LMFDB - Embedded newform 1475.2.a.r.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1475,2,Mod(1,1475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1475, 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("1475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1475 = 5^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1475.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:	$11.7779342981$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 26x^{12} + 269x^{10} - 1417x^{8} + 4023x^{6} - 5940x^{4} + 3872x^{2} - 576 $ x^14 - 26x^12 + 269x^10 - 1417x^8 + 4023x^6 - 5940x^4 + 3872x^2 - 576
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 295)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.48059$ of defining polynomial
Character	$\chi$	$=$	1475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.48059 q^{2} -3.29929 q^{3} +0.192132 q^{4} +4.88487 q^{6} -0.483534 q^{7} +2.67670 q^{8} +7.88529 q^{9} +O(q^{10})$	\(q-1.48059 q^{2} -3.29929 q^{3} +0.192132 q^{4} +4.88487 q^{6} -0.483534 q^{7} +2.67670 q^{8} +7.88529 q^{9} +5.03821 q^{11} -0.633900 q^{12} -0.640884 q^{13} +0.715913 q^{14} -4.34735 q^{16} +7.22253 q^{17} -11.6748 q^{18} +0.688354 q^{19} +1.59532 q^{21} -7.45949 q^{22} +1.36830 q^{23} -8.83121 q^{24} +0.948883 q^{26} -16.1180 q^{27} -0.0929025 q^{28} +7.80828 q^{29} -3.71525 q^{31} +1.08322 q^{32} -16.6225 q^{33} -10.6936 q^{34} +1.51502 q^{36} -1.06088 q^{37} -1.01917 q^{38} +2.11446 q^{39} -4.95641 q^{41} -2.36200 q^{42} +8.36638 q^{43} +0.968003 q^{44} -2.02589 q^{46} -1.69449 q^{47} +14.3432 q^{48} -6.76620 q^{49} -23.8292 q^{51} -0.123135 q^{52} +4.53236 q^{53} +23.8640 q^{54} -1.29428 q^{56} -2.27108 q^{57} -11.5608 q^{58} -1.00000 q^{59} +8.01102 q^{61} +5.50074 q^{62} -3.81280 q^{63} +7.09090 q^{64} +24.6110 q^{66} -6.92537 q^{67} +1.38768 q^{68} -4.51443 q^{69} -4.83180 q^{71} +21.1066 q^{72} +7.68579 q^{73} +1.57073 q^{74} +0.132255 q^{76} -2.43614 q^{77} -3.13064 q^{78} +10.7745 q^{79} +29.5220 q^{81} +7.33839 q^{82} -6.28601 q^{83} +0.306512 q^{84} -12.3871 q^{86} -25.7618 q^{87} +13.4858 q^{88} -4.49592 q^{89} +0.309889 q^{91} +0.262895 q^{92} +12.2577 q^{93} +2.50883 q^{94} -3.57385 q^{96} +12.5022 q^{97} +10.0179 q^{98} +39.7277 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 24 q^{4} + 14 q^{6} + 14 q^{9}+O(q^{10}) $ 14 * q + 24 * q^4 + 14 * q^6 + 14 * q^9	$ 14 q + 24 q^{4} + 14 q^{6} + 14 q^{9} + 10 q^{11} + 34 q^{14} + 20 q^{16} + 24 q^{19} + 26 q^{21} - 20 q^{24} - 34 q^{26} + 46 q^{29} + 22 q^{31} + 10 q^{34} + 46 q^{36} - 14 q^{39} - 16 q^{41} + 50 q^{44} + 52 q^{46} + 8 q^{49} - 34 q^{51} + 110 q^{54} + 24 q^{56} - 14 q^{59} + 10 q^{61} + 66 q^{64} - 74 q^{66} - 52 q^{71} - 24 q^{74} - 2 q^{76} + 68 q^{79} + 38 q^{81} + 16 q^{84} - 12 q^{86} + 30 q^{89} + 48 q^{91} + 50 q^{94} - 112 q^{96} + 14 q^{99}+O(q^{100}) $ 14 * q + 24 * q^4 + 14 * q^6 + 14 * q^9 + 10 * q^11 + 34 * q^14 + 20 * q^16 + 24 * q^19 + 26 * q^21 - 20 * q^24 - 34 * q^26 + 46 * q^29 + 22 * q^31 + 10 * q^34 + 46 * q^36 - 14 * q^39 - 16 * q^41 + 50 * q^44 + 52 * q^46 + 8 * q^49 - 34 * q^51 + 110 * q^54 + 24 * q^56 - 14 * q^59 + 10 * q^61 + 66 * q^64 - 74 * q^66 - 52 * q^71 - 24 * q^74 - 2 * q^76 + 68 * q^79 + 38 * q^81 + 16 * q^84 - 12 * q^86 + 30 * q^89 + 48 * q^91 + 50 * q^94 - 112 * q^96 + 14 * q^99

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 $

$2$	−1.48059	−1.04693	−0.523466	−	0.852047i	$-0.675361\pi$
$2$	−1.48059	−1.04693	−0.523466	+	0.852047i	$0.675361\pi$
$3$	−3.29929	−1.90484	−0.952422	−	0.304782i	$-0.901416\pi$
$3$	−3.29929	−1.90484	−0.952422	+	0.304782i	$0.901416\pi$
$4$	0.192132	0.0960662
$5$	0	0
$5$	0	0
$6$	4.88487	1.99424
$7$	−0.483534	−0.182759	−0.0913793	−	0.995816i	$-0.529128\pi$
$7$	−0.483534	−0.182759	−0.0913793	+	0.995816i	$0.529128\pi$
$8$	2.67670	0.946357
$9$	7.88529	2.62843
$10$	0	0
$11$	5.03821	1.51908	0.759538	−	0.650463i	$-0.225425\pi$
$11$	5.03821	1.51908	0.759538	+	0.650463i	$0.225425\pi$
$12$	−0.633900	−0.182991
$13$	−0.640884	−0.177749	−0.0888746	−	0.996043i	$-0.528327\pi$
$13$	−0.640884	−0.177749	−0.0888746	+	0.996043i	$0.528327\pi$
$14$	0.715913	0.191336
$15$	0	0
$16$	−4.34735	−1.08684
$17$	7.22253	1.75172	0.875861	−	0.482564i	$-0.160294\pi$
$17$	7.22253	1.75172	0.875861	+	0.482564i	$0.160294\pi$
$18$	−11.6748	−2.75179
$19$	0.688354	0.157919	0.0789596	−	0.996878i	$-0.474840\pi$
$19$	0.688354	0.157919	0.0789596	+	0.996878i	$0.474840\pi$
$20$	0	0
$21$	1.59532	0.348127
$22$	−7.45949	−1.59037
$23$	1.36830	0.285311	0.142656	−	0.989772i	$-0.454436\pi$
$23$	1.36830	0.285311	0.142656	+	0.989772i	$0.454436\pi$
$24$	−8.83121	−1.80266
$25$	0	0
$26$	0.948883	0.186091
$27$	−16.1180	−3.10191
$28$	−0.0929025	−0.0175569
$29$	7.80828	1.44996	0.724981	−	0.688769i	$-0.241848\pi$
$29$	7.80828	1.44996	0.724981	+	0.688769i	$0.241848\pi$
$30$	0	0
$31$	−3.71525	−0.667278	−0.333639	−	0.942701i	$-0.608277\pi$
$31$	−3.71525	−0.667278	−0.333639	+	0.942701i	$0.608277\pi$
$32$	1.08322	0.191488
$33$	−16.6225	−2.89360
$34$	−10.6936	−1.83393
$35$	0	0
$36$	1.51502	0.252503
$37$	−1.06088	−0.174408	−0.0872039	−	0.996190i	$-0.527793\pi$
$37$	−1.06088	−0.174408	−0.0872039	+	0.996190i	$0.527793\pi$
$38$	−1.01917	−0.165331
$39$	2.11446	0.338585
$40$	0	0
$41$	−4.95641	−0.774061	−0.387031	−	0.922067i	$-0.626499\pi$
$41$	−4.95641	−0.774061	−0.387031	+	0.922067i	$0.626499\pi$
$42$	−2.36200	−0.364465
$43$	8.36638	1.27586	0.637930	−	0.770094i	$-0.279791\pi$
$43$	8.36638	1.27586	0.637930	+	0.770094i	$0.279791\pi$
