LMFDB - Embedded newform 9025.2.a.bz.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, 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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.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:	$72.0649878242$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.41289040.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 8x^{4} + 21x^{3} + 18x^{2} - 25x - 5 $ x^6 - 3x^5 - 8x^4 + 21x^3 + 18x^2 - 25*x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 475)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.181608$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.48119 q^{2} +2.85674 q^{3} +0.193937 q^{4} -4.23138 q^{6} +3.78541 q^{7} +2.67513 q^{8} +5.16096 q^{9} -5.59460 q^{11} +0.554026 q^{12} -4.90652 q^{13} -5.60693 q^{14} -4.35026 q^{16} +1.75019 q^{17} -7.64438 q^{18} +10.8139 q^{21} +8.28669 q^{22} -0.581537 q^{23} +7.64215 q^{24} +7.26750 q^{26} +6.17328 q^{27} +0.734130 q^{28} +1.66431 q^{29} +7.01680 q^{31} +1.09332 q^{32} -15.9823 q^{33} -2.59237 q^{34} +1.00090 q^{36} -2.36322 q^{37} -14.0166 q^{39} +0.835351 q^{41} -16.0175 q^{42} +1.07133 q^{43} -1.08500 q^{44} +0.861369 q^{46} -3.86756 q^{47} -12.4276 q^{48} +7.32934 q^{49} +4.99984 q^{51} -0.951553 q^{52} +6.79684 q^{53} -9.14383 q^{54} +10.1265 q^{56} -2.46516 q^{58} -0.408564 q^{59} +13.9603 q^{61} -10.3932 q^{62} +19.5363 q^{63} +7.08110 q^{64} +23.6729 q^{66} -0.781406 q^{67} +0.339426 q^{68} -1.66130 q^{69} +6.37838 q^{71} +13.8062 q^{72} -2.88915 q^{73} +3.50038 q^{74} -21.1779 q^{77} +20.7614 q^{78} -12.5027 q^{79} +2.15259 q^{81} -1.23732 q^{82} +10.3903 q^{83} +2.09722 q^{84} -1.58684 q^{86} +4.75449 q^{87} -14.9663 q^{88} +17.8421 q^{89} -18.5732 q^{91} -0.112781 q^{92} +20.0452 q^{93} +5.72861 q^{94} +3.12333 q^{96} -10.9999 q^{97} -10.8562 q^{98} -28.8735 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - q^{6} + 2 q^{7} + 6 q^{8} + 7 q^{9} - q^{11} + 7 q^{12} + 5 q^{13} - 6 q^{14} - 6 q^{16} - 3 q^{17} + 7 q^{18} + 3 q^{21} + 9 q^{22} - 6 q^{23} + 11 q^{24} + 19 q^{26}+ \cdots - 20 q^{99}+O(q^{100}) $ 6 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - q^6 + 2 * q^7 + 6 * q^8 + 7 * q^9 - q^11 + 7 * q^12 + 5 * q^13 - 6 * q^14 - 6 * q^16 - 3 * q^17 + 7 * q^18 + 3 * q^21 + 9 * q^22 - 6 * q^23 + 11 * q^24 + 19 * q^26 + 18 * q^27 - 4 * q^28 + 3 * q^29 - 3 * q^31 - 6 * q^32 - 18 * q^33 - q^34 + 13 * q^36 - 6 * q^37 + 8 * q^39 + 11 * q^41 - 11 * q^42 + 13 * q^43 + 21 * q^44 - 