LMFDB - Embedded newform 405.2.a.i.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,2,Mod(1,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	405.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-1,0,5,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.23394128186$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.08613$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.08613 q^{2} +2.35194 q^{4} -1.00000 q^{5} +4.08613 q^{7} +0.734191 q^{8} -2.08613 q^{10} +1.35194 q^{11} +0.648061 q^{13} +8.52420 q^{14} -3.17226 q^{16} +1.35194 q^{17} +0.648061 q^{19} -2.35194 q^{20} +2.82032 q^{22} -4.79001 q^{23} +1.00000 q^{25} +1.35194 q^{26} +9.61033 q^{28} -3.87614 q^{29} -7.69646 q^{31} -8.08613 q^{32} +2.82032 q^{34} -4.08613 q^{35} +7.52420 q^{37} +1.35194 q^{38} -0.734191 q^{40} +0.179679 q^{41} -0.820321 q^{43} +3.17968 q^{44} -9.99258 q^{46} -10.9065 q^{47} +9.69646 q^{49} +2.08613 q^{50} +1.52420 q^{52} -4.17226 q^{53} -1.35194 q^{55} +3.00000 q^{56} -8.08613 q^{58} -4.17226 q^{59} -3.82032 q^{61} -16.0558 q^{62} -10.5242 q^{64} -0.648061 q^{65} +8.14195 q^{67} +3.17968 q^{68} -8.52420 q^{70} +6.11644 q^{71} -12.3445 q^{73} +15.6965 q^{74} +1.52420 q^{76} +5.52420 q^{77} +10.3445 q^{79} +3.17226 q^{80} +0.374833 q^{82} +12.2584 q^{83} -1.35194 q^{85} -1.71130 q^{86} +0.992582 q^{88} +3.00000 q^{89} +2.64806 q^{91} -11.2658 q^{92} -22.7523 q^{94} -0.648061 q^{95} -13.5800 q^{97} +20.2281 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 5 q^{4} - 3 q^{5} + 5 q^{7} - 3 q^{8} + q^{10} + 2 q^{11} + 4 q^{13} + 9 q^{14} + 5 q^{16} + 2 q^{17} + 4 q^{19} - 5 q^{20} - 4 q^{22} - 3 q^{23} + 3 q^{25} + 2 q^{26} + 5 q^{28} + 7 q^{29}+ \cdots + 40 q^{98}+O(q^{100}) $ 3 * q - q^2 + 5 * q^4 - 3 * q^5 + 5 * q^7 - 3 * q^8 + q^10 + 2 * q^11 + 4 * q^13 + 9 * q^14 + 5 * q^16 + 2 * q^17 + 4 * q^19 - 5 * q^20 - 4 * q^22 - 3 * q^23 + 3 * q^25 + 2 * q^26 + 5 * q^28 + 7 * q^29 + 8 * q^31 - 17 * q^32 - 4 * q^34 - 5 * q^35 + 6 * q^37 + 2 * q^38 + 3 * q^40 + 13 * q^41 + 10 * q^43 + 22 * q^44 - 3 * q^46 - 13 * q^47 - 2 * q^49 - q^50 - 12 * q^52 + 2 * q^53 - 2 * q^55 + 9 * q^56 - 17 * q^58 + 2 * q^59 + q^61 - 42 * q^62 - 15 * q^64 - 4 * q^65 + 11 * q^67 + 22 * q^68 - 9 * q^70 + 10 * q^71 - 8 * q^73 + 16 * q^74 - 12 * q^76 + 2 * q^79 - 5 * q^80 - 29 * q^82 + 15 * q^83 - 2 * q^85 - 28 * q^86 - 24 * q^88 + 9 * q^89 + 10 * q^91 - 39 * q^92 - 31 * q^94 - 4 * q^95 - 18 * q^97 + 40 * 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 $

$2$	2.08613	1.47512	0.737558	−	0.675283i	$-0.235979\pi$
$2$	2.08613	1.47512	0.737558	+	0.675283i	$0.235979\pi$
$3$	0	0
$3$	0	0
$4$	2.35194	1.17597
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.08613	1.54441	0.772206	−	0.635372i	$-0.219153\pi$
$7$	4.08613	1.54441	0.772206	+	0.635372i	$0.219153\pi$
$8$	0.734191	0.259576
$9$	0	0
$10$	−2.08613	−0.659692
$11$	1.35194	0.407625	0.203813	−	0.979010i	$-0.434667\pi$
$11$	1.35194	0.407625	0.203813	+	0.979010i	$0.434667\pi$
$12$	0	0
$13$	0.648061	0.179740	0.0898699	−	0.995954i	$-0.471355\pi$
$13$	0.648061	0.179740	0.0898699	+	0.995954i	$0.471355\pi$
$14$	8.52420	2.27819
$15$	0	0
$16$	−3.17226	−0.793065
$17$	1.35194	0.327893	0.163947	−	0.986469i	$-0.447577\pi$
$17$	1.35194	0.327893	0.163947	+	0.986469i	$0.447577\pi$
$18$	0	0
$19$	0.648061	0.148675	0.0743377	−	0.997233i	$-0.476316\pi$
$19$	0.648061	0.148675	0.0743377	+	0.997233i	$0.476316\pi$
$20$	−2.35194	−0.525910
$21$	0	0
$22$	2.82032	0.601294
$23$	−4.79001	−0.998786	−0.499393	−	0.866376i	$-0.666444\pi$
$23$	−4.79001	−0.998786	−0.499393	+	0.866376i	$0.666444\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.35194	0.265137
$27$	0	0
$28$	9.61033	1.81618
$29$	−3.87614	−0.719781	−0.359890	−	0.932995i	$-0.617186\pi$
$29$	−3.87614	−0.719781	−0.359890	+	0.932995i	$0.617186\pi$
$30$	0	0
$31$	−7.69646	−1.38233	−0.691163	−	0.722699i	$-0.742901\pi$
$31$	−7.69646	−1.38233	−0.691163	+	0.722699i	$0.742901\pi$
$32$	−8.08613	−1.42944
$33$	0	0
$34$	2.82032	0.483681
$35$	−4.08613	−0.690682
$36$	0	0
$37$	7.52420	1.23697	0.618485	−	0.785796i	$-0.287747\pi$
$37$	7.52420	1.23697	0.618485	+	0.785796i	$0.287747\pi$
$38$	1.35194	0.219313
$39$	0	0
$40$	−0.734191	−0.116086
$41$	0.179679	0.0280611	0.0140306	−	0.999902i	$-0.495534\pi$
$41$	0.179679	0.0280611	0.0140306	+	0.999902i	$0.495534\pi$
$42$	0	0
$43$	−0.820321	−0.125098	−0.0625489	−	0.998042i	$-0.519923\pi$
$43$	−0.820321	−0.125098	−0.0625489	+	0.998042i	$0.519923\pi$
$44$	3.17968	0.479355
$45$	0	0
$46$	−9.99258	−1.47333
$47$	−10.9065	−1.59087	−0.795435	−	0.606039i	$-0.792757\pi$
$47$	−10.9065	−1.59087	−0.795435	+	0.606039i	$0.792757\pi$
$48$	0	0
$49$	9.69646	1.38521
$50$	2.08613	0.295023
$51$	0	0
$52$	1.52420	0.211368
$53$	−4.17226	−0.573104	−0.286552	−	0.958065i	$-0.592509\pi$
$53$	−4.17226	−0.573104	−0.286552	+	0.958065i	$0.592509\pi$
$54$	0	0
$55$	−1.35194	−0.182295
$56$	3.00000	0.400892
$57$	0	0
$58$	−8.08613	−1.06176
$59$	−4.17226	−0.543182	−0.271591	−	0.962413i	$-0.587550\pi$
$59$	−4.17226	−0.543182	−0.271591	+	0.962413i	$0.587550\pi$
$60$	0	0
$61$	−3.82032	−0.489142	−0.244571	−	0.969631i	$-0.578647\pi$
$61$	−3.82032	−0.489142	−0.244571	+	0.969631i	$0.578647\pi$
$62$	−16.0558	−2.03909
$63$	0	0
