LMFDB - Embedded newform 405.4.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, 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("405.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.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:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.7032.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 14x + 18 $ x^3 - x^2 - 14*x + 18
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.32681$ 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)

$f(q)$	$=$	\(q+1.32681 q^{2} -6.23958 q^{4} -5.00000 q^{5} -24.1043 q^{7} -18.8932 q^{8} -6.63404 q^{10} -8.27619 q^{11} +87.1145 q^{13} -31.9818 q^{14} +24.8490 q^{16} -51.9166 q^{17} -88.5107 q^{19} +31.1979 q^{20} -10.9809 q^{22} +129.245 q^{23} +25.0000 q^{25} +115.584 q^{26} +150.400 q^{28} +271.109 q^{29} +224.547 q^{31} +184.115 q^{32} -68.8834 q^{34} +120.521 q^{35} -70.5268 q^{37} -117.437 q^{38} +94.4660 q^{40} -366.938 q^{41} -195.547 q^{43} +51.6399 q^{44} +171.483 q^{46} +359.192 q^{47} +238.016 q^{49} +33.1702 q^{50} -543.558 q^{52} -29.4890 q^{53} +41.3810 q^{55} +455.407 q^{56} +359.709 q^{58} +858.104 q^{59} -556.811 q^{61} +297.931 q^{62} +45.4941 q^{64} -435.573 q^{65} -41.8987 q^{67} +323.938 q^{68} +159.909 q^{70} +549.163 q^{71} -185.505 q^{73} -93.5756 q^{74} +552.269 q^{76} +199.492 q^{77} +80.4913 q^{79} -124.245 q^{80} -486.857 q^{82} +576.753 q^{83} +259.583 q^{85} -259.454 q^{86} +156.364 q^{88} -224.516 q^{89} -2099.83 q^{91} -806.433 q^{92} +476.579 q^{94} +442.553 q^{95} -555.016 q^{97} +315.801 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 5 q^{4} - 15 q^{5} - 25 q^{7} - 27 q^{8} - 5 q^{10} + 58 q^{11} - 47 q^{13} + 159 q^{14} - 127 q^{16} + 34 q^{17} - 5 q^{19} - 25 q^{20} + 260 q^{22} - 51 q^{23} + 75 q^{25} + 253 q^{26}+ \cdots + 38 q^{98}+O(q^{100}) $ 3 * q + q^2 + 5 * q^4 - 15 * q^5 - 25 * q^7 - 27 * q^8 - 5 * q^10 + 58 * q^11 - 47 * q^13 + 159 * q^14 - 127 * q^16 + 34 * q^17 - 5 * q^19 - 25 * q^20 + 260 * q^22 - 51 * q^23 + 75 * q^25 + 253 * q^26 + 83 * q^28 + 350 * q^29 + 638 * q^31 + 245 * q^32 - 154 * q^34 + 125 * q^35 - 414 * q^37 + 397 * q^38 + 135 * q^40 + 179 * q^41 - 836 * q^43 + 332 * q^44 + 261 * q^46 + 235 * q^47 + 892 * q^49 + 25 * q^50 - 1335 * q^52 + 505 * q^53 - 290 * q^55 + 15 * q^56 + 1876 * q^58 + 535 * q^59 - 104 * q^61 + 348 * q^62 - 303 * q^64 + 235 * q^65 - 40 * q^67 + 830 * q^68 - 795 * q^70 + 452 * q^71 - 710 * q^73 + 1394 * q^74 + 849 * q^76 + 2148 * q^77 - 634 * q^79 + 635 * q^80 + 613 * q^82 + 1734 * q^83 - 170 * q^85 + 460 * q^86 - 768 * q^88 - 852 * q^89 - 1229 * q^91 - 1839 * q^92 + 1751 * q^94 + 25 * q^95 + 38 * 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 $

$2$	1.32681	0.469098	0.234549	−	0.972104i	$-0.424639\pi$
$2$	1.32681	0.469098	0.234549	+	0.972104i	$0.424639\pi$
$3$	0	0
$3$	0	0
$4$	−6.23958	−0.779947
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−24.1043	−1.30151	−0.650754	−	0.759289i	$-0.725547\pi$
$7$	−24.1043	−1.30151	−0.650754	+	0.759289i	$0.725547\pi$
$8$	−18.8932	−0.834969
$9$	0	0
$10$	−6.63404	−0.209787
$11$	−8.27619	−0.226851	−0.113426	−	0.993546i	$-0.536182\pi$
$11$	−8.27619	−0.226851	−0.113426	+	0.993546i	$0.536182\pi$
$12$	0	0
$13$	87.1145	1.85856	0.929278	−	0.369382i	$-0.120430\pi$
$13$	87.1145	1.85856	0.929278	+	0.369382i	$0.120430\pi$
$14$	−31.9818	−0.610534
$15$	0	0
$16$	24.8490	0.388265
$17$	−51.9166	−0.740684	−0.370342	−	0.928895i	$-0.620760\pi$
$17$	−51.9166	−0.740684	−0.370342	+	0.928895i	$0.620760\pi$
$18$	0	0
$19$	−88.5107	−1.06872	−0.534362	−	0.845256i	$-0.679448\pi$
$19$	−88.5107	−1.06872	−0.534362	+	0.845256i	$0.679448\pi$
$20$	31.1979	0.348803
$21$	0	0
$22$	−10.9809	−0.106415
$23$	129.245	1.17171	0.585856	−	0.810415i	$-0.300758\pi$
$23$	129.245	1.17171	0.585856	+	0.810415i	$0.300758\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	115.584	0.871844
$27$	0	0
$28$	150.400	1.01511
$29$	271.109	1.73599	0.867993	−	0.496576i	$-0.165410\pi$
$29$	271.109	1.73599	0.867993	+	0.496576i	$0.165410\pi$
$30$	0	0
$31$	224.547	1.30096	0.650481	−	0.759523i	$-0.274568\pi$
$31$	224.547	1.30096	0.650481	+	0.759523i	$0.274568\pi$
$32$	184.115	1.01710
$33$	0	0
$34$	−68.8834	−0.347453
$35$	120.521	0.582052
$36$	0	0
$37$	−70.5268	−0.313366	−0.156683	−	0.987649i	$-0.550080\pi$
$37$	−70.5268	−0.313366	−0.156683	+	0.987649i	$0.550080\pi$
$38$	−117.437	−0.501336
$39$	0	0
$40$	94.4660	0.373410
$41$	−366.938	−1.39771	−0.698855	−	0.715263i	$-0.746307\pi$
$41$	−366.938	−1.39771	−0.698855	+	0.715263i	$0.746307\pi$
$42$	0	0
$43$	−195.547	−0.693504	−0.346752	−	0.937957i	$-0.612715\pi$
$43$	−195.547	−0.693504	−0.346752	+	0.937957i	$0.612715\pi$
$44$	51.6399	0.176932
$45$	0	0
$46$	171.483	0.549648
$47$	359.192	1.11476	0.557378	−	0.830259i	$-0.311807\pi$
$47$	359.192	1.11476	0.557378	+	0.830259i	$0.311807\pi$
$48$	0	0
$49$	238.016	0.693923
$50$	33.1702	0.0938195
$51$	0	0
$52$	−543.558	−1.44958
$53$	−29.4890	−0.0764270	−0.0382135	−	0.999270i	$-0.512167\pi$
$53$	−29.4890	−0.0764270	−0.0382135	+	0.999270i	$0.512167\pi$
$54$	0	0
$55$	41.3810	0.101451
$56$	455.407	1.08672
$57$	0	0
$58$	359.709	0.814347
