0.575872	+	0.817540i	$0.695337\pi$
$770$	0	0
$771$	7.53884e6	0.456740
$772$	0	0
$773$	1.11832e7	0.673156	0.336578	−	0.941656i	$-0.390730\pi$
$773$	1.11832e7	0.673156	0.336578	+	0.941656i	$0.390730\pi$
$774$	0	0
$775$	−1.40950e7	−0.842966
$776$	0	0
$777$	−4.47256e6	−0.265769
$778$	0	0
$779$	9.02387e6	0.532782
$780$	0	0
$781$	3.47685e7	2.03966
$782$	0	0
$783$	−6.02958e6	−0.351465
$784$	0	0
$785$	−3.26144e7	−1.88901
$786$	0	0
$787$	−1.23669e7	−0.711742	−0.355871	−	0.934535i	$-0.615816\pi$
$787$	−1.23669e7	−0.711742	−0.355871	+	0.934535i	$0.615816\pi$
$788$	0	0
$789$	−5.66606e6	−0.324033
$790$	0	0
$791$	−4.99118e6	−0.283637
$792$	0	0
$793$	−1.87970e7	−1.06146
$794$	0	0
$795$	5.93475e6	0.333031
$796$	0	0
$797$	−9.24152e6	−0.515345	−0.257672	−	0.966232i	$-0.582956\pi$
$797$	−9.24152e6	−0.515345	−0.257672	+	0.966232i	$0.582956\pi$
$798$	0	0
$799$	1.76745e7	0.979447
$800$	0	0
$801$	6.14250e6	0.338270
$802$	0	0
$803$	1.68552e7	0.922456
$804$	0	0
$805$	−1.08883e7	−0.592202
$806$	0	0
$807$	2.87137e6	0.155205
$808$	0	0
$809$	2.29668e7	1.23376	0.616878	−	0.787059i	$-0.288397\pi$
$809$	2.29668e7	1.23376	0.616878	+	0.787059i	$0.288397\pi$
$810$	0	0
$811$	1.99916e6	0.106732	0.0533662	−	0.998575i	$-0.483005\pi$
$811$	1.99916e6	0.106732	0.0533662	+	0.998575i	$0.483005\pi$
$812$	0	0
$813$	−6.44399e6	−0.341923
$814$	0	0
$815$	−2.68725e7	−1.41714
$816$	0	0
$817$	4.38016e7	2.29580
$818$	0	0
$819$	−3.30123e6	−0.171975
$820$	0	0
$821$	−2.16393e7	−1.12043	−0.560215	−	0.828347i	$-0.689282\pi$
$821$	−2.16393e7	−1.12043	−0.560215	+	0.828347i	$0.689282\pi$
$822$	0	0
$823$	−2.68119e7	−1.37984	−0.689919	−	0.723887i	$-0.742354\pi$
$823$	−2.68119e7	−1.37984	−0.689919	+	0.723887i	$0.742354\pi$
$824$	0	0
$825$	1.02396e7	0.523781
$826$	0	0
$827$	2.76847e7	1.40759	0.703794	−	0.710404i	$-0.251488\pi$
$827$	2.76847e7	1.40759	0.703794	+	0.710404i	$0.251488\pi$
$828$	0	0
$829$	1.51184e7	0.764046	0.382023	−	0.924153i	$-0.375227\pi$
$829$	1.51184e7	0.764046	0.382023	+	0.924153i	$0.375227\pi$
$830$	0	0
$831$	7.82662e6	0.393162
$832$	0	0
$833$	2.13299e6	0.106506
$834$	0	0
$835$	2.00367e7	0.994512
$836$	0	0
$837$	5.12418e6	0.252820
$838$	0	0
$839$	−3.29436e7	−1.61572	−0.807859	−	0.589376i	$-0.799374\pi$
$839$	−3.29436e7	−1.61572	−0.807859	+	0.589376i	$0.799374\pi$
$840$	0	0
$841$	4.78988e7	2.33526
$842$	0	0
$843$	−1.42797e7	−0.692071
$844$	0	0
$845$	2.29577e7	1.10608
$846$	0	0
$847$	−7.88259e6	−0.377538
$848$	0	0
$849$	1.44208e7	0.686626
$850$	0	0
