LMFDB - Embedded newform 3375.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3375,2,Mod(1,3375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3375, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3375.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3375 = 3^{3} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3375.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:	$26.9495106822$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 5x^{6} + 27x^{5} + 8x^{4} - 52x^{3} - 5x^{2} + 24x + 1 $ x^8 - 4x^7 - 5x^6 + 27x^5 + 8x^4 - 52x^3 - 5x^2 + 24*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.99433$ of defining polynomial
Character	$\chi$	$=$	3375.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.99433 q^{2} +1.97736 q^{4} +1.15949 q^{7} +0.0451562 q^{8} +O(q^{10})$	$q-1.99433 q^{2} +1.97736 q^{4} +1.15949 q^{7} +0.0451562 q^{8} +4.70972 q^{11} +3.78280 q^{13} -2.31241 q^{14} -4.04477 q^{16} +7.76990 q^{17} -7.45906 q^{19} -9.39275 q^{22} +0.679356 q^{23} -7.54416 q^{26} +2.29272 q^{28} +7.90871 q^{29} +4.60439 q^{31} +7.97630 q^{32} -15.4957 q^{34} +10.8287 q^{37} +14.8758 q^{38} +3.12265 q^{41} +0.880006 q^{43} +9.31281 q^{44} -1.35486 q^{46} +4.44334 q^{47} -5.65559 q^{49} +7.47995 q^{52} +3.01114 q^{53} +0.0523581 q^{56} -15.7726 q^{58} -8.16983 q^{59} +7.72280 q^{61} -9.18268 q^{62} -7.81785 q^{64} -6.54567 q^{67} +15.3639 q^{68} -9.01503 q^{71} -11.1002 q^{73} -21.5959 q^{74} -14.7492 q^{76} +5.46087 q^{77} -9.78275 q^{79} -6.22760 q^{82} -1.61819 q^{83} -1.75502 q^{86} +0.212673 q^{88} -0.725124 q^{89} +4.38611 q^{91} +1.34333 q^{92} -8.86149 q^{94} +6.47157 q^{97} +11.2791 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} + 10 q^{4} + 2 q^{7} + 15 q^{8}+O(q^{10}) $ 8 * q + 4 * q^2 + 10 * q^4 + 2 * q^7 + 15 * q^8	$ 8 q + 4 q^{2} + 10 q^{4} + 2 q^{7} + 15 q^{8} + q^{11} - 7 q^{13} - 3 q^{14} + 14 q^{16} + 26 q^{17} + 2 q^{19} - 12 q^{22} + 13 q^{23} - 5 q^{26} + 26 q^{28} - q^{29} + 5 q^{31} + 39 q^{32} + q^{34} + 16 q^{37} + 19 q^{38} - 5 q^{41} - 11 q^{43} + 17 q^{44} + q^{46} + 32 q^{47} - 15 q^{52} + 19 q^{53} + 40 q^{56} - 28 q^{58} - 6 q^{59} + 30 q^{62} + 25 q^{64} + 15 q^{67} + 52 q^{68} + 4 q^{71} + 20 q^{73} + 5 q^{74} + 7 q^{76} + 43 q^{77} - 7 q^{79} + 14 q^{82} + 43 q^{83} - 66 q^{86} - 17 q^{88} + 16 q^{89} - 24 q^{91} + 36 q^{92} + 18 q^{94} - 34 q^{97} + 28 q^{98}+O(q^{100}) $ 8 * q + 4 * q^2 + 10 * q^4 + 2 * q^7 + 15 * q^8 + q^11 - 7 * q^13 - 3 * q^14 + 14 * q^16 + 26 * q^17 + 2 * q^19 - 12 * q^22 + 13 * q^23 - 5 * q^26 + 26 * q^28 - q^29 + 5 * q^31 + 39 * q^32 + q^34 + 16 * q^37 + 19 * q^38 - 5 * q^41 - 11 * q^43 + 17 * q^44 + q^46 + 32 * q^47 - 15 * q^52 + 19 * q^53 + 40 * q^56 - 28 * q^58 - 6 * q^59 + 30 * q^62 + 25 * q^64 + 15 * q^67 + 52 * q^68 + 4 * q^71 + 20 * q^73 + 5 * q^74 + 7 * q^76 + 43 * q^77 - 7 * q^79 + 14 * q^82 + 43 * q^83 - 66 * q^86 - 17 * q^88 + 16 * q^89 - 24 * q^91 + 36 * q^92 + 18 * q^94 - 34 * q^97 + 28 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.99433	−1.41021	−0.705103	−	0.709105i	$-0.749099\pi$
$2$	−1.99433	−1.41021	−0.705103	+	0.709105i	$0.749099\pi$
$3$	0	0
$3$	0	0
$4$	1.97736	0.988679
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.15949	0.438246	0.219123	−	0.975697i	$-0.429680\pi$
$7$	1.15949	0.438246	0.219123	+	0.975697i	$0.429680\pi$
$8$	0.0451562	0.0159651
$9$	0	0
$10$	0	0
$11$	4.70972	1.42004	0.710018	−	0.704184i	$-0.248687\pi$
$11$	4.70972	1.42004	0.710018	+	0.704184i	$0.248687\pi$
$12$	0	0
$13$	3.78280	1.04916	0.524580	−	0.851361i	$-0.324222\pi$
$13$	3.78280	1.04916	0.524580	+	0.851361i	$0.324222\pi$
$14$	−2.31241	−0.618016
$15$	0	0
$16$	−4.04477	−1.01119
$17$	7.76990	1.88448	0.942238	−	0.334943i	$-0.108717\pi$
$17$	7.76990	1.88448	0.942238	+	0.334943i	$0.108717\pi$
$18$	0	0
$19$	−7.45906	−1.71123	−0.855613	−	0.517617i	$-0.826819\pi$
$19$	−7.45906	−1.71123	−0.855613	+	0.517617i	$0.826819\pi$
$20$	0	0
$21$	0	0
$22$	−9.39275	−2.00254
$23$	0.679356	0.141656	0.0708278	−	0.997489i	$-0.477436\pi$
$23$	0.679356	0.141656	0.0708278	+	0.997489i	$0.477436\pi$
$24$	0	0
$25$	0	0
$26$	−7.54416	−1.47953
$27$	0	0
$28$	2.29272	0.433284
$29$	7.90871	1.46861	0.734306	−	0.678819i	$-0.237508\pi$
$29$	7.90871	1.46861	0.734306	+	0.678819i	$0.237508\pi$
$30$	0	0
$31$	4.60439	0.826973	0.413486	−	0.910510i	$-0.364311\pi$
$31$	4.60439	0.826973	0.413486	+	0.910510i	$0.364311\pi$
$32$	7.97630	1.41002
$33$	0	0
$34$	−15.4957	−2.65750
$35$	0	0
$36$	0	0
$37$	10.8287	1.78022	0.890110	−	0.455746i	$-0.150627\pi$
$37$	10.8287	1.78022	0.890110	+	0.455746i	$0.150627\pi$
$38$	14.8758	2.41318
$39$	0	0
$40$	0	0
$41$	3.12265	0.487676	0.243838	−	0.969816i	$-0.421593\pi$
$41$	3.12265	0.487676	0.243838	+	0.969816i	$0.421593\pi$
$42$	0	0
$43$	0.880006	0.134200	0.0670998	−	0.997746i	$-0.478625\pi$
$43$	0.880006	0.134200	0.0670998	+	0.997746i	$0.478625\pi$
$44$	9.31281	1.40396
$45$	0	0
$46$	−1.35486	−0.199763
