LMFDB - Embedded newform 2175.2.a.v.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2175,2,Mod(1,2175)] 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("2175.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,3,4,5,0,3,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$17.3674624396$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2225.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.75660$ of defining polynomial
Character	$\chi$	$=$	2175.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.75660 q^{2} +1.00000 q^{3} +5.59883 q^{4} +2.75660 q^{6} -0.393832 q^{7} +9.92054 q^{8} +1.00000 q^{9} -0.393832 q^{11} +5.59883 q^{12} +2.56511 q^{13} -1.08564 q^{14} +16.1493 q^{16} -2.07830 q^{17} +2.75660 q^{18} -0.958939 q^{19} -0.393832 q^{21} -1.08564 q^{22} -6.15661 q^{23} +9.92054 q^{24} +7.07097 q^{26} +1.00000 q^{27} -2.20500 q^{28} -1.00000 q^{29} -10.1566 q^{31} +24.6760 q^{32} -0.393832 q^{33} -5.72905 q^{34} +5.59883 q^{36} +7.34192 q^{37} -2.64341 q^{38} +2.56511 q^{39} -1.65745 q^{41} -1.08564 q^{42} -10.3279 q^{43} -2.20500 q^{44} -16.9713 q^{46} +11.5915 q^{47} +16.1493 q^{48} -6.84490 q^{49} -2.07830 q^{51} +14.3616 q^{52} +12.3279 q^{53} +2.75660 q^{54} -3.90703 q^{56} -0.958939 q^{57} -2.75660 q^{58} -9.54022 q^{59} -6.25340 q^{61} -27.9977 q^{62} -0.393832 q^{63} +35.7232 q^{64} -1.08564 q^{66} +7.42023 q^{67} -11.6361 q^{68} -6.15661 q^{69} +5.98533 q^{71} +9.92054 q^{72} -3.34192 q^{73} +20.2387 q^{74} -5.36894 q^{76} +0.155104 q^{77} +7.07097 q^{78} -2.06745 q^{79} +1.00000 q^{81} -4.56892 q^{82} -6.41000 q^{83} -2.20500 q^{84} -28.4698 q^{86} -1.00000 q^{87} -3.90703 q^{88} +15.8302 q^{89} -1.01022 q^{91} -34.4698 q^{92} -10.1566 q^{93} +31.9531 q^{94} +24.6760 q^{96} +18.4575 q^{97} -18.8686 q^{98} -0.393832 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{2} + 4 q^{3} + 5 q^{4} + 3 q^{6} - 2 q^{7} + 12 q^{8} + 4 q^{9} - 2 q^{11} + 5 q^{12} + 8 q^{13} - 3 q^{14} + 11 q^{16} + 10 q^{17} + 3 q^{18} - 2 q^{19} - 2 q^{21} - 3 q^{22} + 12 q^{23} + 12 q^{24}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q + 3 * q^2 + 4 * q^3 + 5 * q^4 + 3 * q^6 - 2 * q^7 + 12 * q^8 + 4 * q^9 - 2 * q^11 + 5 * q^12 + 8 * q^13 - 3 * q^14 + 11 * q^16 + 10 * q^17 + 3 * q^18 - 2 * q^19 - 2 * q^21 - 3 * q^22 + 12 * q^23 + 12 * q^24 - 7 * q^26 + 4 * q^27 + 9 * q^28 - 4 * q^29 - 4 * q^31 + 17 * q^32 - 2 * q^33 - q^34 + 5 * q^36 + 16 * q^37 + 10 * q^38 + 8 * q^39 - 12 * q^41 - 3 * q^42 - 2 * q^43 + 9 * q^44 - 8 * q^46 + 12 * q^47 + 11 * q^48 + 6 * q^49 + 10 * q^51 + 3 * q^52 + 10 * q^53 + 3 * q^54 - 2 * q^57 - 3 * q^58 + 2 * q^59 - 26 * q^61 - 20 * q^62 - 2 * q^63 + 34 * q^64 - 3 * q^66 - 2 * q^67 - 9 * q^68 + 12 * q^69 - 10 * q^71 + 12 * q^72 + 48 * q^74 + 16 * q^76 + 34 * q^77 - 7 * q^78 + 22 * q^79 + 4 * q^81 - 38 * q^82 + 10 * q^83 + 9 * q^84 - 4 * q^86 - 4 * q^87 - 4 * q^89 - 8 * q^91 - 28 * q^92 - 4 * q^93 + 39 * q^94 + 17 * q^96 + 22 * q^97 - 34 * q^98 - 2 * q^99

Coefficient data

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

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

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

$2$	2.75660	1.94921	0.974605	−	0.223932i	$-0.0718895\pi$
$2$	2.75660	1.94921	0.974605	+	0.223932i	$0.0718895\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.59883	2.79942
$5$	0	0
$5$	0	0
$6$	2.75660	1.12538
$7$	−0.393832	−0.148855	−0.0744273	−	0.997226i	$-0.523713\pi$
$7$	−0.393832	−0.148855	−0.0744273	+	0.997226i	$0.523713\pi$
$8$	9.92054	3.50744
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.393832	−0.118745	−0.0593725	−	0.998236i	$-0.518910\pi$
$11$	−0.393832	−0.118745	−0.0593725	+	0.998236i	$0.518910\pi$
$12$	5.59883	1.61624
$13$	2.56511	0.711433	0.355716	−	0.934594i	$-0.384237\pi$
$13$	2.56511	0.711433	0.355716	+	0.934594i	$0.384237\pi$
$14$	−1.08564	−0.290149
$15$	0	0
$16$	16.1493	4.03732
$17$	−2.07830	−0.504063	−0.252031	−	0.967719i	$-0.581099\pi$
$17$	−2.07830	−0.504063	−0.252031	+	0.967719i	$0.581099\pi$
$18$	2.75660	0.649736
$19$	−0.958939	−0.219996	−0.109998	−	0.993932i	$-0.535084\pi$
$19$	−0.958939	−0.219996	−0.109998	+	0.993932i	$0.535084\pi$
$20$	0	0
$21$	−0.393832	−0.0859413
$22$	−1.08564	−0.231459
$23$	−6.15661	−1.28374	−0.641871	−	0.766813i	$-0.721841\pi$
$23$	−6.15661	−1.28374	−0.641871	+	0.766813i	$0.721841\pi$
$24$	9.92054	2.02502
$25$	0	0
$26$	7.07097	1.38673
$27$	1.00000	0.192450
$28$	−2.20500	−0.416706
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−10.1566	−1.82418	−0.912090	−	0.409989i	$-0.865532\pi$
$31$	−10.1566	−1.82418	−0.912090	+	0.409989i	$0.865532\pi$
$32$	24.6760	4.36214
$33$	−0.393832	−0.0685574
$34$	−5.72905	−0.982524
$35$	0	0
$36$	5.59883	0.933139
$37$	7.34192	1.20700	0.603502	−	0.797361i	$-0.293771\pi$
$37$	7.34192	1.20700	0.603502	+	0.797361i	$0.293771\pi$
$38$	−2.64341	−0.428818
$39$	2.56511	0.410746
$40$	0	0
$41$	−1.65745	−0.258850	−0.129425	−	0.991589i	$-0.541313\pi$
$41$	−1.65745	−0.258850	−0.129425	+	0.991589i	$0.541313\pi$
$42$	−1.08564	−0.167517
$43$	−10.3279	−1.57499	−0.787494	−	0.616323i	$-0.788622\pi$
$43$	−10.3279	−1.57499	−0.787494	+	0.616323i	$0.788622\pi$
$44$	−2.20500	−0.332417
$45$	0	0
$46$	−16.9713	−2.50228
$47$	11.5915	1.69079	0.845397	−	0.534138i	$-0.179364\pi$
$47$	11.5915	1.69079	0.845397	+	0.534138i	$0.179364\pi$
$48$	16.1493	2.33095
$49$	−6.84490	−0.977842
$50$	0	0
$51$	−2.07830	−0.291021
$52$	14.3616	1.99160
