LMFDB - Embedded newform 4275.2.a.bs.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4275,2,Mod(1,4275)] 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("4275.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4275 = 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4275.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,8,0,0,0,0,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$34.1360468641$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.16717036.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} + 26x^{2} - 19 $ x^6 - 10x^4 + 26x^2 - 19
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.13194$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.13194 q^{2} -0.718710 q^{4} +4.11009 q^{7} -3.07742 q^{8} +5.78432 q^{11} -6.78276 q^{13} +4.65238 q^{14} -2.04604 q^{16} -5.41381 q^{17} -1.00000 q^{19} +6.54751 q^{22} -8.04820 q^{23} -7.67769 q^{26} -2.95396 q^{28} -5.34130 q^{29} +0.327327 q^{31} +3.83884 q^{32} -6.12811 q^{34} -10.7828 q^{37} -1.13194 q^{38} +2.70690 q^{41} -0.654655 q^{43} -4.15725 q^{44} -9.11009 q^{46} +7.67769 q^{47} +9.89286 q^{49} +4.87484 q^{52} +0.813537 q^{53} -12.6485 q^{56} -6.04604 q^{58} -4.97079 q^{59} -7.87484 q^{61} +0.370515 q^{62} +8.43742 q^{64} -9.76475 q^{67} +3.89096 q^{68} +4.97079 q^{71} +12.7857 q^{73} -12.2055 q^{74} +0.718710 q^{76} +23.7741 q^{77} -1.01802 q^{79} +3.06406 q^{82} +1.25656 q^{83} -0.741030 q^{86} -17.8008 q^{88} -11.4961 q^{89} -27.8778 q^{91} +5.78432 q^{92} +8.69069 q^{94} -6.78276 q^{97} +11.1981 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{4} - 16 q^{13} - 6 q^{19} - 10 q^{22} - 30 q^{28} + 2 q^{31} - 12 q^{34} - 40 q^{37} - 4 q^{43} - 30 q^{46} + 10 q^{49} - 20 q^{52} - 24 q^{58} + 2 q^{61} + 26 q^{64} - 34 q^{67} - 22 q^{73}+ \cdots - 16 q^{97}+O(q^{100}) $ 6 * q + 8 * q^4 - 16 * q^13 - 6 * q^19 - 10 * q^22 - 30 * q^28 + 2 * q^31 - 12 * q^34 - 40 * q^37 - 4 * q^43 - 30 * q^46 + 10 * q^49 - 20 * q^52 - 24 * q^58 + 2 * q^61 + 26 * q^64 - 34 * q^67 - 22 * q^73 - 8 * q^76 - 6 * q^79 + 6 * q^82 - 82 * q^88 - 44 * q^91 + 52 * q^94 - 16 * q^97

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 $

$2$	1.13194	0.800403	0.400202	−	0.916427i	$-0.368940\pi$
$2$	1.13194	0.800403	0.400202	+	0.916427i	$0.368940\pi$
$3$	0	0
$3$	0	0
$4$	−0.718710	−0.359355
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.11009	1.55347	0.776734	−	0.629828i	$-0.216875\pi$
$7$	4.11009	1.55347	0.776734	+	0.629828i	$0.216875\pi$
$8$	−3.07742	−1.08803
$9$	0	0
$10$	0	0
$11$	5.78432	1.74404	0.872019	−	0.489471i	$-0.162810\pi$
$11$	5.78432	1.74404	0.872019	+	0.489471i	$0.162810\pi$
$12$	0	0
$13$	−6.78276	−1.88120	−0.940600	−	0.339516i	$-0.889737\pi$
$13$	−6.78276	−1.88120	−0.940600	+	0.339516i	$0.889737\pi$
$14$	4.65238	1.24340
$15$	0	0
$16$	−2.04604	−0.511509
$17$	−5.41381	−1.31304	−0.656521	−	0.754308i	$-0.727972\pi$
$17$	−5.41381	−1.31304	−0.656521	+	0.754308i	$0.727972\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	6.54751	1.39593
$23$	−8.04820	−1.67817	−0.839083	−	0.544003i	$-0.816908\pi$
$23$	−8.04820	−1.67817	−0.839083	+	0.544003i	$0.816908\pi$
$24$	0	0
$25$	0	0
$26$	−7.67769	−1.50572
$27$	0	0
$28$	−2.95396	−0.558247
$29$	−5.34130	−0.991855	−0.495927	−	0.868364i	$-0.665172\pi$
$29$	−5.34130	−0.991855	−0.495927	+	0.868364i	$0.665172\pi$
$30$	0	0
$31$	0.327327	0.0587897	0.0293949	−	0.999568i	$-0.490642\pi$
$31$	0.327327	0.0587897	0.0293949	+	0.999568i	$0.490642\pi$
$32$	3.83884	0.678618
$33$	0	0
$34$	−6.12811	−1.05096
$35$	0	0
$36$	0	0
$37$	−10.7828	−1.77268	−0.886338	−	0.463039i	$-0.846759\pi$
$37$	−10.7828	−1.77268	−0.886338	+	0.463039i	$0.846759\pi$
$38$	−1.13194	−0.183625
$39$	0	0
$40$	0	0
$41$	2.70690	0.422747	0.211374	−	0.977405i	$-0.432206\pi$
$41$	2.70690	0.422747	0.211374	+	0.977405i	$0.432206\pi$
$42$	0	0
$43$	−0.654655	−0.0998339	−0.0499169	−	0.998753i	$-0.515896\pi$
$43$	−0.654655	−0.0998339	−0.0499169	+	0.998753i	$0.515896\pi$
$44$	−4.15725	−0.626729
$45$	0	0
$46$	−9.11009	−1.34321
$47$	7.67769	1.11991	0.559953	−	0.828524i	$-0.310819\pi$
$47$	7.67769	1.11991	0.559953	+	0.828524i	$0.310819\pi$
$48$	0	0
$49$	9.89286	1.41327
$50$	0	0
$51$	0	0
$52$	4.87484	0.676019
$53$	0.813537	0.111748	0.0558739	−	0.998438i	$-0.482206\pi$
$53$	0.813537	0.111748	0.0558739	+	0.998438i	$0.482206\pi$
$54$	0	0
$55$	0	0
$56$	−12.6485	−1.69022
$57$	0	0
$58$	−6.04604	−0.793884
$59$	−4.97079	−0.647141	−0.323571	−	0.946204i	$-0.604883\pi$
$59$	−4.97079	−0.647141	−0.323571	+	0.946204i	$0.604883\pi$
$60$	0	0
$61$	−7.87484	−1.00827	−0.504135	−	0.863625i	$-0.668189\pi$
$61$	−7.87484	−1.00827	−0.504135	+	0.863625i	$0.668189\pi$
$62$	0.370515	0.0470555
$63$	0	0
$64$	8.43742	1.05468