12 * q^46 - 6 * q^47 - 19 * q^48 + 4 * q^49 - 17 * q^51 - q^52 + 18 * q^53 + 18 * q^54 + 4 * q^56 + 5 * q^58 + 4 * q^59 + 25 * q^61 - 21 * q^62 + 43 * q^63 - 22 * q^64 + 34 * q^66 + 6 * q^67 - q^68 + 13 * q^69 + 18 * q^71 + 13 * q^72 + q^73 - 6 * q^74 - 11 * q^77 + 72 * q^78 + 3 * q^79 + 2 * q^81 + 31 * q^82 - 23 * q^83 + 37 * q^84 + 9 * q^86 + 11 * q^87 + 11 * q^88 + 12 * q^89 - 11 * q^91 + 28 * q^92 - 13 * q^93 + 8 * q^94 - 13 * q^96 + 3 * q^97 - 22 * q^98 - 20 * 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 $

$2$	−1.48119	−1.04736	−0.523681	−	0.851914i	$-0.675442\pi$
$2$	−1.48119	−1.04736	−0.523681	+	0.851914i	$0.675442\pi$
$3$	2.85674	1.64934	0.824669	−	0.565615i	$-0.191361\pi$
$3$	2.85674	1.64934	0.824669	+	0.565615i	$0.191361\pi$
$4$	0.193937	0.0969683
$5$	0	0
$5$	0	0
$6$	−4.23138	−1.72746
$7$	3.78541	1.43075	0.715375	−	0.698740i	$-0.246256\pi$
$7$	3.78541	1.43075	0.715375	+	0.698740i	$0.246256\pi$
$8$	2.67513	0.945802
$9$	5.16096	1.72032
$10$	0	0
$11$	−5.59460	−1.68684	−0.843418	−	0.537258i	$-0.819460\pi$
$11$	−5.59460	−1.68684	−0.843418	+	0.537258i	$0.819460\pi$
$12$	0.554026	0.159934
$13$	−4.90652	−1.36082	−0.680411	−	0.732830i	$-0.738199\pi$
$13$	−4.90652	−1.36082	−0.680411	+	0.732830i	$0.738199\pi$
$14$	−5.60693	−1.49851
$15$	0	0
$16$	−4.35026	−1.08757
$17$	1.75019	0.424484	0.212242	−	0.977217i	$-0.431924\pi$
$17$	1.75019	0.424484	0.212242	+	0.977217i	$0.431924\pi$
$18$	−7.64438	−1.80180
$19$	0	0
$19$	0	0
$20$	0	0
$21$	10.8139	2.35979
$22$	8.28669	1.76673
$23$	−0.581537	−0.121259	−0.0606294	−	0.998160i	$-0.519311\pi$
$23$	−0.581537	−0.121259	−0.0606294	+	0.998160i	$0.519311\pi$
$24$	7.64215	1.55995
$25$	0	0
$26$	7.26750	1.42527
$27$	6.17328	1.18805
$28$	0.734130	0.138737
$29$	1.66431	0.309054	0.154527	−	0.987989i	$-0.450615\pi$
$29$	1.66431	0.309054	0.154527	+	0.987989i	$0.450615\pi$
$30$	0	0
$31$	7.01680	1.26025	0.630127	−	0.776492i	$-0.283003\pi$
$31$	7.01680	1.26025	0.630127	+	0.776492i	$0.283003\pi$
$32$	1.09332	0.193274
$33$	−15.9823	−2.78216
$34$	−2.59237	−0.444588
$35$	0	0
$36$	1.00090	0.166816
$37$	−2.36322	−0.388510	−0.194255	−	0.980951i	$-0.562229\pi$
$37$	−2.36322	−0.388510	−0.194255	+	0.980951i	$0.562229\pi$
$38$	0	0
$39$	−14.0166	−2.24446
$40$	0	0
$41$	0.835351	0.130460	0.0652300	−	0.997870i	$-0.479222\pi$
$41$	0.835351	0.130460	0.0652300	+	0.997870i	$0.479222\pi$
$42$	−16.0175	−2.47156
$43$	1.07133	0.163376	0.0816880	−	0.996658i	$-0.473969\pi$
$43$	1.07133	0.163376	0.0816880	+	0.996658i	$0.473969\pi$