$59$	858.104	1.89349	0.946743	−	0.321991i	$-0.104352\pi$
$59$	858.104	1.89349	0.946743	+	0.321991i	$0.104352\pi$
$60$	0	0
$61$	−556.811	−1.16873	−0.584364	−	0.811492i	$-0.698656\pi$
$61$	−556.811	−1.16873	−0.584364	+	0.811492i	$0.698656\pi$
$62$	297.931	0.610278
$63$	0	0
$64$	45.4941	0.0888557
$65$	−435.573	−0.831171
$66$	0	0
$67$	−41.8987	−0.0763991	−0.0381995	−	0.999270i	$-0.512162\pi$
$67$	−41.8987	−0.0763991	−0.0381995	+	0.999270i	$0.512162\pi$
$68$	323.938	0.577695
$69$	0	0
$70$	159.909	0.273039
$71$	549.163	0.917939	0.458970	−	0.888452i	$-0.348219\pi$
$71$	549.163	0.917939	0.458970	+	0.888452i	$0.348219\pi$
$72$	0	0
$73$	−185.505	−0.297420	−0.148710	−	0.988881i	$-0.547512\pi$
$73$	−185.505	−0.297420	−0.148710	+	0.988881i	$0.547512\pi$
$74$	−93.5756	−0.146999
$75$	0	0
$76$	552.269	0.833548
$77$	199.492	0.295249
$78$	0	0
$79$	80.4913	0.114633	0.0573163	−	0.998356i	$-0.481746\pi$
$79$	80.4913	0.114633	0.0573163	+	0.998356i	$0.481746\pi$
$80$	−124.245	−0.173637
$81$	0	0
$82$	−486.857	−0.655663
$83$	576.753	0.762734	0.381367	−	0.924424i	$-0.375453\pi$
$83$	576.753	0.762734	0.381367	+	0.924424i	$0.375453\pi$
$84$	0	0
$85$	259.583	0.331244
$86$	−259.454	−0.325321
$87$	0	0
$88$	156.364	0.189414
$89$	−224.516	−0.267401	−0.133700	−	0.991022i	$-0.542686\pi$
$89$	−224.516	−0.267401	−0.133700	+	0.991022i	$0.542686\pi$
$90$	0	0
$91$	−2099.83	−2.41893
$92$	−806.433	−0.913874
$93$	0	0
$94$	476.579	0.522930
$95$	442.553	0.477948
$96$	0	0
$97$	−555.016	−0.580963	−0.290481	−	0.956881i	$-0.593815\pi$
$97$	−555.016	−0.580963	−0.290481	+	0.956881i	$0.593815\pi$
$98$	315.801	0.325518
$99$	0	0
$100$	−155.989	−0.155989
$101$	−227.000	−0.223637	−0.111818	−	0.993729i	$-0.535667\pi$
$101$	−227.000	−0.223637	−0.111818	+	0.993729i	$0.535667\pi$
$102$	0	0
$103$	383.524	0.366891	0.183446	−	0.983030i	$-0.441275\pi$
$103$	383.524	0.366891	0.183446	+	0.983030i	$0.441275\pi$
$104$	−1645.87	−1.55184
$105$	0	0
$106$	−39.1263	−0.0358517
$107$	1775.32	1.60399	0.801993	−	0.597334i	$-0.203773\pi$
$107$	1775.32	1.60399	0.801993	+	0.597334i	$0.203773\pi$
$108$	0	0
$109$	1530.50	1.34491	0.672454	−	0.740139i	$-0.265240\pi$
$109$	1530.50	1.34491	0.672454	+	0.740139i	$0.265240\pi$
$110$	54.9046	0.0475904
$111$	0	0
$112$	−598.966	−0.505330
$113$	840.782	0.699948	0.349974	−	0.936759i	$-0.386190\pi$
$113$	840.782	0.699948	0.349974	+	0.936759i	$0.386190\pi$
$114$	0	0
$115$	−646.224	−0.524006
$116$	−1691.60	−1.35398
$117$	0	0
$118$	1138.54	0.888230
$119$	1251.41	0.964007
$120$	0	0
$121$	−1262.50	−0.948538
$122$	−738.782	−0.548247
$123$	0	0
$124$	−1401.08	−1.01468
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1038.45	0.725572	0.362786	−	0.931873i	$-0.381826\pi$
$127$	1038.45	0.725572	0.362786	+	0.931873i	$0.381826\pi$
$128$	−1412.56	−0.975422
$129$	0	0
$130$	−577.922	−0.389901
$131$	803.410	0.535834	0.267917	−	0.963442i	$-0.413665\pi$
$131$	803.410	0.535834	0.267917	+	0.963442i	$0.413665\pi$
$132$	0	0
$133$	2133.49	1.39095
$134$	−55.5915	−0.0358386
$135$	0	0
$136$	980.871	0.618449
$137$	−869.152	−0.542019	−0.271010	−	0.962577i	$-0.587358\pi$
$137$	−869.152	−0.542019	−0.271010	+	0.962577i	$0.587358\pi$
$138$	0	0
$139$	−196.398	−0.119844	−0.0599218	−	0.998203i	$-0.519085\pi$
$139$	−196.398	−0.119844	−0.0599218	+	0.998203i	$0.519085\pi$
$140$	−752.002	−0.453970
$141$	0	0
$142$	728.635	0.430603
$143$	−720.976	−0.421616
$144$	0	0
$145$	−1355.54	−0.776357
$146$	−246.129	−0.139519
$147$	0	0
$148$	440.057	0.244409
$149$	−754.283	−0.414720	−0.207360	−	0.978265i	$-0.566487\pi$
$149$	−754.283	−0.414720	−0.207360	+	0.978265i	$0.566487\pi$
$150$	0	0
$151$	2057.64	1.10893	0.554464	−	0.832208i	$-0.312923\pi$
$151$	2057.64	1.10893	0.554464	+	0.832208i	$0.312923\pi$
$152$	1672.25	0.892351
$153$	0	0
$154$	264.687	0.138501
$155$	−1122.73	−0.581808
$156$	0	0
$157$	−3358.58	−1.70729	−0.853644	−	0.520857i	$-0.825613\pi$
$157$	−3358.58	−1.70729	−0.853644	+	0.520857i	$0.825613\pi$
$158$	106.797	0.0537739
$159$	0	0
$160$	−920.577	−0.454863
$161$	−3115.35	−1.52499
$162$	0	0
$163$	−710.376	−0.341356	−0.170678	−	0.985327i	$-0.554596\pi$
$163$	−710.376	−0.341356	−0.170678	+	0.985327i	$0.554596\pi$
$164$	2289.54	1.09014
$165$	0	0
$166$	765.241	0.357797
$167$	1148.27	0.532070	0.266035	−	0.963963i	$-0.414286\pi$
$167$	1148.27	0.532070	0.266035	+	0.963963i	$0.414286\pi$
$168$	0	0
$169$	5391.94	2.45423
$170$	344.417	0.155386
$171$	0	0
$172$	1220.13	0.540897
$173$	2925.29	1.28558	0.642790	−	0.766042i	$-0.277777\pi$
$173$	2925.29	1.28558	0.642790	+	0.766042i	$0.277777\pi$
$174$	0	0
$175$	−602.607	−0.260302
$176$	−205.655	−0.0880785
$177$	0	0
$178$	−297.890	−0.125437
$179$	−3422.02	−1.42890	−0.714452	−	0.699685i	$-0.753324\pi$
$179$	−3422.02	−1.42890	−0.714452	+	0.699685i	$0.753324\pi$
$180$	0	0
$181$	−1151.58	−0.472907	−0.236454	−	0.971643i	$-0.575985\pi$
$181$	−1151.58	−0.472907	−0.236454	+	0.971643i	$0.575985\pi$
$182$	−2786.07	−1.13471
$183$	0	0
$184$	−2441.85	−0.978344
$185$	352.634	0.140141
$186$	0	0
$187$	429.672	0.168025
$188$	−2241.21	−0.869451
$189$	0	0
$190$	587.184	0.224204
$191$	−932.120	−0.353120	−0.176560	−	0.984290i	$-0.556497\pi$