$65$	0	0
$66$	0	0
$67$	−9.76475	−1.19295	−0.596477	−	0.802630i	$-0.703433\pi$
$67$	−9.76475	−1.19295	−0.596477	+	0.802630i	$0.703433\pi$
$68$	3.89096	0.471848
$69$	0	0
$70$	0	0
$71$	4.97079	0.589924	0.294962	−	0.955509i	$-0.404693\pi$
$71$	4.97079	0.589924	0.294962	+	0.955509i	$0.404693\pi$
$72$	0	0
$73$	12.7857	1.49645	0.748227	−	0.663442i	$-0.230905\pi$
$73$	12.7857	1.49645	0.748227	+	0.663442i	$0.230905\pi$
$74$	−12.2055	−1.41886
$75$	0	0
$76$	0.718710	0.0824417
$77$	23.7741	2.70931
$78$	0	0
$79$	−1.01802	−0.114536	−0.0572680	−	0.998359i	$-0.518239\pi$
$79$	−1.01802	−0.114536	−0.0572680	+	0.998359i	$0.518239\pi$
$80$	0	0
$81$	0	0
$82$	3.06406	0.338368
$83$	1.25656	0.137925	0.0689626	−	0.997619i	$-0.478031\pi$
$83$	1.25656	0.137925	0.0689626	+	0.997619i	$0.478031\pi$
$84$	0	0
$85$	0	0
$86$	−0.741030	−0.0799074
$87$	0	0
$88$	−17.8008	−1.89757
$89$	−11.4961	−1.21859	−0.609294	−	0.792944i	$-0.708547\pi$
$89$	−11.4961	−1.21859	−0.609294	+	0.792944i	$0.708547\pi$
$90$	0	0
$91$	−27.8778	−2.92239
$92$	5.78432	0.603057
$93$	0	0
$94$	8.69069	0.896376
$95$	0	0
$96$	0	0
$97$	−6.78276	−0.688685	−0.344343	−	0.938844i	$-0.611898\pi$
$97$	−6.78276	−0.688685	−0.344343	+	0.938844i	$0.611898\pi$
$98$	11.1981	1.13118
$99$	0	0
$100$	0	0
$101$	8.56373	0.852123	0.426062	−	0.904694i	$-0.359901\pi$
$101$	8.56373	0.852123	0.426062	+	0.904694i	$0.359901\pi$
$102$	0	0
$103$	−5.76475	−0.568017	−0.284009	−	0.958822i	$-0.591664\pi$
$103$	−5.76475	−0.568017	−0.284009	+	0.958822i	$0.591664\pi$
$104$	20.8734	2.04681
$105$	0	0
$106$	0.920876	0.0894434
$107$	−12.0117	−1.16121	−0.580606	−	0.814185i	$-0.697184\pi$
$107$	−12.0117	−1.16121	−0.580606	+	0.814185i	$0.697184\pi$
$108$	0	0
$109$	2.56258	0.245451	0.122725	−	0.992441i	$-0.460837\pi$
$109$	2.56258	0.245451	0.122725	+	0.992441i	$0.460837\pi$
$110$	0	0
$111$	0	0
$112$	−8.40940	−0.794614
$113$	14.3481	1.34975	0.674876	−	0.737931i	$-0.264197\pi$
$113$	14.3481	1.34975	0.674876	+	0.737931i	$0.264197\pi$
$114$	0	0
$115$	0	0
$116$	3.83884	0.356428
$117$	0	0
$118$	−5.62664	−0.517974
$119$	−22.2512	−2.03977
$120$	0	0
$121$	22.4584	2.04167
$122$	−8.91385	−0.807022
$123$	0	0
$124$	−0.235253	−0.0211264
$125$	0	0
$126$	0	0
$127$	−2.36336	−0.209715	−0.104857	−	0.994487i	$-0.533439\pi$
$127$	−2.36336	−0.209715	−0.104857	+	0.994487i	$0.533439\pi$
$128$	1.87297	0.165549
$129$	0	0
$130$	0	0
$131$	−14.3481	−1.25360	−0.626798	−	0.779182i	$-0.715635\pi$
$131$	−14.3481	−1.25360	−0.626798	+	0.779182i	$0.715635\pi$
$132$	0	0
$133$	−4.11009	−0.356390
$134$	−11.0531	−0.954844
$135$	0	0
$136$	16.6606	1.42863
$137$	−22.3963	−1.91344	−0.956721	−	0.291007i	$-0.906010\pi$
$137$	−22.3963	−1.91344	−0.956721	+	0.291007i	$0.906010\pi$
$138$	0	0
$139$	−6.11009	−0.518251	−0.259126	−	0.965844i	$-0.583434\pi$
$139$	−6.11009	−0.518251	−0.259126	+	0.965844i	$0.583434\pi$
$140$	0	0
$141$	0	0
$142$	5.62664	0.472177
$143$	−39.2337	−3.28089
$144$	0	0
$145$	0	0
$146$	14.4727	1.19777
$147$	0	0
$148$	7.74968	0.637020
$149$	19.2463	1.57672	0.788361	−	0.615213i	$-0.210930\pi$
$149$	19.2463	1.57672	0.788361	+	0.615213i	$0.210930\pi$
$150$	0	0
$151$	−17.7857	−1.44738	−0.723690	−	0.690125i	$-0.757556\pi$
$151$	−17.7857	−1.44738	−0.723690	+	0.690125i	$0.757556\pi$
$152$	3.07742	0.249612
$153$	0	0
$154$	26.9109	2.16854
$155$	0	0
$156$	0	0
$157$	−10.8568	−0.866469	−0.433234	−	0.901281i	$-0.642628\pi$
$157$	−10.8568	−0.866469	−0.433234	+	0.901281i	$0.642628\pi$
$158$	−1.15234	−0.0916750
$159$	0	0
$160$	0	0
$161$	−33.0789	−2.60698
$162$	0	0
$163$	−3.88991	−0.304681	−0.152341	−	0.988328i	$-0.548681\pi$
$163$	−3.88991	−0.304681	−0.152341	+	0.988328i	$0.548681\pi$
$164$	−1.94548	−0.151916
$165$	0	0
$166$	1.42235	0.110396
$167$	14.9124	1.15395	0.576976	−	0.816761i	$-0.304232\pi$
$167$	14.9124	1.15395	0.576976	+	0.816761i	$0.304232\pi$
$168$	0	0
$169$	33.0059	2.53892
$170$	0	0
$171$	0	0
$172$	0.470507	0.0358758
$173$	1.69958	0.129217	0.0646084	−	0.997911i	$-0.479420\pi$
$173$	1.69958	0.129217	0.0646084	+	0.997911i	$0.479420\pi$
$174$	0	0
$175$	0	0
$176$	−11.8349	−0.892092
$177$	0	0
$178$	−13.0129	−0.975362
$179$	1.18405	0.0885002	0.0442501	−	0.999020i	$-0.485910\pi$
$179$	1.18405	0.0885002	0.0442501	+	0.999020i	$0.485910\pi$
$180$	0	0
$181$	6.87484	0.511003	0.255501	−	0.966809i	$-0.417759\pi$
$181$	6.87484	0.511003	0.255501	+	0.966809i	$0.417759\pi$
$182$	−31.5560	−2.33909
$183$	0	0
$184$	24.7677	1.82590
$185$	0	0
$186$	0	0
$187$	−31.3152	−2.28999
$188$	−5.51803	−0.402444
$189$	0	0
$190$	0	0
$191$	−3.52044	−0.254730	−0.127365	−	0.991856i	$-0.540652\pi$
$191$	−3.52044	−0.254730	−0.127365	+	0.991856i	$0.540652\pi$
$192$	0	0
$193$	−17.3453	−1.24855	−0.624273	−	0.781207i	$-0.714605\pi$
