LMFDB - Embedded newform 975.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(1,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("975.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.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:	$7.78541419707$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1821184.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} + 13x - 4 $ x^5 - 8x^3 + 13x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.26036$ of defining polynomial
Character	$\chi$	$=$	975.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.26036 q^{2} -1.00000 q^{3} +3.10922 q^{4} +2.26036 q^{6} -4.96953 q^{7} -2.50723 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.26036 q^{2} -1.00000 q^{3} +3.10922 q^{4} +2.26036 q^{6} -4.96953 q^{7} -2.50723 q^{8} +1.00000 q^{9} -3.21640 q^{11} -3.10922 q^{12} -1.00000 q^{13} +11.2329 q^{14} -0.551191 q^{16} +0.448809 q^{17} -2.26036 q^{18} -7.23291 q^{19} +4.96953 q^{21} +7.27022 q^{22} +6.26338 q^{23} +2.50723 q^{24} +2.26036 q^{26} -1.00000 q^{27} -15.4513 q^{28} -2.21844 q^{29} +2.19148 q^{31} +6.26036 q^{32} +3.21640 q^{33} -1.01447 q^{34} +3.10922 q^{36} -8.26338 q^{37} +16.3490 q^{38} +1.00000 q^{39} +1.79807 q^{41} -11.2329 q^{42} -6.21844 q^{43} -10.0005 q^{44} -14.1575 q^{46} -1.66521 q^{47} +0.551191 q^{48} +17.6962 q^{49} -0.448809 q^{51} -3.10922 q^{52} -1.16100 q^{53} +2.26036 q^{54} +12.4598 q^{56} +7.23291 q^{57} +5.01447 q^{58} -5.88365 q^{59} -1.76963 q^{61} -4.95352 q^{62} -4.96953 q^{63} -13.0483 q^{64} -7.27022 q^{66} +2.73916 q^{67} +1.39545 q^{68} -6.26338 q^{69} +4.11402 q^{71} -2.50723 q^{72} +10.4369 q^{73} +18.6782 q^{74} -22.4887 q^{76} +15.9840 q^{77} -2.26036 q^{78} +1.05336 q^{79} +1.00000 q^{81} -4.06428 q^{82} +5.77601 q^{83} +15.4513 q^{84} +14.0559 q^{86} +2.21844 q^{87} +8.06428 q^{88} +11.1326 q^{89} +4.96953 q^{91} +19.4742 q^{92} -2.19148 q^{93} +3.76398 q^{94} -6.26036 q^{96} +8.37418 q^{97} -39.9997 q^{98} -3.21640 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 6 q^{4} - 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + 6 * q^4 - 5 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} + 6 q^{4} - 5 q^{7} + 5 q^{9} + 5 q^{11} - 6 q^{12} - 5 q^{13} + 12 q^{14} + 5 q^{17} + 8 q^{19} + 5 q^{21} + 8 q^{22} + 7 q^{23} - 5 q^{27} - 14 q^{28} + 8 q^{29} + 12 q^{31} + 20 q^{32} - 5 q^{33} + 20 q^{34} + 6 q^{36} - 17 q^{37} + 24 q^{38} + 5 q^{39} + 5 q^{41} - 12 q^{42} - 12 q^{43} + 18 q^{44} - 12 q^{46} + 10 q^{47} + 22 q^{49} - 5 q^{51} - 6 q^{52} + 13 q^{53} - 8 q^{57} + 8 q^{59} + 13 q^{61} + 40 q^{62} - 5 q^{63} - 16 q^{64} - 8 q^{66} - 28 q^{67} + 14 q^{68} - 7 q^{69} + 5 q^{71} + 14 q^{73} + 12 q^{74} + 35 q^{77} + q^{79} + 5 q^{81} - 16 q^{82} + 6 q^{83} + 14 q^{84} - 8 q^{87} + 36 q^{88} + 19 q^{89} + 5 q^{91} + 10 q^{92} - 12 q^{93} - 12 q^{94} - 20 q^{96} + 13 q^{97} - 28 q^{98} + 5 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 6 * q^4 - 5 * q^7 + 5 * q^9 + 5 * q^11 - 6 * q^12 - 5 * q^13 + 12 * q^14 + 5 * q^17 + 8 * q^19 + 5 * q^21 + 8 * q^22 + 7 * q^23 - 5 * q^27 - 14 * q^28 + 8 * q^29 + 12 * q^31 + 20 * q^32 - 5 * q^33 + 20 * q^34 + 6 * q^36 - 17 * q^37 + 24 * q^38 + 5 * q^39 + 5 * q^41 - 12 * q^42 - 12 * q^43 + 18 * q^44 - 12 * q^46 + 10 * q^47 + 22 * q^49 - 5 * q^51 - 6 * q^52 + 13 * q^53 - 8 * q^57 + 8 * q^59 + 13 * q^61 + 40 * q^62 - 5 * q^63 - 16 * q^64 - 8 * q^66 - 28 * q^67 + 14 * q^68 - 7 * q^69 + 5 * q^71 + 14 * q^73 + 12 * q^74 + 35 * q^77 + q^79 + 5 * q^81 - 16 * q^82 + 6 * q^83 + 14 * q^84 - 8 * q^87 + 36 * q^88 + 19 * q^89 + 5 * q^91 + 10 * q^92 - 12 * q^93 - 12 * q^94 - 20 * q^96 + 13 * q^97 - 28 * q^98 + 5 * 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 $

$2$	−2.26036	−1.59831	−0.799157	−	0.601122i	$-0.794721\pi$
$2$	−2.26036	−1.59831	−0.799157	+	0.601122i	$0.794721\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.10922	1.55461
$5$	0	0
$5$	0	0
$6$	2.26036	0.922787
$7$	−4.96953	−1.87830	−0.939152	−	0.343501i	$-0.888387\pi$
$7$	−4.96953	−1.87830	−0.939152	+	0.343501i	$0.888387\pi$
$8$	−2.50723	−0.886441
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.21640	−0.969782	−0.484891	−	0.874575i	$-0.661141\pi$
$11$	−3.21640	−0.969782	−0.484891	+	0.874575i	$0.661141\pi$
$12$	−3.10922	−0.897555
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	11.2329	3.00212
$15$	0	0
$16$	−0.551191	−0.137798
$17$	0.448809	0.108852	0.0544261	−	0.998518i	$-0.482667\pi$
$17$	0.448809	0.108852	0.0544261	+	0.998518i	$0.482667\pi$
$18$	−2.26036	−0.532772
$19$	−7.23291	−1.65934	−0.829672	−	0.558252i	$-0.811472\pi$
$19$	−7.23291	−1.65934	−0.829672	+	0.558252i	$0.811472\pi$
$20$	0	0
$21$	4.96953	1.08444
$22$	7.27022	1.55002
$23$	6.26338	1.30601	0.653003	−	0.757355i	$-0.273509\pi$
$23$	6.26338	1.30601	0.653003	+	0.757355i	$0.273509\pi$
$24$	2.50723	0.511787
$25$	0	0
$26$	2.26036	0.443293
$27$	−1.00000	−0.192450
$28$	−15.4513	−2.92003
$29$	−2.21844	−0.411954	−0.205977	−	0.978557i	$-0.566037\pi$
$29$	−2.21844	−0.411954	−0.205977	+	0.978557i	$0.566037\pi$
$30$	0	0
$31$	2.19148	0.393601	0.196800	−	0.980444i	$-0.436945\pi$
$31$	2.19148	0.393601	0.196800	+	0.980444i	$0.436945\pi$
$32$	6.26036	1.10669
$33$	3.21640	0.559904
$34$	−1.01447	−0.173980
$35$	0	0
$36$	3.10922	0.518203
$37$	−8.26338	−1.35849	−0.679246	−	0.733911i	$-0.737693\pi$
$37$	−8.26338	−1.35849	−0.679246	+	0.733911i	$0.737693\pi$
$38$	16.3490	2.65215
$39$	1.00000	0.160128
$40$	0	0
$41$	1.79807	0.280811	0.140405	−	0.990094i	$-0.455159\pi$
$41$	1.79807	0.280811	0.140405	+	0.990094i	$0.455159\pi$
$42$	−11.2329	−1.73328
$43$	−6.21844	−0.948303	−0.474152	−	0.880443i	$-0.657245\pi$