$47$	4.44334	0.648128	0.324064	−	0.946035i	$-0.394951\pi$
$47$	4.44334	0.648128	0.324064	+	0.946035i	$0.394951\pi$
$48$	0	0
$49$	−5.65559	−0.807941
$50$	0	0
$51$	0	0
$52$	7.47995	1.03728
$53$	3.01114	0.413612	0.206806	−	0.978382i	$-0.433693\pi$
$53$	3.01114	0.413612	0.206806	+	0.978382i	$0.433693\pi$
$54$	0	0
$55$	0	0
$56$	0.0523581	0.00699665
$57$	0	0
$58$	−15.7726	−2.07104
$59$	−8.16983	−1.06362	−0.531811	−	0.846863i	$-0.678488\pi$
$59$	−8.16983	−1.06362	−0.531811	+	0.846863i	$0.678488\pi$
$60$	0	0
$61$	7.72280	0.988803	0.494401	−	0.869234i	$-0.335387\pi$
$61$	7.72280	0.988803	0.494401	+	0.869234i	$0.335387\pi$
$62$	−9.18268	−1.16620
$63$	0	0
$64$	−7.81785	−0.977231
$65$	0	0
$66$	0	0
$67$	−6.54567	−0.799681	−0.399841	−	0.916585i	$-0.630934\pi$
$67$	−6.54567	−0.799681	−0.399841	+	0.916585i	$0.630934\pi$
$68$	15.3639	1.86314
$69$	0	0
$70$	0	0
$71$	−9.01503	−1.06989	−0.534943	−	0.844888i	$-0.679667\pi$
$71$	−9.01503	−1.06989	−0.534943	+	0.844888i	$0.679667\pi$
$72$	0	0
$73$	−11.1002	−1.29918	−0.649592	−	0.760283i	$-0.725060\pi$
$73$	−11.1002	−1.29918	−0.649592	+	0.760283i	$0.725060\pi$
$74$	−21.5959	−2.51048
$75$	0	0
$76$	−14.7492	−1.69185
$77$	5.46087	0.622324
$78$	0	0
$79$	−9.78275	−1.10065	−0.550323	−	0.834952i	$-0.685495\pi$
$79$	−9.78275	−1.10065	−0.550323	+	0.834952i	$0.685495\pi$
$80$	0	0
$81$	0	0
$82$	−6.22760	−0.687723
$83$	−1.61819	−0.177620	−0.0888100	−	0.996049i	$-0.528306\pi$
$83$	−1.61819	−0.177620	−0.0888100	+	0.996049i	$0.528306\pi$
$84$	0	0
$85$	0	0
$86$	−1.75502	−0.189249
$87$	0	0
$88$	0.212673	0.0226710
$89$	−0.725124	−0.0768630	−0.0384315	−	0.999261i	$-0.512236\pi$
$89$	−0.725124	−0.0768630	−0.0384315	+	0.999261i	$0.512236\pi$
$90$	0	0
$91$	4.38611	0.459790
$92$	1.34333	0.140052
$93$	0	0
$94$	−8.86149	−0.913993
$95$	0	0
$96$	0	0
$97$	6.47157	0.657089	0.328544	−	0.944489i	$-0.393442\pi$
$97$	6.47157	0.657089	0.328544	+	0.944489i	$0.393442\pi$
$98$	11.2791	1.13936
$99$	0	0
$100$	0	0
$101$	6.28218	0.625100	0.312550	−	0.949901i	$-0.398817\pi$
$101$	6.28218	0.625100	0.312550	+	0.949901i	$0.398817\pi$
$102$	0	0
$103$	11.4977	1.13290	0.566452	−	0.824095i	$-0.308316\pi$
$103$	11.4977	1.13290	0.566452	+	0.824095i	$0.308316\pi$
$104$	0.170817	0.0167500
$105$	0	0
$106$	−6.00522	−0.583278
$107$	−7.26586	−0.702417	−0.351209	−	0.936297i	$-0.614229\pi$
$107$	−7.26586	−0.702417	−0.351209	+	0.936297i	$0.614229\pi$
$108$	0	0
$109$	7.18724	0.688413	0.344207	−	0.938894i	$-0.388148\pi$
$109$	7.18724	0.688413	0.344207	+	0.938894i	$0.388148\pi$
$110$	0	0
$111$	0	0
$112$	−4.68987	−0.443151
$113$	−2.92251	−0.274927	−0.137463	−	0.990507i	$-0.543895\pi$
$113$	−2.92251	−0.274927	−0.137463	+	0.990507i	$0.543895\pi$
$114$	0	0
$115$	0	0
$116$	15.6384	1.45198
$117$	0	0
$118$	16.2933	1.49992
$119$	9.00911	0.825864
$120$	0	0
$121$	11.1815	1.01650
$122$	−15.4018	−1.39441
$123$	0	0
$124$	9.10453	0.817611
$125$	0	0
$126$	0	0
$127$	−14.1559	−1.25613	−0.628065	−	0.778161i	$-0.716153\pi$
$127$	−14.1559	−1.25613	−0.628065	+	0.778161i	$0.716153\pi$
$128$	−0.361226	−0.0319282
$129$	0	0
$130$	0	0
$131$	−16.1389	−1.41006	−0.705030	−	0.709178i	$-0.749066\pi$
$131$	−16.1389	−1.41006	−0.705030	+	0.709178i	$0.749066\pi$
$132$	0	0
$133$	−8.64869	−0.749937
$134$	13.0542	1.12771
$135$	0	0
$136$	0.350859	0.0300859
$137$	−1.10953	−0.0947932	−0.0473966	−	0.998876i	$-0.515092\pi$
$137$	−1.10953	−0.0947932	−0.0473966	+	0.998876i	$0.515092\pi$
$138$	0	0
$139$	−14.7440	−1.25057	−0.625285	−	0.780396i	$-0.715017\pi$
$139$	−14.7440	−1.25057	−0.625285	+	0.780396i	$0.715017\pi$
$140$	0	0
$141$	0	0
$142$	17.9789	1.50876
$143$	17.8159	1.48984
$144$	0	0
$145$	0	0
$146$	22.1375	1.83212
$147$	0	0
$148$	21.4121	1.76007
$149$	−5.36636	−0.439629	−0.219814	−	0.975542i	$-0.570545\pi$
$149$	−5.36636	−0.439629	−0.219814	+	0.975542i	$0.570545\pi$
$150$	0	0
$151$	9.30383	0.757135	0.378568	−	0.925574i	$-0.376417\pi$
$151$	9.30383	0.757135	0.378568	+	0.925574i	$0.376417\pi$
$152$	−0.336823	−0.0273199
$153$	0	0
$154$	−10.8908	−0.877605
$155$	0	0
$156$	0	0
$157$	−20.7575	−1.65663	−0.828313	−	0.560266i	$-0.810699\pi$
$157$	−20.7575	−1.65663	−0.828313	+	0.560266i	$0.810699\pi$
$158$	19.5101	1.55214
$159$	0	0
$160$	0	0
$161$	0.787706	0.0620799
$162$	0	0
$163$	0.454845	0.0356262	0.0178131	−	0.999841i	$-0.494330\pi$
$163$	0.454845	0.0356262	0.0178131	+	0.999841i	$0.494330\pi$
$164$	6.17460	0.482155
$165$	0	0
$166$	3.22722	0.250481
$167$	−9.27068	−0.717387	−0.358693	−	0.933455i	$-0.616778\pi$
$167$	−9.27068	−0.717387	−0.358693	+	0.933455i	$0.616778\pi$
$168$	0	0
$169$	1.30958	0.100737
$170$	0	0
$171$	0	0
$172$	1.74009	0.132680
$173$	25.0049	1.90109	0.950544	−	0.310590i	$-0.100527\pi$
$173$	25.0049	1.90109	0.950544	+	0.310590i	$0.100527\pi$
$174$	0	0
$175$	0	0
$176$	−19.0498	−1.43593
$177$	0	0
$178$	1.44614	0.108393
$179$	16.0592	1.20032	0.600160	−	0.799880i	$-0.295103\pi$
$179$	16.0592	1.20032	0.600160	+	0.799880i	$0.295103\pi$
$180$	0	0