$44$	0.968003	0.145932
$45$	0	0
$46$	−2.02589	−0.298701
$47$	−1.69449	−0.247166	−0.123583	−	0.992334i	$-0.539439\pi$
$47$	−1.69449	−0.247166	−0.123583	+	0.992334i	$0.539439\pi$
$48$	14.3432	2.07026
$49$	−6.76620	−0.966599
$50$	0	0
$51$	−23.8292	−3.33676
$52$	−0.123135	−0.0170757
$53$	4.53236	0.622567	0.311283	−	0.950317i	$-0.399241\pi$
$53$	4.53236	0.622567	0.311283	+	0.950317i	$0.399241\pi$
$54$	23.8640	3.24748
$55$	0	0
$56$	−1.29428	−0.172955
$57$	−2.27108	−0.300811
$58$	−11.5608	−1.51801
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	8.01102	1.02571	0.512853	−	0.858476i	$-0.328589\pi$
$61$	8.01102	1.02571	0.512853	+	0.858476i	$0.328589\pi$
$62$	5.50074	0.698595
$63$	−3.81280	−0.480368
$64$	7.09090	0.886363
$65$	0	0
$66$	24.6110	3.02941
$67$	−6.92537	−0.846069	−0.423035	−	0.906114i	$-0.639035\pi$
$67$	−6.92537	−0.846069	−0.423035	+	0.906114i	$0.639035\pi$
$68$	1.38768	0.168281
$69$	−4.51443	−0.543473
$70$	0	0
$71$	−4.83180	−0.573429	−0.286715	−	0.958016i	$-0.592563\pi$
$71$	−4.83180	−0.573429	−0.286715	+	0.958016i	$0.592563\pi$
$72$	21.1066	2.48743
$73$	7.68579	0.899554	0.449777	−	0.893141i	$-0.351503\pi$
$73$	7.68579	0.899554	0.449777	+	0.893141i	$0.351503\pi$
$74$	1.57073	0.182593
$75$	0	0
$76$	0.132255	0.0151707
$77$	−2.43614	−0.277624
$78$	−3.13064	−0.354475
$79$	10.7745	1.21223	0.606115	−	0.795377i	$-0.292727\pi$
$79$	10.7745	1.21223	0.606115	+	0.795377i	$0.292727\pi$
$80$	0	0
$81$	29.5220	3.28022
$82$	7.33839	0.810389
$83$	−6.28601	−0.689979	−0.344989	−	0.938607i	$-0.612118\pi$
$83$	−6.28601	−0.689979	−0.344989	+	0.938607i	$0.612118\pi$
$84$	0.306512	0.0334432
$85$	0	0
$86$	−12.3871	−1.33574
$87$	−25.7618	−2.76195
$88$	13.4858	1.43759
$89$	−4.49592	−0.476567	−0.238283	−	0.971196i	$-0.576585\pi$
$89$	−4.49592	−0.476567	−0.238283	+	0.971196i	$0.576585\pi$
$90$	0	0
$91$	0.309889	0.0324852
$92$	0.262895	0.0274087
$93$	12.2577	1.27106
$94$	2.50883	0.258766
$95$	0	0
$96$	−3.57385	−0.364754
$97$	12.5022	1.26940	0.634701	−	0.772758i	$-0.281123\pi$
$97$	12.5022	1.26940	0.634701	+	0.772758i	$0.281123\pi$
$98$	10.0179	1.01196
$99$	39.7277	3.99279
$100$	0	0
$101$	−11.0743	−1.10193	−0.550966	−	0.834527i	$-0.685741\pi$
$101$	−11.0743	−1.10193	−0.550966	+	0.834527i	$0.685741\pi$
$102$	35.2812	3.49336
$103$	6.71784	0.661928	0.330964	−	0.943643i	$-0.392626\pi$
$103$	6.71784	0.661928	0.330964	+	0.943643i	$0.392626\pi$
$104$	−1.71546	−0.168214
$105$	0	0
$106$	−6.71054	−0.651785
$107$	10.9091	1.05463	0.527314	−	0.849671i	$-0.323199\pi$
$107$	10.9091	1.05463	0.527314	+	0.849671i	$0.323199\pi$
$108$	−3.09679	−0.297988
$109$	3.09187	0.296148	0.148074	−	0.988976i	$-0.452693\pi$
$109$	3.09187	0.296148	0.148074	+	0.988976i	$0.452693\pi$
$110$	0	0
$111$	3.50015	0.332220
$112$	2.10209	0.198629
$113$	−3.70797	−0.348816	−0.174408	−	0.984673i	$-0.555801\pi$
$113$	−3.70797	−0.348816	−0.174408	+	0.984673i	$0.555801\pi$
$114$	3.36252	0.314929
$115$	0	0
$116$	1.50022	0.139292
$117$	−5.05356	−0.467201
$118$	1.48059	0.136299
$119$	−3.49234	−0.320142
$120$	0	0
$121$	14.3835	1.30759
$122$	−11.8610	−1.07384
$123$	16.3526	1.47447
$124$	−0.713819	−0.0641029
$125$	0	0
$126$	5.64518	0.502913
$127$	−17.4799	−1.55109	−0.775544	−	0.631294i	$-0.782524\pi$
$127$	−17.4799	−1.55109	−0.775544	+	0.631294i	$0.782524\pi$
$128$	−12.6651	−1.11945
$129$	−27.6031	−2.43032
$130$	0	0
$131$	−6.28342	−0.548985	−0.274493	−	0.961589i	$-0.588510\pi$
$131$	−6.28342	−0.548985	−0.274493	+	0.961589i	$0.588510\pi$
$132$	−3.19372	−0.277977
$133$	−0.332842	−0.0288611
$134$	10.2536	0.885777
$135$	0	0
$136$	19.3326	1.65775
$137$	16.6905	1.42597	0.712985	−	0.701179i	$-0.247343\pi$
$137$	16.6905	1.42597	0.712985	+	0.701179i	$0.247343\pi$
$138$	6.68399	0.568979
$139$	−16.9592	−1.43846	−0.719232	−	0.694770i	$-0.755506\pi$
$139$	−16.9592	−1.43846	−0.719232	+	0.694770i	$0.755506\pi$
$140$	0	0
$141$	5.59060	0.470813
$142$	7.15389	0.600341
$143$	−3.22891	−0.270015
$144$	−34.2801	−2.85668
$145$	0	0
$146$	−11.3795	−0.941771
$147$	22.3236	1.84122
$148$	−0.203830	−0.0167547
$149$	19.7965	1.62179	0.810895	−	0.585191i	$-0.198981\pi$
$149$	19.7965	1.62179	0.810895	+	0.585191i	$0.198981\pi$
$150$	0	0
$151$	−12.2850	−0.999740	−0.499870	−	0.866100i	$-0.666619\pi$
$151$	−12.2850	−0.999740	−0.499870	+	0.866100i	$0.666619\pi$
$152$	1.84252	0.149448
$153$	56.9518	4.60428
$154$	3.60692	0.290654
$155$	0	0
$156$	0.406256	0.0325265
$157$	11.6395	0.928933	0.464467	−	0.885591i	$-0.346246\pi$
$157$	11.6395	0.928933	0.464467	+	0.885591i	$0.346246\pi$
$158$	−15.9526	−1.26912
$159$	−14.9535	−1.18589
$160$	0	0
$161$	−0.661621	−0.0521430
$162$	−43.7098	−3.43416
$163$	3.95705	0.309940	0.154970	−	0.987919i	$-0.450472\pi$
$163$	3.95705	0.309940	0.154970	+	0.987919i	$0.450472\pi$
$164$	−0.952287	−0.0743611
$165$	0	0
$166$	9.30697	0.722361
$167$	−18.9684	−1.46782	−0.733911	−	0.679246i	$-0.762307\pi$
$167$	−18.9684	−1.46782	−0.733911	+	0.679246i	$0.762307\pi$
$168$	4.27019	0.329452
$169$	−12.5893	−0.968405
$170$	0	0
$171$	5.42787	0.415080
$172$	1.60745	0.122567
$173$	−1.26730	−0.0963511	−0.0481755	−	0.998839i	$-0.515341\pi$
$173$	−1.26730	−0.0963511	−0.0481755	+	0.998839i	$0.515341\pi$
$174$	38.1425	2.89157
$175$	0	0
$176$	−21.9028	−1.65099
$177$	3.29929	0.247990
$178$	6.65660	0.498933
$179$	−7.37203	−0.551011	−0.275506	−	0.961299i	$-0.588845\pi$
$179$	−7.37203	−0.551011	−0.275506	+	0.961299i	$0.588845\pi$
$180$	0	0
$181$	−0.427117	−0.0317473	−0.0158737	−	0.999874i	$-0.505053\pi$