$44$	−1.08500	−0.163570
$45$	0	0
$46$	0.861369	0.127002
$47$	−3.86756	−0.564142	−0.282071	−	0.959394i	$-0.591021\pi$
$47$	−3.86756	−0.564142	−0.282071	+	0.959394i	$0.591021\pi$
$48$	−12.4276	−1.79376
$49$	7.32934	1.04705
$50$	0	0
$51$	4.99984	0.700117
$52$	−0.951553	−0.131957
$53$	6.79684	0.933618	0.466809	−	0.884358i	$-0.345404\pi$
$53$	6.79684	0.933618	0.466809	+	0.884358i	$0.345404\pi$
$54$	−9.14383	−1.24432
$55$	0	0
$56$	10.1265	1.35321
$57$	0	0
$58$	−2.46516	−0.323691
$59$	−0.408564	−0.0531905	−0.0265953	−	0.999646i	$-0.508467\pi$
$59$	−0.408564	−0.0531905	−0.0265953	+	0.999646i	$0.508467\pi$
$60$	0	0
$61$	13.9603	1.78744	0.893718	−	0.448630i	$-0.148088\pi$
$61$	13.9603	1.78744	0.893718	+	0.448630i	$0.148088\pi$
$62$	−10.3932	−1.31994
$63$	19.5363	2.46135
$64$	7.08110	0.885138
$65$	0	0
$66$	23.6729	2.91393
$67$	−0.781406	−0.0954639	−0.0477319	−	0.998860i	$-0.515199\pi$
$67$	−0.781406	−0.0954639	−0.0477319	+	0.998860i	$0.515199\pi$
$68$	0.339426	0.0411614
$69$	−1.66130	−0.199997
$70$	0	0
$71$	6.37838	0.756974	0.378487	−	0.925607i	$-0.376444\pi$
$71$	6.37838	0.756974	0.378487	+	0.925607i	$0.376444\pi$
$72$	13.8062	1.62708
$73$	−2.88915	−0.338150	−0.169075	−	0.985603i	$-0.554078\pi$
$73$	−2.88915	−0.338150	−0.169075	+	0.985603i	$0.554078\pi$
$74$	3.50038	0.406911
$75$	0	0
$76$	0	0
$77$	−21.1779	−2.41344
$78$	20.7614	2.35076
$79$	−12.5027	−1.40666	−0.703331	−	0.710862i	$-0.748305\pi$
$79$	−12.5027	−1.40666	−0.703331	+	0.710862i	$0.748305\pi$
$80$	0	0
$81$	2.15259	0.239177
$82$	−1.23732	−0.136639
$83$	10.3903	1.14048	0.570240	−	0.821478i	$-0.306850\pi$
$83$	10.3903	1.14048	0.570240	+	0.821478i	$0.306850\pi$
$84$	2.09722	0.228825
$85$	0	0
$86$	−1.58684	−0.171114
$87$	4.75449	0.509735
$88$	−14.9663	−1.59541
$89$	17.8421	1.89126	0.945631	−	0.325242i	$-0.105446\pi$
$89$	17.8421	1.89126	0.945631	+	0.325242i	$0.105446\pi$
$90$	0	0
$91$	−18.5732	−1.94700
$92$	−0.112781	−0.0117583
$93$	20.0452	2.07859
$94$	5.72861	0.590861
$95$	0	0
$96$	3.12333	0.318774
$97$	−10.9999	−1.11687	−0.558435	−	0.829548i	$-0.688598\pi$
$97$	−10.9999	−1.11687	−0.558435	+	0.829548i	$0.688598\pi$
$98$	−10.8562	−1.09664
$99$	−28.8735	−2.90189
$100$	0	0
$101$	5.15888	0.513328	0.256664	−	0.966501i	$-0.417377\pi$
$101$	5.15888	0.513328	0.256664	+	0.966501i	$0.417377\pi$
$102$	−7.40573	−0.733277
$103$	7.75851	0.764469	0.382235	−	0.924065i	$-0.375155\pi$
$103$	7.75851	0.764469	0.382235	+	0.924065i	$0.375155\pi$
$104$	−13.1256	−1.28707
$105$	0	0