$64$	−10.5242	−1.31552
$65$	−0.648061	−0.0803820
$66$	0	0
$67$	8.14195	0.994697	0.497349	−	0.867551i	$-0.334307\pi$
$67$	8.14195	0.994697	0.497349	+	0.867551i	$0.334307\pi$
$68$	3.17968	0.385593
$69$	0	0
$70$	−8.52420	−1.01884
$71$	6.11644	0.725888	0.362944	−	0.931811i	$-0.381772\pi$
$71$	6.11644	0.725888	0.362944	+	0.931811i	$0.381772\pi$
$72$	0	0
$73$	−12.3445	−1.44482	−0.722408	−	0.691467i	$-0.756965\pi$
$73$	−12.3445	−1.44482	−0.722408	+	0.691467i	$0.756965\pi$
$74$	15.6965	1.82468
$75$	0	0
$76$	1.52420	0.174838
$77$	5.52420	0.629541
$78$	0	0
$79$	10.3445	1.16385	0.581925	−	0.813243i	$-0.302300\pi$
$79$	10.3445	1.16385	0.581925	+	0.813243i	$0.302300\pi$
$80$	3.17226	0.354669
$81$	0	0
$82$	0.374833	0.0413934
$83$	12.2584	1.34553	0.672767	−	0.739855i	$-0.265106\pi$
$83$	12.2584	1.34553	0.672767	+	0.739855i	$0.265106\pi$
$84$	0	0
$85$	−1.35194	−0.146638
$86$	−1.71130	−0.184534
$87$	0	0
$88$	0.992582	0.105810
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	2.64806	0.277592
$92$	−11.2658	−1.17454
$93$	0	0
$94$	−22.7523	−2.34672
$95$	−0.648061	−0.0664896
$96$	0	0
$97$	−13.5800	−1.37884	−0.689421	−	0.724361i	$-0.742135\pi$
$97$	−13.5800	−1.37884	−0.689421	+	0.724361i	$0.742135\pi$
$98$	20.2281	2.04334
$99$	0	0
$100$	2.35194	0.235194
$101$	1.46838	0.146109	0.0730547	−	0.997328i	$-0.476725\pi$
$101$	1.46838	0.146109	0.0730547	+	0.997328i	$0.476725\pi$
$102$	0	0
$103$	7.52420	0.741381	0.370691	−	0.928756i	$-0.379121\pi$
$103$	7.52420	0.741381	0.370691	+	0.928756i	$0.379121\pi$
$104$	0.475800	0.0466561
$105$	0	0
$106$	−8.70388	−0.845395
$107$	−1.20999	−0.116974	−0.0584871	−	0.998288i	$-0.518628\pi$
$107$	−1.20999	−0.116974	−0.0584871	+	0.998288i	$0.518628\pi$
$108$	0	0
$109$	14.1042	1.35094	0.675469	−	0.737388i	$-0.263941\pi$
$109$	14.1042	1.35094	0.675469	+	0.737388i	$0.263941\pi$
$110$	−2.82032	−0.268907
$111$	0	0
$112$	−12.9623	−1.22482
$113$	11.9245	1.12177	0.560883	−	0.827895i	$-0.310462\pi$
$113$	11.9245	1.12177	0.560883	+	0.827895i	$0.310462\pi$
$114$	0	0
$115$	4.79001	0.446671
$116$	−9.11644	−0.846440
$117$	0	0
$118$	−8.70388	−0.801257
$119$	5.52420	0.506403
$120$	0	0
$121$	−9.17226	−0.833842
$122$	−7.96969	−0.721542
$123$	0	0
$124$	−18.1016	−1.62557
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.07871	−0.628134	−0.314067	−	0.949401i	$-0.601692\pi$
$127$	−7.07871	−0.628134	−0.314067	+	0.949401i	$0.601692\pi$
$128$	−5.78259	−0.511114
$129$	0	0
$130$	−1.35194	−0.118573
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	2.64806	0.229616
$134$	16.9852	1.46729
$135$	0	0
$136$	0.992582	0.0851132
$137$	7.46838	0.638067	0.319033	−	0.947743i	$-0.396642\pi$
$137$	7.46838	0.638067	0.319033	+	0.947743i	$0.396642\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	−9.61033	−0.812221
$141$	0	0
$142$	12.7597	1.07077
$143$	0.876139	0.0732664
$144$	0	0
$145$	3.87614	0.321896
$146$	−25.7523	−2.13127
$147$	0	0
$148$	17.6965	1.45464
$149$	10.5848	0.867143	0.433571	−	0.901119i	$-0.357253\pi$
$149$	10.5848	0.867143	0.433571	+	0.901119i	$0.357253\pi$
$150$	0	0
$151$	17.6965	1.44012	0.720059	−	0.693913i	$-0.244115\pi$
$151$	17.6965	1.44012	0.720059	+	0.693913i	$0.244115\pi$
$152$	0.475800	0.0385925
$153$	0	0
$154$	11.5242	0.928646
$155$	7.69646	0.618195
$156$	0	0
$157$	−2.53162	−0.202045	−0.101023	−	0.994884i	$-0.532211\pi$
$157$	−2.53162	−0.202045	−0.101023	+	0.994884i	$0.532211\pi$
$158$	21.5800	1.71681
$159$	0	0
$160$	8.08613	0.639265
$161$	−19.5726	−1.54254
$162$	0	0
$163$	8.47580	0.663876	0.331938	−	0.943301i	$-0.392298\pi$
$163$	8.47580	0.663876	0.331938	+	0.943301i	$0.392298\pi$
$164$	0.422594	0.0329990
$165$	0	0
$166$	25.5726	1.98482
$167$	−12.7342	−0.985401	−0.492701	−	0.870199i	$-0.663990\pi$
$167$	−12.7342	−0.985401	−0.492701	+	0.870199i	$0.663990\pi$
$168$	0	0
$169$	−12.5800	−0.967694
$170$	−2.82032	−0.216309
$171$	0	0
$172$	−1.92935	−0.147111
$173$	−23.0484	−1.75234	−0.876169	−	0.482005i	$-0.839909\pi$
$173$	−23.0484	−1.75234	−0.876169	+	0.482005i	$0.839909\pi$
$174$	0	0
$175$	4.08613	0.308882
$176$	−4.28870	−0.323273
$177$	0	0
$178$	6.25839	0.469086
$179$	−2.22808	−0.166534	−0.0832672	−	0.996527i	$-0.526535\pi$
$179$	−2.22808	−0.166534	−0.0832672	+	0.996527i	$0.526535\pi$
$180$	0	0
$181$	0.468382	0.0348146	0.0174073	−	0.999848i	$-0.494459\pi$
$181$	0.468382	0.0348146	0.0174073	+	0.999848i	$0.494459\pi$
$182$	5.52420	0.409481
$183$	0	0
$184$	−3.51678	−0.259261
$185$	−7.52420	−0.553190
$186$	0	0
$187$	1.82774	0.133658
$188$	−25.6513	−1.87081
$189$	0	0
$190$	−1.35194	−0.0980800
$191$	20.2281	1.46365	0.731826	−	0.681491i	$-0.238668\pi$
$191$	20.2281	1.46365	0.731826	+	0.681491i	$0.238668\pi$
$192$	0	0
$193$	−19.9293	−1.43455	−0.717273	−	0.696792i	$-0.754610\pi$
$193$	−19.9293	−1.43455	−0.717273	+	0.696792i	$0.754610\pi$
$194$	−28.3297	−2.03395
$195$	0	0
$196$	22.8055	1.62896
$197$	15.5800	1.11003	0.555015	−	0.831840i	$-0.312712\pi$
$197$	15.5800	1.11003	0.555015	+	0.831840i	$0.312712\pi$
$198$	0	0
$199$	3.58482	0.254121	0.127061	−	0.991895i	$-0.459446\pi$
$199$	3.58482	0.254121	0.127061	+	0.991895i	$0.459446\pi$
$200$	0.734191	0.0519151
$201$	0	0
$202$	3.06324	0.215529
$203$	−15.8384	−1.11164
$204$	0	0
$205$	−0.179679	−0.0125493
$206$	15.6965	1.09362