$181$	−21.4601	−1.59512	−0.797560	−	0.603240i	$-0.793876\pi$
$181$	−21.4601	−1.59512	−0.797560	+	0.603240i	$0.793876\pi$
$182$	−8.74737	−0.648398
$183$	0	0
$184$	0.0306771	0.00226155
$185$	0	0
$186$	0	0
$187$	36.5941	2.67602
$188$	8.78607	0.640790
$189$	0	0
$190$	0	0
$191$	13.2055	0.955517	0.477758	−	0.878491i	$-0.341449\pi$
$191$	13.2055	0.955517	0.477758	+	0.878491i	$0.341449\pi$
$192$	0	0
$193$	−15.2039	−1.09440	−0.547201	−	0.837001i	$-0.684307\pi$
$193$	−15.2039	−1.09440	−0.547201	+	0.837001i	$0.684307\pi$
$194$	−12.9065	−0.926630
$195$	0	0
$196$	−11.1831	−0.798794
$197$	18.6049	1.32555	0.662773	−	0.748820i	$-0.269379\pi$
$197$	18.6049	1.32555	0.662773	+	0.748820i	$0.269379\pi$
$198$	0	0
$199$	−13.9242	−0.987062	−0.493531	−	0.869728i	$-0.664294\pi$
$199$	−13.9242	−0.987062	−0.493531	+	0.869728i	$0.664294\pi$
$200$	0	0
$201$	0	0
$202$	−12.5288	−0.881520
$203$	9.17006	0.643612
$204$	0	0
$205$	0	0
$206$	−22.9303	−1.59763
$207$	0	0
$208$	−15.3006	−1.06090
$209$	−35.1301	−2.43000
$210$	0	0
$211$	−5.79884	−0.399209	−0.199604	−	0.979877i	$-0.563966\pi$
$211$	−5.79884	−0.399209	−0.199604	+	0.979877i	$0.563966\pi$
$212$	5.95411	0.408930
$213$	0	0
$214$	14.4905	0.990552
$215$	0	0
$216$	0	0
$217$	5.33874	0.362417
$218$	−14.3337	−0.970804
$219$	0	0
$220$	0	0
$221$	29.3920	1.97712
$222$	0	0
$223$	26.5159	1.77564	0.887818	−	0.460194i	$-0.152220\pi$
$223$	26.5159	1.77564	0.887818	+	0.460194i	$0.152220\pi$
$224$	9.24844	0.617937
$225$	0	0
$226$	5.82845	0.387703
$227$	−7.77524	−0.516061	−0.258030	−	0.966137i	$-0.583073\pi$
$227$	−7.77524	−0.516061	−0.258030	+	0.966137i	$0.583073\pi$
$228$	0	0
$229$	5.39684	0.356633	0.178317	−	0.983973i	$-0.442935\pi$
$229$	5.39684	0.356633	0.178317	+	0.983973i	$0.442935\pi$
$230$	0	0
$231$	0	0
$232$	0.357127	0.0234466
$233$	8.68705	0.569107	0.284554	−	0.958660i	$-0.408155\pi$
$233$	8.68705	0.569107	0.284554	+	0.958660i	$0.408155\pi$
$234$	0	0
$235$	0	0
$236$	−16.1547	−1.05158
$237$	0	0
$238$	−17.9671	−1.16464
$239$	−16.9788	−1.09826	−0.549132	−	0.835735i	$-0.685042\pi$
$239$	−16.9788	−1.09826	−0.549132	+	0.835735i	$0.685042\pi$
$240$	0	0
$241$	21.8053	1.40460	0.702302	−	0.711879i	$-0.252156\pi$
$241$	21.8053	1.40460	0.702302	+	0.711879i	$0.252156\pi$
$242$	−22.2996	−1.43347
$243$	0	0
$244$	15.2707	0.977608
$245$	0	0
$246$	0	0
$247$	−28.2161	−1.79535
$248$	0.207917	0.0132027
$249$	0	0
$250$	0	0
$251$	−24.8071	−1.56581	−0.782907	−	0.622139i	$-0.786264\pi$
$251$	−24.8071	−1.56581	−0.782907	+	0.622139i	$0.786264\pi$
$252$	0	0
$253$	3.19958	0.201156
$254$	28.2315	1.77140
$255$	0	0
$256$	16.3561	1.02226
$257$	0.569905	0.0355497	0.0177748	−	0.999842i	$-0.494342\pi$
$257$	0.569905	0.0355497	0.0177748	+	0.999842i	$0.494342\pi$
$258$	0	0
$259$	12.5557	0.780174
$260$	0	0
$261$	0	0
$262$	32.1862	1.98847
$263$	−2.60602	−0.160694	−0.0803470	−	0.996767i	$-0.525603\pi$
$263$	−2.60602	−0.160694	−0.0803470	+	0.996767i	$0.525603\pi$
$264$	0	0
$265$	0	0
$266$	17.2484	1.05757
$267$	0	0
$268$	−12.9431	−0.790628
$269$	16.6178	1.01320	0.506602	−	0.862180i	$-0.330901\pi$
$269$	16.6178	1.01320	0.506602	+	0.862180i	$0.330901\pi$
$270$	0	0
$271$	19.2846	1.17145	0.585727	−	0.810508i	$-0.300809\pi$
$271$	19.2846	1.17145	0.585727	+	0.810508i	$0.300809\pi$
$272$	−31.4275	−1.90557
$273$	0	0
$274$	2.21276	0.133678
$275$	0	0
$276$	0	0
$277$	−7.07387	−0.425028	−0.212514	−	0.977158i	$-0.568165\pi$
$277$	−7.07387	−0.425028	−0.212514	+	0.977158i	$0.568165\pi$
$278$	29.4044	1.76356
$279$	0	0
$280$	0	0
$281$	−31.0075	−1.84975	−0.924875	−	0.380272i	$-0.875830\pi$
$281$	−31.0075	−1.84975	−0.924875	+	0.380272i	$0.875830\pi$
$282$	0	0
$283$	−3.18858	−0.189542	−0.0947708	−	0.995499i	$-0.530212\pi$
$283$	−3.18858	−0.189542	−0.0947708	+	0.995499i	$0.530212\pi$
$284$	−17.8259	−1.05777
$285$	0	0
$286$	−35.5309	−2.10099
$287$	3.62068	0.213722
$288$	0	0
$289$	43.3713	2.55125
$290$	0	0
$291$	0	0
$292$	−21.9491	−1.28448
$293$	−2.16075	−0.126233	−0.0631163	−	0.998006i	$-0.520104\pi$
$293$	−2.16075	−0.126233	−0.0631163	+	0.998006i	$0.520104\pi$
$294$	0	0
$295$	0	0
$296$	0.488981	0.0284214
$297$	0	0
$298$	10.7023	0.619967
$299$	2.56987	0.148619
$300$	0	0
$301$	1.02036	0.0588124
$302$	−18.5549	−1.06772
$303$	0	0
$304$	30.1702	1.73038
$305$	0	0
$306$	0	0
$307$	1.16229	0.0663357	0.0331678	−	0.999450i	$-0.489440\pi$
$307$	1.16229	0.0663357	0.0331678	+	0.999450i	$0.489440\pi$
$308$	10.7981	0.615279
$309$	0	0
$310$	0	0
$311$	18.4680	1.04723	0.523613	−	0.851956i	$-0.324584\pi$
$311$	18.4680	1.04723	0.523613	+	0.851956i	$0.324584\pi$
$312$	0	0
$313$	−4.09236	−0.231314	−0.115657	−	0.993289i	$-0.536897\pi$
$313$	−4.09236	−0.231314	−0.115657	+	0.993289i	$0.536897\pi$
$314$	41.3973	2.33618
$315$	0	0
$316$	−19.3440	−1.08819
$317$	16.2209	0.911058	0.455529	−	0.890221i	$-0.349450\pi$
$317$	16.2209	0.911058	0.455529	+	0.890221i	$0.349450\pi$
$318$	0	0
$319$	37.2479	2.08548
$320$	0	0
$321$	0	0
$322$	−1.57095	−0.0875454
$323$	−57.9561	−3.22476
