LMFDB - Embedded newform 414.4.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,4,Mod(1,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("414.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	414.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:	$24.4267907424$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 46)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	414.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} -11.4031 q^{5} +9.40312 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} -11.4031 q^{5} +9.40312 q^{7} +8.00000 q^{8} -22.8062 q^{10} -10.3875 q^{11} -33.0891 q^{13} +18.8062 q^{14} +16.0000 q^{16} -138.837 q^{17} +68.6281 q^{19} -45.6125 q^{20} -20.7750 q^{22} +23.0000 q^{23} +5.03124 q^{25} -66.1781 q^{26} +37.6125 q^{28} -215.602 q^{29} +87.9266 q^{31} +32.0000 q^{32} -277.675 q^{34} -107.225 q^{35} -215.109 q^{37} +137.256 q^{38} -91.2250 q^{40} -175.267 q^{41} -40.7125 q^{43} -41.5500 q^{44} +46.0000 q^{46} -405.245 q^{47} -254.581 q^{49} +10.0625 q^{50} -132.356 q^{52} +276.994 q^{53} +118.450 q^{55} +75.2250 q^{56} -431.203 q^{58} -293.550 q^{59} -450.731 q^{61} +175.853 q^{62} +64.0000 q^{64} +377.319 q^{65} +273.675 q^{67} -555.350 q^{68} -214.450 q^{70} +643.842 q^{71} +106.345 q^{73} -430.219 q^{74} +274.512 q^{76} -97.6750 q^{77} +60.0844 q^{79} -182.450 q^{80} -350.534 q^{82} +372.878 q^{83} +1583.18 q^{85} -81.4250 q^{86} -83.1000 q^{88} +543.947 q^{89} -311.141 q^{91} +92.0000 q^{92} -810.491 q^{94} -782.575 q^{95} +550.944 q^{97} -509.163 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} - 10 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 - 10 * q^5 + 6 * q^7 + 16 * q^8	$ 2 q + 4 q^{2} + 8 q^{4} - 10 q^{5} + 6 q^{7} + 16 q^{8} - 20 q^{10} - 72 q^{11} - 111 q^{13} + 12 q^{14} + 32 q^{16} - 124 q^{17} + 22 q^{19} - 40 q^{20} - 144 q^{22} + 46 q^{23} - 118 q^{25} - 222 q^{26} + 24 q^{28} - 15 q^{29} + 67 q^{31} + 64 q^{32} - 248 q^{34} - 112 q^{35} + 18 q^{37} + 44 q^{38} - 80 q^{40} - 485 q^{41} - 440 q^{43} - 288 q^{44} + 92 q^{46} - 215 q^{47} - 586 q^{49} - 236 q^{50} - 444 q^{52} - 240 q^{53} + 32 q^{55} + 48 q^{56} - 30 q^{58} - 792 q^{59} + 456 q^{61} + 134 q^{62} + 128 q^{64} + 268 q^{65} + 240 q^{67} - 496 q^{68} - 224 q^{70} + 705 q^{71} + 27 q^{73} + 36 q^{74} + 88 q^{76} + 112 q^{77} + 594 q^{79} - 160 q^{80} - 970 q^{82} - 394 q^{83} + 1604 q^{85} - 880 q^{86} - 576 q^{88} + 486 q^{89} - 46 q^{91} + 184 q^{92} - 430 q^{94} - 848 q^{95} + 2152 q^{97} - 1172 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 - 10 * q^5 + 6 * q^7 + 16 * q^8 - 20 * q^10 - 72 * q^11 - 111 * q^13 + 12 * q^14 + 32 * q^16 - 124 * q^17 + 22 * q^19 - 40 * q^20 - 144 * q^22 + 46 * q^23 - 118 * q^25 - 222 * q^26 + 24 * q^28 - 15 * q^29 + 67 * q^31 + 64 * q^32 - 248 * q^34 - 112 * q^35 + 18 * q^37 + 44 * q^38 - 80 * q^40 - 485 * q^41 - 440 * q^43 - 288 * q^44 + 92 * q^46 - 215 * q^47 - 586 * q^49 - 236 * q^50 - 444 * q^52 - 240 * q^53 + 32 * q^55 + 48 * q^56 - 30 * q^58 - 792 * q^59 + 456 * q^61 + 134 * q^62 + 128 * q^64 + 268 * q^65 + 240 * q^67 - 496 * q^68 - 224 * q^70 + 705 * q^71 + 27 * q^73 + 36 * q^74 + 88 * q^76 + 112 * q^77 + 594 * q^79 - 160 * q^80 - 970 * q^82 - 394 * q^83 + 1604 * q^85 - 880 * q^86 - 576 * q^88 + 486 * q^89 - 46 * q^91 + 184 * q^92 - 430 * q^94 - 848 * q^95 + 2152 * q^97 - 1172 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−11.4031	−1.01993	−0.509963	−	0.860196i	$-0.670341\pi$
$5$	−11.4031	−1.01993	−0.509963	+	0.860196i	$0.670341\pi$
$6$	0	0
$7$	9.40312	0.507721	0.253860	−	0.967241i	$-0.418300\pi$
$7$	9.40312	0.507721	0.253860	+	0.967241i	$0.418300\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	−22.8062	−0.721197
$11$	−10.3875	−0.284723	−0.142361	−	0.989815i	$-0.545470\pi$
$11$	−10.3875	−0.284723	−0.142361	+	0.989815i	$0.545470\pi$
$12$	0	0
$13$	−33.0891	−0.705943	−0.352971	−	0.935634i	$-0.614829\pi$
$13$	−33.0891	−0.705943	−0.352971	+	0.935634i	$0.614829\pi$
$14$	18.8062	0.359013
$15$	0	0
$16$	16.0000	0.250000
$17$	−138.837	−1.98077	−0.990383	−	0.138350i	$-0.955820\pi$
$17$	−138.837	−1.98077	−0.990383	+	0.138350i	$0.955820\pi$
$18$	0	0
$19$	68.6281	0.828651	0.414326	−	0.910129i	$-0.364018\pi$
$19$	68.6281	0.828651	0.414326	+	0.910129i	$0.364018\pi$
$20$	−45.6125	−0.509963
$21$	0	0
$22$	−20.7750	−0.201329
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	5.03124	0.0402499
$26$	−66.1781	−0.499177
$27$	0	0
$28$	37.6125	0.253860
$29$	−215.602	−1.38056	−0.690279	−	0.723543i	$-0.742512\pi$
$29$	−215.602	−1.38056	−0.690279	+	0.723543i	$0.742512\pi$
$30$	0	0
$31$	87.9266	0.509422	0.254711	−	0.967017i	$-0.418020\pi$
$31$	87.9266	0.509422	0.254711	+	0.967017i	$0.418020\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−277.675	−1.40061
$35$	−107.225	−0.517838
$36$	0	0
$37$	−215.109	−0.955777	−0.477889	−	0.878420i	$-0.658598\pi$
$37$	−215.109	−0.955777	−0.477889	+	0.878420i	$0.658598\pi$
$38$	137.256	0.585945
$39$	0	0
$40$	−91.2250	−0.360598
$41$	−175.267	−0.667613	−0.333807	−	0.942642i	$-0.608333\pi$
$41$	−175.267	−0.667613	−0.333807	+	0.942642i	$0.608333\pi$
$42$	0	0
$43$	−40.7125	−0.144386	−0.0721930	−	0.997391i	$-0.523000\pi$
$43$	−40.7125	−0.144386	−0.0721930	+	0.997391i	$0.523000\pi$
$44$	−41.5500	−0.142361
$45$	0	0
$46$	46.0000	0.147442