$207$	0	0
$208$	−2.05582	−0.142545
$209$	0.876139	0.0606038
$210$	0	0
$211$	14.9926	1.03213	0.516066	−	0.856549i	$-0.327396\pi$
$211$	14.9926	1.03213	0.516066	+	0.856549i	$0.327396\pi$
$212$	−9.81290	−0.673953
$213$	0	0
$214$	−2.52420	−0.172551
$215$	0.820321	0.0559454
$216$	0	0
$217$	−31.4487	−2.13488
$218$	29.4232	1.99279
$219$	0	0
$220$	−3.17968	−0.214374
$221$	0.876139	0.0589355
$222$	0	0
$223$	−26.8310	−1.79674	−0.898368	−	0.439244i	$-0.855246\pi$
$223$	−26.8310	−1.79674	−0.898368	+	0.439244i	$0.855246\pi$
$224$	−33.0410	−2.20764
$225$	0	0
$226$	24.8761	1.65474
$227$	−1.35194	−0.0897314	−0.0448657	−	0.998993i	$-0.514286\pi$
$227$	−1.35194	−0.0897314	−0.0448657	+	0.998993i	$0.514286\pi$
$228$	0	0
$229$	−8.23550	−0.544217	−0.272108	−	0.962267i	$-0.587721\pi$
$229$	−8.23550	−0.544217	−0.272108	+	0.962267i	$0.587721\pi$
$230$	9.99258	0.658891
$231$	0	0
$232$	−2.84583	−0.186838
$233$	−8.58744	−0.562582	−0.281291	−	0.959623i	$-0.590763\pi$
$233$	−8.58744	−0.562582	−0.281291	+	0.959623i	$0.590763\pi$
$234$	0	0
$235$	10.9065	0.711458
$236$	−9.81290	−0.638766
$237$	0	0
$238$	11.5242	0.747003
$239$	23.9245	1.54755	0.773775	−	0.633461i	$-0.218366\pi$
$239$	23.9245	1.54755	0.773775	+	0.633461i	$0.218366\pi$
$240$	0	0
$241$	−6.24030	−0.401973	−0.200987	−	0.979594i	$-0.564415\pi$
$241$	−6.24030	−0.401973	−0.200987	+	0.979594i	$0.564415\pi$
$242$	−19.1345	−1.23001
$243$	0	0
$244$	−8.98516	−0.575216
$245$	−9.69646	−0.619484
$246$	0	0
$247$	0.419983	0.0267229
$248$	−5.65067	−0.358818
$249$	0	0
$250$	−2.08613	−0.131938
$251$	28.5726	1.80349	0.901743	−	0.432272i	$-0.142288\pi$
$251$	28.5726	1.80349	0.901743	+	0.432272i	$0.142288\pi$
$252$	0	0
$253$	−6.47580	−0.407130
$254$	−14.7671	−0.926571
$255$	0	0
$256$	8.98516	0.561573
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	30.7449	1.91039
$260$	−1.52420	−0.0945268
$261$	0	0
$262$	12.5168	0.773289
$263$	−31.8687	−1.96511	−0.982555	−	0.185974i	$-0.940456\pi$
$263$	−31.8687	−1.96511	−0.982555	+	0.185974i	$0.940456\pi$
$264$	0	0
$265$	4.17226	0.256300
$266$	5.52420	0.338710
$267$	0	0
$268$	19.1494	1.16973
$269$	31.4971	1.92041	0.960207	−	0.279289i	$-0.0900987\pi$
$269$	31.4971	1.92041	0.960207	+	0.279289i	$0.0900987\pi$
$270$	0	0
$271$	−3.24030	−0.196834	−0.0984172	−	0.995145i	$-0.531378\pi$
$271$	−3.24030	−0.196834	−0.0984172	+	0.995145i	$0.531378\pi$
$272$	−4.28870	−0.260041
$273$	0	0
$274$	15.5800	0.941223
$275$	1.35194	0.0815250
$276$	0	0
$277$	−5.58482	−0.335560	−0.167780	−	0.985824i	$-0.553660\pi$
$277$	−5.58482	−0.335560	−0.167780	+	0.985824i	$0.553660\pi$
$278$	16.6890	1.00094
$279$	0	0
$280$	−3.00000	−0.179284
$281$	24.1042	1.43794	0.718969	−	0.695043i	$-0.244615\pi$
$281$	24.1042	1.43794	0.718969	+	0.695043i	$0.244615\pi$
$282$	0	0
$283$	−10.5423	−0.626674	−0.313337	−	0.949642i	$-0.601447\pi$
$283$	−10.5423	−0.626674	−0.313337	+	0.949642i	$0.601447\pi$
$284$	14.3855	0.853622
$285$	0	0
$286$	1.82774	0.108077
$287$	0.734191	0.0433379
$288$	0	0
$289$	−15.1723	−0.892486
$290$	8.08613	0.474834
$291$	0	0
$292$	−29.0336	−1.69906
$293$	−18.9926	−1.10956	−0.554779	−	0.831998i	$-0.687197\pi$
$293$	−18.9926	−1.10956	−0.554779	+	0.831998i	$0.687197\pi$
$294$	0	0
$295$	4.17226	0.242918
$296$	5.52420	0.321088
$297$	0	0
$298$	22.0813	1.27914
$299$	−3.10422	−0.179521
$300$	0	0
$301$	−3.35194	−0.193203
$302$	36.9171	2.12434
$303$	0	0
$304$	−2.05582	−0.117909
$305$	3.82032	0.218751
$306$	0	0
$307$	29.4791	1.68246	0.841229	−	0.540679i	$-0.181833\pi$
$307$	29.4791	1.68246	0.841229	+	0.540679i	$0.181833\pi$
$308$	12.9926	0.740321
$309$	0	0
$310$	16.0558	0.911909
$311$	9.41256	0.533738	0.266869	−	0.963733i	$-0.414011\pi$
$311$	9.41256	0.533738	0.266869	+	0.963733i	$0.414011\pi$
$312$	0	0
$313$	11.6210	0.656858	0.328429	−	0.944529i	$-0.393481\pi$
$313$	11.6210	0.656858	0.328429	+	0.944529i	$0.393481\pi$
$314$	−5.28128	−0.298040
$315$	0	0
$316$	24.3297	1.36865
$317$	9.17968	0.515582	0.257791	−	0.966201i	$-0.417005\pi$
$317$	9.17968	0.515582	0.257791	+	0.966201i	$0.417005\pi$
$318$	0	0
$319$	−5.24030	−0.293401
$320$	10.5242	0.588321
$321$	0	0
$322$	−40.8310	−2.27542
$323$	0.876139	0.0487497
$324$	0	0
$325$	0.648061	0.0359479
$326$	17.6816	0.979295
$327$	0	0
$328$	0.131919	0.00728398
$329$	−44.5652	−2.45696
$330$	0	0
$331$	−7.22066	−0.396883	−0.198442	−	0.980113i	$-0.563588\pi$
$331$	−7.22066	−0.396883	−0.198442	+	0.980113i	$0.563588\pi$
$332$	28.8310	1.58231
$333$	0	0
$334$	−26.5652	−1.45358
$335$	−8.14195	−0.444842
$336$	0	0
$337$	2.28390	0.124412	0.0622059	−	0.998063i	$-0.480186\pi$
$337$	2.28390	0.124412	0.0622059	+	0.998063i	$0.480186\pi$
$338$	−26.2436	−1.42746
$339$	0	0
$340$	−3.17968	−0.172442
$341$	−10.4051	−0.563470
$342$	0	0
$343$	11.0181	0.594921
$344$	−0.602272	−0.0324724
$345$	0	0
$346$	−48.0820	−2.58490
$347$	0.708686	0.0380443	0.0190221	−	0.999819i	$-0.493945\pi$
$347$	0.708686	0.0380443	0.0190221	+	0.999819i	$0.493945\pi$
$348$	0	0
$349$	−21.3445	−1.14255	−0.571273	−	0.820760i	$-0.693550\pi$
$349$	−21.3445	−1.14255	−0.571273	+	0.820760i	$0.693550\pi$
$350$	8.52420	0.455638
$351$	0	0
$352$	−10.9320	−0.582675
$353$	10.0968	0.537398	0.268699	−	0.963224i	$-0.413406\pi$
$353$	10.0968	0.537398	0.268699	+	0.963224i	$0.413406\pi$