$324$	0	0
$325$	0	0
$326$	−0.907111	−0.0502402
$327$	0	0
$328$	0.141007	0.00778581
$329$	5.15200	0.284039
$330$	0	0
$331$	21.5601	1.18505	0.592525	−	0.805552i	$-0.298131\pi$
$331$	21.5601	1.18505	0.592525	+	0.805552i	$0.298131\pi$
$332$	−3.19975	−0.175609
$333$	0	0
$334$	18.4888	1.01166
$335$	0	0
$336$	0	0
$337$	−14.3256	−0.780366	−0.390183	−	0.920737i	$-0.627588\pi$
$337$	−14.3256	−0.780366	−0.390183	+	0.920737i	$0.627588\pi$
$338$	−2.61173	−0.142059
$339$	0	0
$340$	0	0
$341$	21.6854	1.17433
$342$	0	0
$343$	−14.6740	−0.792322
$344$	0.0397377	0.00214251
$345$	0	0
$346$	−49.8681	−2.68092
$347$	25.5194	1.36995	0.684977	−	0.728564i	$-0.259812\pi$
$347$	25.5194	1.36995	0.684977	+	0.728564i	$0.259812\pi$
$348$	0	0
$349$	15.5665	0.833256	0.416628	−	0.909077i	$-0.363212\pi$
$349$	15.5665	0.833256	0.416628	+	0.909077i	$0.363212\pi$
$350$	0	0
$351$	0	0
$352$	37.5662	2.00228
$353$	9.89983	0.526915	0.263457	−	0.964671i	$-0.415137\pi$
$353$	9.89983	0.526915	0.263457	+	0.964671i	$0.415137\pi$
$354$	0	0
$355$	0	0
$356$	−1.43383	−0.0759928
$357$	0	0
$358$	−32.0274	−1.69270
$359$	18.3603	0.969021	0.484510	−	0.874786i	$-0.338998\pi$
$359$	18.3603	0.969021	0.484510	+	0.874786i	$0.338998\pi$
$360$	0	0
$361$	36.6375	1.92829
$362$	42.7986	2.24945
$363$	0	0
$364$	8.67292	0.454584
$365$	0	0
$366$	0	0
$367$	16.0571	0.838174	0.419087	−	0.907946i	$-0.362350\pi$
$367$	16.0571	0.838174	0.419087	+	0.907946i	$0.362350\pi$
$368$	−2.74784	−0.143241
$369$	0	0
$370$	0	0
$371$	3.49139	0.181264
$372$	0	0
$373$	−6.15149	−0.318512	−0.159256	−	0.987237i	$-0.550910\pi$
$373$	−6.15149	−0.318512	−0.159256	+	0.987237i	$0.550910\pi$
$374$	−72.9807	−3.77374
$375$	0	0
$376$	0.200644	0.0103474
$377$	29.9171	1.54081
$378$	0	0
$379$	2.29145	0.117704	0.0588519	−	0.998267i	$-0.481256\pi$
$379$	2.29145	0.117704	0.0588519	+	0.998267i	$0.481256\pi$
$380$	0	0
$381$	0	0
$382$	−26.3362	−1.34748
$383$	0.712765	0.0364206	0.0182103	−	0.999834i	$-0.494203\pi$
$383$	0.712765	0.0364206	0.0182103	+	0.999834i	$0.494203\pi$
$384$	0	0
$385$	0	0
$386$	30.3216	1.54333
$387$	0	0
$388$	12.7966	0.649650
$389$	−11.8603	−0.601343	−0.300671	−	0.953728i	$-0.597211\pi$
$389$	−11.8603	−0.601343	−0.300671	+	0.953728i	$0.597211\pi$
$390$	0	0
$391$	5.27853	0.266947
$392$	−0.255385	−0.0128989
$393$	0	0
$394$	−37.1044	−1.86929
$395$	0	0
$396$	0	0
$397$	1.69173	0.0849055	0.0424527	−	0.999098i	$-0.486483\pi$
$397$	1.69173	0.0849055	0.0424527	+	0.999098i	$0.486483\pi$
$398$	27.7695	1.39196
$399$	0	0
$400$	0	0
$401$	−13.6001	−0.679155	−0.339578	−	0.940578i	$-0.610284\pi$
$401$	−13.6001	−0.679155	−0.339578	+	0.940578i	$0.610284\pi$
$402$	0	0
$403$	17.4175	0.867627
$404$	12.4221	0.618024
$405$	0	0
$406$	−18.2881	−0.907626
$407$	51.0000	2.52798
$408$	0	0
$409$	−9.68193	−0.478741	−0.239370	−	0.970928i	$-0.576941\pi$
$409$	−9.68193	−0.478741	−0.239370	+	0.970928i	$0.576941\pi$
$410$	0	0
$411$	0	0
$412$	22.7351	1.12008
$413$	−9.47283	−0.466127
$414$	0	0
$415$	0	0
$416$	30.1728	1.47934
$417$	0	0
$418$	70.0611	3.42680
$419$	25.8192	1.26135	0.630676	−	0.776046i	$-0.282778\pi$
$419$	25.8192	1.26135	0.630676	+	0.776046i	$0.282778\pi$
$420$	0	0
$421$	9.15051	0.445969	0.222984	−	0.974822i	$-0.428420\pi$
$421$	9.15051	0.445969	0.222984	+	0.974822i	$0.428420\pi$
$422$	11.5648	0.562966
$423$	0	0
$424$	0.135972	0.00660337
$425$	0	0
$426$	0	0
$427$	8.95450	0.433338
$428$	−14.3672	−0.694465
$429$	0	0
$430$	0	0
$431$	0.111820	0.00538617	0.00269308	−	0.999996i	$-0.499143\pi$
$431$	0.111820	0.00538617	0.00269308	+	0.999996i	$0.499143\pi$
$432$	0	0
$433$	−34.7641	−1.67066	−0.835329	−	0.549751i	$-0.814723\pi$
$433$	−34.7641	−1.67066	−0.835329	+	0.549751i	$0.814723\pi$
$434$	−10.6472	−0.511083
$435$	0	0
$436$	14.2118	0.680619
$437$	−5.06736	−0.242405
$438$	0	0
$439$	−5.38540	−0.257031	−0.128516	−	0.991707i	$-0.541021\pi$
$439$	−5.38540	−0.257031	−0.128516	+	0.991707i	$0.541021\pi$
$440$	0	0
$441$	0	0
$442$	−58.6173	−2.78814
$443$	−30.1175	−1.43093	−0.715463	−	0.698650i	$-0.753784\pi$
$443$	−30.1175	−1.43093	−0.715463	+	0.698650i	$0.753784\pi$
$444$	0	0
$445$	0	0
$446$	−52.8815	−2.50401
$447$	0	0
$448$	−9.06471	−0.428267
$449$	−25.9554	−1.22491	−0.612456	−	0.790505i	$-0.709818\pi$
$449$	−25.9554	−1.22491	−0.612456	+	0.790505i	$0.709818\pi$
$450$	0	0
$451$	14.7068	0.692517
$452$	−5.77885	−0.271814
$453$	0	0
$454$	15.5064	0.727752
$455$	0	0
$456$	0	0
$457$	−26.3904	−1.23449	−0.617246	−	0.786770i	$-0.711752\pi$
$457$	−26.3904	−1.23449	−0.617246	+	0.786770i	$0.711752\pi$
$458$	−10.7631	−0.502926
$459$	0	0
$460$	0	0
$461$	−25.1396	−1.17087	−0.585434	−	0.810720i	$-0.699076\pi$
$461$	−25.1396	−1.17087	−0.585434	+	0.810720i	$0.699076\pi$
$462$	0	0
$463$	−12.6596	−0.588341	−0.294171	−	0.955753i	$-0.595043\pi$
$463$	−12.6596	−0.588341	−0.294171	+	0.955753i	$0.595043\pi$
$464$	−31.9889	−1.48505
$465$	0	0
$466$	−17.3249	−0.802558
$467$	2.37641	0.109967	0.0549836	−	0.998487i	$-0.482489\pi$