$47$	−405.245	−1.25768	−0.628841	−	0.777534i	$-0.716471\pi$
$47$	−405.245	−1.25768	−0.628841	+	0.777534i	$0.716471\pi$
$48$	0	0
$49$	−254.581	−0.742219
$50$	10.0625	0.0284610
$51$	0	0
$52$	−132.356	−0.352971
$53$	276.994	0.717887	0.358944	−	0.933359i	$-0.383137\pi$
$53$	276.994	0.717887	0.358944	+	0.933359i	$0.383137\pi$
$54$	0	0
$55$	118.450	0.290396
$56$	75.2250	0.179506
$57$	0	0
$58$	−431.203	−0.976202
$59$	−293.550	−0.647745	−0.323873	−	0.946101i	$-0.604985\pi$
$59$	−293.550	−0.647745	−0.323873	+	0.946101i	$0.604985\pi$
$60$	0	0
$61$	−450.731	−0.946069	−0.473035	−	0.881044i	$-0.656841\pi$
$61$	−450.731	−0.946069	−0.473035	+	0.881044i	$0.656841\pi$
$62$	175.853	0.360216
$63$	0	0
$64$	64.0000	0.125000
$65$	377.319	0.720010
$66$	0	0
$67$	273.675	0.499026	0.249513	−	0.968371i	$-0.419730\pi$
$67$	273.675	0.499026	0.249513	+	0.968371i	$0.419730\pi$
$68$	−555.350	−0.990383
$69$	0	0
$70$	−214.450	−0.366167
$71$	643.842	1.07620	0.538099	−	0.842882i	$-0.319143\pi$
$71$	643.842	1.07620	0.538099	+	0.842882i	$0.319143\pi$
$72$	0	0
$73$	106.345	0.170504	0.0852519	−	0.996359i	$-0.472831\pi$
$73$	106.345	0.170504	0.0852519	+	0.996359i	$0.472831\pi$
$74$	−430.219	−0.675837
$75$	0	0
$76$	274.512	0.414326
$77$	−97.6750	−0.144560
$78$	0	0
$79$	60.0844	0.0855699	0.0427850	−	0.999084i	$-0.486377\pi$
$79$	60.0844	0.0855699	0.0427850	+	0.999084i	$0.486377\pi$
$80$	−182.450	−0.254982
$81$	0	0
$82$	−350.534	−0.472074
$83$	372.878	0.493117	0.246558	−	0.969128i	$-0.420700\pi$
$83$	372.878	0.493117	0.246558	+	0.969128i	$0.420700\pi$
$84$	0	0
$85$	1583.18	2.02024
$86$	−81.4250	−0.102096
$87$	0	0
$88$	−83.1000	−0.100665
$89$	543.947	0.647846	0.323923	−	0.946084i	$-0.394998\pi$
$89$	543.947	0.647846	0.323923	+	0.946084i	$0.394998\pi$
$90$	0	0
$91$	−311.141	−0.358422
$92$	92.0000	0.104257
$93$	0	0
$94$	−810.491	−0.889316
$95$	−782.575	−0.845163
$96$	0	0
$97$	550.944	0.576700	0.288350	−	0.957525i	$-0.406893\pi$
$97$	550.944	0.576700	0.288350	+	0.957525i	$0.406893\pi$
$98$	−509.163	−0.524828
$99$	0	0
$100$	20.1250	0.0201250
$101$	−1084.57	−1.06851	−0.534254	−	0.845324i	$-0.679407\pi$
$101$	−1084.57	−1.06851	−0.534254	+	0.845324i	$0.679407\pi$
$102$	0	0
$103$	1369.19	1.30981	0.654905	−	0.755712i	$-0.272709\pi$
$103$	1369.19	1.30981	0.654905	+	0.755712i	$0.272709\pi$
$104$	−264.713	−0.249588
$105$	0	0
$106$	553.987	0.507623
$107$	1193.82	1.07861	0.539304	−	0.842111i	$-0.318687\pi$
$107$	1193.82	1.07861	0.539304	+	0.842111i	$0.318687\pi$
$108$	0	0
$109$	−893.309	−0.784986	−0.392493	−	0.919755i	$-0.628387\pi$
$109$	−893.309	−0.784986	−0.392493	+	0.919755i	$0.628387\pi$
$110$	236.900	0.205341
$111$	0	0
$112$	150.450	0.126930
$113$	−1842.16	−1.53359	−0.766797	−	0.641890i	$-0.778151\pi$
$113$	−1842.16	−1.53359	−0.766797	+	0.641890i	$0.778151\pi$
$114$	0	0
$115$	−262.272	−0.212669
$116$	−862.406	−0.690279
$117$	0	0
$118$	−587.100	−0.458025
$119$	−1305.51	−1.00568
$120$	0	0
$121$	−1223.10	−0.918933
$122$	−901.462	−0.668972
$123$	0	0
$124$	351.706	0.254711
$125$	1368.02	0.978874
$126$	0	0
$127$	599.361	0.418777	0.209389	−	0.977833i	$-0.432853\pi$
$127$	599.361	0.418777	0.209389	+	0.977833i	$0.432853\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	754.637	0.509124
$131$	560.136	0.373582	0.186791	−	0.982400i	$-0.440191\pi$
$131$	560.136	0.373582	0.186791	+	0.982400i	$0.440191\pi$
$132$	0	0
$133$	645.319	0.420724
$134$	547.350	0.352864
$135$	0	0
$136$	−1110.70	−0.700307
$137$	2111.69	1.31689	0.658445	−	0.752629i	$-0.271215\pi$
$137$	2111.69	1.31689	0.658445	+	0.752629i	$0.271215\pi$
$138$	0	0
$139$	−944.952	−0.576617	−0.288308	−	0.957538i	$-0.593093\pi$
$139$	−944.952	−0.576617	−0.288308	+	0.957538i	$0.593093\pi$
$140$	−428.900	−0.258919
$141$	0	0
$142$	1287.68	0.760986
$143$	343.713	0.200998
$144$	0	0
$145$	2458.53	1.40807
$146$	212.691	0.120564
$147$	0	0
$148$	−860.437	−0.477889
$149$	2480.47	1.36381	0.681906	−	0.731440i	$-0.261151\pi$
$149$	2480.47	1.36381	0.681906	+	0.731440i	$0.261151\pi$
$150$	0	0
$151$	1164.53	0.627605	0.313802	−	0.949488i	$-0.398397\pi$
$151$	1164.53	0.627605	0.313802	+	0.949488i	$0.398397\pi$
$152$	549.025	0.292972
$153$	0	0
$154$	−195.350	−0.102219
$155$	−1002.64	−0.519573
$156$	0	0
$157$	−1525.27	−0.775347	−0.387673	−	0.921797i	$-0.626721\pi$
$157$	−1525.27	−0.775347	−0.387673	+	0.921797i	$0.626721\pi$
$158$	120.169	0.0605071
$159$	0	0
$160$	−364.900	−0.180299
$161$	216.272	0.105867
$162$	0	0
$163$	1944.47	0.934371	0.467185	−	0.884159i	$-0.345268\pi$
$163$	1944.47	0.934371	0.467185	+	0.884159i	$0.345268\pi$
$164$	−701.069	−0.333807
$165$	0	0
$166$	745.756	0.348686
$167$	2632.58	1.21985	0.609926	−	0.792458i	$-0.291199\pi$
$167$	2632.58	1.21985	0.609926	+	0.792458i	$0.291199\pi$
$168$	0	0
$169$	−1102.11	−0.501645
$170$	3166.36	1.42852
$171$	0	0
$172$	−162.850	−0.0721930
$173$	−1189.63	−0.522806	−0.261403	−	0.965230i	$-0.584185\pi$
$173$	−1189.63	−0.522806	−0.261403	+	0.965230i	$0.584185\pi$
$174$	0	0
$175$	47.3094	0.0204357
$176$	−166.200	−0.0711807
$177$	0	0
$178$	1087.89	0.458096
$179$	1548.97	0.646790	0.323395	−	0.946264i	$-0.395176\pi$
$179$	1548.97	0.646790	0.323395	+	0.946264i	$0.395176\pi$
$180$	0	0
$181$	−1908.38	−0.783697	−0.391848	−	0.920030i	$-0.628164\pi$
$181$	−1908.38	−0.783697	−0.391848	+	0.920030i	$0.628164\pi$
$182$	−622.281	−0.253443
$183$	0	0
$184$	184.000	0.0737210
$185$	2452.92	0.974823
