LMFDB - Embedded newform 414.4.a.n.1.4

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:	$0$
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} - 2x^{3} - 219x^{2} - 468x + 3240 $ x^4 - 2x^3 - 219x^2 - 468*x + 3240
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-11.4152$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} +4.00000 q^{4} +13.2306 q^{5} -10.2926 q^{7} +8.00000 q^{8} +26.4612 q^{10} +14.6928 q^{11} -23.2916 q^{13} -20.5853 q^{14} +16.0000 q^{16} +119.245 q^{17} +112.461 q^{19} +52.9223 q^{20} +29.3855 q^{22} +23.0000 q^{23} +50.0484 q^{25} -46.5833 q^{26} -41.1705 q^{28} -104.646 q^{29} -27.1038 q^{31} +32.0000 q^{32} +238.490 q^{34} -136.177 q^{35} -24.7531 q^{37} +224.921 q^{38} +105.845 q^{40} -174.061 q^{41} +506.208 q^{43} +58.7710 q^{44} +46.0000 q^{46} +256.367 q^{47} -237.062 q^{49} +100.097 q^{50} -93.1665 q^{52} +200.452 q^{53} +194.394 q^{55} -82.3410 q^{56} -209.292 q^{58} -446.988 q^{59} +419.190 q^{61} -54.2077 q^{62} +64.0000 q^{64} -308.162 q^{65} -70.3675 q^{67} +476.981 q^{68} -272.355 q^{70} +190.743 q^{71} +480.337 q^{73} -49.5063 q^{74} +449.842 q^{76} -151.227 q^{77} +1321.34 q^{79} +211.689 q^{80} -348.123 q^{82} -407.398 q^{83} +1577.68 q^{85} +1012.42 q^{86} +117.542 q^{88} -1435.48 q^{89} +239.732 q^{91} +92.0000 q^{92} +512.734 q^{94} +1487.92 q^{95} -985.582 q^{97} -474.124 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 16 q^{4} + 18 q^{7} + 32 q^{8} + 42 q^{11} + 108 q^{13} + 36 q^{14} + 64 q^{16} + 26 q^{17} + 132 q^{19} + 84 q^{22} + 92 q^{23} + 260 q^{25} + 216 q^{26} + 72 q^{28} + 252 q^{29} + 428 q^{31}+ \cdots + 2152 q^{98}+O(q^{100}) $ 4 * q + 8 * q^2 + 16 * q^4 + 18 * q^7 + 32 * q^8 + 42 * q^11 + 108 * q^13 + 36 * q^14 + 64 * q^16 + 26 * q^17 + 132 * q^19 + 84 * q^22 + 92 * q^23 + 260 * q^25 + 216 * q^26 + 72 * q^28 + 252 * q^29 + 428 * q^31 + 128 * q^32 + 52 * q^34 + 348 * q^35 + 278 * q^37 + 264 * q^38 - 4 * q^41 + 216 * q^43 + 168 * q^44 + 184 * q^46 - 4 * q^47 + 1076 * q^49 + 520 * q^50 + 432 * q^52 - 460 * q^53 + 512 * q^55 + 144 * q^56 + 504 * q^58 - 504 * q^59 + 1086 * q^61 + 856 * q^62 + 256 * q^64 - 1420 * q^65 + 1484 * q^67 + 104 * q^68 + 696 * q^70 - 876 * q^71 + 1072 * q^73 + 556 * q^74 + 528 * q^76 - 1580 * q^77 + 1634 * q^79 - 8 * q^82 - 2246 * q^83 + 872 * q^85 + 432 * q^86 + 336 * q^88 - 474 * q^89 + 40 * q^91 + 368 * q^92 - 8 * q^94 - 2780 * q^95 + 1044 * q^97 + 2152 * q^98

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
$5$	13.2306	1.18338	0.591690	−	0.806166i	$-0.298461\pi$
$5$	13.2306	1.18338	0.591690	+	0.806166i	$0.298461\pi$
$6$	0	0
$7$	−10.2926	−0.555750	−0.277875	−	0.960617i	$-0.589630\pi$
$7$	−10.2926	−0.555750	−0.277875	+	0.960617i	$0.589630\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	26.4612	0.836776
$11$	14.6928	0.402730	0.201365	−	0.979516i	$-0.435462\pi$
$11$	14.6928	0.402730	0.201365	+	0.979516i	$0.435462\pi$
$12$	0	0
$13$	−23.2916	−0.496918	−0.248459	−	0.968642i	$-0.579924\pi$
$13$	−23.2916	−0.496918	−0.248459	+	0.968642i	$0.579924\pi$
$14$	−20.5853	−0.392974
$15$	0	0
$16$	16.0000	0.250000
$17$	119.245	1.70125	0.850623	−	0.525775i	$-0.176225\pi$
$17$	119.245	1.70125	0.850623	+	0.525775i	$0.176225\pi$
$18$	0	0
$19$	112.461	1.35791	0.678953	−	0.734182i	$-0.262434\pi$
$19$	112.461	1.35791	0.678953	+	0.734182i	$0.262434\pi$
$20$	52.9223	0.591690
$21$	0	0
$22$	29.3855	0.284773
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	50.0484	0.400387
$26$	−46.5833	−0.351374
$27$	0	0
$28$	−41.1705	−0.277875
$29$	−104.646	−0.670078	−0.335039	−	0.942204i	$-0.608749\pi$
$29$	−104.646	−0.670078	−0.335039	+	0.942204i	$0.608749\pi$
$30$	0	0
$31$	−27.1038	−0.157032	−0.0785160	−	0.996913i	$-0.525018\pi$
$31$	−27.1038	−0.157032	−0.0785160	+	0.996913i	$0.525018\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	238.490	1.20296
$35$	−136.177	−0.657663
$36$	0	0
$37$	−24.7531	−0.109983	−0.0549917	−	0.998487i	$-0.517513\pi$
$37$	−24.7531	−0.109983	−0.0549917	+	0.998487i	$0.517513\pi$
$38$	224.921	0.960184
$39$	0	0
$40$	105.845	0.418388
$41$	−174.061	−0.663020	−0.331510	−	0.943452i	$-0.607558\pi$
$41$	−174.061	−0.663020	−0.331510	+	0.943452i	$0.607558\pi$
$42$	0	0
$43$	506.208	1.79526	0.897628	−	0.440753i	$-0.145289\pi$
$43$	506.208	1.79526	0.897628	+	0.440753i	$0.145289\pi$
$44$	58.7710	0.201365
$45$	0	0
$46$	46.0000	0.147442
$47$	256.367	0.795637	0.397819	−	0.917464i	$-0.369767\pi$
$47$	256.367	0.795637	0.397819	+	0.917464i	$0.369767\pi$
$48$	0	0
$49$	−237.062	−0.691142
$50$	100.097	0.283117
$51$	0	0
$52$	−93.1665	−0.248459
$53$	200.452	0.519514	0.259757	−	0.965674i	$-0.416358\pi$
$53$	200.452	0.519514	0.259757	+	0.965674i	$0.416358\pi$
$54$	0	0
$55$	194.394	0.476583
$56$	−82.3410	−0.196487
$57$	0	0
$58$	−209.292	−0.473817
$59$	−446.988	−0.986319	−0.493160	−	0.869939i	$-0.664158\pi$
$59$	−446.988	−0.986319	−0.493160	+	0.869939i	$0.664158\pi$
$60$	0	0
$61$	419.190	0.879866	0.439933	−	0.898031i	$-0.355002\pi$
$61$	419.190	0.879866	0.439933	+	0.898031i	$0.355002\pi$
$62$	−54.2077	−0.111038