$193$	−17.3453	−1.24855	−0.624273	+	0.781207i	$0.714605\pi$
$194$	−7.67769	−0.551226
$195$	0	0
$196$	−7.11009	−0.507864
$197$	−3.14993	−0.224423	−0.112211	−	0.993684i	$-0.535793\pi$
$197$	−3.14993	−0.224423	−0.112211	+	0.993684i	$0.535793\pi$
$198$	0	0
$199$	1.88991	0.133972	0.0669860	−	0.997754i	$-0.478662\pi$
$199$	1.88991	0.133972	0.0669860	+	0.997754i	$0.478662\pi$
$200$	0	0
$201$	0	0
$202$	9.69364	0.682042
$203$	−21.9532	−1.54082
$204$	0	0
$205$	0	0
$206$	−6.52535	−0.454643
$207$	0	0
$208$	13.8778	0.962251
$209$	−5.78432	−0.400110
$210$	0	0
$211$	3.01802	0.207769	0.103884	−	0.994589i	$-0.466873\pi$
$211$	3.01802	0.207769	0.103884	+	0.994589i	$0.466873\pi$
$212$	−0.584697	−0.0401571
$213$	0	0
$214$	−13.5965	−0.929437
$215$	0	0
$216$	0	0
$217$	1.34535	0.0913280
$218$	2.90069	0.196460
$219$	0	0
$220$	0	0
$221$	36.7206	2.47009
$222$	0	0
$223$	−15.8929	−1.06426	−0.532132	−	0.846661i	$-0.678609\pi$
$223$	−15.8929	−1.06426	−0.532132	+	0.846661i	$0.678609\pi$
$224$	15.7780	1.05421
$225$	0	0
$226$	16.2412	1.08035
$227$	22.6943	1.50627	0.753136	−	0.657865i	$-0.228540\pi$
$227$	22.6943	1.50627	0.753136	+	0.657865i	$0.228540\pi$
$228$	0	0
$229$	−10.3634	−0.684830	−0.342415	−	0.939549i	$-0.611245\pi$
$229$	−10.3634	−0.684830	−0.342415	+	0.939549i	$0.611245\pi$
$230$	0	0
$231$	0	0
$232$	16.4374	1.07917
$233$	17.6193	1.15428	0.577138	−	0.816647i	$-0.304169\pi$
$233$	17.6193	1.15428	0.577138	+	0.816647i	$0.304169\pi$
$234$	0	0
$235$	0	0
$236$	3.57255	0.232553
$237$	0	0
$238$	−25.1871	−1.63264
$239$	3.74594	0.242305	0.121152	−	0.992634i	$-0.461341\pi$
$239$	3.74594	0.242305	0.121152	+	0.992634i	$0.461341\pi$
$240$	0	0
$241$	−11.5295	−0.742680	−0.371340	−	0.928497i	$-0.621102\pi$
$241$	−11.5295	−0.742680	−0.371340	+	0.928497i	$0.621102\pi$
$242$	25.4216	1.63416
$243$	0	0
$244$	5.65972	0.362327
$245$	0	0
$246$	0	0
$247$	6.78276	0.431577
$248$	−1.00732	−0.0639651
$249$	0	0
$250$	0	0
$251$	−7.04088	−0.444416	−0.222208	−	0.974999i	$-0.571326\pi$
$251$	−7.04088	−0.444416	−0.222208	+	0.974999i	$0.571326\pi$
$252$	0	0
$253$	−46.5534	−2.92679
$254$	−2.67519	−0.167856
$255$	0	0
$256$	−14.7547	−0.922172
$257$	21.4377	1.33725	0.668624	−	0.743601i	$-0.266884\pi$
$257$	21.4377	1.33725	0.668624	+	0.743601i	$0.266884\pi$
$258$	0	0
$259$	−44.3182	−2.75380
$260$	0	0
$261$	0	0
$262$	−16.2412	−1.00338
$263$	−5.78432	−0.356677	−0.178338	−	0.983969i	$-0.557072\pi$
$263$	−5.78432	−0.356677	−0.178338	+	0.983969i	$0.557072\pi$
$264$	0	0
$265$	0	0
$266$	−4.65238	−0.285256
$267$	0	0
$268$	7.01802	0.428694
$269$	−13.1957	−0.804557	−0.402279	−	0.915517i	$-0.631782\pi$
$269$	−13.1957	−0.804557	−0.402279	+	0.915517i	$0.631782\pi$
$270$	0	0
$271$	−21.6756	−1.31670	−0.658350	−	0.752712i	$-0.728745\pi$
$271$	−21.6756	−1.31670	−0.658350	+	0.752712i	$0.728745\pi$
$272$	11.0769	0.671633
$273$	0	0
$274$	−25.3512	−1.53152
$275$	0	0
$276$	0	0
$277$	−14.5295	−0.872993	−0.436496	−	0.899706i	$-0.643781\pi$
$277$	−14.5295	−0.872993	−0.436496	+	0.899706i	$0.643781\pi$
$278$	−6.91626	−0.414810
$279$	0	0
$280$	0	0
$281$	−2.03838	−0.121600	−0.0607998	−	0.998150i	$-0.519365\pi$
$281$	−2.03838	−0.121600	−0.0607998	+	0.998150i	$0.519365\pi$
$282$	0	0
$283$	−3.52949	−0.209807	−0.104903	−	0.994482i	$-0.533453\pi$
$283$	−3.52949	−0.209807	−0.104903	+	0.994482i	$0.533453\pi$
$284$	−3.57255	−0.211992
$285$	0	0
$286$	−44.4102	−2.62603
$287$	11.1256	0.656725
$288$	0	0
$289$	12.3093	0.724077
$290$	0	0
$291$	0	0
$292$	−9.18922	−0.537758
$293$	−7.30717	−0.426890	−0.213445	−	0.976955i	$-0.568468\pi$
$293$	−7.30717	−0.426890	−0.213445	+	0.976955i	$0.568468\pi$
$294$	0	0
$295$	0	0
$296$	33.1831	1.92873
$297$	0	0
$298$	21.7857	1.26201
$299$	54.5891	3.15697
$300$	0	0
$301$	−2.69069	−0.155089
$302$	−20.1324	−1.15849
$303$	0	0
$304$	2.04604	0.117348
$305$	0	0
$306$	0	0
$307$	20.8598	1.19053	0.595265	−	0.803529i	$-0.297047\pi$
$307$	20.8598	1.19053	0.595265	+	0.803529i	$0.297047\pi$
$308$	−17.0867	−0.973604
$309$	0	0
$310$	0	0
$311$	19.7381	1.11925	0.559623	−	0.828747i	$-0.310946\pi$
$311$	19.7381	1.11925	0.559623	+	0.828747i	$0.310946\pi$
$312$	0	0
$313$	−4.32733	−0.244595	−0.122298	−	0.992493i	$-0.539026\pi$
$313$	−4.32733	−0.244595	−0.122298	+	0.992493i	$0.539026\pi$
$314$	−12.2893	−0.693524
$315$	0	0
$316$	0.731659	0.0411591
$317$	−25.9655	−1.45837	−0.729183	−	0.684319i	$-0.760100\pi$
$317$	−25.9655	−1.45837	−0.729183	+	0.684319i	$0.760100\pi$
$318$	0	0
$319$	−30.8958	−1.72983
$320$	0	0
$321$	0	0
$322$	−37.4433	−2.08663
$323$	5.41381	0.301232
$324$	0	0
$325$	0	0
$326$	−4.40315	−0.243868
$327$	0	0
$328$	−8.33028	−0.459963
$329$	31.5560	1.73974
$330$	0	0
$331$	−6.58355	−0.361865	−0.180932	−	0.983496i	$-0.557911\pi$
