LMFDB - Embedded newform 1305.4.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,4,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1305.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1305.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:	$76.9974925575$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 49x^{4} + 27x^{3} + 692x^{2} - 82x - 2588 $ x^6 - x^5 - 49x^4 + 27x^3 + 692x^2 - 82x - 2588
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.42717$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.42717 q^{2} +21.4542 q^{4} +5.00000 q^{5} -10.2400 q^{7} -73.0180 q^{8} -27.1358 q^{10} -12.6318 q^{11} +84.2342 q^{13} +55.5740 q^{14} +224.647 q^{16} -6.89181 q^{17} -79.7045 q^{19} +107.271 q^{20} +68.5548 q^{22} +73.8181 q^{23} +25.0000 q^{25} -457.153 q^{26} -219.690 q^{28} -29.0000 q^{29} +0.254654 q^{31} -635.056 q^{32} +37.4030 q^{34} -51.1998 q^{35} +40.4065 q^{37} +432.570 q^{38} -365.090 q^{40} -208.354 q^{41} -194.427 q^{43} -271.004 q^{44} -400.623 q^{46} +522.168 q^{47} -238.143 q^{49} -135.679 q^{50} +1807.17 q^{52} +294.893 q^{53} -63.1589 q^{55} +747.701 q^{56} +157.388 q^{58} +196.229 q^{59} +189.313 q^{61} -1.38205 q^{62} +1649.38 q^{64} +421.171 q^{65} -570.840 q^{67} -147.858 q^{68} +277.870 q^{70} +1126.87 q^{71} +581.431 q^{73} -219.293 q^{74} -1709.99 q^{76} +129.349 q^{77} -255.631 q^{79} +1123.24 q^{80} +1130.77 q^{82} +983.081 q^{83} -34.4591 q^{85} +1055.19 q^{86} +922.347 q^{88} -678.642 q^{89} -862.555 q^{91} +1583.70 q^{92} -2833.89 q^{94} -398.523 q^{95} -30.8442 q^{97} +1292.44 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} + 51 q^{4} + 30 q^{5} + 47 q^{7} - 51 q^{8} - 5 q^{10} - 81 q^{11} + 169 q^{13} + 30 q^{14} + 131 q^{16} + q^{17} + 116 q^{19} + 255 q^{20} + 90 q^{22} + 52 q^{23} + 150 q^{25} - 294 q^{26}+ \cdots + 1433 q^{98}+O(q^{100}) $ 6 * q - q^2 + 51 * q^4 + 30 * q^5 + 47 * q^7 - 51 * q^8 - 5 * q^10 - 81 * q^11 + 169 * q^13 + 30 * q^14 + 131 * q^16 + q^17 + 116 * q^19 + 255 * q^20 + 90 * q^22 + 52 * q^23 + 150 * q^25 - 294 * q^26 + 344 * q^28 - 174 * q^29 + 340 * q^31 - 499 * q^32 + 920 * q^34 + 235 * q^35 + 332 * q^37 + 378 * q^38 - 255 * q^40 + 616 * q^41 + 334 * q^43 + 52 * q^44 - 158 * q^46 + 85 * q^47 + 879 * q^49 - 25 * q^50 + 2220 * q^52 + 850 * q^53 - 405 * q^55 + 624 * q^56 + 29 * q^58 + 758 * q^59 - 36 * q^61 + 152 * q^62 + 1795 * q^64 + 845 * q^65 + 939 * q^67 + 186 * q^68 + 150 * q^70 + 1388 * q^71 + 1708 * q^73 + 814 * q^74 + 566 * q^76 - 2585 * q^77 + 1250 * q^79 + 655 * q^80 + 1372 * q^82 + 748 * q^83 + 5 * q^85 + 800 * q^86 + 536 * q^88 - 1099 * q^89 + 539 * q^91 - 1698 * q^92 - 4542 * q^94 + 580 * q^95 - 22 * q^97 + 1433 * 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 $

$2$	−5.42717	−1.91879	−0.959397	−	0.282060i	$-0.908982\pi$
$2$	−5.42717	−1.91879	−0.959397	+	0.282060i	$0.908982\pi$
$3$	0	0
$3$	0	0
$4$	21.4542	2.68177
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−10.2400	−0.552906	−0.276453	−	0.961027i	$-0.589159\pi$
$7$	−10.2400	−0.552906	−0.276453	+	0.961027i	$0.589159\pi$
$8$	−73.0180	−3.22697
$9$	0	0
$10$	−27.1358	−0.858111
$11$	−12.6318	−0.346239	−0.173119	−	0.984901i	$-0.555385\pi$
$11$	−12.6318	−0.346239	−0.173119	+	0.984901i	$0.555385\pi$
$12$	0	0
$13$	84.2342	1.79710	0.898552	−	0.438866i	$-0.144620\pi$
$13$	84.2342	1.79710	0.898552	+	0.438866i	$0.144620\pi$
$14$	55.5740	1.06091
$15$	0	0
$16$	224.647	3.51012
$17$	−6.89181	−0.0983241	−0.0491621	−	0.998791i	$-0.515655\pi$
$17$	−6.89181	−0.0983241	−0.0491621	+	0.998791i	$0.515655\pi$
$18$	0	0
$19$	−79.7045	−0.962393	−0.481196	−	0.876613i	$-0.659798\pi$
$19$	−79.7045	−0.962393	−0.481196	+	0.876613i	$0.659798\pi$
$20$	107.271	1.19932
$21$	0	0
$22$	68.5548	0.664361
$23$	73.8181	0.669223	0.334612	−	0.942356i	$-0.391395\pi$
$23$	73.8181	0.669223	0.334612	+	0.942356i	$0.391395\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−457.153	−3.44827
$27$	0	0
$28$	−219.690	−1.48277
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	0.254654	0.00147540	0.000737698	−	1.00000i	$-0.499765\pi$
$31$	0.254654	0.00147540	0.000737698		1.00000i	$0.499765\pi$
$32$	−635.056	−3.50822
$33$	0	0
$34$	37.4030	0.188664
$35$	−51.1998	−0.247267
$36$	0	0
$37$	40.4065	0.179535	0.0897674	−	0.995963i	$-0.471388\pi$
$37$	40.4065	0.179535	0.0897674	+	0.995963i	$0.471388\pi$
$38$	432.570	1.84663
$39$	0	0
$40$	−365.090	−1.44314
$41$	−208.354	−0.793645	−0.396822	−	0.917895i	$-0.629887\pi$
$41$	−208.354	−0.793645	−0.396822	+	0.917895i	$0.629887\pi$
$42$	0	0
$43$	−194.427	−0.689531	−0.344766	−	0.938689i	$-0.612042\pi$
$43$	−194.427	−0.689531	−0.344766	+	0.938689i	$0.612042\pi$
$44$	−271.004	−0.928532
$45$	0	0
$46$	−400.623	−1.28410
$47$	522.168	1.62055	0.810277	−	0.586047i	$-0.199317\pi$
$47$	522.168	1.62055	0.810277	+	0.586047i	$0.199317\pi$
$48$	0	0
$49$	−238.143	−0.694295
$50$	−135.679	−0.383759
$51$	0	0
$52$	1807.17	4.81942
$53$	294.893	0.764276	0.382138	−	0.924105i	$-0.375188\pi$
$53$	294.893	0.764276	0.382138	+	0.924105i	$0.375188\pi$
$54$	0	0
$55$	−63.1589	−0.154843
$56$	747.701	1.78421
$57$	0	0
