LMFDB - Embedded newform 92.6.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [92,6,Mod(1,92)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(92, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("92.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 92 = 2^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	92.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$14.7553114228$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 544x^{2} + 2488x + 27000 $ x^4 - x^3 - 544x^2 + 2488x + 27000
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$11.4167$ of defining polynomial
Character	$\chi$	$=$	92.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+18.4167 q^{3} -57.5774 q^{5} +74.4992 q^{7} +96.1744 q^{9} +479.811 q^{11} +1128.85 q^{13} -1060.39 q^{15} +1508.32 q^{17} -785.781 q^{19} +1372.03 q^{21} -529.000 q^{23} +190.158 q^{25} -2704.04 q^{27} -2247.76 q^{29} +9913.15 q^{31} +8836.53 q^{33} -4289.47 q^{35} +12679.2 q^{37} +20789.7 q^{39} +4947.86 q^{41} -10913.3 q^{43} -5537.47 q^{45} -20307.4 q^{47} -11256.9 q^{49} +27778.3 q^{51} -3219.45 q^{53} -27626.3 q^{55} -14471.5 q^{57} -23969.2 q^{59} -18845.5 q^{61} +7164.91 q^{63} -64996.5 q^{65} +39610.7 q^{67} -9742.43 q^{69} +2132.25 q^{71} -29857.3 q^{73} +3502.08 q^{75} +35745.5 q^{77} +82839.7 q^{79} -73169.9 q^{81} -70505.6 q^{83} -86845.1 q^{85} -41396.2 q^{87} -5277.95 q^{89} +84098.7 q^{91} +182567. q^{93} +45243.2 q^{95} -130.946 q^{97} +46145.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 29 q^{3} + 14 q^{5} + 248 q^{7} + 327 q^{9} - 80 q^{11} + 1331 q^{13} + 2046 q^{15} + 1690 q^{17} + 2752 q^{19} + 3100 q^{21} - 2116 q^{23} + 7192 q^{25} + 4463 q^{27} + 9281 q^{29} + 8105 q^{31}+ \cdots + 290370 q^{99}+O(q^{100}) $ 4 * q + 29 * q^3 + 14 * q^5 + 248 * q^7 + 327 * q^9 - 80 * q^11 + 1331 * q^13 + 2046 * q^15 + 1690 * q^17 + 2752 * q^19 + 3100 * q^21 - 2116 * q^23 + 7192 * q^25 + 4463 * q^27 + 9281 * q^29 + 8105 * q^31 + 16470 * q^33 + 20288 * q^35 + 9834 * q^37 + 8691 * q^39 - 11653 * q^41 + 22358 * q^43 - 552 * q^45 + 2013 * q^47 - 892 * q^49 + 460 * q^51 - 53614 * q^53 - 51604 * q^55 - 44246 * q^57 - 19660 * q^59 - 43446 * q^61 - 50508 * q^63 - 73282 * q^65 - 56544 * q^67 - 15341 * q^69 + 15921 * q^71 - 179843 * q^73 - 1845 * q^75 - 102352 * q^77 + 116422 * q^79 - 66804 * q^81 - 9838 * q^83 - 86752 * q^85 + 115745 * q^87 - 148986 * q^89 + 157444 * q^91 - 12601 * q^93 + 159244 * q^95 + 72030 * q^97 + 290370 * 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
$3$	18.4167	1.18143	0.590716	−	0.806880i	$-0.298846\pi$
$3$	18.4167	1.18143	0.590716	+	0.806880i	$0.298846\pi$
$4$	0	0
$5$	−57.5774	−1.02998	−0.514988	−	0.857197i	$-0.672204\pi$
$5$	−57.5774	−1.02998	−0.514988	+	0.857197i	$0.672204\pi$
$6$	0	0
$7$	74.4992	0.574654	0.287327	−	0.957833i	$-0.407233\pi$
$7$	74.4992	0.574654	0.287327	+	0.957833i	$0.407233\pi$
$8$	0	0
$9$	96.1744	0.395779
$10$	0	0
$11$	479.811	1.19561	0.597803	−	0.801643i	$-0.296040\pi$
$11$	479.811	1.19561	0.597803	+	0.801643i	$0.296040\pi$
$12$	0	0
$13$	1128.85	1.85259	0.926295	−	0.376799i	$-0.122975\pi$
$13$	1128.85	1.85259	0.926295	+	0.376799i	$0.122975\pi$
$14$	0	0
$15$	−1060.39	−1.21685
$16$	0	0
$17$	1508.32	1.26582	0.632909	−	0.774226i	$-0.281861\pi$
$17$	1508.32	1.26582	0.632909	+	0.774226i	$0.281861\pi$
$18$	0	0
$19$	−785.781	−0.499364	−0.249682	−	0.968328i	$-0.580326\pi$
$19$	−785.781	−0.499364	−0.249682	+	0.968328i	$0.580326\pi$
$20$	0	0
$21$	1372.03	0.678914
$22$	0	0
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	190.158	0.0608506
$26$	0	0
$27$	−2704.04	−0.713845
$28$	0	0
$29$	−2247.76	−0.496311	−0.248156	−	0.968720i	$-0.579824\pi$
$29$	−2247.76	−0.496311	−0.248156	+	0.968720i	$0.579824\pi$
$30$	0	0
$31$	9913.15	1.85271	0.926355	−	0.376651i	$-0.122924\pi$
$31$	9913.15	1.85271	0.926355	+	0.376651i	$0.122924\pi$
$32$	0	0
$33$	8836.53	1.41253
$34$	0	0
$35$	−4289.47	−0.591880
$36$	0	0
$37$	12679.2	1.52260	0.761302	−	0.648397i	$-0.224560\pi$
$37$	12679.2	1.52260	0.761302	+	0.648397i	$0.224560\pi$
$38$	0	0
$39$	20789.7	2.18871
$40$	0	0
$41$	4947.86	0.459682	0.229841	−	0.973228i	$-0.426179\pi$
$41$	4947.86	0.459682	0.229841	+	0.973228i	$0.426179\pi$
$42$	0	0
$43$	−10913.3	−0.900085	−0.450043	−	0.893007i	$-0.648591\pi$
$43$	−10913.3	−0.900085	−0.450043	+	0.893007i	$0.648591\pi$
$44$	0	0
$45$	−5537.47	−0.407643
$46$	0	0
$47$	−20307.4	−1.34094	−0.670469	−	0.741938i	$-0.733907\pi$
$47$	−20307.4	−1.34094	−0.670469	+	0.741938i	$0.733907\pi$
$48$	0	0
$49$	−11256.9	−0.669773
$50$	0	0
$51$	27778.3	1.49548
$52$	0	0
$53$	−3219.45	−0.157432	−0.0787158	−	0.996897i	$-0.525082\pi$
$53$	−3219.45	−0.157432	−0.0787158	+	0.996897i	$0.525082\pi$
$54$	0	0
$55$	−27626.3	−1.23145
$56$	0	0
$57$	−14471.5	−0.589964
$58$	0	0
$59$	−23969.2	−0.896443	−0.448222	−	0.893923i	$-0.647942\pi$