$186$	0	0
$187$	1442.17	0.563969
$188$	−1620.98	−0.628841
$189$	0	0
$190$	−1565.15	−0.597621
$191$	−352.625	−0.133587	−0.0667933	−	0.997767i	$-0.521277\pi$
$191$	−352.625	−0.133587	−0.0667933	+	0.997767i	$0.521277\pi$
$192$	0	0
$193$	−1414.28	−0.527472	−0.263736	−	0.964595i	$-0.584955\pi$
$193$	−1414.28	−0.527472	−0.263736	+	0.964595i	$0.584955\pi$
$194$	1101.89	0.407788
$195$	0	0
$196$	−1018.33	−0.371110
$197$	432.530	0.156429	0.0782144	−	0.996937i	$-0.475078\pi$
$197$	432.530	0.156429	0.0782144	+	0.996937i	$0.475078\pi$
$198$	0	0
$199$	−108.519	−0.0386567	−0.0193283	−	0.999813i	$-0.506153\pi$
$199$	−108.519	−0.0386567	−0.0193283	+	0.999813i	$0.506153\pi$
$200$	40.2499	0.0142305
$201$	0	0
$202$	−2169.15	−0.755549
$203$	−2027.33	−0.700939
$204$	0	0
$205$	1998.59	0.680916
$206$	2738.38	0.926175
$207$	0	0
$208$	−529.425	−0.176486
$209$	−712.875	−0.235936
$210$	0	0
$211$	−1082.37	−0.353144	−0.176572	−	0.984288i	$-0.556501\pi$
$211$	−1082.37	−0.353144	−0.176572	+	0.984288i	$0.556501\pi$
$212$	1107.97	0.358944
$213$	0	0
$214$	2387.64	0.762691
$215$	464.250	0.147263
$216$	0	0
$217$	826.784	0.258644
$218$	−1786.62	−0.555069
$219$	0	0
$220$	473.800	0.145198
$221$	4594.00	1.39831
$222$	0	0
$223$	−1619.44	−0.486303	−0.243151	−	0.969988i	$-0.578181\pi$
$223$	−1619.44	−0.486303	−0.243151	+	0.969988i	$0.578181\pi$
$224$	300.900	0.0897532
$225$	0	0
$226$	−3684.32	−1.08441
$227$	−6746.10	−1.97249	−0.986244	−	0.165299i	$-0.947141\pi$
$227$	−6746.10	−1.97249	−0.986244	+	0.165299i	$0.947141\pi$
$228$	0	0
$229$	1192.63	0.344155	0.172077	−	0.985083i	$-0.444952\pi$
$229$	1192.63	0.344155	0.172077	+	0.985083i	$0.444952\pi$
$230$	−524.544	−0.150380
$231$	0	0
$232$	−1724.81	−0.488101
$233$	2860.16	0.804187	0.402093	−	0.915599i	$-0.368283\pi$
$233$	2860.16	0.804187	0.402093	+	0.915599i	$0.368283\pi$
$234$	0	0
$235$	4621.06	1.28274
$236$	−1174.20	−0.323873
$237$	0	0
$238$	−2611.01	−0.711121
$239$	594.373	0.160865	0.0804327	−	0.996760i	$-0.474370\pi$
$239$	594.373	0.160865	0.0804327	+	0.996760i	$0.474370\pi$
$240$	0	0
$241$	3930.15	1.05047	0.525234	−	0.850958i	$-0.323978\pi$
$241$	3930.15	1.05047	0.525234	+	0.850958i	$0.323978\pi$
$242$	−2446.20	−0.649784
$243$	0	0
$244$	−1802.92	−0.473035
$245$	2903.02	0.757009
$246$	0	0
$247$	−2270.84	−0.584980
$248$	703.412	0.180108
$249$	0	0
$250$	2736.04	0.692169
$251$	−2122.07	−0.533641	−0.266821	−	0.963746i	$-0.585973\pi$
$251$	−2122.07	−0.533641	−0.266821	+	0.963746i	$0.585973\pi$
$252$	0	0
$253$	−238.913	−0.0593688
$254$	1198.72	0.296120
$255$	0	0
$256$	256.000	0.0625000
$257$	−1143.68	−0.277591	−0.138795	−	0.990321i	$-0.544323\pi$
$257$	−1143.68	−0.277591	−0.138795	+	0.990321i	$0.544323\pi$
$258$	0	0
$259$	−2022.70	−0.485268
$260$	1509.27	0.360005
$261$	0	0
$262$	1120.27	0.264163
$263$	947.534	0.222158	0.111079	−	0.993812i	$-0.464569\pi$
$263$	947.534	0.222158	0.111079	+	0.993812i	$0.464569\pi$
$264$	0	0
$265$	−3158.59	−0.732192
$266$	1290.64	0.297496
$267$	0	0
$268$	1094.70	0.249513
$269$	−3217.70	−0.729317	−0.364659	−	0.931141i	$-0.618814\pi$
$269$	−3217.70	−0.729317	−0.364659	+	0.931141i	$0.618814\pi$
$270$	0	0
$271$	−7284.05	−1.63275	−0.816374	−	0.577524i	$-0.804019\pi$
$271$	−7284.05	−1.63275	−0.816374	+	0.577524i	$0.804019\pi$
$272$	−2221.40	−0.495192
$273$	0	0
$274$	4223.38	0.931182
$275$	−52.2620	−0.0114601
$276$	0	0
$277$	2524.08	0.547498	0.273749	−	0.961801i	$-0.411736\pi$
$277$	2524.08	0.547498	0.273749	+	0.961801i	$0.411736\pi$
$278$	−1889.90	−0.407730
$279$	0	0
$280$	−857.800	−0.183083
$281$	6277.44	1.33267	0.666336	−	0.745652i	$-0.267862\pi$
$281$	6277.44	1.33267	0.666336	+	0.745652i	$0.267862\pi$
$282$	0	0
$283$	−8.34415	−0.00175268	−0.000876340	−	1.00000i	$-0.500279\pi$
$283$	−8.34415	−0.00175268	−0.000876340		1.00000i	$0.500279\pi$
$284$	2575.37	0.538099
$285$	0	0
$286$	687.426	0.142127
$287$	−1648.06	−0.338961
$288$	0	0
$289$	14362.8	2.92344
$290$	4917.06	0.995655
$291$	0	0
$292$	425.381	0.0852519
$293$	−3075.15	−0.613147	−0.306573	−	0.951847i	$-0.599182\pi$
$293$	−3075.15	−0.613147	−0.306573	+	0.951847i	$0.599182\pi$
$294$	0	0
$295$	3347.39	0.660652
$296$	−1720.87	−0.337918
$297$	0	0
$298$	4960.94	0.964360
$299$	−761.048	−0.147199
$300$	0	0
$301$	−382.825	−0.0733078
$302$	2329.07	0.443784
$303$	0	0
$304$	1098.05	0.207163
$305$	5139.74	0.964921
$306$	0	0
$307$	−1039.90	−0.193323	−0.0966616	−	0.995317i	$-0.530816\pi$
$307$	−1039.90	−0.193323	−0.0966616	+	0.995317i	$0.530816\pi$
$308$	−390.700	−0.0722799
$309$	0	0
$310$	−2005.27	−0.367393
$311$	−5917.12	−1.07887	−0.539436	−	0.842027i	$-0.681362\pi$
$311$	−5917.12	−1.07887	−0.539436	+	0.842027i	$0.681362\pi$
$312$	0	0
$313$	2581.67	0.466213	0.233107	−	0.972451i	$-0.425111\pi$
$313$	2581.67	0.466213	0.233107	+	0.972451i	$0.425111\pi$
$314$	−3050.53	−0.548253
$315$	0	0
$316$	240.338	0.0427850
$317$	−7203.26	−1.27626	−0.638132	−	0.769927i	$-0.720292\pi$
$317$	−7203.26	−1.27626	−0.638132	+	0.769927i	$0.720292\pi$
$318$	0	0
$319$	2239.56	0.393076
$320$	−729.800	−0.127491
$321$	0	0
$322$	432.544	0.0748594
$323$	−9528.16	−1.64136
$324$	0	0
$325$	−166.479	−0.0284142
$326$	3888.93	0.660700
$327$	0	0
$328$	−1402.14	−0.236037
$329$	−3810.57	−0.638552
$330$	0	0
$331$	−11007.6	−1.82789	−0.913943	−	0.405843i	$-0.866978\pi$
$331$	−11007.6	−1.82789	−0.913943	+	0.405843i	$0.866978\pi$
$332$	1491.51	0.246558