$53$	12.3279	1.69336	0.846682	−	0.532099i	$-0.178596\pi$
$53$	12.3279	1.69336	0.846682	+	0.532099i	$0.178596\pi$
$54$	2.75660	0.375126
$55$	0	0
$56$	−3.90703	−0.522099
$57$	−0.958939	−0.127015
$58$	−2.75660	−0.361959
$59$	−9.54022	−1.24203	−0.621015	−	0.783798i	$-0.713280\pi$
$59$	−9.54022	−1.24203	−0.621015	+	0.783798i	$0.713280\pi$
$60$	0	0
$61$	−6.25340	−0.800665	−0.400333	−	0.916370i	$-0.631105\pi$
$61$	−6.25340	−0.800665	−0.400333	+	0.916370i	$0.631105\pi$
$62$	−27.9977	−3.55571
$63$	−0.393832	−0.0496182
$64$	35.7232	4.46540
$65$	0	0
$66$	−1.08564	−0.133633
$67$	7.42023	0.906525	0.453262	−	0.891377i	$-0.350260\pi$
$67$	7.42023	0.906525	0.453262	+	0.891377i	$0.350260\pi$
$68$	−11.6361	−1.41108
$69$	−6.15661	−0.741168
$70$	0	0
$71$	5.98533	0.710328	0.355164	−	0.934804i	$-0.384425\pi$
$71$	5.98533	0.710328	0.355164	+	0.934804i	$0.384425\pi$
$72$	9.92054	1.16915
$73$	−3.34192	−0.391142	−0.195571	−	0.980690i	$-0.562656\pi$
$73$	−3.34192	−0.391142	−0.195571	+	0.980690i	$0.562656\pi$
$74$	20.2387	2.35270
$75$	0	0
$76$	−5.36894	−0.615860
$77$	0.155104	0.0176757
$78$	7.07097	0.800630
$79$	−2.06745	−0.232607	−0.116303	−	0.993214i	$-0.537104\pi$
$79$	−2.06745	−0.232607	−0.116303	+	0.993214i	$0.537104\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−4.56892	−0.504553
$83$	−6.41000	−0.703589	−0.351795	−	0.936077i	$-0.614428\pi$
$83$	−6.41000	−0.703589	−0.351795	+	0.936077i	$0.614428\pi$
$84$	−2.20500	−0.240585
$85$	0	0
$86$	−28.4698	−3.06998
$87$	−1.00000	−0.107211
$88$	−3.90703	−0.416491
$89$	15.8302	1.67800	0.839000	−	0.544131i	$-0.183140\pi$
$89$	15.8302	1.67800	0.839000	+	0.544131i	$0.183140\pi$
$90$	0	0
$91$	−1.01022	−0.105900
$92$	−34.4698	−3.59373
$93$	−10.1566	−1.05319
$94$	31.9531	3.29571
$95$	0	0
$96$	24.6760	2.51848
$97$	18.4575	1.87407	0.937036	−	0.349233i	$-0.113558\pi$
$97$	18.4575	1.87407	0.937036	+	0.349233i	$0.113558\pi$
$98$	−18.8686	−1.90602
$99$	−0.393832	−0.0395816
$100$	0	0
$101$	−12.8038	−1.27403	−0.637015	−	0.770852i	$-0.719831\pi$
$101$	−12.8038	−1.27403	−0.637015	+	0.770852i	$0.719831\pi$
$102$	−5.72905	−0.567260
$103$	−4.86979	−0.479834	−0.239917	−	0.970793i	$-0.577120\pi$
$103$	−4.86979	−0.479834	−0.239917	+	0.970793i	$0.577120\pi$
$104$	25.4472	2.49531
$105$	0	0
$106$	33.9830	3.30072
$107$	2.34255	0.226463	0.113231	−	0.993569i	$-0.463880\pi$
$107$	2.34255	0.226463	0.113231	+	0.993569i	$0.463880\pi$
$108$	5.59883	0.538748
$109$	8.55044	0.818984	0.409492	−	0.912314i	$-0.365706\pi$
$109$	8.55044	0.818984	0.409492	+	0.912314i	$0.365706\pi$
$110$	0	0
$111$	7.34192	0.696864
$112$	−6.36011	−0.600973
$113$	11.2085	1.05441	0.527204	−	0.849739i	$-0.323240\pi$
$113$	11.2085	1.05441	0.527204	+	0.849739i	$0.323240\pi$
$114$	−2.64341	−0.247578
$115$	0	0
$116$	−5.59883	−0.519839
$117$	2.56511	0.237144
$118$	−26.2985	−2.42098
$119$	0.818503	0.0750321
$120$	0	0
$121$	−10.8449	−0.985900
$122$	−17.2381	−1.56066
$123$	−1.65745	−0.149447
$124$	−56.8652	−5.10664
$125$	0	0
$126$	−1.08564	−0.0967163
$127$	−20.1566	−1.78861	−0.894305	−	0.447458i	$-0.852329\pi$
$127$	−20.1566	−1.78861	−0.894305	+	0.447458i	$0.852329\pi$
$128$	49.1226	4.34187
$129$	−10.3279	−0.909319
$130$	0	0
$131$	−5.50235	−0.480742	−0.240371	−	0.970681i	$-0.577269\pi$
$131$	−5.50235	−0.480742	−0.240371	+	0.970681i	$0.577269\pi$
$132$	−2.20500	−0.191921
$133$	0.377661	0.0327474
$134$	20.4546	1.76701
$135$	0	0
$136$	−20.6179	−1.76797
$137$	−7.49853	−0.640643	−0.320321	−	0.947309i	$-0.603791\pi$
$137$	−7.49853	−0.640643	−0.320321	+	0.947309i	$0.603791\pi$
$138$	−16.9713	−1.44469
$139$	−9.35277	−0.793292	−0.396646	−	0.917972i	$-0.629826\pi$
$139$	−9.35277	−0.793292	−0.396646	+	0.917972i	$0.629826\pi$
$140$	0	0
$141$	11.5915	0.976180
$142$	16.4992	1.38458
$143$	−1.01022	−0.0844790
$144$	16.1493	1.34577
$145$	0	0
$146$	−9.21234	−0.762418
$147$	−6.84490	−0.564558
$148$	41.1062	3.37891
$149$	−11.5402	−0.945411	−0.472706	−	0.881220i	$-0.656723\pi$
$149$	−11.5402	−0.945411	−0.472706	+	0.881220i	$0.656723\pi$
$150$	0	0
$151$	6.08212	0.494956	0.247478	−	0.968894i	$-0.420398\pi$
$151$	6.08212	0.494956	0.247478	+	0.968894i	$0.420398\pi$
$152$	−9.51320	−0.771622
$153$	−2.07830	−0.168021
$154$	0.427559	0.0344537
$155$	0	0
$156$	14.3616	1.14985
$157$	11.5953	0.925407	0.462704	−	0.886513i	$-0.346879\pi$
$157$	11.5953	0.925407	0.462704	+	0.886513i	$0.346879\pi$
$158$	−5.69914	−0.453399
$159$	12.3279	0.977665
$160$	0	0
$161$	2.42467	0.191091
$162$	2.75660	0.216579
$163$	0.855118	0.0669780	0.0334890	−	0.999439i	$-0.489338\pi$
$163$	0.855118	0.0669780	0.0334890	+	0.999439i	$0.489338\pi$
$164$	−9.27979	−0.724630
$165$	0	0
$166$	−17.6698	−1.37144
$167$	−7.73194	−0.598315	−0.299158	−	0.954204i	$-0.596706\pi$
$167$	−7.73194	−0.598315	−0.299158	+	0.954204i	$0.596706\pi$
$168$	−3.90703	−0.301434
$169$	−6.42023	−0.493863
$170$	0	0
$171$	−0.958939	−0.0733319
$172$	−57.8241	−4.40905
$173$	−8.07449	−0.613892	−0.306946	−	0.951727i	$-0.599307\pi$
$173$	−8.07449	−0.613892	−0.306946	+	0.951727i	$0.599307\pi$
$174$	−2.75660	−0.208977
$175$	0	0
$176$	−6.36011	−0.479411
$177$	−9.54022	−0.717087
$178$	43.6376	3.27077
$179$	12.4100	0.927567	0.463784	−	0.885949i	$-0.346492\pi$
$179$	12.4100	0.927567	0.463784	+	0.885949i	$0.346492\pi$
$180$	0	0
$181$	−2.49765	−0.185649	−0.0928246	−	0.995682i	$-0.529590\pi$
$181$	−2.49765	−0.185649	−0.0928246	+	0.995682i	$0.529590\pi$
$182$	−2.78478	−0.206421
$183$	−6.25340	−0.462264
$184$	−61.0769	−4.50265