$43$	−6.21844	−0.948303	−0.474152	+	0.880443i	$0.657245\pi$
$44$	−10.0005	−1.50763
$45$	0	0
$46$	−14.1575	−2.08741
$47$	−1.66521	−0.242896	−0.121448	−	0.992598i	$-0.538754\pi$
$47$	−1.66521	−0.242896	−0.121448	+	0.992598i	$0.538754\pi$
$48$	0.551191	0.0795575
$49$	17.6962	2.52803
$50$	0	0
$51$	−0.448809	−0.0628459
$52$	−3.10922	−0.431171
$53$	−1.16100	−0.159476	−0.0797380	−	0.996816i	$-0.525408\pi$
$53$	−1.16100	−0.159476	−0.0797380	+	0.996816i	$0.525408\pi$
$54$	2.26036	0.307596
$55$	0	0
$56$	12.4598	1.66501
$57$	7.23291	0.958022
$58$	5.01447	0.658432
$59$	−5.88365	−0.765986	−0.382993	−	0.923751i	$-0.625107\pi$
$59$	−5.88365	−0.765986	−0.382993	+	0.923751i	$0.625107\pi$
$60$	0	0
$61$	−1.76963	−0.226578	−0.113289	−	0.993562i	$-0.536139\pi$
$61$	−1.76963	−0.226578	−0.113289	+	0.993562i	$0.536139\pi$
$62$	−4.95352	−0.629098
$63$	−4.96953	−0.626101
$64$	−13.0483	−1.63103
$65$	0	0
$66$	−7.27022	−0.894903
$67$	2.73916	0.334641	0.167321	−	0.985903i	$-0.446488\pi$
$67$	2.73916	0.334641	0.167321	+	0.985903i	$0.446488\pi$
$68$	1.39545	0.169223
$69$	−6.26338	−0.754023
$70$	0	0
$71$	4.11402	0.488244	0.244122	−	0.969744i	$-0.421500\pi$
$71$	4.11402	0.488244	0.244122	+	0.969744i	$0.421500\pi$
$72$	−2.50723	−0.295480
$73$	10.4369	1.22154	0.610772	−	0.791806i	$-0.290859\pi$
$73$	10.4369	1.22154	0.610772	+	0.791806i	$0.290859\pi$
$74$	18.6782	2.17130
$75$	0	0
$76$	−22.4887	−2.57963
$77$	15.9840	1.82155
$78$	−2.26036	−0.255935
$79$	1.05336	0.118513	0.0592563	−	0.998243i	$-0.481127\pi$
$79$	1.05336	0.118513	0.0592563	+	0.998243i	$0.481127\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−4.06428	−0.448824
$83$	5.77601	0.634000	0.317000	−	0.948426i	$-0.397325\pi$
$83$	5.77601	0.634000	0.317000	+	0.948426i	$0.397325\pi$
$84$	15.4513	1.68588
$85$	0	0
$86$	14.0559	1.51569
$87$	2.21844	0.237842
$88$	8.06428	0.859655
$89$	11.1326	1.18005	0.590025	−	0.807385i	$-0.299118\pi$
$89$	11.1326	1.18005	0.590025	+	0.807385i	$0.299118\pi$
$90$	0	0
$91$	4.96953	0.520948
$92$	19.4742	2.03033
$93$	−2.19148	−0.227246
$94$	3.76398	0.388224
$95$	0	0
$96$	−6.26036	−0.638945
$97$	8.37418	0.850270	0.425135	−	0.905130i	$-0.360227\pi$
$97$	8.37418	0.850270	0.425135	+	0.905130i	$0.360227\pi$
$98$	−39.9997	−4.04058
$99$	−3.21640	−0.323261
$100$	0	0
$101$	5.50625	0.547892	0.273946	−	0.961745i	$-0.411671\pi$
$101$	5.50625	0.547892	0.273946	+	0.961745i	$0.411671\pi$
$102$	1.01447	0.100447
$103$	1.32082	0.130144	0.0650722	−	0.997881i	$-0.479272\pi$
$103$	1.32082	0.130144	0.0650722	+	0.997881i	$0.479272\pi$
$104$	2.50723	0.245855
$105$	0	0
$106$	2.62428	0.254893
$107$	7.49024	0.724109	0.362055	−	0.932157i	$-0.382075\pi$
$107$	7.49024	0.724109	0.362055	+	0.932157i	$0.382075\pi$
$108$	−3.10922	−0.299185
$109$	14.0330	1.34412	0.672060	−	0.740497i	$-0.265410\pi$
$109$	14.0330	1.34412	0.672060	+	0.740497i	$0.265410\pi$
$110$	0	0
$111$	8.26338	0.784326
$112$	2.73916	0.258826
$113$	11.2269	1.05613	0.528067	−	0.849203i	$-0.322917\pi$
$113$	11.2269	1.05613	0.528067	+	0.849203i	$0.322917\pi$
$114$	−16.3490	−1.53122
$115$	0	0
$116$	−6.89762	−0.640428
$117$	−1.00000	−0.0924500
$118$	13.2992	1.22429
$119$	−2.23037	−0.204458
$120$	0	0
$121$	−0.654755	−0.0595232
$122$	4.00000	0.362143
$123$	−1.79807	−0.162126
$124$	6.81378	0.611896
$125$	0	0
$126$	11.2329	1.00071
$127$	1.50217	0.133296	0.0666481	−	0.997777i	$-0.478770\pi$
$127$	1.50217	0.133296	0.0666481	+	0.997777i	$0.478770\pi$
$128$	16.9731	1.50022
$129$	6.21844	0.547503
$130$	0	0
$131$	12.5477	1.09630	0.548148	−	0.836381i	$-0.315333\pi$
$131$	12.5477	1.09630	0.548148	+	0.836381i	$0.315333\pi$
$132$	10.0005	0.870432
$133$	35.9441	3.11675
$134$	−6.19148	−0.534862
$135$	0	0
$136$	−1.12527	−0.0964911
$137$	10.6797	0.912427	0.456213	−	0.889870i	$-0.349205\pi$
$137$	10.6797	0.912427	0.456213	+	0.889870i	$0.349205\pi$
$138$	14.1575	1.20517
$139$	−20.6962	−1.75543	−0.877714	−	0.479185i	$-0.840932\pi$
$139$	−20.6962	−1.75543	−0.877714	+	0.479185i	$0.840932\pi$
$140$	0	0
$141$	1.66521	0.140236
$142$	−9.29916	−0.780368
$143$	3.21640	0.268969
$144$	−0.551191	−0.0459326
$145$	0	0
$146$	−23.5911	−1.95241
$147$	−17.6962	−1.45956
$148$	−25.6927	−2.11193
$149$	−10.3457	−0.847557	−0.423778	−	0.905766i	$-0.639296\pi$
$149$	−10.3457	−0.847557	−0.423778	+	0.905766i	$0.639296\pi$
$150$	0	0
$151$	9.34779	0.760712	0.380356	−	0.924840i	$-0.375801\pi$
$151$	9.34779	0.760712	0.380356	+	0.924840i	$0.375801\pi$
$152$	18.1346	1.47091
$153$	0.448809	0.0362841
$154$	−36.1296	−2.91140
$155$	0	0
$156$	3.10922	0.248937
$157$	−0.987506	−0.0788115	−0.0394058	−	0.999223i	$-0.512546\pi$
$157$	−0.987506	−0.0788115	−0.0394058	+	0.999223i	$0.512546\pi$
$158$	−2.38098	−0.189420
$159$	1.16100	0.0920735
$160$	0	0
$161$	−31.1260	−2.45308
$162$	−2.26036	−0.177591
$163$	−11.0635	−0.866559	−0.433280	−	0.901260i	$-0.642644\pi$
$163$	−11.0635	−0.866559	−0.433280	+	0.901260i	$0.642644\pi$
$164$	5.59059	0.436551
$165$	0	0
$166$	−13.0559	−1.01333
$167$	−19.6373	−1.51958	−0.759789	−	0.650170i	$-0.774698\pi$
$167$	−19.6373	−1.51958	−0.759789	+	0.650170i	$0.774698\pi$
$168$	−12.4598	−0.961292
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.23291	−0.553114
$172$	−19.3345	−1.47424
$173$	−15.4783	−1.17679	−0.588397	−	0.808572i	$-0.700241\pi$
$173$	−15.4783	−1.17679	−0.588397	+	0.808572i	$0.700241\pi$
$174$	−5.01447	−0.380146
$175$	0	0
$176$	1.77285	0.133634
$177$	5.88365	0.442242
$178$	−25.1636	−1.88609
$179$	−3.10238	−0.231883	−0.115941	−	0.993256i	$-0.536988\pi$
$179$	−3.10238	−0.231883	−0.115941	+	0.993256i	$0.536988\pi$
$180$	0	0
$181$	12.8111	0.952239	0.476119	−	0.879381i	$-0.342043\pi$