$354$	0	0
$355$	−6.11644	−0.324627
$356$	7.05582	0.373958
$357$	0	0
$358$	−4.64806	−0.245658
$359$	−30.5578	−1.61278	−0.806388	−	0.591386i	$-0.798581\pi$
$359$	−30.5578	−1.61278	−0.806388	+	0.591386i	$0.798581\pi$
$360$	0	0
$361$	−18.5800	−0.977896
$362$	0.977106	0.0513555
$363$	0	0
$364$	6.22808	0.326440
$365$	12.3445	0.646142
$366$	0	0
$367$	−7.17968	−0.374776	−0.187388	−	0.982286i	$-0.560002\pi$
$367$	−7.17968	−0.374776	−0.187388	+	0.982286i	$0.560002\pi$
$368$	15.1952	0.792102
$369$	0	0
$370$	−15.6965	−0.816020
$371$	−17.0484	−0.885109
$372$	0	0
$373$	−21.9245	−1.13521	−0.567605	−	0.823301i	$-0.692130\pi$
$373$	−21.9245	−1.13521	−0.567605	+	0.823301i	$0.692130\pi$
$374$	3.81290	0.197161
$375$	0	0
$376$	−8.00742	−0.412951
$377$	−2.51197	−0.129373
$378$	0	0
$379$	−17.3929	−0.893414	−0.446707	−	0.894680i	$-0.647403\pi$
$379$	−17.3929	−0.893414	−0.446707	+	0.894680i	$0.647403\pi$
$380$	−1.52420	−0.0781898
$381$	0	0
$382$	42.1984	2.15906
$383$	−0.475800	−0.0243123	−0.0121561	−	0.999926i	$-0.503870\pi$
$383$	−0.475800	−0.0243123	−0.0121561	+	0.999926i	$0.503870\pi$
$384$	0	0
$385$	−5.52420	−0.281539
$386$	−41.5752	−2.11612
$387$	0	0
$388$	−31.9394	−1.62148
$389$	5.58744	0.283294	0.141647	−	0.989917i	$-0.454760\pi$
$389$	5.58744	0.283294	0.141647	+	0.989917i	$0.454760\pi$
$390$	0	0
$391$	−6.47580	−0.327495
$392$	7.11905	0.359567
$393$	0	0
$394$	32.5019	1.63742
$395$	−10.3445	−0.520489
$396$	0	0
$397$	3.75228	0.188321	0.0941607	−	0.995557i	$-0.469983\pi$
$397$	3.75228	0.188321	0.0941607	+	0.995557i	$0.469983\pi$
$398$	7.47841	0.374859
$399$	0	0
$400$	−3.17226	−0.158613
$401$	−23.5652	−1.17679	−0.588394	−	0.808574i	$-0.700240\pi$
$401$	−23.5652	−1.17679	−0.588394	+	0.808574i	$0.700240\pi$
$402$	0	0
$403$	−4.98777	−0.248459
$404$	3.45355	0.171820
$405$	0	0
$406$	−33.0410	−1.63980
$407$	10.1723	0.504220
$408$	0	0
$409$	1.04840	0.0518400	0.0259200	−	0.999664i	$-0.491748\pi$
$409$	1.04840	0.0518400	0.0259200	+	0.999664i	$0.491748\pi$
$410$	−0.374833	−0.0185117
$411$	0	0
$412$	17.6965	0.871842
$413$	−17.0484	−0.838897
$414$	0	0
$415$	−12.2584	−0.601741
$416$	−5.24030	−0.256927
$417$	0	0
$418$	1.82774	0.0893977
$419$	25.9197	1.26626	0.633131	−	0.774045i	$-0.281769\pi$
$419$	25.9197	1.26626	0.633131	+	0.774045i	$0.281769\pi$
$420$	0	0
$421$	7.64064	0.372382	0.186191	−	0.982514i	$-0.440386\pi$
$421$	7.64064	0.372382	0.186191	+	0.982514i	$0.440386\pi$
$422$	31.2765	1.52252
$423$	0	0
$424$	−3.06324	−0.148764
$425$	1.35194	0.0655787
$426$	0	0
$427$	−15.6103	−0.755437
$428$	−2.84583	−0.137558
$429$	0	0
$430$	1.71130	0.0825261
$431$	−7.98516	−0.384632	−0.192316	−	0.981333i	$-0.561600\pi$
$431$	−7.98516	−0.384632	−0.192316	+	0.981333i	$0.561600\pi$
$432$	0	0
$433$	−12.5120	−0.601287	−0.300644	−	0.953737i	$-0.597201\pi$
$433$	−12.5120	−0.601287	−0.300644	+	0.953737i	$0.597201\pi$
$434$	−65.6062	−3.14920
$435$	0	0
$436$	33.1723	1.58866
$437$	−3.10422	−0.148495
$438$	0	0
$439$	−8.76450	−0.418307	−0.209153	−	0.977883i	$-0.567071\pi$
$439$	−8.76450	−0.418307	−0.209153	+	0.977883i	$0.567071\pi$
$440$	−0.992582	−0.0473195
$441$	0	0
$442$	1.82774	0.0869367
$443$	−3.67095	−0.174412	−0.0872062	−	0.996190i	$-0.527794\pi$
$443$	−3.67095	−0.174412	−0.0872062	+	0.996190i	$0.527794\pi$
$444$	0	0
$445$	−3.00000	−0.142214
$446$	−55.9729	−2.65040
$447$	0	0
$448$	−43.0032	−2.03171
$449$	−28.1723	−1.32953	−0.664766	−	0.747052i	$-0.731469\pi$
$449$	−28.1723	−1.32953	−0.664766	+	0.747052i	$0.731469\pi$
$450$	0	0
$451$	0.242915	0.0114384
$452$	28.0458	1.31916
$453$	0	0
$454$	−2.82032	−0.132364
$455$	−2.64806	−0.124143
$456$	0	0
$457$	35.2616	1.64947	0.824735	−	0.565519i	$-0.191324\pi$
$457$	35.2616	1.64947	0.824735	+	0.565519i	$0.191324\pi$
$458$	−17.1803	−0.802784
$459$	0	0
$460$	11.2658	0.525271
$461$	−34.6768	−1.61506	−0.807530	−	0.589826i	$-0.799196\pi$
$461$	−34.6768	−1.61506	−0.807530	+	0.589826i	$0.799196\pi$
$462$	0	0
$463$	7.44874	0.346172	0.173086	−	0.984907i	$-0.444626\pi$
$463$	7.44874	0.346172	0.173086	+	0.984907i	$0.444626\pi$
$464$	12.2961	0.570833
$465$	0	0
$466$	−17.9145	−0.829874
$467$	−29.9655	−1.38664	−0.693319	−	0.720630i	$-0.743852\pi$
$467$	−29.9655	−1.38664	−0.693319	+	0.720630i	$0.743852\pi$
$468$	0	0
$469$	33.2691	1.53622
$470$	22.7523	1.04948
$471$	0	0
$472$	−3.06324	−0.140997
$473$	−1.10902	−0.0509930
$474$	0	0
$475$	0.648061	0.0297351
$476$	12.9926	0.595514
$477$	0	0
$478$	49.9097	2.28282
$479$	7.98516	0.364851	0.182426	−	0.983220i	$-0.441605\pi$
$479$	7.98516	0.364851	0.182426	+	0.983220i	$0.441605\pi$
$480$	0	0
$481$	4.87614	0.222333
$482$	−13.0181	−0.592958
$483$	0	0
$484$	−21.5726	−0.980573
$485$	13.5800	0.616637
$486$	0	0
$487$	−11.9442	−0.541243	−0.270621	−	0.962686i	$-0.587229\pi$
$487$	−11.9442	−0.541243	−0.270621	+	0.962686i	$0.587229\pi$
$488$	−2.80485	−0.126969
$489$	0	0
$490$	−20.2281	−0.913811
$491$	9.22066	0.416123	0.208061	−	0.978116i	$-0.433285\pi$
$491$	9.22066	0.416123	0.208061	+	0.978116i	$0.433285\pi$
$492$	0	0
$493$	−5.24030	−0.236011
$494$	0.876139	0.0394193
$495$	0	0
$496$	24.4152	1.09627
$497$	24.9926	1.12107
$498$	0	0
$499$	30.1723	1.35070	0.675348	−	0.737499i	$-0.263993\pi$
$499$	30.1723	1.35070	0.675348	+	0.737499i	$0.263993\pi$
$500$	−2.35194	−0.105182