$333$	0	0
$334$	5265.16	0.862565
$335$	−3120.75	−0.508969
$336$	0	0
$337$	1436.15	0.232143	0.116072	−	0.993241i	$-0.462970\pi$
$337$	1436.15	0.232143	0.116072	+	0.993241i	$0.462970\pi$
$338$	−2204.23	−0.354716
$339$	0	0
$340$	6332.72	1.01012
$341$	−913.337	−0.145044
$342$	0	0
$343$	−5619.13	−0.884561
$344$	−325.700	−0.0510482
$345$	0	0
$346$	−2379.25	−0.369680
$347$	−4296.61	−0.664710	−0.332355	−	0.943154i	$-0.607843\pi$
$347$	−4296.61	−0.664710	−0.332355	+	0.943154i	$0.607843\pi$
$348$	0	0
$349$	3496.28	0.536250	0.268125	−	0.963384i	$-0.413596\pi$
$349$	3496.28	0.536250	0.268125	+	0.963384i	$0.413596\pi$
$350$	94.6188	0.0144502
$351$	0	0
$352$	−332.400	−0.0503323
$353$	−2884.90	−0.434980	−0.217490	−	0.976063i	$-0.569787\pi$
$353$	−2884.90	−0.434980	−0.217490	+	0.976063i	$0.569787\pi$
$354$	0	0
$355$	−7341.81	−1.09764
$356$	2175.79	0.323923
$357$	0	0
$358$	3097.94	0.457350
$359$	9774.21	1.43694	0.718472	−	0.695556i	$-0.244842\pi$
$359$	9774.21	1.43694	0.718472	+	0.695556i	$0.244842\pi$
$360$	0	0
$361$	−2149.18	−0.313337
$362$	−3816.77	−0.554157
$363$	0	0
$364$	−1244.56	−0.179211
$365$	−1212.67	−0.173901
$366$	0	0
$367$	−5245.42	−0.746073	−0.373036	−	0.927817i	$-0.621683\pi$
$367$	−5245.42	−0.746073	−0.373036	+	0.927817i	$0.621683\pi$
$368$	368.000	0.0521286
$369$	0	0
$370$	4905.84	0.689304
$371$	2604.61	0.364486
$372$	0	0
$373$	−12320.9	−1.71032	−0.855160	−	0.518364i	$-0.826541\pi$
$373$	−12320.9	−1.71032	−0.855160	+	0.518364i	$0.826541\pi$
$374$	2884.35	0.398787
$375$	0	0
$376$	−3241.96	−0.444658
$377$	7134.05	0.974595
$378$	0	0
$379$	−1216.68	−0.164899	−0.0824495	−	0.996595i	$-0.526274\pi$
$379$	−1216.68	−0.164899	−0.0824495	+	0.996595i	$0.526274\pi$
$380$	−3130.30	−0.422582
$381$	0	0
$382$	−705.250	−0.0944600
$383$	3989.13	0.532206	0.266103	−	0.963945i	$-0.414264\pi$
$383$	3989.13	0.532206	0.266103	+	0.963945i	$0.414264\pi$
$384$	0	0
$385$	1113.80	0.147440
$386$	−2828.56	−0.372979
$387$	0	0
$388$	2203.78	0.288350
$389$	−15077.7	−1.96522	−0.982608	−	0.185693i	$-0.940547\pi$
$389$	−15077.7	−1.96522	−0.982608	+	0.185693i	$0.940547\pi$
$390$	0	0
$391$	−3193.26	−0.413018
$392$	−2036.65	−0.262414
$393$	0	0
$394$	865.060	0.110612
$395$	−685.150	−0.0872750
$396$	0	0
$397$	−12767.4	−1.61405	−0.807026	−	0.590515i	$-0.798925\pi$
$397$	−12767.4	−1.61405	−0.807026	+	0.590515i	$0.798925\pi$
$398$	−217.037	−0.0273344
$399$	0	0
$400$	80.4999	0.0100625
$401$	8846.67	1.10170	0.550850	−	0.834604i	$-0.314304\pi$
$401$	8846.67	1.10170	0.550850	+	0.834604i	$0.314304\pi$
$402$	0	0
$403$	−2909.41	−0.359623
$404$	−4338.30	−0.534254
$405$	0	0
$406$	−4054.66	−0.495638
$407$	2234.45	0.272132
$408$	0	0
$409$	−3945.27	−0.476970	−0.238485	−	0.971146i	$-0.576651\pi$
$409$	−3945.27	−0.476970	−0.238485	+	0.971146i	$0.576651\pi$
$410$	3997.19	0.481481
$411$	0	0
$412$	5476.76	0.654905
$413$	−2760.29	−0.328874
$414$	0	0
$415$	−4251.97	−0.502943
$416$	−1058.85	−0.124794
$417$	0	0
$418$	−1425.75	−0.166832
$419$	−13937.2	−1.62500	−0.812502	−	0.582958i	$-0.801895\pi$
$419$	−13937.2	−1.62500	−0.812502	+	0.582958i	$0.801895\pi$
$420$	0	0
$421$	−2471.55	−0.286119	−0.143059	−	0.989714i	$-0.545694\pi$
$421$	−2471.55	−0.286119	−0.143059	+	0.989714i	$0.545694\pi$
$422$	−2164.74	−0.249710
$423$	0	0
$424$	2215.95	0.253811
$425$	−698.525	−0.0797257
$426$	0	0
$427$	−4238.28	−0.480339
$428$	4775.29	0.539304
$429$	0	0
$430$	928.500	0.104131
$431$	−7340.81	−0.820404	−0.410202	−	0.911995i	$-0.634542\pi$
$431$	−7340.81	−0.820404	−0.410202	+	0.911995i	$0.634542\pi$
$432$	0	0
$433$	−8838.13	−0.980909	−0.490455	−	0.871467i	$-0.663169\pi$
$433$	−8838.13	−0.980909	−0.490455	+	0.871467i	$0.663169\pi$
$434$	1653.57	0.182889
$435$	0	0
$436$	−3573.24	−0.392493
$437$	1578.45	0.172786
$438$	0	0
$439$	−10403.7	−1.13107	−0.565534	−	0.824725i	$-0.691330\pi$
$439$	−10403.7	−1.13107	−0.565534	+	0.824725i	$0.691330\pi$
$440$	947.600	0.102671
$441$	0	0
$442$	9188.01	0.988753
$443$	4415.27	0.473535	0.236767	−	0.971566i	$-0.423912\pi$
$443$	4415.27	0.473535	0.236767	+	0.971566i	$0.423912\pi$
$444$	0	0
$445$	−6202.69	−0.660755
$446$	−3238.88	−0.343868
$447$	0	0
$448$	601.800	0.0634651
$449$	−3711.34	−0.390087	−0.195044	−	0.980795i	$-0.562485\pi$
$449$	−3711.34	−0.390087	−0.195044	+	0.980795i	$0.562485\pi$
$450$	0	0
$451$	1820.59	0.190085
$452$	−7368.65	−0.766797
$453$	0	0
$454$	−13492.2	−1.39476
$455$	3547.97	0.365564
$456$	0	0
$457$	15711.9	1.60825	0.804124	−	0.594461i	$-0.202635\pi$
$457$	15711.9	1.60825	0.804124	+	0.594461i	$0.202635\pi$
$458$	2385.27	0.243354
$459$	0	0
$460$	−1049.09	−0.106335
$461$	−3177.04	−0.320975	−0.160488	−	0.987038i	$-0.551307\pi$
$461$	−3177.04	−0.320975	−0.160488	+	0.987038i	$0.551307\pi$
$462$	0	0
$463$	−14774.8	−1.48303	−0.741516	−	0.670935i	$-0.765893\pi$
$463$	−14774.8	−1.48303	−0.741516	+	0.670935i	$0.765893\pi$
$464$	−3449.62	−0.345140
$465$	0	0
$466$	5720.33	0.568646
$467$	2408.12	0.238618	0.119309	−	0.992857i	$-0.461932\pi$
$467$	2408.12	0.238618	0.119309	+	0.992857i	$0.461932\pi$
$468$	0	0
$469$	2573.40	0.253366
$470$	9242.12	0.907037
$471$	0	0
$472$	−2348.40	−0.229012
$473$	422.901	0.0411100
$474$	0	0
$475$	345.285	0.0333532
$476$	−5222.02	−0.502838
$477$	0	0
$478$	1188.75	0.113749
$479$	12528.3	1.19506	0.597528	−	0.801848i	$-0.296150\pi$
$479$	12528.3	1.19506	0.597528	+	0.801848i	$0.296150\pi$