$851$	−3.14638e7	−1.48932
$852$	0	0
$853$	1.37677e7	0.647869	0.323935	−	0.946079i	$-0.394994\pi$
$853$	1.37677e7	0.647869	0.323935	+	0.946079i	$0.394994\pi$
$854$	0	0
$855$	1.69120e7	0.791189
$856$	0	0
$857$	1.52437e7	0.708986	0.354493	−	0.935059i	$-0.384654\pi$
$857$	1.52437e7	0.708986	0.354493	+	0.935059i	$0.384654\pi$
$858$	0	0
$859$	8.37545e6	0.387280	0.193640	−	0.981073i	$-0.437971\pi$
$859$	8.37545e6	0.387280	0.193640	+	0.981073i	$0.437971\pi$
$860$	0	0
$861$	1.36518e6	0.0627599
$862$	0	0
$863$	1.92437e7	0.879550	0.439775	−	0.898108i	$-0.355058\pi$
$863$	1.92437e7	0.879550	0.439775	+	0.898108i	$0.355058\pi$
$864$	0	0
$865$	−1.16080e7	−0.527495
$866$	0	0
$867$	5.67583e6	0.256438
$868$	0	0
$869$	8.86861e6	0.398388
$870$	0	0
$871$	5.33911e6	0.238464
$872$	0	0
$873$	−7.93913e6	−0.352563
$874$	0	0
$875$	3.92996e6	0.173527
$876$	0	0
$877$	−1.04613e7	−0.459290	−0.229645	−	0.973275i	$-0.573756\pi$
$877$	−1.04613e7	−0.459290	−0.229645	+	0.973275i	$0.573756\pi$
$878$	0	0
$879$	−1.06845e7	−0.466426
$880$	0	0
$881$	2.71859e7	1.18006	0.590030	−	0.807381i	$-0.299116\pi$
$881$	2.71859e7	1.18006	0.590030	+	0.807381i	$0.299116\pi$
$882$	0	0
$883$	2.07191e7	0.894272	0.447136	−	0.894466i	$-0.352444\pi$
$883$	2.07191e7	0.894272	0.447136	+	0.894466i	$0.352444\pi$
$884$	0	0
$885$	6.64053e6	0.285000
$886$	0	0
$887$	1.29309e7	0.551850	0.275925	−	0.961179i	$-0.411016\pi$
$887$	1.29309e7	0.551850	0.275925	+	0.961179i	$0.411016\pi$
$888$	0	0
$889$	7.54523e6	0.320197
$890$	0	0
$891$	−3.72258e6	−0.157090
$892$	0	0
$893$	5.79953e7	2.43369
$894$	0	0
$895$	−1.09857e7	−0.458426
$896$	0	0
$897$	−2.32237e7	−0.963720
$898$	0	0
$899$	−5.81375e7	−2.39915
$900$	0	0
$901$	−8.17875e6	−0.335641
$902$	0	0
$903$	6.62654e6	0.270438
$904$	0	0
$905$	−2.31375e7	−0.939065
$906$	0	0
$907$	3.43055e7	1.38467	0.692334	−	0.721578i	$-0.256583\pi$
$907$	3.43055e7	1.38467	0.692334	+	0.721578i	$0.256583\pi$
$908$	0	0
$909$	4.41182e6	0.177096
$910$	0	0
$911$	−2.23945e7	−0.894017	−0.447009	−	0.894530i	$-0.647511\pi$
$911$	−2.23945e7	−0.894017	−0.447009	+	0.894530i	$0.647511\pi$
$912$	0	0
$913$	−946523.	−0.0375798
$914$	0	0
$915$	1.45682e7	0.575244
$916$	0	0
$917$	−1.20114e7	−0.471704
$918$	0	0
$919$	2.86863e7	1.12043	0.560216	−	0.828346i	$-0.310718\pi$
$919$	2.86863e7	1.12043	0.560216	+	0.828346i	$0.310718\pi$
$920$	0	0
$921$	2.02060e7	0.784930
$922$	0	0
$923$	−5.09691e7	−1.96926
$924$	0	0
$925$	−2.03369e7	−0.781503
$926$	0	0
$927$	−1.44883e7	−0.553754