$58$	157.388	0.356311
$59$	196.229	0.432997	0.216498	−	0.976283i	$-0.430536\pi$
$59$	196.229	0.432997	0.216498	+	0.976283i	$0.430536\pi$
$60$	0	0
$61$	189.313	0.397360	0.198680	−	0.980064i	$-0.436334\pi$
$61$	189.313	0.397360	0.198680	+	0.980064i	$0.436334\pi$
$62$	−1.38205	−0.00283098
$63$	0	0
$64$	1649.38	3.22144
$65$	421.171	0.803690
$66$	0	0
$67$	−570.840	−1.04088	−0.520442	−	0.853897i	$-0.674233\pi$
$67$	−570.840	−1.04088	−0.520442	+	0.853897i	$0.674233\pi$
$68$	−147.858	−0.263683
$69$	0	0
$70$	277.870	0.474454
$71$	1126.87	1.88358	0.941792	−	0.336196i	$-0.109141\pi$
$71$	1126.87	1.88358	0.941792	+	0.336196i	$0.109141\pi$
$72$	0	0
$73$	581.431	0.932210	0.466105	−	0.884729i	$-0.345657\pi$
$73$	581.431	0.932210	0.466105	+	0.884729i	$0.345657\pi$
$74$	−219.293	−0.344490
$75$	0	0
$76$	−1709.99	−2.58092
$77$	129.349	0.191437
$78$	0	0
$79$	−255.631	−0.364060	−0.182030	−	0.983293i	$-0.558267\pi$
$79$	−255.631	−0.364060	−0.182030	+	0.983293i	$0.558267\pi$
$80$	1123.24	1.56977
$81$	0	0
$82$	1130.77	1.52284
$83$	983.081	1.30009	0.650043	−	0.759897i	$-0.274751\pi$
$83$	983.081	1.30009	0.650043	+	0.759897i	$0.274751\pi$
$84$	0	0
$85$	−34.4591	−0.0439719
$86$	1055.19	1.32307
$87$	0	0
$88$	922.347	1.11730
$89$	−678.642	−0.808268	−0.404134	−	0.914700i	$-0.632427\pi$
$89$	−678.642	−0.808268	−0.404134	+	0.914700i	$0.632427\pi$
$90$	0	0
$91$	−862.555	−0.993630
$92$	1583.70	1.79470
$93$	0	0
$94$	−2833.89	−3.10951
$95$	−398.523	−0.430395
$96$	0	0
$97$	−30.8442	−0.0322861	−0.0161431	−	0.999870i	$-0.505139\pi$
$97$	−30.8442	−0.0322861	−0.0161431	+	0.999870i	$0.505139\pi$
$98$	1292.44	1.33221
$99$	0	0
$100$	536.354	0.536354
$101$	−1749.78	−1.72386	−0.861930	−	0.507028i	$-0.830744\pi$
$101$	−1749.78	−1.72386	−0.861930	+	0.507028i	$0.830744\pi$
$102$	0	0
$103$	1912.26	1.82933	0.914663	−	0.404217i	$-0.132456\pi$
$103$	1912.26	1.82933	0.914663	+	0.404217i	$0.132456\pi$
$104$	−6150.61	−5.79920
$105$	0	0
$106$	−1600.43	−1.46649
$107$	−1719.25	−1.55333	−0.776663	−	0.629916i	$-0.783089\pi$
$107$	−1719.25	−1.55333	−0.776663	+	0.629916i	$0.783089\pi$
$108$	0	0
$109$	−779.425	−0.684911	−0.342456	−	0.939534i	$-0.611259\pi$
$109$	−779.425	−0.684911	−0.342456	+	0.939534i	$0.611259\pi$
$110$	342.774	0.297111
$111$	0	0
$112$	−2300.38	−1.94076
$113$	−766.529	−0.638132	−0.319066	−	0.947732i	$-0.603369\pi$
$113$	−766.529	−0.638132	−0.319066	+	0.947732i	$0.603369\pi$
$114$	0	0
$115$	369.090	0.299286
$116$	−622.170	−0.497992
$117$	0	0
$118$	−1064.97	−0.830831
$119$	70.5719	0.0543640
$120$	0	0
$121$	−1171.44	−0.880119
$122$	−1027.43	−0.762453
$123$	0	0
$124$	5.46339	0.00395667
$125$	125.000	0.0894427
$126$	0	0
$127$	−1363.54	−0.952715	−0.476358	−	0.879252i	$-0.658043\pi$
$127$	−1363.54	−0.952715	−0.476358	+	0.879252i	$0.658043\pi$
$128$	−3870.99	−2.67305
$129$	0	0
$130$	−2285.77	−1.54211
$131$	1346.83	0.898269	0.449134	−	0.893464i	$-0.351732\pi$
$131$	1346.83	0.898269	0.449134	+	0.893464i	$0.351732\pi$
$132$	0	0
$133$	816.171	0.532113
$134$	3098.05	1.99724
$135$	0	0
$136$	503.226	0.317289
$137$	2508.42	1.56430	0.782148	−	0.623093i	$-0.214124\pi$
$137$	2508.42	1.56430	0.782148	+	0.623093i	$0.214124\pi$
$138$	0	0
$139$	1363.95	0.832292	0.416146	−	0.909298i	$-0.363380\pi$
$139$	1363.95	0.832292	0.416146	+	0.909298i	$0.363380\pi$
$140$	−1098.45	−0.663113
$141$	0	0
$142$	−6115.69	−3.61421
$143$	−1064.03	−0.622227
$144$	0	0
$145$	−145.000	−0.0830455
$146$	−3155.52	−1.78872
$147$	0	0
$148$	866.887	0.481471
$149$	−2792.25	−1.53523	−0.767617	−	0.640909i	$-0.778558\pi$
$149$	−2792.25	−1.53523	−0.767617	+	0.640909i	$0.778558\pi$
$150$	0	0
$151$	306.493	0.165179	0.0825897	−	0.996584i	$-0.473681\pi$
$151$	306.493	0.165179	0.0825897	+	0.996584i	$0.473681\pi$
$152$	5819.86	3.10561
$153$	0	0
$154$	−701.999	−0.367329
$155$	1.27327	0.000659817 0
$156$	0	0
$157$	−1828.15	−0.929315	−0.464657	−	0.885491i	$-0.653822\pi$
$157$	−1828.15	−0.929315	−0.464657	+	0.885491i	$0.653822\pi$
$158$	1387.35	0.698556
$159$	0	0
$160$	−3175.28	−1.56892
$161$	−755.894	−0.370018
$162$	0	0
$163$	2101.15	1.00966	0.504830	−	0.863219i	$-0.331555\pi$
$163$	2101.15	1.00966	0.504830	+	0.863219i	$0.331555\pi$
$164$	−4470.06	−2.12837
$165$	0	0
$166$	−5335.35	−2.49460
$167$	−1564.53	−0.724950	−0.362475	−	0.931994i	$-0.618068\pi$
$167$	−1564.53	−0.724950	−0.362475	+	0.931994i	$0.618068\pi$
$168$	0	0
$169$	4898.40	2.22959
$170$	187.015	0.0843730
$171$	0	0
$172$	−4171.27	−1.84916
$173$	2695.23	1.18448	0.592239	−	0.805762i	$-0.298244\pi$
$173$	2695.23	1.18448	0.592239	+	0.805762i	$0.298244\pi$
$174$	0	0
$175$	−255.999	−0.110581
$176$	−2837.70	−1.21534
$177$	0	0
$178$	3683.10	1.55090
$179$	3604.16	1.50496	0.752480	−	0.658615i	$-0.228857\pi$
$179$	3604.16	1.50496	0.752480	+	0.658615i	$0.228857\pi$
$180$	0	0
$181$	−2658.32	−1.09166	−0.545832	−	0.837894i	$-0.683786\pi$
$181$	−2658.32	−1.09166	−0.545832	+	0.837894i	$0.683786\pi$
$182$	4681.23	1.90657
$183$	0	0
$184$	−5390.05	−2.15956
$185$	202.033	0.0802904
$186$	0	0
$187$	87.0559	0.0340436
$188$	11202.7	4.34595
$189$	0	0
$190$	2162.85	0.825840