$181$	−0.427117	−0.0317473	−0.0158737	+	0.999874i	$0.505053\pi$
$182$	−0.458817	−0.0340098
$183$	−26.4307	−1.95381
$184$	3.66254	0.270006
$185$	0	0
$186$	−18.1485	−1.33071
$187$	36.3886	2.66100
$188$	−0.325566	−0.0237443
$189$	7.79359	0.566900
$190$	0	0
$191$	19.3373	1.39920	0.699600	−	0.714535i	$-0.253362\pi$
$191$	19.3373	1.39920	0.699600	+	0.714535i	$0.253362\pi$
$192$	−23.3949	−1.68838
$193$	−7.72682	−0.556189	−0.278094	−	0.960554i	$-0.589703\pi$
$193$	−7.72682	−0.556189	−0.278094	+	0.960554i	$0.589703\pi$
$194$	−18.5105	−1.32898
$195$	0	0
$196$	−1.30001	−0.0928575
$197$	12.7736	0.910080	0.455040	−	0.890471i	$-0.349625\pi$
$197$	12.7736	0.910080	0.455040	+	0.890471i	$0.349625\pi$
$198$	−58.8203	−4.18018
$199$	2.96109	0.209906	0.104953	−	0.994477i	$-0.466531\pi$
$199$	2.96109	0.209906	0.104953	+	0.994477i	$0.466531\pi$
$200$	0	0
$201$	22.8488	1.61163
$202$	16.3964	1.15365
$203$	−3.77557	−0.264993
$204$	−4.57836	−0.320550
$205$	0	0
$206$	−9.94633	−0.692994
$207$	10.7895	0.749920
$208$	2.78615	0.193185
$209$	3.46807	0.239891
$210$	0	0
$211$	16.5014	1.13601	0.568003	−	0.823027i	$-0.307716\pi$
$211$	16.5014	1.13601	0.568003	+	0.823027i	$0.307716\pi$
$212$	0.870812	0.0598076
$213$	15.9415	1.09229
$214$	−16.1519	−1.10412
$215$	0	0
$216$	−43.1430	−2.93551
$217$	1.79645	0.121951
$218$	−4.57778	−0.310047
$219$	−25.3576	−1.71351
$220$	0	0
$221$	−4.62881	−0.311367
$222$	−5.18227	−0.347811
$223$	−5.67650	−0.380127	−0.190063	−	0.981772i	$-0.560869\pi$
$223$	−5.67650	−0.380127	−0.190063	+	0.981772i	$0.560869\pi$
$224$	−0.523772	−0.0349960
$225$	0	0
$226$	5.48996	0.365187
$227$	21.0089	1.39441	0.697205	−	0.716872i	$-0.254427\pi$
$227$	21.0089	1.39441	0.697205	+	0.716872i	$0.254427\pi$
$228$	−0.436347	−0.0288978
$229$	16.5097	1.09099	0.545495	−	0.838114i	$-0.316342\pi$
$229$	16.5097	1.09099	0.545495	+	0.838114i	$0.316342\pi$
$230$	0	0
$231$	8.03753	0.528831
$232$	20.9005	1.37218
$233$	−27.0228	−1.77033	−0.885163	−	0.465282i	$-0.845953\pi$
$233$	−27.0228	−1.77033	−0.885163	+	0.465282i	$0.845953\pi$
$234$	7.48222	0.489128
$235$	0	0
$236$	−0.192132	−0.0125068
$237$	−35.5483	−2.30911
$238$	5.17070	0.335167
$239$	−5.84745	−0.378240	−0.189120	−	0.981954i	$-0.560564\pi$
$239$	−5.84745	−0.378240	−0.189120	+	0.981954i	$0.560564\pi$
$240$	0	0
$241$	−2.01053	−0.129510	−0.0647549	−	0.997901i	$-0.520627\pi$
$241$	−2.01053	−0.129510	−0.0647549	+	0.997901i	$0.520627\pi$
$242$	−21.2960	−1.36896
$243$	−49.0474	−3.14640
$244$	1.53918	0.0985357
$245$	0	0
$246$	−24.2114	−1.54367
$247$	−0.441155	−0.0280700
$248$	−9.94461	−0.631483
$249$	20.7393	1.31430
$250$	0	0
$251$	−8.00041	−0.504981	−0.252491	−	0.967599i	$-0.581250\pi$
$251$	−8.00041	−0.504981	−0.252491	+	0.967599i	$0.581250\pi$
$252$	−0.732563	−0.0461471
$253$	6.89380	0.433409
$254$	25.8804	1.62388
$255$	0	0
$256$	4.56998	0.285624
$257$	11.7580	0.733446	0.366723	−	0.930330i	$-0.380480\pi$
$257$	11.7580	0.733446	0.366723	+	0.930330i	$0.380480\pi$
$258$	40.8687	2.54437
$259$	0.512972	0.0318745
$260$	0	0
$261$	61.5706	3.81112
$262$	9.30314	0.574750
$263$	−12.2421	−0.754880	−0.377440	−	0.926034i	$-0.623196\pi$
$263$	−12.2421	−0.754880	−0.377440	+	0.926034i	$0.623196\pi$
$264$	−44.4934	−2.73838
$265$	0	0
$266$	0.492801	0.0302156
$267$	14.8333	0.907785
$268$	−1.33059	−0.0812786
$269$	−22.3827	−1.36470	−0.682349	−	0.731026i	$-0.739042\pi$
$269$	−22.3827	−1.36470	−0.682349	+	0.731026i	$0.739042\pi$
$270$	0	0
$271$	11.8058	0.717153	0.358577	−	0.933500i	$-0.383262\pi$
$271$	11.8058	0.717153	0.358577	+	0.933500i	$0.383262\pi$
$272$	−31.3989	−1.90384
$273$	−1.02241	−0.0618792
$274$	−24.7118	−1.49289
$275$	0	0
$276$	−0.867367	−0.0522094
$277$	−4.75691	−0.285815	−0.142907	−	0.989736i	$-0.545645\pi$
$277$	−4.75691	−0.285815	−0.142907	+	0.989736i	$0.545645\pi$
$278$	25.1096	1.50597
$279$	−29.2958	−1.75389
$280$	0	0
$281$	−5.83292	−0.347963	−0.173982	−	0.984749i	$-0.555663\pi$
$281$	−5.83292	−0.347963	−0.173982	+	0.984749i	$0.555663\pi$
$282$	−8.27736	−0.492909
$283$	−2.83617	−0.168593	−0.0842963	−	0.996441i	$-0.526864\pi$
$283$	−2.83617	−0.168593	−0.0842963	+	0.996441i	$0.526864\pi$
$284$	−0.928345	−0.0550872
$285$	0	0
$286$	4.78067	0.282687
$287$	2.39659	0.141466
$288$	8.54149	0.503312
$289$	35.1650	2.06853
$290$	0	0
$291$	−41.2482	−2.41801
$292$	1.47669	0.0864167
$293$	17.0563	0.996442	0.498221	−	0.867050i	$-0.333987\pi$
$293$	17.0563	0.996442	0.498221	+	0.867050i	$0.333987\pi$
$294$	−33.0520	−1.92763
$295$	0	0
$296$	−2.83966	−0.165052
$297$	−81.2057	−4.71203
$298$	−29.3104	−1.69790
$299$	−0.876924	−0.0507138
$300$	0	0
$301$	−4.04543	−0.233174
$302$	18.1890	1.04666
$303$	36.5372	2.09901
$304$	−2.99251	−0.171633
$305$	0	0
$306$	−84.3220	−4.82037
$307$	−15.4270	−0.880464	−0.440232	−	0.897884i	$-0.645104\pi$
$307$	−15.4270	−0.880464	−0.440232	+	0.897884i	$0.645104\pi$
$308$	−0.468062	−0.0266703
$309$	−22.1641	−1.26087
$310$	0	0
$311$	24.4645	1.38725	0.693627	−	0.720334i	$-0.256012\pi$
$311$	24.4645	1.38725	0.693627	+	0.720334i	$0.256012\pi$
$312$	5.65978	0.320422
$313$	−17.5512	−0.992055	−0.496027	−	0.868307i	$-0.665208\pi$
$313$	−17.5512	−0.992055	−0.496027	+	0.868307i	$0.665208\pi$
$314$	−17.2333	−0.972530
$315$	0	0
$316$	2.07014	0.116454
$317$	15.2581	0.856982	0.428491	−	0.903546i	$-0.359045\pi$
$317$	15.2581	0.856982	0.428491	+	0.903546i	$0.359045\pi$
$318$	22.1400	1.24155
$319$	39.3398	2.20260
$320$	0	0
$321$	−35.9924	−2.00890
$322$	0.979586	0.0545902
$323$	4.97166	0.276630
$324$	5.67212	0.315118
$325$	0	0