$480$	0	0
$481$	7117.77	0.674724
$482$	7860.29	0.742794
$483$	0	0
$484$	−4892.40	−0.459466
$485$	−6282.48	−0.588191
$486$	0	0
$487$	16798.6	1.56307	0.781537	−	0.623858i	$-0.214436\pi$
$487$	16798.6	1.56307	0.781537	+	0.623858i	$0.214436\pi$
$488$	−3605.85	−0.334486
$489$	0	0
$490$	5806.04	0.535286
$491$	18007.7	1.65514	0.827571	−	0.561361i	$-0.189722\pi$
$491$	18007.7	1.65514	0.827571	+	0.561361i	$0.189722\pi$
$492$	0	0
$493$	29933.6	2.73456
$494$	−4541.68	−0.413643
$495$	0	0
$496$	1406.82	0.127355
$497$	6054.13	0.546408
$498$	0	0
$499$	696.627	0.0624956	0.0312478	−	0.999512i	$-0.490052\pi$
$499$	696.627	0.0624956	0.0312478	+	0.999512i	$0.490052\pi$
$500$	5472.07	0.489437
$501$	0	0
$502$	−4244.14	−0.377341
$503$	20469.0	1.81444	0.907222	−	0.420652i	$-0.138199\pi$
$503$	20469.0	1.81444	0.907222	+	0.420652i	$0.138199\pi$
$504$	0	0
$505$	12367.5	1.08980
$506$	−477.825	−0.0419801
$507$	0	0
$508$	2397.44	0.209389
$509$	−13437.5	−1.17015	−0.585073	−	0.810980i	$-0.698934\pi$
$509$	−13437.5	−1.17015	−0.585073	+	0.810980i	$0.698934\pi$
$510$	0	0
$511$	999.978	0.0865683
$512$	512.000	0.0441942
$513$	0	0
$514$	−2287.36	−0.196286
$515$	−15613.0	−1.33591
$516$	0	0
$517$	4209.49	0.358091
$518$	−4045.40	−0.343136
$519$	0	0
$520$	3018.55	0.254562
$521$	4917.10	0.413478	0.206739	−	0.978396i	$-0.433715\pi$
$521$	4917.10	0.413478	0.206739	+	0.978396i	$0.433715\pi$
$522$	0	0
$523$	15959.4	1.33434	0.667168	−	0.744908i	$-0.267507\pi$
$523$	15959.4	1.33434	0.667168	+	0.744908i	$0.267507\pi$
$524$	2240.54	0.186791
$525$	0	0
$526$	1895.07	0.157089
$527$	−12207.5	−1.00905
$528$	0	0
$529$	529.000	0.0434783
$530$	−6317.19	−0.517738
$531$	0	0
$532$	2581.27	0.210362
$533$	5799.43	0.471297
$534$	0	0
$535$	−13613.3	−1.10010
$536$	2189.40	0.176432
$537$	0	0
$538$	−6435.39	−0.515705
$539$	2644.46	0.211327
$540$	0	0
$541$	−8018.32	−0.637217	−0.318608	−	0.947886i	$-0.603216\pi$
$541$	−8018.32	−0.637217	−0.318608	+	0.947886i	$0.603216\pi$
$542$	−14568.1	−1.15453
$543$	0	0
$544$	−4442.80	−0.350153
$545$	10186.5	0.800628
$546$	0	0
$547$	13255.0	1.03609	0.518047	−	0.855352i	$-0.326659\pi$
$547$	13255.0	1.03609	0.518047	+	0.855352i	$0.326659\pi$
$548$	8446.76	0.658445
$549$	0	0
$550$	−104.524	−0.00810349
$551$	−14796.3	−1.14400
$552$	0	0
$553$	564.981	0.0434456
$554$	5048.15	0.387140
$555$	0	0
$556$	−3779.81	−0.288308
$557$	10410.3	0.791918	0.395959	−	0.918268i	$-0.370412\pi$
$557$	10410.3	0.791918	0.395959	+	0.918268i	$0.370412\pi$
$558$	0	0
$559$	1347.14	0.101928
$560$	−1715.60	−0.129460
$561$	0	0
$562$	12554.9	0.942341
$563$	−19009.9	−1.42304	−0.711520	−	0.702666i	$-0.751993\pi$
$563$	−19009.9	−1.42304	−0.711520	+	0.702666i	$0.751993\pi$
$564$	0	0
$565$	21006.4	1.56415
$566$	−16.6883	−0.00123933
$567$	0	0
$568$	5150.74	0.380493
$569$	−11671.7	−0.859934	−0.429967	−	0.902845i	$-0.641475\pi$
$569$	−11671.7	−0.859934	−0.429967	+	0.902845i	$0.641475\pi$
$570$	0	0
$571$	−21359.8	−1.56546	−0.782732	−	0.622359i	$-0.786174\pi$
$571$	−21359.8	−1.56546	−0.782732	+	0.622359i	$0.786174\pi$
$572$	1374.85	0.100499
$573$	0	0
$574$	−3296.12	−0.239682
$575$	115.719	0.00839269
$576$	0	0
$577$	13416.4	0.967996	0.483998	−	0.875069i	$-0.339184\pi$
$577$	13416.4	0.967996	0.483998	+	0.875069i	$0.339184\pi$
$578$	28725.7	2.06718
$579$	0	0
$580$	9834.12	0.704034
$581$	3506.22	0.250366
$582$	0	0
$583$	−2877.27	−0.204399
$584$	850.762	0.0602822
$585$	0	0
$586$	−6150.29	−0.433560
$587$	11542.3	0.811586	0.405793	−	0.913965i	$-0.366995\pi$
$587$	11542.3	0.811586	0.405793	+	0.913965i	$0.366995\pi$
$588$	0	0
$589$	6034.23	0.422133
$590$	6694.77	0.467152
$591$	0	0
$592$	−3441.75	−0.238944
$593$	−3151.02	−0.218207	−0.109104	−	0.994030i	$-0.534798\pi$
$593$	−3151.02	−0.218207	−0.109104	+	0.994030i	$0.534798\pi$
$594$	0	0
$595$	14886.8	1.02572
$596$	9921.87	0.681906
$597$	0	0
$598$	−1522.10	−0.104086
$599$	293.676	0.0200322	0.0100161	−	0.999950i	$-0.496812\pi$
$599$	293.676	0.0200322	0.0100161	+	0.999950i	$0.496812\pi$
$600$	0	0
$601$	−19297.6	−1.30976	−0.654881	−	0.755732i	$-0.727281\pi$
$601$	−19297.6	−1.30976	−0.654881	+	0.755732i	$0.727281\pi$
$602$	−765.650	−0.0518365
$603$	0	0
$604$	4658.13	0.313802
$605$	13947.2	0.937244
$606$	0	0
$607$	−24573.1	−1.64315	−0.821573	−	0.570103i	$-0.806903\pi$
$607$	−24573.1	−1.64315	−0.821573	+	0.570103i	$0.806903\pi$
$608$	2196.10	0.146486
$609$	0	0
$610$	10279.5	0.682302
$611$	13409.2	0.887852
$612$	0	0
$613$	14171.5	0.933738	0.466869	−	0.884326i	$-0.345382\pi$
$613$	14171.5	0.933738	0.466869	+	0.884326i	$0.345382\pi$
$614$	−2079.80	−0.136700
$615$	0	0
$616$	−781.400	−0.0511096
$617$	3693.48	0.240995	0.120498	−	0.992714i	$-0.461551\pi$
$617$	3693.48	0.240995	0.120498	+	0.992714i	$0.461551\pi$
$618$	0	0
$619$	29215.7	1.89705	0.948527	−	0.316697i	$-0.102574\pi$
$619$	29215.7	1.89705	0.948527	+	0.316697i	$0.102574\pi$
$620$	−4010.55	−0.259786
$621$	0	0
$622$	−11834.2	−0.762878
$623$	5114.80	0.328925
$624$	0	0
$625$	−16228.6	−1.03863
$626$	5163.34	0.329663
$627$	0	0
$628$	−6101.06	−0.387673
$629$	29865.2	1.89317
$630$	0	0
$631$	−10216.8	−0.644571	−0.322286	−	0.946643i	$-0.604451\pi$
$631$	−10216.8	−0.644571	−0.322286	+	0.946643i	$0.604451\pi$
$632$	480.675	0.0302535
$633$	0	0
$634$	−14406.5	−0.902454
$635$	−6834.59	−0.427122
$636$	0	0
$637$	8423.86	0.523964
$638$	4479.12	0.277947
$639$	0	0