$59$	−23969.2	−0.896443	−0.448222	+	0.893923i	$0.647942\pi$
$60$	0	0
$61$	−18845.5	−0.648459	−0.324229	−	0.945979i	$-0.605105\pi$
$61$	−18845.5	−0.648459	−0.324229	+	0.945979i	$0.605105\pi$
$62$	0	0
$63$	7164.91	0.227436
$64$	0	0
$65$	−64996.5	−1.90812
$66$	0	0
$67$	39610.7	1.07802	0.539008	−	0.842300i	$-0.318799\pi$
$67$	39610.7	1.07802	0.539008	+	0.842300i	$0.318799\pi$
$68$	0	0
$69$	−9742.43	−0.246345
$70$	0	0
$71$	2132.25	0.0501987	0.0250993	−	0.999685i	$-0.492010\pi$
$71$	2132.25	0.0501987	0.0250993	+	0.999685i	$0.492010\pi$
$72$	0	0
$73$	−29857.3	−0.655758	−0.327879	−	0.944720i	$-0.606334\pi$
$73$	−29857.3	−0.655758	−0.327879	+	0.944720i	$0.606334\pi$
$74$	0	0
$75$	3502.08	0.0718907
$76$	0	0
$77$	35745.5	0.687060
$78$	0	0
$79$	82839.7	1.49338	0.746691	−	0.665171i	$-0.231641\pi$
$79$	82839.7	1.49338	0.746691	+	0.665171i	$0.231641\pi$
$80$	0	0
$81$	−73169.9	−1.23914
$82$	0	0
$83$	−70505.6	−1.12338	−0.561692	−	0.827346i	$-0.689850\pi$
$83$	−70505.6	−1.12338	−0.561692	+	0.827346i	$0.689850\pi$
$84$	0	0
$85$	−86845.1	−1.30376
$86$	0	0
$87$	−41396.2	−0.586358
$88$	0	0
$89$	−5277.95	−0.0706301	−0.0353151	−	0.999376i	$-0.511243\pi$
$89$	−5277.95	−0.0706301	−0.0353151	+	0.999376i	$0.511243\pi$
$90$	0	0
$91$	84098.7	1.06460
$92$	0	0
$93$	182567.	2.18885
$94$	0	0
$95$	45243.2	0.514333
$96$	0	0
$97$	−130.946	−0.00141306	−0.000706531	−	1.00000i	$-0.500225\pi$
$97$	−130.946	−0.00141306	−0.000706531		1.00000i	$0.500225\pi$
$98$	0	0
$99$	46145.5	0.473197
$100$	0	0
$101$	6833.94	0.0666604	0.0333302	−	0.999444i	$-0.489389\pi$
$101$	6833.94	0.0666604	0.0333302	+	0.999444i	$0.489389\pi$
$102$	0	0
$103$	−50741.8	−0.471274	−0.235637	−	0.971841i	$-0.575718\pi$
$103$	−50741.8	−0.471274	−0.235637	+	0.971841i	$0.575718\pi$
$104$	0	0
$105$	−78997.8	−0.699265
$106$	0	0
$107$	77381.9	0.653401	0.326701	−	0.945128i	$-0.394063\pi$
$107$	77381.9	0.653401	0.326701	+	0.945128i	$0.394063\pi$
$108$	0	0
$109$	−212663.	−1.71445	−0.857225	−	0.514942i	$-0.827813\pi$
$109$	−212663.	−1.71445	−0.857225	+	0.514942i	$0.827813\pi$
$110$	0	0
$111$	233509.	1.79885
$112$	0	0
$113$	−133453.	−0.983177	−0.491589	−	0.870827i	$-0.663584\pi$
$113$	−133453.	−0.983177	−0.491589	+	0.870827i	$0.663584\pi$
$114$	0	0
$115$	30458.4	0.214765
$116$	0	0
$117$	108567.	0.733217
$118$	0	0
$119$	112369.	0.727407
$120$	0	0
$121$	69167.5	0.429476
$122$	0	0
$123$	91123.2	0.543083
$124$	0	0
$125$	168981.	0.967301
$126$	0	0
$127$	10618.2	0.0584171	0.0292086	−	0.999573i	$-0.490701\pi$
$127$	10618.2	0.0584171	0.0292086	+	0.999573i	$0.490701\pi$
$128$	0	0
$129$	−200986.	−1.06339
$130$	0	0
$131$	−240217.	−1.22300	−0.611498	−	0.791246i	$-0.709433\pi$
$131$	−240217.	−1.22300	−0.611498	+	0.791246i	$0.709433\pi$
$132$	0	0
$133$	−58540.0	−0.286962
$134$	0	0
$135$	155692.	0.735243
$136$	0	0
$137$	14162.8	0.0644684	0.0322342	−	0.999480i	$-0.489738\pi$
$137$	14162.8	0.0644684	0.0322342	+	0.999480i	$0.489738\pi$
$138$	0	0
$139$	214580.	0.942005	0.471002	−	0.882132i	$-0.343892\pi$
$139$	214580.	0.942005	0.471002	+	0.882132i	$0.343892\pi$
$140$	0	0
$141$	−373994.	−1.58423
$142$	0	0
$143$	541636.	2.21497
$144$	0	0
$145$	129420.	0.511189
$146$	0	0
$147$	−207314.	−0.791290
$148$	0	0
$149$	439806.	1.62291	0.811457	−	0.584412i	$-0.198675\pi$
$149$	439806.	1.62291	0.811457	+	0.584412i	$0.198675\pi$
$150$	0	0
$151$	381239.	1.36068	0.680338	−	0.732898i	$-0.261833\pi$
$151$	381239.	1.36068	0.680338	+	0.732898i	$0.261833\pi$
$152$	0	0
$153$	145062.	0.500985
$154$	0	0
$155$	−570774.	−1.90825
$156$	0	0
$157$	208496.	0.675071	0.337535	−	0.941313i	$-0.390407\pi$
$157$	208496.	0.675071	0.337535	+	0.941313i	$0.390407\pi$
$158$	0	0
$159$	−59291.6	−0.185995
$160$	0	0
$161$	−39410.1	−0.119824
$162$	0	0
$163$	−424768.	−1.25223	−0.626113	−	0.779732i	$-0.715355\pi$
$163$	−424768.	−1.25223	−0.626113	+	0.779732i	$0.715355\pi$
$164$	0	0
$165$	−508784.	−1.45487
$166$	0	0
$167$	−570020.	−1.58161	−0.790805	−	0.612069i	$-0.790338\pi$
$167$	−570020.	−1.58161	−0.790805	+	0.612069i	$0.790338\pi$
$168$	0	0
$169$	903018.	2.43209
$170$	0	0
$171$	−75572.0	−0.197638
$172$	0	0
$173$	−694964.	−1.76542	−0.882708	−	0.469922i	$-0.844282\pi$
$173$	−694964.	−1.76542	−0.882708	+	0.469922i	$0.844282\pi$
$174$	0	0
$175$	14166.6	0.0349680
$176$	0	0
$177$	−441432.	−1.05909
$178$	0	0
$179$	−215546.	−0.502815	−0.251407	−	0.967881i	$-0.580893\pi$
$179$	−215546.	−0.502815	−0.251407	+	0.967881i	$0.580893\pi$
$180$	0	0
$181$	278115.	0.630998	0.315499	−	0.948926i	$-0.397828\pi$
$181$	278115.	0.630998	0.315499	+	0.948926i	$0.397828\pi$
$182$	0	0
$183$	−347071.	−0.766109
$184$	0	0
$185$	−730035.	−1.56825
$186$	0	0
$187$	723708.	1.51342
$188$	0	0
$189$	−201449.	−0.410214
$190$	0	0
$191$	522817.	1.03697	0.518485	−	0.855087i	$-0.326496\pi$
$191$	522817.	1.03697	0.518485	+	0.855087i	$0.326496\pi$
$192$	0	0
$193$	−534054.	−1.03203	−0.516014	−	0.856580i	$-0.672585\pi$