$106$	−10.0674	−0.977837
$107$	12.6521	1.22313	0.611564	−	0.791195i	$-0.290541\pi$
$107$	12.6521	1.22313	0.611564	+	0.791195i	$0.290541\pi$
$108$	1.19723	0.115203
$109$	−8.19394	−0.784837	−0.392418	−	0.919787i	$-0.628361\pi$
$109$	−8.19394	−0.784837	−0.392418	+	0.919787i	$0.628361\pi$
$110$	0	0
$111$	−6.75109	−0.640785
$112$	−16.4675	−1.55604
$113$	12.5551	1.18108	0.590541	−	0.807007i	$-0.298914\pi$
$113$	12.5551	1.18108	0.590541	+	0.807007i	$0.298914\pi$
$114$	0	0
$115$	0	0
$116$	0.322770	0.0299684
$117$	−25.3223	−2.34105
$118$	0.605163	0.0557098
$119$	6.62519	0.607330
$120$	0	0
$121$	20.2996	1.84541
$122$	−20.6779	−1.87209
$123$	2.38638	0.215173
$124$	1.36081	0.122205
$125$	0	0
$126$	−28.9371	−2.57792
$127$	8.15913	0.724006	0.362003	−	0.932177i	$-0.382093\pi$
$127$	8.15913	0.724006	0.362003	+	0.932177i	$0.382093\pi$
$128$	−12.6751	−1.12033
$129$	3.06050	0.269462
$130$	0	0
$131$	19.2498	1.68186	0.840932	−	0.541141i	$-0.182007\pi$
$131$	19.2498	1.68186	0.840932	+	0.541141i	$0.182007\pi$
$132$	−3.09955	−0.269782
$133$	0	0
$134$	1.15741	0.0999853
$135$	0	0
$136$	4.68199	0.401477
$137$	1.57191	0.134297	0.0671487	−	0.997743i	$-0.478610\pi$
$137$	1.57191	0.134297	0.0671487	+	0.997743i	$0.478610\pi$
$138$	2.46070	0.209469
$139$	−2.04918	−0.173809	−0.0869047	−	0.996217i	$-0.527698\pi$
$139$	−2.04918	−0.173809	−0.0869047	+	0.996217i	$0.527698\pi$
$140$	0	0
$141$	−11.0486	−0.930461
$142$	−9.44762	−0.792826
$143$	27.4500	2.29548
$144$	−22.4515	−1.87096
$145$	0	0
$146$	4.27940	0.354165
$147$	20.9380	1.72694
$148$	−0.458314	−0.0376732
$149$	−1.47568	−0.120892	−0.0604461	−	0.998171i	$-0.519252\pi$
$149$	−1.47568	−0.120892	−0.0604461	+	0.998171i	$0.519252\pi$
$150$	0	0
$151$	6.42543	0.522894	0.261447	−	0.965218i	$-0.415800\pi$
$151$	6.42543	0.522894	0.261447	+	0.965218i	$0.415800\pi$
$152$	0	0
$153$	9.03266	0.730247
$154$	31.3685	2.52775
$155$	0	0
$156$	−2.71834	−0.217641
$157$	22.2889	1.77885	0.889425	−	0.457081i	$-0.151105\pi$
$157$	22.2889	1.77885	0.889425	+	0.457081i	$0.151105\pi$
$158$	18.5189	1.47329
$159$	19.4168	1.53985
$160$	0	0
$161$	−2.20135	−0.173491
$162$	−3.18841	−0.250505
$163$	−15.0321	−1.17741	−0.588703	−	0.808350i	$-0.700361\pi$
$163$	−15.0321	−1.17741	−0.588703	+	0.808350i	$0.700361\pi$
$164$	0.162005	0.0126505
$165$	0	0
$166$	−15.3900	−1.19450
$167$	9.59857	0.742759	0.371380	−	0.928481i	$-0.378885\pi$
$167$	9.59857	0.742759	0.371380	+	0.928481i	$0.378885\pi$
$168$	28.9287	2.23190
$169$	11.0739	0.851838
$170$	0	0
$171$	0	0