$928$	0	0
$929$	−4.36199e7	−1.65823	−0.829116	−	0.559076i	$-0.811156\pi$
$929$	−4.36199e7	−1.65823	−0.829116	+	0.559076i	$0.811156\pi$
$930$	0	0
$931$	6.99896e6	0.264642
$932$	0	0
$933$	−5.05820e6	−0.190236
$934$	0	0
$935$	−3.61027e7	−1.35055
$936$	0	0
$937$	−2.15646e7	−0.802401	−0.401201	−	0.915990i	$-0.631407\pi$
$937$	−2.15646e7	−0.802401	−0.401201	+	0.915990i	$0.631407\pi$
$938$	0	0
$939$	−2.88782e6	−0.106882
$940$	0	0
$941$	−1.30720e7	−0.481246	−0.240623	−	0.970619i	$-0.577352\pi$
$941$	−1.30720e7	−0.481246	−0.240623	+	0.970619i	$0.577352\pi$
$942$	0	0
$943$	9.60386e6	0.351695
$944$	0	0
$945$	2.55854e6	0.0931994
$946$	0	0
$947$	−5.81492e6	−0.210702	−0.105351	−	0.994435i	$-0.533597\pi$
$947$	−5.81492e6	−0.210702	−0.105351	+	0.994435i	$0.533597\pi$
$948$	0	0
$949$	−2.47090e7	−0.890615
$950$	0	0
$951$	1.96793e7	0.705600
$952$	0	0
$953$	−1.70378e7	−0.607689	−0.303845	−	0.952722i	$-0.598270\pi$
$953$	−1.70378e7	−0.607689	−0.303845	+	0.952722i	$0.598270\pi$
$954$	0	0
$955$	4.69019e7	1.66411
$956$	0	0
$957$	4.22354e7	1.49072
$958$	0	0
$959$	−4.19153e6	−0.147172
$960$	0	0
$961$	2.07784e7	0.725779
$962$	0	0
$963$	−533944.	−0.0185537
$964$	0	0
$965$	5.55858e7	1.92152
$966$	0	0
$967$	−3.39209e7	−1.16654	−0.583272	−	0.812277i	$-0.698228\pi$
$967$	−3.39209e7	−1.16654	−0.583272	+	0.812277i	$0.698228\pi$
$968$	0	0
$969$	−2.33066e7	−0.797389
$970$	0	0
$971$	−3.89258e7	−1.32492	−0.662460	−	0.749097i	$-0.730487\pi$
$971$	−3.89258e7	−1.32492	−0.662460	+	0.749097i	$0.730487\pi$
$972$	0	0
$973$	9.59154e6	0.324793
$974$	0	0
$975$	−1.50108e7	−0.505701
$976$	0	0
$977$	−4.06514e7	−1.36251	−0.681253	−	0.732048i	$-0.738565\pi$
$977$	−4.06514e7	−1.36251	−0.681253	+	0.732048i	$0.738565\pi$
$978$	0	0
$979$	−4.30263e7	−1.43475
$980$	0	0
$981$	1.44136e7	0.478190
$982$	0	0
$983$	1.55210e7	0.512315	0.256158	−	0.966635i	$-0.417543\pi$
$983$	1.55210e7	0.512315	0.256158	+	0.966635i	$0.417543\pi$
$984$	0	0
$985$	−2.85990e7	−0.939204
$986$	0	0
$987$	8.77386e6	0.286680
$988$	0	0
$989$	4.66168e7	1.51549
$990$	0	0
$991$	−1.73682e7	−0.561784	−0.280892	−	0.959739i	$-0.590630\pi$
$991$	−1.73682e7	−0.561784	−0.280892	+	0.959739i	$0.590630\pi$
$992$	0	0
$993$	−1.90259e6	−0.0612312
$994$	0	0
$995$	682219.	0.0218457
$996$	0	0
$997$	2.42265e7	0.771885	0.385943	−	0.922523i	$-0.373876\pi$
$997$	2.42265e7	0.771885	0.385943	+	0.922523i	$0.373876\pi$
$998$	0	0
$999$	7.39342e6	0.234386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.6.a.c.1.2	✓	2