$191$	−932.120	−0.353120	−0.176560	+	0.984290i	$0.556497\pi$
$192$	0	0
$193$	−4272.81	−1.59359	−0.796797	−	0.604247i	$-0.793474\pi$
$193$	−4272.81	−1.59359	−0.796797	+	0.604247i	$0.793474\pi$
$194$	−736.400	−0.272528
$195$	0	0
$196$	−1485.12	−0.541224
$197$	1924.15	0.695888	0.347944	−	0.937515i	$-0.386880\pi$
$197$	1924.15	0.695888	0.347944	+	0.937515i	$0.386880\pi$
$198$	0	0
$199$	1738.84	0.619414	0.309707	−	0.950832i	$-0.399769\pi$
$199$	1738.84	0.619414	0.309707	+	0.950832i	$0.399769\pi$
$200$	−472.330	−0.166994
$201$	0	0
$202$	−301.185	−0.104907
$203$	−6534.87	−2.25940
$204$	0	0
$205$	1834.69	0.625075
$206$	508.864	0.172108
$207$	0	0
$208$	2164.71	0.721613
$209$	732.531	0.242441
$210$	0	0
$211$	3202.35	1.04483	0.522414	−	0.852692i	$-0.325031\pi$
$211$	3202.35	1.04483	0.522414	+	0.852692i	$0.325031\pi$
$212$	183.999	0.0596090
$213$	0	0
$214$	2355.51	0.752426
$215$	977.736	0.310144
$216$	0	0
$217$	−5412.54	−1.69321
$218$	2030.68	0.630893
$219$	0	0
$220$	−258.200	−0.0791265
$221$	−4522.69	−1.37660
$222$	0	0
$223$	−1404.99	−0.421906	−0.210953	−	0.977496i	$-0.567657\pi$
$223$	−1404.99	−0.421906	−0.210953	+	0.977496i	$0.567657\pi$
$224$	−4437.97	−1.32377
$225$	0	0
$226$	1115.56	0.328344
$227$	−6238.35	−1.82402	−0.912012	−	0.410163i	$-0.865472\pi$
$227$	−6238.35	−1.82402	−0.912012	+	0.410163i	$0.865472\pi$
$228$	0	0
$229$	6630.31	1.91329	0.956644	−	0.291259i	$-0.0940742\pi$
$229$	6630.31	1.91329	0.956644	+	0.291259i	$0.0940742\pi$
$230$	−857.415	−0.245810
$231$	0	0
$232$	−5122.11	−1.44949
$233$	−2453.48	−0.689840	−0.344920	−	0.938632i	$-0.612094\pi$
$233$	−2453.48	−0.689840	−0.344920	+	0.938632i	$0.612094\pi$
$234$	0	0
$235$	−1795.96	−0.498534
$236$	−5354.21	−1.47682
$237$	0	0
$238$	1660.38	0.452213
$239$	6949.74	1.88093	0.940463	−	0.339895i	$-0.110392\pi$
$239$	6949.74	1.88093	0.940463	+	0.339895i	$0.110392\pi$
$240$	0	0
$241$	6335.19	1.69330	0.846651	−	0.532149i	$-0.178615\pi$
$241$	6335.19	1.69330	0.846651	+	0.532149i	$0.178615\pi$
$242$	−1675.10	−0.444957
$243$	0	0
$244$	3474.27	0.911546
$245$	−1190.08	−0.310332
$246$	0	0
$247$	−7710.57	−1.98628
$248$	−4242.41	−1.08626
$249$	0	0
$250$	−165.851	−0.0419574
$251$	4022.65	1.01158	0.505792	−	0.862656i	$-0.331200\pi$
$251$	4022.65	1.01158	0.505792	+	0.862656i	$0.331200\pi$
$252$	0	0
$253$	−1069.65	−0.265805
$254$	1377.83	0.340364
$255$	0	0
$256$	−2238.15	−0.546424
$257$	4997.74	1.21304	0.606519	−	0.795069i	$-0.292565\pi$
$257$	4997.74	1.21304	0.606519	+	0.795069i	$0.292565\pi$
$258$	0	0
$259$	1700.00	0.407848
$260$	2717.79	0.648270
$261$	0	0
$262$	1065.97	0.251359
$263$	3992.47	0.936069	0.468035	−	0.883710i	$-0.344962\pi$
$263$	3992.47	0.936069	0.468035	+	0.883710i	$0.344962\pi$
$264$	0	0
$265$	147.445	0.0341792
$266$	2830.73	0.652493
$267$	0	0
$268$	261.430	0.0595872
$269$	2188.89	0.496131	0.248065	−	0.968743i	$-0.420205\pi$
$269$	2188.89	0.496131	0.248065	+	0.968743i	$0.420205\pi$
$270$	0	0
$271$	4280.26	0.959437	0.479718	−	0.877423i	$-0.340739\pi$
$271$	4280.26	0.959437	0.479718	+	0.877423i	$0.340739\pi$
$272$	−1290.08	−0.287582
$273$	0	0
$274$	−1153.20	−0.254260
$275$	−206.905	−0.0453703
$276$	0	0
$277$	−3879.97	−0.841606	−0.420803	−	0.907152i	$-0.638252\pi$
$277$	−3879.97	−0.841606	−0.420803	+	0.907152i	$0.638252\pi$
$278$	−260.583	−0.0562184
$279$	0	0
$280$	−2277.03	−0.485996
$281$	6397.43	1.35814	0.679072	−	0.734071i	$-0.262382\pi$
$281$	6397.43	1.35814	0.679072	+	0.734071i	$0.262382\pi$
$282$	0	0
$283$	−342.869	−0.0720193	−0.0360096	−	0.999351i	$-0.511465\pi$
$283$	−342.869	−0.0720193	−0.0360096	+	0.999351i	$0.511465\pi$
$284$	−3426.55	−0.715944
$285$	0	0
$286$	−956.598	−0.197779
$287$	8844.78	1.81913
$288$	0	0
$289$	−2217.66	−0.451387
$290$	−1798.55	−0.364187
$291$	0	0
$292$	1157.47	0.231972
$293$	7333.43	1.46220	0.731098	−	0.682272i	$-0.239008\pi$
$293$	7333.43	1.46220	0.731098	+	0.682272i	$0.239008\pi$
$294$	0	0
$295$	−4290.52	−0.846793
$296$	1332.48	0.261651
$297$	0	0
$298$	−1000.79	−0.194544
$299$	11259.1	2.17769
$300$	0	0
$301$	4713.52	0.902601
$302$	2730.09	0.520195
$303$	0	0
$304$	−2199.40	−0.414948
$305$	2784.06	0.522671
$306$	0	0
$307$	−7965.33	−1.48080	−0.740399	−	0.672167i	$-0.765364\pi$
$307$	−7965.33	−1.48080	−0.740399	+	0.672167i	$0.765364\pi$
$308$	−1244.74	−0.230279
$309$	0	0
$310$	−1489.65	−0.272925
$311$	2186.19	0.398609	0.199305	−	0.979938i	$-0.436132\pi$
$311$	2186.19	0.398609	0.199305	+	0.979938i	$0.436132\pi$
$312$	0	0
$313$	38.6303	0.00697609	0.00348805	−	0.999994i	$-0.498890\pi$
$313$	38.6303	0.00697609	0.00348805	+	0.999994i	$0.498890\pi$
$314$	−4456.20	−0.800885
$315$	0	0
$316$	−502.232	−0.0894074
$317$	−9535.81	−1.68954	−0.844770	−	0.535129i	$-0.820263\pi$
$317$	−9535.81	−1.68954	−0.844770	+	0.535129i	$0.820263\pi$
$318$	0	0
$319$	−2243.75	−0.393811
$320$	−227.471	−0.0397375
$321$	0	0
$322$	−4133.47	−0.715371
$323$	4595.18	0.791587
$324$	0	0
$325$	2177.86	0.371711
$326$	−942.533	−0.160129
$327$	0	0
$328$	6932.63	1.16704
$329$	−8658.06	−1.45086
$330$	0	0
$331$	−3010.58	−0.499929	−0.249964	−	0.968255i	$-0.580419\pi$
$331$	−3010.58	−0.499929	−0.249964	+	0.968255i	$0.580419\pi$
$332$	−3598.70	−0.594892
$333$	0	0
$334$	1523.53	0.249593