$467$	2.37641	0.109967	0.0549836	+	0.998487i	$0.482489\pi$
$468$	0	0
$469$	−7.58963	−0.350457
$470$	0	0
$471$	0	0
$472$	−0.368918	−0.0169808
$473$	4.14458	0.190568
$474$	0	0
$475$	0	0
$476$	17.8142	0.816514
$477$	0	0
$478$	33.8613	1.54878
$479$	10.1435	0.463470	0.231735	−	0.972779i	$-0.425560\pi$
$479$	10.1435	0.463470	0.231735	+	0.972779i	$0.425560\pi$
$480$	0	0
$481$	40.9626	1.86774
$482$	−43.4870	−1.98078
$483$	0	0
$484$	22.1098	1.00499
$485$	0	0
$486$	0	0
$487$	−15.3107	−0.693795	−0.346897	−	0.937903i	$-0.612765\pi$
$487$	−15.3107	−0.693795	−0.346897	+	0.937903i	$0.612765\pi$
$488$	0.348732	0.0157864
$489$	0	0
$490$	0	0
$491$	11.2044	0.505648	0.252824	−	0.967512i	$-0.418641\pi$
$491$	11.2044	0.505648	0.252824	+	0.967512i	$0.418641\pi$
$492$	0	0
$493$	61.4499	2.76756
$494$	56.2723	2.53181
$495$	0	0
$496$	−18.6237	−0.836229
$497$	−10.4528	−0.468873
$498$	0	0
$499$	41.4418	1.85519	0.927595	−	0.373587i	$-0.121872\pi$
$499$	41.4418	1.85519	0.927595	+	0.373587i	$0.121872\pi$
$500$	0	0
$501$	0	0
$502$	49.4737	2.20812
$503$	0.268738	0.0119824	0.00599121	−	0.999982i	$-0.498093\pi$
$503$	0.268738	0.0119824	0.00599121	+	0.999982i	$0.498093\pi$
$504$	0	0
$505$	0	0
$506$	−6.38102	−0.283671
$507$	0	0
$508$	−27.9912	−1.24191
$509$	−17.6865	−0.783943	−0.391971	−	0.919977i	$-0.628207\pi$
$509$	−17.6865	−0.783943	−0.391971	+	0.919977i	$0.628207\pi$
$510$	0	0
$511$	−12.8706	−0.569362
$512$	−31.8970	−1.40966
$513$	0	0
$514$	−1.13658	−0.0501324
$515$	0	0
$516$	0	0
$517$	20.9269	0.920364
$518$	−25.0402	−1.10020
$519$	0	0
$520$	0	0
$521$	34.6951	1.52002	0.760010	−	0.649911i	$-0.225194\pi$
$521$	34.6951	1.52002	0.760010	+	0.649911i	$0.225194\pi$
$522$	0	0
$523$	28.9268	1.26488	0.632441	−	0.774609i	$-0.282053\pi$
$523$	28.9268	1.26488	0.632441	+	0.774609i	$0.282053\pi$
$524$	−31.9123	−1.39410
$525$	0	0
$526$	5.19727	0.226612
$527$	35.7756	1.55841
$528$	0	0
$529$	−22.5385	−0.979934
$530$	0	0
$531$	0	0
$532$	−17.1016	−0.741447
$533$	11.8124	0.511650
$534$	0	0
$535$	0	0
$536$	−0.295578	−0.0127670
$537$	0	0
$538$	−33.1414	−1.42883
$539$	−26.6362	−1.14730
$540$	0	0
$541$	−20.7170	−0.890693	−0.445347	−	0.895358i	$-0.646920\pi$
$541$	−20.7170	−0.890693	−0.445347	+	0.895358i	$0.646920\pi$
$542$	−38.4598	−1.65199
$543$	0	0
$544$	61.9750	2.65716
$545$	0	0
$546$	0	0
$547$	14.6395	0.625942	0.312971	−	0.949763i	$-0.398676\pi$
$547$	14.6395	0.625942	0.312971	+	0.949763i	$0.398676\pi$
$548$	−2.19393	−0.0937200
$549$	0	0
$550$	0	0
$551$	−58.9915	−2.51312
$552$	0	0
$553$	−11.3430	−0.482353
$554$	14.1076	0.599377
$555$	0	0
$556$	−29.1542	−1.23641
$557$	−23.8600	−1.01098	−0.505491	−	0.862832i	$-0.668689\pi$
$557$	−23.8600	−1.01098	−0.505491	+	0.862832i	$0.668689\pi$
$558$	0	0
$559$	3.32889	0.140797
$560$	0	0
$561$	0	0
$562$	61.8391	2.60853
$563$	−0.949926	−0.0400346	−0.0200173	−	0.999800i	$-0.506372\pi$
$563$	−0.949926	−0.0400346	−0.0200173	+	0.999800i	$0.506372\pi$
$564$	0	0
$565$	0	0
$566$	6.35909	0.267292
$567$	0	0
$568$	−0.407084	−0.0170809
$569$	10.8077	0.453080	0.226540	−	0.974002i	$-0.427259\pi$
$569$	10.8077	0.453080	0.226540	+	0.974002i	$0.427259\pi$
$570$	0	0
$571$	−3.39741	−0.142177	−0.0710887	−	0.997470i	$-0.522647\pi$
$571$	−3.39741	−0.142177	−0.0710887	+	0.997470i	$0.522647\pi$
$572$	35.2285	1.47298
$573$	0	0
$574$	−7.22083	−0.301392
$575$	0	0
$576$	0	0
$577$	38.1374	1.58768	0.793841	−	0.608126i	$-0.208078\pi$
$577$	38.1374	1.58768	0.793841	+	0.608126i	$0.208078\pi$
$578$	−86.4967	−3.59779
$579$	0	0
$580$	0	0
$581$	−1.87628	−0.0778412
$582$	0	0
$583$	14.1816	0.587344
$584$	−0.501244	−0.0207416
$585$	0	0
$586$	4.30926	0.178014
$587$	28.3968	1.17206	0.586031	−	0.810288i	$-0.300690\pi$
$587$	28.3968	1.17206	0.586031	+	0.810288i	$0.300690\pi$
$588$	0	0
$589$	−34.3444	−1.41514
$590$	0	0
$591$	0	0
$592$	−43.7994	−1.80015
$593$	37.4581	1.53822	0.769109	−	0.639117i	$-0.220700\pi$
$593$	37.4581	1.53822	0.769109	+	0.639117i	$0.220700\pi$
$594$	0	0
$595$	0	0
$596$	−10.6112	−0.434652
$597$	0	0
$598$	−5.12517	−0.209584
$599$	3.92835	0.160508	0.0802539	−	0.996774i	$-0.474427\pi$
$599$	3.92835	0.160508	0.0802539	+	0.996774i	$0.474427\pi$
$600$	0	0
$601$	−22.4919	−0.917464	−0.458732	−	0.888575i	$-0.651696\pi$
$601$	−22.4919	−0.917464	−0.458732	+	0.888575i	$0.651696\pi$
$602$	−2.03493	−0.0829375
$603$	0	0
$604$	18.3970	0.748563
$605$	0	0
$606$	0	0
$607$	−2.71474	−0.110188	−0.0550939	−	0.998481i	$-0.517546\pi$
$607$	−2.71474	−0.110188	−0.0550939	+	0.998481i	$0.517546\pi$
$608$	−59.4957	−2.41287
$609$	0	0
$610$	0	0
$611$	16.8083	0.679990
$612$	0	0
$613$	−22.0275	−0.889681	−0.444840	−	0.895610i	$-0.646740\pi$
$613$	−22.0275	−0.889681	−0.444840	+	0.895610i	$0.646740\pi$
$614$	−2.31800	−0.0935469
$615$	0	0
$616$	0.246592	0.00993548
$617$	43.6836	1.75864	0.879318	−	0.476234i	$-0.157999\pi$
$617$	43.6836	1.75864	0.879318	+	0.476234i	$0.157999\pi$
$618$	0	0
$619$	14.9432	0.600619	0.300309	−	0.953842i	$-0.402910\pi$