$326$	−5.85875	−0.324486
$327$	−10.2010	−0.564116
$328$	−13.2668	−0.732538
$329$	0.819342	0.0451718
$330$	0	0
$331$	17.7533	0.975812	0.487906	−	0.872896i	$-0.337761\pi$
$331$	17.7533	0.975812	0.487906	+	0.872896i	$0.337761\pi$
$332$	−1.20775	−0.0662837
$333$	−8.36536	−0.458419
$334$	28.0844	1.53671
$335$	0	0
$336$	−6.93540	−0.378357
$337$	−23.5449	−1.28257	−0.641286	−	0.767302i	$-0.721599\pi$
$337$	−23.5449	−1.28257	−0.641286	+	0.767302i	$0.721599\pi$
$338$	18.6395	1.01385
$339$	12.2337	0.664441
$340$	0	0
$341$	−18.7182	−1.01365
$342$	−8.03642	−0.434560
$343$	6.65642	0.359413
$344$	22.3943	1.20742
$345$	0	0
$346$	1.87635	0.100873
$347$	−23.9407	−1.28520	−0.642601	−	0.766201i	$-0.722145\pi$
$347$	−23.9407	−1.28520	−0.642601	+	0.766201i	$0.722145\pi$
$348$	−4.94967	−0.265330
$349$	−6.97093	−0.373145	−0.186573	−	0.982441i	$-0.559738\pi$
$349$	−6.97093	−0.373145	−0.186573	+	0.982441i	$0.559738\pi$
$350$	0	0
$351$	10.3298	0.551361
$352$	5.45748	0.290884
$353$	−5.80728	−0.309090	−0.154545	−	0.987986i	$-0.549391\pi$
$353$	−5.80728	−0.309090	−0.154545	+	0.987986i	$0.549391\pi$
$354$	−4.88487	−0.259628
$355$	0	0
$356$	−0.863812	−0.0457820
$357$	11.5222	0.609821
$358$	10.9149	0.576871
$359$	2.67466	0.141163	0.0705815	−	0.997506i	$-0.477515\pi$
$359$	2.67466	0.141163	0.0705815	+	0.997506i	$0.477515\pi$
$360$	0	0
$361$	−18.5262	−0.975062
$362$	0.632382	0.0332373
$363$	−47.4554	−2.49076
$364$	0.0595397	0.00312073
$365$	0	0
$366$	39.1328	2.04551
$367$	1.28931	0.0673016	0.0336508	−	0.999434i	$-0.489287\pi$
$367$	1.28931	0.0673016	0.0336508	+	0.999434i	$0.489287\pi$
$368$	−5.94850	−0.310087
$369$	−39.0827	−2.03457
$370$	0	0
$371$	−2.19155	−0.113779
$372$	2.35509	0.122106
$373$	4.18716	0.216803	0.108401	−	0.994107i	$-0.465427\pi$
$373$	4.18716	0.216803	0.108401	+	0.994107i	$0.465427\pi$
$374$	−53.8764	−2.78588
$375$	0	0
$376$	−4.53564	−0.233908
$377$	−5.00420	−0.257730
$378$	−11.5391	−0.593506
$379$	17.1502	0.880947	0.440474	−	0.897766i	$-0.354811\pi$
$379$	17.1502	0.880947	0.440474	+	0.897766i	$0.354811\pi$
$380$	0	0
$381$	57.6711	2.95458
$382$	−28.6306	−1.46487
$383$	−15.9039	−0.812654	−0.406327	−	0.913728i	$-0.633191\pi$
$383$	−15.9039	−0.812654	−0.406327	+	0.913728i	$0.633191\pi$
$384$	41.7859	2.13238
$385$	0	0
$386$	11.4402	0.582292
$387$	65.9713	3.35351
$388$	2.40207	0.121947
$389$	17.7504	0.899979	0.449989	−	0.893034i	$-0.351428\pi$
$389$	17.7504	0.899979	0.449989	+	0.893034i	$0.351428\pi$
$390$	0	0
$391$	9.88262	0.499786
$392$	−18.1111	−0.914748
$393$	20.7308	1.04573
$394$	−18.9124	−0.952791
$395$	0	0
$396$	7.63298	0.383572
$397$	32.0239	1.60723	0.803617	−	0.595146i	$-0.202906\pi$
$397$	32.0239	1.60723	0.803617	+	0.595146i	$0.202906\pi$
$398$	−4.38415	−0.219757
$399$	1.09814	0.0549759
$400$	0	0
$401$	21.9395	1.09561	0.547804	−	0.836607i	$-0.315464\pi$
$401$	21.9395	1.09561	0.547804	+	0.836607i	$0.315464\pi$
$402$	−33.8296	−1.68727
$403$	2.38104	0.118608
$404$	−2.12773	−0.105858
$405$	0	0
$406$	5.59005	0.277430
$407$	−5.34494	−0.264939
$408$	−63.7837	−3.15776
$409$	23.3638	1.15527	0.577634	−	0.816296i	$-0.303976\pi$
$409$	23.3638	1.15527	0.577634	+	0.816296i	$0.303976\pi$
$410$	0	0
$411$	−55.0669	−2.71625
$412$	1.29071	0.0635889
$413$	0.483534	0.0237931
$414$	−15.9747	−0.785115
$415$	0	0
$416$	−0.694217	−0.0340368
$417$	55.9534	2.74005
$418$	−5.13477	−0.251150
$419$	17.6680	0.863137	0.431569	−	0.902080i	$-0.357960\pi$
$419$	17.6680	0.863137	0.431569	+	0.902080i	$0.357960\pi$
$420$	0	0
$421$	28.9472	1.41080	0.705400	−	0.708810i	$-0.250768\pi$
$421$	28.9472	1.41080	0.705400	+	0.708810i	$0.250768\pi$
$422$	−24.4318	−1.18932
$423$	−13.3615	−0.649660
$424$	12.1318	0.589171
$425$	0	0
$426$	−23.6027	−1.14356
$427$	−3.87360	−0.187457
$428$	2.09600	0.101314
$429$	10.6531	0.514336
$430$	0	0
$431$	5.00932	0.241290	0.120645	−	0.992696i	$-0.461504\pi$
$431$	5.00932	0.241290	0.120645	+	0.992696i	$0.461504\pi$
$432$	70.0705	3.37127
$433$	19.1536	0.920464	0.460232	−	0.887799i	$-0.347766\pi$
$433$	19.1536	0.920464	0.460232	+	0.887799i	$0.347766\pi$
$434$	−2.65979	−0.127674
$435$	0	0
$436$	0.594049	0.0284498
$437$	0.941877	0.0450561
$438$	37.5441	1.79393
$439$	13.9666	0.666587	0.333294	−	0.942823i	$-0.391840\pi$
$439$	13.9666	0.666587	0.333294	+	0.942823i	$0.391840\pi$
$440$	0	0
$441$	−53.3534	−2.54064
$442$	6.85334	0.325980
$443$	14.0196	0.666093	0.333046	−	0.942910i	$-0.391923\pi$
$443$	14.0196	0.666093	0.333046	+	0.942910i	$0.391923\pi$
$444$	0.672493	0.0319151
$445$	0	0
$446$	8.40454	0.397967
$447$	−65.3142	−3.08926
$448$	−3.42869	−0.161990
$449$	−16.1511	−0.762217	−0.381108	−	0.924530i	$-0.624458\pi$
$449$	−16.1511	−0.762217	−0.381108	+	0.924530i	$0.624458\pi$
$450$	0	0
$451$	−24.9714	−1.17586
$452$	−0.712421	−0.0335095
$453$	40.5318	1.90435
$454$	−31.1055	−1.45985
$455$	0	0
$456$	−6.07899	−0.284675
$457$	−4.82597	−0.225750	−0.112875	−	0.993609i	$-0.536006\pi$
$457$	−4.82597	−0.225750	−0.112875	+	0.993609i	$0.536006\pi$
$458$	−24.4440	−1.14219
$459$	−116.413	−5.43368
$460$	0	0
$461$	12.1560	0.566160	0.283080	−	0.959096i	$-0.408644\pi$
$461$	12.1560	0.566160	0.283080	+	0.959096i	$0.408644\pi$
$462$	−11.9003	−0.553650
$463$	14.8285	0.689140	0.344570	−	0.938761i	$-0.388025\pi$
$463$	14.8285	0.689140	0.344570	+	0.938761i	$0.388025\pi$
$464$	−33.9453	−1.57587
$465$	0	0
$466$	40.0096	1.85341
$467$	−15.1804	−0.702464	−0.351232	−	0.936289i	$-0.614237\pi$
$467$	−15.1804	−0.702464	−0.351232	+	0.936289i	$0.614237\pi$
$468$	−0.970952	−0.0448823
$469$	3.34865	0.154626