$63$	0	0
$64$	64.0000	0.125000
$65$	−308.162	−0.588043
$66$	0	0
$67$	−70.3675	−0.128310	−0.0641549	−	0.997940i	$-0.520435\pi$
$67$	−70.3675	−0.128310	−0.0641549	+	0.997940i	$0.520435\pi$
$68$	476.981	0.850623
$69$	0	0
$70$	−272.355	−0.465038
$71$	190.743	0.318832	0.159416	−	0.987211i	$-0.449039\pi$
$71$	190.743	0.318832	0.159416	+	0.987211i	$0.449039\pi$
$72$	0	0
$73$	480.337	0.770125	0.385063	−	0.922890i	$-0.374180\pi$
$73$	480.337	0.770125	0.385063	+	0.922890i	$0.374180\pi$
$74$	−49.5063	−0.0777701
$75$	0	0
$76$	449.842	0.678953
$77$	−151.227	−0.223817
$78$	0	0
$79$	1321.34	1.88180	0.940901	−	0.338681i	$-0.109981\pi$
$79$	1321.34	1.88180	0.940901	+	0.338681i	$0.109981\pi$
$80$	211.689	0.295845
$81$	0	0
$82$	−348.123	−0.468826
$83$	−407.398	−0.538767	−0.269384	−	0.963033i	$-0.586820\pi$
$83$	−407.398	−0.538767	−0.269384	+	0.963033i	$0.586820\pi$
$84$	0	0
$85$	1577.68	2.01322
$86$	1012.42	1.26944
$87$	0	0
$88$	117.542	0.142387
$89$	−1435.48	−1.70967	−0.854835	−	0.518899i	$-0.826342\pi$
$89$	−1435.48	−1.70967	−0.854835	+	0.518899i	$0.826342\pi$
$90$	0	0
$91$	239.732	0.276162
$92$	92.0000	0.104257
$93$	0	0
$94$	512.734	0.562601
$95$	1487.92	1.60692
$96$	0	0
$97$	−985.582	−1.03166	−0.515828	−	0.856692i	$-0.672516\pi$
$97$	−985.582	−1.03166	−0.515828	+	0.856692i	$0.672516\pi$
$98$	−474.124	−0.488711
$99$	0	0
$100$	200.194	0.200194
$101$	−1107.79	−1.09138	−0.545689	−	0.837988i	$-0.683732\pi$
$101$	−1107.79	−1.09138	−0.545689	+	0.837988i	$0.683732\pi$
$102$	0	0
$103$	225.199	0.215432	0.107716	−	0.994182i	$-0.465646\pi$
$103$	225.199	0.215432	0.107716	+	0.994182i	$0.465646\pi$
$104$	−186.333	−0.175687
$105$	0	0
$106$	400.904	0.367352
$107$	−1020.49	−0.922005	−0.461002	−	0.887399i	$-0.652510\pi$
$107$	−1020.49	−0.922005	−0.461002	+	0.887399i	$0.652510\pi$
$108$	0	0
$109$	−706.032	−0.620418	−0.310209	−	0.950668i	$-0.600399\pi$
$109$	−706.032	−0.620418	−0.310209	+	0.950668i	$0.600399\pi$
$110$	388.788	0.336995
$111$	0	0
$112$	−164.682	−0.138937
$113$	−1639.57	−1.36494	−0.682468	−	0.730915i	$-0.739094\pi$
$113$	−1639.57	−1.36494	−0.682468	+	0.730915i	$0.739094\pi$
$114$	0	0
$115$	304.303	0.246752
$116$	−418.584	−0.335039
$117$	0	0
$118$	−893.975	−0.697433
$119$	−1227.35	−0.945467
$120$	0	0
$121$	−1115.12	−0.837808
$122$	838.381	0.622159
$123$	0	0
$124$	−108.415	−0.0785160
$125$	−991.653	−0.709569
$126$	0	0
$127$	−1639.53	−1.14555	−0.572774	−	0.819714i	$-0.694133\pi$
$127$	−1639.53	−1.14555	−0.572774	+	0.819714i	$0.694133\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−616.324	−0.415809
$131$	629.817	0.420056	0.210028	−	0.977695i	$-0.432644\pi$
$131$	629.817	0.420056	0.210028	+	0.977695i	$0.432644\pi$
$132$	0	0
$133$	−1157.51	−0.754656
$134$	−140.735	−0.0907287
$135$	0	0
$136$	953.961	0.601482
$137$	1140.97	0.711528	0.355764	−	0.934576i	$-0.384221\pi$
$137$	1140.97	0.711528	0.355764	+	0.934576i	$0.384221\pi$
$138$	0	0
$139$	−136.295	−0.0831682	−0.0415841	−	0.999135i	$-0.513240\pi$
$139$	−136.295	−0.0831682	−0.0415841	+	0.999135i	$0.513240\pi$
$140$	−544.710	−0.328831
$141$	0	0
$142$	381.487	0.225448
$143$	−342.218	−0.200124
$144$	0	0
$145$	−1384.53	−0.792957
$146$	960.673	0.544561
$147$	0	0
$148$	−99.0125	−0.0549917
$149$	2087.11	1.14754	0.573768	−	0.819018i	$-0.305481\pi$
$149$	2087.11	1.14754	0.573768	+	0.819018i	$0.305481\pi$
$150$	0	0
$151$	1944.71	1.04807	0.524034	−	0.851697i	$-0.324427\pi$
$151$	1944.71	1.04807	0.524034	+	0.851697i	$0.324427\pi$
$152$	899.684	0.480092
$153$	0	0
$154$	−302.454	−0.158263
$155$	−358.600	−0.185828
$156$	0	0
$157$	−734.564	−0.373405	−0.186702	−	0.982417i	$-0.559780\pi$
$157$	−734.564	−0.373405	−0.186702	+	0.982417i	$0.559780\pi$
$158$	2642.68	1.33064
$159$	0	0
$160$	423.379	0.209194
$161$	−236.730	−0.115882
$162$	0	0
$163$	−2737.81	−1.31559	−0.657796	−	0.753196i	$-0.728511\pi$
$163$	−2737.81	−1.31559	−0.657796	+	0.753196i	$0.728511\pi$
$164$	−696.245	−0.331510
$165$	0	0
$166$	−814.795	−0.380966
$167$	−1702.98	−0.789103	−0.394551	−	0.918874i	$-0.629100\pi$
$167$	−1702.98	−0.789103	−0.394551	+	0.918874i	$0.629100\pi$
$168$	0	0
$169$	−1654.50	−0.753072
$170$	3155.37	1.42356
$171$	0	0
$172$	2024.83	0.897628
$173$	−139.362	−0.0612456	−0.0306228	−	0.999531i	$-0.509749\pi$
$173$	−139.362	−0.0612456	−0.0306228	+	0.999531i	$0.509749\pi$
$174$	0	0
$175$	−515.130	−0.222515
$176$	235.084	0.100683
$177$	0	0
$178$	−2870.96	−1.20892
$179$	982.417	0.410219	0.205110	−	0.978739i	$-0.434245\pi$
$179$	982.417	0.410219	0.205110	+	0.978739i	$0.434245\pi$
$180$	0	0
$181$	−3593.96	−1.47590	−0.737948	−	0.674858i	$-0.764205\pi$
$181$	−3593.96	−1.47590	−0.737948	+	0.674858i	$0.764205\pi$
$182$	479.464	0.195276
$183$	0	0
$184$	184.000	0.0737210
$185$	−327.498	−0.130152
$186$	0	0
$187$	1752.04	0.685144
$188$	1025.47	0.397819
$189$	0	0
$190$	2975.84	1.13626
$191$	51.6405	0.0195632	0.00978161	−	0.999952i	$-0.496886\pi$
$191$	51.6405	0.0195632	0.00978161	+	0.999952i	$0.496886\pi$
$192$	0	0
$193$	1176.44	0.438768	0.219384	−	0.975639i	$-0.429595\pi$
$193$	1176.44	0.438768	0.219384	+	0.975639i	$0.429595\pi$