$191$	145.942	0.0552877	0.0276439	−	0.999618i	$-0.491200\pi$
$191$	145.942	0.0552877	0.0276439	+	0.999618i	$0.491200\pi$
$192$	0	0
$193$	1151.46	0.429451	0.214725	−	0.976674i	$-0.431114\pi$
$193$	1151.46	0.429451	0.214725	+	0.976674i	$0.431114\pi$
$194$	167.397	0.0619504
$195$	0	0
$196$	−5109.16	−1.86194
$197$	−4328.08	−1.56529	−0.782647	−	0.622466i	$-0.786131\pi$
$197$	−4328.08	−1.56529	−0.782647	+	0.622466i	$0.786131\pi$
$198$	0	0
$199$	1532.72	0.545988	0.272994	−	0.962016i	$-0.411986\pi$
$199$	1532.72	0.545988	0.272994	+	0.962016i	$0.411986\pi$
$200$	−1825.45	−0.645394
$201$	0	0
$202$	9496.36	3.30773
$203$	296.959	0.102672
$204$	0	0
$205$	−1041.77	−0.354929
$206$	−10378.2	−3.51010
$207$	0	0
$208$	18923.0	6.30805
$209$	1006.81	0.333218
$210$	0	0
$211$	5345.39	1.74404	0.872019	−	0.489472i	$-0.162810\pi$
$211$	5345.39	1.74404	0.872019	+	0.489472i	$0.162810\pi$
$212$	6326.67	2.04961
$213$	0	0
$214$	9330.65	2.98051
$215$	−972.135	−0.308368
$216$	0	0
$217$	−2.60765	−0.000815755 0
$218$	4230.07	1.31420
$219$	0	0
$220$	−1355.02	−0.415252
$221$	−580.526	−0.176699
$222$	0	0
$223$	3783.05	1.13602	0.568008	−	0.823023i	$-0.307714\pi$
$223$	3783.05	1.13602	0.568008	+	0.823023i	$0.307714\pi$
$224$	6502.95	1.93972
$225$	0	0
$226$	4160.08	1.22444
$227$	4348.46	1.27144	0.635721	−	0.771919i	$-0.280703\pi$
$227$	4348.46	1.27144	0.635721	+	0.771919i	$0.280703\pi$
$228$	0	0
$229$	−3068.67	−0.885517	−0.442759	−	0.896641i	$-0.646000\pi$
$229$	−3068.67	−0.885517	−0.442759	+	0.896641i	$0.646000\pi$
$230$	−2003.12	−0.574268
$231$	0	0
$232$	2117.52	0.599233
$233$	275.696	0.0775170	0.0387585	−	0.999249i	$-0.487660\pi$
$233$	275.696	0.0775170	0.0387585	+	0.999249i	$0.487660\pi$
$234$	0	0
$235$	2610.84	0.724734
$236$	4209.92	1.16120
$237$	0	0
$238$	−383.006	−0.104313
$239$	−3132.87	−0.847901	−0.423950	−	0.905685i	$-0.639357\pi$
$239$	−3132.87	−0.847901	−0.423950	+	0.905685i	$0.639357\pi$
$240$	0	0
$241$	5699.61	1.52342	0.761710	−	0.647918i	$-0.224360\pi$
$241$	5699.61	1.52342	0.761710	+	0.647918i	$0.224360\pi$
$242$	6357.59	1.68877
$243$	0	0
$244$	4061.54	1.06563
$245$	−1190.72	−0.310498
$246$	0	0
$247$	−6713.85	−1.72952
$248$	−18.5943	−0.00476105
$249$	0	0
$250$	−678.396	−0.171622
$251$	−1714.12	−0.431054	−0.215527	−	0.976498i	$-0.569147\pi$
$251$	−1714.12	−0.431054	−0.215527	+	0.976498i	$0.569147\pi$
$252$	0	0
$253$	−932.454	−0.231711
$254$	7400.17	1.82806
$255$	0	0
$256$	7813.52	1.90760
$257$	803.085	0.194922	0.0974612	−	0.995239i	$-0.468928\pi$
$257$	803.085	0.194922	0.0974612	+	0.995239i	$0.468928\pi$
$258$	0	0
$259$	−413.761	−0.0992659
$260$	9035.87	2.15531
$261$	0	0
$262$	−7309.48	−1.72359
$263$	1694.24	0.397230	0.198615	−	0.980078i	$-0.436356\pi$
$263$	1694.24	0.397230	0.198615	+	0.980078i	$0.436356\pi$
$264$	0	0
$265$	1474.46	0.341794
$266$	−4429.50	−1.02101
$267$	0	0
$268$	−12246.9	−2.79141
$269$	−3364.95	−0.762693	−0.381347	−	0.924432i	$-0.624540\pi$
$269$	−3364.95	−0.762693	−0.381347	+	0.924432i	$0.624540\pi$
$270$	0	0
$271$	8044.10	1.80312	0.901558	−	0.432658i	$-0.142424\pi$
$271$	8044.10	1.80312	0.901558	+	0.432658i	$0.142424\pi$
$272$	−1548.23	−0.345129
$273$	0	0
$274$	−13613.6	−3.00156
$275$	−315.795	−0.0692477
$276$	0	0
$277$	−6971.54	−1.51220	−0.756100	−	0.654456i	$-0.772898\pi$
$277$	−6971.54	−1.51220	−0.756100	+	0.654456i	$0.772898\pi$
$278$	−7402.38	−1.59700
$279$	0	0
$280$	3738.51	0.797923
$281$	1916.94	0.406957	0.203478	−	0.979079i	$-0.434775\pi$
$281$	1916.94	0.406957	0.203478	+	0.979079i	$0.434775\pi$
$282$	0	0
$283$	2036.63	0.427792	0.213896	−	0.976856i	$-0.431385\pi$
$283$	2036.63	0.427792	0.213896	+	0.976856i	$0.431385\pi$
$284$	24176.0	5.05134
$285$	0	0
$286$	5774.66	1.19393
$287$	2133.54	0.438811
$288$	0	0
$289$	−4865.50	−0.990332
$290$	786.939	0.159347
$291$	0	0
$292$	12474.1	2.49997
$293$	1754.55	0.349836	0.174918	−	0.984583i	$-0.444034\pi$
$293$	1754.55	0.349836	0.174918	+	0.984583i	$0.444034\pi$
$294$	0	0
$295$	981.143	0.193642
$296$	−2950.40	−0.579353
$297$	0	0
$298$	15154.0	2.94580
$299$	6218.01	1.20266
$300$	0	0
$301$	1990.92	0.381246
$302$	−1663.39	−0.316945
$303$	0	0
$304$	−17905.4	−3.37811
$305$	946.563	0.177705
$306$	0	0
$307$	8909.41	1.65631	0.828154	−	0.560500i	$-0.189391\pi$
$307$	8909.41	1.65631	0.828154	+	0.560500i	$0.189391\pi$
$308$	2775.07	0.513391
$309$	0	0
$310$	−6.91026	−0.00126605
$311$	7422.88	1.35342	0.676709	−	0.736251i	$-0.263405\pi$
$311$	7422.88	1.35342	0.676709	+	0.736251i	$0.263405\pi$
$312$	0	0
$313$	−1453.62	−0.262504	−0.131252	−	0.991349i	$-0.541900\pi$
$313$	−1453.62	−0.262504	−0.131252	+	0.991349i	$0.541900\pi$
$314$	9921.69	1.78316
$315$	0	0
$316$	−5484.35	−0.976326
$317$	−5203.30	−0.921912	−0.460956	−	0.887423i	$-0.652493\pi$
$317$	−5203.30	−0.921912	−0.460956	+	0.887423i	$0.652493\pi$
$318$	0	0
$319$	366.322	0.0642949
$320$	8246.88	1.44067
$321$	0	0
$322$	4102.37	0.709987
$323$	549.309	0.0946264
$324$	0	0
$325$	2105.86	0.359421
$326$	−11403.3	−1.93733
$327$	0	0
$328$	15213.6	2.56107
$329$	−5346.98	−0.896014
$330$	0	0
$331$	−10985.0	−1.82414	−0.912070	−	0.410035i	$-0.865516\pi$
$331$	−10985.0	−1.82414	−0.912070	+	0.410035i	$0.865516\pi$