$470$	0	0
$471$	−38.4020	−1.76947
$472$	−2.67670	−0.123205
$473$	42.1515	1.93813
$474$	52.6323	2.41748
$475$	0	0
$476$	−0.670991	−0.0307548
$477$	35.7389	1.63637
$478$	8.65765	0.395992
$479$	38.0119	1.73681	0.868405	−	0.495856i	$-0.165146\pi$
$479$	38.0119	1.73681	0.868405	+	0.495856i	$0.165146\pi$
$480$	0	0
$481$	0.679902	0.0310009
$482$	2.97677	0.135588
$483$	2.18288	0.0993244
$484$	2.76354	0.125616
$485$	0	0
$486$	72.6189	3.29406
$487$	18.7562	0.849924	0.424962	−	0.905211i	$-0.360287\pi$
$487$	18.7562	0.849924	0.424962	+	0.905211i	$0.360287\pi$
$488$	21.4431	0.970684
$489$	−13.0554	−0.590387
$490$	0	0
$491$	23.2982	1.05143	0.525716	−	0.850660i	$-0.323798\pi$
$491$	23.2982	1.05143	0.525716	+	0.850660i	$0.323798\pi$
$492$	3.14187	0.141646
$493$	56.3956	2.53993
$494$	0.653167	0.0293874
$495$	0	0
$496$	16.1515	0.725223
$497$	2.33634	0.104799
$498$	−30.7064	−1.37598
$499$	−14.7486	−0.660240	−0.330120	−	0.943939i	$-0.607089\pi$
$499$	−14.7486	−0.660240	−0.330120	+	0.943939i	$0.607089\pi$
$500$	0	0
$501$	62.5823	2.79597
$502$	11.8453	0.528681
$503$	−26.6428	−1.18794	−0.593972	−	0.804486i	$-0.702441\pi$
$503$	−26.6428	−1.18794	−0.593972	+	0.804486i	$0.702441\pi$
$504$	−10.2057	−0.454600
$505$	0	0
$506$	−10.2069	−0.453750
$507$	41.5356	1.84466
$508$	−3.35845	−0.149007
$509$	−7.96271	−0.352941	−0.176470	−	0.984306i	$-0.556468\pi$
$509$	−7.96271	−0.352941	−0.176470	+	0.984306i	$0.556468\pi$
$510$	0	0
$511$	−3.71634	−0.164401
$512$	18.5640	0.820420
$513$	−11.0949	−0.489851
$514$	−17.4088	−0.767868
$515$	0	0
$516$	−5.30345	−0.233471
$517$	−8.53718	−0.375465
$518$	−0.759499	−0.0333705
$519$	4.18119	0.183534
$520$	0	0
$521$	4.93098	0.216030	0.108015	−	0.994149i	$-0.465551\pi$
$521$	4.93098	0.216030	0.108015	+	0.994149i	$0.465551\pi$
$522$	−91.1605	−3.98999
$523$	21.0503	0.920467	0.460233	−	0.887798i	$-0.347766\pi$
$523$	21.0503	0.920467	0.460233	+	0.887798i	$0.347766\pi$
$524$	−1.20725	−0.0527389
$525$	0	0
$526$	18.1255	0.790308
$527$	−26.8335	−1.16889
$528$	72.2638	3.14488
$529$	−21.1277	−0.918598
$530$	0	0
$531$	−7.88529	−0.342193
$532$	−0.0639498	−0.00277257
$533$	3.17648	0.137589
$534$	−21.9620	−0.950389
$535$	0	0
$536$	−18.5372	−0.800683
$537$	24.3224	1.04959
$538$	33.1395	1.42875
$539$	−34.0895	−1.46834
$540$	0	0
$541$	1.11767	0.0480523	0.0240262	−	0.999711i	$-0.492352\pi$
$541$	1.11767	0.0480523	0.0240262	+	0.999711i	$0.492352\pi$
$542$	−17.4795	−0.750810
$543$	1.40918	0.0604737
$544$	7.82358	0.335433
$545$	0	0
$546$	1.51377	0.0647833
$547$	−1.26426	−0.0540558	−0.0270279	−	0.999635i	$-0.508604\pi$
$547$	−1.26426	−0.0540558	−0.0270279	+	0.999635i	$0.508604\pi$
$548$	3.20680	0.136987
$549$	63.1693	2.69600
$550$	0	0
$551$	5.37486	0.228977
$552$	−12.0838	−0.514320
$553$	−5.20985	−0.221545
$554$	7.04301	0.299229
$555$	0	0
$556$	−3.25842	−0.138188
$557$	23.8410	1.01018	0.505088	−	0.863068i	$-0.331460\pi$
$557$	23.8410	1.01018	0.505088	+	0.863068i	$0.331460\pi$
$558$	43.3749	1.83621
$559$	−5.36188	−0.226783
$560$	0	0
$561$	−120.056	−5.06879
$562$	8.63614	0.364294
$563$	−2.84195	−0.119774	−0.0598869	−	0.998205i	$-0.519074\pi$
$563$	−2.84195	−0.119774	−0.0598869	+	0.998205i	$0.519074\pi$
$564$	1.07414	0.0452292
$565$	0	0
$566$	4.19918	0.176505
$567$	−14.2749	−0.599488
$568$	−12.9333	−0.542669
$569$	−0.419856	−0.0176013	−0.00880065	−	0.999961i	$-0.502801\pi$
$569$	−0.419856	−0.0176013	−0.00880065	+	0.999961i	$0.502801\pi$
$570$	0	0
$571$	28.1515	1.17810	0.589051	−	0.808096i	$-0.299502\pi$
$571$	28.1515	1.17810	0.589051	+	0.808096i	$0.299502\pi$
$572$	−0.620377	−0.0259393
$573$	−63.7994	−2.66526
$574$	−3.54836	−0.148106
$575$	0	0
$576$	55.9138	2.32974
$577$	−33.4028	−1.39058	−0.695288	−	0.718732i	$-0.744723\pi$
$577$	−33.4028	−1.39058	−0.695288	+	0.718732i	$0.744723\pi$
$578$	−52.0648	−2.16561
$579$	25.4930	1.05945
$580$	0	0
$581$	3.03950	0.126100
$582$	61.0714	2.53149
$583$	22.8349	0.945727
$584$	20.5726	0.851299
$585$	0	0
$586$	−25.2534	−1.04321
$587$	1.81855	0.0750594	0.0375297	−	0.999296i	$-0.488051\pi$
$587$	1.81855	0.0750594	0.0375297	+	0.999296i	$0.488051\pi$
$588$	4.28909	0.176879
$589$	−2.55740	−0.105376
$590$	0	0
$591$	−42.1437	−1.73356
$592$	4.61202	0.189553
$593$	−0.0210180	−0.000863106 0	−0.000431553	−	1.00000i	$-0.500137\pi$
$593$	−0.0210180	−0.000863106 0	−0.000431553		1.00000i	$0.500137\pi$
$594$	120.232	4.93318
$595$	0	0
$596$	3.80354	0.155799
$597$	−9.76948	−0.399838
$598$	1.29836	0.0530939
$599$	−26.1503	−1.06847	−0.534236	−	0.845335i	$-0.679401\pi$
$599$	−26.1503	−1.06847	−0.534236	+	0.845335i	$0.679401\pi$
$600$	0	0
$601$	−2.09310	−0.0853795	−0.0426898	−	0.999088i	$-0.513593\pi$
$601$	−2.09310	−0.0853795	−0.0426898	+	0.999088i	$0.513593\pi$
$602$	5.98960	0.244118
$603$	−54.6086	−2.22383
$604$	−2.36035	−0.0960412
$605$	0	0
$606$	−54.0965	−2.19752
$607$	−20.5690	−0.834870	−0.417435	−	0.908707i	$-0.637071\pi$
$607$	−20.5690	−0.834870	−0.417435	+	0.908707i	$0.637071\pi$
$608$	0.745637	0.0302396
$609$	12.4567	0.504770
$610$	0	0
$611$	1.08597	0.0439336
$612$	10.9423	0.442316
$613$	−33.9280	−1.37034	−0.685169	−	0.728384i	$-0.740272\pi$
$613$	−33.9280	−1.37034	−0.685169	+	0.728384i	$0.740272\pi$
$614$	22.8409	0.921786
$615$	0	0
$616$	−6.52083	−0.262732
$617$	3.03117	0.122030	0.0610152	−	0.998137i	$-0.480566\pi$
$617$	3.03117	0.122030	0.0610152	+	0.998137i	$0.480566\pi$
$618$	32.8158	1.32005
$619$	8.66615	0.348322	0.174161	−	0.984717i	$-0.444279\pi$
$619$	8.66615	0.348322	0.174161	+	0.984717i	$0.444279\pi$
$620$	0	0