$501$	0	0
$502$	59.6062	2.66035
$503$	10.5981	0.472546	0.236273	−	0.971687i	$-0.424074\pi$
$503$	10.5981	0.472546	0.236273	+	0.971687i	$0.424074\pi$
$504$	0	0
$505$	−1.46838	−0.0653421
$506$	−13.5094	−0.600564
$507$	0	0
$508$	−16.6487	−0.738667
$509$	−28.7523	−1.27442	−0.637211	−	0.770689i	$-0.719912\pi$
$509$	−28.7523	−1.27442	−0.637211	+	0.770689i	$0.719912\pi$
$510$	0	0
$511$	−50.4413	−2.23139
$512$	30.3094	1.33950
$513$	0	0
$514$	37.5503	1.65627
$515$	−7.52420	−0.331556
$516$	0	0
$517$	−14.7449	−0.648478
$518$	64.1378	2.81805
$519$	0	0
$520$	−0.475800	−0.0208652
$521$	36.0942	1.58132	0.790658	−	0.612259i	$-0.209739\pi$
$521$	36.0942	1.58132	0.790658	+	0.612259i	$0.209739\pi$
$522$	0	0
$523$	11.1297	0.486669	0.243334	−	0.969942i	$-0.421759\pi$
$523$	11.1297	0.486669	0.243334	+	0.969942i	$0.421759\pi$
$524$	14.1116	0.616470
$525$	0	0
$526$	−66.4823	−2.89877
$527$	−10.4051	−0.453255
$528$	0	0
$529$	−0.0558176	−0.00242685
$530$	8.70388	0.378072
$531$	0	0
$532$	6.22808	0.270021
$533$	0.116443	0.00504370
$534$	0	0
$535$	1.20999	0.0523125
$536$	5.97774	0.258199
$537$	0	0
$538$	65.7071	2.83284
$539$	13.1090	0.564646
$540$	0	0
$541$	−34.7374	−1.49348	−0.746740	−	0.665116i	$-0.768382\pi$
$541$	−34.7374	−1.49348	−0.746740	+	0.665116i	$0.768382\pi$
$542$	−6.75970	−0.290354
$543$	0	0
$544$	−10.9320	−0.468704
$545$	−14.1042	−0.604158
$546$	0	0
$547$	−2.71455	−0.116066	−0.0580328	−	0.998315i	$-0.518483\pi$
$547$	−2.71455	−0.116066	−0.0580328	+	0.998315i	$0.518483\pi$
$548$	17.5652	0.750347
$549$	0	0
$550$	2.82032	0.120259
$551$	−2.51197	−0.107014
$552$	0	0
$553$	42.2691	1.79746
$554$	−11.6507	−0.494990
$555$	0	0
$556$	18.8155	0.797956
$557$	8.93676	0.378663	0.189331	−	0.981913i	$-0.439368\pi$
$557$	8.93676	0.378663	0.189331	+	0.981913i	$0.439368\pi$
$558$	0	0
$559$	−0.531618	−0.0224850
$560$	12.9623	0.547756
$561$	0	0
$562$	50.2845	2.12113
$563$	9.36261	0.394587	0.197293	−	0.980344i	$-0.436785\pi$
$563$	9.36261	0.394587	0.197293	+	0.980344i	$0.436785\pi$
$564$	0	0
$565$	−11.9245	−0.501669
$566$	−21.9926	−0.924417
$567$	0	0
$568$	4.49064	0.188423
$569$	−35.8735	−1.50390	−0.751948	−	0.659222i	$-0.770886\pi$
$569$	−35.8735	−1.50390	−0.751948	+	0.659222i	$0.770886\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	2.06063	0.0861591
$573$	0	0
$574$	1.53162	0.0639285
$575$	−4.79001	−0.199757
$576$	0	0
$577$	1.35675	0.0564821	0.0282411	−	0.999601i	$-0.491009\pi$
$577$	1.35675	0.0564821	0.0282411	+	0.999601i	$0.491009\pi$
$578$	−31.6513	−1.31652
$579$	0	0
$580$	9.11644	0.378540
$581$	50.0894	2.07806
$582$	0	0
$583$	−5.64064	−0.233612
$584$	−9.06324	−0.375039
$585$	0	0
$586$	−39.6210	−1.63673
$587$	−28.7900	−1.18829	−0.594145	−	0.804358i	$-0.702510\pi$
$587$	−28.7900	−1.18829	−0.594145	+	0.804358i	$0.702510\pi$
$588$	0	0
$589$	−4.98777	−0.205518
$590$	8.70388	0.358333
$591$	0	0
$592$	−23.8687	−0.980998
$593$	−30.9171	−1.26961	−0.634807	−	0.772671i	$-0.718920\pi$
$593$	−30.9171	−1.26961	−0.634807	+	0.772671i	$0.718920\pi$
$594$	0	0
$595$	−5.52420	−0.226470
$596$	24.8949	1.01973
$597$	0	0
$598$	−6.47580	−0.264815
$599$	−1.39292	−0.0569132	−0.0284566	−	0.999595i	$-0.509059\pi$
$599$	−1.39292	−0.0569132	−0.0284566	+	0.999595i	$0.509059\pi$
$600$	0	0
$601$	8.82513	0.359985	0.179992	−	0.983668i	$-0.442393\pi$
$601$	8.82513	0.359985	0.179992	+	0.983668i	$0.442393\pi$
$602$	−6.99258	−0.284996
$603$	0	0
$604$	41.6210	1.69353
$605$	9.17226	0.372905
$606$	0	0
$607$	−2.15678	−0.0875412	−0.0437706	−	0.999042i	$-0.513937\pi$
$607$	−2.15678	−0.0875412	−0.0437706	+	0.999042i	$0.513937\pi$
$608$	−5.24030	−0.212522
$609$	0	0
$610$	7.96969	0.322683
$611$	−7.06804	−0.285942
$612$	0	0
$613$	−9.57521	−0.386739	−0.193370	−	0.981126i	$-0.561942\pi$
$613$	−9.57521	−0.386739	−0.193370	+	0.981126i	$0.561942\pi$
$614$	61.4971	2.48182
$615$	0	0
$616$	4.05582	0.163414
$617$	37.6768	1.51681	0.758406	−	0.651783i	$-0.225979\pi$
$617$	37.6768	1.51681	0.758406	+	0.651783i	$0.225979\pi$
$618$	0	0
$619$	17.1042	0.687477	0.343738	−	0.939065i	$-0.388307\pi$
$619$	17.1042	0.687477	0.343738	+	0.939065i	$0.388307\pi$
$620$	18.1016	0.726978
$621$	0	0
$622$	19.6358	0.787325
$623$	12.2584	0.491122
$624$	0	0
$625$	1.00000	0.0400000
$626$	24.2429	0.968942
$627$	0	0
$628$	−5.95421	−0.237599
$629$	10.1723	0.405595
$630$	0	0
$631$	33.1090	1.31805	0.659025	−	0.752121i	$-0.270969\pi$
$631$	33.1090	1.31805	0.659025	+	0.752121i	$0.270969\pi$
$632$	7.59485	0.302107
$633$	0	0
$634$	19.1500	0.760544
$635$	7.07871	0.280910
$636$	0	0
$637$	6.28390	0.248977
$638$	−10.9320	−0.432800
$639$	0	0
$640$	5.78259	0.228577
$641$	23.1526	0.914473	0.457237	−	0.889345i	$-0.348839\pi$
$641$	23.1526	0.914473	0.457237	+	0.889345i	$0.348839\pi$
$642$	0	0
$643$	43.0639	1.69827	0.849137	−	0.528173i	$-0.177123\pi$
$643$	43.0639	1.69827	0.849137	+	0.528173i	$0.177123\pi$
$644$	−46.0336	−1.81398
$645$	0	0
$646$	1.82774	0.0719115
$647$	20.6439	0.811595	0.405798	−	0.913963i	$-0.366994\pi$
$647$	20.6439	0.811595	0.405798	+	0.913963i	$0.366994\pi$
$648$	0	0
$649$	−5.64064	−0.221415
$650$	1.35194	0.0530274
$651$	0	0
$652$	19.9346	0.780698
$653$	−6.83516	−0.267480	−0.133740	−	0.991016i	$-0.542699\pi$