$185$	0	0
$186$	−27.9977	−2.05289
$187$	0.818503	0.0598549
$188$	64.8989	4.73324
$189$	−0.393832	−0.0286471
$190$	0	0
$191$	−6.32085	−0.457361	−0.228680	−	0.973502i	$-0.573441\pi$
$191$	−6.32085	−0.457361	−0.228680	+	0.973502i	$0.573441\pi$
$192$	35.7232	2.57810
$193$	−25.2921	−1.82057	−0.910284	−	0.413984i	$-0.864137\pi$
$193$	−25.2921	−1.82057	−0.910284	+	0.413984i	$0.864137\pi$
$194$	50.8798	3.65296
$195$	0	0
$196$	−38.3234	−2.73739
$197$	21.2047	1.51077	0.755386	−	0.655280i	$-0.227449\pi$
$197$	21.2047	1.51077	0.755386	+	0.655280i	$0.227449\pi$
$198$	−1.08564	−0.0771529
$199$	−2.64723	−0.187657	−0.0938285	−	0.995588i	$-0.529911\pi$
$199$	−2.64723	−0.187657	−0.0938285	+	0.995588i	$0.529911\pi$
$200$	0	0
$201$	7.42023	0.523382
$202$	−35.2950	−2.48335
$203$	0.393832	0.0276416
$204$	−11.6361	−0.814688
$205$	0	0
$206$	−13.4240	−0.935297
$207$	−6.15661	−0.427914
$208$	41.4246	2.87228
$209$	0.377661	0.0261234
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	69.0218	4.74043
$213$	5.98533	0.410108
$214$	6.45747	0.441423
$215$	0	0
$216$	9.92054	0.675007
$217$	4.00000	0.271538
$218$	23.5701	1.59637
$219$	−3.34192	−0.225826
$220$	0	0
$221$	−5.33107	−0.358607
$222$	20.2387	1.35833
$223$	−12.5504	−0.840440	−0.420220	−	0.907422i	$-0.638047\pi$
$223$	−12.5504	−0.840440	−0.420220	+	0.907422i	$0.638047\pi$
$224$	−9.71820	−0.649324
$225$	0	0
$226$	30.8974	2.05526
$227$	−1.30149	−0.0863829	−0.0431914	−	0.999067i	$-0.513753\pi$
$227$	−1.30149	−0.0863829	−0.0431914	+	0.999067i	$0.513753\pi$
$228$	−5.36894	−0.355567
$229$	3.28682	0.217199	0.108600	−	0.994086i	$-0.465363\pi$
$229$	3.28682	0.217199	0.108600	+	0.994086i	$0.465363\pi$
$230$	0	0
$231$	0.155104	0.0102051
$232$	−9.92054	−0.651315
$233$	−17.7115	−1.16032	−0.580159	−	0.814503i	$-0.697010\pi$
$233$	−17.7115	−1.16032	−0.580159	+	0.814503i	$0.697010\pi$
$234$	7.07097	0.462244
$235$	0	0
$236$	−53.4141	−3.47696
$237$	−2.06745	−0.134296
$238$	2.25628	0.146253
$239$	21.2651	1.37553	0.687763	−	0.725935i	$-0.258593\pi$
$239$	21.2651	1.37553	0.687763	+	0.725935i	$0.258593\pi$
$240$	0	0
$241$	15.3177	0.986697	0.493349	−	0.869832i	$-0.335773\pi$
$241$	15.3177	0.986697	0.493349	+	0.869832i	$0.335773\pi$
$242$	−29.8950	−1.92172
$243$	1.00000	0.0641500
$244$	−35.0117	−2.24140
$245$	0	0
$246$	−4.56892	−0.291304
$247$	−2.45978	−0.156512
$248$	−100.759	−6.39820
$249$	−6.41000	−0.406217
$250$	0	0
$251$	26.8917	1.69739	0.848696	−	0.528882i	$-0.177388\pi$
$251$	26.8917	1.69739	0.848696	+	0.528882i	$0.177388\pi$
$252$	−2.20500	−0.138902
$253$	2.42467	0.152438
$254$	−55.5637	−3.48637
$255$	0	0
$256$	63.9648	3.99780
$257$	6.85512	0.427611	0.213805	−	0.976876i	$-0.431414\pi$
$257$	6.85512	0.427611	0.213805	+	0.976876i	$0.431414\pi$
$258$	−28.4698	−1.77245
$259$	−2.89149	−0.179668
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	−15.1678	−0.937067
$263$	13.6294	0.840423	0.420212	−	0.907426i	$-0.361956\pi$
$263$	13.6294	0.840423	0.420212	+	0.907426i	$0.361956\pi$
$264$	−3.90703	−0.240461
$265$	0	0
$266$	1.04106	0.0638315
$267$	15.8302	0.968794
$268$	41.5446	2.53774
$269$	−8.46129	−0.515894	−0.257947	−	0.966159i	$-0.583046\pi$
$269$	−8.46129	−0.515894	−0.257947	+	0.966159i	$0.583046\pi$
$270$	0	0
$271$	6.34255	0.385282	0.192641	−	0.981269i	$-0.438295\pi$
$271$	6.34255	0.385282	0.192641	+	0.981269i	$0.438295\pi$
$272$	−33.5631	−2.03506
$273$	−1.01022	−0.0611414
$274$	−20.6704	−1.24875
$275$	0	0
$276$	−34.4698	−2.07484
$277$	17.1464	1.03023	0.515113	−	0.857122i	$-0.327750\pi$
$277$	17.1464	1.03023	0.515113	+	0.857122i	$0.327750\pi$
$278$	−25.7818	−1.54629
$279$	−10.1566	−0.608060
$280$	0	0
$281$	−12.2985	−0.733670	−0.366835	−	0.930286i	$-0.619559\pi$
$281$	−12.2985	−0.733670	−0.366835	+	0.930286i	$0.619559\pi$
$282$	31.9531	1.90278
$283$	−25.1830	−1.49697	−0.748487	−	0.663149i	$-0.769219\pi$
$283$	−25.1830	−1.49697	−0.748487	+	0.663149i	$0.769219\pi$
$284$	33.5109	1.98851
$285$	0	0
$286$	−2.78478	−0.164667
$287$	0.652757	0.0385311
$288$	24.6760	1.45405
$289$	−12.6807	−0.745921
$290$	0	0
$291$	18.4575	1.08200
$292$	−18.7109	−1.09497
$293$	−12.3170	−0.719569	−0.359784	−	0.933035i	$-0.617150\pi$
$293$	−12.3170	−0.719569	−0.359784	+	0.933035i	$0.617150\pi$
$294$	−18.8686	−1.10044
$295$	0	0
$296$	72.8358	4.23350
$297$	−0.393832	−0.0228525
$298$	−31.8117	−1.84280
$299$	−15.7924	−0.913296
$300$	0	0
$301$	4.06745	0.234444
$302$	16.7660	0.964773
$303$	−12.8038	−0.735561
$304$	−15.4862	−0.888193
$305$	0	0
$306$	−5.72905	−0.327508
$307$	22.7379	1.29772	0.648860	−	0.760908i	$-0.275246\pi$
$307$	22.7379	1.29772	0.648860	+	0.760908i	$0.275246\pi$
$308$	0.868401	0.0494817
$309$	−4.86979	−0.277032
$310$	0	0
$311$	−5.33810	−0.302696	−0.151348	−	0.988481i	$-0.548361\pi$
$311$	−5.33810	−0.302696	−0.151348	+	0.988481i	$0.548361\pi$
$312$	25.4472	1.44067
$313$	1.96338	0.110977	0.0554885	−	0.998459i	$-0.482328\pi$
$313$	1.96338	0.110977	0.0554885	+	0.998459i	$0.482328\pi$
$314$	31.9636	1.80381
$315$	0	0
$316$	−11.5753	−0.651163
$317$	−1.29064	−0.0724895	−0.0362448	−	0.999343i	$-0.511540\pi$
$317$	−1.29064	−0.0724895	−0.0362448	+	0.999343i	$0.511540\pi$
$318$	33.9830	1.90567
$319$	0.393832	0.0220504
$320$	0	0
$321$	2.34255	0.130748
$322$	6.68384	0.372476
$323$	1.99297	0.110892
$324$	5.59883	0.311046
$325$	0	0
$326$	2.35722	0.130554
$327$	8.55044	0.472840
$328$	−16.4428	−0.907902
$329$	−4.56511	−0.251683
$330$	0	0
$331$	28.9971	1.59382	0.796911	−	0.604096i	$-0.206466\pi$