$332$	21091.2	3.48653
$333$	0	0
$334$	8490.94	1.39103
$335$	−2854.20	−0.465498
$336$	0	0
$337$	2295.27	0.371013	0.185507	−	0.982643i	$-0.440607\pi$
$337$	2295.27	0.371013	0.185507	+	0.982643i	$0.440607\pi$
$338$	−26584.4	−4.27812
$339$	0	0
$340$	−739.290	−0.117922
$341$	−3.21674	−0.000510839 0
$342$	0	0
$343$	5950.88	0.936786
$344$	14196.7	2.22510
$345$	0	0
$346$	−14627.5	−2.27277
$347$	5313.01	0.821952	0.410976	−	0.911646i	$-0.365188\pi$
$347$	5313.01	0.821952	0.410976	+	0.911646i	$0.365188\pi$
$348$	0	0
$349$	2086.08	0.319958	0.159979	−	0.987120i	$-0.448857\pi$
$349$	2086.08	0.319958	0.159979	+	0.987120i	$0.448857\pi$
$350$	1389.35	0.212182
$351$	0	0
$352$	8021.89	1.21468
$353$	−5382.37	−0.811543	−0.405772	−	0.913975i	$-0.632997\pi$
$353$	−5382.37	−0.811543	−0.405772	+	0.913975i	$0.632997\pi$
$354$	0	0
$355$	5634.33	0.842364
$356$	−14559.7	−2.16759
$357$	0	0
$358$	−19560.4	−2.88771
$359$	7990.36	1.17469	0.587347	−	0.809335i	$-0.300173\pi$
$359$	7990.36	1.17469	0.587347	+	0.809335i	$0.300173\pi$
$360$	0	0
$361$	−506.193	−0.0737998
$362$	14427.1	2.09468
$363$	0	0
$364$	−18505.4	−2.66469
$365$	2907.16	0.416897
$366$	0	0
$367$	−429.201	−0.0610466	−0.0305233	−	0.999534i	$-0.509717\pi$
$367$	−429.201	−0.0610466	−0.0305233	+	0.999534i	$0.509717\pi$
$368$	16583.0	2.34905
$369$	0	0
$370$	−1096.46	−0.154061
$371$	−3019.69	−0.422572
$372$	0	0
$373$	6270.80	0.870482	0.435241	−	0.900314i	$-0.356663\pi$
$373$	6270.80	0.870482	0.435241	+	0.900314i	$0.356663\pi$
$374$	−472.467	−0.0653227
$375$	0	0
$376$	−38127.7	−5.22948
$377$	−2442.79	−0.333714
$378$	0	0
$379$	10417.0	1.41183	0.705917	−	0.708294i	$-0.250535\pi$
$379$	10417.0	1.41183	0.705917	+	0.708294i	$0.250535\pi$
$380$	−8549.96	−1.15422
$381$	0	0
$382$	−792.049	−0.106086
$383$	−10769.3	−1.43677	−0.718387	−	0.695644i	$-0.755119\pi$
$383$	−10769.3	−1.43677	−0.718387	+	0.695644i	$0.755119\pi$
$384$	0	0
$385$	646.745	0.0856134
$386$	−6249.18	−0.824028
$387$	0	0
$388$	−661.736	−0.0865839
$389$	11121.7	1.44960	0.724798	−	0.688961i	$-0.241933\pi$
$389$	11121.7	1.44960	0.724798	+	0.688961i	$0.241933\pi$
$390$	0	0
$391$	−508.740	−0.0658008
$392$	17388.7	2.24047
$393$	0	0
$394$	23489.2	3.00348
$395$	−1278.16	−0.162813
$396$	0	0
$397$	11286.7	1.42686	0.713432	−	0.700724i	$-0.247140\pi$
$397$	11286.7	1.42686	0.713432	+	0.700724i	$0.247140\pi$
$398$	−8318.33	−1.04764
$399$	0	0
$400$	5616.19	0.702023
$401$	15418.0	1.92005	0.960023	−	0.279922i	$-0.0903086\pi$
$401$	15418.0	1.92005	0.960023	+	0.279922i	$0.0903086\pi$
$402$	0	0
$403$	21.4506	0.00265144
$404$	−37540.1	−4.62299
$405$	0	0
$406$	−1611.65	−0.197006
$407$	−510.406	−0.0621619
$408$	0	0
$409$	13849.5	1.67437	0.837183	−	0.546922i	$-0.184201\pi$
$409$	13849.5	1.67437	0.837183	+	0.546922i	$0.184201\pi$
$410$	5653.86	0.681035
$411$	0	0
$412$	41025.9	4.90583
$413$	−2009.37	−0.239406
$414$	0	0
$415$	4915.40	0.581416
$416$	−53493.4	−6.30464
$417$	0	0
$418$	−5464.13	−0.639376
$419$	2793.33	0.325688	0.162844	−	0.986652i	$-0.447933\pi$
$419$	2793.33	0.325688	0.162844	+	0.986652i	$0.447933\pi$
$420$	0	0
$421$	−3643.79	−0.421823	−0.210911	−	0.977505i	$-0.567643\pi$
$421$	−3643.79	−0.421823	−0.210911	+	0.977505i	$0.567643\pi$
$422$	−29010.4	−3.34645
$423$	0	0
$424$	−21532.4	−2.46629
$425$	−172.295	−0.0196648
$426$	0	0
$427$	−1938.55	−0.219703
$428$	−36885.0	−4.16566
$429$	0	0
$430$	5275.94	0.591694
$431$	−5834.03	−0.652007	−0.326004	−	0.945369i	$-0.605702\pi$
$431$	−5834.03	−0.652007	−0.326004	+	0.945369i	$0.605702\pi$
$432$	0	0
$433$	7919.15	0.878915	0.439457	−	0.898263i	$-0.355171\pi$
$433$	7919.15	0.878915	0.439457	+	0.898263i	$0.355171\pi$
$434$	14.1522	0.00156527
$435$	0	0
$436$	−16721.9	−1.83677
$437$	−5883.63	−0.644056
$438$	0	0
$439$	11696.0	1.27158	0.635788	−	0.771864i	$-0.280675\pi$
$439$	11696.0	1.27158	0.635788	+	0.771864i	$0.280675\pi$
$440$	4611.73	0.499672
$441$	0	0
$442$	3150.61	0.339049
$443$	2786.16	0.298814	0.149407	−	0.988776i	$-0.452264\pi$
$443$	2786.16	0.298814	0.149407	+	0.988776i	$0.452264\pi$
$444$	0	0
$445$	−3393.21	−0.361468
$446$	−20531.2	−2.17978
$447$	0	0
$448$	−16889.5	−1.78115
$449$	−11013.8	−1.15762	−0.578811	−	0.815462i	$-0.696483\pi$
$449$	−11013.8	−1.15762	−0.578811	+	0.815462i	$0.696483\pi$
$450$	0	0
$451$	2631.88	0.274790
$452$	−16445.2	−1.71132
$453$	0	0
$454$	−23599.8	−2.43963
$455$	−4312.77	−0.444365
$456$	0	0
$457$	12773.5	1.30748	0.653739	−	0.756720i	$-0.273199\pi$
$457$	12773.5	1.30748	0.653739	+	0.756720i	$0.273199\pi$
$458$	16654.2	1.69912
$459$	0	0
$460$	7918.52	0.802615
$461$	13343.4	1.34808	0.674039	−	0.738696i	$-0.264558\pi$
$461$	13343.4	1.34808	0.674039	+	0.738696i	$0.264558\pi$
$462$	0	0
$463$	17265.4	1.73303	0.866514	−	0.499153i	$-0.166355\pi$
$463$	17265.4	1.73303	0.866514	+	0.499153i	$0.166355\pi$
$464$	−6514.78	−0.651812
$465$	0	0
$466$	−1496.25	−0.148739
$467$	16016.7	1.58708	0.793538	−	0.608521i	$-0.208237\pi$
$467$	16016.7	1.58708	0.793538	+	0.608521i	$0.208237\pi$
$468$	0	0
$469$	5845.38	0.575511
$470$	−14169.5	−1.39062
$471$	0	0
$472$	−14328.2	−1.39727
$473$	2455.96	0.238742