$619$	14.9432	0.600619	0.300309	+	0.953842i	$0.402910\pi$
$620$	0	0
$621$	0	0
$622$	−36.8314	−1.47680
$623$	−0.840774	−0.0336849
$624$	0	0
$625$	0	0
$626$	8.16153	0.326200
$627$	0	0
$628$	−41.0449	−1.63787
$629$	84.1375	3.35478
$630$	0	0
$631$	8.66652	0.345009	0.172504	−	0.985009i	$-0.444814\pi$
$631$	8.66652	0.345009	0.172504	+	0.985009i	$0.444814\pi$
$632$	−0.441752	−0.0175719
$633$	0	0
$634$	−32.3499	−1.28478
$635$	0	0
$636$	0	0
$637$	−21.3939	−0.847659
$638$	−74.2846	−2.94095
$639$	0	0
$640$	0	0
$641$	−48.9618	−1.93388	−0.966938	−	0.255011i	$-0.917921\pi$
$641$	−48.9618	−1.93388	−0.966938	+	0.255011i	$0.917921\pi$
$642$	0	0
$643$	−33.8097	−1.33332	−0.666662	−	0.745360i	$-0.732278\pi$
$643$	−33.8097	−1.33332	−0.666662	+	0.745360i	$0.732278\pi$
$644$	1.55758	0.0613771
$645$	0	0
$646$	115.584	4.54758
$647$	27.6759	1.08805	0.544027	−	0.839068i	$-0.316899\pi$
$647$	27.6759	1.08805	0.544027	+	0.839068i	$0.316899\pi$
$648$	0	0
$649$	−38.4776	−1.51038
$650$	0	0
$651$	0	0
$652$	0.899391	0.0352229
$653$	26.2456	1.02707	0.513535	−	0.858069i	$-0.328336\pi$
$653$	26.2456	1.02707	0.513535	+	0.858069i	$0.328336\pi$
$654$	0	0
$655$	0	0
$656$	−12.6304	−0.493135
$657$	0	0
$658$	−10.2748	−0.400554
$659$	9.02787	0.351676	0.175838	−	0.984419i	$-0.443737\pi$
$659$	9.02787	0.351676	0.175838	+	0.984419i	$0.443737\pi$
$660$	0	0
$661$	−30.9143	−1.20243	−0.601214	−	0.799088i	$-0.705316\pi$
$661$	−30.9143	−1.20243	−0.601214	+	0.799088i	$0.705316\pi$
$662$	−42.9980	−1.67116
$663$	0	0
$664$	−0.0730715	−0.00283572
$665$	0	0
$666$	0	0
$667$	5.37283	0.208037
$668$	−18.3315	−0.709265
$669$	0	0
$670$	0	0
$671$	36.3722	1.40413
$672$	0	0
$673$	37.6224	1.45024	0.725119	−	0.688623i	$-0.241785\pi$
$673$	37.6224	1.45024	0.725119	+	0.688623i	$0.241785\pi$
$674$	28.5700	1.10048
$675$	0	0
$676$	2.58950	0.0995962
$677$	−31.9800	−1.22909	−0.614545	−	0.788882i	$-0.710660\pi$
$677$	−31.9800	−1.22909	−0.614545	+	0.788882i	$0.710660\pi$
$678$	0	0
$679$	7.50372	0.287966
$680$	0	0
$681$	0	0
$682$	−43.2479	−1.65605
$683$	−34.6768	−1.32687	−0.663436	−	0.748233i	$-0.730903\pi$
$683$	−34.6768	−1.32687	−0.663436	+	0.748233i	$0.730903\pi$
$684$	0	0
$685$	0	0
$686$	29.2648	1.11734
$687$	0	0
$688$	−3.55942	−0.135702
$689$	11.3905	0.433945
$690$	0	0
$691$	−1.34502	−0.0511669	−0.0255834	−	0.999673i	$-0.508144\pi$
$691$	−1.34502	−0.0511669	−0.0255834	+	0.999673i	$0.508144\pi$
$692$	49.4437	1.87957
$693$	0	0
$694$	−50.8942	−1.93192
$695$	0	0
$696$	0	0
$697$	24.2627	0.919014
$698$	−31.0448	−1.17506
$699$	0	0
$700$	0	0
$701$	−20.5966	−0.777922	−0.388961	−	0.921254i	$-0.627166\pi$
$701$	−20.5966	−0.777922	−0.388961	+	0.921254i	$0.627166\pi$
$702$	0	0
$703$	−80.7716	−3.04636
$704$	−36.8199	−1.38770
$705$	0	0
$706$	−19.7435	−0.743058
$707$	7.28412	0.273948
$708$	0	0
$709$	−8.23015	−0.309090	−0.154545	−	0.987986i	$-0.549391\pi$
$709$	−8.23015	−0.309090	−0.154545	+	0.987986i	$0.549391\pi$
$710$	0	0
$711$	0	0
$712$	−0.0327438	−0.00122713
$713$	3.12802	0.117145
$714$	0	0
$715$	0	0
$716$	31.7548	1.18673
$717$	0	0
$718$	−36.6166	−1.36652
$719$	−38.5799	−1.43879	−0.719394	−	0.694602i	$-0.755580\pi$
$719$	−38.5799	−1.43879	−0.719394	+	0.694602i	$0.755580\pi$
$720$	0	0
$721$	13.3315	0.496490
$722$	−73.0674	−2.71929
$723$	0	0
$724$	−42.4344	−1.57706
$725$	0	0
$726$	0	0
$727$	−8.08144	−0.299724	−0.149862	−	0.988707i	$-0.547883\pi$
$727$	−8.08144	−0.299724	−0.149862	+	0.988707i	$0.547883\pi$
$728$	0.198060	0.00734060
$729$	0	0
$730$	0	0
$731$	6.83755	0.252896
$732$	0	0
$733$	19.2744	0.711916	0.355958	−	0.934502i	$-0.384155\pi$
$733$	19.2744	0.711916	0.355958	+	0.934502i	$0.384155\pi$
$734$	−32.0232	−1.18200
$735$	0	0
$736$	5.41875	0.199738
$737$	−30.8283	−1.13558
$738$	0	0
$739$	−18.9840	−0.698338	−0.349169	−	0.937060i	$-0.613536\pi$
$739$	−18.9840	−0.698338	−0.349169	+	0.937060i	$0.613536\pi$
$740$	0	0
$741$	0	0
$742$	−6.96298	−0.255619
$743$	−3.46470	−0.127108	−0.0635538	−	0.997978i	$-0.520243\pi$
$743$	−3.46470	−0.127108	−0.0635538	+	0.997978i	$0.520243\pi$
$744$	0	0
$745$	0	0
$746$	12.2681	0.449167
$747$	0	0
$748$	72.3596	2.64573
$749$	−8.42468	−0.307831
$750$	0	0
$751$	−18.4317	−0.672582	−0.336291	−	0.941758i	$-0.609173\pi$
$751$	−18.4317	−0.672582	−0.336291	+	0.941758i	$0.609173\pi$
$752$	−17.9723	−0.655382
$753$	0	0
$754$	−59.6646	−2.17286
$755$	0	0
$756$	0	0
$757$	−11.6972	−0.425141	−0.212571	−	0.977146i	$-0.568184\pi$
$757$	−11.6972	−0.425141	−0.212571	+	0.977146i	$0.568184\pi$
$758$	−4.56991	−0.165987
$759$	0	0
$760$	0	0
$761$	30.4169	1.10261	0.551307	−	0.834303i	$-0.314129\pi$
$761$	30.4169	1.10261	0.551307	+	0.834303i	$0.314129\pi$
$762$	0	0
$763$	8.33353	0.301694
$764$	26.1120	0.944699
$765$	0	0
$766$	−1.42149	−0.0513605
$767$	−30.9048	−1.11591
$768$	0	0
$769$	−17.3723	−0.626460	−0.313230	−	0.949677i	$-0.601411\pi$
$769$	−17.3723	−0.626460	−0.313230	+	0.949677i	$0.601411\pi$
$770$	0	0
$771$	0	0
$772$	−30.0636	−1.08201
$773$	0.989321	0.0355834	0.0177917	−	0.999842i	$-0.494336\pi$