$331$	28.9971	1.59382	0.796911	+	0.604096i	$0.206466\pi$
$332$	−35.8885	−1.96964
$333$	7.34192	0.402335
$334$	−21.3138	−1.16624
$335$	0	0
$336$	−6.36011	−0.346972
$337$	16.7854	0.914356	0.457178	−	0.889375i	$-0.348860\pi$
$337$	16.7854	0.914356	0.457178	+	0.889375i	$0.348860\pi$
$338$	−17.6980	−0.962643
$339$	11.2085	0.608763
$340$	0	0
$341$	4.00000	0.216612
$342$	−2.64341	−0.142939
$343$	5.45257	0.294411
$344$	−102.458	−5.52417
$345$	0	0
$346$	−22.2581	−1.19660
$347$	17.8681	0.959210	0.479605	−	0.877485i	$-0.340780\pi$
$347$	17.8681	0.959210	0.479605	+	0.877485i	$0.340780\pi$
$348$	−5.59883	−0.300129
$349$	8.73789	0.467728	0.233864	−	0.972269i	$-0.424863\pi$
$349$	8.73789	0.467728	0.233864	+	0.972269i	$0.424863\pi$
$350$	0	0
$351$	2.56511	0.136915
$352$	−9.71820	−0.517982
$353$	22.6017	1.20297	0.601484	−	0.798885i	$-0.294576\pi$
$353$	22.6017	1.20297	0.601484	+	0.798885i	$0.294576\pi$
$354$	−26.2985	−1.39775
$355$	0	0
$356$	88.6308	4.69742
$357$	0.818503	0.0433198
$358$	34.2094	1.80802
$359$	−13.1830	−0.695772	−0.347886	−	0.937537i	$-0.613100\pi$
$359$	−13.1830	−0.695772	−0.347886	+	0.937537i	$0.613100\pi$
$360$	0	0
$361$	−18.0804	−0.951602
$362$	−6.88503	−0.361869
$363$	−10.8449	−0.569209
$364$	−5.65607	−0.296458
$365$	0	0
$366$	−17.2381	−0.901050
$367$	−17.4511	−0.910938	−0.455469	−	0.890252i	$-0.650528\pi$
$367$	−17.4511	−0.910938	−0.455469	+	0.890252i	$0.650528\pi$
$368$	−99.4247	−5.18287
$369$	−1.65745	−0.0862834
$370$	0	0
$371$	−4.85512	−0.252065
$372$	−56.8652	−2.94832
$373$	32.2018	1.66734	0.833672	−	0.552260i	$-0.186234\pi$
$373$	32.2018	1.66734	0.833672	+	0.552260i	$0.186234\pi$
$374$	2.25628	0.116670
$375$	0	0
$376$	114.994	5.93036
$377$	−2.56511	−0.132110
$378$	−1.08564	−0.0558392
$379$	32.3660	1.66253	0.831265	−	0.555876i	$-0.187617\pi$
$379$	32.3660	1.66253	0.831265	+	0.555876i	$0.187617\pi$
$380$	0	0
$381$	−20.1566	−1.03265
$382$	−17.4240	−0.891492
$383$	25.8739	1.32209	0.661047	−	0.750345i	$-0.270113\pi$
$383$	25.8739	1.32209	0.661047	+	0.750345i	$0.270113\pi$
$384$	49.1226	2.50678
$385$	0	0
$386$	−69.7203	−3.54867
$387$	−10.3279	−0.524996
$388$	103.340	5.24631
$389$	32.6103	1.65341	0.826703	−	0.562639i	$-0.190214\pi$
$389$	32.6103	1.65341	0.826703	+	0.562639i	$0.190214\pi$
$390$	0	0
$391$	12.7953	0.647086
$392$	−67.9051	−3.42972
$393$	−5.50235	−0.277557
$394$	58.4528	2.94481
$395$	0	0
$396$	−2.20500	−0.110806
$397$	−4.39534	−0.220596	−0.110298	−	0.993899i	$-0.535180\pi$
$397$	−4.39534	−0.220596	−0.110298	+	0.993899i	$0.535180\pi$
$398$	−7.29734	−0.365783
$399$	0.377661	0.0189067
$400$	0	0
$401$	−19.0658	−0.952099	−0.476049	−	0.879418i	$-0.657932\pi$
$401$	−19.0658	−0.952099	−0.476049	+	0.879418i	$0.657932\pi$
$402$	20.4546	1.02018
$403$	−26.0528	−1.29778
$404$	−71.6865	−3.56654
$405$	0	0
$406$	1.08564	0.0538793
$407$	−2.89149	−0.143326
$408$	−20.6179	−1.02074
$409$	−1.38235	−0.0683530	−0.0341765	−	0.999416i	$-0.510881\pi$
$409$	−1.38235	−0.0683530	−0.0341765	+	0.999416i	$0.510881\pi$
$410$	0	0
$411$	−7.49853	−0.369875
$412$	−27.2651	−1.34326
$413$	3.75725	0.184882
$414$	−16.9713	−0.834094
$415$	0	0
$416$	63.2965	3.10337
$417$	−9.35277	−0.458007
$418$	1.04106	0.0509199
$419$	2.86979	0.140198	0.0700991	−	0.997540i	$-0.477668\pi$
$419$	2.86979	0.140198	0.0700991	+	0.997540i	$0.477668\pi$
$420$	0	0
$421$	13.6440	0.664970	0.332485	−	0.943109i	$-0.392113\pi$
$421$	13.6440	0.664970	0.332485	+	0.943109i	$0.392113\pi$
$422$	5.51320	0.268378
$423$	11.5915	0.563598
$424$	122.299	5.93938
$425$	0	0
$426$	16.4992	0.799387
$427$	2.46279	0.119183
$428$	13.1155	0.633964
$429$	−1.01022	−0.0487740
$430$	0	0
$431$	−11.9707	−0.576607	−0.288303	−	0.957539i	$-0.593091\pi$
$431$	−11.9707	−0.576607	−0.288303	+	0.957539i	$0.593091\pi$
$432$	16.1493	0.776982
$433$	−17.9907	−0.864576	−0.432288	−	0.901736i	$-0.642294\pi$
$433$	−17.9907	−0.864576	−0.432288	+	0.901736i	$0.642294\pi$
$434$	11.0264	0.529284
$435$	0	0
$436$	47.8725	2.29268
$437$	5.90381	0.282418
$438$	−9.21234	−0.440182
$439$	−16.2226	−0.774260	−0.387130	−	0.922025i	$-0.626534\pi$
$439$	−16.2226	−0.774260	−0.387130	+	0.922025i	$0.626534\pi$
$440$	0	0
$441$	−6.84490	−0.325947
$442$	−14.6956	−0.698999
$443$	35.3762	1.68078	0.840388	−	0.541986i	$-0.182327\pi$
$443$	35.3762	1.68078	0.840388	+	0.541986i	$0.182327\pi$
$444$	41.1062	1.95081
$445$	0	0
$446$	−34.5965	−1.63819
$447$	−11.5402	−0.545834
$448$	−14.0690	−0.664696
$449$	41.6971	1.96781	0.983903	−	0.178702i	$-0.0571898\pi$
$449$	41.6971	1.96781	0.983903	+	0.178702i	$0.0571898\pi$
$450$	0	0
$451$	0.652757	0.0307371
$452$	62.7546	2.95173
$453$	6.08212	0.285763
$454$	−3.58768	−0.168378
$455$	0	0
$456$	−9.51320	−0.445496
$457$	16.1111	0.753646	0.376823	−	0.926285i	$-0.377017\pi$
$457$	16.1111	0.753646	0.376823	+	0.926285i	$0.377017\pi$
$458$	9.06045	0.423367
$459$	−2.07830	−0.0970069
$460$	0	0
$461$	17.3396	0.807586	0.403793	−	0.914850i	$-0.367692\pi$
$461$	17.3396	0.807586	0.403793	+	0.914850i	$0.367692\pi$
$462$	0.427559	0.0198918
$463$	−19.3177	−0.897768	−0.448884	−	0.893590i	$-0.648178\pi$
$463$	−19.3177	−0.897768	−0.448884	+	0.893590i	$0.648178\pi$
$464$	−16.1493	−0.749711
$465$	0	0
$466$	−48.8235	−2.26170
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	14.3616	0.663866
$469$	−2.92232	−0.134940
$470$	0	0
$471$	11.5953	0.534284
$472$	−94.6441	−4.35635
$473$	4.06745	0.187022
$474$	−5.69914	−0.261770
$475$	0	0
$476$	4.58266	0.210046
$477$	12.3279	0.564455