$474$	0	0
$475$	−1992.61	−0.192479
$476$	1514.06	0.145792
$477$	0	0
$478$	17002.6	1.62695
$479$	5847.01	0.557738	0.278869	−	0.960329i	$-0.410040\pi$
$479$	5847.01	0.557738	0.278869	+	0.960329i	$0.410040\pi$
$480$	0	0
$481$	3403.61	0.322643
$482$	−30932.7	−2.92313
$483$	0	0
$484$	−25132.2	−2.36028
$485$	−154.221	−0.0144388
$486$	0	0
$487$	−342.662	−0.0318840	−0.0159420	−	0.999873i	$-0.505075\pi$
$487$	−342.662	−0.0318840	−0.0159420	+	0.999873i	$0.505075\pi$
$488$	−13823.2	−1.28227
$489$	0	0
$490$	6462.22	0.595782
$491$	1673.48	0.153815	0.0769073	−	0.997038i	$-0.475495\pi$
$491$	1673.48	0.153815	0.0769073	+	0.997038i	$0.475495\pi$
$492$	0	0
$493$	199.863	0.0182583
$494$	36437.2	3.31859
$495$	0	0
$496$	57.2074	0.00517881
$497$	−11539.1	−1.04144
$498$	0	0
$499$	10918.2	0.979487	0.489743	−	0.871867i	$-0.337090\pi$
$499$	10918.2	0.979487	0.489743	+	0.871867i	$0.337090\pi$
$500$	2681.77	0.239865
$501$	0	0
$502$	9302.84	0.827103
$503$	−16957.6	−1.50319	−0.751595	−	0.659625i	$-0.770715\pi$
$503$	−16957.6	−1.50319	−0.751595	+	0.659625i	$0.770715\pi$
$504$	0	0
$505$	−8748.91	−0.770933
$506$	5060.58	0.444606
$507$	0	0
$508$	−29253.6	−2.55496
$509$	226.179	0.0196959	0.00984795	−	0.999952i	$-0.496865\pi$
$509$	226.179	0.0196959	0.00984795	+	0.999952i	$0.496865\pi$
$510$	0	0
$511$	−5953.83	−0.515425
$512$	−11437.3	−0.987234
$513$	0	0
$514$	−4358.48	−0.374016
$515$	9561.30	0.818100
$516$	0	0
$517$	−6595.91	−0.561099
$518$	2245.55	0.190471
$519$	0	0
$520$	−30753.0	−2.59348
$521$	2051.24	0.172489	0.0862443	−	0.996274i	$-0.472513\pi$
$521$	2051.24	0.172489	0.0862443	+	0.996274i	$0.472513\pi$
$522$	0	0
$523$	−11922.9	−0.996852	−0.498426	−	0.866932i	$-0.666088\pi$
$523$	−11922.9	−0.996852	−0.498426	+	0.866932i	$0.666088\pi$
$524$	28895.1	2.40895
$525$	0	0
$526$	−9194.94	−0.762203
$527$	−1.75503	−0.000145067 0
$528$	0	0
$529$	−6717.89	−0.552140
$530$	−8002.16	−0.655833
$531$	0	0
$532$	17510.3	1.42700
$533$	−17550.5	−1.42626
$534$	0	0
$535$	−8596.24	−0.694669
$536$	41681.6	3.35890
$537$	0	0
$538$	18262.1	1.46345
$539$	3008.17	0.240392
$540$	0	0
$541$	12876.2	1.02327	0.511635	−	0.859203i	$-0.329040\pi$
$541$	12876.2	1.02327	0.511635	+	0.859203i	$0.329040\pi$
$542$	−43656.7	−3.45981
$543$	0	0
$544$	4376.69	0.344943
$545$	−3897.12	−0.306302
$546$	0	0
$547$	9065.39	0.708607	0.354304	−	0.935130i	$-0.384718\pi$
$547$	9065.39	0.708607	0.354304	+	0.935130i	$0.384718\pi$
$548$	53816.0	4.19508
$549$	0	0
$550$	1713.87	0.132872
$551$	2311.43	0.178712
$552$	0	0
$553$	2617.65	0.201291
$554$	37835.7	2.90160
$555$	0	0
$556$	29262.4	2.23202
$557$	19520.2	1.48492	0.742458	−	0.669893i	$-0.233660\pi$
$557$	19520.2	1.48492	0.742458	+	0.669893i	$0.233660\pi$
$558$	0	0
$559$	−16377.4	−1.23916
$560$	−11501.9	−0.867936
$561$	0	0
$562$	−10403.5	−0.780866
$563$	−9786.46	−0.732593	−0.366297	−	0.930498i	$-0.619374\pi$
$563$	−9786.46	−0.732593	−0.366297	+	0.930498i	$0.619374\pi$
$564$	0	0
$565$	−3832.64	−0.285381
$566$	−11053.1	−0.820844
$567$	0	0
$568$	−82281.5	−6.07826
$569$	−13236.9	−0.975257	−0.487628	−	0.873051i	$-0.662138\pi$
$569$	−13236.9	−0.975257	−0.487628	+	0.873051i	$0.662138\pi$
$570$	0	0
$571$	−21385.2	−1.56733	−0.783663	−	0.621186i	$-0.786651\pi$
$571$	−21385.2	−1.56733	−0.783663	+	0.621186i	$0.786651\pi$
$572$	−22827.8	−1.66867
$573$	0	0
$574$	−11579.1	−0.841987
$575$	1845.45	0.133845
$576$	0	0
$577$	1531.13	0.110471	0.0552356	−	0.998473i	$-0.482409\pi$
$577$	1531.13	0.110471	0.0552356	+	0.998473i	$0.482409\pi$
$578$	26405.9	1.90024
$579$	0	0
$580$	−3110.85	−0.222709
$581$	−10066.7	−0.718825
$582$	0	0
$583$	−3725.02	−0.264622
$584$	−42454.9	−3.00821
$585$	0	0
$586$	−9522.25	−0.671263
$587$	−2742.12	−0.192810	−0.0964050	−	0.995342i	$-0.530734\pi$
$587$	−2742.12	−0.192810	−0.0964050	+	0.995342i	$0.530734\pi$
$588$	0	0
$589$	−20.2971	−0.00141991
$590$	−5324.83	−0.371559
$591$	0	0
$592$	9077.22	0.630188
$593$	8476.47	0.586993	0.293497	−	0.955960i	$-0.405181\pi$
$593$	8476.47	0.586993	0.293497	+	0.955960i	$0.405181\pi$
$594$	0	0
$595$	352.859	0.0243123
$596$	−59905.3	−4.11714
$597$	0	0
$598$	−33746.2	−2.30767
$599$	14839.1	1.01220	0.506100	−	0.862475i	$-0.331087\pi$
$599$	14839.1	1.01220	0.506100	+	0.862475i	$0.331087\pi$
$600$	0	0
$601$	−13198.6	−0.895814	−0.447907	−	0.894080i	$-0.647830\pi$
$601$	−13198.6	−0.895814	−0.447907	+	0.894080i	$0.647830\pi$
$602$	−10805.1	−0.731532
$603$	0	0
$604$	6575.56	0.442973
$605$	−5857.19	−0.393601
$606$	0	0
$607$	2310.67	0.154509	0.0772547	−	0.997011i	$-0.475385\pi$
$607$	2310.67	0.154509	0.0772547	+	0.997011i	$0.475385\pi$
$608$	50616.8	3.37629
$609$	0	0
$610$	−5137.15	−0.340979
$611$	43984.4	2.91231
$612$	0	0
$613$	−3944.92	−0.259925	−0.129962	−	0.991519i	$-0.541486\pi$
$613$	−3944.92	−0.259925	−0.129962	+	0.991519i	$0.541486\pi$
$614$	−48352.9	−3.17811
$615$	0	0
$616$	−9444.80	−0.617762
$617$	26587.7	1.73482	0.867408	−	0.497598i	$-0.165784\pi$
$617$	26587.7	1.73482	0.867408	+	0.497598i	$0.165784\pi$
$618$	0	0
$619$	8691.35	0.564353	0.282177	−	0.959362i	$-0.408944\pi$
$619$	8691.35	0.564353	0.282177	+	0.959362i	$0.408944\pi$