$172$	0.207770	0.0158423
$173$	−8.83223	−0.671502	−0.335751	−	0.941951i	$-0.608990\pi$
$173$	−8.83223	−0.671502	−0.335751	+	0.941951i	$0.608990\pi$
$174$	−7.04232	−0.533877
$175$	0	0
$176$	24.3380	1.83454
$177$	−1.16716	−0.0877292
$178$	−26.4277	−1.98084
$179$	10.8178	0.808564	0.404282	−	0.914634i	$-0.367522\pi$
$179$	10.8178	0.808564	0.404282	+	0.914634i	$0.367522\pi$
$180$	0	0
$181$	10.7472	0.798831	0.399416	−	0.916770i	$-0.369213\pi$
$181$	10.7472	0.798831	0.399416	+	0.916770i	$0.369213\pi$
$182$	27.5105	2.03921
$183$	39.8810	2.94809
$184$	−1.55569	−0.114687
$185$	0	0
$186$	−29.6908	−2.17703
$187$	−9.79162	−0.716034
$188$	−0.750062	−0.0547039
$189$	23.3684	1.69980
$190$	0	0
$191$	−6.28362	−0.454667	−0.227334	−	0.973817i	$-0.573001\pi$
$191$	−6.28362	−0.454667	−0.227334	+	0.973817i	$0.573001\pi$
$192$	20.2289	1.45989
$193$	17.8292	1.28337	0.641686	−	0.766967i	$-0.278235\pi$
$193$	17.8292	1.28337	0.641686	+	0.766967i	$0.278235\pi$
$194$	16.2930	1.16977
$195$	0	0
$196$	1.42143	0.101530
$197$	−16.9925	−1.21066	−0.605332	−	0.795973i	$-0.706959\pi$
$197$	−16.9925	−1.21066	−0.605332	+	0.795973i	$0.706959\pi$
$198$	42.7672	3.03934
$199$	16.4442	1.16570	0.582850	−	0.812580i	$-0.301937\pi$
$199$	16.4442	1.16570	0.582850	+	0.812580i	$0.301937\pi$
$200$	0	0
$201$	−2.23227	−0.157452
$202$	−7.64131	−0.537641
$203$	6.30008	0.442179
$204$	0.969651	0.0678892
$205$	0	0
$206$	−11.4919	−0.800676
$207$	−3.00128	−0.208604
$208$	21.3446	1.47998
$209$	0	0
$210$	0	0
$211$	15.5926	1.07344	0.536721	−	0.843760i	$-0.319663\pi$
$211$	15.5926	1.07344	0.536721	+	0.843760i	$0.319663\pi$
$212$	1.31816	0.0905313
$213$	18.2214	1.24851
$214$	−18.7403	−1.28106
$215$	0	0
$216$	16.5143	1.12366
$217$	26.5615	1.80311
$218$	12.1368	0.822009
$219$	−8.25355	−0.557723
$220$	0	0
$221$	−8.58734	−0.577647
$222$	9.99967	0.671134
$223$	7.80626	0.522746	0.261373	−	0.965238i	$-0.415825\pi$
$223$	7.80626	0.522746	0.261373	+	0.965238i	$0.415825\pi$
$224$	4.13867	0.276527
$225$	0	0
$226$	−18.5965	−1.23702
$227$	16.8511	1.11845	0.559223	−	0.829017i	$-0.311100\pi$
$227$	16.8511	1.11845	0.559223	+	0.829017i	$0.311100\pi$
$228$	0	0
$229$	−0.251385	−0.0166120	−0.00830600	−	0.999966i	$-0.502644\pi$
$229$	−0.251385	−0.0166120	−0.00830600	+	0.999966i	$0.502644\pi$
$230$	0	0
$231$	−60.4996	−3.98058
$232$	4.45224	0.292304
$233$	23.2029	1.52007	0.760035	−	0.649882i	$-0.225182\pi$
$233$	23.2029	1.52007	0.760035	+	0.649882i	$0.225182\pi$
$234$	37.5073	2.45193
$235$	0	0
$236$	−0.0792355	−0.00515779