$181$	12.8111	0.952239	0.476119	+	0.879381i	$0.342043\pi$
$182$	−11.2329	−0.832639
$183$	1.76963	0.130815
$184$	−15.7038	−1.15770
$185$	0	0
$186$	4.95352	0.363210
$187$	−1.44355	−0.105563
$188$	−5.17751	−0.377609
$189$	4.96953	0.361480
$190$	0	0
$191$	15.1780	1.09824	0.549121	−	0.835743i	$-0.314963\pi$
$191$	15.1780	1.09824	0.549121	+	0.835743i	$0.314963\pi$
$192$	13.0483	0.941678
$193$	0.173496	0.0124885	0.00624427	−	0.999981i	$-0.498012\pi$
$193$	0.173496	0.0124885	0.00624427	+	0.999981i	$0.498012\pi$
$194$	−18.9287	−1.35900
$195$	0	0
$196$	55.0213	3.93010
$197$	4.27989	0.304930	0.152465	−	0.988309i	$-0.451279\pi$
$197$	4.27989	0.304930	0.152465	+	0.988309i	$0.451279\pi$
$198$	7.27022	0.516672
$199$	16.6843	1.18272	0.591358	−	0.806409i	$-0.298592\pi$
$199$	16.6843	1.18272	0.591358	+	0.806409i	$0.298592\pi$
$200$	0	0
$201$	−2.73916	−0.193205
$202$	−12.4461	−0.875704
$203$	11.0246	0.773775
$204$	−1.39545	−0.0977008
$205$	0	0
$206$	−2.98553	−0.208012
$207$	6.26338	0.435335
$208$	0.551191	0.0382182
$209$	23.2639	1.60920
$210$	0	0
$211$	−24.3013	−1.67297	−0.836485	−	0.547989i	$-0.815394\pi$
$211$	−24.3013	−1.67297	−0.836485	+	0.547989i	$0.815394\pi$
$212$	−3.60981	−0.247923
$213$	−4.11402	−0.281888
$214$	−16.9306	−1.15735
$215$	0	0
$216$	2.50723	0.170596
$217$	−10.8906	−0.739302
$218$	−31.7196	−2.14833
$219$	−10.4369	−0.705259
$220$	0	0
$221$	−0.448809	−0.0301902
$222$	−18.6782	−1.25360
$223$	−0.821018	−0.0549794	−0.0274897	−	0.999622i	$-0.508751\pi$
$223$	−0.821018	−0.0549794	−0.0274897	+	0.999622i	$0.508751\pi$
$224$	−31.1110	−2.07869
$225$	0	0
$226$	−25.3767	−1.68803
$227$	−6.24591	−0.414555	−0.207278	−	0.978282i	$-0.566460\pi$
$227$	−6.24591	−0.414555	−0.207278	+	0.978282i	$0.566460\pi$
$228$	22.4887	1.48935
$229$	20.5766	1.35974	0.679871	−	0.733332i	$-0.262036\pi$
$229$	20.5766	1.35974	0.679871	+	0.733332i	$0.262036\pi$
$230$	0	0
$231$	−15.9840	−1.05167
$232$	5.56215	0.365173
$233$	−5.19402	−0.340271	−0.170136	−	0.985421i	$-0.554421\pi$
$233$	−5.19402	−0.340271	−0.170136	+	0.985421i	$0.554421\pi$
$234$	2.26036	0.147764
$235$	0	0
$236$	−18.2936	−1.19081
$237$	−1.05336	−0.0684233
$238$	5.04143	0.326788
$239$	−12.3229	−0.797100	−0.398550	−	0.917147i	$-0.630486\pi$
$239$	−12.3229	−0.797100	−0.398550	+	0.917147i	$0.630486\pi$
$240$	0	0
$241$	−2.41597	−0.155626	−0.0778131	−	0.996968i	$-0.524794\pi$
$241$	−2.41597	−0.155626	−0.0778131	+	0.996968i	$0.524794\pi$
$242$	1.47998	0.0951368
$243$	−1.00000	−0.0641500
$244$	−5.50217	−0.352240
$245$	0	0
$246$	4.06428	0.259129
$247$	7.23291	0.460219
$248$	−5.49455	−0.348904
$249$	−5.77601	−0.366040
$250$	0	0
$251$	21.6428	1.36608	0.683042	−	0.730380i	$-0.260657\pi$
$251$	21.6428	1.36608	0.683042	+	0.730380i	$0.260657\pi$
$252$	−15.4513	−0.973344
$253$	−20.1456	−1.26654
$254$	−3.39545	−0.213049
$255$	0	0
$256$	−12.2686	−0.766790
$257$	−16.9027	−1.05436	−0.527181	−	0.849753i	$-0.676751\pi$
$257$	−16.9027	−1.05436	−0.527181	+	0.849753i	$0.676751\pi$
$258$	−14.0559	−0.875082
$259$	41.0651	2.55166
$260$	0	0
$261$	−2.21844	−0.137318
$262$	−28.3623	−1.75223
$263$	−20.1864	−1.24475	−0.622374	−	0.782720i	$-0.713832\pi$
$263$	−20.1864	−1.24475	−0.622374	+	0.782720i	$0.713832\pi$
$264$	−8.06428	−0.496322
$265$	0	0
$266$	−81.2466	−4.98155
$267$	−11.1326	−0.681302
$268$	8.51664	0.520237
$269$	−29.6277	−1.80643	−0.903215	−	0.429188i	$-0.858800\pi$
$269$	−29.6277	−1.80643	−0.903215	+	0.429188i	$0.858800\pi$
$270$	0	0
$271$	22.2703	1.35282	0.676411	−	0.736524i	$-0.263534\pi$
$271$	22.2703	1.35282	0.676411	+	0.736524i	$0.263534\pi$
$272$	−0.247380	−0.0149996
$273$	−4.96953	−0.300769
$274$	−24.1399	−1.45835
$275$	0	0
$276$	−19.4742	−1.17221
$277$	1.88512	0.113266	0.0566331	−	0.998395i	$-0.481963\pi$
$277$	1.88512	0.113266	0.0566331	+	0.998395i	$0.481963\pi$
$278$	46.7808	2.80573
$279$	2.19148	0.131200
$280$	0	0
$281$	−8.84827	−0.527843	−0.263922	−	0.964544i	$-0.585016\pi$
$281$	−8.84827	−0.527843	−0.263922	+	0.964544i	$0.585016\pi$
$282$	−3.76398	−0.224141
$283$	24.0219	1.42795	0.713977	−	0.700169i	$-0.246892\pi$
$283$	24.0219	1.42795	0.713977	+	0.700169i	$0.246892\pi$
$284$	12.7914	0.759030
$285$	0	0
$286$	−7.27022	−0.429897
$287$	−8.93554	−0.527448
$288$	6.26036	0.368895
$289$	−16.7986	−0.988151
$290$	0	0
$291$	−8.37418	−0.490903
$292$	32.4506	1.89903
$293$	22.6019	1.32042	0.660208	−	0.751082i	$-0.270468\pi$
$293$	22.6019	1.32042	0.660208	+	0.751082i	$0.270468\pi$
$294$	39.9997	2.33283
$295$	0	0
$296$	20.7182	1.20422
$297$	3.21640	0.186635
$298$	23.3851	1.35466
$299$	−6.26338	−0.362221
$300$	0	0
$301$	30.9027	1.78120
$302$	−21.1293	−1.21586
$303$	−5.50625	−0.316326
$304$	3.98671	0.228654
$305$	0	0
$306$	−1.01447	−0.0579934
$307$	2.90858	0.166001	0.0830007	−	0.996549i	$-0.473550\pi$
$307$	2.90858	0.166001	0.0830007	+	0.996549i	$0.473550\pi$
$308$	49.6978	2.83179
$309$	−1.32082	−0.0751389
$310$	0	0
$311$	7.13290	0.404469	0.202235	−	0.979337i	$-0.435180\pi$
$311$	7.13290	0.404469	0.202235	+	0.979337i	$0.435180\pi$
$312$	−2.50723	−0.141944
$313$	−26.8418	−1.51719	−0.758593	−	0.651565i	$-0.774113\pi$
$313$	−26.8418	−1.51719	−0.758593	+	0.651565i	$0.774113\pi$
$314$	2.23212	0.125966
$315$	0	0
$316$	3.27514	0.184241
$317$	8.19000	0.459996	0.229998	−	0.973191i	$-0.426128\pi$
$317$	8.19000	0.459996	0.229998	+	0.973191i	$0.426128\pi$
$318$	−2.62428	−0.147162
$319$	7.13540	0.399505
$320$	0	0
$321$	−7.49024	−0.418065
$322$	70.3560	3.92079
$323$	−3.24620	−0.180623
$324$	3.10922	0.172734
$325$	0	0
$326$	25.0074	1.38503
$327$	−14.0330	−0.776027
$328$	−4.50818	−0.248922
$329$	8.27531	0.456233
$330$	0	0
$331$	−4.75599	−0.261413	−0.130707	−	0.991421i	$-0.541725\pi$