$194$	−1971.16	−0.729492
$195$	0	0
$196$	−948.247	−0.345571
$197$	1092.50	0.395113	0.197557	−	0.980291i	$-0.436699\pi$
$197$	1092.50	0.395113	0.197557	+	0.980291i	$0.436699\pi$
$198$	0	0
$199$	155.379	0.0553495	0.0276747	−	0.999617i	$-0.491190\pi$
$199$	155.379	0.0553495	0.0276747	+	0.999617i	$0.491190\pi$
$200$	400.387	0.141558
$201$	0	0
$202$	−2215.58	−0.771721
$203$	1077.08	0.372396
$204$	0	0
$205$	−2302.93	−0.784604
$206$	450.398	0.152334
$207$	0	0
$208$	−372.666	−0.124230
$209$	1652.36	0.546870
$210$	0	0
$211$	−5365.96	−1.75075	−0.875374	−	0.483447i	$-0.839385\pi$
$211$	−5365.96	−1.75075	−0.875374	+	0.483447i	$0.839385\pi$
$212$	801.809	0.259757
$213$	0	0
$214$	−2040.98	−0.651956
$215$	6697.43	2.12447
$216$	0	0
$217$	278.970	0.0872705
$218$	−1412.06	−0.438702
$219$	0	0
$220$	777.575	0.238291
$221$	−2777.41	−0.845380
$222$	0	0
$223$	446.221	0.133996	0.0669982	−	0.997753i	$-0.478658\pi$
$223$	446.221	0.133996	0.0669982	+	0.997753i	$0.478658\pi$
$224$	−329.364	−0.0982436
$225$	0	0
$226$	−3279.14	−0.965156
$227$	−2263.94	−0.661950	−0.330975	−	0.943639i	$-0.607378\pi$
$227$	−2263.94	−0.661950	−0.330975	+	0.943639i	$0.607378\pi$
$228$	0	0
$229$	1722.70	0.497116	0.248558	−	0.968617i	$-0.420043\pi$
$229$	1722.70	0.497116	0.248558	+	0.968617i	$0.420043\pi$
$230$	608.607	0.174480
$231$	0	0
$232$	−837.167	−0.236908
$233$	−4122.28	−1.15905	−0.579527	−	0.814953i	$-0.696763\pi$
$233$	−4122.28	−1.15905	−0.579527	+	0.814953i	$0.696763\pi$
$234$	0	0
$235$	3391.88	0.941541
$236$	−1787.95	−0.493160
$237$	0	0
$238$	−2454.69	−0.668546
$239$	1199.74	0.324707	0.162353	−	0.986733i	$-0.448092\pi$
$239$	1199.74	0.324707	0.162353	+	0.986733i	$0.448092\pi$
$240$	0	0
$241$	7262.37	1.94112	0.970561	−	0.240855i	$-0.0774278\pi$
$241$	7262.37	1.94112	0.970561	+	0.240855i	$0.0774278\pi$
$242$	−2230.25	−0.592420
$243$	0	0
$244$	1676.76	0.439933
$245$	−3136.47	−0.817884
$246$	0	0
$247$	−2619.39	−0.674768
$248$	−216.831	−0.0555192
$249$	0	0
$250$	−1983.31	−0.501741
$251$	−1662.11	−0.417974	−0.208987	−	0.977918i	$-0.567017\pi$
$251$	−1662.11	−0.417974	−0.208987	+	0.977918i	$0.567017\pi$
$252$	0	0
$253$	337.933	0.0839751
$254$	−3279.05	−0.810024
$255$	0	0
$256$	256.000	0.0625000
$257$	−3216.11	−0.780605	−0.390302	−	0.920687i	$-0.627630\pi$
$257$	−3216.11	−0.780605	−0.390302	+	0.920687i	$0.627630\pi$
$258$	0	0
$259$	254.775	0.0611233
$260$	−1232.65	−0.294021
$261$	0	0
$262$	1259.63	0.297025
$263$	−1552.10	−0.363903	−0.181951	−	0.983308i	$-0.558241\pi$
$263$	−1552.10	−0.363903	−0.181951	+	0.983308i	$0.558241\pi$
$264$	0	0
$265$	2652.10	0.614782
$266$	−2315.03	−0.533622
$267$	0	0
$268$	−281.470	−0.0641549
$269$	4575.50	1.03707	0.518537	−	0.855055i	$-0.326477\pi$
$269$	4575.50	1.03707	0.518537	+	0.855055i	$0.326477\pi$
$270$	0	0
$271$	−2277.78	−0.510574	−0.255287	−	0.966865i	$-0.582170\pi$
$271$	−2277.78	−0.510574	−0.255287	+	0.966865i	$0.582170\pi$
$272$	1907.92	0.425312
$273$	0	0
$274$	2281.93	0.503127
$275$	735.350	0.161248
$276$	0	0
$277$	780.955	0.169397	0.0846986	−	0.996407i	$-0.473007\pi$
$277$	780.955	0.169397	0.0846986	+	0.996407i	$0.473007\pi$
$278$	−272.590	−0.0588088
$279$	0	0
$280$	−1089.42	−0.232519
$281$	52.7391	0.0111963	0.00559813	−	0.999984i	$-0.498218\pi$
$281$	52.7391	0.0111963	0.00559813	+	0.999984i	$0.498218\pi$
$282$	0	0
$283$	−6200.42	−1.30239	−0.651196	−	0.758910i	$-0.725732\pi$
$283$	−6200.42	−1.30239	−0.651196	+	0.758910i	$0.725732\pi$
$284$	762.973	0.159416
$285$	0	0
$286$	−684.437	−0.141509
$287$	1791.55	0.368473
$288$	0	0
$289$	9306.41	1.89424
$290$	−2769.05	−0.560705
$291$	0	0
$292$	1921.35	0.385063
$293$	−6131.11	−1.22247	−0.611234	−	0.791450i	$-0.709327\pi$
$293$	−6131.11	−1.22247	−0.611234	+	0.791450i	$0.709327\pi$
$294$	0	0
$295$	−5913.91	−1.16719
$296$	−198.025	−0.0388850
$297$	0	0
$298$	4174.23	0.811431
$299$	−535.707	−0.103615
$300$	0	0
$301$	−5210.21	−0.997713
$302$	3889.42	0.741096
$303$	0	0
$304$	1799.37	0.339476
$305$	5546.13	1.04122
$306$	0	0
$307$	−7056.18	−1.31178	−0.655892	−	0.754855i	$-0.727707\pi$
$307$	−7056.18	−1.31178	−0.655892	+	0.754855i	$0.727707\pi$
$308$	−604.908	−0.111909
$309$	0	0
$310$	−717.199	−0.131401
$311$	7368.26	1.34346	0.671730	−	0.740796i	$-0.265552\pi$
$311$	7368.26	1.34346	0.671730	+	0.740796i	$0.265552\pi$
$312$	0	0
$313$	9068.44	1.63763	0.818816	−	0.574056i	$-0.194631\pi$
$313$	9068.44	1.63763	0.818816	+	0.574056i	$0.194631\pi$
$314$	−1469.13	−0.264037
$315$	0	0
$316$	5285.36	0.940901
$317$	−1952.28	−0.345902	−0.172951	−	0.984930i	$-0.555330\pi$
$317$	−1952.28	−0.345902	−0.172951	+	0.984930i	$0.555330\pi$
$318$	0	0
$319$	−1537.54	−0.269861
$320$	846.758	0.147922
$321$	0	0
$322$	−473.461	−0.0819408
$323$	13410.4	2.31013
$324$	0	0
$325$	−1165.71	−0.198960
$326$	−5475.61	−0.930264
$327$	0	0
$328$	−1392.49	−0.234413
$329$	−2638.69	−0.442175
$330$	0	0
$331$	11508.2	1.91102	0.955511	−	0.294956i	$-0.0953051\pi$
$331$	11508.2	1.91102	0.955511	+	0.294956i	$0.0953051\pi$
$332$	−1629.59	−0.269384
$333$	0	0
$334$	−3405.95	−0.557980