$193$	−534054.	−1.03203	−0.516014	+	0.856580i	$0.672585\pi$
$194$	0	0
$195$	−1.19702e6	−2.25432
$196$	0	0
$197$	−204755.	−0.375897	−0.187949	−	0.982179i	$-0.560184\pi$
$197$	−204755.	−0.375897	−0.187949	+	0.982179i	$0.560184\pi$
$198$	0	0
$199$	−260531.	−0.466366	−0.233183	−	0.972433i	$-0.574914\pi$
$199$	−260531.	−0.466366	−0.233183	+	0.972433i	$0.574914\pi$
$200$	0	0
$201$	729498.	1.27360
$202$	0	0
$203$	−167456.	−0.285207
$204$	0	0
$205$	−284885.	−0.473462
$206$	0	0
$207$	−50876.3	−0.0825257
$208$	0	0
$209$	−377026.	−0.597043
$210$	0	0
$211$	173321.	0.268006	0.134003	−	0.990981i	$-0.457217\pi$
$211$	173321.	0.268006	0.134003	+	0.990981i	$0.457217\pi$
$212$	0	0
$213$	39269.0	0.0593063
$214$	0	0
$215$	628358.	0.927066
$216$	0	0
$217$	738522.	1.06467
$218$	0	0
$219$	−549873.	−0.774733
$220$	0	0
$221$	1.70267e6	2.34504
$222$	0	0
$223$	489602.	0.659297	0.329649	−	0.944104i	$-0.393070\pi$
$223$	489602.	0.659297	0.329649	+	0.944104i	$0.393070\pi$
$224$	0	0
$225$	18288.3	0.0240834
$226$	0	0
$227$	913101.	1.17613	0.588064	−	0.808815i	$-0.299890\pi$
$227$	913101.	1.17613	0.588064	+	0.808815i	$0.299890\pi$
$228$	0	0
$229$	775624.	0.977378	0.488689	−	0.872458i	$-0.337475\pi$
$229$	775624.	0.977378	0.488689	+	0.872458i	$0.337475\pi$
$230$	0	0
$231$	658314.	0.811715
$232$	0	0
$233$	−1.21804e6	−1.46984	−0.734921	−	0.678152i	$-0.762781\pi$
$233$	−1.21804e6	−1.46984	−0.734921	+	0.678152i	$0.762781\pi$
$234$	0	0
$235$	1.16924e6	1.38113
$236$	0	0
$237$	1.52563e6	1.76433
$238$	0	0
$239$	−1.15864e6	−1.31207	−0.656033	−	0.754732i	$-0.727767\pi$
$239$	−1.15864e6	−1.31207	−0.656033	+	0.754732i	$0.727767\pi$
$240$	0	0
$241$	−318710.	−0.353471	−0.176735	−	0.984258i	$-0.556554\pi$
$241$	−318710.	−0.353471	−0.176735	+	0.984258i	$0.556554\pi$
$242$	0	0
$243$	−690464.	−0.750111
$244$	0	0
$245$	648141.	0.689850
$246$	0	0
$247$	−887031.	−0.925117
$248$	0	0
$249$	−1.29848e6	−1.32720
$250$	0	0
$251$	233096.	0.233535	0.116767	−	0.993159i	$-0.462747\pi$
$251$	233096.	0.233535	0.116767	+	0.993159i	$0.462747\pi$
$252$	0	0
$253$	−253820.	−0.249301
$254$	0	0
$255$	−1.59940e6	−1.54030
$256$	0	0
$257$	1.54900e6	1.46292	0.731458	−	0.681886i	$-0.238840\pi$
$257$	1.54900e6	1.46292	0.731458	+	0.681886i	$0.238840\pi$
$258$	0	0
$259$	944589.	0.874971
$260$	0	0
$261$	−216177.	−0.196430
$262$	0	0
$263$	−1.04023e6	−0.927340	−0.463670	−	0.886008i	$-0.653468\pi$
$263$	−1.04023e6	−0.927340	−0.463670	+	0.886008i	$0.653468\pi$
$264$	0	0
$265$	185368.	0.162151
$266$	0	0
$267$	−97202.3	−0.0834446
$268$	0	0
$269$	−1.29876e6	−1.09433	−0.547166	−	0.837024i	$-0.684293\pi$
$269$	−1.29876e6	−1.09433	−0.547166	+	0.837024i	$0.684293\pi$
$270$	0	0
$271$	1.46692e6	1.21334	0.606672	−	0.794952i	$-0.292504\pi$
$271$	1.46692e6	1.21334	0.606672	+	0.794952i	$0.292504\pi$
$272$	0	0
$273$	1.54882e6	1.25775
$274$	0	0
$275$	91239.9	0.0727534
$276$	0	0
$277$	1.61836e6	1.26729	0.633646	−	0.773623i	$-0.281557\pi$
$277$	1.61836e6	1.26729	0.633646	+	0.773623i	$0.281557\pi$
$278$	0	0
$279$	953391.	0.733265
$280$	0	0
$281$	−1.91098e6	−1.44374	−0.721871	−	0.692027i	$-0.756718\pi$
$281$	−1.91098e6	−1.44374	−0.721871	+	0.692027i	$0.756718\pi$
$282$	0	0
$283$	762121.	0.565663	0.282831	−	0.959170i	$-0.408726\pi$
$283$	762121.	0.565663	0.282831	+	0.959170i	$0.408726\pi$
$284$	0	0
$285$	833230.	0.607649
$286$	0	0
$287$	368612.	0.264158
$288$	0	0
$289$	855172.	0.602294
$290$	0	0
$291$	−2411.58	−0.00166944
$292$	0	0
$293$	−1.83275e6	−1.24719	−0.623597	−	0.781746i	$-0.714329\pi$
$293$	−1.83275e6	−1.24719	−0.623597	+	0.781746i	$0.714329\pi$
$294$	0	0
$295$	1.38008e6	0.923315
$296$	0	0
$297$	−1.29743e6	−0.853478
$298$	0	0
$299$	−597164.	−0.386292
$300$	0	0
$301$	−813030.	−0.517238
$302$	0	0
$303$	125859.	0.0787546
$304$	0	0
$305$	1.08507e6	0.667897
$306$	0	0
$307$	−399979.	−0.242209	−0.121105	−	0.992640i	$-0.538644\pi$
$307$	−399979.	−0.242209	−0.121105	+	0.992640i	$0.538644\pi$
$308$	0	0
$309$	−934497.	−0.556777
$310$	0	0
$311$	−863087.	−0.506004	−0.253002	−	0.967466i	$-0.581418\pi$
$311$	−863087.	−0.506004	−0.253002	+	0.967466i	$0.581418\pi$
$312$	0	0
$313$	2.42238e6	1.39760	0.698799	−	0.715318i	$-0.253718\pi$
$313$	2.42238e6	1.39760	0.698799	+	0.715318i	$0.253718\pi$
$314$	0	0
$315$	−412537.	−0.234254
$316$	0	0
$317$	1.95151e6	1.09074	0.545371	−	0.838195i	$-0.316389\pi$
$317$	1.95151e6	1.09074	0.545371	+	0.838195i	$0.316389\pi$
$318$	0	0
$319$	−1.07850e6	−0.593393
$320$	0	0
$321$	1.42512e6	0.771948
$322$	0	0
$323$	−1.18521e6	−0.632104
$324$	0	0
$325$	214661.	0.112731
$326$	0	0
$327$	−3.91654e6	−2.02550
$328$	0	0
$329$	−1.51288e6	−0.770575
$330$	0	0
$331$	1.06373e6	0.533655	0.266827	−	0.963744i	$-0.414025\pi$
$331$	1.06373e6	0.533655	0.266827	+	0.963744i	$0.414025\pi$
$332$	0	0
$333$	1.21941e6	0.602615
$334$	0	0
$335$	−2.28068e6	−1.11033
$336$	0	0