$237$	−35.7169	−2.32006
$238$	−9.81319	−0.636095
$239$	−7.58778	−0.490813	−0.245406	−	0.969420i	$-0.578921\pi$
$239$	−7.58778	−0.490813	−0.245406	+	0.969420i	$0.578921\pi$
$240$	0	0
$241$	−5.51617	−0.355328	−0.177664	−	0.984091i	$-0.556854\pi$
$241$	−5.51617	−0.355328	−0.177664	+	0.984091i	$0.556854\pi$
$242$	−30.0676	−1.93282
$243$	−12.3705	−0.793565
$244$	2.70742	0.173325
$245$	0	0
$246$	−3.53469	−0.225364
$247$	0	0
$248$	18.7708	1.19195
$249$	29.6823	1.88104
$250$	0	0
$251$	−18.6587	−1.17772	−0.588862	−	0.808234i	$-0.700424\pi$
$251$	−18.6587	−1.17772	−0.588862	+	0.808234i	$0.700424\pi$
$252$	3.78881	0.238673
$253$	3.25346	0.204544
$254$	−12.0853	−0.758297
$255$	0	0
$256$	4.61213	0.288258
$257$	0.692703	0.0432097	0.0216048	−	0.999767i	$-0.493122\pi$
$257$	0.692703	0.0432097	0.0216048	+	0.999767i	$0.493122\pi$
$258$	−4.53320	−0.282225
$259$	−8.94574	−0.555861
$260$	0	0
$261$	8.58941	0.531671
$262$	−28.5127	−1.76152
$263$	11.6268	0.716936	0.358468	−	0.933542i	$-0.383299\pi$
$263$	11.6268	0.716936	0.358468	+	0.933542i	$0.383299\pi$
$264$	−42.7548	−2.63137
$265$	0	0
$266$	0	0
$267$	50.9703	3.11933
$268$	−0.151543	−0.00925697
$269$	−23.6115	−1.43962	−0.719809	−	0.694173i	$-0.755770\pi$
$269$	−23.6115	−1.43962	−0.719809	+	0.694173i	$0.755770\pi$
$270$	0	0
$271$	−7.63990	−0.464091	−0.232046	−	0.972705i	$-0.574542\pi$
$271$	−7.63990	−0.464091	−0.232046	+	0.972705i	$0.574542\pi$
$272$	−7.61379	−0.461654
$273$	−53.0587	−3.21126
$274$	−2.32830	−0.140658
$275$	0	0
$276$	−0.322186	−0.0193933
$277$	−18.6537	−1.12079	−0.560397	−	0.828224i	$-0.689352\pi$
$277$	−18.6537	−1.12079	−0.560397	+	0.828224i	$0.689352\pi$
$278$	3.03524	0.182041
$279$	36.2134	2.16804
$280$	0	0
$281$	13.9898	0.834560	0.417280	−	0.908778i	$-0.362983\pi$
$281$	13.9898	0.834560	0.417280	+	0.908778i	$0.362983\pi$
$282$	16.3651	0.974530
$283$	−24.6668	−1.46629	−0.733143	−	0.680074i	$-0.761948\pi$
$283$	−24.6668	−1.46629	−0.733143	+	0.680074i	$0.761948\pi$
$284$	1.23700	0.0734025
$285$	0	0
$286$	−40.6588	−2.40420
$287$	3.16215	0.186656
$288$	5.64258	0.332492
$289$	−13.9368	−0.819814
$290$	0	0
$291$	−31.4238	−1.84210
$292$	−0.560312	−0.0327898
$293$	10.5784	0.617994	0.308997	−	0.951063i	$-0.400007\pi$
$293$	10.5784	0.617994	0.308997	+	0.951063i	$0.400007\pi$
$294$	−31.0132	−1.80873
$295$	0	0
$296$	−6.32191	−0.367454
$297$	−34.5371	−2.00404
$298$	2.18576	0.126618
$299$	2.85332	0.165012
$300$	0	0
$301$	4.05542	0.233750
$302$	−9.51731	−0.547660
$303$	14.7376	0.846652
$304$	0	0