$335$	−931.003	−0.151839
$336$	0	0
$337$	−425.493	−0.0687776	−0.0343888	−	0.999409i	$-0.510948\pi$
$337$	−425.493	−0.0687776	−0.0343888	+	0.999409i	$0.510948\pi$
$338$	−3309.00	−0.532503
$339$	0	0
$340$	6310.73	1.00661
$341$	−398.230	−0.0632416
$342$	0	0
$343$	5970.36	0.939852
$344$	4049.67	0.634719
$345$	0	0
$346$	−278.724	−0.0433072
$347$	1438.94	0.222612	0.111306	−	0.993786i	$-0.464497\pi$
$347$	1438.94	0.222612	0.111306	+	0.993786i	$0.464497\pi$
$348$	0	0
$349$	−681.032	−0.104455	−0.0522275	−	0.998635i	$-0.516632\pi$
$349$	−681.032	−0.104455	−0.0522275	+	0.998635i	$0.516632\pi$
$350$	−1030.26	−0.157342
$351$	0	0
$352$	470.168	0.0711933
$353$	−6608.06	−0.996350	−0.498175	−	0.867076i	$-0.665996\pi$
$353$	−6608.06	−0.996350	−0.498175	+	0.867076i	$0.665996\pi$
$354$	0	0
$355$	2523.65	0.377299
$356$	−5741.92	−0.854835
$357$	0	0
$358$	1964.83	0.290069
$359$	−2801.68	−0.411886	−0.205943	−	0.978564i	$-0.566026\pi$
$359$	−2801.68	−0.411886	−0.205943	+	0.978564i	$0.566026\pi$
$360$	0	0
$361$	5788.36	0.843908
$362$	−7187.93	−1.04362
$363$	0	0
$364$	958.928	0.138081
$365$	6355.14	0.911351
$366$	0	0
$367$	8642.79	1.22929	0.614645	−	0.788803i	$-0.289299\pi$
$367$	8642.79	1.22929	0.614645	+	0.788803i	$0.289299\pi$
$368$	368.000	0.0521286
$369$	0	0
$370$	−654.997	−0.0920315
$371$	−2063.18	−0.288720
$372$	0	0
$373$	−9923.81	−1.37757	−0.688787	−	0.724963i	$-0.741857\pi$
$373$	−9923.81	−1.37757	−0.688787	+	0.724963i	$0.741857\pi$
$374$	3504.08	0.484470
$375$	0	0
$376$	2050.94	0.281300
$377$	2437.37	0.332974
$378$	0	0
$379$	5030.01	0.681727	0.340863	−	0.940113i	$-0.389281\pi$
$379$	5030.01	0.681727	0.340863	+	0.940113i	$0.389281\pi$
$380$	5951.67	0.803459
$381$	0	0
$382$	103.281	0.0138333
$383$	2494.31	0.332776	0.166388	−	0.986060i	$-0.446790\pi$
$383$	2494.31	0.332776	0.166388	+	0.986060i	$0.446790\pi$
$384$	0	0
$385$	−2000.82	−0.264861
$386$	2352.89	0.310256
$387$	0	0
$388$	−3942.33	−0.515828
$389$	−6948.47	−0.905659	−0.452830	−	0.891597i	$-0.649585\pi$
$389$	−6948.47	−0.905659	−0.452830	+	0.891597i	$0.649585\pi$
$390$	0	0
$391$	2742.64	0.354735
$392$	−1896.49	−0.244356
$393$	0	0
$394$	2185.00	0.279387
$395$	17482.1	2.22689
$396$	0	0
$397$	7921.85	1.00148	0.500738	−	0.865599i	$-0.333062\pi$
$397$	7921.85	1.00148	0.500738	+	0.865599i	$0.333062\pi$
$398$	310.759	0.0391380
$399$	0	0
$400$	800.775	0.100097
$401$	−10902.0	−1.35765	−0.678826	−	0.734299i	$-0.737511\pi$
$401$	−10902.0	−1.35765	−0.678826	+	0.734299i	$0.737511\pi$
$402$	0	0
$403$	631.292	0.0780320
$404$	−4431.16	−0.545689
$405$	0	0
$406$	2154.16	0.263323
$407$	−363.692	−0.0442937
$408$	0	0
$409$	−3070.16	−0.371172	−0.185586	−	0.982628i	$-0.559418\pi$
$409$	−3070.16	−0.371172	−0.185586	+	0.982628i	$0.559418\pi$
$410$	−4605.87	−0.554799
$411$	0	0
$412$	900.796	0.107716
$413$	4600.68	0.548146
$414$	0	0
$415$	−5390.11	−0.637566
$416$	−745.332	−0.0878435
$417$	0	0
$418$	3304.71	0.386695
$419$	−9682.14	−1.12889	−0.564443	−	0.825472i	$-0.690909\pi$
$419$	−9682.14	−1.12889	−0.564443	+	0.825472i	$0.690909\pi$
$420$	0	0
$421$	7481.37	0.866080	0.433040	−	0.901375i	$-0.357441\pi$
$421$	7481.37	0.866080	0.433040	+	0.901375i	$0.357441\pi$
$422$	−10731.9	−1.23797
$423$	0	0
$424$	1603.62	0.183676
$425$	5968.03	0.681158
$426$	0	0
$427$	−4314.57	−0.488985
$428$	−4081.96	−0.461002
$429$	0	0
$430$	13394.9	1.50223
$431$	16513.5	1.84554	0.922768	−	0.385356i	$-0.125921\pi$
$431$	16513.5	1.84554	0.922768	+	0.385356i	$0.125921\pi$
$432$	0	0
$433$	11765.8	1.30583	0.652917	−	0.757429i	$-0.273545\pi$
$433$	11765.8	1.30583	0.652917	+	0.757429i	$0.273545\pi$
$434$	557.939	0.0617095
$435$	0	0
$436$	−2824.13	−0.310209
$437$	2586.59	0.283143
$438$	0	0
$439$	16715.0	1.81722	0.908612	−	0.417640i	$-0.137143\pi$
$439$	16715.0	1.81722	0.908612	+	0.417640i	$0.137143\pi$
$440$	1555.15	0.168498
$441$	0	0
$442$	−5554.83	−0.597774
$443$	12687.1	1.36068	0.680341	−	0.732895i	$-0.261832\pi$
$443$	12687.1	1.36068	0.680341	+	0.732895i	$0.261832\pi$
$444$	0	0
$445$	−18992.3	−2.02319
$446$	892.442	0.0947497
$447$	0	0
$448$	−658.728	−0.0694687
$449$	−16077.7	−1.68987	−0.844937	−	0.534866i	$-0.820362\pi$
$449$	−16077.7	−1.68987	−0.844937	+	0.534866i	$0.820362\pi$
$450$	0	0
$451$	−2557.44	−0.267018
$452$	−6558.28	−0.682468
$453$	0	0
$454$	−4527.87	−0.468070
$455$	3171.80	0.326805
$456$	0	0
$457$	7930.19	0.811726	0.405863	−	0.913934i	$-0.366971\pi$
$457$	7930.19	0.811726	0.405863	+	0.913934i	$0.366971\pi$
$458$	3445.41	0.351514
$459$	0	0
$460$	1217.21	0.123376
$461$	11530.0	1.16487	0.582436	−	0.812877i	$-0.302100\pi$
$461$	11530.0	1.16487	0.582436	+	0.812877i	$0.302100\pi$
$462$	0	0
$463$	−7590.41	−0.761893	−0.380947	−	0.924597i	$-0.624402\pi$
$463$	−7590.41	−0.761893	−0.380947	+	0.924597i	$0.624402\pi$
$464$	−1674.33	−0.167519
$465$	0	0
$466$	−8244.56	−0.819575
$467$	6686.26	0.662534	0.331267	−	0.943537i	$-0.392524\pi$
$467$	6686.26	0.662534	0.331267	+	0.943537i	$0.392524\pi$
$468$	0	0
$469$	724.266	0.0713081
$470$	6783.77	0.665770
$471$	0	0
$472$	−3575.90	−0.348716
$473$	7437.60	0.723004
$474$	0	0