$337$	−2.17731e6	−1.04435	−0.522175	−	0.852839i	$-0.674879\pi$
$337$	−2.17731e6	−1.04435	−0.522175	+	0.852839i	$0.674879\pi$
$338$	0	0
$339$	−2.45776e6	−1.16156
$340$	0	0
$341$	4.75644e6	2.21511
$342$	0	0
$343$	−2.09074e6	−0.959542
$344$	0	0
$345$	560944.	0.253730
$346$	0	0
$347$	−241090.	−0.107487	−0.0537434	−	0.998555i	$-0.517115\pi$
$347$	−241090.	−0.107487	−0.0537434	+	0.998555i	$0.517115\pi$
$348$	0	0
$349$	−4.02835e6	−1.77037	−0.885184	−	0.465241i	$-0.845968\pi$
$349$	−4.02835e6	−1.77037	−0.885184	+	0.465241i	$0.845968\pi$
$350$	0	0
$351$	−3.05247e6	−1.32246
$352$	0	0
$353$	−386302.	−0.165002	−0.0825012	−	0.996591i	$-0.526291\pi$
$353$	−386302.	−0.165002	−0.0825012	+	0.996591i	$0.526291\pi$
$354$	0	0
$355$	−122769.	−0.0517034
$356$	0	0
$357$	2.06946e6	0.859382
$358$	0	0
$359$	−1.72014e6	−0.704414	−0.352207	−	0.935922i	$-0.614569\pi$
$359$	−1.72014e6	−0.704414	−0.352207	+	0.935922i	$0.614569\pi$
$360$	0	0
$361$	−1.85865e6	−0.750635
$362$	0	0
$363$	1.27384e6	0.507396
$364$	0	0
$365$	1.71911e6	0.675415
$366$	0	0
$367$	−1.62104e6	−0.628243	−0.314122	−	0.949383i	$-0.601710\pi$
$367$	−1.62104e6	−0.628243	−0.314122	+	0.949383i	$0.601710\pi$
$368$	0	0
$369$	475858.	0.181933
$370$	0	0
$371$	−239846.	−0.0904687
$372$	0	0
$373$	2.38681e6	0.888271	0.444135	−	0.895960i	$-0.353511\pi$
$373$	2.38681e6	0.888271	0.444135	+	0.895960i	$0.353511\pi$
$374$	0	0
$375$	3.11206e6	1.14280
$376$	0	0
$377$	−2.53739e6	−0.919462
$378$	0	0
$379$	1.76498e6	0.631165	0.315582	−	0.948898i	$-0.397800\pi$
$379$	1.76498e6	0.631165	0.315582	+	0.948898i	$0.397800\pi$
$380$	0	0
$381$	195551.	0.0690158
$382$	0	0
$383$	379150.	0.132073	0.0660365	−	0.997817i	$-0.478965\pi$
$383$	379150.	0.132073	0.0660365	+	0.997817i	$0.478965\pi$
$384$	0	0
$385$	−2.05813e6	−0.707656
$386$	0	0
$387$	−1.04958e6	−0.356235
$388$	0	0
$389$	5.37869e6	1.80220	0.901098	−	0.433615i	$-0.142762\pi$
$389$	5.37869e6	1.80220	0.901098	+	0.433615i	$0.142762\pi$
$390$	0	0
$391$	−797901.	−0.263941
$392$	0	0
$393$	−4.42399e6	−1.44488
$394$	0	0
$395$	−4.76970e6	−1.53815
$396$	0	0
$397$	−5.39076e6	−1.71662	−0.858309	−	0.513133i	$-0.828485\pi$
$397$	−5.39076e6	−1.71662	−0.858309	+	0.513133i	$0.828485\pi$
$398$	0	0
$399$	−1.07811e6	−0.339025
$400$	0	0
$401$	−6.41022e6	−1.99073	−0.995364	−	0.0961832i	$-0.969337\pi$
$401$	−6.41022e6	−1.99073	−0.995364	+	0.0961832i	$0.969337\pi$
$402$	0	0
$403$	1.11905e7	3.43231
$404$	0	0
$405$	4.21293e6	1.27628
$406$	0	0
$407$	6.08361e6	1.82044
$408$	0	0
$409$	−13428.8	−0.00396942	−0.00198471	−	0.999998i	$-0.500632\pi$
$409$	−13428.8	−0.00396942	−0.00198471	+	0.999998i	$0.500632\pi$
$410$	0	0
$411$	260831.	0.0761650
$412$	0	0
$413$	−1.78568e6	−0.515145
$414$	0	0
$415$	4.05953e6	1.15706
$416$	0	0
$417$	3.95186e6	1.11291
$418$	0	0
$419$	687589.	0.191335	0.0956674	−	0.995413i	$-0.469501\pi$
$419$	687589.	0.191335	0.0956674	+	0.995413i	$0.469501\pi$
$420$	0	0
$421$	29609.6	0.00814194	0.00407097	−	0.999992i	$-0.498704\pi$
$421$	29609.6	0.00814194	0.00407097	+	0.999992i	$0.498704\pi$
$422$	0	0
$423$	−1.95305e6	−0.530716
$424$	0	0
$425$	286819.	0.0770257
$426$	0	0
$427$	−1.40397e6	−0.372639
$428$	0	0
$429$	9.97515e6	2.61683
$430$	0	0
$431$	3.54961e6	0.920423	0.460212	−	0.887809i	$-0.347773\pi$
$431$	3.54961e6	0.920423	0.460212	+	0.887809i	$0.347773\pi$
$432$	0	0
$433$	−5.01538e6	−1.28554	−0.642768	−	0.766061i	$-0.722214\pi$
$433$	−5.01538e6	−1.28554	−0.642768	+	0.766061i	$0.722214\pi$
$434$	0	0
$435$	2.38349e6	0.603934
$436$	0	0
$437$	415678.	0.104125
$438$	0	0
$439$	2.56785e6	0.635928	0.317964	−	0.948103i	$-0.397001\pi$
$439$	2.56785e6	0.635928	0.317964	+	0.948103i	$0.397001\pi$
$440$	0	0
$441$	−1.08262e6	−0.265082
$442$	0	0
$443$	7.42432e6	1.79741	0.898706	−	0.438552i	$-0.144508\pi$
$443$	7.42432e6	1.79741	0.898706	+	0.438552i	$0.144508\pi$
$444$	0	0
$445$	303890.	0.0727473
$446$	0	0
$447$	8.09977e6	1.91736
$448$	0	0
$449$	−6.24917e6	−1.46287	−0.731436	−	0.681910i	$-0.761150\pi$
$449$	−6.24917e6	−1.46287	−0.731436	+	0.681910i	$0.761150\pi$
$450$	0	0
$451$	2.37404e6	0.549600
$452$	0	0
$453$	7.02116e6	1.60755
$454$	0	0
$455$	−4.84218e6	−1.09651
$456$	0	0
$457$	−4.67168e6	−1.04636	−0.523182	−	0.852221i	$-0.675255\pi$
$457$	−4.67168e6	−1.04636	−0.523182	+	0.852221i	$0.675255\pi$
$458$	0	0
$459$	−4.07856e6	−0.903598
$460$	0	0
$461$	−2.59168e6	−0.567975	−0.283987	−	0.958828i	$-0.591657\pi$
$461$	−2.59168e6	−0.567975	−0.283987	+	0.958828i	$0.591657\pi$
$462$	0	0
$463$	4.52109e6	0.980147	0.490073	−	0.871681i	$-0.336970\pi$
$463$	4.52109e6	0.980147	0.490073	+	0.871681i	$0.336970\pi$
$464$	0	0
$465$	−1.05118e7	−2.25446
$466$	0	0
$467$	−2.96158e6	−0.628392	−0.314196	−	0.949358i	$-0.601735\pi$
$467$	−2.96158e6	−0.628392	−0.314196	+	0.949358i	$0.601735\pi$
$468$	0	0
$469$	2.95096e6	0.619487
$470$	0	0
$471$	3.83981e6	0.797550
$472$	0	0
$473$	−5.23631e6	−1.07615
$474$	0	0