$305$	0	0
$306$	−13.3791	−0.764833
$307$	−12.8244	−0.731930	−0.365965	−	0.930629i	$-0.619261\pi$
$307$	−12.8244	−0.731930	−0.365965	+	0.930629i	$0.619261\pi$
$308$	−4.10716	−0.234027
$309$	22.1640	1.26087
$310$	0	0
$311$	3.42706	0.194331	0.0971655	−	0.995268i	$-0.469022\pi$
$311$	3.42706	0.194331	0.0971655	+	0.995268i	$0.469022\pi$
$312$	−37.4963	−2.12281
$313$	2.84448	0.160779	0.0803896	−	0.996764i	$-0.474384\pi$
$313$	2.84448	0.160779	0.0803896	+	0.996764i	$0.474384\pi$
$314$	−33.0142	−1.86310
$315$	0	0
$316$	−2.42473	−0.136402
$317$	4.66145	0.261813	0.130907	−	0.991395i	$-0.458211\pi$
$317$	4.66145	0.261813	0.130907	+	0.991395i	$0.458211\pi$
$318$	−28.7601	−1.61278
$319$	−9.31113	−0.521323
$320$	0	0
$321$	36.1438	2.01735
$322$	3.26063	0.181708
$323$	0	0
$324$	0.417467	0.0231926
$325$	0	0
$326$	22.2655	1.23317
$327$	−23.4079	−1.29446
$328$	2.23467	0.123389
$329$	−14.6403	−0.807147
$330$	0	0
$331$	−7.00260	−0.384898	−0.192449	−	0.981307i	$-0.561643\pi$
$331$	−7.00260	−0.384898	−0.192449	+	0.981307i	$0.561643\pi$
$332$	2.01505	0.110590
$333$	−12.1964	−0.668361
$334$	−14.2173	−0.777938
$335$	0	0
$336$	−47.0434	−2.56643
$337$	13.5376	0.737441	0.368721	−	0.929540i	$-0.379796\pi$
$337$	13.5376	0.737441	0.368721	+	0.929540i	$0.379796\pi$
$338$	−16.4026	−0.892183
$339$	35.8666	1.94801
$340$	0	0
$341$	−39.2562	−2.12584
$342$	0	0
$343$	1.24667	0.0673140
$344$	2.86594	0.154521
$345$	0	0
$346$	13.0822	0.703306
$347$	−21.9633	−1.17905	−0.589527	−	0.807749i	$-0.700686\pi$
$347$	−21.9633	−1.17905	−0.589527	+	0.807749i	$0.700686\pi$
$348$	0.922069	0.0494281
$349$	−22.1930	−1.18796	−0.593981	−	0.804479i	$-0.702445\pi$
$349$	−22.1930	−1.18796	−0.593981	+	0.804479i	$0.702445\pi$
$350$	0	0
$351$	−30.2893	−1.61672
$352$	−6.11669	−0.326021
$353$	0.679013	0.0361402	0.0180701	−	0.999837i	$-0.494248\pi$
$353$	0.679013	0.0361402	0.0180701	+	0.999837i	$0.494248\pi$
$354$	1.72879	0.0918843
$355$	0	0
$356$	3.46024	0.183392
$357$	18.9264	1.00169
$358$	−16.0233	−0.846859
$359$	−25.3255	−1.33663	−0.668314	−	0.743879i	$-0.732984\pi$
$359$	−25.3255	−1.33663	−0.668314	+	0.743879i	$0.732984\pi$
$360$	0	0
$361$	0	0
$362$	−15.9187	−0.836666
$363$	57.9905	3.04371
$364$	−3.60202	−0.188797
$365$	0	0
$366$	−59.0715	−3.08772
$367$	−11.6631	−0.608807	−0.304404	−	0.952543i	$-0.598457\pi$
$367$	−11.6631	−0.608807	−0.304404	+	0.952543i	$0.598457\pi$
$368$	2.52984	0.131877
$369$	4.31121	0.224433
$370$	0	0
$371$	25.7288	1.33577
$372$	3.88749	0.201557