$620$	27.3170	0.00176948
$621$	0	0
$622$	−40285.2	−2.59693
$623$	6949.26	0.446896
$624$	0	0
$625$	625.000	0.0400000
$626$	7889.06	0.503690
$627$	0	0
$628$	−39221.5	−2.49221
$629$	−278.474	−0.0176526
$630$	0	0
$631$	−19396.9	−1.22374	−0.611869	−	0.790959i	$-0.709582\pi$
$631$	−19396.9	−1.22374	−0.611869	+	0.790959i	$0.709582\pi$
$632$	18665.7	1.17481
$633$	0	0
$634$	28239.2	1.76896
$635$	−6817.71	−0.426067
$636$	0	0
$637$	−20059.8	−1.24772
$638$	−1988.09	−0.123369
$639$	0	0
$640$	−19355.0	−1.19543
$641$	11088.7	0.683273	0.341636	−	0.939832i	$-0.389019\pi$
$641$	11088.7	0.683273	0.341636	+	0.939832i	$0.389019\pi$
$642$	0	0
$643$	−3896.33	−0.238968	−0.119484	−	0.992836i	$-0.538124\pi$
$643$	−3896.33	−0.238968	−0.119484	+	0.992836i	$0.538124\pi$
$644$	−16217.1	−0.992302
$645$	0	0
$646$	−2981.19	−0.181569
$647$	−14565.2	−0.885034	−0.442517	−	0.896760i	$-0.645914\pi$
$647$	−14565.2	−0.885034	−0.442517	+	0.896760i	$0.645914\pi$
$648$	0	0
$649$	−2478.72	−0.149920
$650$	−11428.8	−0.689655
$651$	0	0
$652$	45078.3	2.70767
$653$	−27443.4	−1.64463	−0.822314	−	0.569034i	$-0.807317\pi$
$653$	−27443.4	−1.64463	−0.822314	+	0.569034i	$0.807317\pi$
$654$	0	0
$655$	6734.16	0.401718
$656$	−46806.2	−2.78579
$657$	0	0
$658$	29019.0	1.71927
$659$	−10355.6	−0.612135	−0.306068	−	0.952010i	$-0.599013\pi$
$659$	−10355.6	−0.612135	−0.306068	+	0.952010i	$0.599013\pi$
$660$	0	0
$661$	9704.01	0.571017	0.285508	−	0.958376i	$-0.407838\pi$
$661$	9704.01	0.571017	0.285508	+	0.958376i	$0.407838\pi$
$662$	59617.4	3.50015
$663$	0	0
$664$	−71782.6	−4.19534
$665$	4080.85	0.237968
$666$	0	0
$667$	−2140.72	−0.124272
$668$	−33565.6	−1.94415
$669$	0	0
$670$	15490.2	0.893194
$671$	−2391.35	−0.137582
$672$	0	0
$673$	16876.5	0.966629	0.483315	−	0.875447i	$-0.339433\pi$
$673$	16876.5	0.966629	0.483315	+	0.875447i	$0.339433\pi$
$674$	−12456.8	−0.711898
$675$	0	0
$676$	105091.	5.97924
$677$	1095.44	0.0621880	0.0310940	−	0.999516i	$-0.490101\pi$
$677$	1095.44	0.0621880	0.0310940	+	0.999516i	$0.490101\pi$
$678$	0	0
$679$	315.843	0.0178512
$680$	2516.13	0.141896
$681$	0	0
$682$	17.4578	0.000980195 0
$683$	23004.5	1.28879	0.644394	−	0.764693i	$-0.277110\pi$
$683$	23004.5	1.28879	0.644394	+	0.764693i	$0.277110\pi$
$684$	0	0
$685$	12542.1	0.699574
$686$	−32296.4	−1.79750
$687$	0	0
$688$	−43677.5	−2.42034
$689$	24840.0	1.37348
$690$	0	0
$691$	4973.91	0.273830	0.136915	−	0.990583i	$-0.456281\pi$
$691$	4973.91	0.273830	0.136915	+	0.990583i	$0.456281\pi$
$692$	57824.0	3.17650
$693$	0	0
$694$	−28834.6	−1.57716
$695$	6819.75	0.372212
$696$	0	0
$697$	1435.94	0.0780344
$698$	−11321.5	−0.613933
$699$	0	0
$700$	−5492.24	−0.296553
$701$	−19281.6	−1.03888	−0.519440	−	0.854507i	$-0.673859\pi$
$701$	−19281.6	−1.03888	−0.519440	+	0.854507i	$0.673859\pi$
$702$	0	0
$703$	−3220.58	−0.172783
$704$	−20834.6	−1.11539
$705$	0	0
$706$	29211.0	1.55718
$707$	17917.7	0.953132
$708$	0	0
$709$	35391.4	1.87468	0.937342	−	0.348412i	$-0.113279\pi$
$709$	35391.4	1.87468	0.937342	+	0.348412i	$0.113279\pi$
$710$	−30578.5	−1.61632
$711$	0	0
$712$	49553.0	2.60826
$713$	18.7981	0.000987369 0
$714$	0	0
$715$	−5320.14	−0.278268
$716$	77324.3	4.03596
$717$	0	0
$718$	−43365.0	−2.25399
$719$	19325.0	1.00237	0.501183	−	0.865341i	$-0.332898\pi$
$719$	19325.0	1.00237	0.501183	+	0.865341i	$0.332898\pi$
$720$	0	0
$721$	−19581.5	−1.01145
$722$	2747.19	0.141607
$723$	0	0
$724$	−57032.0	−2.92759
$725$	−725.000	−0.0371391
$726$	0	0
$727$	16887.9	0.861537	0.430768	−	0.902462i	$-0.358243\pi$
$727$	16887.9	0.861537	0.430768	+	0.902462i	$0.358243\pi$
$728$	62982.0	3.20641
$729$	0	0
$730$	−15777.6	−0.799940
$731$	1339.95	0.0677975
$732$	0	0
$733$	6698.16	0.337520	0.168760	−	0.985657i	$-0.446024\pi$
$733$	6698.16	0.337520	0.168760	+	0.985657i	$0.446024\pi$
$734$	2329.35	0.117136
$735$	0	0
$736$	−46878.6	−2.34778
$737$	7210.73	0.360394
$738$	0	0
$739$	4934.50	0.245627	0.122814	−	0.992430i	$-0.460808\pi$
$739$	4934.50	0.245627	0.122814	+	0.992430i	$0.460808\pi$
$740$	4334.44	0.215320
$741$	0	0
$742$	16388.4	0.810829
$743$	5140.65	0.253825	0.126913	−	0.991914i	$-0.459493\pi$
$743$	5140.65	0.253825	0.126913	+	0.991914i	$0.459493\pi$
$744$	0	0
$745$	−13961.2	−0.686578
$746$	−34032.7	−1.67027
$747$	0	0
$748$	1867.71	0.0912971
$749$	17605.0	0.858843
$750$	0	0
$751$	9625.37	0.467690	0.233845	−	0.972274i	$-0.424869\pi$
$751$	9625.37	0.467690	0.233845	+	0.972274i	$0.424869\pi$
$752$	117304.	5.68834
$753$	0	0
$754$	13257.4	0.640328
$755$	1532.47	0.0738704
$756$	0	0
$757$	−5128.83	−0.246249	−0.123125	−	0.992391i	$-0.539291\pi$
$757$	−5128.83	−0.246249	−0.123125	+	0.992391i	$0.539291\pi$
$758$	−56534.8	−2.70902
$759$	0	0
$760$	29099.3	1.38887
$761$	25024.7	1.19204	0.596022	−	0.802968i	$-0.296747\pi$
$761$	25024.7	1.19204	0.596022	+	0.802968i	$0.296747\pi$
$762$	0	0
$763$	7981.28	0.378692
$764$	3131.05	0.148269
$765$	0	0
$766$	58446.7	2.75687
$767$	16529.2	0.778140
$768$	0	0
$769$	−26474.8	−1.24149	−0.620746	−	0.784012i	$-0.713170\pi$
$769$	−26474.8	−1.24149	−0.620746	+	0.784012i	$0.713170\pi$
$770$	−3509.99	−0.164274
$771$	0	0
$772$	24703.6	1.15169