$331$	−4.75599	−0.261413	−0.130707	+	0.991421i	$0.541725\pi$
$332$	17.9589	0.985622
$333$	−8.26338	−0.452831
$334$	44.3873	2.42876
$335$	0	0
$336$	−2.73916	−0.149433
$337$	−4.54361	−0.247506	−0.123753	−	0.992313i	$-0.539493\pi$
$337$	−4.54361	−0.247506	−0.123753	+	0.992313i	$0.539493\pi$
$338$	−2.26036	−0.122947
$339$	−11.2269	−0.609759
$340$	0	0
$341$	−7.04867	−0.381707
$342$	16.3490	0.884051
$343$	−53.1550	−2.87010
$344$	15.5911	0.840615
$345$	0	0
$346$	34.9865	1.88089
$347$	33.2648	1.78575	0.892874	−	0.450307i	$-0.148686\pi$
$347$	33.2648	1.78575	0.892874	+	0.450307i	$0.148686\pi$
$348$	6.89762	0.369751
$349$	19.6460	1.05163	0.525813	−	0.850600i	$-0.323761\pi$
$349$	19.6460	1.05163	0.525813	+	0.850600i	$0.323761\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−20.1358	−1.07324
$353$	−12.1609	−0.647262	−0.323631	−	0.946183i	$-0.604904\pi$
$353$	−12.1609	−0.647262	−0.323631	+	0.946183i	$0.604904\pi$
$354$	−13.2992	−0.706842
$355$	0	0
$356$	34.6136	1.83452
$357$	2.23037	0.118044
$358$	7.01249	0.370622
$359$	−23.5378	−1.24228	−0.621139	−	0.783701i	$-0.713330\pi$
$359$	−23.5378	−1.24228	−0.621139	+	0.783701i	$0.713330\pi$
$360$	0	0
$361$	33.3150	1.75342
$362$	−28.9576	−1.52198
$363$	0.654755	0.0343657
$364$	15.4513	0.809871
$365$	0	0
$366$	−4.00000	−0.209083
$367$	12.5944	0.657421	0.328710	−	0.944431i	$-0.393386\pi$
$367$	12.5944	0.657421	0.328710	+	0.944431i	$0.393386\pi$
$368$	−3.45232	−0.179965
$369$	1.79807	0.0936036
$370$	0	0
$371$	5.76963	0.299544
$372$	−6.81378	−0.353278
$373$	−10.6585	−0.551875	−0.275938	−	0.961176i	$-0.588988\pi$
$373$	−10.6585	−0.551875	−0.275938	+	0.961176i	$0.588988\pi$
$374$	3.26294	0.168723
$375$	0	0
$376$	4.17508	0.215313
$377$	2.21844	0.114255
$378$	−11.2329	−0.577759
$379$	−22.5065	−1.15608	−0.578040	−	0.816009i	$-0.696182\pi$
$379$	−22.5065	−1.15608	−0.578040	+	0.816009i	$0.696182\pi$
$380$	0	0
$381$	−1.50217	−0.0769586
$382$	−34.3077	−1.75534
$383$	6.22399	0.318031	0.159015	−	0.987276i	$-0.449168\pi$
$383$	6.22399	0.318031	0.159015	+	0.987276i	$0.449168\pi$
$384$	−16.9731	−0.866153
$385$	0	0
$386$	−0.392164	−0.0199606
$387$	−6.21844	−0.316101
$388$	26.0372	1.32184
$389$	−6.49059	−0.329086	−0.164543	−	0.986370i	$-0.552615\pi$
$389$	−6.49059	−0.329086	−0.164543	+	0.986370i	$0.552615\pi$
$390$	0	0
$391$	2.81106	0.142162
$392$	−44.3685	−2.24095
$393$	−12.5477	−0.632947
$394$	−9.67409	−0.487374
$395$	0	0
$396$	−10.0005	−0.502544
$397$	−13.0832	−0.656628	−0.328314	−	0.944569i	$-0.606480\pi$
$397$	−13.0832	−0.656628	−0.328314	+	0.944569i	$0.606480\pi$
$398$	−37.7124	−1.89035
$399$	−35.9441	−1.79946
$400$	0	0
$401$	24.2326	1.21012	0.605060	−	0.796180i	$-0.293149\pi$
$401$	24.2326	1.21012	0.605060	+	0.796180i	$0.293149\pi$
$402$	6.19148	0.308803
$403$	−2.19148	−0.109165
$404$	17.1201	0.851758
$405$	0	0
$406$	−24.9195	−1.23674
$407$	26.5784	1.31744
$408$	1.12527	0.0557092
$409$	2.52576	0.124891	0.0624454	−	0.998048i	$-0.480110\pi$
$409$	2.52576	0.124891	0.0624454	+	0.998048i	$0.480110\pi$
$410$	0	0
$411$	−10.6797	−0.526790
$412$	4.10673	0.202324
$413$	29.2390	1.43876
$414$	−14.1575	−0.695803
$415$	0	0
$416$	−6.26036	−0.306939
$417$	20.6962	1.01350
$418$	−52.5849	−2.57201
$419$	29.1917	1.42611	0.713054	−	0.701109i	$-0.247312\pi$
$419$	29.1917	1.42611	0.713054	+	0.701109i	$0.247312\pi$
$420$	0	0
$421$	−22.3719	−1.09034	−0.545169	−	0.838326i	$-0.683534\pi$
$421$	−22.3719	−1.09034	−0.545169	+	0.838326i	$0.683534\pi$
$422$	54.9297	2.67393
$423$	−1.66521	−0.0809654
$424$	2.91091	0.141366
$425$	0	0
$426$	9.29916	0.450546
$427$	8.79423	0.425582
$428$	23.2888	1.12571
$429$	−3.21640	−0.155289
$430$	0	0
$431$	−26.6111	−1.28181	−0.640906	−	0.767619i	$-0.721441\pi$
$431$	−26.6111	−1.28181	−0.640906	+	0.767619i	$0.721441\pi$
$432$	0.551191	0.0265192
$433$	10.4369	0.501564	0.250782	−	0.968044i	$-0.419312\pi$
$433$	10.4369	0.501564	0.250782	+	0.968044i	$0.419312\pi$
$434$	24.6167	1.18164
$435$	0	0
$436$	43.6317	2.08958
$437$	−45.3025	−2.16711
$438$	23.5911	1.12723
$439$	−28.0916	−1.34074	−0.670370	−	0.742027i	$-0.733865\pi$
$439$	−28.0916	−1.34074	−0.670370	+	0.742027i	$0.733865\pi$
$440$	0	0
$441$	17.6962	0.842676
$442$	1.01447	0.0482534
$443$	28.9146	1.37378	0.686888	−	0.726764i	$-0.258976\pi$
$443$	28.9146	1.37378	0.686888	+	0.726764i	$0.258976\pi$
$444$	25.6927	1.21932
$445$	0	0
$446$	1.85579	0.0878744
$447$	10.3457	0.489337
$448$	64.8437	3.06358
$449$	27.9652	1.31976	0.659879	−	0.751372i	$-0.270608\pi$
$449$	27.9652	1.31976	0.659879	+	0.751372i	$0.270608\pi$
$450$	0	0
$451$	−5.78331	−0.272325
$452$	34.9068	1.64188
$453$	−9.34779	−0.439197
$454$	14.1180	0.662590
$455$	0	0
$456$	−18.1346	−0.849231
$457$	−28.7251	−1.34370	−0.671852	−	0.740685i	$-0.734501\pi$
$457$	−28.7251	−1.34370	−0.671852	+	0.740685i	$0.734501\pi$
$458$	−46.5105	−2.17329
$459$	−0.448809	−0.0209486
$460$	0	0
$461$	5.81017	0.270607	0.135303	−	0.990804i	$-0.456799\pi$
$461$	5.81017	0.270607	0.135303	+	0.990804i	$0.456799\pi$
$462$	36.1296	1.68090
$463$	−27.1743	−1.26290	−0.631448	−	0.775418i	$-0.717539\pi$
$463$	−27.1743	−1.26290	−0.631448	+	0.775418i	$0.717539\pi$
$464$	1.22278	0.0567663
$465$	0	0
$466$	11.7403	0.543861
$467$	23.1290	1.07028	0.535141	−	0.844763i	$-0.320258\pi$
$467$	23.1290	1.07028	0.535141	+	0.844763i	$0.320258\pi$
$468$	−3.10922	−0.143724
$469$	−13.6123	−0.628558
$470$	0	0
$471$	0.987506	0.0455019
$472$	14.7517	0.679002
$473$	20.0010	0.919647
$474$	2.38098	0.109362
$475$	0	0
$476$	−6.93471	−0.317852
$477$	−1.16100	−0.0531586
$478$	27.8541	1.27402
$479$	23.3243	1.06571	0.532856	−	0.846206i	$-0.321119\pi$
$479$	23.3243	1.06571	0.532856	+	0.846206i	$0.321119\pi$