$331$	−6.58355	−0.361865	−0.180932	+	0.983496i	$0.557911\pi$
$332$	−0.903101	−0.0495641
$333$	0	0
$334$	16.8799	0.923627
$335$	0	0
$336$	0	0
$337$	12.4404	0.677670	0.338835	−	0.940846i	$-0.389967\pi$
$337$	12.4404	0.677670	0.338835	+	0.940846i	$0.389967\pi$
$338$	37.3607	2.03216
$339$	0	0
$340$	0	0
$341$	1.89337	0.102532
$342$	0	0
$343$	11.8899	0.641995
$344$	2.01465	0.108622
$345$	0	0
$346$	1.92382	0.103425
$347$	−12.2055	−0.655223	−0.327612	−	0.944813i	$-0.606244\pi$
$347$	−12.2055	−0.655223	−0.327612	+	0.944813i	$0.606244\pi$
$348$	0	0
$349$	11.6606	0.624175	0.312088	−	0.950053i	$-0.398972\pi$
$349$	11.6606	0.624175	0.312088	+	0.950053i	$0.398972\pi$
$350$	0	0
$351$	0	0
$352$	22.2051	1.18354
$353$	33.8199	1.80005	0.900026	−	0.435837i	$-0.143548\pi$
$353$	33.8199	1.80005	0.900026	+	0.435837i	$0.143548\pi$
$354$	0	0
$355$	0	0
$356$	8.26239	0.437906
$357$	0	0
$358$	1.34028	0.0708358
$359$	12.2055	0.644179	0.322090	−	0.946709i	$-0.395615\pi$
$359$	12.2055	0.644179	0.322090	+	0.946709i	$0.395615\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	7.78191	0.409008
$363$	0	0
$364$	20.0360	1.05017
$365$	0	0
$366$	0	0
$367$	14.4764	0.755662	0.377831	−	0.925875i	$-0.376670\pi$
$367$	14.4764	0.755662	0.377831	+	0.925875i	$0.376670\pi$
$368$	16.4669	0.858398
$369$	0	0
$370$	0	0
$371$	3.34371	0.173597
$372$	0	0
$373$	−13.7798	−0.713492	−0.356746	−	0.934201i	$-0.616114\pi$
$373$	−13.7798	−0.713492	−0.356746	+	0.934201i	$0.616114\pi$
$374$	−35.4470	−1.83292
$375$	0	0
$376$	−23.6275	−1.21849
$377$	36.2288	1.86588
$378$	0	0
$379$	−23.8778	−1.22652	−0.613260	−	0.789881i	$-0.710142\pi$
$379$	−23.8778	−1.22652	−0.613260	+	0.789881i	$0.710142\pi$
$380$	0	0
$381$	0	0
$382$	−3.98493	−0.203887
$383$	18.1665	0.928265	0.464133	−	0.885766i	$-0.346366\pi$
$383$	18.1665	0.928265	0.464133	+	0.885766i	$0.346366\pi$
$384$	0	0
$385$	0	0
$386$	−19.6339	−0.999340
$387$	0	0
$388$	4.87484	0.247482
$389$	10.0458	0.509342	0.254671	−	0.967028i	$-0.418033\pi$
$389$	10.0458	0.509342	0.254671	+	0.967028i	$0.418033\pi$
$390$	0	0
$391$	43.5714	2.20350
$392$	−30.4445	−1.53768
$393$	0	0
$394$	−3.56553	−0.179629
$395$	0	0
$396$	0	0
$397$	8.76770	0.440038	0.220019	−	0.975496i	$-0.429388\pi$
$397$	8.76770	0.440038	0.220019	+	0.975496i	$0.429388\pi$
$398$	2.13926	0.107232
$399$	0	0
$400$	0	0
$401$	37.7279	1.88404	0.942021	−	0.335554i	$-0.108924\pi$
$401$	37.7279	1.88404	0.942021	+	0.335554i	$0.108924\pi$
$402$	0	0
$403$	−2.22018	−0.110595
$404$	−6.15484	−0.306215
$405$	0	0
$406$	−24.8498	−1.23327
$407$	−62.3710	−3.09162
$408$	0	0
$409$	15.9138	0.786888	0.393444	−	0.919349i	$-0.371284\pi$
$409$	15.9138	0.786888	0.393444	+	0.919349i	$0.371284\pi$
$410$	0	0
$411$	0	0
$412$	4.14318	0.204120
$413$	−20.4304	−1.00531
$414$	0	0
$415$	0	0
$416$	−26.0380	−1.27662
$417$	0	0
$418$	−6.54751	−0.320249
$419$	5.01956	0.245221	0.122611	−	0.992455i	$-0.460873\pi$
$419$	5.01956	0.245221	0.122611	+	0.992455i	$0.460873\pi$
$420$	0	0
$421$	−9.90793	−0.482883	−0.241441	−	0.970415i	$-0.577620\pi$
$421$	−9.90793	−0.482883	−0.241441	+	0.970415i	$0.577620\pi$
$422$	3.41622	0.166299
$423$	0	0
$424$	−2.50359	−0.121585
$425$	0	0
$426$	0	0
$427$	−32.3663	−1.56632
$428$	8.63290	0.417287
$429$	0	0
$430$	0	0
$431$	5.11580	0.246419	0.123210	−	0.992381i	$-0.460681\pi$
$431$	5.11580	0.246419	0.123210	+	0.992381i	$0.460681\pi$
$432$	0	0
$433$	−13.4374	−0.645761	−0.322881	−	0.946440i	$-0.604651\pi$
$433$	−13.4374	−0.645761	−0.322881	+	0.946440i	$0.604651\pi$
$434$	1.52285	0.0730992
$435$	0	0
$436$	−1.84175	−0.0882039
$437$	8.04820	0.384998
$438$	0	0
$439$	8.76770	0.418459	0.209230	−	0.977867i	$-0.432904\pi$
$439$	8.76770	0.418459	0.209230	+	0.977867i	$0.432904\pi$
$440$	0	0
$441$	0	0
$442$	41.5655	1.97707
$443$	39.3550	1.86981	0.934906	−	0.354896i	$-0.115484\pi$
$443$	39.3550	1.86981	0.934906	+	0.354896i	$0.115484\pi$
$444$	0	0
$445$	0	0
$446$	−17.9898	−0.851841
$447$	0	0
$448$	34.6786	1.63841
$449$	−25.7229	−1.21394	−0.606970	−	0.794725i	$-0.707615\pi$
$449$	−25.7229	−1.21394	−0.606970	+	0.794725i	$0.707615\pi$
$450$	0	0
$451$	15.6576	0.737288
$452$	−10.3121	−0.485040
$453$	0	0
$454$	25.6886	1.20562
$455$	0	0
$456$	0	0
$457$	−31.3693	−1.46739	−0.733696	−	0.679478i	$-0.762206\pi$
$457$	−31.3693	−1.46739	−0.733696	+	0.679478i	$0.762206\pi$
$458$	−11.7307	−0.548140
$459$	0	0
$460$	0	0
$461$	−12.8015	−0.596224	−0.298112	−	0.954531i	$-0.596357\pi$
$461$	−12.8015	−0.596224	−0.298112	+	0.954531i	$0.596357\pi$
$462$	0	0
$463$	−21.1671	−0.983718	−0.491859	−	0.870675i	$-0.663683\pi$
$463$	−21.1671	−0.983718	−0.491859	+	0.870675i	$0.663683\pi$
$464$	10.9285	0.507343
$465$	0	0
$466$	19.9440	0.923886
$467$	−1.11155	−0.0514362	−0.0257181	−	0.999669i	$-0.508187\pi$