$478$	58.6194	2.68119
$479$	−40.3877	−1.84536	−0.922681	−	0.385565i	$-0.874006\pi$
$479$	−40.3877	−1.84536	−0.922681	+	0.385565i	$0.874006\pi$
$480$	0	0
$481$	18.8328	0.858702
$482$	42.2246	1.92328
$483$	2.42467	0.110326
$484$	−60.7188	−2.75994
$485$	0	0
$486$	2.75660	0.125042
$487$	−2.84171	−0.128770	−0.0643850	−	0.997925i	$-0.520509\pi$
$487$	−2.84171	−0.128770	−0.0643850	+	0.997925i	$0.520509\pi$
$488$	−62.0371	−2.80829
$489$	0.855118	0.0386698
$490$	0	0
$491$	0.157863	0.00712428	0.00356214	−	0.999994i	$-0.498866\pi$
$491$	0.157863	0.00712428	0.00356214	+	0.999994i	$0.498866\pi$
$492$	−9.27979	−0.418365
$493$	2.07830	0.0936021
$494$	−6.78063	−0.305075
$495$	0	0
$496$	−164.022	−7.36480
$497$	−2.35722	−0.105736
$498$	−17.6698	−0.791803
$499$	2.64723	0.118506	0.0592531	−	0.998243i	$-0.481128\pi$
$499$	2.64723	0.118506	0.0592531	+	0.998243i	$0.481128\pi$
$500$	0	0
$501$	−7.73194	−0.345437
$502$	74.1297	3.30857
$503$	−13.9545	−0.622200	−0.311100	−	0.950377i	$-0.600697\pi$
$503$	−13.9545	−0.622200	−0.311100	+	0.950377i	$0.600697\pi$
$504$	−3.90703	−0.174033
$505$	0	0
$506$	6.68384	0.297133
$507$	−6.42023	−0.285132
$508$	−112.853	−5.00706
$509$	−16.4921	−0.731001	−0.365500	−	0.930811i	$-0.619102\pi$
$509$	−16.4921	−0.731001	−0.365500	+	0.930811i	$0.619102\pi$
$510$	0	0
$511$	1.31616	0.0582233
$512$	78.0802	3.45069
$513$	−0.958939	−0.0423382
$514$	18.8968	0.833502
$515$	0	0
$516$	−57.8241	−2.54556
$517$	−4.56511	−0.200773
$518$	−7.97066	−0.350211
$519$	−8.07449	−0.354431
$520$	0	0
$521$	−12.2253	−0.535601	−0.267800	−	0.963474i	$-0.586297\pi$
$521$	−12.2253	−0.535601	−0.267800	+	0.963474i	$0.586297\pi$
$522$	−2.75660	−0.120653
$523$	−1.66659	−0.0728748	−0.0364374	−	0.999336i	$-0.511601\pi$
$523$	−1.66659	−0.0728748	−0.0364374	+	0.999336i	$0.511601\pi$
$524$	−30.8067	−1.34580
$525$	0	0
$526$	37.5707	1.63816
$527$	21.1085	0.919501
$528$	−6.36011	−0.276788
$529$	14.9038	0.647992
$530$	0	0
$531$	−9.54022	−0.414010
$532$	2.11446	0.0916736
$533$	−4.25154	−0.184155
$534$	43.6376	1.88838
$535$	0	0
$536$	73.6126	3.17958
$537$	12.4100	0.535531
$538$	−23.3244	−1.00558
$539$	2.69574	0.116114
$540$	0	0
$541$	−34.6558	−1.48997	−0.744984	−	0.667083i	$-0.767543\pi$
$541$	−34.6558	−1.48997	−0.744984	+	0.667083i	$0.767543\pi$
$542$	17.4839	0.750996
$543$	−2.49765	−0.107185
$544$	−51.2842	−2.19879
$545$	0	0
$546$	−2.78478	−0.119177
$547$	−38.2530	−1.63558	−0.817791	−	0.575515i	$-0.804801\pi$
$547$	−38.2530	−1.63558	−0.817791	+	0.575515i	$0.804801\pi$
$548$	−41.9830	−1.79343
$549$	−6.25340	−0.266888
$550$	0	0
$551$	0.958939	0.0408522
$552$	−61.0769	−2.59960
$553$	0.814230	0.0346246
$554$	47.2657	2.00813
$555$	0	0
$556$	−52.3646	−2.22075
$557$	28.7760	1.21928	0.609639	−	0.792679i	$-0.291314\pi$
$557$	28.7760	1.21928	0.609639	+	0.792679i	$0.291314\pi$
$558$	−27.9977	−1.18524
$559$	−26.4921	−1.12050
$560$	0	0
$561$	0.818503	0.0345572
$562$	−33.9022	−1.43008
$563$	32.9809	1.38998	0.694989	−	0.719020i	$-0.255409\pi$
$563$	32.9809	1.38998	0.694989	+	0.719020i	$0.255409\pi$
$564$	64.8989	2.73274
$565$	0	0
$566$	−69.4194	−2.91792
$567$	−0.393832	−0.0165394
$568$	59.3777	2.49143
$569$	16.8038	0.704453	0.352227	−	0.935915i	$-0.385425\pi$
$569$	16.8038	0.704453	0.352227	+	0.935915i	$0.385425\pi$
$570$	0	0
$571$	−6.21703	−0.260175	−0.130087	−	0.991503i	$-0.541526\pi$
$571$	−6.21703	−0.260175	−0.130087	+	0.991503i	$0.541526\pi$
$572$	−5.65607	−0.236492
$573$	−6.32085	−0.264057
$574$	1.79939	0.0751051
$575$	0	0
$576$	35.7232	1.48847
$577$	−11.9977	−0.499470	−0.249735	−	0.968314i	$-0.580344\pi$
$577$	−11.9977	−0.499470	−0.249735	+	0.968314i	$0.580344\pi$
$578$	−34.9555	−1.45396
$579$	−25.2921	−1.05111
$580$	0	0
$581$	2.52447	0.104733
$582$	50.8798	2.10904
$583$	−4.85512	−0.201078
$584$	−33.1537	−1.37191
$585$	0	0
$586$	−33.9531	−1.40259
$587$	11.2194	0.463073	0.231536	−	0.972826i	$-0.425625\pi$
$587$	11.2194	0.463073	0.231536	+	0.972826i	$0.425625\pi$
$588$	−38.3234	−1.58043
$589$	9.73957	0.401312
$590$	0	0
$591$	21.2047	0.872245
$592$	118.567	4.87306
$593$	7.69682	0.316071	0.158035	−	0.987433i	$-0.449484\pi$
$593$	7.69682	0.316071	0.158035	+	0.987433i	$0.449484\pi$
$594$	−1.08564	−0.0445442
$595$	0	0
$596$	−64.6118	−2.64660
$597$	−2.64723	−0.108344
$598$	−43.5332	−1.78020
$599$	20.0543	0.819396	0.409698	−	0.912221i	$-0.365634\pi$
$599$	20.0543	0.819396	0.409698	+	0.912221i	$0.365634\pi$
$600$	0	0
$601$	3.10851	0.126799	0.0633995	−	0.997988i	$-0.479806\pi$
$601$	3.10851	0.126799	0.0633995	+	0.997988i	$0.479806\pi$
$602$	11.2123	0.456981
$603$	7.42023	0.302175
$604$	34.0528	1.38559
$605$	0	0
$606$	−35.2950	−1.43376
$607$	−37.0441	−1.50357	−0.751786	−	0.659407i	$-0.770807\pi$
$607$	−37.0441	−1.50357	−0.751786	+	0.659407i	$0.770807\pi$
$608$	−23.6628	−0.959652
$609$	0.393832	0.0159589
$610$	0	0
$611$	29.7334	1.20289
$612$	−11.6361	−0.470361
$613$	13.3839	0.540569	0.270284	−	0.962781i	$-0.412882\pi$
$613$	13.3839	0.540569	0.270284	+	0.962781i	$0.412882\pi$
$614$	62.6792	2.52953
$615$	0	0
$616$	1.53871	0.0619966
$617$	12.6364	0.508721	0.254361	−	0.967109i	$-0.418135\pi$
$617$	12.6364	0.508721	0.254361	+	0.967109i	$0.418135\pi$
$618$	−13.4240	−0.539994
$619$	−41.2447	−1.65776	−0.828882	−	0.559424i	$-0.811022\pi$
$619$	−41.2447	−1.65776	−0.828882	+	0.559424i	$0.811022\pi$
$620$	0	0
$621$	−6.15661	−0.247056
$622$	−14.7150	−0.590018
$623$	−6.23446	−0.249778
$624$	41.4246	1.65831
$625$	0	0
$626$	5.41226	0.216318