$335$	209.493	0.0341667
$336$	0	0
$337$	2812.72	0.454655	0.227327	−	0.973818i	$-0.427001\pi$
$337$	2812.72	0.454655	0.227327	+	0.973818i	$0.427001\pi$
$338$	7154.07	1.15127
$339$	0	0
$340$	−1619.69	−0.258353
$341$	−1858.39	−0.295125
$342$	0	0
$343$	2530.57	0.398361
$344$	3694.51	0.579054
$345$	0	0
$346$	3881.30	0.603063
$347$	139.354	0.0215588	0.0107794	−	0.999942i	$-0.496569\pi$
$347$	139.354	0.0215588	0.0107794	+	0.999942i	$0.496569\pi$
$348$	0	0
$349$	−5210.29	−0.799141	−0.399571	−	0.916702i	$-0.630841\pi$
$349$	−5210.29	−0.799141	−0.399571	+	0.916702i	$0.630841\pi$
$350$	−799.544	−0.122107
$351$	0	0
$352$	−1523.77	−0.230731
$353$	3410.46	0.514223	0.257112	−	0.966382i	$-0.417229\pi$
$353$	3410.46	0.514223	0.257112	+	0.966382i	$0.417229\pi$
$354$	0	0
$355$	−2745.82	−0.410515
$356$	1400.89	0.208558
$357$	0	0
$358$	−4540.36	−0.670295
$359$	−8131.85	−1.19549	−0.597747	−	0.801685i	$-0.703937\pi$
$359$	−8131.85	−1.19549	−0.597747	+	0.801685i	$0.703937\pi$
$360$	0	0
$361$	975.143	0.142170
$362$	−1527.93	−0.221840
$363$	0	0
$364$	13102.1	1.88663
$365$	927.523	0.133010
$366$	0	0
$367$	132.151	0.0187963	0.00939816	−	0.999956i	$-0.497008\pi$
$367$	132.151	0.0187963	0.00939816	+	0.999956i	$0.497008\pi$
$368$	3211.60	0.454935
$369$	0	0
$370$	467.878	0.0657400
$371$	710.812	0.0994703
$372$	0	0
$373$	9348.88	1.29777	0.648883	−	0.760888i	$-0.275236\pi$
$373$	9348.88	1.29777	0.648883	+	0.760888i	$0.275236\pi$
$374$	570.092	0.0788203
$375$	0	0
$376$	−6786.29	−0.930787
$377$	23617.5	3.22643
$378$	0	0
$379$	6164.17	0.835441	0.417720	−	0.908576i	$-0.362829\pi$
$379$	6164.17	0.835441	0.417720	+	0.908576i	$0.362829\pi$
$380$	−2761.35	−0.372774
$381$	0	0
$382$	−1236.75	−0.165648
$383$	2601.27	0.347047	0.173523	−	0.984830i	$-0.444485\pi$
$383$	2601.27	0.347047	0.173523	+	0.984830i	$0.444485\pi$
$384$	0	0
$385$	−997.458	−0.132039
$386$	−5669.20	−0.747552
$387$	0	0
$388$	3463.07	0.453120
$389$	4250.57	0.554017	0.277008	−	0.960867i	$-0.410657\pi$
$389$	4250.57	0.554017	0.277008	+	0.960867i	$0.410657\pi$
$390$	0	0
$391$	−6709.95	−0.867869
$392$	−4496.88	−0.579405
$393$	0	0
$394$	2552.98	0.326440
$395$	−402.456	−0.0512653
$396$	0	0
$397$	−6088.35	−0.769687	−0.384843	−	0.922982i	$-0.625745\pi$
$397$	−6088.35	−0.769687	−0.384843	+	0.922982i	$0.625745\pi$
$398$	2307.11	0.290566
$399$	0	0
$400$	621.224	0.0776530
$401$	7725.79	0.962114	0.481057	−	0.876689i	$-0.340253\pi$
$401$	7725.79	0.962114	0.481057	+	0.876689i	$0.340253\pi$
$402$	0	0
$403$	19561.3	2.41791
$404$	1416.38	0.174425
$405$	0	0
$406$	−8670.53	−1.05988
$407$	583.693	0.0710875
$408$	0	0
$409$	16321.1	1.97318	0.986588	−	0.163232i	$-0.0521920\pi$
$409$	16321.1	1.97318	0.986588	+	0.163232i	$0.0521920\pi$
$410$	2434.28	0.293221
$411$	0	0
$412$	−2393.03	−0.286156
$413$	−20684.0	−2.46439
$414$	0	0
$415$	−2883.77	−0.341105
$416$	16039.1	1.89034
$417$	0	0
$418$	971.929	0.113729
$419$	1152.46	0.134370	0.0671851	−	0.997741i	$-0.478598\pi$
$419$	1152.46	0.134370	0.0671851	+	0.997741i	$0.478598\pi$
$420$	0	0
$421$	−3531.23	−0.408792	−0.204396	−	0.978888i	$-0.565523\pi$
$421$	−3531.23	−0.408792	−0.204396	+	0.978888i	$0.565523\pi$
$422$	4248.91	0.490127
$423$	0	0
$424$	557.142	0.0638142
$425$	−1297.92	−0.148137
$426$	0	0
$427$	13421.5	1.52111
$428$	−11077.2	−1.25102
$429$	0	0
$430$	1297.27	0.145488
$431$	10230.6	1.14337	0.571683	−	0.820475i	$-0.306291\pi$
$431$	10230.6	1.14337	0.571683	+	0.820475i	$0.306291\pi$
$432$	0	0
$433$	−7311.31	−0.811453	−0.405726	−	0.913995i	$-0.632981\pi$
$433$	−7311.31	−0.811453	−0.405726	+	0.913995i	$0.632981\pi$
$434$	−7181.40	−0.794282
$435$	0	0
$436$	−9549.65	−1.04896
$437$	−11439.5	−1.25224
$438$	0	0
$439$	7053.18	0.766811	0.383405	−	0.923580i	$-0.374751\pi$
$439$	7053.18	0.766811	0.383405	+	0.923580i	$0.374751\pi$
$440$	−781.818	−0.0847085
$441$	0	0
$442$	−6000.75	−0.645761
$443$	−167.790	−0.0179954	−0.00899770	−	0.999960i	$-0.502864\pi$
$443$	−167.790	−0.0179954	−0.00899770	+	0.999960i	$0.502864\pi$
$444$	0	0
$445$	1122.58	0.119585
$446$	−1864.15	−0.197915
$447$	0	0
$448$	−1096.60	−0.115646
$449$	3949.94	0.415165	0.207582	−	0.978218i	$-0.433440\pi$
$449$	3949.94	0.415165	0.207582	+	0.978218i	$0.433440\pi$
$450$	0	0
$451$	3036.85	0.317072
$452$	−5246.13	−0.545922
$453$	0	0
$454$	−8277.09	−0.855646
$455$	10499.2	1.08178
$456$	0	0
$457$	−7447.60	−0.762328	−0.381164	−	0.924507i	$-0.624477\pi$
$457$	−7447.60	−0.762328	−0.381164	+	0.924507i	$0.624477\pi$
$458$	8797.15	0.897519
$459$	0	0
$460$	4032.16	0.408697
$461$	10887.6	1.09997	0.549987	−	0.835173i	$-0.314633\pi$
$461$	10887.6	1.09997	0.549987	+	0.835173i	$0.314633\pi$
$462$	0	0
$463$	−3171.27	−0.318318	−0.159159	−	0.987253i	$-0.550878\pi$
$463$	−3171.27	−0.318318	−0.159159	+	0.987253i	$0.550878\pi$
$464$	6736.77	0.674023
$465$	0	0
$466$	−3255.29	−0.323602
$467$	−16348.5	−1.61996	−0.809978	−	0.586461i	$-0.800521\pi$
$467$	−16348.5	−1.61996	−0.809978	+	0.586461i	$0.800521\pi$
$468$	0	0
$469$	1009.94	0.0994340
$470$	−2382.90	−0.233861
$471$	0	0
$472$	−16212.3	−1.58100
$473$	1618.39	0.157322
$474$	0	0
$475$	−2212.77	−0.213745
$476$	−7808.29	−0.751874
$477$	0	0
$478$	9220.98	0.882338