$467$	−1.11155	−0.0514362	−0.0257181	+	0.999669i	$0.508187\pi$
$468$	0	0
$469$	−40.1340	−1.85322
$470$	0	0
$471$	0	0
$472$	15.2972	0.704110
$473$	−3.78673	−0.174114
$474$	0	0
$475$	0	0
$476$	15.9922	0.733001
$477$	0	0
$478$	4.24019	0.193942
$479$	3.41622	0.156091	0.0780455	−	0.996950i	$-0.475132\pi$
$479$	3.41622	0.156091	0.0780455	+	0.996950i	$0.475132\pi$
$480$	0	0
$481$	73.1370	3.33476
$482$	−13.0507	−0.594443
$483$	0	0
$484$	−16.1411	−0.733685
$485$	0	0
$486$	0	0
$487$	−16.3424	−0.740545	−0.370272	−	0.928923i	$-0.620736\pi$
$487$	−16.3424	−0.740545	−0.370272	+	0.928923i	$0.620736\pi$
$488$	24.2342	1.09703
$489$	0	0
$490$	0	0
$491$	−3.39916	−0.153402	−0.0767010	−	0.997054i	$-0.524439\pi$
$491$	−3.39916	−0.153402	−0.0767010	+	0.997054i	$0.524439\pi$
$492$	0	0
$493$	28.9168	1.30235
$494$	7.67769	0.345436
$495$	0	0
$496$	−0.669724	−0.0300715
$497$	20.4304	0.916428
$498$	0	0
$499$	35.2110	1.57626	0.788131	−	0.615508i	$-0.211049\pi$
$499$	35.2110	1.57626	0.788131	+	0.615508i	$0.211049\pi$
$500$	0	0
$501$	0	0
$502$	−7.96986	−0.355712
$503$	18.2153	0.812179	0.406090	−	0.913833i	$-0.366892\pi$
$503$	18.2153	0.812179	0.406090	+	0.913833i	$0.366892\pi$
$504$	0	0
$505$	0	0
$506$	−52.6957	−2.34261
$507$	0	0
$508$	1.69857	0.0753620
$509$	4.31024	0.191048	0.0955241	−	0.995427i	$-0.469547\pi$
$509$	4.31024	0.191048	0.0955241	+	0.995427i	$0.469547\pi$
$510$	0	0
$511$	52.5505	2.32470
$512$	−20.4474	−0.903658
$513$	0	0
$514$	24.2662	1.07034
$515$	0	0
$516$	0	0
$517$	44.4102	1.95316
$518$	−50.1655	−2.20415
$519$	0	0
$520$	0	0
$521$	44.5750	1.95287	0.976433	−	0.215820i	$-0.0692426\pi$
$521$	44.5750	1.95287	0.976433	+	0.215820i	$0.0692426\pi$
$522$	0	0
$523$	9.90793	0.433243	0.216622	−	0.976256i	$-0.430496\pi$
$523$	9.90793	0.433243	0.216622	+	0.976256i	$0.430496\pi$
$524$	10.3121	0.450486
$525$	0	0
$526$	−6.54751	−0.285485
$527$	−1.77209	−0.0771933
$528$	0	0
$529$	41.7736	1.81624
$530$	0	0
$531$	0	0
$532$	2.95396	0.128071
$533$	−18.3603	−0.795272
$534$	0	0
$535$	0	0
$536$	30.0502	1.29797
$537$	0	0
$538$	−14.9368	−0.643970
$539$	57.2235	2.46479
$540$	0	0
$541$	−36.2972	−1.56054	−0.780269	−	0.625444i	$-0.784918\pi$
$541$	−36.2972	−1.56054	−0.780269	+	0.625444i	$0.784918\pi$
$542$	−24.5355	−1.05389
$543$	0	0
$544$	−20.7828	−0.891054
$545$	0	0
$546$	0	0
$547$	21.8067	0.932386	0.466193	−	0.884683i	$-0.345625\pi$
$547$	21.8067	0.932386	0.466193	+	0.884683i	$0.345625\pi$
$548$	16.0964	0.687605
$549$	0	0
$550$	0	0
$551$	5.34130	0.227547
$552$	0	0
$553$	−4.18415	−0.177928
$554$	−16.4465	−0.698746
$555$	0	0
$556$	4.39138	0.186236
$557$	−23.7333	−1.00561	−0.502806	−	0.864399i	$-0.667699\pi$
$557$	−23.7333	−1.00561	−0.502806	+	0.864399i	$0.667699\pi$
$558$	0	0
$559$	4.44037	0.187808
$560$	0	0
$561$	0	0
$562$	−2.30733	−0.0973287
$563$	−33.2319	−1.40056	−0.700278	−	0.713870i	$-0.746941\pi$
$563$	−33.2319	−1.40056	−0.700278	+	0.713870i	$0.746941\pi$
$564$	0	0
$565$	0	0
$566$	−3.99518	−0.167930
$567$	0	0
$568$	−15.2972	−0.641856
$569$	−19.5443	−0.819341	−0.409671	−	0.912233i	$-0.634356\pi$
$569$	−19.5443	−0.819341	−0.409671	+	0.912233i	$0.634356\pi$
$570$	0	0
$571$	−38.5564	−1.61353	−0.806767	−	0.590870i	$-0.798785\pi$
$571$	−38.5564	−1.61353	−0.806767	+	0.590870i	$0.798785\pi$
$572$	28.1976	1.17900
$573$	0	0
$574$	12.5935	0.525645
$575$	0	0
$576$	0	0
$577$	−35.6425	−1.48382	−0.741909	−	0.670501i	$-0.766079\pi$
$577$	−35.6425	−1.48382	−0.741909	+	0.670501i	$0.766079\pi$
$578$	13.9334	0.579554
$579$	0	0
$580$	0	0
$581$	5.16457	0.214263
$582$	0	0
$583$	4.70576	0.194893
$584$	−39.3470	−1.62819
$585$	0	0
$586$	−8.27129	−0.341684
$587$	17.7472	0.732506	0.366253	−	0.930515i	$-0.380640\pi$
$587$	17.7472	0.732506	0.366253	+	0.930515i	$0.380640\pi$
$588$	0	0
$589$	−0.327327	−0.0134873
$590$	0	0
$591$	0	0
$592$	22.0619	0.906740
$593$	3.89096	0.159782	0.0798912	−	0.996804i	$-0.474543\pi$
$593$	3.89096	0.159782	0.0798912	+	0.996804i	$0.474543\pi$
$594$	0	0
$595$	0	0
$596$	−13.8325	−0.566602
$597$	0	0
$598$	61.7916	2.52685
$599$	−7.04088	−0.287683	−0.143841	−	0.989601i	$-0.545945\pi$
$599$	−7.04088	−0.287683	−0.143841	+	0.989601i	$0.545945\pi$
$600$	0	0
$601$	41.4433	1.69051	0.845254	−	0.534365i	$-0.179449\pi$
$601$	41.4433	1.69051	0.845254	+	0.534365i	$0.179449\pi$
$602$	−3.04570	−0.124134
$603$	0	0
$604$	12.7828	0.520123
$605$	0	0
$606$	0	0
$607$	−20.4194	−0.828798	−0.414399	−	0.910095i	$-0.636008\pi$
$607$	−20.4194	−0.828798	−0.414399	+	0.910095i	$0.636008\pi$
$608$	−3.83884	−0.155686
$609$	0	0
$610$	0	0
$611$	−52.0760	−2.10677
$612$	0	0
$613$	7.07700	0.285838	0.142919	−	0.989734i	$-0.454351\pi$
$613$	7.07700	0.285838	0.142919	+	0.989734i	$0.454351\pi$
$614$	23.6120	0.952904
$615$	0	0