$627$	0.377661	0.0150823
$628$	64.9203	2.59060
$629$	−15.2587	−0.608406
$630$	0	0
$631$	29.8009	1.18635	0.593177	−	0.805072i	$-0.297873\pi$
$631$	29.8009	1.18635	0.593177	+	0.805072i	$0.297873\pi$
$632$	−20.5103	−0.815854
$633$	2.00000	0.0794929
$634$	−3.55777	−0.141297
$635$	0	0
$636$	69.0218	2.73689
$637$	−17.5579	−0.695669
$638$	1.08564	0.0429808
$639$	5.98533	0.236776
$640$	0	0
$641$	17.7774	0.702167	0.351083	−	0.936344i	$-0.385813\pi$
$641$	17.7774	0.702167	0.351083	+	0.936344i	$0.385813\pi$
$642$	6.45747	0.254856
$643$	44.5534	1.75702	0.878508	−	0.477727i	$-0.158539\pi$
$643$	44.5534	1.75702	0.878508	+	0.477727i	$0.158539\pi$
$644$	13.5753	0.534943
$645$	0	0
$646$	5.49381	0.216151
$647$	−42.9339	−1.68790	−0.843952	−	0.536418i	$-0.819777\pi$
$647$	−42.9339	−1.68790	−0.843952	+	0.536418i	$0.819777\pi$
$648$	9.92054	0.389716
$649$	3.75725	0.147485
$650$	0	0
$651$	4.00000	0.156772
$652$	4.78766	0.187499
$653$	19.8426	0.776500	0.388250	−	0.921554i	$-0.373080\pi$
$653$	19.8426	0.776500	0.388250	+	0.921554i	$0.373080\pi$
$654$	23.5701	0.921665
$655$	0	0
$656$	−26.7666	−1.04506
$657$	−3.34192	−0.130381
$658$	−12.5842	−0.490582
$659$	23.0483	0.897836	0.448918	−	0.893573i	$-0.351810\pi$
$659$	23.0483	0.897836	0.448918	+	0.893573i	$0.351810\pi$
$660$	0	0
$661$	−36.2079	−1.40832	−0.704162	−	0.710040i	$-0.748677\pi$
$661$	−36.2079	−1.40832	−0.704162	+	0.710040i	$0.748677\pi$
$662$	79.9332	3.10669
$663$	−5.33107	−0.207042
$664$	−63.5907	−2.46780
$665$	0	0
$666$	20.2387	0.784235
$667$	6.15661	0.238385
$668$	−43.2898	−1.67493
$669$	−12.5504	−0.485228
$670$	0	0
$671$	2.46279	0.0950749
$672$	−9.71820	−0.374888
$673$	−22.9000	−0.882731	−0.441365	−	0.897327i	$-0.645506\pi$
$673$	−22.9000	−0.882731	−0.441365	+	0.897327i	$0.645506\pi$
$674$	46.2705	1.78227
$675$	0	0
$676$	−35.9458	−1.38253
$677$	−36.0068	−1.38385	−0.691927	−	0.721967i	$-0.743238\pi$
$677$	−36.0068	−1.38385	−0.691927	+	0.721967i	$0.743238\pi$
$678$	30.8974	1.18661
$679$	−7.26915	−0.278964
$680$	0	0
$681$	−1.30149	−0.0498732
$682$	11.0264	0.422222
$683$	−9.12896	−0.349310	−0.174655	−	0.984630i	$-0.555881\pi$
$683$	−9.12896	−0.349310	−0.174655	+	0.984630i	$0.555881\pi$
$684$	−5.36894	−0.205287
$685$	0	0
$686$	15.0305	0.573869
$687$	3.28682	0.125400
$688$	−166.788	−6.35873
$689$	31.6223	1.20472
$690$	0	0
$691$	5.85956	0.222908	0.111454	−	0.993770i	$-0.464449\pi$
$691$	5.85956	0.222908	0.111454	+	0.993770i	$0.464449\pi$
$692$	−45.2077	−1.71854
$693$	0.155104	0.00589191
$694$	49.2552	1.86970
$695$	0	0
$696$	−9.92054	−0.376037
$697$	3.44469	0.130477
$698$	24.0868	0.911700
$699$	−17.7115	−0.669910
$700$	0	0
$701$	−7.56955	−0.285898	−0.142949	−	0.989730i	$-0.545658\pi$
$701$	−7.56955	−0.285898	−0.142949	+	0.989730i	$0.545658\pi$
$702$	7.07097	0.266877
$703$	−7.04046	−0.265536
$704$	−14.0690	−0.530244
$705$	0	0
$706$	62.3039	2.34484
$707$	5.04256	0.189645
$708$	−53.4141	−2.00742
$709$	14.5209	0.545342	0.272671	−	0.962107i	$-0.412093\pi$
$709$	14.5209	0.545342	0.272671	+	0.962107i	$0.412093\pi$
$710$	0	0
$711$	−2.06745	−0.0775356
$712$	157.044	5.88549
$713$	62.5302	2.34178
$714$	2.25628	0.0844393
$715$	0	0
$716$	69.4815	2.59665
$717$	21.2651	0.794161
$718$	−36.3402	−1.35621
$719$	12.2164	0.455596	0.227798	−	0.973708i	$-0.426847\pi$
$719$	12.2164	0.455596	0.227798	+	0.973708i	$0.426847\pi$
$720$	0	0
$721$	1.91788	0.0714255
$722$	−49.8405	−1.85487
$723$	15.3177	0.569670
$724$	−13.9839	−0.519709
$725$	0	0
$726$	−29.8950	−1.10951
$727$	40.8856	1.51636	0.758182	−	0.652044i	$-0.226088\pi$
$727$	40.8856	1.51636	0.758182	+	0.652044i	$0.226088\pi$
$728$	−10.0219	−0.371438
$729$	1.00000	0.0370370
$730$	0	0
$731$	21.4645	0.793892
$732$	−35.0117	−1.29407
$733$	−25.4915	−0.941550	−0.470775	−	0.882253i	$-0.656026\pi$
$733$	−25.4915	−0.941550	−0.470775	+	0.882253i	$0.656026\pi$
$734$	−48.1056	−1.77561
$735$	0	0
$736$	−151.920	−5.59986
$737$	−2.92232	−0.107645
$738$	−4.56892	−0.168184
$739$	9.56830	0.351975	0.175988	−	0.984392i	$-0.443688\pi$
$739$	9.56830	0.351975	0.175988	+	0.984392i	$0.443688\pi$
$740$	0	0
$741$	−2.45978	−0.0903624
$742$	−13.3836	−0.491328
$743$	18.0455	0.662025	0.331013	−	0.943626i	$-0.392610\pi$
$743$	18.0455	0.662025	0.331013	+	0.943626i	$0.392610\pi$
$744$	−100.759	−3.69400
$745$	0	0
$746$	88.7673	3.25000
$747$	−6.41000	−0.234530
$748$	4.58266	0.167559
$749$	−0.922572	−0.0337100
$750$	0	0
$751$	11.9255	0.435168	0.217584	−	0.976042i	$-0.430182\pi$
$751$	11.9255	0.435168	0.217584	+	0.976042i	$0.430182\pi$
$752$	187.194	6.82627
$753$	26.8917	0.979989
$754$	−7.07097	−0.257510
$755$	0	0
$756$	−2.20500	−0.0801951
$757$	−5.86152	−0.213041	−0.106520	−	0.994311i	$-0.533971\pi$
$757$	−5.86152	−0.213041	−0.106520	+	0.994311i	$0.533971\pi$
$758$	89.2201	3.24062
$759$	2.42467	0.0880100
$760$	0	0
$761$	49.8739	1.80793	0.903963	−	0.427610i	$-0.140644\pi$
$761$	49.8739	1.80793	0.903963	+	0.427610i	$0.140644\pi$
$762$	−55.5637	−2.01286
$763$	−3.36744	−0.121909
$764$	−35.3894	−1.28034
$765$	0	0
$766$	71.3239	2.57704
$767$	−24.4717	−0.883621
$768$	63.9648	2.30813
$769$	29.5121	1.06423	0.532117	−	0.846671i	$-0.321396\pi$
$769$	29.5121	1.06423	0.532117	+	0.846671i	$0.321396\pi$
$770$	0	0
$771$	6.85512	0.246881
$772$	−141.607	−5.09653
$773$	2.41935	0.0870180	0.0435090	−	0.999053i	$-0.486146\pi$
$773$	2.41935	0.0870180	0.0435090	+	0.999053i	$0.486146\pi$
$774$	−28.4698	−1.02333
$775$	0	0
$776$	183.108	6.57320
$777$	−2.89149	−0.103731