$475$	−149422.	−0.0303866
$476$	0	0
$477$	−309629.	−0.0623082
$478$	0	0
$479$	1.36968e6	0.272760	0.136380	−	0.990657i	$-0.456453\pi$
$479$	1.36968e6	0.272760	0.136380	+	0.990657i	$0.456453\pi$
$480$	0	0
$481$	1.43129e7	2.82076
$482$	0	0
$483$	−725803.	−0.141563
$484$	0	0
$485$	7539.50	0.00145542
$486$	0	0
$487$	−9.09542e6	−1.73780	−0.868901	−	0.494985i	$-0.835173\pi$
$487$	−9.09542e6	−1.73780	−0.868901	+	0.494985i	$0.835173\pi$
$488$	0	0
$489$	−7.82282e6	−1.47942
$490$	0	0
$491$	1.14589e6	0.214506	0.107253	−	0.994232i	$-0.465795\pi$
$491$	1.14589e6	0.214506	0.107253	+	0.994232i	$0.465795\pi$
$492$	0	0
$493$	−3.39034e6	−0.628240
$494$	0	0
$495$	−2.65694e6	−0.487381
$496$	0	0
$497$	158851.	0.0288469
$498$	0	0
$499$	−985286.	−0.177138	−0.0885688	−	0.996070i	$-0.528229\pi$
$499$	−985286.	−0.177138	−0.0885688	+	0.996070i	$0.528229\pi$
$500$	0	0
$501$	−1.04979e7	−1.86856
$502$	0	0
$503$	2.25682e6	0.397720	0.198860	−	0.980028i	$-0.436276\pi$
$503$	2.25682e6	0.397720	0.198860	+	0.980028i	$0.436276\pi$
$504$	0	0
$505$	−393480.	−0.0686586
$506$	0	0
$507$	1.66306e7	2.87335
$508$	0	0
$509$	−5.89974e6	−1.00934	−0.504671	−	0.863312i	$-0.668386\pi$
$509$	−5.89974e6	−1.00934	−0.504671	+	0.863312i	$0.668386\pi$
$510$	0	0
$511$	−2.22435e6	−0.376834
$512$	0	0
$513$	2.12478e6	0.356469
$514$	0	0
$515$	2.92158e6	0.485401
$516$	0	0
$517$	−9.74369e6	−1.60323
$518$	0	0
$519$	−1.27989e7	−2.08572
$520$	0	0
$521$	4.76461e6	0.769012	0.384506	−	0.923123i	$-0.374372\pi$
$521$	4.76461e6	0.769012	0.384506	+	0.923123i	$0.374372\pi$
$522$	0	0
$523$	−6.87556e6	−1.09914	−0.549571	−	0.835447i	$-0.685209\pi$
$523$	−6.87556e6	−1.09914	−0.549571	+	0.835447i	$0.685209\pi$
$524$	0	0
$525$	260902.	0.0413123
$526$	0	0
$527$	1.49522e7	2.34519
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	−2.30522e6	−0.354794
$532$	0	0
$533$	5.58541e6	0.851603
$534$	0	0
$535$	−4.45545e6	−0.672987
$536$	0	0
$537$	−3.96965e6	−0.594041
$538$	0	0
$539$	−5.40117e6	−0.800785
$540$	0	0
$541$	3.42122e6	0.502560	0.251280	−	0.967914i	$-0.419149\pi$
$541$	3.42122e6	0.502560	0.251280	+	0.967914i	$0.419149\pi$
$542$	0	0
$543$	5.12196e6	0.745481
$544$	0	0
$545$	1.22446e7	1.76584
$546$	0	0
$547$	−804933.	−0.115025	−0.0575124	−	0.998345i	$-0.518317\pi$
$547$	−804933.	−0.115025	−0.0575124	+	0.998345i	$0.518317\pi$
$548$	0	0
$549$	−1.81245e6	−0.256647
$550$	0	0
$551$	1.76624e6	0.247840
$552$	0	0
$553$	6.17149e6	0.858178
$554$	0	0
$555$	−1.34448e7	−1.85277
$556$	0	0
$557$	−5.46048e6	−0.745749	−0.372875	−	0.927882i	$-0.621628\pi$
$557$	−5.46048e6	−0.745749	−0.372875	+	0.927882i	$0.621628\pi$
$558$	0	0
$559$	−1.23195e7	−1.66749
$560$	0	0
$561$	1.33283e7	1.78800
$562$	0	0
$563$	1.34002e7	1.78172	0.890860	−	0.454278i	$-0.150103\pi$
$563$	1.34002e7	1.78172	0.890860	+	0.454278i	$0.150103\pi$
$564$	0	0
$565$	7.68387e6	1.01265
$566$	0	0
$567$	−5.45110e6	−0.712076
$568$	0	0
$569$	−1.34909e7	−1.74687	−0.873437	−	0.486937i	$-0.838114\pi$
$569$	−1.34909e7	−1.74687	−0.873437	+	0.486937i	$0.838114\pi$
$570$	0	0
$571$	−3.26977e6	−0.419688	−0.209844	−	0.977735i	$-0.567296\pi$
$571$	−3.26977e6	−0.419688	−0.209844	+	0.977735i	$0.567296\pi$
$572$	0	0
$573$	9.62856e6	1.22511
$574$	0	0
$575$	−100594.	−0.0126882
$576$	0	0
$577$	−1.45614e7	−1.82081	−0.910404	−	0.413720i	$-0.864229\pi$
$577$	−1.45614e7	−1.82081	−0.910404	+	0.413720i	$0.864229\pi$
$578$	0	0
$579$	−9.83550e6	−1.21927
$580$	0	0
$581$	−5.25261e6	−0.645558
$582$	0	0
$583$	−1.54473e6	−0.188226
$584$	0	0
$585$	−6.25100e6	−0.755196
$586$	0	0
$587$	6.11740e6	0.732776	0.366388	−	0.930462i	$-0.380594\pi$
$587$	6.11740e6	0.732776	0.366388	+	0.930462i	$0.380594\pi$
$588$	0	0
$589$	−7.78956e6	−0.925177
$590$	0	0
$591$	−3.77091e6	−0.444097
$592$	0	0
$593$	1.58731e7	1.85364	0.926819	−	0.375509i	$-0.122532\pi$
$593$	1.58731e7	1.85364	0.926819	+	0.375509i	$0.122532\pi$
$594$	0	0
$595$	−6.46989e6	−0.749212
$596$	0	0
$597$	−4.79812e6	−0.550979
$598$	0	0
$599$	2.08386e6	0.237302	0.118651	−	0.992936i	$-0.462143\pi$
$599$	2.08386e6	0.237302	0.118651	+	0.992936i	$0.462143\pi$
$600$	0	0
$601$	−6.87476e6	−0.776375	−0.388187	−	0.921580i	$-0.626899\pi$
$601$	−6.87476e6	−0.776375	−0.388187	+	0.921580i	$0.626899\pi$
$602$	0	0
$603$	3.80953e6	0.426657
$604$	0	0
$605$	−3.98249e6	−0.442350
$606$	0	0
$607$	1.69724e7	1.86970	0.934849	−	0.355047i	$-0.115535\pi$
$607$	1.69724e7	1.86970	0.934849	+	0.355047i	$0.115535\pi$
$608$	0	0
$609$	−3.08399e6	−0.336953
$610$	0	0
$611$	−2.29240e7	−2.48421
$612$	0	0
$613$	1.82930e7	1.96622	0.983112	−	0.183006i	$-0.0585828\pi$
$613$	1.82930e7	1.96622	0.983112	+	0.183006i	$0.0585828\pi$
$614$	0	0
$615$	−5.24664e6	−0.559363
$616$	0	0
$617$	2.49607e6	0.263963	0.131982	−	0.991252i	$-0.457866\pi$
$617$	2.49607e6	0.263963	0.131982	+	0.991252i	$0.457866\pi$
$618$	0	0
$619$	1.07584e7	1.12855	0.564273	−	0.825588i	$-0.309156\pi$