$479$	9413.69	0.897959	0.448980	−	0.893542i	$-0.351788\pi$
$479$	9413.69	0.897959	0.448980	+	0.893542i	$0.351788\pi$
$480$	0	0
$481$	−6143.91	−0.582408
$482$	8405.59	0.794324
$483$	0	0
$484$	7877.50	0.739810
$485$	2775.08	0.259814
$486$	0	0
$487$	13482.3	1.25450	0.627250	−	0.778818i	$-0.284180\pi$
$487$	13482.3	1.25450	0.627250	+	0.778818i	$0.284180\pi$
$488$	10519.9	0.975852
$489$	0	0
$490$	−1579.01	−0.145576
$491$	−9219.74	−0.847416	−0.423708	−	0.905799i	$-0.639272\pi$
$491$	−9219.74	−0.847416	−0.423708	+	0.905799i	$0.639272\pi$
$492$	0	0
$493$	−14075.0	−1.28582
$494$	−10230.4	−0.931760
$495$	0	0
$496$	5579.76	0.505118
$497$	−13237.2	−1.19471
$498$	0	0
$499$	104.346	0.00936108	0.00468054	−	0.999989i	$-0.498510\pi$
$499$	104.346	0.00936108	0.00468054	+	0.999989i	$0.498510\pi$
$500$	779.947	0.0697606
$501$	0	0
$502$	5337.29	0.474531
$503$	490.652	0.0434933	0.0217466	−	0.999764i	$-0.493077\pi$
$503$	490.652	0.0434933	0.0217466	+	0.999764i	$0.493077\pi$
$504$	0	0
$505$	1135.00	0.100013
$506$	−1419.23	−0.124688
$507$	0	0
$508$	−6479.50	−0.565908
$509$	−3412.81	−0.297191	−0.148595	−	0.988898i	$-0.547475\pi$
$509$	−3412.81	−0.297191	−0.148595	+	0.988898i	$0.547475\pi$
$510$	0	0
$511$	4471.45	0.387095
$512$	8330.89	0.719095
$513$	0	0
$514$	6631.05	0.569033
$515$	−1917.62	−0.164079
$516$	0	0
$517$	−2972.74	−0.252884
$518$	2255.57	0.191321
$519$	0	0
$520$	8229.36	0.694002
$521$	−3486.31	−0.293163	−0.146582	−	0.989199i	$-0.546827\pi$
$521$	−3486.31	−0.293163	−0.146582	+	0.989199i	$0.546827\pi$
$522$	0	0
$523$	14465.6	1.20943	0.604717	−	0.796440i	$-0.293286\pi$
$523$	14465.6	1.20943	0.604717	+	0.796440i	$0.293286\pi$
$524$	−5012.94	−0.417923
$525$	0	0
$526$	5297.24	0.439108
$527$	−11657.7	−0.963602
$528$	0	0
$529$	4537.21	0.372911
$530$	195.632	0.0160334
$531$	0	0
$532$	−13312.1	−1.08487
$533$	−31965.6	−2.59772
$534$	0	0
$535$	−8876.59	−0.717324
$536$	791.600	0.0637909
$537$	0	0
$538$	2904.24	0.232734
$539$	−1969.86	−0.157417
$540$	0	0
$541$	6602.78	0.524724	0.262362	−	0.964970i	$-0.415499\pi$
$541$	6602.78	0.524724	0.262362	+	0.964970i	$0.415499\pi$
$542$	5679.09	0.450070
$543$	0	0
$544$	−9558.65	−0.753353
$545$	−7652.48	−0.601461
$546$	0	0
$547$	6358.56	0.497024	0.248512	−	0.968629i	$-0.420058\pi$
$547$	6358.56	0.497024	0.248512	+	0.968629i	$0.420058\pi$
$548$	5423.14	0.422746
$549$	0	0
$550$	−274.523	−0.0212831
$551$	−23996.0	−1.85529
$552$	0	0
$553$	−1940.18	−0.149195
$554$	−5147.98	−0.394795
$555$	0	0
$556$	1225.44	0.0934717
$557$	−20782.3	−1.58092	−0.790461	−	0.612513i	$-0.790159\pi$
$557$	−20782.3	−1.58092	−0.790461	+	0.612513i	$0.790159\pi$
$558$	0	0
$559$	−17035.0	−1.28892
$560$	2994.83	0.225991
$561$	0	0
$562$	8488.17	0.637103
$563$	16470.0	1.23291	0.616453	−	0.787392i	$-0.288569\pi$
$563$	16470.0	1.23291	0.616453	+	0.787392i	$0.288569\pi$
$564$	0	0
$565$	−4203.91	−0.313026
$566$	−454.922	−0.0337841
$567$	0	0
$568$	−10375.4	−0.766451
$569$	5425.39	0.399726	0.199863	−	0.979824i	$-0.435950\pi$
$569$	5425.39	0.399726	0.199863	+	0.979824i	$0.435950\pi$
$570$	0	0
$571$	−14689.7	−1.07661	−0.538306	−	0.842749i	$-0.680936\pi$
$571$	−14689.7	−1.07661	−0.538306	+	0.842749i	$0.680936\pi$
$572$	4498.59	0.328838
$573$	0	0
$574$	11735.3	0.853350
$575$	3231.12	0.234343
$576$	0	0
$577$	−17933.1	−1.29387	−0.646935	−	0.762545i	$-0.723949\pi$
$577$	−17933.1	−1.29387	−0.646935	+	0.762545i	$0.723949\pi$
$578$	−2942.41	−0.211745
$579$	0	0
$580$	8458.02	0.605517
$581$	−13902.2	−0.992704
$582$	0	0
$583$	244.057	0.0173376
$584$	3504.78	0.248337
$585$	0	0
$586$	9730.06	0.685913
$587$	4801.16	0.337589	0.168795	−	0.985651i	$-0.446013\pi$
$587$	4801.16	0.337589	0.168795	+	0.985651i	$0.446013\pi$
$588$	0	0
$589$	−19874.8	−1.39037
$590$	−5692.70	−0.397228
$591$	0	0
$592$	−1752.52	−0.121669
$593$	−16803.8	−1.16366	−0.581830	−	0.813311i	$-0.697663\pi$
$593$	−16803.8	−1.16366	−0.581830	+	0.813311i	$0.697663\pi$
$594$	0	0
$595$	−6257.06	−0.431117
$596$	4706.41	0.323460
$597$	0	0
$598$	14938.7	1.02155
$599$	−17220.2	−1.17462	−0.587310	−	0.809362i	$-0.699813\pi$
$599$	−17220.2	−1.17462	−0.587310	+	0.809362i	$0.699813\pi$
$600$	0	0
$601$	21536.7	1.46173	0.730865	−	0.682522i	$-0.239117\pi$
$601$	21536.7	1.46173	0.730865	+	0.682522i	$0.239117\pi$
$602$	6253.94	0.423408
$603$	0	0
$604$	−12838.8	−0.864905
$605$	6312.52	0.424199
$606$	0	0
$607$	−24050.9	−1.60823	−0.804117	−	0.594472i	$-0.797361\pi$
$607$	−24050.9	−1.60823	−0.804117	+	0.594472i	$0.797361\pi$
$608$	−16296.2	−1.08700
$609$	0	0
$610$	3693.91	0.245184
$611$	31290.8	2.07184
$612$	0	0
$613$	−3554.49	−0.234200	−0.117100	−	0.993120i	$-0.537360\pi$
$613$	−3554.49	−0.234200	−0.117100	+	0.993120i	$0.537360\pi$
$614$	−10568.5	−0.694639
$615$	0	0
$616$	−3769.03	−0.246524
$617$	9747.42	0.636007	0.318003	−	0.948090i	$-0.396988\pi$
$617$	9747.42	0.636007	0.318003	+	0.948090i	$0.396988\pi$
$618$	0	0
$619$	−20765.2	−1.34834	−0.674171	−	0.738576i	$-0.735499\pi$
$619$	−20765.2	−1.34834	−0.674171	+	0.738576i	$0.735499\pi$
$620$	7005.39	0.453779
$621$	0	0
$622$	2900.66	0.186987
$623$	5411.80	0.348024
$624$	0	0
$625$	625.000	0.0400000
$626$	51.2551	0.00327247
$627$	0	0
$628$	20956.2	1.33159