$480$	0	0
$481$	8.26338	0.376778
$482$	5.46095	0.248740
$483$	31.1260	1.41628
$484$	−2.03578	−0.0925354
$485$	0	0
$486$	2.26036	0.102532
$487$	−7.23466	−0.327834	−0.163917	−	0.986474i	$-0.552413\pi$
$487$	−7.23466	−0.327834	−0.163917	+	0.986474i	$0.552413\pi$
$488$	4.43688	0.200848
$489$	11.0635	0.500308
$490$	0	0
$491$	−42.9453	−1.93809	−0.969047	−	0.246874i	$-0.920597\pi$
$491$	−42.9453	−1.93809	−0.969047	+	0.246874i	$0.920597\pi$
$492$	−5.59059	−0.252043
$493$	−0.995656	−0.0448421
$494$	−16.3490	−0.735575
$495$	0	0
$496$	−1.20792	−0.0542373
$497$	−20.4447	−0.917072
$498$	13.0559	0.585047
$499$	−18.0487	−0.807969	−0.403985	−	0.914766i	$-0.632375\pi$
$499$	−18.0487	−0.807969	−0.403985	+	0.914766i	$0.632375\pi$
$500$	0	0
$501$	19.6373	0.877329
$502$	−48.9205	−2.18343
$503$	10.3252	0.460376	0.230188	−	0.973146i	$-0.426066\pi$
$503$	10.3252	0.460376	0.230188	+	0.973146i	$0.426066\pi$
$504$	12.4598	0.555002
$505$	0	0
$506$	45.5362	2.02433
$507$	−1.00000	−0.0444116
$508$	4.67058	0.207224
$509$	−8.57786	−0.380207	−0.190104	−	0.981764i	$-0.560882\pi$
$509$	−8.57786	−0.380207	−0.190104	+	0.981764i	$0.560882\pi$
$510$	0	0
$511$	−51.8663	−2.29443
$512$	−6.21458	−0.274648
$513$	7.23291	0.319341
$514$	38.2062	1.68520
$515$	0	0
$516$	19.3345	0.851154
$517$	5.35599	0.235556
$518$	−92.8218	−4.07836
$519$	15.4783	0.679423
$520$	0	0
$521$	8.44706	0.370072	0.185036	−	0.982732i	$-0.440760\pi$
$521$	8.44706	0.370072	0.185036	+	0.982732i	$0.440760\pi$
$522$	5.01447	0.219477
$523$	−31.5650	−1.38024	−0.690121	−	0.723694i	$-0.742443\pi$
$523$	−31.5650	−1.38024	−0.690121	+	0.723694i	$0.742443\pi$
$524$	39.0135	1.70431
$525$	0	0
$526$	45.6286	1.98950
$527$	0.983555	0.0428443
$528$	−1.77285	−0.0771535
$529$	16.2300	0.705651
$530$	0	0
$531$	−5.88365	−0.255329
$532$	111.758	4.84533
$533$	−1.79807	−0.0778829
$534$	25.1636	1.08893
$535$	0	0
$536$	−6.86771	−0.296640
$537$	3.10238	0.133878
$538$	66.9691	2.88724
$539$	−56.9181	−2.45163
$540$	0	0
$541$	24.7478	1.06399	0.531995	−	0.846747i	$-0.321442\pi$
$541$	24.7478	1.06399	0.531995	+	0.846747i	$0.321442\pi$
$542$	−50.3388	−2.16224
$543$	−12.8111	−0.549775
$544$	2.80971	0.120465
$545$	0	0
$546$	11.2329	0.480724
$547$	25.9207	1.10829	0.554145	−	0.832420i	$-0.313045\pi$
$547$	25.9207	1.10829	0.554145	+	0.832420i	$0.313045\pi$
$548$	33.2055	1.41847
$549$	−1.76963	−0.0755260
$550$	0	0
$551$	16.0458	0.683573
$552$	15.7038	0.668397
$553$	−5.23471	−0.222603
$554$	−4.26106	−0.181035
$555$	0	0
$556$	−64.3490	−2.72901
$557$	35.3538	1.49799	0.748993	−	0.662577i	$-0.230537\pi$
$557$	35.3538	1.49799	0.748993	+	0.662577i	$0.230537\pi$
$558$	−4.95352	−0.209699
$559$	6.21844	0.263012
$560$	0	0
$561$	1.44355	0.0609468
$562$	20.0003	0.843660
$563$	22.3919	0.943708	0.471854	−	0.881677i	$-0.343585\pi$
$563$	22.3919	0.943708	0.471854	+	0.881677i	$0.343585\pi$
$564$	5.17751	0.218012
$565$	0	0
$566$	−54.2981	−2.28232
$567$	−4.96953	−0.208700
$568$	−10.3148	−0.432800
$569$	41.1034	1.72314	0.861572	−	0.507636i	$-0.169480\pi$
$569$	41.1034	1.72314	0.861572	+	0.507636i	$0.169480\pi$
$570$	0	0
$571$	28.3848	1.18787	0.593933	−	0.804514i	$-0.297574\pi$
$571$	28.3848	1.18787	0.593933	+	0.804514i	$0.297574\pi$
$572$	10.0005	0.418142
$573$	−15.1780	−0.634071
$574$	20.1975	0.843028
$575$	0	0
$576$	−13.0483	−0.543678
$577$	5.44592	0.226716	0.113358	−	0.993554i	$-0.463839\pi$
$577$	5.44592	0.226716	0.113358	+	0.993554i	$0.463839\pi$
$578$	37.9708	1.57938
$579$	−0.173496	−0.00721026
$580$	0	0
$581$	−28.7040	−1.19084
$582$	18.9287	0.784618
$583$	3.73425	0.154657
$584$	−26.1677	−1.08283
$585$	0	0
$586$	−51.0884	−2.11044
$587$	−18.7717	−0.774790	−0.387395	−	0.921914i	$-0.626625\pi$
$587$	−18.7717	−0.774790	−0.387395	+	0.921914i	$0.626625\pi$
$588$	−55.0213	−2.26904
$589$	−15.8508	−0.653119
$590$	0	0
$591$	−4.27989	−0.176051
$592$	4.55470	0.187197
$593$	−39.9300	−1.63973	−0.819863	−	0.572559i	$-0.805951\pi$
$593$	−39.9300	−1.63973	−0.819863	+	0.572559i	$0.805951\pi$
$594$	−7.27022	−0.298301
$595$	0	0
$596$	−32.1672	−1.31762
$597$	−16.6843	−0.682841
$598$	14.1575	0.578943
$599$	36.7310	1.50079	0.750393	−	0.660992i	$-0.229864\pi$
$599$	36.7310	1.50079	0.750393	+	0.660992i	$0.229864\pi$
$600$	0	0
$601$	−25.7376	−1.04986	−0.524930	−	0.851146i	$-0.675908\pi$
$601$	−25.7376	−1.04986	−0.524930	+	0.851146i	$0.675908\pi$
$602$	−69.8512	−2.84692
$603$	2.73916	0.111547
$604$	29.0643	1.18261
$605$	0	0
$606$	12.4461	0.505588
$607$	3.95975	0.160721	0.0803606	−	0.996766i	$-0.474393\pi$
$607$	3.95975	0.160721	0.0803606	+	0.996766i	$0.474393\pi$
$608$	−45.2806	−1.83637
$609$	−11.0246	−0.446739
$610$	0	0
$611$	1.66521	0.0673673
$612$	1.39545	0.0564076
$613$	14.0707	0.568311	0.284156	−	0.958778i	$-0.408287\pi$
$613$	14.0707	0.568311	0.284156	+	0.958778i	$0.408287\pi$
$614$	−6.57443	−0.265322
$615$	0	0
$616$	−40.0756	−1.61469
$617$	−31.6852	−1.27560	−0.637798	−	0.770204i	$-0.720155\pi$
$617$	−31.6852	−1.27560	−0.637798	+	0.770204i	$0.720155\pi$
$618$	2.98553	0.120096
$619$	12.4566	0.500673	0.250337	−	0.968159i	$-0.419459\pi$
$619$	12.4566	0.500673	0.250337	+	0.968159i	$0.419459\pi$
$620$	0	0
$621$	−6.26338	−0.251341
$622$	−16.1229	−0.646469
$623$	−55.3236	−2.21649
$624$	−0.551191	−0.0220653
$625$	0	0
$626$	60.6720	2.42494
$627$	−23.2639	−0.929073
$628$	−3.07037	−0.122521
$629$	−3.70868	−0.147875
$630$	0	0
$631$	−14.8500	−0.591167	−0.295584	−	0.955317i	$-0.595514\pi$
$631$	−14.8500	−0.591167	−0.295584	+	0.955317i	$0.595514\pi$
$632$	−2.64103	−0.105054
$633$	24.3013	0.965890
$634$	−18.5123	−0.735219
$635$	0	0
$636$	3.60981	0.143138
$637$	−17.6962	−0.701149