$619$	1.07584e7	1.12855	0.564273	+	0.825588i	$0.309156\pi$
$620$	0	0
$621$	1.43044e6	0.148847
$622$	0	0
$623$	−393203.	−0.0405879
$624$	0	0
$625$	−1.03237e7	−1.05715
$626$	0	0
$627$	−6.94357e6	−0.705365
$628$	0	0
$629$	1.91243e7	1.92734
$630$	0	0
$631$	−4.23020e6	−0.422948	−0.211474	−	0.977384i	$-0.567826\pi$
$631$	−4.23020e6	−0.422948	−0.211474	+	0.977384i	$0.567826\pi$
$632$	0	0
$633$	3.19200e6	0.316631
$634$	0	0
$635$	−611366.	−0.0601682
$636$	0	0
$637$	−1.27074e7	−1.24081
$638$	0	0
$639$	205068.	0.0198676
$640$	0	0
$641$	−446244.	−0.0428971	−0.0214485	−	0.999770i	$-0.506828\pi$
$641$	−446244.	−0.0428971	−0.0214485	+	0.999770i	$0.506828\pi$
$642$	0	0
$643$	−6.65817e6	−0.635079	−0.317539	−	0.948245i	$-0.602857\pi$
$643$	−6.65817e6	−0.635079	−0.317539	+	0.948245i	$0.602857\pi$
$644$	0	0
$645$	1.15723e7	1.09526
$646$	0	0
$647$	1.13575e7	1.06665	0.533323	−	0.845912i	$-0.320943\pi$
$647$	1.13575e7	1.06665	0.533323	+	0.845912i	$0.320943\pi$
$648$	0	0
$649$	−1.15007e7	−1.07179
$650$	0	0
$651$	1.36011e7	1.25783
$652$	0	0
$653$	5.69583e6	0.522726	0.261363	−	0.965241i	$-0.415828\pi$
$653$	5.69583e6	0.522726	0.261363	+	0.965241i	$0.415828\pi$
$654$	0	0
$655$	1.38311e7	1.25966
$656$	0	0
$657$	−2.87151e6	−0.259536
$658$	0	0
$659$	7.16511e6	0.642702	0.321351	−	0.946960i	$-0.395863\pi$
$659$	7.16511e6	0.642702	0.321351	+	0.946960i	$0.395863\pi$
$660$	0	0
$661$	3.66045e6	0.325859	0.162930	−	0.986638i	$-0.447906\pi$
$661$	3.66045e6	0.325859	0.162930	+	0.986638i	$0.447906\pi$
$662$	0	0
$663$	3.13576e7	2.77050
$664$	0	0
$665$	3.37058e6	0.295564
$666$	0	0
$667$	1.18906e6	0.103488
$668$	0	0
$669$	9.01685e6	0.778914
$670$	0	0
$671$	−9.04226e6	−0.775302
$672$	0	0
$673$	−1.70068e6	−0.144739	−0.0723694	−	0.997378i	$-0.523056\pi$
$673$	−1.70068e6	−0.144739	−0.0723694	+	0.997378i	$0.523056\pi$
$674$	0	0
$675$	−514195.	−0.0434379
$676$	0	0
$677$	2.22659e6	0.186711	0.0933553	−	0.995633i	$-0.470241\pi$
$677$	2.22659e6	0.186711	0.0933553	+	0.995633i	$0.470241\pi$
$678$	0	0
$679$	−9755.34	−0.000812022 0
$680$	0	0
$681$	1.68163e7	1.38951
$682$	0	0
$683$	1.52068e7	1.24734	0.623672	−	0.781686i	$-0.285640\pi$
$683$	1.52068e7	1.24734	0.623672	+	0.781686i	$0.285640\pi$
$684$	0	0
$685$	−815456.	−0.0664009
$686$	0	0
$687$	1.42844e7	1.15470
$688$	0	0
$689$	−3.63429e6	−0.291656
$690$	0	0
$691$	−3.12898e6	−0.249292	−0.124646	−	0.992201i	$-0.539779\pi$
$691$	−3.12898e6	−0.249292	−0.124646	+	0.992201i	$0.539779\pi$
$692$	0	0
$693$	3.43780e6	0.271924
$694$	0	0
$695$	−1.23550e7	−0.970242
$696$	0	0
$697$	7.46296e6	0.581874
$698$	0	0
$699$	−2.24322e7	−1.73652
$700$	0	0
$701$	1.63658e7	1.25789	0.628943	−	0.777452i	$-0.283488\pi$
$701$	1.63658e7	1.25789	0.628943	+	0.777452i	$0.283488\pi$
$702$	0	0
$703$	−9.96306e6	−0.760334
$704$	0	0
$705$	2.15336e7	1.63171
$706$	0	0
$707$	509123.	0.0383066
$708$	0	0
$709$	−9.06223e6	−0.677048	−0.338524	−	0.940958i	$-0.609928\pi$
$709$	−9.06223e6	−0.677048	−0.338524	+	0.940958i	$0.609928\pi$
$710$	0	0
$711$	7.96706e6	0.591050
$712$	0	0
$713$	−5.24406e6	−0.386317
$714$	0	0
$715$	−3.11860e7	−2.28137
$716$	0	0
$717$	−2.13384e7	−1.55011
$718$	0	0
$719$	−1.77767e7	−1.28241	−0.641207	−	0.767368i	$-0.721566\pi$
$719$	−1.77767e7	−1.28241	−0.641207	+	0.767368i	$0.721566\pi$
$720$	0	0
$721$	−3.78023e6	−0.270819
$722$	0	0
$723$	−5.86959e6	−0.417601
$724$	0	0
$725$	−427429.	−0.0302008
$726$	0	0
$727$	1.54462e7	1.08389	0.541946	−	0.840413i	$-0.317688\pi$
$727$	1.54462e7	1.08389	0.541946	+	0.840413i	$0.317688\pi$
$728$	0	0
$729$	5.06421e6	0.352933
$730$	0	0
$731$	−1.64607e7	−1.13934
$732$	0	0
$733$	1.99063e7	1.36845	0.684226	−	0.729270i	$-0.260140\pi$
$733$	1.99063e7	1.36845	0.684226	+	0.729270i	$0.260140\pi$
$734$	0	0
$735$	1.19366e7	0.815010
$736$	0	0
$737$	1.90056e7	1.28888
$738$	0	0
$739$	1.06607e7	0.718083	0.359041	−	0.933322i	$-0.383104\pi$
$739$	1.06607e7	0.718083	0.359041	+	0.933322i	$0.383104\pi$
$740$	0	0
$741$	−1.63362e7	−1.09296
$742$	0	0
$743$	1.43627e7	0.954476	0.477238	−	0.878774i	$-0.341638\pi$
$743$	1.43627e7	0.954476	0.477238	+	0.878774i	$0.341638\pi$
$744$	0	0
$745$	−2.53229e7	−1.67156
$746$	0	0
$747$	−6.78084e6	−0.444613
$748$	0	0
$749$	5.76489e6	0.375480
$750$	0	0
$751$	−1.67371e7	−1.08288	−0.541439	−	0.840740i	$-0.682120\pi$
$751$	−1.67371e7	−1.08288	−0.541439	+	0.840740i	$0.682120\pi$
$752$	0	0
$753$	4.29287e6	0.275905
$754$	0	0
$755$	−2.19507e7	−1.40146
$756$	0	0
$757$	−1.02546e7	−0.650398	−0.325199	−	0.945646i	$-0.605431\pi$
$757$	−1.02546e7	−0.650398	−0.325199	+	0.945646i	$0.605431\pi$
$758$	0	0
$759$	−4.67452e6	−0.294532
$760$	0	0
$761$	−6.43275e6	−0.402657	−0.201328	−	0.979524i	$-0.564526\pi$
$761$	−6.43275e6	−0.402657	−0.201328	+	0.979524i	$0.564526\pi$
$762$	0	0
$763$	−1.58432e7	−0.985216
$764$	0	0
$765$	−8.35228e6	−0.516002
$766$	0	0
$767$	−2.70577e7	−1.66074
$768$	0	0