$616$	−73.1629	−2.94782
$617$	−8.27371	−0.333087	−0.166543	−	0.986034i	$-0.553261\pi$
$617$	−8.27371	−0.333087	−0.166543	+	0.986034i	$0.553261\pi$
$618$	0	0
$619$	25.0210	1.00568	0.502839	−	0.864380i	$-0.332289\pi$
$619$	25.0210	1.00568	0.502839	+	0.864380i	$0.332289\pi$
$620$	0	0
$621$	0	0
$622$	22.3424	0.895848
$623$	−47.2502	−1.89304
$624$	0	0
$625$	0	0
$626$	−4.89828	−0.195775
$627$	0	0
$628$	7.80290	0.311370
$629$	58.3758	2.32760
$630$	0	0
$631$	24.6966	0.983156	0.491578	−	0.870834i	$-0.336420\pi$
$631$	24.6966	0.983156	0.491578	+	0.870834i	$0.336420\pi$
$632$	3.13287	0.124619
$633$	0	0
$634$	−29.3914	−1.16728
$635$	0	0
$636$	0	0
$637$	−67.1009	−2.65864
$638$	−34.9722	−1.38456
$639$	0	0
$640$	0	0
$641$	−7.18589	−0.283826	−0.141913	−	0.989879i	$-0.545325\pi$
$641$	−7.18589	−0.283826	−0.141913	+	0.989879i	$0.545325\pi$
$642$	0	0
$643$	19.8598	0.783193	0.391596	−	0.920137i	$-0.371923\pi$
$643$	19.8598	0.783193	0.391596	+	0.920137i	$0.371923\pi$
$644$	23.7741	0.936831
$645$	0	0
$646$	6.12811	0.241107
$647$	−19.7619	−0.776919	−0.388459	−	0.921466i	$-0.626993\pi$
$647$	−19.7619	−0.776919	−0.388459	+	0.921466i	$0.626993\pi$
$648$	0	0
$649$	−28.7526	−1.12864
$650$	0	0
$651$	0	0
$652$	2.79571	0.109489
$653$	27.0282	1.05770	0.528848	−	0.848716i	$-0.322624\pi$
$653$	27.0282	1.05770	0.528848	+	0.848716i	$0.322624\pi$
$654$	0	0
$655$	0	0
$656$	−5.53843	−0.216239
$657$	0	0
$658$	35.7195	1.39249
$659$	−4.52776	−0.176377	−0.0881883	−	0.996104i	$-0.528108\pi$
$659$	−4.52776	−0.176377	−0.0881883	+	0.996104i	$0.528108\pi$
$660$	0	0
$661$	24.1841	0.940654	0.470327	−	0.882492i	$-0.344136\pi$
$661$	24.1841	0.940654	0.470327	+	0.882492i	$0.344136\pi$
$662$	−7.45219	−0.289637
$663$	0	0
$664$	−3.86696	−0.150067
$665$	0	0
$666$	0	0
$667$	42.9879	1.66450
$668$	−10.7177	−0.414679
$669$	0	0
$670$	0	0
$671$	−45.5506	−1.75846
$672$	0	0
$673$	27.8418	1.07322	0.536610	−	0.843830i	$-0.319705\pi$
$673$	27.8418	1.07322	0.536610	+	0.843830i	$0.319705\pi$
$674$	14.0818	0.542409
$675$	0	0
$676$	−23.7217	−0.912371
$677$	41.3209	1.58809	0.794045	−	0.607859i	$-0.207972\pi$
$677$	41.3209	1.58809	0.794045	+	0.607859i	$0.207972\pi$
$678$	0	0
$679$	−27.8778	−1.06985
$680$	0	0
$681$	0	0
$682$	2.14318	0.0820666
$683$	7.33889	0.280815	0.140407	−	0.990094i	$-0.455159\pi$
$683$	7.33889	0.280815	0.140407	+	0.990094i	$0.455159\pi$
$684$	0	0
$685$	0	0
$686$	13.4587	0.513855
$687$	0	0
$688$	1.33945	0.0510660
$689$	−5.51803	−0.210220
$690$	0	0
$691$	31.3453	1.19243	0.596217	−	0.802824i	$-0.296670\pi$
$691$	31.3453	1.19243	0.596217	+	0.802824i	$0.296670\pi$
$692$	−1.22150	−0.0464346
$693$	0	0
$694$	−13.8159	−0.524443
$695$	0	0
$696$	0	0
$697$	−14.6547	−0.555085
$698$	13.1991	0.499592
$699$	0	0
$700$	0	0
$701$	33.2239	1.25485	0.627424	−	0.778678i	$-0.284109\pi$
$701$	33.2239	1.25485	0.627424	+	0.778678i	$0.284109\pi$
$702$	0	0
$703$	10.7828	0.406680
$704$	48.8048	1.83940
$705$	0	0
$706$	38.2821	1.44077
$707$	35.1977	1.32375
$708$	0	0
$709$	9.00590	0.338224	0.169112	−	0.985597i	$-0.445910\pi$
$709$	9.00590	0.338224	0.169112	+	0.985597i	$0.445910\pi$
$710$	0	0
$711$	0	0
$712$	35.3784	1.32586
$713$	−2.63440	−0.0986589
$714$	0	0
$715$	0	0
$716$	−0.850990	−0.0318030
$717$	0	0
$718$	13.8159	0.515603
$719$	−44.9138	−1.67500	−0.837501	−	0.546436i	$-0.815984\pi$
$719$	−44.9138	−1.67500	−0.837501	+	0.546436i	$0.815984\pi$
$720$	0	0
$721$	−23.6936	−0.882397
$722$	1.13194	0.0421265
$723$	0	0
$724$	−4.94101	−0.183631
$725$	0	0
$726$	0	0
$727$	−15.2772	−0.566600	−0.283300	−	0.959031i	$-0.591429\pi$
$727$	−15.2772	−0.566600	−0.283300	+	0.959031i	$0.591429\pi$
$728$	85.7916	3.17965
$729$	0	0
$730$	0	0
$731$	3.54417	0.131086
$732$	0	0
$733$	−10.3814	−0.383445	−0.191723	−	0.981449i	$-0.561407\pi$
$733$	−10.3814	−0.383445	−0.191723	+	0.981449i	$0.561407\pi$
$734$	16.3864	0.604834
$735$	0	0
$736$	−30.8958	−1.13883
$737$	−56.4824	−2.08056
$738$	0	0
$739$	−46.9548	−1.72726	−0.863630	−	0.504126i	$-0.831815\pi$
$739$	−46.9548	−1.72726	−0.863630	+	0.504126i	$0.831815\pi$
$740$	0	0
$741$	0	0
$742$	3.78488	0.138947
$743$	5.26081	0.193000	0.0965002	−	0.995333i	$-0.469235\pi$
$743$	5.26081	0.193000	0.0965002	+	0.995333i	$0.469235\pi$
$744$	0	0
$745$	0	0
$746$	−15.5979	−0.571081
$747$	0	0
$748$	22.5065	0.822921
$749$	−49.3691	−1.80391
$750$	0	0
$751$	1.21134	0.0442023	0.0221012	−	0.999756i	$-0.492964\pi$
$751$	1.21134	0.0442023	0.0221012	+	0.999756i	$0.492964\pi$
$752$	−15.7088	−0.572842
$753$	0	0
$754$	41.0088	1.49345
$755$	0	0
$756$	0	0
$757$	−15.0360	−0.546494	−0.273247	−	0.961944i	$-0.588098\pi$
$757$	−15.0360	−0.546494	−0.273247	+	0.961944i	$0.588098\pi$
$758$	−27.0282	−0.981710
$759$	0	0
$760$	0	0