$475$	5628.47	0.543688
$476$	−4909.38	−0.472734
$477$	0	0
$478$	2399.49	0.229602
$479$	12072.0	1.15153	0.575766	−	0.817614i	$-0.304704\pi$
$479$	12072.0	1.15153	0.575766	+	0.817614i	$0.304704\pi$
$480$	0	0
$481$	576.541	0.0546528
$482$	14524.7	1.37258
$483$	0	0
$484$	−4460.49	−0.418904
$485$	−13039.8	−1.22084
$486$	0	0
$487$	−1155.32	−0.107500	−0.0537499	−	0.998554i	$-0.517117\pi$
$487$	−1155.32	−0.107500	−0.0537499	+	0.998554i	$0.517117\pi$
$488$	3353.52	0.311080
$489$	0	0
$490$	−6272.93	−0.578331
$491$	7251.93	0.666548	0.333274	−	0.942830i	$-0.391847\pi$
$491$	7251.93	0.666548	0.333274	+	0.942830i	$0.391847\pi$
$492$	0	0
$493$	−12478.5	−1.13997
$494$	−5238.78	−0.477133
$495$	0	0
$496$	−433.661	−0.0392580
$497$	−1963.25	−0.177191
$498$	0	0
$499$	11437.9	1.02612	0.513059	−	0.858354i	$-0.328512\pi$
$499$	11437.9	1.02612	0.513059	+	0.858354i	$0.328512\pi$
$500$	−3966.61	−0.354785
$501$	0	0
$502$	−3324.22	−0.295552
$503$	5624.43	0.498570	0.249285	−	0.968430i	$-0.419804\pi$
$503$	5624.43	0.498570	0.249285	+	0.968430i	$0.419804\pi$
$504$	0	0
$505$	−14656.7	−1.29152
$506$	675.867	0.0593794
$507$	0	0
$508$	−6558.11	−0.572774
$509$	−14983.3	−1.30476	−0.652381	−	0.757891i	$-0.726230\pi$
$509$	−14983.3	−1.30476	−0.652381	+	0.757891i	$0.726230\pi$
$510$	0	0
$511$	−4943.93	−0.427997
$512$	512.000	0.0441942
$513$	0	0
$514$	−6432.22	−0.551971
$515$	2979.51	0.254938
$516$	0	0
$517$	3766.74	0.320427
$518$	509.549	0.0432207
$519$	0	0
$520$	−2465.30	−0.207904
$521$	11536.9	0.970139	0.485069	−	0.874476i	$-0.338794\pi$
$521$	11536.9	0.970139	0.485069	+	0.874476i	$0.338794\pi$
$522$	0	0
$523$	12093.5	1.01111	0.505555	−	0.862794i	$-0.331288\pi$
$523$	12093.5	1.01111	0.505555	+	0.862794i	$0.331288\pi$
$524$	2519.27	0.210028
$525$	0	0
$526$	−3104.20	−0.257318
$527$	−3232.00	−0.267150
$528$	0	0
$529$	529.000	0.0434783
$530$	5304.20	0.434716
$531$	0	0
$532$	−4630.06	−0.377328
$533$	4054.17	0.329467
$534$	0	0
$535$	−13501.7	−1.09108
$536$	−562.940	−0.0453644
$537$	0	0
$538$	9150.99	0.733322
$539$	−3483.09	−0.278344
$540$	0	0
$541$	5566.83	0.442397	0.221198	−	0.975229i	$-0.429003\pi$
$541$	5566.83	0.442397	0.221198	+	0.975229i	$0.429003\pi$
$542$	−4555.57	−0.361030
$543$	0	0
$544$	3815.85	0.300741
$545$	−9341.21	−0.734190
$546$	0	0
$547$	−19574.6	−1.53007	−0.765034	−	0.643989i	$-0.777278\pi$
$547$	−19574.6	−1.53007	−0.765034	+	0.643989i	$0.777278\pi$
$548$	4563.87	0.355764
$549$	0	0
$550$	1470.70	0.114020
$551$	−11768.5	−0.909903
$552$	0	0
$553$	−13600.1	−1.04581
$554$	1561.91	0.119782
$555$	0	0
$556$	−545.180	−0.0415841
$557$	−18416.6	−1.40096	−0.700481	−	0.713671i	$-0.747031\pi$
$557$	−18416.6	−1.40096	−0.700481	+	0.713671i	$0.747031\pi$
$558$	0	0
$559$	−11790.4	−0.892095
$560$	−2178.84	−0.164416
$561$	0	0
$562$	105.478	0.00791695
$563$	−3944.28	−0.295260	−0.147630	−	0.989043i	$-0.547165\pi$
$563$	−3944.28	−0.295260	−0.147630	+	0.989043i	$0.547165\pi$
$564$	0	0
$565$	−21692.5	−1.61524
$566$	−12400.8	−0.920930
$567$	0	0
$568$	1525.95	0.112724
$569$	−22904.6	−1.68754	−0.843769	−	0.536707i	$-0.819668\pi$
$569$	−22904.6	−1.68754	−0.843769	+	0.536707i	$0.819668\pi$
$570$	0	0
$571$	9613.06	0.704543	0.352272	−	0.935898i	$-0.385409\pi$
$571$	9613.06	0.704543	0.352272	+	0.935898i	$0.385409\pi$
$572$	−1368.87	−0.100062
$573$	0	0
$574$	3583.10	0.260550
$575$	1151.11	0.0834865
$576$	0	0
$577$	7910.77	0.570762	0.285381	−	0.958414i	$-0.407880\pi$
$577$	7910.77	0.570762	0.285381	+	0.958414i	$0.407880\pi$
$578$	18612.8	1.33943
$579$	0	0
$580$	−5538.11	−0.396478
$581$	4193.19	0.299420
$582$	0	0
$583$	2945.20	0.209224
$584$	3842.69	0.272280
$585$	0	0
$586$	−12262.2	−0.864416
$587$	−3351.56	−0.235662	−0.117831	−	0.993034i	$-0.537594\pi$
$587$	−3351.56	−0.235662	−0.117831	+	0.993034i	$0.537594\pi$
$588$	0	0
$589$	−3048.11	−0.213235
$590$	−11827.8	−0.825328
$591$	0	0
$592$	−396.050	−0.0274959
$593$	−12072.3	−0.836002	−0.418001	−	0.908447i	$-0.637269\pi$
$593$	−12072.3	−0.836002	−0.418001	+	0.908447i	$0.637269\pi$
$594$	0	0
$595$	−16238.5	−1.11885
$596$	8348.45	0.573768
$597$	0	0
$598$	−1071.41	−0.0732666
$599$	13998.0	0.954826	0.477413	−	0.878679i	$-0.341575\pi$
$599$	13998.0	0.954826	0.477413	+	0.878679i	$0.341575\pi$
$600$	0	0
$601$	7796.28	0.529146	0.264573	−	0.964366i	$-0.414769\pi$
$601$	7796.28	0.529146	0.264573	+	0.964366i	$0.414769\pi$
$602$	−10420.4	−0.705490
$603$	0	0
$604$	7778.84	0.524034
$605$	−14753.7	−0.991445
$606$	0	0
$607$	−5202.79	−0.347899	−0.173950	−	0.984755i	$-0.555653\pi$
$607$	−5202.79	−0.347899	−0.173950	+	0.984755i	$0.555653\pi$
$608$	3598.74	0.240046
$609$	0	0
$610$	11092.3	0.736251
$611$	−5971.20	−0.395367
$612$	0	0
$613$	−15697.6	−1.03429	−0.517144	−	0.855898i	$-0.673005\pi$
$613$	−15697.6	−1.03429	−0.517144	+	0.855898i	$0.673005\pi$
$614$	−14112.4	−0.927571
$615$	0	0
$616$	−1209.82	−0.0791313
$617$	26268.4	1.71398	0.856991	−	0.515331i	$-0.172331\pi$
$617$	26268.4	1.71398	0.856991	+	0.515331i	$0.172331\pi$
$618$	0	0
$619$	1486.34	0.0965124	0.0482562	−	0.998835i	$-0.484634\pi$
$619$	1486.34	0.0965124	0.0482562	+	0.998835i	$0.484634\pi$