$638$	−16.1286	−0.638536
$639$	4.11402	0.162748
$640$	0	0
$641$	13.2878	0.524837	0.262418	−	0.964954i	$-0.415480\pi$
$641$	13.2878	0.524837	0.262418	+	0.964954i	$0.415480\pi$
$642$	16.9306	0.668199
$643$	18.7409	0.739069	0.369535	−	0.929217i	$-0.379517\pi$
$643$	18.7409	0.739069	0.369535	+	0.929217i	$0.379517\pi$
$644$	−96.7777	−3.81358
$645$	0	0
$646$	7.33757	0.288693
$647$	30.9106	1.21522	0.607610	−	0.794236i	$-0.292129\pi$
$647$	30.9106	1.21522	0.607610	+	0.794236i	$0.292129\pi$
$648$	−2.50723	−0.0984935
$649$	18.9242	0.742840
$650$	0	0
$651$	10.8906	0.426836
$652$	−34.3988	−1.34716
$653$	−1.99040	−0.0778903	−0.0389452	−	0.999241i	$-0.512400\pi$
$653$	−1.99040	−0.0778903	−0.0389452	+	0.999241i	$0.512400\pi$
$654$	31.7196	1.24034
$655$	0	0
$656$	−0.991078	−0.0386951
$657$	10.4369	0.407181
$658$	−18.7052	−0.729204
$659$	38.5954	1.50346	0.751731	−	0.659470i	$-0.229219\pi$
$659$	38.5954	1.50346	0.751731	+	0.659470i	$0.229219\pi$
$660$	0	0
$661$	−10.3150	−0.401206	−0.200603	−	0.979673i	$-0.564290\pi$
$661$	−10.3150	−0.401206	−0.200603	+	0.979673i	$0.564290\pi$
$662$	10.7503	0.417820
$663$	0.448809	0.0174303
$664$	−14.4818	−0.562004
$665$	0	0
$666$	18.6782	0.723766
$667$	−13.8949	−0.538014
$668$	−61.0566	−2.36235
$669$	0.821018	0.0317424
$670$	0	0
$671$	5.69185	0.219731
$672$	31.1110	1.20013
$673$	19.1024	0.736343	0.368171	−	0.929758i	$-0.379984\pi$
$673$	19.1024	0.736343	0.368171	+	0.929758i	$0.379984\pi$
$674$	10.2702	0.395592
$675$	0	0
$676$	3.10922	0.119585
$677$	−18.2482	−0.701336	−0.350668	−	0.936500i	$-0.614045\pi$
$677$	−18.2482	−0.701336	−0.350668	+	0.936500i	$0.614045\pi$
$678$	25.3767	0.974587
$679$	−41.6157	−1.59706
$680$	0	0
$681$	6.24591	0.239344
$682$	15.9325	0.610088
$683$	45.2264	1.73054	0.865270	−	0.501306i	$-0.167147\pi$
$683$	45.2264	1.73054	0.865270	+	0.501306i	$0.167147\pi$
$684$	−22.4887	−0.859877
$685$	0	0
$686$	120.149	4.58732
$687$	−20.5766	−0.785047
$688$	3.42755	0.130674
$689$	1.16100	0.0442307
$690$	0	0
$691$	12.4293	0.472831	0.236416	−	0.971652i	$-0.424027\pi$
$691$	12.4293	0.472831	0.236416	+	0.971652i	$0.424027\pi$
$692$	−48.1255	−1.82946
$693$	15.9840	0.607182
$694$	−75.1903	−2.85419
$695$	0	0
$696$	−5.56215	−0.210833
$697$	0.806989	0.0305669
$698$	−44.4070	−1.68083
$699$	5.19402	0.196456
$700$	0	0
$701$	−30.9611	−1.16939	−0.584693	−	0.811254i	$-0.698785\pi$
$701$	−30.9611	−1.16939	−0.584693	+	0.811254i	$0.698785\pi$
$702$	−2.26036	−0.0853117
$703$	59.7683	2.25420
$704$	41.9685	1.58175
$705$	0	0
$706$	27.4881	1.03453
$707$	−27.3634	−1.02911
$708$	18.2936	0.687514
$709$	2.20069	0.0826486	0.0413243	−	0.999146i	$-0.486842\pi$
$709$	2.20069	0.0826486	0.0413243	+	0.999146i	$0.486842\pi$
$710$	0	0
$711$	1.05336	0.0395042
$712$	−27.9120	−1.04604
$713$	13.7261	0.514045
$714$	−5.04143	−0.188671
$715$	0	0
$716$	−9.64599	−0.360487
$717$	12.3229	0.460206
$718$	53.2038	1.98555
$719$	−44.5913	−1.66297	−0.831487	−	0.555543i	$-0.812510\pi$
$719$	−44.5913	−1.66297	−0.831487	+	0.555543i	$0.812510\pi$
$720$	0	0
$721$	−6.56386	−0.244451
$722$	−75.3038	−2.80252
$723$	2.41597	0.0898508
$724$	39.8324	1.48036
$725$	0	0
$726$	−1.47998	−0.0549273
$727$	−34.7808	−1.28995	−0.644974	−	0.764204i	$-0.723132\pi$
$727$	−34.7808	−1.28995	−0.644974	+	0.764204i	$0.723132\pi$
$728$	−12.4598	−0.461790
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.79089	−0.103225
$732$	5.50217	0.203366
$733$	−31.0158	−1.14560	−0.572798	−	0.819697i	$-0.694142\pi$
$733$	−31.0158	−1.14560	−0.572798	+	0.819697i	$0.694142\pi$
$734$	−28.4678	−1.05077
$735$	0	0
$736$	39.2110	1.44534
$737$	−8.81023	−0.324529
$738$	−4.06428	−0.149608
$739$	37.9029	1.39428	0.697141	−	0.716934i	$-0.254455\pi$
$739$	37.9029	1.39428	0.697141	+	0.716934i	$0.254455\pi$
$740$	0	0
$741$	−7.23291	−0.265708
$742$	−13.0414	−0.478766
$743$	14.6939	0.539066	0.269533	−	0.962991i	$-0.413131\pi$
$743$	14.6939	0.539066	0.269533	+	0.962991i	$0.413131\pi$
$744$	5.49455	0.201440
$745$	0	0
$746$	24.0920	0.882070
$747$	5.77601	0.211333
$748$	−4.48832	−0.164109
$749$	−37.2230	−1.36010
$750$	0	0
$751$	−32.5622	−1.18821	−0.594106	−	0.804387i	$-0.702494\pi$
$751$	−32.5622	−1.18821	−0.594106	+	0.804387i	$0.702494\pi$
$752$	0.917849	0.0334705
$753$	−21.6428	−0.788708
$754$	−5.01447	−0.182616
$755$	0	0
$756$	15.4513	0.561960
$757$	−15.2365	−0.553779	−0.276889	−	0.960902i	$-0.589304\pi$
$757$	−15.2365	−0.553779	−0.276889	+	0.960902i	$0.589304\pi$
$758$	50.8727	1.84778
$759$	20.1456	0.731238
$760$	0	0
$761$	51.5327	1.86806	0.934030	−	0.357194i	$-0.116266\pi$
$761$	51.5327	1.86806	0.934030	+	0.357194i	$0.116266\pi$
$762$	3.39545	0.123004
$763$	−69.7374	−2.52466
$764$	47.1918	1.70734
$765$	0	0
$766$	−14.0684	−0.508314
$767$	5.88365	0.212446
$768$	12.2686	0.442706
$769$	28.7148	1.03548	0.517741	−	0.855538i	$-0.326773\pi$
$769$	28.7148	1.03548	0.517741	+	0.855538i	$0.326773\pi$
$770$	0	0
$771$	16.9027	0.608736
$772$	0.539439	0.0194148
$773$	−34.5120	−1.24131	−0.620655	−	0.784084i	$-0.713133\pi$
$773$	−34.5120	−1.24131	−0.620655	+	0.784084i	$0.713133\pi$
$774$	14.0559	0.505229
$775$	0	0
$776$	−20.9960	−0.753714
$777$	−41.0651	−1.47320
$778$	14.6711	0.525983
$779$	−13.0053	−0.465962
$780$	0	0
$781$	−13.2323	−0.473491
$782$	−6.35401	−0.227219
$783$	2.21844	0.0792806
$784$	−9.75398	−0.348356
$785$	0	0
$786$	28.3623	1.01165
$787$	34.2046	1.21926	0.609631	−	0.792685i	$-0.291318\pi$
$787$	34.2046	1.21926	0.609631	+	0.792685i	$0.291318\pi$
$788$	13.3071	0.474047
$789$	20.1864	0.718656
$790$	0	0
$791$	−55.7922	−1.98374
$792$	8.06428	0.286552
$793$	1.76963	0.0628414
$794$	29.5728	1.04950
$795$	0	0
$796$	51.8750	1.83866