$621$	−22.0543	−0.885008
$622$	−36.2218	−1.45236
$623$	2.17393	0.0870967
$624$	−9.19230	−0.367986
$625$	0	0
$626$	25.9861	1.03861
$627$	−11.4422	−0.456956
$628$	2.23632	0.0892391
$629$	−7.66225	−0.305514
$630$	0	0
$631$	−29.2507	−1.16445	−0.582225	−	0.813028i	$-0.697818\pi$
$631$	−29.2507	−1.16445	−0.582225	+	0.813028i	$0.697818\pi$
$632$	28.8402	1.14720
$633$	−54.4430	−2.16391
$634$	−22.5910	−0.897202
$635$	0	0
$636$	−2.87306	−0.113924
$637$	4.33635	0.171812
$638$	−58.2459	−2.30598
$639$	−38.1002	−1.50722
$640$	0	0
$641$	−36.0936	−1.42561	−0.712807	−	0.701361i	$-0.752576\pi$
$641$	−36.0936	−1.42561	−0.712807	+	0.701361i	$0.752576\pi$
$642$	53.2898	2.10318
$643$	−23.6649	−0.933254	−0.466627	−	0.884454i	$-0.654531\pi$
$643$	−23.6649	−0.933254	−0.466627	+	0.884454i	$0.654531\pi$
$644$	−0.127119	−0.00500918
$645$	0	0
$646$	−7.36096	−0.289613
$647$	43.3567	1.70453	0.852263	−	0.523113i	$-0.175230\pi$
$647$	43.3567	1.70453	0.852263	+	0.523113i	$0.175230\pi$
$648$	79.0215	3.10426
$649$	−5.03821	−0.197767
$650$	0	0
$651$	−5.92699	−0.232297
$652$	0.760277	0.0297747
$653$	38.6146	1.51111	0.755553	−	0.655087i	$-0.227368\pi$
$653$	38.6146	1.51111	0.755553	+	0.655087i	$0.227368\pi$
$654$	15.1034	0.590590
$655$	0	0
$656$	21.5473	0.841279
$657$	60.6047	2.36441
$658$	−1.21311	−0.0472918
$659$	−50.9432	−1.98447	−0.992233	−	0.124397i	$-0.960300\pi$
$659$	−50.9432	−1.98447	−0.992233	+	0.124397i	$0.960300\pi$
$660$	0	0
$661$	39.6687	1.54293	0.771466	−	0.636270i	$-0.219524\pi$
$661$	39.6687	1.54293	0.771466	+	0.636270i	$0.219524\pi$
$662$	−26.2853	−1.02161
$663$	15.2718	0.593106
$664$	−16.8258	−0.652966
$665$	0	0
$666$	12.3856	0.479933
$667$	10.6841	0.413690
$668$	−3.64445	−0.141008
$669$	18.7284	0.724082
$670$	0	0
$671$	40.3612	1.55813
$672$	1.72807	0.0666619
$673$	18.3917	0.708949	0.354475	−	0.935066i	$-0.384660\pi$
$673$	18.3917	0.708949	0.354475	+	0.935066i	$0.384660\pi$
$674$	34.8602	1.34277
$675$	0	0
$676$	−2.41881	−0.0930310
$677$	22.9225	0.880983	0.440492	−	0.897757i	$-0.354804\pi$
$677$	22.9225	0.880983	0.440492	+	0.897757i	$0.354804\pi$
$678$	−18.1130	−0.695624
$679$	−6.04521	−0.231994
$680$	0	0
$681$	−69.3144	−2.65613
$682$	27.7139	1.06122
$683$	−29.8337	−1.14155	−0.570777	−	0.821105i	$-0.693358\pi$
$683$	−29.8337	−1.14155	−0.570777	+	0.821105i	$0.693358\pi$
$684$	1.04287	0.0398751
$685$	0	0
$686$	−9.85540	−0.376281
$687$	−54.4701	−2.07816
$688$	−36.3716	−1.38665
$689$	−2.90471	−0.110661
$690$	0	0
$691$	−6.55202	−0.249251	−0.124625	−	0.992204i	$-0.539773\pi$
$691$	−6.55202	−0.249251	−0.124625	+	0.992204i	$0.539773\pi$
$692$	−0.243489	−0.00925608
$693$	−19.2097	−0.729716
$694$	35.4462	1.34552
$695$	0	0
$696$	−68.9566	−2.61379
$697$	−35.7978	−1.35594
$698$	10.3211	0.390658
$699$	89.1561	3.37219
$700$	0	0
$701$	9.51866	0.359515	0.179757	−	0.983711i	$-0.442469\pi$
$701$	9.51866	0.359515	0.179757	+	0.983711i	$0.442469\pi$
$702$	−15.2941	−0.577238
$703$	−0.730262	−0.0275423
$704$	35.7254	1.34645
$705$	0	0
$706$	8.59817	0.323596
$707$	5.35479	0.201388
$708$	0.633900	0.0238234
$709$	44.5658	1.67370	0.836851	−	0.547431i	$-0.184394\pi$
$709$	44.5658	1.67370	0.836851	+	0.547431i	$0.184394\pi$
$710$	0	0
$711$	84.9604	3.18626
$712$	−12.0342	−0.451002
$713$	−5.08359	−0.190382
$714$	−17.0596	−0.638441
$715$	0	0
$716$	−1.41641	−0.0529336
$717$	19.2924	0.720489
$718$	−3.96006	−0.147788
$719$	25.7177	0.959110	0.479555	−	0.877512i	$-0.340798\pi$
$719$	25.7177	0.959110	0.479555	+	0.877512i	$0.340798\pi$
$720$	0	0
$721$	−3.24830	−0.120973
$722$	27.4296	1.02082
$723$	6.63333	0.246696
$724$	−0.0820629	−0.00304984
$725$	0	0
$726$	70.2617	2.60766
$727$	−24.4977	−0.908569	−0.454284	−	0.890857i	$-0.650105\pi$
$727$	−24.4977	−0.908569	−0.454284	+	0.890857i	$0.650105\pi$
$728$	0.829480	0.0307426
$729$	73.2557	2.71317
$730$	0	0
$731$	60.4265	2.23495
$732$	−5.07819	−0.187695
$733$	30.8105	1.13801	0.569007	−	0.822333i	$-0.307328\pi$
$733$	30.8105	1.13801	0.569007	+	0.822333i	$0.307328\pi$
$734$	−1.90894	−0.0704602
$735$	0	0
$736$	1.48217	0.0546336
$737$	−34.8915	−1.28524
$738$	57.8653	2.13005
$739$	39.6549	1.45873	0.729364	−	0.684126i	$-0.239816\pi$
$739$	39.6549	1.45873	0.729364	+	0.684126i	$0.239816\pi$
$740$	0	0
$741$	1.45550	0.0534690
$742$	3.24477	0.119119
$743$	−19.2677	−0.706864	−0.353432	−	0.935460i	$-0.614985\pi$
$743$	−19.2677	−0.706864	−0.353432	+	0.935460i	$0.614985\pi$
$744$	32.8101	1.20288
$745$	0	0
$746$	−6.19945	−0.226978
$747$	−49.5670	−1.81356
$748$	6.99143	0.255632
$749$	−5.27494	−0.192742
$750$	0	0
$751$	44.6414	1.62899	0.814495	−	0.580171i	$-0.197014\pi$
$751$	44.6414	1.62899	0.814495	+	0.580171i	$0.197014\pi$
$752$	7.36653	0.268630
$753$	26.3956	0.961911
$754$	7.40915	0.269825
$755$	0	0
$756$	1.49740	0.0544599
$757$	−15.0566	−0.547240	−0.273620	−	0.961838i	$-0.588221\pi$
$757$	−15.0566	−0.547240	−0.273620	+	0.961838i	$0.588221\pi$
$758$	−25.3923	−0.922292
$759$	−22.7446	−0.825577
$760$	0	0
$761$	−13.5614	−0.491601	−0.245800	−	0.969320i	$-0.579051\pi$
$761$	−13.5614	−0.491601	−0.245800	+	0.969320i	$0.579051\pi$
$762$	−85.3870	−3.09324
$763$	−1.49503	−0.0541236
$764$	3.71533	0.134416
$765$	0	0
$766$	23.5472	0.850793
$767$	0.640884	0.0231410
$768$	−15.0777	−0.544069
$769$	21.9904	0.792996	0.396498	−	0.918036i	$-0.370225\pi$
$769$	21.9904	0.792996	0.396498	+	0.918036i	$0.370225\pi$
$770$	0	0
$771$	−38.7931	−1.39710
$772$	−1.48457	−0.0534309
$773$	9.89721	0.355978	0.177989	−	0.984032i	$-0.443041\pi$
$773$	9.89721	0.355978	0.177989	+	0.984032i	$0.443041\pi$