$653$	−6.83516	−0.267480	−0.133740	+	0.991016i	$0.542699\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	−0.569988	−0.0222543
$657$	0	0
$658$	−92.9688	−3.62430
$659$	26.8613	1.04637	0.523184	−	0.852220i	$-0.324744\pi$
$659$	26.8613	1.04637	0.523184	+	0.852220i	$0.324744\pi$
$660$	0	0
$661$	−2.12125	−0.0825071	−0.0412535	−	0.999149i	$-0.513135\pi$
$661$	−2.12125	−0.0825071	−0.0412535	+	0.999149i	$0.513135\pi$
$662$	−15.0632	−0.585449
$663$	0	0
$664$	9.00000	0.349268
$665$	−2.64806	−0.102687
$666$	0	0
$667$	18.5667	0.718907
$668$	−29.9500	−1.15880
$669$	0	0
$670$	−16.9852	−0.656194
$671$	−5.16484	−0.199387
$672$	0	0
$673$	34.8203	1.34222	0.671112	−	0.741356i	$-0.265817\pi$
$673$	34.8203	1.34222	0.671112	+	0.741356i	$0.265817\pi$
$674$	4.76450	0.183522
$675$	0	0
$676$	−29.5874	−1.13798
$677$	24.6842	0.948692	0.474346	−	0.880338i	$-0.342685\pi$
$677$	24.6842	0.948692	0.474346	+	0.880338i	$0.342685\pi$
$678$	0	0
$679$	−55.4897	−2.12950
$680$	−0.992582	−0.0380638
$681$	0	0
$682$	−21.7065	−0.831184
$683$	38.4610	1.47167	0.735834	−	0.677162i	$-0.236790\pi$
$683$	38.4610	1.47167	0.735834	+	0.677162i	$0.236790\pi$
$684$	0	0
$685$	−7.46838	−0.285352
$686$	22.9852	0.877578
$687$	0	0
$688$	2.60227	0.0992107
$689$	−2.70388	−0.103010
$690$	0	0
$691$	0.480608	0.0182832	0.00914159	−	0.999958i	$-0.497090\pi$
$691$	0.480608	0.0182832	0.00914159	+	0.999958i	$0.497090\pi$
$692$	−54.2084	−2.06070
$693$	0	0
$694$	1.47841	0.0561197
$695$	−8.00000	−0.303457
$696$	0	0
$697$	0.242915	0.00920105
$698$	−44.5274	−1.68539
$699$	0	0
$700$	9.61033	0.363236
$701$	18.1797	0.686637	0.343318	−	0.939219i	$-0.388449\pi$
$701$	18.1797	0.686637	0.343318	+	0.939219i	$0.388449\pi$
$702$	0	0
$703$	4.87614	0.183907
$704$	−14.2281	−0.536241
$705$	0	0
$706$	21.0632	0.792725
$707$	6.00000	0.225653
$708$	0	0
$709$	7.18710	0.269917	0.134959	−	0.990851i	$-0.456910\pi$
$709$	7.18710	0.269917	0.134959	+	0.990851i	$0.456910\pi$
$710$	−12.7597	−0.478863
$711$	0	0
$712$	2.20257	0.0825449
$713$	36.8661	1.38065
$714$	0	0
$715$	−0.876139	−0.0327657
$716$	−5.24030	−0.195839
$717$	0	0
$718$	−63.7475	−2.37903
$719$	−12.5168	−0.466797	−0.233399	−	0.972381i	$-0.574985\pi$
$719$	−12.5168	−0.466797	−0.233399	+	0.972381i	$0.574985\pi$
$720$	0	0
$721$	30.7449	1.14500
$722$	−38.7603	−1.44251
$723$	0	0
$724$	1.10161	0.0409409
$725$	−3.87614	−0.143956
$726$	0	0
$727$	8.42584	0.312497	0.156249	−	0.987718i	$-0.450060\pi$
$727$	8.42584	0.312497	0.156249	+	0.987718i	$0.450060\pi$
$728$	1.94418	0.0720562
$729$	0	0
$730$	25.7523	0.953135
$731$	−1.10902	−0.0410187
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	−14.9777	−0.552839
$735$	0	0
$736$	38.7326	1.42770
$737$	11.0074	0.405463
$738$	0	0
$739$	−1.81290	−0.0666887	−0.0333444	−	0.999444i	$-0.510616\pi$
$739$	−1.81290	−0.0666887	−0.0333444	+	0.999444i	$0.510616\pi$
$740$	−17.6965	−0.650535
$741$	0	0
$742$	−35.5652	−1.30564
$743$	−20.1371	−0.738760	−0.369380	−	0.929278i	$-0.620430\pi$
$743$	−20.1371	−0.738760	−0.369380	+	0.929278i	$0.620430\pi$
$744$	0	0
$745$	−10.5848	−0.387798
$746$	−45.7374	−1.67457
$747$	0	0
$748$	4.29873	0.157177
$749$	−4.94418	−0.180656
$750$	0	0
$751$	12.2132	0.445668	0.222834	−	0.974856i	$-0.428469\pi$
$751$	12.2132	0.445668	0.222834	+	0.974856i	$0.428469\pi$
$752$	34.5981	1.26166
$753$	0	0
$754$	−5.24030	−0.190841
$755$	−17.6965	−0.644040
$756$	0	0
$757$	52.9533	1.92462	0.962310	−	0.271955i	$-0.0876701\pi$
$757$	52.9533	1.92462	0.962310	+	0.271955i	$0.0876701\pi$
$758$	−36.2839	−1.31789
$759$	0	0
$760$	−0.475800	−0.0172591
$761$	18.4535	0.668940	0.334470	−	0.942406i	$-0.391443\pi$
$761$	18.4535	0.668940	0.334470	+	0.942406i	$0.391443\pi$
$762$	0	0
$763$	57.6317	2.08641
$764$	47.5752	1.72121
$765$	0	0
$766$	−0.992582	−0.0358634
$767$	−2.70388	−0.0976314
$768$	0	0
$769$	−4.45355	−0.160599	−0.0802995	−	0.996771i	$-0.525588\pi$
$769$	−4.45355	−0.160599	−0.0802995	+	0.996771i	$0.525588\pi$
$770$	−11.5242	−0.415303
$771$	0	0
$772$	−46.8726	−1.68698
$773$	−38.9368	−1.40046	−0.700229	−	0.713918i	$-0.746919\pi$
$773$	−38.9368	−1.40046	−0.700229	+	0.713918i	$0.746919\pi$
$774$	0	0
$775$	−7.69646	−0.276465
$776$	−9.97033	−0.357914
$777$	0	0
$778$	11.6561	0.417892
$779$	0.116443	0.00417200
$780$	0	0
$781$	8.26906	0.295890
$782$	−13.5094	−0.483094
$783$	0	0
$784$	−30.7597	−1.09856
$785$	2.53162	0.0903573
$786$	0	0
$787$	34.2281	1.22010	0.610050	−	0.792363i	$-0.291150\pi$
$787$	34.2281	1.22010	0.610050	+	0.792363i	$0.291150\pi$
$788$	36.6433	1.30536
$789$	0	0
$790$	−21.5800	−0.767783
$791$	48.7252	1.73247
$792$	0	0
$793$	−2.47580	−0.0879183
$794$	7.82774	0.277796
$795$	0	0
$796$	8.43129	0.298839
$797$	−23.9655	−0.848902	−0.424451	−	0.905451i	$-0.639533\pi$
$797$	−23.9655	−0.848902	−0.424451	+	0.905451i	$0.639533\pi$
$798$	0	0
$799$	−14.7449	−0.521636
$800$	−8.08613	−0.285888
$801$	0	0
$802$	−49.1600	−1.73590
$803$	−16.6890	−0.588943
$804$	0	0
$805$	19.5726	0.689843
$806$	−10.4051	−0.366506
$807$	0	0
$808$	1.07807	0.0379265
$809$	0.283896	0.00998124	0.00499062	−	0.999988i	$-0.498411\pi$
$809$	0.283896	0.00998124	0.00499062	+	0.999988i	$0.498411\pi$
$810$	0	0
$811$	32.4413	1.13917	0.569584	−	0.821933i	$-0.307104\pi$
$811$	32.4413	1.13917	0.569584	+	0.821933i	$0.307104\pi$
$812$	−37.2510	−1.30725
$813$	0	0