$773$	24685.1	1.14859	0.574295	−	0.818648i	$-0.305276\pi$
$773$	24685.1	1.14859	0.574295	+	0.818648i	$0.305276\pi$
$774$	0	0
$775$	6.36636	0.000295079 0
$776$	2252.18	0.104186
$777$	0	0
$778$	−60359.4	−2.78148
$779$	16606.8	0.763798
$780$	0	0
$781$	−14234.3	−0.652170
$782$	2761.02	0.126258
$783$	0	0
$784$	−53498.3	−2.43706
$785$	−9140.76	−0.415602
$786$	0	0
$787$	−35383.4	−1.60265	−0.801323	−	0.598232i	$-0.795870\pi$
$787$	−35383.4	−1.60265	−0.801323	+	0.598232i	$0.795870\pi$
$788$	−92855.3	−4.19776
$789$	0	0
$790$	6936.77	0.312404
$791$	7849.22	0.352827
$792$	0	0
$793$	15946.6	0.714098
$794$	−61255.0	−2.73786
$795$	0	0
$796$	32883.2	1.46421
$797$	24869.4	1.10529	0.552647	−	0.833415i	$-0.313618\pi$
$797$	24869.4	1.10529	0.552647	+	0.833415i	$0.313618\pi$
$798$	0	0
$799$	−3598.69	−0.159340
$800$	−15876.4	−0.701644
$801$	0	0
$802$	−83676.1	−3.68417
$803$	−7344.51	−0.322767
$804$	0	0
$805$	−3779.47	−0.165477
$806$	−116.416	−0.00508757
$807$	0	0
$808$	127765.	5.56284
$809$	−834.568	−0.0362693	−0.0181346	−	0.999836i	$-0.505773\pi$
$809$	−834.568	−0.0362693	−0.0181346	+	0.999836i	$0.505773\pi$
$810$	0	0
$811$	−17419.5	−0.754232	−0.377116	−	0.926166i	$-0.623084\pi$
$811$	−17419.5	−0.754232	−0.377116	+	0.926166i	$0.623084\pi$
$812$	6371.00	0.275343
$813$	0	0
$814$	2770.06	0.119276
$815$	10505.7	0.451533
$816$	0	0
$817$	15496.7	0.663600
$818$	−75163.8	−3.21276
$819$	0	0
$820$	−22350.3	−0.951837
$821$	−20163.7	−0.857148	−0.428574	−	0.903507i	$-0.640984\pi$
$821$	−20163.7	−0.857148	−0.428574	+	0.903507i	$0.640984\pi$
$822$	0	0
$823$	10456.9	0.442898	0.221449	−	0.975172i	$-0.428921\pi$
$823$	10456.9	0.442898	0.221449	+	0.975172i	$0.428921\pi$
$824$	−139629.	−5.90318
$825$	0	0
$826$	10905.2	0.459371
$827$	10943.6	0.460154	0.230077	−	0.973172i	$-0.426102\pi$
$827$	10943.6	0.460154	0.230077	+	0.973172i	$0.426102\pi$
$828$	0	0
$829$	−33775.8	−1.41506	−0.707528	−	0.706685i	$-0.750190\pi$
$829$	−33775.8	−1.41506	−0.707528	+	0.706685i	$0.750190\pi$
$830$	−26676.7	−1.11562
$831$	0	0
$832$	138934.	5.78926
$833$	1641.24	0.0682660
$834$	0	0
$835$	−7822.63	−0.324207
$836$	21600.3	0.893613
$837$	0	0
$838$	−15159.9	−0.624927
$839$	−39087.3	−1.60840	−0.804198	−	0.594361i	$-0.797405\pi$
$839$	−39087.3	−1.60840	−0.804198	+	0.594361i	$0.797405\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	19775.4	0.809391
$843$	0	0
$844$	114681.	4.67711
$845$	24492.0	0.997101
$846$	0	0
$847$	11995.5	0.486623
$848$	66246.9	2.68270
$849$	0	0
$850$	935.076	0.0377327
$851$	2982.73	0.120149
$852$	0	0
$853$	47274.9	1.89761	0.948806	−	0.315859i	$-0.102293\pi$
$853$	47274.9	1.89761	0.948806	+	0.315859i	$0.102293\pi$
$854$	10520.9	0.421565
$855$	0	0
$856$	125536.	5.01254
$857$	−15482.0	−0.617099	−0.308550	−	0.951208i	$-0.599844\pi$
$857$	−15482.0	−0.617099	−0.308550	+	0.951208i	$0.599844\pi$
$858$	0	0
$859$	−32719.1	−1.29960	−0.649802	−	0.760103i	$-0.725148\pi$
$859$	−32719.1	−1.29960	−0.649802	+	0.760103i	$0.725148\pi$
$860$	−20856.3	−0.826971
$861$	0	0
$862$	31662.2	1.25107
$863$	1741.34	0.0686858	0.0343429	−	0.999410i	$-0.489066\pi$
$863$	1741.34	0.0686858	0.0343429	+	0.999410i	$0.489066\pi$
$864$	0	0
$865$	13476.2	0.529715
$866$	−42978.6	−1.68646
$867$	0	0
$868$	−55.9449	−0.00218767
$869$	3229.08	0.126052
$870$	0	0
$871$	−48084.3	−1.87058
$872$	56912.0	2.21019
$873$	0	0
$874$	31931.5	1.23581
$875$	−1280.00	−0.0494534
$876$	0	0
$877$	11848.6	0.456213	0.228106	−	0.973636i	$-0.426747\pi$
$877$	11848.6	0.456213	0.228106	+	0.973636i	$0.426747\pi$
$878$	−63476.4	−2.43989
$879$	0	0
$880$	−14188.5	−0.543516
$881$	29332.3	1.12171	0.560857	−	0.827913i	$-0.310472\pi$
$881$	29332.3	1.12171	0.560857	+	0.827913i	$0.310472\pi$
$882$	0	0
$883$	−25723.3	−0.980360	−0.490180	−	0.871621i	$-0.663069\pi$
$883$	−25723.3	−0.980360	−0.490180	+	0.871621i	$0.663069\pi$
$884$	−12454.7	−0.473865
$885$	0	0
$886$	−15121.0	−0.573362
$887$	−10959.8	−0.414875	−0.207437	−	0.978248i	$-0.566512\pi$
$887$	−10959.8	−0.414875	−0.207437	+	0.978248i	$0.566512\pi$
$888$	0	0
$889$	13962.6	0.526762
$890$	18415.5	0.693583
$891$	0	0
$892$	81162.1	3.04653
$893$	−41619.2	−1.55961
$894$	0	0
$895$	18020.8	0.673039
$896$	39638.8	1.47795
$897$	0	0
$898$	59773.6	2.22124
$899$	−7.38497	−0.000273974 0
$900$	0	0
$901$	−2032.34	−0.0751467
$902$	−14283.7	−0.527266
$903$	0	0
$904$	55970.3	2.05923
$905$	−13291.6	−0.488207
$906$	0	0
$907$	−3053.42	−0.111783	−0.0558915	−	0.998437i	$-0.517800\pi$
$907$	−3053.42	−0.111783	−0.0558915	+	0.998437i	$0.517800\pi$
$908$	93292.4	3.40971
$909$	0	0
$910$	23406.2	0.852644
$911$	32880.5	1.19581	0.597903	−	0.801568i	$-0.296001\pi$
$911$	32880.5	1.19581	0.597903	+	0.801568i	$0.296001\pi$
$912$	0	0
$913$	−12418.1	−0.450140
$914$	−69323.8	−2.50878
$915$	0	0
$916$	−65835.7	−2.37475
$917$	−13791.5	−0.496658
$918$	0	0
$919$	−49962.9	−1.79339	−0.896694	−	0.442650i	$-0.854038\pi$
$919$	−49962.9	−1.79339	−0.896694	+	0.442650i	$0.854038\pi$
$920$	−26950.2	−0.965786
$921$	0	0
$922$	−72416.8	−2.58668
$923$	94920.7	3.38500
$924$	0	0
$925$	1010.16	0.0359070
$926$	−93702.3	−3.32532
$927$	0	0
$928$	18416.6	0.651460