$761$	10.2884	0.372953	0.186476	−	0.982459i	$-0.440293\pi$
$761$	10.2884	0.372953	0.186476	+	0.982459i	$0.440293\pi$
$762$	0	0
$763$	10.5324	0.381300
$764$	2.53017	0.0915385
$765$	0	0
$766$	20.5634	0.742986
$767$	33.7157	1.21740
$768$	0	0
$769$	8.77982	0.316608	0.158304	−	0.987390i	$-0.449397\pi$
$769$	8.77982	0.316608	0.158304	+	0.987390i	$0.449397\pi$
$770$	0	0
$771$	0	0
$772$	12.4663	0.448671
$773$	−10.4888	−0.377256	−0.188628	−	0.982049i	$-0.560404\pi$
$773$	−10.4888	−0.377256	−0.188628	+	0.982049i	$0.560404\pi$
$774$	0	0
$775$	0	0
$776$	20.8734	0.749312
$777$	0	0
$778$	11.3712	0.407679
$779$	−2.70690	−0.0969849
$780$	0	0
$781$	28.7526	1.02885
$782$	49.3203	1.76369
$783$	0	0
$784$	−20.2412	−0.722898
$785$	0	0
$786$	0	0
$787$	−28.4613	−1.01454	−0.507269	−	0.861788i	$-0.669345\pi$
$787$	−28.4613	−1.01454	−0.507269	+	0.861788i	$0.669345\pi$
$788$	2.26388	0.0806475
$789$	0	0
$790$	0	0
$791$	58.9718	2.09680
$792$	0	0
$793$	53.4132	1.89676
$794$	9.92451	0.352208
$795$	0	0
$796$	−1.35830	−0.0481435
$797$	7.83068	0.277377	0.138689	−	0.990336i	$-0.455711\pi$
$797$	7.83068	0.277377	0.138689	+	0.990336i	$0.455711\pi$
$798$	0	0
$799$	−41.5655	−1.47048
$800$	0	0
$801$	0	0
$802$	42.7058	1.50799
$803$	73.9567	2.60987
$804$	0	0
$805$	0	0
$806$	−2.51312	−0.0885208
$807$	0	0
$808$	−26.3542	−0.927137
$809$	5.37302	0.188905	0.0944526	−	0.995529i	$-0.469890\pi$
$809$	5.37302	0.188905	0.0944526	+	0.995529i	$0.469890\pi$
$810$	0	0
$811$	−36.2470	−1.27281	−0.636403	−	0.771357i	$-0.719578\pi$
$811$	−36.2470	−1.27281	−0.636403	+	0.771357i	$0.719578\pi$
$812$	15.7780	0.553699
$813$	0	0
$814$	−70.6003	−2.47454
$815$	0	0
$816$	0	0
$817$	0.654655	0.0229035
$818$	18.0135	0.629828
$819$	0	0
$820$	0	0
$821$	−11.9562	−0.417275	−0.208637	−	0.977993i	$-0.566903\pi$
$821$	−11.9562	−0.417275	−0.208637	+	0.977993i	$0.566903\pi$
$822$	0	0
$823$	−7.42530	−0.258830	−0.129415	−	0.991591i	$-0.541310\pi$
$823$	−7.42530	−0.258830	−0.129415	+	0.991591i	$0.541310\pi$
$824$	17.7405	0.618021
$825$	0	0
$826$	−23.1260	−0.804656
$827$	22.1062	0.768709	0.384355	−	0.923186i	$-0.374424\pi$
$827$	22.1062	0.768709	0.384355	+	0.923186i	$0.374424\pi$
$828$	0	0
$829$	47.6635	1.65542	0.827711	−	0.561155i	$-0.189643\pi$
$829$	47.6635	1.65542	0.827711	+	0.561155i	$0.189643\pi$
$830$	0	0
$831$	0	0
$832$	−57.2290	−1.98406
$833$	−53.5580	−1.85568
$834$	0	0
$835$	0	0
$836$	4.15725	0.143781
$837$	0	0
$838$	5.68184	0.196276
$839$	−29.9777	−1.03495	−0.517473	−	0.855700i	$-0.673127\pi$
$839$	−29.9777	−1.03495	−0.517473	+	0.855700i	$0.673127\pi$
$840$	0	0
$841$	−0.470507	−0.0162244
$842$	−11.2152	−0.386501
$843$	0	0
$844$	−2.16908	−0.0746628
$845$	0	0
$846$	0	0
$847$	92.3060	3.17167
$848$	−1.66453	−0.0571601
$849$	0	0
$850$	0	0
$851$	86.7819	2.97485
$852$	0	0
$853$	2.41645	0.0827378	0.0413689	−	0.999144i	$-0.486828\pi$
$853$	2.41645	0.0827378	0.0413689	+	0.999144i	$0.486828\pi$
$854$	−36.6368	−1.25368
$855$	0	0
$856$	36.9649	1.26344
$857$	0.0487732	0.00166606	0.000833030	−	1.00000i	$-0.499735\pi$
$857$	0.0487732	0.00166606	0.000833030		1.00000i	$0.499735\pi$
$858$	0	0
$859$	−16.0741	−0.548440	−0.274220	−	0.961667i	$-0.588420\pi$
$859$	−16.0741	−0.548440	−0.274220	+	0.961667i	$0.588420\pi$
$860$	0	0
$861$	0	0
$862$	5.79078	0.197235
$863$	32.7809	1.11587	0.557937	−	0.829884i	$-0.311593\pi$
$863$	32.7809	1.11587	0.557937	+	0.829884i	$0.311593\pi$
$864$	0	0
$865$	0	0
$866$	−15.2104	−0.516869
$867$	0	0
$868$	−0.966913	−0.0328192
$869$	−5.88854	−0.199755
$870$	0	0
$871$	66.2320	2.24419
$872$	−7.88613	−0.267058
$873$	0	0
$874$	9.11009	0.308153
$875$	0	0
$876$	0	0
$877$	−30.2762	−1.02236	−0.511178	−	0.859475i	$-0.670791\pi$
$877$	−30.2762	−1.02236	−0.511178	+	0.859475i	$0.670791\pi$
$878$	9.92451	0.334936
$879$	0	0
$880$	0	0
$881$	49.2795	1.66027	0.830134	−	0.557564i	$-0.188264\pi$
$881$	49.2795	1.66027	0.830134	+	0.557564i	$0.188264\pi$
$882$	0	0
$883$	34.0800	1.14688	0.573441	−	0.819247i	$-0.305608\pi$
$883$	34.0800	1.14688	0.573441	+	0.819247i	$0.305608\pi$
$884$	−26.3914	−0.887640
$885$	0	0
$886$	44.5475	1.49660
$887$	−36.7206	−1.23296	−0.616478	−	0.787372i	$-0.711441\pi$
$887$	−36.7206	−1.23296	−0.616478	+	0.787372i	$0.711441\pi$
$888$	0	0
$889$	−9.71364	−0.325785
$890$	0	0
$891$	0	0
$892$	11.4224	0.382449
$893$	−7.67769	−0.256924
$894$	0	0
$895$	0	0
$896$	7.69808	0.257175
$897$	0	0
$898$	−29.1168	−0.971641
$899$	−1.74835	−0.0583109
$900$	0	0
$901$	−4.40433	−0.146730
$902$	17.7235	0.590127
$903$	0	0
$904$	−44.1550	−1.46857
$905$	0	0
$906$	0	0
$907$	5.56553	0.184800	0.0924002	−	0.995722i	$-0.470546\pi$
$907$	5.56553	0.184800	0.0924002	+	0.995722i	$0.470546\pi$
$908$	−16.3106	−0.541286
$909$	0	0
$910$	0	0
$911$	34.5529	1.14479	0.572395	−	0.819978i	$-0.306014\pi$