$814$	21.2207	0.743784
$815$	−8.47580	−0.296894
$816$	0	0
$817$	−0.531618	−0.0185990
$818$	2.18710	0.0764701
$819$	0	0
$820$	−0.422594	−0.0147576
$821$	−41.6694	−1.45427	−0.727136	−	0.686493i	$-0.759149\pi$
$821$	−41.6694	−1.45427	−0.727136	+	0.686493i	$0.759149\pi$
$822$	0	0
$823$	−19.3626	−0.674938	−0.337469	−	0.941337i	$-0.609571\pi$
$823$	−19.3626	−0.674938	−0.337469	+	0.941337i	$0.609571\pi$
$824$	5.52420	0.192445
$825$	0	0
$826$	−35.5652	−1.23747
$827$	−18.8097	−0.654076	−0.327038	−	0.945011i	$-0.606050\pi$
$827$	−18.8097	−0.654076	−0.327038	+	0.945011i	$0.606050\pi$
$828$	0	0
$829$	−33.1016	−1.14967	−0.574833	−	0.818271i	$-0.694933\pi$
$829$	−33.1016	−1.14967	−0.574833	+	0.818271i	$0.694933\pi$
$830$	−25.5726	−0.887638
$831$	0	0
$832$	−6.82032	−0.236452
$833$	13.1090	0.454201
$834$	0	0
$835$	12.7342	0.440685
$836$	2.06063	0.0712682
$837$	0	0
$838$	54.0719	1.86788
$839$	39.6965	1.37047	0.685237	−	0.728320i	$-0.259699\pi$
$839$	39.6965	1.37047	0.685237	+	0.728320i	$0.259699\pi$
$840$	0	0
$841$	−13.9755	−0.481915
$842$	15.9394	0.549307
$843$	0	0
$844$	35.2616	1.21376
$845$	12.5800	0.432766
$846$	0	0
$847$	−37.4791	−1.28780
$848$	13.2355	0.454509
$849$	0	0
$850$	2.82032	0.0967362
$851$	−36.0410	−1.23547
$852$	0	0
$853$	−4.11644	−0.140944	−0.0704722	−	0.997514i	$-0.522451\pi$
$853$	−4.11644	−0.140944	−0.0704722	+	0.997514i	$0.522451\pi$
$854$	−32.5652	−1.11436
$855$	0	0
$856$	−0.888365	−0.0303637
$857$	−14.7449	−0.503675	−0.251837	−	0.967770i	$-0.581035\pi$
$857$	−14.7449	−0.503675	−0.251837	+	0.967770i	$0.581035\pi$
$858$	0	0
$859$	−37.8539	−1.29156	−0.645779	−	0.763524i	$-0.723467\pi$
$859$	−37.8539	−1.29156	−0.645779	+	0.763524i	$0.723467\pi$
$860$	1.92935	0.0657901
$861$	0	0
$862$	−16.6581	−0.567377
$863$	−26.7704	−0.911274	−0.455637	−	0.890166i	$-0.650588\pi$
$863$	−26.7704	−0.911274	−0.455637	+	0.890166i	$0.650588\pi$
$864$	0	0
$865$	23.0484	0.783669
$866$	−26.1016	−0.886969
$867$	0	0
$868$	−73.9655	−2.51055
$869$	13.9852	0.474414
$870$	0	0
$871$	5.27648	0.178787
$872$	10.3552	0.350671
$873$	0	0
$874$	−6.47580	−0.219047
$875$	−4.08613	−0.138136
$876$	0	0
$877$	−46.9681	−1.58600	−0.793001	−	0.609221i	$-0.791482\pi$
$877$	−46.9681	−1.58600	−0.793001	+	0.609221i	$0.791482\pi$
$878$	−18.2839	−0.617052
$879$	0	0
$880$	4.28870	0.144572
$881$	19.8055	0.667264	0.333632	−	0.942703i	$-0.391726\pi$
$881$	19.8055	0.667264	0.333632	+	0.942703i	$0.391726\pi$
$882$	0	0
$883$	−6.20257	−0.208733	−0.104367	−	0.994539i	$-0.533282\pi$
$883$	−6.20257	−0.208733	−0.104367	+	0.994539i	$0.533282\pi$
$884$	2.06063	0.0693063
$885$	0	0
$886$	−7.65809	−0.257279
$887$	−13.4274	−0.450848	−0.225424	−	0.974261i	$-0.572377\pi$
$887$	−13.4274	−0.450848	−0.225424	+	0.974261i	$0.572377\pi$
$888$	0	0
$889$	−28.9245	−0.970098
$890$	−6.25839	−0.209782
$891$	0	0
$892$	−63.1049	−2.11291
$893$	−7.06804	−0.236523
$894$	0	0
$895$	2.22808	0.0744764
$896$	−23.6284	−0.789370
$897$	0	0
$898$	−58.7710	−1.96121
$899$	29.8325	0.994971
$900$	0	0
$901$	−5.64064	−0.187917
$902$	0.506752	0.0168730
$903$	0	0
$904$	8.75489	0.291183
$905$	−0.468382	−0.0155695
$906$	0	0
$907$	0.673566	0.0223654	0.0111827	−	0.999937i	$-0.496440\pi$
$907$	0.673566	0.0223654	0.0111827	+	0.999937i	$0.496440\pi$
$908$	−3.17968	−0.105521
$909$	0	0
$910$	−5.52420	−0.183125
$911$	7.90970	0.262060	0.131030	−	0.991378i	$-0.458172\pi$
$911$	7.90970	0.262060	0.131030	+	0.991378i	$0.458172\pi$
$912$	0	0
$913$	16.5726	0.548473
$914$	73.5604	2.43316
$915$	0	0
$916$	−19.3694	−0.639983
$917$	24.5168	0.809615
$918$	0	0
$919$	−8.58263	−0.283115	−0.141557	−	0.989930i	$-0.545211\pi$
$919$	−8.58263	−0.283115	−0.141557	+	0.989930i	$0.545211\pi$
$920$	3.51678	0.115945
$921$	0	0
$922$	−72.3404	−2.38240
$923$	3.96383	0.130471
$924$	0	0
$925$	7.52420	0.247394
$926$	15.5390	0.510644
$927$	0	0
$928$	31.3430	1.02888
$929$	−29.6162	−0.971676	−0.485838	−	0.874049i	$-0.661485\pi$
$929$	−29.6162	−0.971676	−0.485838	+	0.874049i	$0.661485\pi$
$930$	0	0
$931$	6.28390	0.205946
$932$	−20.1971	−0.661579
$933$	0	0
$934$	−62.5120	−2.04545
$935$	−1.82774	−0.0597735
$936$	0	0
$937$	−15.2058	−0.496753	−0.248376	−	0.968664i	$-0.579897\pi$
$937$	−15.2058	−0.496753	−0.248376	+	0.968664i	$0.579897\pi$
$938$	69.4036	2.26611
$939$	0	0
$940$	25.6513	0.836654
$941$	−5.65287	−0.184278	−0.0921391	−	0.995746i	$-0.529370\pi$
$941$	−5.65287	−0.184278	−0.0921391	+	0.995746i	$0.529370\pi$
$942$	0	0
$943$	−0.860663	−0.0280270
$944$	13.2355	0.430779
$945$	0	0
$946$	−2.31357	−0.0752206
$947$	40.3962	1.31270	0.656350	−	0.754457i	$-0.272100\pi$
$947$	40.3962	1.31270	0.656350	+	0.754457i	$0.272100\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	1.35194	0.0438627
$951$	0	0
$952$	4.05582	0.131450
$953$	22.9320	0.742839	0.371419	−	0.928465i	$-0.378871\pi$
$953$	22.9320	0.742839	0.371419	+	0.928465i	$0.378871\pi$
$954$	0	0
$955$	−20.2281	−0.654565
$956$	56.2691	1.81987
$957$	0	0
$958$	16.6581	0.538198
$959$	30.5168	0.985438
$960$	0	0
$961$	28.2355	0.910822
$962$	10.1723	0.327967
$963$	0	0
$964$	−14.6768	−0.472708
$965$	19.9293	0.641548
$966$	0	0
$967$	10.3700	0.333478	0.166739	−	0.986001i	$-0.446676\pi$
$967$	10.3700	0.333478	0.166739	+	0.986001i	$0.446676\pi$
$968$	−6.73419	−0.216445