$773$	0.989321	0.0355834	0.0177917	+	0.999842i	$0.494336\pi$
$774$	0	0
$775$	0	0
$776$	0.292232	0.0104905
$777$	0	0
$778$	23.6534	0.848017
$779$	−23.2920	−0.834524
$780$	0	0
$781$	−42.4583	−1.51928
$782$	−10.5271	−0.376449
$783$	0	0
$784$	22.8756	0.816984
$785$	0	0
$786$	0	0
$787$	41.0851	1.46453	0.732263	−	0.681022i	$-0.238464\pi$
$787$	41.0851	1.46453	0.732263	+	0.681022i	$0.238464\pi$
$788$	36.7886	1.31054
$789$	0	0
$790$	0	0
$791$	−3.38862	−0.120485
$792$	0	0
$793$	29.2138	1.03741
$794$	−3.37387	−0.119734
$795$	0	0
$796$	−27.5332	−0.975887
$797$	−32.8501	−1.16361	−0.581805	−	0.813328i	$-0.697653\pi$
$797$	−32.8501	−1.16361	−0.581805	+	0.813328i	$0.697653\pi$
$798$	0	0
$799$	34.5243	1.22138
$800$	0	0
$801$	0	0
$802$	27.1231	0.957749
$803$	−52.2791	−1.84489
$804$	0	0
$805$	0	0
$806$	−34.7362	−1.22353
$807$	0	0
$808$	0.283679	0.00997981
$809$	−16.2879	−0.572651	−0.286326	−	0.958132i	$-0.592434\pi$
$809$	−16.2879	−0.572651	−0.286326	+	0.958132i	$0.592434\pi$
$810$	0	0
$811$	−32.9085	−1.15557	−0.577787	−	0.816188i	$-0.696083\pi$
$811$	−32.9085	−1.15557	−0.577787	+	0.816188i	$0.696083\pi$
$812$	18.1325	0.636326
$813$	0	0
$814$	−101.711	−3.56496
$815$	0	0
$816$	0	0
$817$	−6.56401	−0.229646
$818$	19.3090	0.675123
$819$	0	0
$820$	0	0
$821$	−6.46628	−0.225675	−0.112837	−	0.993613i	$-0.535994\pi$
$821$	−6.46628	−0.225675	−0.112837	+	0.993613i	$0.535994\pi$
$822$	0	0
$823$	50.2971	1.75325	0.876623	−	0.481178i	$-0.159791\pi$
$823$	50.2971	1.75325	0.876623	+	0.481178i	$0.159791\pi$
$824$	0.519193	0.0180869
$825$	0	0
$826$	18.8920	0.657335
$827$	10.2503	0.356437	0.178219	−	0.983991i	$-0.442967\pi$
$827$	10.2503	0.356437	0.178219	+	0.983991i	$0.442967\pi$
$828$	0	0
$829$	2.72665	0.0947004	0.0473502	−	0.998878i	$-0.484922\pi$
$829$	2.72665	0.0947004	0.0473502	+	0.998878i	$0.484922\pi$
$830$	0	0
$831$	0	0
$832$	−29.5734	−1.02527
$833$	−43.9433	−1.52255
$834$	0	0
$835$	0	0
$836$	−69.4648	−2.40249
$837$	0	0
$838$	−51.4921	−1.77876
$839$	1.85046	0.0638849	0.0319424	−	0.999490i	$-0.489831\pi$
$839$	1.85046	0.0638849	0.0319424	+	0.999490i	$0.489831\pi$
$840$	0	0
$841$	33.5477	1.15682
$842$	−18.2491	−0.628907
$843$	0	0
$844$	−11.4664	−0.394689
$845$	0	0
$846$	0	0
$847$	12.9648	0.445477
$848$	−12.1794	−0.418242
$849$	0	0
$850$	0	0
$851$	7.35651	0.252178
$852$	0	0
$853$	−30.3689	−1.03981	−0.519906	−	0.854223i	$-0.674033\pi$
$853$	−30.3689	−1.03981	−0.519906	+	0.854223i	$0.674033\pi$
$854$	−17.8582	−0.611096
$855$	0	0
$856$	−0.328099	−0.0112142
$857$	52.7816	1.80298	0.901492	−	0.432795i	$-0.142473\pi$
$857$	52.7816	1.80298	0.901492	+	0.432795i	$0.142473\pi$
$858$	0	0
$859$	−51.6669	−1.76285	−0.881425	−	0.472323i	$-0.843415\pi$
$859$	−51.6669	−1.76285	−0.881425	+	0.472323i	$0.843415\pi$
$860$	0	0
$861$	0	0
$862$	−0.223006	−0.00759560
$863$	45.6540	1.55408	0.777040	−	0.629451i	$-0.216720\pi$
$863$	45.6540	1.55408	0.777040	+	0.629451i	$0.216720\pi$
$864$	0	0
$865$	0	0
$866$	69.3312	2.35597
$867$	0	0
$868$	10.5566	0.358314
$869$	−46.0741	−1.56296
$870$	0	0
$871$	−24.7610	−0.838993
$872$	0.324549	0.0109906
$873$	0	0
$874$	10.1060	0.341840
$875$	0	0
$876$	0	0
$877$	38.1248	1.28738	0.643692	−	0.765285i	$-0.277402\pi$
$877$	38.1248	1.28738	0.643692	+	0.765285i	$0.277402\pi$
$878$	10.7403	0.362467
$879$	0	0
$880$	0	0
$881$	12.3129	0.414831	0.207415	−	0.978253i	$-0.433495\pi$
$881$	12.3129	0.414831	0.207415	+	0.978253i	$0.433495\pi$
$882$	0	0
$883$	12.9509	0.435832	0.217916	−	0.975968i	$-0.430074\pi$
$883$	12.9509	0.435832	0.217916	+	0.975968i	$0.430074\pi$
$884$	58.1184	1.95473
$885$	0	0
$886$	60.0643	2.01790
$887$	0.325195	0.0109190	0.00545949	−	0.999985i	$-0.498262\pi$
$887$	0.325195	0.0109190	0.00545949	+	0.999985i	$0.498262\pi$
$888$	0	0
$889$	−16.4136	−0.550493
$890$	0	0
$891$	0	0
$892$	52.4314	1.75553
$893$	−33.1431	−1.10909
$894$	0	0
$895$	0	0
$896$	−0.418838	−0.0139924
$897$	0	0
$898$	51.7637	1.72738
$899$	36.4148	1.21450
$900$	0	0
$901$	23.3963	0.779442
$902$	−29.3303	−0.976592
$903$	0	0
$904$	−0.131969	−0.00438924
$905$	0	0
$906$	0	0
$907$	−22.1150	−0.734317	−0.367158	−	0.930158i	$-0.619669\pi$
$907$	−22.1150	−0.734317	−0.367158	+	0.930158i	$0.619669\pi$
$908$	−15.3744	−0.510218
$909$	0	0
$910$	0	0
$911$	50.1070	1.66012	0.830060	−	0.557674i	$-0.188306\pi$
$911$	50.1070	1.66012	0.830060	+	0.557674i	$0.188306\pi$
$912$	0	0
$913$	−7.62125	−0.252227
$914$	52.6312	1.74089
$915$	0	0
$916$	10.6715	0.352596
$917$	−18.7128	−0.617952
$918$	0	0
$919$	−40.9980	−1.35240	−0.676200	−	0.736718i	$-0.736375\pi$
$919$	−40.9980	−1.35240	−0.676200	+	0.736718i	$0.736375\pi$
$920$	0	0
$921$	0	0
$922$	50.1367	1.65116
$923$	−34.1020	−1.12248
$924$	0	0
$925$	0	0
$926$	25.2474	0.829682
$927$	0	0
$928$	63.0823	2.07078
$929$	9.21792	0.302430	0.151215	−	0.988501i	$-0.451681\pi$
$929$	9.21792	0.302430	0.151215	+	0.988501i	$0.451681\pi$
$930$	0	0
$931$	42.1853	1.38257
$932$	17.1774	0.562664
$933$	0	0
$934$	−4.73935	−0.155076