$640$	−1459.60	−0.0901496
$641$	−11330.1	−0.698148	−0.349074	−	0.937095i	$-0.613504\pi$
$641$	−11330.1	−0.698148	−0.349074	+	0.937095i	$0.613504\pi$
$642$	0	0
$643$	−18572.6	−1.13908	−0.569542	−	0.821962i	$-0.692879\pi$
$643$	−18572.6	−1.13908	−0.569542	+	0.821962i	$0.692879\pi$
$644$	865.087	0.0529336
$645$	0	0
$646$	−19056.3	−1.16062
$647$	13995.8	0.850436	0.425218	−	0.905091i	$-0.360198\pi$
$647$	13995.8	0.850436	0.425218	+	0.905091i	$0.360198\pi$
$648$	0	0
$649$	3049.25	0.184428
$650$	−332.958	−0.0200918
$651$	0	0
$652$	7777.87	0.467185
$653$	−20473.1	−1.22691	−0.613456	−	0.789729i	$-0.710221\pi$
$653$	−20473.1	−1.22691	−0.613456	+	0.789729i	$0.710221\pi$
$654$	0	0
$655$	−6387.30	−0.381027
$656$	−2804.28	−0.166903
$657$	0	0
$658$	−7621.14	−0.451524
$659$	27951.8	1.65227	0.826135	−	0.563473i	$-0.190535\pi$
$659$	27951.8	1.65227	0.826135	+	0.563473i	$0.190535\pi$
$660$	0	0
$661$	20698.3	1.21796	0.608980	−	0.793185i	$-0.291579\pi$
$661$	20698.3	1.21796	0.608980	+	0.793185i	$0.291579\pi$
$662$	−22015.1	−1.29251
$663$	0	0
$664$	2983.02	0.174343
$665$	−7358.65	−0.429107
$666$	0	0
$667$	−4958.84	−0.287866
$668$	10530.3	0.609926
$669$	0	0
$670$	−6241.50	−0.359896
$671$	4681.97	0.269367
$672$	0	0
$673$	28428.7	1.62830	0.814150	−	0.580654i	$-0.197203\pi$
$673$	28428.7	1.62830	0.814150	+	0.580654i	$0.197203\pi$
$674$	2872.31	0.164150
$675$	0	0
$676$	−4408.46	−0.250822
$677$	7179.44	0.407575	0.203787	−	0.979015i	$-0.434675\pi$
$677$	7179.44	0.407575	0.203787	+	0.979015i	$0.434675\pi$
$678$	0	0
$679$	5180.59	0.292803
$680$	12665.4	0.714261
$681$	0	0
$682$	−1826.67	−0.102562
$683$	−5146.87	−0.288345	−0.144172	−	0.989553i	$-0.546052\pi$
$683$	−5146.87	−0.288345	−0.144172	+	0.989553i	$0.546052\pi$
$684$	0	0
$685$	−24079.9	−1.34313
$686$	−11238.3	−0.625479
$687$	0	0
$688$	−651.400	−0.0360965
$689$	−9165.46	−0.506787
$690$	0	0
$691$	29679.6	1.63396	0.816978	−	0.576669i	$-0.195648\pi$
$691$	29679.6	1.63396	0.816978	+	0.576669i	$0.195648\pi$
$692$	−4758.50	−0.261403
$693$	0	0
$694$	−8593.23	−0.470021
$695$	10775.4	0.588107
$696$	0	0
$697$	24333.7	1.32239
$698$	6992.55	0.379186
$699$	0	0
$700$	189.238	0.0102179
$701$	−9627.78	−0.518739	−0.259370	−	0.965778i	$-0.583515\pi$
$701$	−9627.78	−0.518739	−0.259370	+	0.965778i	$0.583515\pi$
$702$	0	0
$703$	−14762.5	−0.792006
$704$	−664.800	−0.0355903
$705$	0	0
$706$	−5769.80	−0.307577
$707$	−10198.4	−0.542504
$708$	0	0
$709$	11527.4	0.610606	0.305303	−	0.952255i	$-0.401242\pi$
$709$	11527.4	0.610606	0.305303	+	0.952255i	$0.401242\pi$
$710$	−14683.6	−0.776150
$711$	0	0
$712$	4351.57	0.229048
$713$	2022.31	0.106222
$714$	0	0
$715$	−3919.40	−0.205003
$716$	6195.88	0.323395
$717$	0	0
$718$	19548.4	1.01607
$719$	7837.87	0.406542	0.203271	−	0.979123i	$-0.434843\pi$
$719$	7837.87	0.406542	0.203271	+	0.979123i	$0.434843\pi$
$720$	0	0
$721$	12874.7	0.665018
$722$	−4298.36	−0.221563
$723$	0	0
$724$	−7633.54	−0.391848
$725$	−1084.74	−0.0555674
$726$	0	0
$727$	726.845	0.0370800	0.0185400	−	0.999828i	$-0.494098\pi$
$727$	726.845	0.0370800	0.0185400	+	0.999828i	$0.494098\pi$
$728$	−2489.12	−0.126721
$729$	0	0
$730$	−2425.34	−0.122967
$731$	5652.42	0.285995
$732$	0	0
$733$	−378.674	−0.0190814	−0.00954069	−	0.999954i	$-0.503037\pi$
$733$	−378.674	−0.0190814	−0.00954069	+	0.999954i	$0.503037\pi$
$734$	−10490.8	−0.527553
$735$	0	0
$736$	736.000	0.0368605
$737$	−2842.80	−0.142084
$738$	0	0
$739$	18952.9	0.943428	0.471714	−	0.881752i	$-0.343636\pi$
$739$	18952.9	0.943428	0.471714	+	0.881752i	$0.343636\pi$
$740$	9811.67	0.487411
$741$	0	0
$742$	5209.21	0.257731
$743$	19317.8	0.953838	0.476919	−	0.878947i	$-0.341754\pi$
$743$	19317.8	0.953838	0.476919	+	0.878947i	$0.341754\pi$
$744$	0	0
$745$	−28285.1	−1.39099
$746$	−24641.7	−1.20938
$747$	0	0
$748$	5768.70	0.281985
$749$	11225.7	0.547632
$750$	0	0
$751$	−25592.6	−1.24353	−0.621763	−	0.783206i	$-0.713583\pi$
$751$	−25592.6	−1.24353	−0.621763	+	0.783206i	$0.713583\pi$
$752$	−6483.92	−0.314421
$753$	0	0
$754$	14268.1	0.689143
$755$	−13279.3	−0.640111
$756$	0	0
$757$	−36790.6	−1.76642	−0.883208	−	0.468981i	$-0.844621\pi$
$757$	−36790.6	−1.76642	−0.883208	+	0.468981i	$0.844621\pi$
$758$	−2433.36	−0.116601
$759$	0	0
$760$	−6260.60	−0.298810
$761$	−2257.71	−0.107545	−0.0537726	−	0.998553i	$-0.517125\pi$
$761$	−2257.71	−0.107545	−0.0537726	+	0.998553i	$0.517125\pi$
$762$	0	0
$763$	−8399.90	−0.398554
$764$	−1410.50	−0.0667933
$765$	0	0
$766$	7978.26	0.376327
$767$	9713.30	0.457271
$768$	0	0
$769$	6903.74	0.323739	0.161869	−	0.986812i	$-0.448248\pi$
$769$	6903.74	0.323739	0.161869	+	0.986812i	$0.448248\pi$
$770$	2227.60	0.104256
$771$	0	0
$772$	−5657.12	−0.263736
$773$	30096.1	1.40036	0.700182	−	0.713965i	$-0.253102\pi$
$773$	30096.1	1.40036	0.700182	+	0.713965i	$0.253102\pi$
$774$	0	0
$775$	442.380	0.0205042
$776$	4407.55	0.203894
$777$	0	0
$778$	−30155.4	−1.38962
$779$	−12028.3	−0.553218
$780$	0	0
$781$	−6687.91	−0.306418
$782$	−6386.52	−0.292048
$783$	0	0
$784$	−4073.30	−0.185555
$785$	17392.8	0.790797
$786$	0	0
$787$	−6753.37	−0.305885	−0.152943	−	0.988235i	$-0.548875\pi$
$787$	−6753.37	−0.305885	−0.152943	+	0.988235i	$0.548875\pi$
$788$	1730.12	0.0782144
$789$	0	0
$790$	−1370.30	−0.0617128
$791$	−17322.1	−0.778638
$792$	0	0
$793$	14914.3	0.667871
$794$	−25534.9	−1.14131
$795$	0	0
$796$	−434.074	−0.0193283