$774$	−97.6762	−3.51090
$775$	0	0
$776$	33.4645	1.20131
$777$	−1.69244	−0.0607160
$778$	−26.2809	−0.942216
$779$	−3.41176	−0.122239
$780$	0	0
$781$	−24.3436	−0.871083
$782$	−14.6321	−0.523241
$783$	−125.854	−4.49765
$784$	29.4150	1.05054
$785$	0	0
$786$	−30.6937	−1.09481
$787$	31.2273	1.11313	0.556566	−	0.830803i	$-0.312119\pi$
$787$	31.2273	1.11313	0.556566	+	0.830803i	$0.312119\pi$
$788$	2.45422	0.0874279
$789$	40.3902	1.43793
$790$	0	0
$791$	1.79293	0.0637492
$792$	106.339	3.77860
$793$	−5.13414	−0.182318
$794$	−47.4142	−1.68267
$795$	0	0
$796$	0.568921	0.0201649
$797$	−43.9688	−1.55746	−0.778728	−	0.627361i	$-0.784135\pi$
$797$	−43.9688	−1.55746	−0.778728	+	0.627361i	$0.784135\pi$
$798$	−1.62589	−0.0575560
$799$	−12.2385	−0.432967
$800$	0	0
$801$	−35.4517	−1.25262
$802$	−32.4834	−1.14703
$803$	38.7226	1.36649
$804$	4.38999	0.154823
$805$	0	0
$806$	−3.52534	−0.124175
$807$	73.8470	2.59954
$808$	−29.6426	−1.04282
$809$	4.99256	0.175529	0.0877646	−	0.996141i	$-0.472028\pi$
$809$	4.99256	0.175529	0.0877646	+	0.996141i	$0.472028\pi$
$810$	0	0
$811$	−7.58004	−0.266171	−0.133086	−	0.991105i	$-0.542489\pi$
$811$	−7.58004	−0.266171	−0.133086	+	0.991105i	$0.542489\pi$
$812$	−0.725409	−0.0254569
$813$	−38.9508	−1.36606
$814$	7.91364	0.277373
$815$	0	0
$816$	103.594	3.62651
$817$	5.75903	0.201483
$818$	−34.5921	−1.20949
$819$	2.44356	0.0853851
$820$	0	0
$821$	−20.7577	−0.724449	−0.362224	−	0.932091i	$-0.617983\pi$
$821$	−20.7577	−0.724449	−0.362224	+	0.932091i	$0.617983\pi$
$822$	81.5312	2.84373
$823$	40.1469	1.39943	0.699717	−	0.714420i	$-0.253309\pi$
$823$	40.1469	1.39943	0.699717	+	0.714420i	$0.253309\pi$
$824$	17.9817	0.626421
$825$	0	0
$826$	−0.715913	−0.0249098
$827$	38.9942	1.35596	0.677981	−	0.735080i	$-0.262855\pi$
$827$	38.9942	1.35596	0.677981	+	0.735080i	$0.262855\pi$
$828$	2.07301	0.0720420
$829$	40.9609	1.42263	0.711315	−	0.702874i	$-0.248100\pi$
$829$	40.9609	1.42263	0.711315	+	0.702874i	$0.248100\pi$
$830$	0	0
$831$	15.6944	0.544433
$832$	−4.54445	−0.157550
$833$	−48.8691	−1.69321
$834$	−82.8438	−2.86864
$835$	0	0
$836$	0.666328	0.0230454
$837$	59.8823	2.06983
$838$	−26.1590	−0.903646
$839$	−47.3640	−1.63519	−0.817593	−	0.575796i	$-0.804692\pi$
$839$	−47.3640	−1.63519	−0.817593	+	0.575796i	$0.804692\pi$
$840$	0	0
$841$	31.9693	1.10239
$842$	−42.8588	−1.47701
$843$	19.2445	0.662815
$844$	3.17046	0.109132
$845$	0	0
$846$	19.7829	0.680149
$847$	−6.95492	−0.238974
$848$	−19.7037	−0.676629
$849$	9.35732	0.321142
$850$	0	0
$851$	−1.45161	−0.0497605
$852$	3.06288	0.104932
$853$	−45.0228	−1.54155	−0.770776	−	0.637107i	$-0.780131\pi$
$853$	−45.0228	−1.54155	−0.770776	+	0.637107i	$0.780131\pi$
$854$	5.73519	0.196254
$855$	0	0
$856$	29.2005	0.998054
$857$	−31.9289	−1.09067	−0.545335	−	0.838218i	$-0.683597\pi$
$857$	−31.9289	−1.09067	−0.545335	+	0.838218i	$0.683597\pi$
$858$	−15.7728	−0.538474
$859$	−36.3982	−1.24189	−0.620945	−	0.783854i	$-0.713251\pi$
$859$	−36.3982	−1.24189	−0.620945	+	0.783854i	$0.713251\pi$
$860$	0	0
$861$	−7.90704	−0.269471
$862$	−7.41672	−0.252614
$863$	53.4822	1.82055	0.910277	−	0.414000i	$-0.135869\pi$
$863$	53.4822	1.82055	0.910277	+	0.414000i	$0.135869\pi$
$864$	−17.4593	−0.593977
$865$	0	0
$866$	−28.3586	−0.963663
$867$	−116.019	−3.94023
$868$	0.345156	0.0117153
$869$	54.2844	1.84147
$870$	0	0
$871$	4.43836	0.150388
$872$	8.27603	0.280262
$873$	98.5831	3.33653
$874$	−1.39453	−0.0471707
$875$	0	0
$876$	−4.87202	−0.164610
$877$	9.30361	0.314161	0.157080	−	0.987586i	$-0.449792\pi$
$877$	9.30361	0.314161	0.157080	+	0.987586i	$0.449792\pi$
$878$	−20.6787	−0.697871
$879$	−56.2737	−1.89807
$880$	0	0
$881$	15.0561	0.507252	0.253626	−	0.967302i	$-0.418377\pi$
$881$	15.0561	0.507252	0.253626	+	0.967302i	$0.418377\pi$
$882$	78.9943	2.65988
$883$	9.35705	0.314890	0.157445	−	0.987528i	$-0.449674\pi$
$883$	9.35705	0.314890	0.157445	+	0.987528i	$0.449674\pi$
$884$	−0.889343	−0.0299119
$885$	0	0
$886$	−20.7573	−0.697354
$887$	−18.6720	−0.626943	−0.313472	−	0.949598i	$-0.601492\pi$
$887$	−18.6720	−0.626943	−0.313472	+	0.949598i	$0.601492\pi$
$888$	9.36886	0.314399
$889$	8.45210	0.283474
$890$	0	0
$891$	148.738	4.98290
$892$	−1.09064	−0.0365173
$893$	−1.16641	−0.0390323
$894$	96.7033	3.23424
$895$	0	0
$896$	6.12401	0.204589
$897$	2.89322	0.0966019
$898$	23.9131	0.797989
$899$	−29.0097	−0.967528
$900$	0	0
$901$	32.7351	1.09056
$902$	36.9723	1.23104
$903$	13.3470	0.444161
$904$	−9.92513	−0.330105
$905$	0	0
$906$	−60.0107	−1.99372
$907$	−7.29564	−0.242248	−0.121124	−	0.992637i	$-0.538650\pi$
$907$	−7.29564	−0.242248	−0.121124	+	0.992637i	$0.538650\pi$
$908$	4.03649	0.133956
$909$	−87.3240	−2.89635
$910$	0	0
$911$	−7.90399	−0.261871	−0.130936	−	0.991391i	$-0.541798\pi$
$911$	−7.90399	−0.261871	−0.130936	+	0.991391i	$0.541798\pi$
$912$	9.87316	0.326933
$913$	−31.6702	−1.04813
$914$	7.14527	0.236344
$915$	0	0
$916$	3.17204	0.104807
$917$	3.03825	0.100332
$918$	172.359	5.68869
$919$	−8.76451	−0.289114	−0.144557	−	0.989496i	$-0.546176\pi$
$919$	−8.76451	−0.289114	−0.144557	+	0.989496i	$0.546176\pi$
$920$	0	0
$921$	50.8980	1.67715
$922$	−17.9980	−0.592731
$923$	3.09662	0.101927
$924$	1.54427	0.0508028
$925$	0	0
$926$	−21.9549	−0.721483
$927$	52.9721	1.73983
$928$	8.45807	0.277650
$929$	−43.8310	−1.43805	−0.719024	−	0.694985i	$-0.755411\pi$
$929$	−43.8310	−1.43805	−0.719024	+	0.694985i	$0.755411\pi$
$930$	0	0
$931$	−4.65754	−0.152645
$932$	−5.19196	−0.170068
$933$	−80.7154	−2.64250
$934$	22.4758	0.735431