3.2	odd	2		252.6.a.h.1.1		2
4.3	odd	2		336.6.a.x.1.2		2
7.2	even	3		588.6.i.l.361.1		4
7.3	odd	6		588.6.i.i.373.2		4
7.4	even	3		588.6.i.l.373.1		4
7.5	odd	6		588.6.i.i.361.2		4
7.6	odd	2		588.6.a.k.1.1		2
12.11	even	2		1008.6.a.bo.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.a.c.1.2	✓	2	1.1	even	1	trivial
252.6.a.h.1.1		2	3.2	odd	2
336.6.a.x.1.2		2	4.3	odd	2
588.6.a.k.1.1		2	7.6	odd	2
588.6.i.i.361.2		4	7.5	odd	6
588.6.i.i.373.2		4	7.3	odd	6
588.6.i.l.361.1		4	7.2	even	3
588.6.i.l.373.1		4	7.4	even	3
1008.6.a.bo.1.1		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	84.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +71.6257 q^{5} -49.0000 q^{7} +81.0000 q^{9} -567.380 q^{11} +831.754 q^{13} -644.632 q^{15} +888.374 q^{17} +2915.02 q^{19} +441.000 q^{21} +3102.37 q^{23} +2005.25 q^{25} -729.000 q^{27} +8271.03 q^{29} -7029.05 q^{31} +5106.42 q^{33} -3509.66 q^{35} -10141.9 q^{37} -7485.79 q^{39} +3095.65 q^{41} +15026.2 q^{43} +5801.68 q^{45} +19895.4 q^{47} +2401.00 q^{49} -7995.37 q^{51} -9206.42 q^{53} -40639.0 q^{55} -26235.2 q^{57} -10301.3 q^{59} -22599.2 q^{61} -3969.00 q^{63} +59575.0 q^{65} +6419.09 q^{67} -27921.4 q^{69} -61279.0 q^{71} -29707.1 q^{73} -18047.2 q^{75} +27801.6 q^{77} -15630.8 q^{79} +6561.00 q^{81} +1668.23 q^{83} +63630.5 q^{85} -74439.3 q^{87} +75833.3 q^{89} -40756.0 q^{91} +63261.5 q^{93} +208790. q^{95} -98013.9 q^{97} -45957.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{3} - 6 q^{5} - 98 q^{7} + 162 q^{9} - 90 q^{11} + 768 q^{13} + 54 q^{15} + 1926 q^{17} + 2248 q^{19} + 882 q^{21} + 6354 q^{23} + 4906 q^{25} - 1458 q^{27} + 10572 q^{29} - 3312 q^{31} + 810 q^{33}+ \cdots - 7290 q^{99}+O(q^{100}) \) 2 * q - 18 * q^3 - 6 * q^5 - 98 * q^7 + 162 * q^9 - 90 * q^11 + 768 * q^13 + 54 * q^15 + 1926 * q^17 + 2248 * q^19 + 882 * q^21 + 6354 * q^23 + 4906 * q^25 - 1458 * q^27 + 10572 * q^29 - 3312 * q^31 + 810 * q^33 + 294 * q^35 + 2104 * q^37 - 6912 * q^39 + 1266 * q^41 - 5768 * q^43 - 486 * q^45 + 15612 * q^47 + 4802 * q^49 - 17334 * q^51 + 16512 * q^53 - 77696 * q^55 - 20232 * q^57 - 13140 * q^59 - 5796 * q^61 - 7938 * q^63 + 64524 * q^65 - 56116 * q^67 - 57186 * q^69 + 11022 * q^71 - 85384 * q^73 - 44154 * q^75 + 4410 * q^77 - 19620 * q^79 + 13122 * q^81 - 44424 * q^83 - 16916 * q^85 - 95148 * q^87 + 211218 * q^89 - 37632 * q^91 + 29808 * q^93 + 260568 * q^95 + 44864 * q^97 - 7290 * q^99

Introduction

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