$629$	3661.51	0.232105
$630$	0	0
$631$	23740.6	1.49778	0.748890	−	0.662694i	$-0.230587\pi$
$631$	23740.6	1.49778	0.748890	+	0.662694i	$0.230587\pi$
$632$	−1520.74	−0.0957147
$633$	0	0
$634$	−12652.2	−0.792560
$635$	−5192.25	−0.324486
$636$	0	0
$637$	20734.6	1.28970
$638$	−2977.02	−0.184736
$639$	0	0
$640$	7062.81	0.436222
$641$	−26108.7	−1.60879	−0.804394	−	0.594097i	$-0.797510\pi$
$641$	−26108.7	−1.60879	−0.804394	+	0.594097i	$0.797510\pi$
$642$	0	0
$643$	−24017.9	−1.47305	−0.736526	−	0.676409i	$-0.763535\pi$
$643$	−24017.9	−1.47305	−0.736526	+	0.676409i	$0.763535\pi$
$644$	19438.5	1.18941
$645$	0	0
$646$	6096.92	0.371331
$647$	−18588.3	−1.12949	−0.564747	−	0.825264i	$-0.691026\pi$
$647$	−18588.3	−1.12949	−0.564747	+	0.825264i	$0.691026\pi$
$648$	0	0
$649$	−7101.83	−0.429540
$650$	2889.61	0.174369
$651$	0	0
$652$	4432.45	0.266239
$653$	−12171.9	−0.729441	−0.364721	−	0.931117i	$-0.618835\pi$
$653$	−12171.9	−0.729441	−0.364721	+	0.931117i	$0.618835\pi$
$654$	0	0
$655$	−4017.05	−0.239632
$656$	−9118.04	−0.542682
$657$	0	0
$658$	−11487.6	−0.680597
$659$	−8589.64	−0.507746	−0.253873	−	0.967238i	$-0.581705\pi$
$659$	−8589.64	−0.507746	−0.253873	+	0.967238i	$0.581705\pi$
$660$	0	0
$661$	24995.3	1.47081	0.735405	−	0.677628i	$-0.236992\pi$
$661$	24995.3	1.47081	0.735405	+	0.677628i	$0.236992\pi$
$662$	−3994.46	−0.234515
$663$	0	0
$664$	−10896.7	−0.636859
$665$	−10667.4	−0.622053
$666$	0	0
$667$	35039.4	2.03408
$668$	−7164.71	−0.414987
$669$	0	0
$670$	277.958	0.0160275
$671$	4608.28	0.265127
$672$	0	0
$673$	−2540.17	−0.145492	−0.0727461	−	0.997350i	$-0.523176\pi$
$673$	−2540.17	−0.145492	−0.0727461	+	0.997350i	$0.523176\pi$
$674$	3731.94	0.213277
$675$	0	0
$676$	−33643.4	−1.91417
$677$	−4197.79	−0.238307	−0.119154	−	0.992876i	$-0.538018\pi$
$677$	−4197.79	−0.238307	−0.119154	+	0.992876i	$0.538018\pi$
$678$	0	0
$679$	13378.3	0.756128
$680$	−4904.36	−0.276579
$681$	0	0
$682$	−2465.73	−0.138442
$683$	−8523.02	−0.477488	−0.238744	−	0.971083i	$-0.576736\pi$
$683$	−8523.02	−0.477488	−0.238744	+	0.971083i	$0.576736\pi$
$684$	0	0
$685$	4345.76	0.242398
$686$	3357.58	0.186870
$687$	0	0
$688$	−4859.15	−0.269263
$689$	−2568.92	−0.142044
$690$	0	0
$691$	−20292.0	−1.11714	−0.558571	−	0.829457i	$-0.688650\pi$
$691$	−20292.0	−1.11714	−0.558571	+	0.829457i	$0.688650\pi$
$692$	−18252.6	−1.00269
$693$	0	0
$694$	184.896	0.0101132
$695$	981.990	0.0535957
$696$	0	0
$697$	19050.2	1.03526
$698$	−6913.05	−0.374875
$699$	0	0
$700$	3760.01	0.203022
$701$	−11223.6	−0.604721	−0.302361	−	0.953194i	$-0.597775\pi$
$701$	−11223.6	−0.604721	−0.302361	+	0.953194i	$0.597775\pi$
$702$	0	0
$703$	6242.38	0.334901
$704$	−376.518	−0.0201570
$705$	0	0
$706$	4525.03	0.241221
$707$	5471.66	0.291065
$708$	0	0
$709$	−19931.0	−1.05575	−0.527873	−	0.849324i	$-0.677010\pi$
$709$	−19931.0	−1.05575	−0.527873	+	0.849324i	$0.677010\pi$
$710$	−3643.17	−0.192572
$711$	0	0
$712$	4241.83	0.223271
$713$	29021.5	1.52435
$714$	0	0
$715$	3604.88	0.188552
$716$	21352.0	1.11447
$717$	0	0
$718$	−10789.4	−0.560804
$719$	−3186.13	−0.165261	−0.0826305	−	0.996580i	$-0.526332\pi$
$719$	−3186.13	−0.165261	−0.0826305	+	0.996580i	$0.526332\pi$
$720$	0	0
$721$	−9244.58	−0.477512
$722$	1293.83	0.0666916
$723$	0	0
$724$	7185.37	0.368843
$725$	6777.71	0.347197
$726$	0	0
$727$	−23088.2	−1.17785	−0.588923	−	0.808189i	$-0.700448\pi$
$727$	−23088.2	−1.17785	−0.588923	+	0.808189i	$0.700448\pi$
$728$	39672.5	2.01973
$729$	0	0
$730$	1230.65	0.0623949
$731$	10152.2	0.513667
$732$	0	0
$733$	−29801.4	−1.50169	−0.750846	−	0.660477i	$-0.770354\pi$
$733$	−29801.4	−1.50169	−0.750846	+	0.660477i	$0.770354\pi$
$734$	175.340	0.00881731
$735$	0	0
$736$	23796.0	1.19175
$737$	346.761	0.0173312
$738$	0	0
$739$	39741.5	1.97823	0.989117	−	0.147134i	$-0.0470047\pi$
$739$	39741.5	1.97823	0.989117	+	0.147134i	$0.0470047\pi$
$740$	−2200.29	−0.109303
$741$	0	0
$742$	943.111	0.0466613
$743$	9713.53	0.479616	0.239808	−	0.970820i	$-0.422915\pi$
$743$	9713.53	0.479616	0.239808	+	0.970820i	$0.422915\pi$
$744$	0	0
$745$	3771.42	0.185468
$746$	12404.2	0.608779
$747$	0	0
$748$	−2680.97	−0.131051
$749$	−42792.7	−2.08760
$750$	0	0
$751$	−2509.28	−0.121924	−0.0609619	−	0.998140i	$-0.519417\pi$
$751$	−2509.28	−0.121924	−0.0609619	+	0.998140i	$0.519417\pi$
$752$	8925.55	0.432821
$753$	0	0
$754$	31335.9	1.51351
$755$	−10288.2	−0.495927
$756$	0	0
$757$	9705.73	0.465998	0.232999	−	0.972477i	$-0.425146\pi$
$757$	9705.73	0.465998	0.232999	+	0.972477i	$0.425146\pi$
$758$	8178.67	0.391903
$759$	0	0
$760$	−8361.25	−0.399072
$761$	−7778.44	−0.370523	−0.185262	−	0.982689i	$-0.559313\pi$
$761$	−7778.44	−0.370523	−0.185262	+	0.982689i	$0.559313\pi$
$762$	0	0
$763$	−36891.5	−1.75041
$764$	5816.04	0.275415
$765$	0	0
$766$	3451.39	0.162799
$767$	74753.3	3.51915
$768$	0	0
$769$	1387.90	0.0650833	0.0325416	−	0.999470i	$-0.489640\pi$
$769$	1387.90	0.0650833	0.0325416	+	0.999470i	$0.489640\pi$
$770$	−1323.44	−0.0619393
$771$	0	0
$772$	26660.5	1.24292
$773$	20692.0	0.962796	0.481398	−	0.876502i	$-0.340129\pi$
$773$	20692.0	0.962796	0.481398	+	0.876502i	$0.340129\pi$
$774$	0	0
$775$	5613.67	0.260192
$776$	10486.0	0.485086
$777$	0	0