$778$	89.8934	3.22283
$779$	1.58939	0.0569460
$780$	0	0
$781$	−2.35722	−0.0843479
$782$	35.2715	1.26131
$783$	−1.00000	−0.0357371
$784$	−110.540	−3.94786
$785$	0	0
$786$	−15.1678	−0.541016
$787$	51.1549	1.82348	0.911738	−	0.410772i	$-0.134741\pi$
$787$	51.1549	1.82348	0.911738	+	0.410772i	$0.134741\pi$
$788$	118.722	4.22928
$789$	13.6294	0.485218
$790$	0	0
$791$	−4.41428	−0.156954
$792$	−3.90703	−0.138830
$793$	−16.0406	−0.569619
$794$	−12.1162	−0.429987
$795$	0	0
$796$	−14.8214	−0.525330
$797$	−28.8662	−1.02249	−0.511247	−	0.859434i	$-0.670816\pi$
$797$	−28.8662	−1.02249	−0.511247	+	0.859434i	$0.670816\pi$
$798$	1.04106	0.0368531
$799$	−24.0907	−0.852266
$800$	0	0
$801$	15.8302	0.559334
$802$	−52.5567	−1.85584
$803$	1.31616	0.0464462
$804$	41.5446	1.46517
$805$	0	0
$806$	−71.8171	−2.52965
$807$	−8.46129	−0.297851
$808$	−127.021	−4.46858
$809$	19.0426	0.669501	0.334750	−	0.942307i	$-0.391348\pi$
$809$	19.0426	0.669501	0.334750	+	0.942307i	$0.391348\pi$
$810$	0	0
$811$	−20.8783	−0.733137	−0.366569	−	0.930391i	$-0.619467\pi$
$811$	−20.8783	−0.733137	−0.366569	+	0.930391i	$0.619467\pi$
$812$	2.20500	0.0773804
$813$	6.34255	0.222443
$814$	−7.97066	−0.279372
$815$	0	0
$816$	−33.5631	−1.17494
$817$	9.90381	0.346491
$818$	−3.81060	−0.133234
$819$	−1.01022	−0.0353000
$820$	0	0
$821$	3.60466	0.125804	0.0629018	−	0.998020i	$-0.479965\pi$
$821$	3.60466	0.125804	0.0629018	+	0.998020i	$0.479965\pi$
$822$	−20.6704	−0.720964
$823$	−27.2288	−0.949135	−0.474567	−	0.880219i	$-0.657395\pi$
$823$	−27.2288	−0.949135	−0.474567	+	0.880219i	$0.657395\pi$
$824$	−48.3109	−1.68299
$825$	0	0
$826$	10.3572	0.360374
$827$	−47.3936	−1.64804	−0.824019	−	0.566562i	$-0.808273\pi$
$827$	−47.3936	−1.64804	−0.824019	+	0.566562i	$0.808273\pi$
$828$	−34.4698	−1.19791
$829$	−41.8388	−1.45312	−0.726560	−	0.687103i	$-0.758882\pi$
$829$	−41.8388	−1.45312	−0.726560	+	0.687103i	$0.758882\pi$
$830$	0	0
$831$	17.1464	0.594802
$832$	91.6339	3.17683
$833$	14.2258	0.492894
$834$	−25.7818	−0.892752
$835$	0	0
$836$	2.11446	0.0731302
$837$	−10.1566	−0.351064
$838$	7.91085	0.273276
$839$	−31.6385	−1.09228	−0.546141	−	0.837693i	$-0.683904\pi$
$839$	−31.6385	−1.09228	−0.546141	+	0.837693i	$0.683904\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	37.6111	1.29617
$843$	−12.2985	−0.423584
$844$	11.1977	0.385440
$845$	0	0
$846$	31.9531	1.09857
$847$	4.27107	0.146756
$848$	199.086	6.83665
$849$	−25.1830	−0.864278
$850$	0	0
$851$	−45.2013	−1.54948
$852$	33.5109	1.14806
$853$	−44.2000	−1.51338	−0.756690	−	0.653773i	$-0.773185\pi$
$853$	−44.2000	−1.51338	−0.756690	+	0.653773i	$0.773185\pi$
$854$	6.78892	0.232312
$855$	0	0
$856$	23.2394	0.794305
$857$	−14.4775	−0.494541	−0.247270	−	0.968947i	$-0.579534\pi$
$857$	−14.4775	−0.494541	−0.247270	+	0.968947i	$0.579534\pi$
$858$	−2.78478	−0.0950707
$859$	22.5883	0.770703	0.385352	−	0.922770i	$-0.374080\pi$
$859$	22.5883	0.770703	0.385352	+	0.922770i	$0.374080\pi$
$860$	0	0
$861$	0.652757	0.0222459
$862$	−32.9983	−1.12393
$863$	22.9900	0.782590	0.391295	−	0.920265i	$-0.372027\pi$
$863$	22.9900	0.782590	0.391295	+	0.920265i	$0.372027\pi$
$864$	24.6760	0.839494
$865$	0	0
$866$	−49.5930	−1.68524
$867$	−12.6807	−0.430658
$868$	22.3953	0.760147
$869$	0.814230	0.0276209
$870$	0	0
$871$	19.0337	0.644931
$872$	84.8250	2.87254
$873$	18.4575	0.624691
$874$	16.2744	0.550491
$875$	0	0
$876$	−18.7109	−0.632182
$877$	−13.5741	−0.458364	−0.229182	−	0.973384i	$-0.573605\pi$
$877$	−13.5741	−0.458364	−0.229182	+	0.973384i	$0.573605\pi$
$878$	−44.7191	−1.50920
$879$	−12.3170	−0.415443
$880$	0	0
$881$	9.91235	0.333956	0.166978	−	0.985961i	$-0.446599\pi$
$881$	9.91235	0.333956	0.166978	+	0.985961i	$0.446599\pi$
$882$	−18.8686	−0.635340
$883$	−39.3192	−1.32320	−0.661598	−	0.749859i	$-0.730121\pi$
$883$	−39.3192	−1.32320	−0.661598	+	0.749859i	$0.730121\pi$
$884$	−29.8478	−1.00389
$885$	0	0
$886$	97.5180	3.27618
$887$	59.2520	1.98949	0.994743	−	0.102403i	$-0.0326531\pi$
$887$	59.2520	1.98949	0.994743	+	0.102403i	$0.0326531\pi$
$888$	72.8358	2.44421
$889$	7.93832	0.266243
$890$	0	0
$891$	−0.393832	−0.0131939
$892$	−70.2678	−2.35274
$893$	−11.1155	−0.371968
$894$	−31.8117	−1.06394
$895$	0	0
$896$	−19.3461	−0.646307
$897$	−15.7924	−0.527291
$898$	114.942	3.83567
$899$	10.1566	0.338742
$900$	0	0
$901$	−25.6211	−0.853562
$902$	1.79939	0.0599131
$903$	4.06745	0.135356
$904$	111.195	3.69828
$905$	0	0
$906$	16.7660	0.557012
$907$	5.91210	0.196308	0.0981541	−	0.995171i	$-0.468706\pi$
$907$	5.91210	0.196308	0.0981541	+	0.995171i	$0.468706\pi$
$908$	−7.28682	−0.241822
$909$	−12.8038	−0.424676
$910$	0	0
$911$	−12.4185	−0.411445	−0.205722	−	0.978610i	$-0.565954\pi$
$911$	−12.4185	−0.411445	−0.205722	+	0.978610i	$0.565954\pi$
$912$	−15.4862	−0.512799
$913$	2.52447	0.0835476
$914$	44.4118	1.46901
$915$	0	0
$916$	18.4024	0.608031
$917$	2.16700	0.0715607
$918$	−5.72905	−0.189087
$919$	16.4332	0.542081	0.271041	−	0.962568i	$-0.412632\pi$
$919$	16.4332	0.542081	0.271041	+	0.962568i	$0.412632\pi$
$920$	0	0
$921$	22.7379	0.749239
$922$	47.7983	1.57415
$923$	15.3530	0.505351
$924$	0.868401	0.0285683
$925$	0	0
$926$	−53.2510	−1.74994
$927$	−4.86979	−0.159945
$928$	−24.6760	−0.810029
$929$	13.8358	0.453936	0.226968	−	0.973902i	$-0.427119\pi$
$929$	13.8358	0.453936	0.226968	+	0.973902i	$0.427119\pi$
$930$	0	0
$931$	6.56384	0.215121
$932$	−99.1637	−3.24822
$933$	−5.33810	−0.174762
$934$	22.0528	0.721589
$935$	0	0
$936$	25.4472	0.831769