$969$	0	0
$970$	28.3297	0.909611
$971$	−48.0410	−1.54171	−0.770854	−	0.637012i	$-0.780170\pi$
$971$	−48.0410	−1.54171	−0.770854	+	0.637012i	$0.780170\pi$
$972$	0	0
$973$	32.6890	1.04796
$974$	−24.9171	−0.798396
$975$	0	0
$976$	12.1191	0.387921
$977$	−27.0532	−0.865509	−0.432754	−	0.901512i	$-0.642458\pi$
$977$	−27.0532	−0.865509	−0.432754	+	0.901512i	$0.642458\pi$
$978$	0	0
$979$	4.05582	0.129624
$980$	−22.8055	−0.728494
$981$	0	0
$982$	19.2355	0.613829
$983$	22.4817	0.717054	0.358527	−	0.933519i	$-0.383279\pi$
$983$	22.4817	0.717054	0.358527	+	0.933519i	$0.383279\pi$
$984$	0	0
$985$	−15.5800	−0.496421
$986$	−10.9320	−0.348144
$987$	0	0
$988$	0.987774	0.0314253
$989$	3.92935	0.124946
$990$	0	0
$991$	−26.5316	−0.842805	−0.421402	−	0.906874i	$-0.638462\pi$
$991$	−26.5316	−0.842805	−0.421402	+	0.906874i	$0.638462\pi$
$992$	62.2346	1.97595
$993$	0	0
$994$	52.1378	1.65371
$995$	−3.58482	−0.113647
$996$	0	0
$997$	−28.3659	−0.898356	−0.449178	−	0.893442i	$-0.648283\pi$
$997$	−28.3659	−0.898356	−0.449178	+	0.893442i	$0.648283\pi$
$998$	62.9433	1.99243
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.2.a.i.1.3		3
3.2	odd	2		405.2.a.j.1.1		3
4.3	odd	2		6480.2.a.bs.1.1		3
5.2	odd	4		2025.2.b.m.649.5		6
5.3	odd	4		2025.2.b.m.649.2		6
5.4	even	2		2025.2.a.o.1.1		3
9.2	odd	6		45.2.e.b.31.3	yes	6
9.4	even	3		135.2.e.b.46.1		6
9.5	odd	6		45.2.e.b.16.3	✓	6
9.7	even	3		135.2.e.b.91.1		6
12.11	even	2		6480.2.a.bv.1.1		3
15.2	even	4		2025.2.b.l.649.2		6
15.8	even	4		2025.2.b.l.649.5		6
15.14	odd	2		2025.2.a.n.1.3		3
36.7	odd	6		2160.2.q.k.1441.3		6
36.11	even	6		720.2.q.i.481.3		6
36.23	even	6		720.2.q.i.241.3		6
36.31	odd	6		2160.2.q.k.721.3		6
45.2	even	12		225.2.k.b.49.2		12
45.4	even	6		675.2.e.b.451.3		6
45.7	odd	12		675.2.k.b.199.5		12
45.13	odd	12		675.2.k.b.424.5		12
45.14	odd	6		225.2.e.b.151.1		6
45.22	odd	12		675.2.k.b.424.2		12
45.23	even	12		225.2.k.b.124.2		12
45.29	odd	6		225.2.e.b.76.1		6
45.32	even	12		225.2.k.b.124.5		12
45.34	even	6		675.2.e.b.226.3		6
45.38	even	12		225.2.k.b.49.5		12
45.43	odd	12		675.2.k.b.199.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.e.b.16.3	✓	6	9.5	odd	6
45.2.e.b.31.3	yes	6	9.2	odd	6
135.2.e.b.46.1		6	9.4	even	3
135.2.e.b.91.1		6	9.7	even	3
225.2.e.b.76.1		6	45.29	odd	6
225.2.e.b.151.1		6	45.14	odd	6
225.2.k.b.49.2		12	45.2	even	12
225.2.k.b.49.5		12	45.38	even	12
225.2.k.b.124.2		12	45.23	even	12
225.2.k.b.124.5		12	45.32	even	12
405.2.a.i.1.3		3	1.1	even	1	trivial
405.2.a.j.1.1		3	3.2	odd	2
675.2.e.b.226.3		6	45.34	even	6
675.2.e.b.451.3		6	45.4	even	6
675.2.k.b.199.2		12	45.43	odd	12
675.2.k.b.199.5		12	45.7	odd	12
675.2.k.b.424.2		12	45.22	odd	12
675.2.k.b.424.5		12	45.13	odd	12
720.2.q.i.241.3		6	36.23	even	6
720.2.q.i.481.3		6	36.11	even	6
2025.2.a.n.1.3		3	15.14	odd	2
2025.2.a.o.1.1		3	5.4	even	2
2025.2.b.l.649.2		6	15.2	even	4
2025.2.b.l.649.5		6	15.8	even	4
2025.2.b.m.649.2		6	5.3	odd	4
2025.2.b.m.649.5		6	5.2	odd	4
2160.2.q.k.721.3		6	36.31	odd	6
2160.2.q.k.1441.3		6	36.7	odd	6
6480.2.a.bs.1.1		3	4.3	odd	2
6480.2.a.bv.1.1		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q+2.08613 q^{2} +2.35194 q^{4} -1.00000 q^{5} +4.08613 q^{7} +0.734191 q^{8} -2.08613 q^{10} +1.35194 q^{11} +0.648061 q^{13} +8.52420 q^{14} -3.17226 q^{16} +1.35194 q^{17} +0.648061 q^{19} -2.35194 q^{20} +2.82032 q^{22} -4.79001 q^{23} +1.00000 q^{25} +1.35194 q^{26} +9.61033 q^{28} -3.87614 q^{29} -7.69646 q^{31} -8.08613 q^{32} +2.82032 q^{34} -4.08613 q^{35} +7.52420 q^{37} +1.35194 q^{38} -0.734191 q^{40} +0.179679 q^{41} -0.820321 q^{43} +3.17968 q^{44} -9.99258 q^{46} -10.9065 q^{47} +9.69646 q^{49} +2.08613 q^{50} +1.52420 q^{52} -4.17226 q^{53} -1.35194 q^{55} +3.00000 q^{56} -8.08613 q^{58} -4.17226 q^{59} -3.82032 q^{61} -16.0558 q^{62} -10.5242 q^{64} -0.648061 q^{65} +8.14195 q^{67} +3.17968 q^{68} -8.52420 q^{70} +6.11644 q^{71} -12.3445 q^{73} +15.6965 q^{74} +1.52420 q^{76} +5.52420 q^{77} +10.3445 q^{79} +3.17226 q^{80} +0.374833 q^{82} +12.2584 q^{83} -1.35194 q^{85} -1.71130 q^{86} +0.992582 q^{88} +3.00000 q^{89} +2.64806 q^{91} -11.2658 q^{92} -22.7523 q^{94} -0.648061 q^{95} -13.5800 q^{97} +20.2281 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 5 q^{4} - 3 q^{5} + 5 q^{7} - 3 q^{8} + q^{10} + 2 q^{11} + 4 q^{13} + 9 q^{14} + 5 q^{16} + 2 q^{17} + 4 q^{19} - 5 q^{20} - 4 q^{22} - 3 q^{23} + 3 q^{25} + 2 q^{26} + 5 q^{28} + 7 q^{29}+ \cdots + 40 q^{98}+O(q^{100}) \) 3 * q - q^2 + 5 * q^4 - 3 * q^5 + 5 * q^7 - 3 * q^8 + q^10 + 2 * q^11 + 4 * q^13 + 9 * q^14 + 5 * q^16 + 2 * q^17 + 4 * q^19 - 5 * q^20 - 4 * q^22 - 3 * q^23 + 3 * q^25 + 2 * q^26 + 5 * q^28 + 7 * q^29 + 8 * q^31 - 17 * q^32 - 4 * q^34 - 5 * q^35 + 6 * q^37 + 2 * q^38 + 3 * q^40 + 13 * q^41 + 10 * q^43 + 22 * q^44 - 3 * q^46 - 13 * q^47 - 2 * q^49 - q^50 - 12 * q^52 + 2 * q^53 - 2 * q^55 + 9 * q^56 - 17 * q^58 + 2 * q^59 + q^61 - 42 * q^62 - 15 * q^64 - 4 * q^65 + 11 * q^67 + 22 * q^68 - 9 * q^70 + 10 * q^71 - 8 * q^73 + 16 * q^74 - 12 * q^76 + 2 * q^79 - 5 * q^80 - 29 * q^82 + 15 * q^83 - 2 * q^85 - 28 * q^86 - 24 * q^88 + 9 * q^89 + 10 * q^91 - 39 * q^92 - 31 * q^94 - 4 * q^95 - 18 * q^97 + 40 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more