$797$	42.9365	1.52089	0.760445	−	0.649402i	$-0.224981\pi$
$797$	42.9365	1.52089	0.760445	+	0.649402i	$0.224981\pi$
$798$	81.2466	2.87610
$799$	−0.747362	−0.0264398
$800$	0	0
$801$	11.1326	0.393350
$802$	−54.7744	−1.93415
$803$	−33.5692	−1.18463
$804$	−8.51664	−0.300359
$805$	0	0
$806$	4.95352	0.174480
$807$	29.6277	1.04294
$808$	−13.8055	−0.485674
$809$	−8.42320	−0.296144	−0.148072	−	0.988977i	$-0.547307\pi$
$809$	−8.42320	−0.296144	−0.148072	+	0.988977i	$0.547307\pi$
$810$	0	0
$811$	39.8090	1.39788	0.698941	−	0.715180i	$-0.253655\pi$
$811$	39.8090	1.39788	0.698941	+	0.715180i	$0.253655\pi$
$812$	34.2779	1.20292
$813$	−22.2703	−0.781052
$814$	−60.0766	−2.10569
$815$	0	0
$816$	0.247380	0.00866001
$817$	44.9774	1.57356
$818$	−5.70913	−0.199615
$819$	4.96953	0.173649
$820$	0	0
$821$	−29.3494	−1.02430	−0.512151	−	0.858895i	$-0.671151\pi$
$821$	−29.3494	−1.02430	−0.512151	+	0.858895i	$0.671151\pi$
$822$	24.1399	0.841976
$823$	−4.65419	−0.162235	−0.0811174	−	0.996705i	$-0.525849\pi$
$823$	−4.65419	−0.162235	−0.0811174	+	0.996705i	$0.525849\pi$
$824$	−3.31161	−0.115365
$825$	0	0
$826$	−66.0905	−2.29958
$827$	−53.8430	−1.87231	−0.936153	−	0.351593i	$-0.885640\pi$
$827$	−53.8430	−1.87231	−0.936153	+	0.351593i	$0.885640\pi$
$828$	19.4742	0.676777
$829$	−4.58070	−0.159094	−0.0795471	−	0.996831i	$-0.525347\pi$
$829$	−4.58070	−0.159094	−0.0795471	+	0.996831i	$0.525347\pi$
$830$	0	0
$831$	−1.88512	−0.0653942
$832$	13.0483	0.452367
$833$	7.94221	0.275181
$834$	−46.7808	−1.61989
$835$	0	0
$836$	72.3327	2.50168
$837$	−2.19148	−0.0757485
$838$	−65.9837	−2.27937
$839$	21.2754	0.734509	0.367254	−	0.930121i	$-0.380298\pi$
$839$	21.2754	0.734509	0.367254	+	0.930121i	$0.380298\pi$
$840$	0	0
$841$	−24.0785	−0.830294
$842$	50.5684	1.74270
$843$	8.84827	0.304751
$844$	−75.5581	−2.60082
$845$	0	0
$846$	3.76398	0.129408
$847$	3.25382	0.111803
$848$	0.639934	0.0219754
$849$	−24.0219	−0.824430
$850$	0	0
$851$	−51.7567	−1.77420
$852$	−12.7914	−0.438226
$853$	−15.5689	−0.533070	−0.266535	−	0.963825i	$-0.585879\pi$
$853$	−15.5689	−0.533070	−0.266535	+	0.963825i	$0.585879\pi$
$854$	−19.8781	−0.680215
$855$	0	0
$856$	−18.7798	−0.641880
$857$	−18.5524	−0.633737	−0.316869	−	0.948469i	$-0.602631\pi$
$857$	−18.5524	−0.633737	−0.316869	+	0.948469i	$0.602631\pi$
$858$	7.27022	0.248201
$859$	−26.2980	−0.897275	−0.448638	−	0.893714i	$-0.648091\pi$
$859$	−26.2980	−0.897275	−0.448638	+	0.893714i	$0.648091\pi$
$860$	0	0
$861$	8.93554	0.304522
$862$	60.1506	2.04874
$863$	−0.537842	−0.0183083	−0.00915417	−	0.999958i	$-0.502914\pi$
$863$	−0.537842	−0.0183083	−0.00915417	+	0.999958i	$0.502914\pi$
$864$	−6.26036	−0.212982
$865$	0	0
$866$	−23.5911	−0.801658
$867$	16.7986	0.570509
$868$	−33.8613	−1.14933
$869$	−3.38804	−0.114931
$870$	0	0
$871$	−2.73916	−0.0928128
$872$	−35.1841	−1.19148
$873$	8.37418	0.283423
$874$	102.400	3.46373
$875$	0	0
$876$	−32.4506	−1.09640
$877$	49.9065	1.68522	0.842611	−	0.538523i	$-0.181018\pi$
$877$	49.9065	1.68522	0.842611	+	0.538523i	$0.181018\pi$
$878$	63.4972	2.14293
$879$	−22.6019	−0.762343
$880$	0	0
$881$	41.0327	1.38243	0.691213	−	0.722651i	$-0.257076\pi$
$881$	41.0327	1.38243	0.691213	+	0.722651i	$0.257076\pi$
$882$	−39.9997	−1.34686
$883$	6.59753	0.222025	0.111012	−	0.993819i	$-0.464591\pi$
$883$	6.59753	0.222025	0.111012	+	0.993819i	$0.464591\pi$
$884$	−1.39545	−0.0469339
$885$	0	0
$886$	−65.3574	−2.19573
$887$	9.02521	0.303037	0.151519	−	0.988454i	$-0.451584\pi$
$887$	9.02521	0.303037	0.151519	+	0.988454i	$0.451584\pi$
$888$	−20.7182	−0.695259
$889$	−7.46508	−0.250371
$890$	0	0
$891$	−3.21640	−0.107754
$892$	−2.55273	−0.0854716
$893$	12.0443	0.403048
$894$	−23.3851	−0.782115
$895$	0	0
$896$	−84.3480	−2.81787
$897$	6.26338	0.209128
$898$	−63.2113	−2.10939
$899$	−4.86166	−0.162145
$900$	0	0
$901$	−0.521069	−0.0173593
$902$	13.0723	0.435262
$903$	−30.9027	−1.02838
$904$	−28.1484	−0.936201
$905$	0	0
$906$	21.1293	0.701975
$907$	15.5850	0.517493	0.258746	−	0.965945i	$-0.416691\pi$
$907$	15.5850	0.517493	0.258746	+	0.965945i	$0.416691\pi$
$908$	−19.4199	−0.644472
$909$	5.50625	0.182631
$910$	0	0
$911$	2.86303	0.0948564	0.0474282	−	0.998875i	$-0.484897\pi$
$911$	2.86303	0.0948564	0.0474282	+	0.998875i	$0.484897\pi$
$912$	−3.98671	−0.132013
$913$	−18.5780	−0.614841
$914$	64.9291	2.14766
$915$	0	0
$916$	63.9772	2.11387
$917$	−62.3560	−2.05918
$918$	1.01447	0.0334825
$919$	−53.0874	−1.75119	−0.875596	−	0.483045i	$-0.839531\pi$
$919$	−53.0874	−1.75119	−0.875596	+	0.483045i	$0.839531\pi$
$920$	0	0
$921$	−2.90858	−0.0958409
$922$	−13.1331	−0.432514
$923$	−4.11402	−0.135415
$924$	−49.6978	−1.63494
$925$	0	0
$926$	61.4236	2.01851
$927$	1.32082	0.0433815
$928$	−13.8882	−0.455903
$929$	25.0925	0.823259	0.411630	−	0.911351i	$-0.364960\pi$
$929$	25.0925	0.823259	0.411630	+	0.911351i	$0.364960\pi$
$930$	0	0
$931$	−127.995	−4.19486
$932$	−16.1493	−0.528989
$933$	−7.13290	−0.233521
$934$	−52.2798	−1.71065
$935$	0	0
$936$	2.50723	0.0819515
$937$	54.5029	1.78053	0.890266	−	0.455441i	$-0.150518\pi$
$937$	54.5029	1.78053	0.890266	+	0.455441i	$0.150518\pi$
$938$	30.7687	1.00463
$939$	26.8418	0.875947
$940$	0	0
$941$	32.3538	1.05470	0.527351	−	0.849647i	$-0.323185\pi$
$941$	32.3538	1.05470	0.527351	+	0.849647i	$0.323185\pi$
$942$	−2.23212	−0.0727263
$943$	11.2620	0.366741
$944$	3.24301	0.105551
$945$	0	0
$946$	−45.2094	−1.46989
$947$	36.3069	1.17981	0.589907	−	0.807471i	$-0.299164\pi$
$947$	36.3069	1.17981	0.589907	+	0.807471i	$0.299164\pi$
$948$	−3.27514	−0.106371
$949$	−10.4369	−0.338795
$950$	0	0
$951$	−8.19000	−0.265579
$952$	5.59206	0.181240
$953$	−13.3176	−0.431399	−0.215699	−	0.976460i	$-0.569203\pi$