$797$	−26666.0	−1.18514	−0.592570	−	0.805519i	$-0.701887\pi$
$797$	−26666.0	−1.18514	−0.592570	+	0.805519i	$0.701887\pi$
$798$	0	0
$799$	56263.2	2.49118
$800$	161.000	0.00711525
$801$	0	0
$802$	17693.3	0.779019
$803$	−1104.66	−0.0485463
$804$	0	0
$805$	−2466.17	−0.107977
$806$	−5818.82	−0.254292
$807$	0	0
$808$	−8676.60	−0.377774
$809$	16430.2	0.714035	0.357018	−	0.934098i	$-0.383794\pi$
$809$	16430.2	0.714035	0.357018	+	0.934098i	$0.383794\pi$
$810$	0	0
$811$	−23394.6	−1.01294	−0.506470	−	0.862257i	$-0.669050\pi$
$811$	−23394.6	−1.01294	−0.506470	+	0.862257i	$0.669050\pi$
$812$	−8109.31	−0.350469
$813$	0	0
$814$	4468.90	0.192426
$815$	−22173.0	−0.952989
$816$	0	0
$817$	−2794.02	−0.119646
$818$	−7890.54	−0.337269
$819$	0	0
$820$	7994.37	0.340458
$821$	−30615.7	−1.30145	−0.650727	−	0.759312i	$-0.725536\pi$
$821$	−30615.7	−1.30145	−0.650727	+	0.759312i	$0.725536\pi$
$822$	0	0
$823$	−959.744	−0.0406496	−0.0203248	−	0.999793i	$-0.506470\pi$
$823$	−959.744	−0.0406496	−0.0203248	+	0.999793i	$0.506470\pi$
$824$	10953.5	0.463087
$825$	0	0
$826$	−5520.57	−0.232549
$827$	−37507.5	−1.57710	−0.788551	−	0.614970i	$-0.789168\pi$
$827$	−37507.5	−1.57710	−0.788551	+	0.614970i	$0.789168\pi$
$828$	0	0
$829$	−27029.9	−1.13243	−0.566217	−	0.824256i	$-0.691594\pi$
$829$	−27029.9	−1.13243	−0.566217	+	0.824256i	$0.691594\pi$
$830$	−8503.95	−0.355634
$831$	0	0
$832$	−2117.70	−0.0882428
$833$	35345.4	1.47016
$834$	0	0
$835$	−30019.6	−1.24416
$836$	−2851.50	−0.117968
$837$	0	0
$838$	−27874.4	−1.14905
$839$	−38982.8	−1.60409	−0.802047	−	0.597261i	$-0.796256\pi$
$839$	−38982.8	−1.60409	−0.802047	+	0.597261i	$0.796256\pi$
$840$	0	0
$841$	22095.0	0.905942
$842$	−4943.10	−0.202316
$843$	0	0
$844$	−4329.48	−0.176572
$845$	12567.5	0.511641
$846$	0	0
$847$	−11501.0	−0.466562
$848$	4431.90	0.179472
$849$	0	0
$850$	−1397.05	−0.0563746
$851$	−4947.52	−0.199293
$852$	0	0
$853$	−16634.7	−0.667715	−0.333858	−	0.942624i	$-0.608350\pi$
$853$	−16634.7	−0.667715	−0.333858	+	0.942624i	$0.608350\pi$
$854$	−8476.56	−0.339651
$855$	0	0
$856$	9550.57	0.381346
$857$	15234.4	0.607230	0.303615	−	0.952795i	$-0.401806\pi$
$857$	15234.4	0.607230	0.303615	+	0.952795i	$0.401806\pi$
$858$	0	0
$859$	31925.2	1.26807	0.634036	−	0.773303i	$-0.281397\pi$
$859$	31925.2	1.26807	0.634036	+	0.773303i	$0.281397\pi$
$860$	1857.00	0.0736316
$861$	0	0
$862$	−14681.6	−0.580113
$863$	19518.9	0.769910	0.384955	−	0.922935i	$-0.374217\pi$
$863$	19518.9	0.769910	0.384955	+	0.922935i	$0.374217\pi$
$864$	0	0
$865$	13565.4	0.533224
$866$	−17676.3	−0.693607
$867$	0	0
$868$	3307.14	0.129322
$869$	−624.127	−0.0243637
$870$	0	0
$871$	−9055.65	−0.352284
$872$	−7146.48	−0.277535
$873$	0	0
$874$	3156.89	0.122178
$875$	12863.6	0.496995
$876$	0	0
$877$	16853.3	0.648913	0.324456	−	0.945901i	$-0.394819\pi$
$877$	16853.3	0.648913	0.324456	+	0.945901i	$0.394819\pi$
$878$	−20807.3	−0.799786
$879$	0	0
$880$	1895.20	0.0725991
$881$	−10834.3	−0.414320	−0.207160	−	0.978307i	$-0.566422\pi$
$881$	−10834.3	−0.414320	−0.207160	+	0.978307i	$0.566422\pi$
$882$	0	0
$883$	−23506.2	−0.895863	−0.447931	−	0.894068i	$-0.647839\pi$
$883$	−23506.2	−0.895863	−0.447931	+	0.894068i	$0.647839\pi$
$884$	18376.0	0.699154
$885$	0	0
$886$	8830.54	0.334840
$887$	−34287.2	−1.29792	−0.648958	−	0.760824i	$-0.724795\pi$
$887$	−34287.2	−1.29792	−0.648958	+	0.760824i	$0.724795\pi$
$888$	0	0
$889$	5635.87	0.212622
$890$	−12405.4	−0.467224
$891$	0	0
$892$	−6477.75	−0.243151
$893$	−27811.2	−1.04218
$894$	0	0
$895$	−17663.1	−0.659679
$896$	1203.60	0.0448766
$897$	0	0
$898$	−7422.69	−0.275833
$899$	−18957.1	−0.703287
$900$	0	0
$901$	−38457.1	−1.42197
$902$	3641.18	0.134410
$903$	0	0
$904$	−14737.3	−0.542207
$905$	21761.5	0.799313
$906$	0	0
$907$	47930.9	1.75471	0.877354	−	0.479844i	$-0.159307\pi$
$907$	47930.9	1.75471	0.877354	+	0.479844i	$0.159307\pi$
$908$	−26984.4	−0.986244
$909$	0	0
$910$	7095.95	0.258493
$911$	−9787.98	−0.355972	−0.177986	−	0.984033i	$-0.556958\pi$
$911$	−9787.98	−0.355972	−0.177986	+	0.984033i	$0.556958\pi$
$912$	0	0
$913$	−3873.27	−0.140402
$914$	31423.7	1.13720
$915$	0	0
$916$	4770.54	0.172077
$917$	5267.03	0.189676
$918$	0	0
$919$	25226.6	0.905495	0.452748	−	0.891639i	$-0.350444\pi$
$919$	25226.6	0.905495	0.452748	+	0.891639i	$0.350444\pi$
$920$	−2098.17	−0.0751900
$921$	0	0
$922$	−6354.08	−0.226964
$923$	−21304.1	−0.759734
$924$	0	0
$925$	−1082.27	−0.0384700
$926$	−29549.6	−1.04866
$927$	0	0
$928$	−6899.25	−0.244051
$929$	−38214.5	−1.34960	−0.674800	−	0.738001i	$-0.735770\pi$
$929$	−38214.5	−1.34960	−0.674800	+	0.738001i	$0.735770\pi$
$930$	0	0
$931$	−17471.4	−0.615041
$932$	11440.7	0.402093
$933$	0	0
$934$	4816.25	0.168728
$935$	−16445.3	−0.575207
$936$	0	0
$937$	−19742.9	−0.688338	−0.344169	−	0.938908i	$-0.611839\pi$
$937$	−19742.9	−0.688338	−0.344169	+	0.938908i	$0.611839\pi$
$938$	5146.80	0.179157
$939$	0	0
$940$	18484.2	0.641372
$941$	39767.5	1.37767	0.688833	−	0.724920i	$-0.258123\pi$
$941$	39767.5	1.37767	0.688833	+	0.724920i	$0.258123\pi$
$942$	0	0
$943$	−4031.15	−0.139207
$944$	−4696.80	−0.161936
$945$	0	0
$946$	845.803	0.0290692
$947$	−32884.3	−1.12840	−0.564200	−	0.825638i	$-0.690815\pi$
$947$	−32884.3	−1.12840	−0.564200	+	0.825638i	$0.690815\pi$
$948$	0	0
$949$	−3518.87	−0.120366
$950$	690.569	0.0235842
$951$	0	0
$952$	−10444.0	−0.355560
$953$	−4674.79	−0.158899	−0.0794497	−	0.996839i	$-0.525316\pi$