$937$	−20.7528	−0.677964	−0.338982	−	0.940793i	$-0.610083\pi$
$937$	−20.7528	−0.677964	−0.338982	+	0.940793i	$0.610083\pi$
$938$	−8.05567	−0.263027
$939$	1.96338	0.0640726
$940$	0	0
$941$	40.1666	1.30940	0.654698	−	0.755891i	$-0.272796\pi$
$941$	40.1666	1.30940	0.654698	+	0.755891i	$0.272796\pi$
$942$	31.9636	1.04143
$943$	10.2043	0.332297
$944$	−154.068	−5.01447
$945$	0	0
$946$	11.2123	0.364544
$947$	17.4144	0.565894	0.282947	−	0.959136i	$-0.408688\pi$
$947$	17.4144	0.565894	0.282947	+	0.959136i	$0.408688\pi$
$948$	−11.5753	−0.375949
$949$	−8.57239	−0.278271
$950$	0	0
$951$	−1.29064	−0.0418518
$952$	8.11999	0.263170
$953$	−13.5121	−0.437701	−0.218851	−	0.975758i	$-0.570231\pi$
$953$	−13.5121	−0.437701	−0.218851	+	0.975758i	$0.570231\pi$
$954$	33.9830	1.10024
$955$	0	0
$956$	119.060	3.85067
$957$	0.393832	0.0127308
$958$	−111.333	−3.59700
$959$	2.95316	0.0953626
$960$	0	0
$961$	72.1567	2.32763
$962$	51.9145	1.67379
$963$	2.34255	0.0754876
$964$	85.7610	2.76218
$965$	0	0
$966$	6.68384	0.215049
$967$	24.0598	0.773712	0.386856	−	0.922140i	$-0.373561\pi$
$967$	24.0598	0.773712	0.386856	+	0.922140i	$0.373561\pi$
$968$	−107.587	−3.45798
$969$	1.99297	0.0640233
$970$	0	0
$971$	34.7877	1.11639	0.558195	−	0.829710i	$-0.311494\pi$
$971$	34.7877	1.11639	0.558195	+	0.829710i	$0.311494\pi$
$972$	5.59883	0.179583
$973$	3.68342	0.118085
$974$	−7.83344	−0.251000
$975$	0	0
$976$	−100.988	−3.23254
$977$	−35.7566	−1.14396	−0.571978	−	0.820269i	$-0.693824\pi$
$977$	−35.7566	−1.14396	−0.571978	+	0.820269i	$0.693824\pi$
$978$	2.35722	0.0753755
$979$	−6.23446	−0.199254
$980$	0	0
$981$	8.55044	0.272995
$982$	0.435166	0.0138867
$983$	7.19064	0.229346	0.114673	−	0.993403i	$-0.463418\pi$
$983$	7.19064	0.229346	0.114673	+	0.993403i	$0.463418\pi$
$984$	−16.4428	−0.524177
$985$	0	0
$986$	5.72905	0.182450
$987$	−4.56511	−0.145309
$988$	−13.7719	−0.438143
$989$	63.5847	2.02188
$990$	0	0
$991$	−19.9123	−0.632537	−0.316268	−	0.948670i	$-0.602430\pi$
$991$	−19.9123	−0.632537	−0.316268	+	0.948670i	$0.602430\pi$
$992$	−250.624	−7.95733
$993$	28.9971	0.920194
$994$	−6.49790	−0.206101
$995$	0	0
$996$	−35.8885	−1.13717
$997$	1.57302	0.0498179	0.0249089	−	0.999690i	$-0.492070\pi$
$997$	1.57302	0.0498179	0.0249089	+	0.999690i	$0.492070\pi$
$998$	7.29734	0.230993
$999$	7.34192	0.232288

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.a.v.1.4		4
3.2	odd	2		6525.2.a.bi.1.1		4
5.2	odd	4		2175.2.c.n.349.8		8
5.3	odd	4		2175.2.c.n.349.1		8
5.4	even	2		435.2.a.j.1.1	✓	4
15.14	odd	2		1305.2.a.r.1.4		4
20.19	odd	2		6960.2.a.co.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.j.1.1	✓	4	5.4	even	2
1305.2.a.r.1.4		4	15.14	odd	2
2175.2.a.v.1.4		4	1.1	even	1	trivial
2175.2.c.n.349.1		8	5.3	odd	4
2175.2.c.n.349.8		8	5.2	odd	4
6525.2.a.bi.1.1		4	3.2	odd	2
6960.2.a.co.1.2		4	20.19	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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q+2.75660 q^{2} +1.00000 q^{3} +5.59883 q^{4} +2.75660 q^{6} -0.393832 q^{7} +9.92054 q^{8} +1.00000 q^{9} -0.393832 q^{11} +5.59883 q^{12} +2.56511 q^{13} -1.08564 q^{14} +16.1493 q^{16} -2.07830 q^{17} +2.75660 q^{18} -0.958939 q^{19} -0.393832 q^{21} -1.08564 q^{22} -6.15661 q^{23} +9.92054 q^{24} +7.07097 q^{26} +1.00000 q^{27} -2.20500 q^{28} -1.00000 q^{29} -10.1566 q^{31} +24.6760 q^{32} -0.393832 q^{33} -5.72905 q^{34} +5.59883 q^{36} +7.34192 q^{37} -2.64341 q^{38} +2.56511 q^{39} -1.65745 q^{41} -1.08564 q^{42} -10.3279 q^{43} -2.20500 q^{44} -16.9713 q^{46} +11.5915 q^{47} +16.1493 q^{48} -6.84490 q^{49} -2.07830 q^{51} +14.3616 q^{52} +12.3279 q^{53} +2.75660 q^{54} -3.90703 q^{56} -0.958939 q^{57} -2.75660 q^{58} -9.54022 q^{59} -6.25340 q^{61} -27.9977 q^{62} -0.393832 q^{63} +35.7232 q^{64} -1.08564 q^{66} +7.42023 q^{67} -11.6361 q^{68} -6.15661 q^{69} +5.98533 q^{71} +9.92054 q^{72} -3.34192 q^{73} +20.2387 q^{74} -5.36894 q^{76} +0.155104 q^{77} +7.07097 q^{78} -2.06745 q^{79} +1.00000 q^{81} -4.56892 q^{82} -6.41000 q^{83} -2.20500 q^{84} -28.4698 q^{86} -1.00000 q^{87} -3.90703 q^{88} +15.8302 q^{89} -1.01022 q^{91} -34.4698 q^{92} -10.1566 q^{93} +31.9531 q^{94} +24.6760 q^{96} +18.4575 q^{97} -18.8686 q^{98} -0.393832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{2} + 4 q^{3} + 5 q^{4} + 3 q^{6} - 2 q^{7} + 12 q^{8} + 4 q^{9} - 2 q^{11} + 5 q^{12} + 8 q^{13} - 3 q^{14} + 11 q^{16} + 10 q^{17} + 3 q^{18} - 2 q^{19} - 2 q^{21} - 3 q^{22} + 12 q^{23} + 12 q^{24}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q + 3 * q^2 + 4 * q^3 + 5 * q^4 + 3 * q^6 - 2 * q^7 + 12 * q^8 + 4 * q^9 - 2 * q^11 + 5 * q^12 + 8 * q^13 - 3 * q^14 + 11 * q^16 + 10 * q^17 + 3 * q^18 - 2 * q^19 - 2 * q^21 - 3 * q^22 + 12 * q^23 + 12 * q^24 - 7 * q^26 + 4 * q^27 + 9 * q^28 - 4 * q^29 - 4 * q^31 + 17 * q^32 - 2 * q^33 - q^34 + 5 * q^36 + 16 * q^37 + 10 * q^38 + 8 * q^39 - 12 * q^41 - 3 * q^42 - 2 * q^43 + 9 * q^44 - 8 * q^46 + 12 * q^47 + 11 * q^48 + 6 * q^49 + 10 * q^51 + 3 * q^52 + 10 * q^53 + 3 * q^54 - 2 * q^57 - 3 * q^58 + 2 * q^59 - 26 * q^61 - 20 * q^62 - 2 * q^63 + 34 * q^64 - 3 * q^66 - 2 * q^67 - 9 * q^68 + 12 * q^69 - 10 * q^71 + 12 * q^72 + 48 * q^74 + 16 * q^76 + 34 * q^77 - 7 * q^78 + 22 * q^79 + 4 * q^81 - 38 * q^82 + 10 * q^83 + 9 * q^84 - 4 * q^86 - 4 * q^87 - 4 * q^89 - 8 * q^91 - 28 * q^92 - 4 * q^93 + 39 * q^94 + 17 * q^96 + 22 * q^97 - 34 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more