$935$	0	0
$936$	0	0
$937$	−26.8366	−0.876712	−0.438356	−	0.898801i	$-0.644439\pi$
$937$	−26.8366	−0.876712	−0.438356	+	0.898801i	$0.644439\pi$
$938$	15.1362	0.494216
$939$	0	0
$940$	0	0
$941$	−22.0491	−0.718781	−0.359390	−	0.933187i	$-0.617015\pi$
$941$	−22.0491	−0.718781	−0.359390	+	0.933187i	$0.617015\pi$
$942$	0	0
$943$	2.12139	0.0690820
$944$	33.0451	1.07553
$945$	0	0
$946$	−8.26568	−0.268740
$947$	−32.3480	−1.05117	−0.525584	−	0.850742i	$-0.676153\pi$
$947$	−32.3480	−1.05117	−0.525584	+	0.850742i	$0.676153\pi$
$948$	0	0
$949$	−41.9900	−1.36305
$950$	0	0
$951$	0	0
$952$	0.406817	0.0131850
$953$	−18.6610	−0.604489	−0.302245	−	0.953230i	$-0.597736\pi$
$953$	−18.6610	−0.604489	−0.302245	+	0.953230i	$0.597736\pi$
$954$	0	0
$955$	0	0
$956$	−33.5731	−1.08583
$957$	0	0
$958$	−20.2296	−0.653588
$959$	−1.28648	−0.0415427
$960$	0	0
$961$	−9.79959	−0.316116
$962$	−81.6931	−2.63389
$963$	0	0
$964$	43.1169	1.38870
$965$	0	0
$966$	0	0
$967$	−36.8947	−1.18645	−0.593226	−	0.805036i	$-0.702146\pi$
$967$	−36.8947	−1.18645	−0.593226	+	0.805036i	$0.702146\pi$
$968$	0.504914	0.0162286
$969$	0	0
$970$	0	0
$971$	7.64508	0.245342	0.122671	−	0.992447i	$-0.460854\pi$
$971$	7.64508	0.245342	0.122671	+	0.992447i	$0.460854\pi$
$972$	0	0
$973$	−17.0955	−0.548057
$974$	30.5346	0.978393
$975$	0	0
$976$	−31.2369	−0.999870
$977$	−11.1069	−0.355342	−0.177671	−	0.984090i	$-0.556856\pi$
$977$	−11.1069	−0.355342	−0.177671	+	0.984090i	$0.556856\pi$
$978$	0	0
$979$	−3.41514	−0.109148
$980$	0	0
$981$	0	0
$982$	−22.3453	−0.713067
$983$	2.19235	0.0699252	0.0349626	−	0.999389i	$-0.488869\pi$
$983$	2.19235	0.0699252	0.0349626	+	0.999389i	$0.488869\pi$
$984$	0	0
$985$	0	0
$986$	−122.551	−3.90283
$987$	0	0
$988$	−55.7934	−1.77502
$989$	0.597837	0.0190101
$990$	0	0
$991$	−15.9622	−0.507057	−0.253529	−	0.967328i	$-0.581591\pi$
$991$	−15.9622	−0.507057	−0.253529	+	0.967328i	$0.581591\pi$
$992$	36.7260	1.16605
$993$	0	0
$994$	20.8464	0.661207
$995$	0	0
$996$	0	0
$997$	14.7485	0.467090	0.233545	−	0.972346i	$-0.424967\pi$
$997$	14.7485	0.467090	0.233545	+	0.972346i	$0.424967\pi$
$998$	−82.6487	−2.61620
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3375.2.a.q.1.1	yes	8
3.2	odd	2		3375.2.a.g.1.8	yes	8
5.4	even	2		3375.2.a.f.1.8	✓	8
15.14	odd	2		3375.2.a.p.1.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3375.2.a.f.1.8	✓	8	5.4	even	2
3375.2.a.g.1.8	yes	8	3.2	odd	2
3375.2.a.p.1.1	yes	8	15.14	odd	2
3375.2.a.q.1.1	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3375 = 3^{3} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3375.a (trivial)

\(f(q)\)	\(=\)	\(q-1.99433 q^{2} +1.97736 q^{4} +1.15949 q^{7} +0.0451562 q^{8} +O(q^{10})\)	\(q-1.99433 q^{2} +1.97736 q^{4} +1.15949 q^{7} +0.0451562 q^{8} +4.70972 q^{11} +3.78280 q^{13} -2.31241 q^{14} -4.04477 q^{16} +7.76990 q^{17} -7.45906 q^{19} -9.39275 q^{22} +0.679356 q^{23} -7.54416 q^{26} +2.29272 q^{28} +7.90871 q^{29} +4.60439 q^{31} +7.97630 q^{32} -15.4957 q^{34} +10.8287 q^{37} +14.8758 q^{38} +3.12265 q^{41} +0.880006 q^{43} +9.31281 q^{44} -1.35486 q^{46} +4.44334 q^{47} -5.65559 q^{49} +7.47995 q^{52} +3.01114 q^{53} +0.0523581 q^{56} -15.7726 q^{58} -8.16983 q^{59} +7.72280 q^{61} -9.18268 q^{62} -7.81785 q^{64} -6.54567 q^{67} +15.3639 q^{68} -9.01503 q^{71} -11.1002 q^{73} -21.5959 q^{74} -14.7492 q^{76} +5.46087 q^{77} -9.78275 q^{79} -6.22760 q^{82} -1.61819 q^{83} -1.75502 q^{86} +0.212673 q^{88} -0.725124 q^{89} +4.38611 q^{91} +1.34333 q^{92} -8.86149 q^{94} +6.47157 q^{97} +11.2791 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 10 q^{4} + 2 q^{7} + 15 q^{8}+O(q^{10}) \) 8 * q + 4 * q^2 + 10 * q^4 + 2 * q^7 + 15 * q^8	\( 8 q + 4 q^{2} + 10 q^{4} + 2 q^{7} + 15 q^{8} + q^{11} - 7 q^{13} - 3 q^{14} + 14 q^{16} + 26 q^{17} + 2 q^{19} - 12 q^{22} + 13 q^{23} - 5 q^{26} + 26 q^{28} - q^{29} + 5 q^{31} + 39 q^{32} + q^{34} + 16 q^{37} + 19 q^{38} - 5 q^{41} - 11 q^{43} + 17 q^{44} + q^{46} + 32 q^{47} - 15 q^{52} + 19 q^{53} + 40 q^{56} - 28 q^{58} - 6 q^{59} + 30 q^{62} + 25 q^{64} + 15 q^{67} + 52 q^{68} + 4 q^{71} + 20 q^{73} + 5 q^{74} + 7 q^{76} + 43 q^{77} - 7 q^{79} + 14 q^{82} + 43 q^{83} - 66 q^{86} - 17 q^{88} + 16 q^{89} - 24 q^{91} + 36 q^{92} + 18 q^{94} - 34 q^{97} + 28 q^{98}+O(q^{100}) \) 8 * q + 4 * q^2 + 10 * q^4 + 2 * q^7 + 15 * q^8 + q^11 - 7 * q^13 - 3 * q^14 + 14 * q^16 + 26 * q^17 + 2 * q^19 - 12 * q^22 + 13 * q^23 - 5 * q^26 + 26 * q^28 - q^29 + 5 * q^31 + 39 * q^32 + q^34 + 16 * q^37 + 19 * q^38 - 5 * q^41 - 11 * q^43 + 17 * q^44 + q^46 + 32 * q^47 - 15 * q^52 + 19 * q^53 + 40 * q^56 - 28 * q^58 - 6 * q^59 + 30 * q^62 + 25 * q^64 + 15 * q^67 + 52 * q^68 + 4 * q^71 + 20 * q^73 + 5 * q^74 + 7 * q^76 + 43 * q^77 - 7 * q^79 + 14 * q^82 + 43 * q^83 - 66 * q^86 - 17 * q^88 + 16 * q^89 - 24 * q^91 + 36 * q^92 + 18 * q^94 - 34 * q^97 + 28 * q^98

Introduction

L-functions

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