$769$	−1.87869e7	−1.14561	−0.572807	−	0.819690i	$-0.694146\pi$
$769$	−1.87869e7	−1.14561	−0.572807	+	0.819690i	$0.694146\pi$
$770$	0	0
$771$	2.85275e7	1.72833
$772$	0	0
$773$	1.91258e7	1.15126	0.575628	−	0.817712i	$-0.304758\pi$
$773$	1.91258e7	1.15126	0.575628	+	0.817712i	$0.304758\pi$
$774$	0	0
$775$	1.88507e6	0.112738
$776$	0	0
$777$	1.73962e7	1.03372
$778$	0	0
$779$	−3.88793e6	−0.229549
$780$	0	0
$781$	1.02308e6	0.0600179
$782$	0	0
$783$	6.07803e6	0.354289
$784$	0	0
$785$	−1.20047e7	−0.695307
$786$	0	0
$787$	2.51823e7	1.44930	0.724649	−	0.689118i	$-0.242002\pi$
$787$	2.51823e7	1.44930	0.724649	+	0.689118i	$0.242002\pi$
$788$	0	0
$789$	−1.91575e7	−1.09559
$790$	0	0
$791$	−9.94213e6	−0.564987
$792$	0	0
$793$	−2.12738e7	−1.20133
$794$	0	0
$795$	3.41386e6	0.191570
$796$	0	0
$797$	2.77755e7	1.54887	0.774437	−	0.632651i	$-0.218033\pi$
$797$	2.77755e7	1.54887	0.774437	+	0.632651i	$0.218033\pi$
$798$	0	0
$799$	−3.06300e7	−1.69738
$800$	0	0
$801$	−507603.	−0.0279539
$802$	0	0
$803$	−1.43259e7	−0.784029
$804$	0	0
$805$	2.26913e6	0.123415
$806$	0	0
$807$	−2.39189e7	−1.29288
$808$	0	0
$809$	9.12918e6	0.490411	0.245206	−	0.969471i	$-0.421145\pi$
$809$	9.12918e6	0.490411	0.245206	+	0.969471i	$0.421145\pi$
$810$	0	0
$811$	2.85495e7	1.52422	0.762109	−	0.647449i	$-0.224164\pi$
$811$	2.85495e7	1.52422	0.762109	+	0.647449i	$0.224164\pi$
$812$	0	0
$813$	2.70159e7	1.43348
$814$	0	0
$815$	2.44570e7	1.28976
$816$	0	0
$817$	8.57544e6	0.449470
$818$	0	0
$819$	8.08814e6	0.421346
$820$	0	0
$821$	−7.96909e6	−0.412621	−0.206310	−	0.978487i	$-0.566146\pi$
$821$	−7.96909e6	−0.412621	−0.206310	+	0.978487i	$0.566146\pi$
$822$	0	0
$823$	−2.63742e7	−1.35731	−0.678657	−	0.734456i	$-0.737438\pi$
$823$	−2.63742e7	−1.35731	−0.678657	+	0.734456i	$0.737438\pi$
$824$	0	0
$825$	1.68034e6	0.0859531
$826$	0	0
$827$	−1.97623e7	−1.00479	−0.502393	−	0.864639i	$-0.667547\pi$
$827$	−1.97623e7	−1.00479	−0.502393	+	0.864639i	$0.667547\pi$
$828$	0	0
$829$	−1.76929e7	−0.894155	−0.447078	−	0.894495i	$-0.647535\pi$
$829$	−1.76929e7	−0.894155	−0.447078	+	0.894495i	$0.647535\pi$
$830$	0	0
$831$	2.98049e7	1.49722
$832$	0	0
$833$	−1.69790e7	−0.847810
$834$	0	0
$835$	3.28203e7	1.62902
$836$	0	0
$837$	−2.68056e7	−1.32255
$838$	0	0
$839$	−3.17557e7	−1.55746	−0.778729	−	0.627360i	$-0.784135\pi$
$839$	−3.17557e7	−1.55746	−0.778729	+	0.627360i	$0.784135\pi$
$840$	0	0
$841$	−1.54587e7	−0.753675
$842$	0	0
$843$	−3.51939e7	−1.70568
$844$	0	0
$845$	−5.19934e7	−2.50499
$846$	0	0
$847$	5.15293e6	0.246800
$848$	0	0
$849$	1.40357e7	0.668292
$850$	0	0
$851$	−6.70729e6	−0.317485
$852$	0	0
$853$	−1.92334e6	−0.0905075	−0.0452537	−	0.998976i	$-0.514410\pi$
$853$	−1.92334e6	−0.0905075	−0.0452537	+	0.998976i	$0.514410\pi$
$854$	0	0
$855$	4.35124e6	0.203562
$856$	0	0
$857$	1.68379e7	0.783133	0.391566	−	0.920150i	$-0.371933\pi$
$857$	1.68379e7	0.783133	0.391566	+	0.920150i	$0.371933\pi$
$858$	0	0
$859$	1.29887e7	0.600596	0.300298	−	0.953845i	$-0.402914\pi$
$859$	1.29887e7	0.600596	0.300298	+	0.953845i	$0.402914\pi$
$860$	0	0
$861$	6.78861e6	0.312085
$862$	0	0
$863$	−3.47187e7	−1.58685	−0.793426	−	0.608667i	$-0.791704\pi$
$863$	−3.47187e7	−1.58685	−0.793426	+	0.608667i	$0.791704\pi$
$864$	0	0
$865$	4.00142e7	1.81834
$866$	0	0
$867$	1.57494e7	0.711569
$868$	0	0
$869$	3.97474e7	1.78550
$870$	0	0
$871$	4.47147e7	1.99712
$872$	0	0
$873$	−12593.6	−0.000559261 0
$874$	0	0
$875$	1.25889e7	0.555864
$876$	0	0
$877$	5.84624e6	0.256672	0.128336	−	0.991731i	$-0.459036\pi$
$877$	5.84624e6	0.256672	0.128336	+	0.991731i	$0.459036\pi$
$878$	0	0
$879$	−3.37532e7	−1.47347
$880$	0	0
$881$	4.26374e7	1.85076	0.925381	−	0.379038i	$-0.123745\pi$
$881$	4.26374e7	1.85076	0.925381	+	0.379038i	$0.123745\pi$
$882$	0	0
$883$	4.33814e7	1.87241	0.936206	−	0.351452i	$-0.114312\pi$
$883$	4.33814e7	1.87241	0.936206	+	0.351452i	$0.114312\pi$
$884$	0	0
$885$	2.54165e7	1.09083
$886$	0	0
$887$	1.09428e7	0.467005	0.233502	−	0.972356i	$-0.424981\pi$
$887$	1.09428e7	0.467005	0.233502	+	0.972356i	$0.424981\pi$
$888$	0	0
$889$	791045.	0.0335696
$890$	0	0
$891$	−3.51077e7	−1.48152
$892$	0	0
$893$	1.59571e7	0.669616
$894$	0	0
$895$	1.24106e7	0.517887
$896$	0	0
$897$	−1.09978e7	−0.456377
$898$	0	0
$899$	−2.22824e7	−0.919522
$900$	0	0
$901$	−4.85596e6	−0.199280
$902$	0	0
$903$	−1.49733e7	−0.611081
$904$	0	0
$905$	−1.60132e7	−0.649913
$906$	0	0
$907$	−3.69879e7	−1.49294	−0.746468	−	0.665422i	$-0.768252\pi$
$907$	−3.69879e7	−1.49294	−0.746468	+	0.665422i	$0.768252\pi$
$908$	0	0
$909$	657250.	0.0263828
$910$	0	0
$911$	3.50997e6	0.140122	0.0700612	−	0.997543i	$-0.477681\pi$
$911$	3.50997e6	0.140122	0.0700612	+	0.997543i	$0.477681\pi$
$912$	0	0
$913$	−3.38294e7	−1.34313
$914$	0	0
$915$	1.99835e7	0.789074
$916$	0	0
$917$	−1.78959e7	−0.702799
$918$	0	0
$919$	3.64427e7	1.42338	0.711691	−	0.702492i	$-0.247930\pi$
$919$	3.64427e7	1.42338	0.711691	+	0.702492i	$0.247930\pi$