$911$	34.5529	1.14479	0.572395	+	0.819978i	$0.306014\pi$
$912$	0	0
$913$	7.26834	0.240547
$914$	−35.5082	−1.17451
$915$	0	0
$916$	7.44825	0.246097
$917$	−58.9718	−1.94742
$918$	0	0
$919$	−18.8008	−0.620181	−0.310090	−	0.950707i	$-0.600359\pi$
$919$	−18.8008	−0.620181	−0.310090	+	0.950707i	$0.600359\pi$
$920$	0	0
$921$	0	0
$922$	−14.4905	−0.477219
$923$	−33.7157	−1.10976
$924$	0	0
$925$	0	0
$926$	−23.9599	−0.787371
$927$	0	0
$928$	−20.5044	−0.673091
$929$	−9.05553	−0.297102	−0.148551	−	0.988905i	$-0.547461\pi$
$929$	−9.05553	−0.297102	−0.148551	+	0.988905i	$0.547461\pi$
$930$	0	0
$931$	−9.89286	−0.324225
$932$	−12.6631	−0.414795
$933$	0	0
$934$	−1.25820	−0.0411697
$935$	0	0
$936$	0	0
$937$	36.4584	1.19104	0.595522	−	0.803339i	$-0.296945\pi$
$937$	36.4584	1.19104	0.595522	+	0.803339i	$0.296945\pi$
$938$	−45.4293	−1.48332
$939$	0	0
$940$	0	0
$941$	10.8039	0.352196	0.176098	−	0.984373i	$-0.443652\pi$
$941$	10.8039	0.352196	0.176098	+	0.984373i	$0.443652\pi$
$942$	0	0
$943$	−21.7857	−0.709440
$944$	10.1704	0.331019
$945$	0	0
$946$	−4.28636	−0.139362
$947$	9.79656	0.318345	0.159173	−	0.987251i	$-0.449117\pi$
$947$	9.79656	0.318345	0.159173	+	0.987251i	$0.449117\pi$
$948$	0	0
$949$	−86.7225	−2.81513
$950$	0	0
$951$	0	0
$952$	68.4764	2.21933
$953$	−53.5330	−1.73410	−0.867052	−	0.498218i	$-0.833988\pi$
$953$	−53.5330	−1.73410	−0.867052	+	0.498218i	$0.833988\pi$
$954$	0	0
$955$	0	0
$956$	−2.69225	−0.0870734
$957$	0	0
$958$	3.86696	0.124936
$959$	−92.0507	−2.97247
$960$	0	0
$961$	−30.8929	−0.996544
$962$	82.7867	2.66915
$963$	0	0
$964$	8.28636	0.266886
$965$	0	0
$966$	0	0
$967$	−7.38337	−0.237433	−0.118717	−	0.992928i	$-0.537878\pi$
$967$	−7.38337	−0.237433	−0.118717	+	0.992928i	$0.537878\pi$
$968$	−69.1139	−2.22140
$969$	0	0
$970$	0	0
$971$	−27.1245	−0.870466	−0.435233	−	0.900318i	$-0.643334\pi$
$971$	−27.1245	−0.870466	−0.435233	+	0.900318i	$0.643334\pi$
$972$	0	0
$973$	−25.1130	−0.805087
$974$	−18.4986	−0.592734
$975$	0	0
$976$	16.1122	0.515739
$977$	23.5249	0.752627	0.376314	−	0.926492i	$-0.377192\pi$
$977$	23.5249	0.752627	0.376314	+	0.926492i	$0.377192\pi$
$978$	0	0
$979$	−66.4974	−2.12527
$980$	0	0
$981$	0	0
$982$	−3.84765	−0.122783
$983$	−20.3341	−0.648559	−0.324279	−	0.945961i	$-0.605122\pi$
$983$	−20.3341	−0.648559	−0.324279	+	0.945961i	$0.605122\pi$
$984$	0	0
$985$	0	0
$986$	32.7321	1.04240
$987$	0	0
$988$	−4.87484	−0.155089
$989$	5.26879	0.167538
$990$	0	0
$991$	33.3001	1.05781	0.528907	−	0.848680i	$-0.322602\pi$
$991$	33.3001	1.05781	0.528907	+	0.848680i	$0.322602\pi$
$992$	1.25656	0.0398958
$993$	0	0
$994$	23.1260	0.733512
$995$	0	0
$996$	0	0
$997$	39.1720	1.24059	0.620295	−	0.784368i	$-0.287013\pi$
$997$	39.1720	1.24059	0.620295	+	0.784368i	$0.287013\pi$
$998$	39.8568	1.26164
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bs.1.4	yes	6
3.2	odd	2	inner	4275.2.a.bs.1.3	✓	6
5.4	even	2		4275.2.a.bt.1.3	yes	6
15.14	odd	2		4275.2.a.bt.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4275.2.a.bs.1.3	✓	6	3.2	odd	2	inner
4275.2.a.bs.1.4	yes	6	1.1	even	1	trivial
4275.2.a.bt.1.3	yes	6	5.4	even	2
4275.2.a.bt.1.4	yes	6	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q+1.13194 q^{2} -0.718710 q^{4} +4.11009 q^{7} -3.07742 q^{8} +5.78432 q^{11} -6.78276 q^{13} +4.65238 q^{14} -2.04604 q^{16} -5.41381 q^{17} -1.00000 q^{19} +6.54751 q^{22} -8.04820 q^{23} -7.67769 q^{26} -2.95396 q^{28} -5.34130 q^{29} +0.327327 q^{31} +3.83884 q^{32} -6.12811 q^{34} -10.7828 q^{37} -1.13194 q^{38} +2.70690 q^{41} -0.654655 q^{43} -4.15725 q^{44} -9.11009 q^{46} +7.67769 q^{47} +9.89286 q^{49} +4.87484 q^{52} +0.813537 q^{53} -12.6485 q^{56} -6.04604 q^{58} -4.97079 q^{59} -7.87484 q^{61} +0.370515 q^{62} +8.43742 q^{64} -9.76475 q^{67} +3.89096 q^{68} +4.97079 q^{71} +12.7857 q^{73} -12.2055 q^{74} +0.718710 q^{76} +23.7741 q^{77} -1.01802 q^{79} +3.06406 q^{82} +1.25656 q^{83} -0.741030 q^{86} -17.8008 q^{88} -11.4961 q^{89} -27.8778 q^{91} +5.78432 q^{92} +8.69069 q^{94} -6.78276 q^{97} +11.1981 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{4} - 16 q^{13} - 6 q^{19} - 10 q^{22} - 30 q^{28} + 2 q^{31} - 12 q^{34} - 40 q^{37} - 4 q^{43} - 30 q^{46} + 10 q^{49} - 20 q^{52} - 24 q^{58} + 2 q^{61} + 26 q^{64} - 34 q^{67} - 22 q^{73}+ \cdots - 16 q^{97}+O(q^{100}) \) 6 * q + 8 * q^4 - 16 * q^13 - 6 * q^19 - 10 * q^22 - 30 * q^28 + 2 * q^31 - 12 * q^34 - 40 * q^37 - 4 * q^43 - 30 * q^46 + 10 * q^49 - 20 * q^52 - 24 * q^58 + 2 * q^61 + 26 * q^64 - 34 * q^67 - 22 * q^73 - 8 * q^76 - 6 * q^79 + 6 * q^82 - 82 * q^88 - 44 * q^91 + 52 * q^94 - 16 * q^97

Introduction

L-functions

Modular forms

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