$778$	5639.70	0.259888
$779$	32478.0	1.49377
$780$	0	0
$781$	−4544.98	−0.208236
$782$	−8902.82	−0.407115
$783$	0	0
$784$	5914.45	0.269426
$785$	16792.9	0.763523
$786$	0	0
$787$	33896.8	1.53531	0.767655	−	0.640863i	$-0.221423\pi$
$787$	33896.8	1.53531	0.767655	+	0.640863i	$0.221423\pi$
$788$	−12005.9	−0.542756
$789$	0	0
$790$	−533.983	−0.0240484
$791$	−20266.4	−0.910988
$792$	0	0
$793$	−48506.4	−2.17215
$794$	−8078.08	−0.361058
$795$	0	0
$796$	−10849.7	−0.483110
$797$	−11954.5	−0.531307	−0.265653	−	0.964069i	$-0.585588\pi$
$797$	−11954.5	−0.531307	−0.265653	+	0.964069i	$0.585588\pi$
$798$	0	0
$799$	−18648.0	−0.825682
$800$	4602.89	0.203421
$801$	0	0
$802$	10250.7	0.451325
$803$	1535.27	0.0674702
$804$	0	0
$805$	15576.8	0.681998
$806$	25954.1	1.13424
$807$	0	0
$808$	4288.75	0.186730
$809$	27377.3	1.18978	0.594891	−	0.803807i	$-0.297195\pi$
$809$	27377.3	1.18978	0.594891	+	0.803807i	$0.297195\pi$
$810$	0	0
$811$	−713.034	−0.0308730	−0.0154365	−	0.999881i	$-0.504914\pi$
$811$	−713.034	−0.0308730	−0.0154365	+	0.999881i	$0.504914\pi$
$812$	40774.9	1.76221
$813$	0	0
$814$	774.449	0.0333470
$815$	3551.88	0.152659
$816$	0	0
$817$	17308.0	0.741164
$818$	21655.0	0.925612
$819$	0	0
$820$	−11447.7	−0.487526
$821$	6384.12	0.271385	0.135693	−	0.990751i	$-0.456674\pi$
$821$	6384.12	0.271385	0.135693	+	0.990751i	$0.456674\pi$
$822$	0	0
$823$	19636.3	0.831686	0.415843	−	0.909436i	$-0.363487\pi$
$823$	19636.3	0.831686	0.415843	+	0.909436i	$0.363487\pi$
$824$	−7246.00	−0.306343
$825$	0	0
$826$	−27443.7	−1.15604
$827$	43578.4	1.83237	0.916185	−	0.400756i	$-0.131253\pi$
$827$	43578.4	1.83237	0.916185	+	0.400756i	$0.131253\pi$
$828$	0	0
$829$	−20418.8	−0.855458	−0.427729	−	0.903907i	$-0.640686\pi$
$829$	−20418.8	−0.855458	−0.427729	+	0.903907i	$0.640686\pi$
$830$	−3826.21	−0.160012
$831$	0	0
$832$	3963.20	0.165143
$833$	−12357.0	−0.513978
$834$	0	0
$835$	−5741.34	−0.237949
$836$	−4570.69	−0.189092
$837$	0	0
$838$	1529.09	0.0630328
$839$	−10962.0	−0.451073	−0.225536	−	0.974235i	$-0.572413\pi$
$839$	−10962.0	−0.451073	−0.225536	+	0.974235i	$0.572413\pi$
$840$	0	0
$841$	49110.9	2.01365
$842$	−4685.26	−0.191763
$843$	0	0
$844$	−19981.3	−0.814911
$845$	−26959.7	−1.09756
$846$	0	0
$847$	30431.8	1.23453
$848$	−732.772	−0.0296739
$849$	0	0
$850$	−1722.09	−0.0694907
$851$	−9115.22	−0.367175
$852$	0	0
$853$	−5246.41	−0.210590	−0.105295	−	0.994441i	$-0.533579\pi$
$853$	−5246.41	−0.210590	−0.105295	+	0.994441i	$0.533579\pi$
$854$	17807.8	0.713548
$855$	0	0
$856$	−33541.4	−1.33928
$857$	−27283.1	−1.08748	−0.543741	−	0.839253i	$-0.682993\pi$
$857$	−27283.1	−1.08748	−0.543741	+	0.839253i	$0.682993\pi$
$858$	0	0
$859$	3581.05	0.142240	0.0711198	−	0.997468i	$-0.477343\pi$
$859$	3581.05	0.142240	0.0711198	+	0.997468i	$0.477343\pi$
$860$	−6100.66	−0.241896
$861$	0	0
$862$	13574.0	0.536350
$863$	−41710.9	−1.64526	−0.822629	−	0.568579i	$-0.807493\pi$
$863$	−41710.9	−1.64526	−0.822629	+	0.568579i	$0.807493\pi$
$864$	0	0
$865$	−14626.4	−0.574929
$866$	−9700.71	−0.380651
$867$	0	0
$868$	33771.9	1.32062
$869$	−666.161	−0.0260046
$870$	0	0
$871$	−3649.98	−0.141992
$872$	−28916.0	−1.12296
$873$	0	0
$874$	−15178.1	−0.587422
$875$	3013.03	0.116410
$876$	0	0
$877$	21533.6	0.829119	0.414560	−	0.910022i	$-0.363936\pi$
$877$	21533.6	0.829119	0.414560	+	0.910022i	$0.363936\pi$
$878$	9358.22	0.359709
$879$	0	0
$880$	1028.27	0.0393899
$881$	−30919.1	−1.18240	−0.591199	−	0.806526i	$-0.701345\pi$
$881$	−30919.1	−1.18240	−0.591199	+	0.806526i	$0.701345\pi$
$882$	0	0
$883$	5341.72	0.203582	0.101791	−	0.994806i	$-0.467543\pi$
$883$	5341.72	0.203582	0.101791	+	0.994806i	$0.467543\pi$
$884$	28219.7	1.07368
$885$	0	0
$886$	−222.626	−0.00844160
$887$	−11210.3	−0.424358	−0.212179	−	0.977231i	$-0.568056\pi$
$887$	−11210.3	−0.424358	−0.212179	+	0.977231i	$0.568056\pi$
$888$	0	0
$889$	−25031.1	−0.944337
$890$	1489.45	0.0560972
$891$	0	0
$892$	8766.54	0.329064
$893$	−31792.3	−1.19137
$894$	0	0
$895$	17110.1	0.639025
$896$	34048.8	1.26952
$897$	0	0
$898$	5240.81	0.194753
$899$	60876.6	2.25845
$900$	0	0
$901$	1530.97	0.0566083
$902$	4029.32	0.148738
$903$	0	0
$904$	−15885.1	−0.584435
$905$	5757.90	0.211491
$906$	0	0
$907$	20690.8	0.757473	0.378737	−	0.925505i	$-0.376359\pi$
$907$	20690.8	0.757473	0.378737	+	0.925505i	$0.376359\pi$
$908$	38924.6	1.42264
$909$	0	0
$910$	13930.4	0.507459
$911$	−19471.6	−0.708147	−0.354073	−	0.935218i	$-0.615204\pi$
$911$	−19471.6	−0.708147	−0.354073	+	0.935218i	$0.615204\pi$
$912$	0	0
$913$	−4773.32	−0.173027
$914$	−9881.53	−0.357606
$915$	0	0
$916$	−41370.3	−1.49226
$917$	−19365.6	−0.697393
$918$	0	0
$919$	−30002.0	−1.07690	−0.538452	−	0.842656i	$-0.680991\pi$
$919$	−30002.0	−1.07690	−0.538452	+	0.842656i	$0.680991\pi$
$920$	12209.2	0.437529
$921$	0	0
$922$	14445.8	0.515996
$923$	47840.1	1.70604
$924$	0	0
$925$	−1763.17	−0.0626732
$926$	−4207.66	−0.149322
$927$	0	0
$928$	49915.3	1.76568
$929$	−22262.8	−0.786241	−0.393121	−	0.919487i	$-0.628605\pi$
$929$	−22262.8	−0.786241	−0.393121	+	0.919487i	$0.628605\pi$
$930$	0	0
$931$	−21066.9	−0.741612
$932$	15308.7	0.538039
$933$	0	0
$934$	−21691.4	−0.759917
$935$	−2148.36	−0.0751432
$936$	0	0