$920$	0	0
$921$	−7.36628e6	−0.286154
$922$	0	0
$923$	2.40700e6	0.0929976
$924$	0	0
$925$	2.41105e6	0.0926513
$926$	0	0
$927$	−4.88007e6	−0.186520
$928$	0	0
$929$	−2.35875e7	−0.896692	−0.448346	−	0.893860i	$-0.647987\pi$
$929$	−2.35875e7	−0.896692	−0.448346	+	0.893860i	$0.647987\pi$
$930$	0	0
$931$	8.84543e6	0.334460
$932$	0	0
$933$	−1.58952e7	−0.597808
$934$	0	0
$935$	−4.16693e7	−1.55879
$936$	0	0
$937$	7.72449e6	0.287423	0.143711	−	0.989620i	$-0.454096\pi$
$937$	7.72449e6	0.287423	0.143711	+	0.989620i	$0.454096\pi$
$938$	0	0
$939$	4.46123e7	1.65116
$940$	0	0
$941$	388339.	0.0142967	0.00714837	−	0.999974i	$-0.497725\pi$
$941$	388339.	0.0142967	0.00714837	+	0.999974i	$0.497725\pi$
$942$	0	0
$943$	−2.61742e6	−0.0958504
$944$	0	0
$945$	1.15989e7	0.422511
$946$	0	0
$947$	3.43933e7	1.24623	0.623115	−	0.782130i	$-0.285867\pi$
$947$	3.43933e7	1.24623	0.623115	+	0.782130i	$0.285867\pi$
$948$	0	0
$949$	−3.37046e7	−1.21485
$950$	0	0
$951$	3.59403e7	1.28864
$952$	0	0
$953$	1.09806e7	0.391646	0.195823	−	0.980639i	$-0.437262\pi$
$953$	1.09806e7	0.391646	0.195823	+	0.980639i	$0.437262\pi$
$954$	0	0
$955$	−3.01025e7	−1.06805
$956$	0	0
$957$	−1.98624e7	−0.701053
$958$	0	0
$959$	1.05512e6	0.0370470
$960$	0	0
$961$	6.96415e7	2.43254
$962$	0	0
$963$	7.44216e6	0.258603
$964$	0	0
$965$	3.07494e7	1.06296
$966$	0	0
$967$	2.78573e7	0.958017	0.479008	−	0.877810i	$-0.340996\pi$
$967$	2.78573e7	0.958017	0.479008	+	0.877810i	$0.340996\pi$
$968$	0	0
$969$	−2.18276e7	−0.746787
$970$	0	0
$971$	5.08273e7	1.73001	0.865005	−	0.501763i	$-0.167315\pi$
$971$	5.08273e7	1.73001	0.865005	+	0.501763i	$0.167315\pi$
$972$	0	0
$973$	1.59861e7	0.541327
$974$	0	0
$975$	3.95334e6	0.133184
$976$	0	0
$977$	−2.75054e7	−0.921894	−0.460947	−	0.887428i	$-0.652490\pi$
$977$	−2.75054e7	−0.921894	−0.460947	+	0.887428i	$0.652490\pi$
$978$	0	0
$979$	−2.53242e6	−0.0844459
$980$	0	0
$981$	−2.04527e7	−0.678544
$982$	0	0
$983$	3.07264e7	1.01421	0.507105	−	0.861884i	$-0.330715\pi$
$983$	3.07264e7	1.01421	0.507105	+	0.861884i	$0.330715\pi$
$984$	0	0
$985$	1.17893e7	0.387165
$986$	0	0
$987$	−2.78623e7	−0.910382
$988$	0	0
$989$	5.77312e6	0.187681
$990$	0	0
$991$	−2.22331e7	−0.719143	−0.359571	−	0.933118i	$-0.617077\pi$
$991$	−2.22331e7	−0.719143	−0.359571	+	0.933118i	$0.617077\pi$
$992$	0	0
$993$	1.95903e7	0.630476
$994$	0	0
$995$	1.50007e7	0.480345
$996$	0	0
$997$	1.57756e7	0.502629	0.251315	−	0.967905i	$-0.419137\pi$
$997$	1.57756e7	0.502629	0.251315	+	0.967905i	$0.419137\pi$
$998$	0	0
$999$	−3.42850e7	−1.08690

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	92.6.a.b.1.3	✓	4
3.2	odd	2		828.6.a.b.1.3		4
4.3	odd	2		368.6.a.f.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
92.6.a.b.1.3	✓	4	1.1	even	1	trivial
368.6.a.f.1.2		4	4.3	odd	2
828.6.a.b.1.3		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 92 = 2^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	92.a (trivial)

\(f(q)\)	\(=\)	\(q+18.4167 q^{3} -57.5774 q^{5} +74.4992 q^{7} +96.1744 q^{9} +479.811 q^{11} +1128.85 q^{13} -1060.39 q^{15} +1508.32 q^{17} -785.781 q^{19} +1372.03 q^{21} -529.000 q^{23} +190.158 q^{25} -2704.04 q^{27} -2247.76 q^{29} +9913.15 q^{31} +8836.53 q^{33} -4289.47 q^{35} +12679.2 q^{37} +20789.7 q^{39} +4947.86 q^{41} -10913.3 q^{43} -5537.47 q^{45} -20307.4 q^{47} -11256.9 q^{49} +27778.3 q^{51} -3219.45 q^{53} -27626.3 q^{55} -14471.5 q^{57} -23969.2 q^{59} -18845.5 q^{61} +7164.91 q^{63} -64996.5 q^{65} +39610.7 q^{67} -9742.43 q^{69} +2132.25 q^{71} -29857.3 q^{73} +3502.08 q^{75} +35745.5 q^{77} +82839.7 q^{79} -73169.9 q^{81} -70505.6 q^{83} -86845.1 q^{85} -41396.2 q^{87} -5277.95 q^{89} +84098.7 q^{91} +182567. q^{93} +45243.2 q^{95} -130.946 q^{97} +46145.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 29 q^{3} + 14 q^{5} + 248 q^{7} + 327 q^{9} - 80 q^{11} + 1331 q^{13} + 2046 q^{15} + 1690 q^{17} + 2752 q^{19} + 3100 q^{21} - 2116 q^{23} + 7192 q^{25} + 4463 q^{27} + 9281 q^{29} + 8105 q^{31}+ \cdots + 290370 q^{99}+O(q^{100}) \) 4 * q + 29 * q^3 + 14 * q^5 + 248 * q^7 + 327 * q^9 - 80 * q^11 + 1331 * q^13 + 2046 * q^15 + 1690 * q^17 + 2752 * q^19 + 3100 * q^21 - 2116 * q^23 + 7192 * q^25 + 4463 * q^27 + 9281 * q^29 + 8105 * q^31 + 16470 * q^33 + 20288 * q^35 + 9834 * q^37 + 8691 * q^39 - 11653 * q^41 + 22358 * q^43 - 552 * q^45 + 2013 * q^47 - 892 * q^49 + 460 * q^51 - 53614 * q^53 - 51604 * q^55 - 44246 * q^57 - 19660 * q^59 - 43446 * q^61 - 50508 * q^63 - 73282 * q^65 - 56544 * q^67 - 15341 * q^69 + 15921 * q^71 - 179843 * q^73 - 1845 * q^75 - 102352 * q^77 + 116422 * q^79 - 66804 * q^81 - 9838 * q^83 - 86752 * q^85 + 115745 * q^87 - 148986 * q^89 + 157444 * q^91 - 12601 * q^93 + 159244 * q^95 + 72030 * q^97 + 290370 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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