$620$	−1434.40	−0.0929142
$621$	0	0
$622$	14736.5	0.949969
$623$	14774.9	0.950149
$624$	0	0
$625$	−19376.2	−1.24008
$626$	18136.9	1.15798
$627$	0	0
$628$	−2938.25	−0.186702
$629$	−2951.69	−0.187109
$630$	0	0
$631$	9554.18	0.602767	0.301384	−	0.953503i	$-0.402552\pi$
$631$	9554.18	0.602767	0.301384	+	0.953503i	$0.402552\pi$
$632$	10570.7	0.665318
$633$	0	0
$634$	−3904.55	−0.244589
$635$	−21691.9	−1.35562
$636$	0	0
$637$	5521.56	0.343441
$638$	−3075.07	−0.190820
$639$	0	0
$640$	1693.52	0.104597
$641$	−28526.3	−1.75776	−0.878879	−	0.477045i	$-0.841708\pi$
$641$	−28526.3	−1.75776	−0.878879	+	0.477045i	$0.841708\pi$
$642$	0	0
$643$	−21360.4	−1.31006	−0.655032	−	0.755601i	$-0.727345\pi$
$643$	−21360.4	−1.31006	−0.655032	+	0.755601i	$0.727345\pi$
$644$	−946.922	−0.0579409
$645$	0	0
$646$	26820.7	1.63351
$647$	−17343.0	−1.05382	−0.526910	−	0.849921i	$-0.676650\pi$
$647$	−17343.0	−1.05382	−0.526910	+	0.849921i	$0.676650\pi$
$648$	0	0
$649$	−6567.48	−0.397221
$650$	−2331.42	−0.140686
$651$	0	0
$652$	−10951.2	−0.657796
$653$	6044.46	0.362233	0.181116	−	0.983462i	$-0.442029\pi$
$653$	6044.46	0.362233	0.181116	+	0.983462i	$0.442029\pi$
$654$	0	0
$655$	8332.85	0.497086
$656$	−2784.98	−0.165755
$657$	0	0
$658$	−5277.38	−0.312665
$659$	−29374.0	−1.73634	−0.868172	−	0.496264i	$-0.834705\pi$
$659$	−29374.0	−1.73634	−0.868172	+	0.496264i	$0.834705\pi$
$660$	0	0
$661$	16562.1	0.974572	0.487286	−	0.873242i	$-0.337987\pi$
$661$	16562.1	0.974572	0.487286	+	0.873242i	$0.337987\pi$
$662$	23016.4	1.35130
$663$	0	0
$664$	−3259.18	−0.190483
$665$	−15314.6	−0.893044
$666$	0	0
$667$	−2406.86	−0.139721
$668$	−6811.90	−0.394551
$669$	0	0
$670$	−1862.01	−0.107367
$671$	6159.06	0.354349
$672$	0	0
$673$	−1304.30	−0.0747059	−0.0373529	−	0.999302i	$-0.511893\pi$
$673$	−1304.30	−0.0747059	−0.0373529	+	0.999302i	$0.511893\pi$
$674$	−850.985	−0.0486331
$675$	0	0
$676$	−6618.00	−0.376536
$677$	8890.43	0.504708	0.252354	−	0.967635i	$-0.418795\pi$
$677$	8890.43	0.504708	0.252354	+	0.967635i	$0.418795\pi$
$678$	0	0
$679$	10144.2	0.573343
$680$	12621.5	0.711781
$681$	0	0
$682$	−796.460	−0.0447185
$683$	−16635.8	−0.931991	−0.465995	−	0.884787i	$-0.654304\pi$
$683$	−16635.8	−0.931991	−0.465995	+	0.884787i	$0.654304\pi$
$684$	0	0
$685$	15095.7	0.842008
$686$	11940.7	0.664576
$687$	0	0
$688$	8099.33	0.448814
$689$	−4668.86	−0.258156
$690$	0	0
$691$	22690.2	1.24917	0.624585	−	0.780957i	$-0.285268\pi$
$691$	22690.2	1.24917	0.624585	+	0.780957i	$0.285268\pi$
$692$	−557.448	−0.0306228
$693$	0	0
$694$	2877.88	0.157411
$695$	−1803.26	−0.0984196
$696$	0	0
$697$	−20756.0	−1.12796
$698$	−1362.06	−0.0738608
$699$	0	0
$700$	−2060.52	−0.111258
$701$	32623.6	1.75774	0.878872	−	0.477059i	$-0.158297\pi$
$701$	32623.6	1.75774	0.878872	+	0.477059i	$0.158297\pi$
$702$	0	0
$703$	−2783.75	−0.149347
$704$	940.337	0.0503413
$705$	0	0
$706$	−13216.1	−0.704526
$707$	11402.1	0.606533
$708$	0	0
$709$	−26739.1	−1.41637	−0.708187	−	0.706025i	$-0.750486\pi$
$709$	−26739.1	−1.41637	−0.708187	+	0.706025i	$0.750486\pi$
$710$	5047.29	0.266791
$711$	0	0
$712$	−11483.8	−0.604460
$713$	−623.388	−0.0327434
$714$	0	0
$715$	−4527.75	−0.236823
$716$	3929.67	0.205110
$717$	0	0
$718$	−5603.36	−0.291248
$719$	−20628.0	−1.06995	−0.534976	−	0.844867i	$-0.679679\pi$
$719$	−20628.0	−1.06995	−0.534976	+	0.844867i	$0.679679\pi$
$720$	0	0
$721$	−2317.89	−0.119726
$722$	11576.7	0.596733
$723$	0	0
$724$	−14375.9	−0.737948
$725$	−5237.36	−0.268291
$726$	0	0
$727$	−35027.0	−1.78691	−0.893453	−	0.449158i	$-0.851724\pi$
$727$	−35027.0	−1.78691	−0.893453	+	0.449158i	$0.851724\pi$
$728$	1917.86	0.0976380
$729$	0	0
$730$	12710.3	0.644422
$731$	60362.9	3.05417
$732$	0	0
$733$	−17949.2	−0.904460	−0.452230	−	0.891901i	$-0.649371\pi$
$733$	−17949.2	−0.904460	−0.452230	+	0.891901i	$0.649371\pi$
$734$	17285.6	0.869240
$735$	0	0
$736$	736.000	0.0368605
$737$	−1033.89	−0.0516742
$738$	0	0
$739$	−9531.78	−0.474469	−0.237234	−	0.971452i	$-0.576241\pi$
$739$	−9531.78	−0.474469	−0.237234	+	0.971452i	$0.576241\pi$
$740$	−1309.99	−0.0650761
$741$	0	0
$742$	−4126.36	−0.204156
$743$	12341.6	0.609378	0.304689	−	0.952452i	$-0.401448\pi$
$743$	12341.6	0.609378	0.304689	+	0.952452i	$0.401448\pi$
$744$	0	0
$745$	27613.7	1.35797
$746$	−19847.6	−0.974092
$747$	0	0
$748$	7008.16	0.342572
$749$	10503.5	0.512404
$750$	0	0
$751$	20118.7	0.977553	0.488777	−	0.872409i	$-0.337443\pi$
$751$	20118.7	0.977553	0.488777	+	0.872409i	$0.337443\pi$
$752$	4101.87	0.198909
$753$	0	0
$754$	4874.75	0.235448
$755$	25729.6	1.24026
$756$	0	0
$757$	40167.2	1.92854	0.964269	−	0.264926i	$-0.0853476\pi$
$757$	40167.2	1.92854	0.964269	+	0.264926i	$0.0853476\pi$
$758$	10060.0	0.482054
$759$	0	0
$760$	11903.3	0.568131
$761$	−21242.6	−1.01188	−0.505941	−	0.862568i	$-0.668855\pi$
$761$	−21242.6	−1.01188	−0.505941	+	0.862568i	$0.668855\pi$
$762$	0	0
$763$	7266.92	0.344797
$764$	206.562	0.00978161
$765$	0	0
$766$	4988.62	0.235308
$767$	10411.1	0.490120
$768$	0	0
$769$	−40261.8	−1.88801	−0.944003	−	0.329937i	$-0.892973\pi$