$953$	−13.3176	−0.431399	−0.215699	+	0.976460i	$0.569203\pi$
$954$	2.62428	0.0849642
$955$	0	0
$956$	−38.3145	−1.23918
$957$	−7.13540	−0.230655
$958$	−52.7212	−1.70334
$959$	−53.0730	−1.71382
$960$	0	0
$961$	−26.1974	−0.845078
$962$	−18.6782	−0.602210
$963$	7.49024	0.241370
$964$	−7.51177	−0.241938
$965$	0	0
$966$	−70.3560	−2.26367
$967$	−46.9352	−1.50933	−0.754667	−	0.656108i	$-0.772202\pi$
$967$	−46.9352	−1.50933	−0.754667	+	0.656108i	$0.772202\pi$
$968$	1.64162	0.0527638
$969$	3.24620	0.104283
$970$	0	0
$971$	−22.5340	−0.723151	−0.361575	−	0.932343i	$-0.617761\pi$
$971$	−22.5340	−0.723151	−0.361575	+	0.932343i	$0.617761\pi$
$972$	−3.10922	−0.0997283
$973$	102.850	3.29723
$974$	16.3529	0.523981
$975$	0	0
$976$	0.975404	0.0312219
$977$	59.9345	1.91747	0.958737	−	0.284296i	$-0.0917598\pi$
$977$	59.9345	1.91747	0.958737	+	0.284296i	$0.0917598\pi$
$978$	−25.0074	−0.799650
$979$	−35.8068	−1.14439
$980$	0	0
$981$	14.0330	0.448040
$982$	97.0718	3.09769
$983$	52.0879	1.66135	0.830673	−	0.556760i	$-0.187956\pi$
$983$	52.0879	1.66135	0.830673	+	0.556760i	$0.187956\pi$
$984$	4.50818	0.143715
$985$	0	0
$986$	2.25054	0.0716718
$987$	−8.27531	−0.263406
$988$	22.4887	0.715461
$989$	−38.9485	−1.23849
$990$	0	0
$991$	18.6973	0.593940	0.296970	−	0.954887i	$-0.404024\pi$
$991$	18.6973	0.593940	0.296970	+	0.954887i	$0.404024\pi$
$992$	13.7194	0.435592
$993$	4.75599	0.150927
$994$	46.2124	1.46577
$995$	0	0
$996$	−17.9589	−0.569049
$997$	−18.9805	−0.601118	−0.300559	−	0.953763i	$-0.597173\pi$
$997$	−18.9805	−0.601118	−0.300559	+	0.953763i	$0.597173\pi$
$998$	40.7965	1.29139
$999$	8.26338	0.261442

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.a.r.1.1		5
3.2	odd	2		2925.2.a.bl.1.5		5
5.2	odd	4		195.2.c.b.79.2	✓	10
5.3	odd	4		195.2.c.b.79.9	yes	10
5.4	even	2		975.2.a.s.1.5		5
15.2	even	4		585.2.c.c.469.9		10
15.8	even	4		585.2.c.c.469.2		10
15.14	odd	2		2925.2.a.bm.1.1		5
20.3	even	4		3120.2.l.p.1249.6		10
20.7	even	4		3120.2.l.p.1249.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.c.b.79.2	✓	10	5.2	odd	4
195.2.c.b.79.9	yes	10	5.3	odd	4
585.2.c.c.469.2		10	15.8	even	4
585.2.c.c.469.9		10	15.2	even	4
975.2.a.r.1.1		5	1.1	even	1	trivial
975.2.a.s.1.5		5	5.4	even	2
2925.2.a.bl.1.5		5	3.2	odd	2
2925.2.a.bm.1.1		5	15.14	odd	2
3120.2.l.p.1249.1		10	20.7	even	4
3120.2.l.p.1249.6		10	20.3	even	4

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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.a (trivial)

\(f(q)\)	\(=\)	\(q-2.26036 q^{2} -1.00000 q^{3} +3.10922 q^{4} +2.26036 q^{6} -4.96953 q^{7} -2.50723 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.26036 q^{2} -1.00000 q^{3} +3.10922 q^{4} +2.26036 q^{6} -4.96953 q^{7} -2.50723 q^{8} +1.00000 q^{9} -3.21640 q^{11} -3.10922 q^{12} -1.00000 q^{13} +11.2329 q^{14} -0.551191 q^{16} +0.448809 q^{17} -2.26036 q^{18} -7.23291 q^{19} +4.96953 q^{21} +7.27022 q^{22} +6.26338 q^{23} +2.50723 q^{24} +2.26036 q^{26} -1.00000 q^{27} -15.4513 q^{28} -2.21844 q^{29} +2.19148 q^{31} +6.26036 q^{32} +3.21640 q^{33} -1.01447 q^{34} +3.10922 q^{36} -8.26338 q^{37} +16.3490 q^{38} +1.00000 q^{39} +1.79807 q^{41} -11.2329 q^{42} -6.21844 q^{43} -10.0005 q^{44} -14.1575 q^{46} -1.66521 q^{47} +0.551191 q^{48} +17.6962 q^{49} -0.448809 q^{51} -3.10922 q^{52} -1.16100 q^{53} +2.26036 q^{54} +12.4598 q^{56} +7.23291 q^{57} +5.01447 q^{58} -5.88365 q^{59} -1.76963 q^{61} -4.95352 q^{62} -4.96953 q^{63} -13.0483 q^{64} -7.27022 q^{66} +2.73916 q^{67} +1.39545 q^{68} -6.26338 q^{69} +4.11402 q^{71} -2.50723 q^{72} +10.4369 q^{73} +18.6782 q^{74} -22.4887 q^{76} +15.9840 q^{77} -2.26036 q^{78} +1.05336 q^{79} +1.00000 q^{81} -4.06428 q^{82} +5.77601 q^{83} +15.4513 q^{84} +14.0559 q^{86} +2.21844 q^{87} +8.06428 q^{88} +11.1326 q^{89} +4.96953 q^{91} +19.4742 q^{92} -2.19148 q^{93} +3.76398 q^{94} -6.26036 q^{96} +8.37418 q^{97} -39.9997 q^{98} -3.21640 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 6 q^{4} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + 6 * q^4 - 5 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + 6 q^{4} - 5 q^{7} + 5 q^{9} + 5 q^{11} - 6 q^{12} - 5 q^{13} + 12 q^{14} + 5 q^{17} + 8 q^{19} + 5 q^{21} + 8 q^{22} + 7 q^{23} - 5 q^{27} - 14 q^{28} + 8 q^{29} + 12 q^{31} + 20 q^{32} - 5 q^{33} + 20 q^{34} + 6 q^{36} - 17 q^{37} + 24 q^{38} + 5 q^{39} + 5 q^{41} - 12 q^{42} - 12 q^{43} + 18 q^{44} - 12 q^{46} + 10 q^{47} + 22 q^{49} - 5 q^{51} - 6 q^{52} + 13 q^{53} - 8 q^{57} + 8 q^{59} + 13 q^{61} + 40 q^{62} - 5 q^{63} - 16 q^{64} - 8 q^{66} - 28 q^{67} + 14 q^{68} - 7 q^{69} + 5 q^{71} + 14 q^{73} + 12 q^{74} + 35 q^{77} + q^{79} + 5 q^{81} - 16 q^{82} + 6 q^{83} + 14 q^{84} - 8 q^{87} + 36 q^{88} + 19 q^{89} + 5 q^{91} + 10 q^{92} - 12 q^{93} - 12 q^{94} - 20 q^{96} + 13 q^{97} - 28 q^{98} + 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 6 * q^4 - 5 * q^7 + 5 * q^9 + 5 * q^11 - 6 * q^12 - 5 * q^13 + 12 * q^14 + 5 * q^17 + 8 * q^19 + 5 * q^21 + 8 * q^22 + 7 * q^23 - 5 * q^27 - 14 * q^28 + 8 * q^29 + 12 * q^31 + 20 * q^32 - 5 * q^33 + 20 * q^34 + 6 * q^36 - 17 * q^37 + 24 * q^38 + 5 * q^39 + 5 * q^41 - 12 * q^42 - 12 * q^43 + 18 * q^44 - 12 * q^46 + 10 * q^47 + 22 * q^49 - 5 * q^51 - 6 * q^52 + 13 * q^53 - 8 * q^57 + 8 * q^59 + 13 * q^61 + 40 * q^62 - 5 * q^63 - 16 * q^64 - 8 * q^66 - 28 * q^67 + 14 * q^68 - 7 * q^69 + 5 * q^71 + 14 * q^73 + 12 * q^74 + 35 * q^77 + q^79 + 5 * q^81 - 16 * q^82 + 6 * q^83 + 14 * q^84 - 8 * q^87 + 36 * q^88 + 19 * q^89 + 5 * q^91 + 10 * q^92 - 12 * q^93 - 12 * q^94 - 20 * q^96 + 13 * q^97 - 28 * q^98 + 5 * q^99

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