$953$	−4674.79	−0.158899	−0.0794497	+	0.996839i	$0.525316\pi$
$954$	0	0
$955$	4021.02	0.136248
$956$	2377.49	0.0804327
$957$	0	0
$958$	25056.6	0.845032
$959$	19856.5	0.668613
$960$	0	0
$961$	−22059.9	−0.740489
$962$	14235.5	0.477102
$963$	0	0
$964$	15720.6	0.525234
$965$	16127.2	0.537982
$966$	0	0
$967$	−27165.3	−0.903390	−0.451695	−	0.892173i	$-0.649180\pi$
$967$	−27165.3	−0.903390	−0.451695	+	0.892173i	$0.649180\pi$
$968$	−9784.80	−0.324892
$969$	0	0
$970$	−12565.0	−0.415914
$971$	−14597.4	−0.482443	−0.241221	−	0.970470i	$-0.577548\pi$
$971$	−14597.4	−0.482443	−0.241221	+	0.970470i	$0.577548\pi$
$972$	0	0
$973$	−8885.50	−0.292760
$974$	33597.2	1.10526
$975$	0	0
$976$	−7211.70	−0.236517
$977$	−33327.5	−1.09134	−0.545670	−	0.838000i	$-0.683725\pi$
$977$	−33327.5	−1.09134	−0.545670	+	0.838000i	$0.683725\pi$
$978$	0	0
$979$	−5650.25	−0.184456
$980$	11612.1	0.378505
$981$	0	0
$982$	36015.4	1.17036
$983$	21834.8	0.708465	0.354232	−	0.935157i	$-0.384742\pi$
$983$	21834.8	0.708465	0.354232	+	0.935157i	$0.384742\pi$
$984$	0	0
$985$	−4932.19	−0.159546
$986$	59867.2	1.93363
$987$	0	0
$988$	−9083.36	−0.292490
$989$	−936.388	−0.0301066
$990$	0	0
$991$	45191.9	1.44861	0.724303	−	0.689482i	$-0.242162\pi$
$991$	45191.9	1.44861	0.724303	+	0.689482i	$0.242162\pi$
$992$	2813.65	0.0900539
$993$	0	0
$994$	12108.3	0.386369
$995$	1237.45	0.0394269
$996$	0	0
$997$	18760.0	0.595924	0.297962	−	0.954578i	$-0.403693\pi$
$997$	18760.0	0.595924	0.297962	+	0.954578i	$0.403693\pi$
$998$	1393.25	0.0441911
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.4.a.j.1.1		2
3.2	odd	2		46.4.a.c.1.1	✓	2
12.11	even	2		368.4.a.g.1.2		2
15.2	even	4		1150.4.b.i.599.2		4
15.8	even	4		1150.4.b.i.599.3		4
15.14	odd	2		1150.4.a.k.1.2		2
21.20	even	2		2254.4.a.d.1.2		2
24.5	odd	2		1472.4.a.m.1.2		2
24.11	even	2		1472.4.a.l.1.1		2
69.68	even	2		1058.4.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.4.a.c.1.1	✓	2	3.2	odd	2
368.4.a.g.1.2		2	12.11	even	2
414.4.a.j.1.1		2	1.1	even	1	trivial
1058.4.a.f.1.1		2	69.68	even	2
1150.4.a.k.1.2		2	15.14	odd	2
1150.4.b.i.599.2		4	15.2	even	4
1150.4.b.i.599.3		4	15.8	even	4
1472.4.a.l.1.1		2	24.11	even	2
1472.4.a.m.1.2		2	24.5	odd	2
2254.4.a.d.1.2		2	21.20	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	414.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} -11.4031 q^{5} +9.40312 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} -11.4031 q^{5} +9.40312 q^{7} +8.00000 q^{8} -22.8062 q^{10} -10.3875 q^{11} -33.0891 q^{13} +18.8062 q^{14} +16.0000 q^{16} -138.837 q^{17} +68.6281 q^{19} -45.6125 q^{20} -20.7750 q^{22} +23.0000 q^{23} +5.03124 q^{25} -66.1781 q^{26} +37.6125 q^{28} -215.602 q^{29} +87.9266 q^{31} +32.0000 q^{32} -277.675 q^{34} -107.225 q^{35} -215.109 q^{37} +137.256 q^{38} -91.2250 q^{40} -175.267 q^{41} -40.7125 q^{43} -41.5500 q^{44} +46.0000 q^{46} -405.245 q^{47} -254.581 q^{49} +10.0625 q^{50} -132.356 q^{52} +276.994 q^{53} +118.450 q^{55} +75.2250 q^{56} -431.203 q^{58} -293.550 q^{59} -450.731 q^{61} +175.853 q^{62} +64.0000 q^{64} +377.319 q^{65} +273.675 q^{67} -555.350 q^{68} -214.450 q^{70} +643.842 q^{71} +106.345 q^{73} -430.219 q^{74} +274.512 q^{76} -97.6750 q^{77} +60.0844 q^{79} -182.450 q^{80} -350.534 q^{82} +372.878 q^{83} +1583.18 q^{85} -81.4250 q^{86} -83.1000 q^{88} +543.947 q^{89} -311.141 q^{91} +92.0000 q^{92} -810.491 q^{94} -782.575 q^{95} +550.944 q^{97} -509.163 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} - 10 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 - 10 * q^5 + 6 * q^7 + 16 * q^8	\( 2 q + 4 q^{2} + 8 q^{4} - 10 q^{5} + 6 q^{7} + 16 q^{8} - 20 q^{10} - 72 q^{11} - 111 q^{13} + 12 q^{14} + 32 q^{16} - 124 q^{17} + 22 q^{19} - 40 q^{20} - 144 q^{22} + 46 q^{23} - 118 q^{25} - 222 q^{26} + 24 q^{28} - 15 q^{29} + 67 q^{31} + 64 q^{32} - 248 q^{34} - 112 q^{35} + 18 q^{37} + 44 q^{38} - 80 q^{40} - 485 q^{41} - 440 q^{43} - 288 q^{44} + 92 q^{46} - 215 q^{47} - 586 q^{49} - 236 q^{50} - 444 q^{52} - 240 q^{53} + 32 q^{55} + 48 q^{56} - 30 q^{58} - 792 q^{59} + 456 q^{61} + 134 q^{62} + 128 q^{64} + 268 q^{65} + 240 q^{67} - 496 q^{68} - 224 q^{70} + 705 q^{71} + 27 q^{73} + 36 q^{74} + 88 q^{76} + 112 q^{77} + 594 q^{79} - 160 q^{80} - 970 q^{82} - 394 q^{83} + 1604 q^{85} - 880 q^{86} - 576 q^{88} + 486 q^{89} - 46 q^{91} + 184 q^{92} - 430 q^{94} - 848 q^{95} + 2152 q^{97} - 1172 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 - 10 * q^5 + 6 * q^7 + 16 * q^8 - 20 * q^10 - 72 * q^11 - 111 * q^13 + 12 * q^14 + 32 * q^16 - 124 * q^17 + 22 * q^19 - 40 * q^20 - 144 * q^22 + 46 * q^23 - 118 * q^25 - 222 * q^26 + 24 * q^28 - 15 * q^29 + 67 * q^31 + 64 * q^32 - 248 * q^34 - 112 * q^35 + 18 * q^37 + 44 * q^38 - 80 * q^40 - 485 * q^41 - 440 * q^43 - 288 * q^44 + 92 * q^46 - 215 * q^47 - 586 * q^49 - 236 * q^50 - 444 * q^52 - 240 * q^53 + 32 * q^55 + 48 * q^56 - 30 * q^58 - 792 * q^59 + 456 * q^61 + 134 * q^62 + 128 * q^64 + 268 * q^65 + 240 * q^67 - 496 * q^68 - 224 * q^70 + 705 * q^71 + 27 * q^73 + 36 * q^74 + 88 * q^76 + 112 * q^77 + 594 * q^79 - 160 * q^80 - 970 * q^82 - 394 * q^83 + 1604 * q^85 - 880 * q^86 - 576 * q^88 + 486 * q^89 - 46 * q^91 + 184 * q^92 - 430 * q^94 - 848 * q^95 + 2152 * q^97 - 1172 * q^98

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