$937$	23361.5	0.814500	0.407250	−	0.913317i	$-0.366488\pi$
$937$	23361.5	0.814500	0.407250	+	0.913317i	$0.366488\pi$
$938$	1339.99	0.0466443
$939$	0	0
$940$	11206.0	0.388830
$941$	−39330.9	−1.36254	−0.681271	−	0.732031i	$-0.738572\pi$
$941$	−39330.9	−1.36254	−0.681271	+	0.732031i	$0.738572\pi$
$942$	0	0
$943$	−47424.8	−1.63771
$944$	21323.0	0.735175
$945$	0	0
$946$	2147.29	0.0737995
$947$	26869.9	0.922021	0.461011	−	0.887395i	$-0.347487\pi$
$947$	26869.9	0.922021	0.461011	+	0.887395i	$0.347487\pi$
$948$	0	0
$949$	−16160.1	−0.552772
$950$	−2935.92	−0.100267
$951$	0	0
$952$	−23643.2	−0.804916
$953$	−16422.6	−0.558218	−0.279109	−	0.960259i	$-0.590039\pi$
$953$	−16422.6	−0.558218	−0.279109	+	0.960259i	$0.590039\pi$
$954$	0	0
$955$	4660.60	0.157920
$956$	−43363.5	−1.46702
$957$	0	0
$958$	12490.2	0.421231
$959$	20950.3	0.705442
$960$	0	0
$961$	20630.3	0.692500
$962$	−8151.79	−0.273206
$963$	0	0
$964$	−39528.9	−1.32069
$965$	21364.1	0.712677
$966$	0	0
$967$	10304.4	0.342677	0.171339	−	0.985212i	$-0.445191\pi$
$967$	10304.4	0.342677	0.171339	+	0.985212i	$0.445191\pi$
$968$	23852.7	0.792000
$969$	0	0
$970$	3682.00	0.121878
$971$	44153.1	1.45926	0.729630	−	0.683842i	$-0.239692\pi$
$971$	44153.1	1.45926	0.729630	+	0.683842i	$0.239692\pi$
$972$	0	0
$973$	4734.03	0.155977
$974$	17888.4	0.588483
$975$	0	0
$976$	−13836.2	−0.453776
$977$	−16115.6	−0.527722	−0.263861	−	0.964561i	$-0.584996\pi$
$977$	−16115.6	−0.527722	−0.263861	+	0.964561i	$0.584996\pi$
$978$	0	0
$979$	1858.14	0.0606602
$980$	7425.59	0.242043
$981$	0	0
$982$	−12232.8	−0.397521
$983$	−38748.6	−1.25726	−0.628630	−	0.777704i	$-0.716384\pi$
$983$	−38748.6	−1.25726	−0.628630	+	0.777704i	$0.716384\pi$
$984$	0	0
$985$	−9620.75	−0.311211
$986$	−18674.9	−0.603174
$987$	0	0
$988$	48110.7	1.54920
$989$	−25273.5	−0.812587
$990$	0	0
$991$	−42906.4	−1.37534	−0.687672	−	0.726022i	$-0.741367\pi$
$991$	−42906.4	−1.37534	−0.687672	+	0.726022i	$0.741367\pi$
$992$	41342.5	1.32321
$993$	0	0
$994$	−17563.2	−0.560434
$995$	−8694.22	−0.277010
$996$	0	0
$997$	−19395.3	−0.616104	−0.308052	−	0.951369i	$-0.599677\pi$
$997$	−19395.3	−0.616104	−0.308052	+	0.951369i	$0.599677\pi$
$998$	138.447	0.00439126
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.i.1.2	yes	3
3.2	odd	2		405.4.a.g.1.2	✓	3
5.4	even	2		2025.4.a.p.1.2		3
9.2	odd	6		405.4.e.u.271.2		6
9.4	even	3		405.4.e.s.136.2		6
9.5	odd	6		405.4.e.u.136.2		6
9.7	even	3		405.4.e.s.271.2		6
15.14	odd	2		2025.4.a.r.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.g.1.2	✓	3	3.2	odd	2
405.4.a.i.1.2	yes	3	1.1	even	1	trivial
405.4.e.s.136.2		6	9.4	even	3
405.4.e.s.271.2		6	9.7	even	3
405.4.e.u.136.2		6	9.5	odd	6
405.4.e.u.271.2		6	9.2	odd	6
2025.4.a.p.1.2		3	5.4	even	2
2025.4.a.r.1.2		3	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q+1.32681 q^{2} -6.23958 q^{4} -5.00000 q^{5} -24.1043 q^{7} -18.8932 q^{8} -6.63404 q^{10} -8.27619 q^{11} +87.1145 q^{13} -31.9818 q^{14} +24.8490 q^{16} -51.9166 q^{17} -88.5107 q^{19} +31.1979 q^{20} -10.9809 q^{22} +129.245 q^{23} +25.0000 q^{25} +115.584 q^{26} +150.400 q^{28} +271.109 q^{29} +224.547 q^{31} +184.115 q^{32} -68.8834 q^{34} +120.521 q^{35} -70.5268 q^{37} -117.437 q^{38} +94.4660 q^{40} -366.938 q^{41} -195.547 q^{43} +51.6399 q^{44} +171.483 q^{46} +359.192 q^{47} +238.016 q^{49} +33.1702 q^{50} -543.558 q^{52} -29.4890 q^{53} +41.3810 q^{55} +455.407 q^{56} +359.709 q^{58} +858.104 q^{59} -556.811 q^{61} +297.931 q^{62} +45.4941 q^{64} -435.573 q^{65} -41.8987 q^{67} +323.938 q^{68} +159.909 q^{70} +549.163 q^{71} -185.505 q^{73} -93.5756 q^{74} +552.269 q^{76} +199.492 q^{77} +80.4913 q^{79} -124.245 q^{80} -486.857 q^{82} +576.753 q^{83} +259.583 q^{85} -259.454 q^{86} +156.364 q^{88} -224.516 q^{89} -2099.83 q^{91} -806.433 q^{92} +476.579 q^{94} +442.553 q^{95} -555.016 q^{97} +315.801 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 5 q^{4} - 15 q^{5} - 25 q^{7} - 27 q^{8} - 5 q^{10} + 58 q^{11} - 47 q^{13} + 159 q^{14} - 127 q^{16} + 34 q^{17} - 5 q^{19} - 25 q^{20} + 260 q^{22} - 51 q^{23} + 75 q^{25} + 253 q^{26}+ \cdots + 38 q^{98}+O(q^{100}) \) 3 * q + q^2 + 5 * q^4 - 15 * q^5 - 25 * q^7 - 27 * q^8 - 5 * q^10 + 58 * q^11 - 47 * q^13 + 159 * q^14 - 127 * q^16 + 34 * q^17 - 5 * q^19 - 25 * q^20 + 260 * q^22 - 51 * q^23 + 75 * q^25 + 253 * q^26 + 83 * q^28 + 350 * q^29 + 638 * q^31 + 245 * q^32 - 154 * q^34 + 125 * q^35 - 414 * q^37 + 397 * q^38 + 135 * q^40 + 179 * q^41 - 836 * q^43 + 332 * q^44 + 261 * q^46 + 235 * q^47 + 892 * q^49 + 25 * q^50 - 1335 * q^52 + 505 * q^53 - 290 * q^55 + 15 * q^56 + 1876 * q^58 + 535 * q^59 - 104 * q^61 + 348 * q^62 - 303 * q^64 + 235 * q^65 - 40 * q^67 + 830 * q^68 - 795 * q^70 + 452 * q^71 - 710 * q^73 + 1394 * q^74 + 849 * q^76 + 2148 * q^77 - 634 * q^79 + 635 * q^80 + 613 * q^82 + 1734 * q^83 - 170 * q^85 + 460 * q^86 - 768 * q^88 - 852 * q^89 - 1229 * q^91 - 1839 * q^92 + 1751 * q^94 + 25 * q^95 + 38 * q^98

Introduction

L-functions

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Varieties

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Database

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