$769$	−40261.8	−1.88801	−0.944003	+	0.329937i	$0.892973\pi$
$770$	−4001.65	−0.187285
$771$	0	0
$772$	4705.77	0.219384
$773$	−3861.68	−0.179683	−0.0898415	−	0.995956i	$-0.528636\pi$
$773$	−3861.68	−0.179683	−0.0898415	+	0.995956i	$0.528636\pi$
$774$	0	0
$775$	−1356.50	−0.0628736
$776$	−7884.66	−0.364746
$777$	0	0
$778$	−13896.9	−0.640398
$779$	−19575.0	−0.900318
$780$	0	0
$781$	2802.55	0.128403
$782$	5485.28	0.250835
$783$	0	0
$784$	−3792.99	−0.172786
$785$	−9718.71	−0.441880
$786$	0	0
$787$	40478.2	1.83341	0.916704	−	0.399568i	$-0.130840\pi$
$787$	40478.2	1.83341	0.916704	+	0.399568i	$0.130840\pi$
$788$	4370.00	0.197557
$789$	0	0
$790$	34964.2	1.57465
$791$	16875.5	0.758563
$792$	0	0
$793$	−9763.62	−0.437221
$794$	15843.7	0.708151
$795$	0	0
$796$	621.517	0.0276747
$797$	−7456.91	−0.331415	−0.165707	−	0.986175i	$-0.552991\pi$
$797$	−7456.91	−0.331415	−0.165707	+	0.986175i	$0.552991\pi$
$798$	0	0
$799$	30570.5	1.35358
$800$	1601.55	0.0707792
$801$	0	0
$802$	−21804.0	−0.960005
$803$	7057.47	0.310153
$804$	0	0
$805$	−3132.08	−0.137132
$806$	1262.58	0.0551770
$807$	0	0
$808$	−8862.32	−0.385861
$809$	4767.72	0.207199	0.103600	−	0.994619i	$-0.466964\pi$
$809$	4767.72	0.207199	0.103600	+	0.994619i	$0.466964\pi$
$810$	0	0
$811$	19199.4	0.831298	0.415649	−	0.909525i	$-0.363554\pi$
$811$	19199.4	0.831298	0.415649	+	0.909525i	$0.363554\pi$
$812$	4308.33	0.186198
$813$	0	0
$814$	−727.384	−0.0313204
$815$	−36222.8	−1.55685
$816$	0	0
$817$	56928.4	2.43779
$818$	−6140.31	−0.262458
$819$	0	0
$820$	−9211.73	−0.392302
$821$	−1595.59	−0.0678277	−0.0339138	−	0.999425i	$-0.510797\pi$
$821$	−1595.59	−0.0678277	−0.0339138	+	0.999425i	$0.510797\pi$
$822$	0	0
$823$	2041.80	0.0864795	0.0432398	−	0.999065i	$-0.486232\pi$
$823$	2041.80	0.0864795	0.0432398	+	0.999065i	$0.486232\pi$
$824$	1801.59	0.0761668
$825$	0	0
$826$	9201.35	0.387598
$827$	−13117.5	−0.551559	−0.275780	−	0.961221i	$-0.588936\pi$
$827$	−13117.5	−0.551559	−0.275780	+	0.961221i	$0.588936\pi$
$828$	0	0
$829$	20788.1	0.870930	0.435465	−	0.900206i	$-0.356584\pi$
$829$	20788.1	0.870930	0.435465	+	0.900206i	$0.356584\pi$
$830$	−10780.2	−0.450828
$831$	0	0
$832$	−1490.66	−0.0621148
$833$	−28268.5	−1.17580
$834$	0	0
$835$	−22531.4	−0.933808
$836$	6609.42	0.273435
$837$	0	0
$838$	−19364.3	−0.798243
$839$	27202.7	1.11936	0.559680	−	0.828709i	$-0.310924\pi$
$839$	27202.7	1.11936	0.559680	+	0.828709i	$0.310924\pi$
$840$	0	0
$841$	−13438.2	−0.550996
$842$	14962.7	0.612411
$843$	0	0
$844$	−21463.8	−0.875374
$845$	−21890.0	−0.891171
$846$	0	0
$847$	11477.5	0.465612
$848$	3207.24	0.129878
$849$	0	0
$850$	11936.1	0.481651
$851$	−569.322	−0.0229331
$852$	0	0
$853$	21562.1	0.865501	0.432750	−	0.901514i	$-0.357543\pi$
$853$	21562.1	0.865501	0.432750	+	0.901514i	$0.357543\pi$
$854$	−8629.14	−0.345765
$855$	0	0
$856$	−8163.92	−0.325978
$857$	−15694.8	−0.625581	−0.312790	−	0.949822i	$-0.601264\pi$
$857$	−15694.8	−0.625581	−0.312790	+	0.949822i	$0.601264\pi$
$858$	0	0
$859$	12667.3	0.503144	0.251572	−	0.967839i	$-0.419052\pi$
$859$	12667.3	0.503144	0.251572	+	0.967839i	$0.419052\pi$
$860$	26789.7	1.06224
$861$	0	0
$862$	33026.9	1.30499
$863$	5118.43	0.201893	0.100946	−	0.994892i	$-0.467813\pi$
$863$	5118.43	0.201893	0.100946	+	0.994892i	$0.467813\pi$
$864$	0	0
$865$	−1843.84	−0.0724768
$866$	23531.5	0.923364
$867$	0	0
$868$	1115.88	0.0436352
$869$	19414.1	0.757859
$870$	0	0
$871$	1638.97	0.0637595
$872$	−5648.25	−0.219351
$873$	0	0
$874$	5173.18	0.200212
$875$	10206.7	0.394343
$876$	0	0
$877$	46908.3	1.80614	0.903068	−	0.429498i	$-0.141309\pi$
$877$	46908.3	1.80614	0.903068	+	0.429498i	$0.141309\pi$
$878$	33429.9	1.28497
$879$	0	0
$880$	3110.30	0.119146
$881$	8248.69	0.315443	0.157722	−	0.987484i	$-0.449585\pi$
$881$	8248.69	0.315443	0.157722	+	0.987484i	$0.449585\pi$
$882$	0	0
$883$	12437.1	0.473999	0.237000	−	0.971510i	$-0.423836\pi$
$883$	12437.1	0.473999	0.237000	+	0.971510i	$0.423836\pi$
$884$	−11109.7	−0.422690
$885$	0	0
$886$	25374.2	0.962148
$887$	−43105.2	−1.63172	−0.815858	−	0.578253i	$-0.803735\pi$
$887$	−43105.2	−1.63172	−0.815858	+	0.578253i	$0.803735\pi$
$888$	0	0
$889$	16875.0	0.636637
$890$	−37984.5	−1.43061
$891$	0	0
$892$	1784.88	0.0669982
$893$	28831.1	1.08040
$894$	0	0
$895$	12997.9	0.485445
$896$	−1317.46	−0.0491218
$897$	0	0
$898$	−32155.4	−1.19492
$899$	2836.31	0.105224
$900$	0	0
$901$	23903.0	0.883821
$902$	−5114.88	−0.188810
$903$	0	0
$904$	−13116.6	−0.482578
$905$	−47550.2	−1.74655
$906$	0	0
$907$	−9236.78	−0.338150	−0.169075	−	0.985603i	$-0.554078\pi$
$907$	−9236.78	−0.338150	−0.169075	+	0.985603i	$0.554078\pi$
$908$	−9055.74	−0.330975
$909$	0	0
$910$	6343.59	0.231086
$911$	−34284.3	−1.24686	−0.623431	−	0.781879i	$-0.714262\pi$
$911$	−34284.3	−1.24686	−0.623431	+	0.781879i	$0.714262\pi$
$912$	0	0
$913$	−5985.80	−0.216978
$914$	15860.4	0.573977
$915$	0	0
$916$	6890.82	0.248558
$917$	−6482.47	−0.233446
$918$	0	0
$919$	−47474.8	−1.70408	−0.852039	−	0.523478i	$-0.824634\pi$
$919$	−47474.8	−1.70408	−0.852039	+	0.523478i	$0.824634\pi$