$935$	0	0
$936$	−13.5269	−0.442139
$937$	−8.61088	−0.281305	−0.140653	−	0.990059i	$-0.544920\pi$
$937$	−8.61088	−0.281305	−0.140653	+	0.990059i	$0.544920\pi$
$938$	−4.95796	−0.161883
$939$	57.9066	1.88971
$940$	0	0
$941$	−52.3119	−1.70532	−0.852659	−	0.522468i	$-0.825012\pi$
$941$	−52.3119	−1.70532	−0.852659	+	0.522468i	$0.825012\pi$
$942$	56.8575	1.85252
$943$	−6.78188	−0.220848
$944$	4.34735	0.141494
$945$	0	0
$946$	−62.4090	−2.02909
$947$	−7.80722	−0.253701	−0.126850	−	0.991922i	$-0.540487\pi$
$947$	−7.80722	−0.253701	−0.126850	+	0.991922i	$0.540487\pi$
$948$	−6.82998	−0.221827
$949$	−4.92570	−0.159895
$950$	0	0
$951$	−50.3410	−1.63242
$952$	−9.34795	−0.302969
$953$	10.3130	0.334071	0.167036	−	0.985951i	$-0.446581\pi$
$953$	10.3130	0.334071	0.167036	+	0.985951i	$0.446581\pi$
$954$	−52.9145	−1.71317
$955$	0	0
$956$	−1.12349	−0.0363361
$957$	−129.793	−4.19562
$958$	−56.2799	−1.81832
$959$	−8.07044	−0.260608
$960$	0	0
$961$	−17.1969	−0.554740
$962$	−1.00665	−0.0324558
$963$	86.0218	2.77201
$964$	−0.386289	−0.0124415
$965$	0	0
$966$	−3.23194	−0.103986
$967$	−40.8884	−1.31488	−0.657442	−	0.753505i	$-0.728362\pi$
$967$	−40.8884	−1.31488	−0.657442	+	0.753505i	$0.728362\pi$
$968$	38.5004	1.23745
$969$	−16.4029	−0.526938
$970$	0	0
$971$	−54.5534	−1.75070	−0.875352	−	0.483487i	$-0.839370\pi$
$971$	−54.5534	−1.75070	−0.875352	+	0.483487i	$0.839370\pi$
$972$	−9.42360	−0.302262
$973$	8.20036	0.262892
$974$	−27.7701	−0.889813
$975$	0	0
$976$	−34.8267	−1.11478
$977$	30.3067	0.969598	0.484799	−	0.874626i	$-0.338893\pi$
$977$	30.3067	0.969598	0.484799	+	0.874626i	$0.338893\pi$
$978$	19.3297	0.618095
$979$	−22.6514	−0.723941
$980$	0	0
$981$	24.3803	0.778404
$982$	−34.4949	−1.10078
$983$	0.723212	0.0230669	0.0115334	−	0.999933i	$-0.496329\pi$
$983$	0.723212	0.0230669	0.0115334	+	0.999933i	$0.496329\pi$
$984$	43.7711	1.39537
$985$	0	0
$986$	−83.4985	−2.65913
$987$	−2.70324	−0.0860452
$988$	−0.0847601	−0.00269658
$989$	11.4477	0.364017
$990$	0	0
$991$	−24.0051	−0.762549	−0.381274	−	0.924462i	$-0.624515\pi$
$991$	−24.0051	−0.762549	−0.381274	+	0.924462i	$0.624515\pi$
$992$	−4.02442	−0.127776
$993$	−58.5734	−1.85877
$994$	−3.45915	−0.109718
$995$	0	0
$996$	3.98470	0.126260
$997$	35.6472	1.12896	0.564479	−	0.825447i	$-0.309077\pi$
$997$	35.6472	1.12896	0.564479	+	0.825447i	$0.309077\pi$
$998$	21.8366	0.691226
$999$	17.0993	0.540997

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1475.2.a.r.1.5		14
5.2	odd	4		295.2.b.d.119.5	✓	14
5.3	odd	4		295.2.b.d.119.10	yes	14
5.4	even	2	inner	1475.2.a.r.1.10		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
295.2.b.d.119.5	✓	14	5.2	odd	4
295.2.b.d.119.10	yes	14	5.3	odd	4
1475.2.a.r.1.5		14	1.1	even	1	trivial
1475.2.a.r.1.10		14	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 1475 = 5^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1475.a (trivial)

\(f(q)\)	\(=\)	\(q-1.48059 q^{2} -3.29929 q^{3} +0.192132 q^{4} +4.88487 q^{6} -0.483534 q^{7} +2.67670 q^{8} +7.88529 q^{9} +O(q^{10})\)	\(q-1.48059 q^{2} -3.29929 q^{3} +0.192132 q^{4} +4.88487 q^{6} -0.483534 q^{7} +2.67670 q^{8} +7.88529 q^{9} +5.03821 q^{11} -0.633900 q^{12} -0.640884 q^{13} +0.715913 q^{14} -4.34735 q^{16} +7.22253 q^{17} -11.6748 q^{18} +0.688354 q^{19} +1.59532 q^{21} -7.45949 q^{22} +1.36830 q^{23} -8.83121 q^{24} +0.948883 q^{26} -16.1180 q^{27} -0.0929025 q^{28} +7.80828 q^{29} -3.71525 q^{31} +1.08322 q^{32} -16.6225 q^{33} -10.6936 q^{34} +1.51502 q^{36} -1.06088 q^{37} -1.01917 q^{38} +2.11446 q^{39} -4.95641 q^{41} -2.36200 q^{42} +8.36638 q^{43} +0.968003 q^{44} -2.02589 q^{46} -1.69449 q^{47} +14.3432 q^{48} -6.76620 q^{49} -23.8292 q^{51} -0.123135 q^{52} +4.53236 q^{53} +23.8640 q^{54} -1.29428 q^{56} -2.27108 q^{57} -11.5608 q^{58} -1.00000 q^{59} +8.01102 q^{61} +5.50074 q^{62} -3.81280 q^{63} +7.09090 q^{64} +24.6110 q^{66} -6.92537 q^{67} +1.38768 q^{68} -4.51443 q^{69} -4.83180 q^{71} +21.1066 q^{72} +7.68579 q^{73} +1.57073 q^{74} +0.132255 q^{76} -2.43614 q^{77} -3.13064 q^{78} +10.7745 q^{79} +29.5220 q^{81} +7.33839 q^{82} -6.28601 q^{83} +0.306512 q^{84} -12.3871 q^{86} -25.7618 q^{87} +13.4858 q^{88} -4.49592 q^{89} +0.309889 q^{91} +0.262895 q^{92} +12.2577 q^{93} +2.50883 q^{94} -3.57385 q^{96} +12.5022 q^{97} +10.0179 q^{98} +39.7277 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 24 q^{4} + 14 q^{6} + 14 q^{9}+O(q^{10}) \) 14 * q + 24 * q^4 + 14 * q^6 + 14 * q^9	\( 14 q + 24 q^{4} + 14 q^{6} + 14 q^{9} + 10 q^{11} + 34 q^{14} + 20 q^{16} + 24 q^{19} + 26 q^{21} - 20 q^{24} - 34 q^{26} + 46 q^{29} + 22 q^{31} + 10 q^{34} + 46 q^{36} - 14 q^{39} - 16 q^{41} + 50 q^{44} + 52 q^{46} + 8 q^{49} - 34 q^{51} + 110 q^{54} + 24 q^{56} - 14 q^{59} + 10 q^{61} + 66 q^{64} - 74 q^{66} - 52 q^{71} - 24 q^{74} - 2 q^{76} + 68 q^{79} + 38 q^{81} + 16 q^{84} - 12 q^{86} + 30 q^{89} + 48 q^{91} + 50 q^{94} - 112 q^{96} + 14 q^{99}+O(q^{100}) \) 14 * q + 24 * q^4 + 14 * q^6 + 14 * q^9 + 10 * q^11 + 34 * q^14 + 20 * q^16 + 24 * q^19 + 26 * q^21 - 20 * q^24 - 34 * q^26 + 46 * q^29 + 22 * q^31 + 10 * q^34 + 46 * q^36 - 14 * q^39 - 16 * q^41 + 50 * q^44 + 52 * q^46 + 8 * q^49 - 34 * q^51 + 110 * q^54 + 24 * q^56 - 14 * q^59 + 10 * q^61 + 66 * q^64 - 74 * q^66 - 52 * q^71 - 24 * q^74 - 2 * q^76 + 68 * q^79 + 38 * q^81 + 16 * q^84 - 12 * q^86 + 30 * q^89 + 48 * q^91 + 50 * q^94 - 112 * q^96 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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