$929$	48163.6	1.70096	0.850482	−	0.526004i	$-0.176310\pi$
$929$	48163.6	1.70096	0.850482	+	0.526004i	$0.176310\pi$
$930$	0	0
$931$	18981.1	0.668185
$932$	5914.83	0.207883
$933$	0	0
$934$	−86925.3	−3.04527
$935$	435.279	0.0152248
$936$	0	0
$937$	32244.1	1.12419	0.562097	−	0.827071i	$-0.309995\pi$
$937$	32244.1	1.12419	0.562097	+	0.827071i	$0.309995\pi$
$938$	−31723.9	−1.10429
$939$	0	0
$940$	56013.4	1.94357
$941$	−37391.5	−1.29535	−0.647677	−	0.761915i	$-0.724260\pi$
$941$	−37391.5	−1.29535	−0.647677	+	0.761915i	$0.724260\pi$
$942$	0	0
$943$	−15380.3	−0.531125
$944$	44082.3	1.51987
$945$	0	0
$946$	−13328.9	−0.458097
$947$	−17887.8	−0.613807	−0.306903	−	0.951741i	$-0.599293\pi$
$947$	−17887.8	−0.613807	−0.306903	+	0.951741i	$0.599293\pi$
$948$	0	0
$949$	48976.4	1.67528
$950$	10814.2	0.369327
$951$	0	0
$952$	−5153.02	−0.175431
$953$	−20302.3	−0.690090	−0.345045	−	0.938586i	$-0.612136\pi$
$953$	−20302.3	−0.690090	−0.345045	+	0.938586i	$0.612136\pi$
$954$	0	0
$955$	729.708	0.0247254
$956$	−67213.0	−2.27387
$957$	0	0
$958$	−31732.7	−1.07018
$959$	−25686.1	−0.864908
$960$	0	0
$961$	−29790.9	−0.999998
$962$	−18472.0	−0.619085
$963$	0	0
$964$	122280.	4.08546
$965$	5757.31	0.192056
$966$	0	0
$967$	−19313.4	−0.642273	−0.321137	−	0.947033i	$-0.604065\pi$
$967$	−19313.4	−0.642273	−0.321137	+	0.947033i	$0.604065\pi$
$968$	85536.0	2.84012
$969$	0	0
$970$	836.983	0.0277050
$971$	25297.3	0.836077	0.418038	−	0.908429i	$-0.362718\pi$
$971$	25297.3	0.836077	0.418038	+	0.908429i	$0.362718\pi$
$972$	0	0
$973$	−13966.8	−0.460179
$974$	1859.68	0.0611787
$975$	0	0
$976$	42528.6	1.39478
$977$	−21461.4	−0.702776	−0.351388	−	0.936230i	$-0.614290\pi$
$977$	−21461.4	−0.702776	−0.351388	+	0.936230i	$0.614290\pi$
$978$	0	0
$979$	8572.45	0.279854
$980$	−25545.8	−0.832684
$981$	0	0
$982$	−9082.24	−0.295138
$983$	−14059.8	−0.456193	−0.228096	−	0.973639i	$-0.573250\pi$
$983$	−14059.8	−0.456193	−0.228096	+	0.973639i	$0.573250\pi$
$984$	0	0
$985$	−21640.4	−0.700021
$986$	−1084.69	−0.0350340
$987$	0	0
$988$	−144040.	−4.63818
$989$	−14352.2	−0.461450
$990$	0	0
$991$	−60932.3	−1.95316	−0.976578	−	0.215162i	$-0.930972\pi$
$991$	−60932.3	−1.95316	−0.976578	+	0.215162i	$0.930972\pi$
$992$	−161.720	−0.00517601
$993$	0	0
$994$	62624.5	1.99832
$995$	7663.60	0.244173
$996$	0	0
$997$	−4183.32	−0.132886	−0.0664428	−	0.997790i	$-0.521165\pi$
$997$	−4183.32	−0.132886	−0.0664428	+	0.997790i	$0.521165\pi$
$998$	−59254.7	−1.87943
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.4.a.h.1.1		6
3.2	odd	2		435.4.a.h.1.6	✓	6
15.14	odd	2		2175.4.a.k.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.h.1.6	✓	6	3.2	odd	2
1305.4.a.h.1.1		6	1.1	even	1	trivial
2175.4.a.k.1.1		6	15.14	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:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q-5.42717 q^{2} +21.4542 q^{4} +5.00000 q^{5} -10.2400 q^{7} -73.0180 q^{8} -27.1358 q^{10} -12.6318 q^{11} +84.2342 q^{13} +55.5740 q^{14} +224.647 q^{16} -6.89181 q^{17} -79.7045 q^{19} +107.271 q^{20} +68.5548 q^{22} +73.8181 q^{23} +25.0000 q^{25} -457.153 q^{26} -219.690 q^{28} -29.0000 q^{29} +0.254654 q^{31} -635.056 q^{32} +37.4030 q^{34} -51.1998 q^{35} +40.4065 q^{37} +432.570 q^{38} -365.090 q^{40} -208.354 q^{41} -194.427 q^{43} -271.004 q^{44} -400.623 q^{46} +522.168 q^{47} -238.143 q^{49} -135.679 q^{50} +1807.17 q^{52} +294.893 q^{53} -63.1589 q^{55} +747.701 q^{56} +157.388 q^{58} +196.229 q^{59} +189.313 q^{61} -1.38205 q^{62} +1649.38 q^{64} +421.171 q^{65} -570.840 q^{67} -147.858 q^{68} +277.870 q^{70} +1126.87 q^{71} +581.431 q^{73} -219.293 q^{74} -1709.99 q^{76} +129.349 q^{77} -255.631 q^{79} +1123.24 q^{80} +1130.77 q^{82} +983.081 q^{83} -34.4591 q^{85} +1055.19 q^{86} +922.347 q^{88} -678.642 q^{89} -862.555 q^{91} +1583.70 q^{92} -2833.89 q^{94} -398.523 q^{95} -30.8442 q^{97} +1292.44 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} + 51 q^{4} + 30 q^{5} + 47 q^{7} - 51 q^{8} - 5 q^{10} - 81 q^{11} + 169 q^{13} + 30 q^{14} + 131 q^{16} + q^{17} + 116 q^{19} + 255 q^{20} + 90 q^{22} + 52 q^{23} + 150 q^{25} - 294 q^{26}+ \cdots + 1433 q^{98}+O(q^{100}) \) 6 * q - q^2 + 51 * q^4 + 30 * q^5 + 47 * q^7 - 51 * q^8 - 5 * q^10 - 81 * q^11 + 169 * q^13 + 30 * q^14 + 131 * q^16 + q^17 + 116 * q^19 + 255 * q^20 + 90 * q^22 + 52 * q^23 + 150 * q^25 - 294 * q^26 + 344 * q^28 - 174 * q^29 + 340 * q^31 - 499 * q^32 + 920 * q^34 + 235 * q^35 + 332 * q^37 + 378 * q^38 - 255 * q^40 + 616 * q^41 + 334 * q^43 + 52 * q^44 - 158 * q^46 + 85 * q^47 + 879 * q^49 - 25 * q^50 + 2220 * q^52 + 850 * q^53 - 405 * q^55 + 624 * q^56 + 29 * q^58 + 758 * q^59 - 36 * q^61 + 152 * q^62 + 1795 * q^64 + 845 * q^65 + 939 * q^67 + 186 * q^68 + 150 * q^70 + 1388 * q^71 + 1708 * q^73 + 814 * q^74 + 566 * q^76 - 2585 * q^77 + 1250 * q^79 + 655 * q^80 + 1372 * q^82 + 748 * q^83 + 5 * q^85 + 800 * q^86 + 536 * q^88 - 1099 * q^89 + 539 * q^91 - 1698 * q^92 - 4542 * q^94 + 580 * q^95 - 22 * q^97 + 1433 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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