$920$	2434.43	0.0872399
$921$	0	0
$922$	23060.0	0.823689
$923$	−4442.72	−0.158433
$924$	0	0
$925$	−1238.86	−0.0440360
$926$	−15180.8	−0.538740
$927$	0	0
$928$	−3348.67	−0.118454
$929$	10436.4	0.368577	0.184288	−	0.982872i	$-0.441002\pi$
$929$	10436.4	0.368577	0.184288	+	0.982872i	$0.441002\pi$
$930$	0	0
$931$	−26660.1	−0.938506
$932$	−16489.1	−0.579527
$933$	0	0
$934$	13372.5	0.468482
$935$	23180.5	0.810785
$936$	0	0
$937$	24205.7	0.843935	0.421967	−	0.906611i	$-0.361340\pi$
$937$	24205.7	0.843935	0.421967	+	0.906611i	$0.361340\pi$
$938$	1448.53	0.0504224
$939$	0	0
$940$	13567.5	0.470771
$941$	−50113.7	−1.73609	−0.868045	−	0.496485i	$-0.834624\pi$
$941$	−50113.7	−1.73609	−0.868045	+	0.496485i	$0.834624\pi$
$942$	0	0
$943$	−4003.41	−0.138249
$944$	−7151.80	−0.246580
$945$	0	0
$946$	14875.2	0.511241
$947$	30633.5	1.05117	0.525583	−	0.850742i	$-0.323847\pi$
$947$	30633.5	1.05117	0.525583	+	0.850742i	$0.323847\pi$
$948$	0	0
$949$	−11187.8	−0.382689
$950$	11256.9	0.384446
$951$	0	0
$952$	−9818.77	−0.334273
$953$	15558.4	0.528842	0.264421	−	0.964407i	$-0.414819\pi$
$953$	15558.4	0.528842	0.264421	+	0.964407i	$0.414819\pi$
$954$	0	0
$955$	683.234	0.0231507
$956$	4798.97	0.162353
$957$	0	0
$958$	24144.0	0.814256
$959$	−11743.5	−0.395432
$960$	0	0
$961$	−29056.4	−0.975341
$962$	1153.08	0.0386454
$963$	0	0
$964$	29049.5	0.970561
$965$	15565.0	0.519229
$966$	0	0
$967$	6567.58	0.218406	0.109203	−	0.994019i	$-0.465170\pi$
$967$	6567.58	0.218406	0.109203	+	0.994019i	$0.465170\pi$
$968$	−8920.98	−0.296210
$969$	0	0
$970$	−26079.7	−0.863266
$971$	41067.3	1.35727	0.678636	−	0.734474i	$-0.262571\pi$
$971$	41067.3	1.35727	0.678636	+	0.734474i	$0.262571\pi$
$972$	0	0
$973$	1402.83	0.0462207
$974$	−2310.63	−0.0760139
$975$	0	0
$976$	6707.05	0.219966
$977$	55639.7	1.82198	0.910989	−	0.412430i	$-0.135320\pi$
$977$	55639.7	1.82198	0.910989	+	0.412430i	$0.135320\pi$
$978$	0	0
$979$	−21091.2	−0.688536
$980$	−12545.9	−0.408942
$981$	0	0
$982$	14503.9	0.471320
$983$	−60200.2	−1.95329	−0.976646	−	0.214854i	$-0.931073\pi$
$983$	−60200.2	−1.95329	−0.976646	+	0.214854i	$0.931073\pi$
$984$	0	0
$985$	14454.4	0.467569
$986$	−24957.0	−0.806079
$987$	0	0
$988$	−10477.6	−0.337384
$989$	11642.8	0.374337
$990$	0	0
$991$	−12965.4	−0.415599	−0.207800	−	0.978171i	$-0.566630\pi$
$991$	−12965.4	−0.415599	−0.207800	+	0.978171i	$0.566630\pi$
$992$	−867.323	−0.0277596
$993$	0	0
$994$	−3926.50	−0.125293
$995$	2055.76	0.0654995
$996$	0	0
$997$	−5425.12	−0.172332	−0.0861661	−	0.996281i	$-0.527462\pi$
$997$	−5425.12	−0.172332	−0.0861661	+	0.996281i	$0.527462\pi$
$998$	22875.9	0.725574
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.4.a.n.1.4	yes	4
3.2	odd	2		414.4.a.m.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.4.a.m.1.1	✓	4	3.2	odd	2
414.4.a.n.1.4	yes	4	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +13.2306 q^{5} -10.2926 q^{7} +8.00000 q^{8} +26.4612 q^{10} +14.6928 q^{11} -23.2916 q^{13} -20.5853 q^{14} +16.0000 q^{16} +119.245 q^{17} +112.461 q^{19} +52.9223 q^{20} +29.3855 q^{22} +23.0000 q^{23} +50.0484 q^{25} -46.5833 q^{26} -41.1705 q^{28} -104.646 q^{29} -27.1038 q^{31} +32.0000 q^{32} +238.490 q^{34} -136.177 q^{35} -24.7531 q^{37} +224.921 q^{38} +105.845 q^{40} -174.061 q^{41} +506.208 q^{43} +58.7710 q^{44} +46.0000 q^{46} +256.367 q^{47} -237.062 q^{49} +100.097 q^{50} -93.1665 q^{52} +200.452 q^{53} +194.394 q^{55} -82.3410 q^{56} -209.292 q^{58} -446.988 q^{59} +419.190 q^{61} -54.2077 q^{62} +64.0000 q^{64} -308.162 q^{65} -70.3675 q^{67} +476.981 q^{68} -272.355 q^{70} +190.743 q^{71} +480.337 q^{73} -49.5063 q^{74} +449.842 q^{76} -151.227 q^{77} +1321.34 q^{79} +211.689 q^{80} -348.123 q^{82} -407.398 q^{83} +1577.68 q^{85} +1012.42 q^{86} +117.542 q^{88} -1435.48 q^{89} +239.732 q^{91} +92.0000 q^{92} +512.734 q^{94} +1487.92 q^{95} -985.582 q^{97} -474.124 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 16 q^{4} + 18 q^{7} + 32 q^{8} + 42 q^{11} + 108 q^{13} + 36 q^{14} + 64 q^{16} + 26 q^{17} + 132 q^{19} + 84 q^{22} + 92 q^{23} + 260 q^{25} + 216 q^{26} + 72 q^{28} + 252 q^{29} + 428 q^{31}+ \cdots + 2152 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 + 16 * q^4 + 18 * q^7 + 32 * q^8 + 42 * q^11 + 108 * q^13 + 36 * q^14 + 64 * q^16 + 26 * q^17 + 132 * q^19 + 84 * q^22 + 92 * q^23 + 260 * q^25 + 216 * q^26 + 72 * q^28 + 252 * q^29 + 428 * q^31 + 128 * q^32 + 52 * q^34 + 348 * q^35 + 278 * q^37 + 264 * q^38 - 4 * q^41 + 216 * q^43 + 168 * q^44 + 184 * q^46 - 4 * q^47 + 1076 * q^49 + 520 * q^50 + 432 * q^52 - 460 * q^53 + 512 * q^55 + 144 * q^56 + 504 * q^58 - 504 * q^59 + 1086 * q^61 + 856 * q^62 + 256 * q^64 - 1420 * q^65 + 1484 * q^67 + 104 * q^68 + 696 * q^70 - 876 * q^71 + 1072 * q^73 + 556 * q^74 + 528 * q^76 - 1580 * q^77 + 1634 * q^79 - 8 * q^82 - 2246 * q^83 + 872 * q^85 + 432 * q^86 + 336 * q^88 - 474 * q^89 + 40 * q^91 + 368 * q^92 - 8 * q^94 - 2780 * q^95 + 1044 * q^97 + 2152 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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