$373$	13.0913	0.677843	0.338921	−	0.940815i	$-0.389938\pi$
$373$	13.0913	0.677843	0.338921	+	0.940815i	$0.389938\pi$
$374$	14.5033	0.749947
$375$	0	0
$376$	−10.3462	−0.533566
$377$	−8.16594	−0.420568
$378$	−34.6132	−1.78031
$379$	−32.7566	−1.68259	−0.841297	−	0.540573i	$-0.818208\pi$
$379$	−32.7566	−1.68259	−0.841297	+	0.540573i	$0.818208\pi$
$380$	0	0
$381$	23.3085	1.19413
$382$	9.30727	0.476201
$383$	25.6455	1.31042	0.655211	−	0.755446i	$-0.272580\pi$
$383$	25.6455	1.31042	0.655211	+	0.755446i	$0.272580\pi$
$384$	−36.2095	−1.84781
$385$	0	0
$386$	−26.4085	−1.34416
$387$	5.52907	0.281059
$388$	−2.13328	−0.108301
$389$	−18.5466	−0.940349	−0.470174	−	0.882574i	$-0.655809\pi$
$389$	−18.5466	−0.940349	−0.470174	+	0.882574i	$0.655809\pi$
$390$	0	0
$391$	−1.01780	−0.0514723
$392$	19.6069	0.990300
$393$	54.9917	2.77396
$394$	25.1691	1.26800
$395$	0	0
$396$	−5.59962	−0.281392
$397$	−15.3961	−0.772710	−0.386355	−	0.922350i	$-0.626266\pi$
$397$	−15.3961	−0.772710	−0.386355	+	0.922350i	$0.626266\pi$
$398$	−24.3571	−1.22091
$399$	0	0
$400$	0	0
$401$	−20.5913	−1.02828	−0.514140	−	0.857706i	$-0.671889\pi$
$401$	−20.5913	−1.02828	−0.514140	+	0.857706i	$0.671889\pi$
$402$	3.30643	0.164910
$403$	−34.4280	−1.71498
$404$	1.00050	0.0497765
$405$	0	0
$406$	−9.33165	−0.463122
$407$	13.2212	0.655353
$408$	13.3752	0.662172
$409$	−1.62384	−0.0802936	−0.0401468	−	0.999194i	$-0.512783\pi$
$409$	−1.62384	−0.0802936	−0.0401468	+	0.999194i	$0.512783\pi$
$410$	0	0
$411$	4.49054	0.221502
$412$	1.50466	0.0741293
$413$	−1.54658	−0.0761024
$414$	4.44548	0.218484
$415$	0	0
$416$	−5.36440	−0.263011
$417$	−5.85398	−0.286671
$418$	0	0
$419$	16.8087	0.821161	0.410580	−	0.911824i	$-0.365326\pi$
$419$	16.8087	0.821161	0.410580	+	0.911824i	$0.365326\pi$
$420$	0	0
$421$	20.9436	1.02073	0.510365	−	0.859958i	$-0.329510\pi$
$421$	20.9436	1.02073	0.510365	+	0.859958i	$0.329510\pi$
$422$	−23.0957	−1.12428
$423$	−19.9603	−0.970504
$424$	18.1824	0.883017
$425$	0	0
$426$	−26.9894	−1.30764
$427$	52.8455	2.55737
$428$	2.45371	0.118605
$429$	78.4175	3.78603
$430$	0	0
$431$	33.7815	1.62720	0.813600	−	0.581425i	$-0.197505\pi$
$431$	33.7815	1.62720	0.813600	+	0.581425i	$0.197505\pi$
$432$	−26.8554	−1.29208
$433$	−15.0289	−0.722241	−0.361120	−	0.932519i	$-0.617606\pi$
$433$	−15.0289	−0.722241	−0.361120	+	0.932519i	$0.617606\pi$
$434$	−39.3427	−1.88851
$435$	0	0
$436$	−1.58910	−0.0761043
$437$	0	0
$438$	12.2251	0.584139
$439$	37.7214	1.80035	0.900173	−	0.435532i	$-0.143440\pi$
$439$	37.7214	1.80035	0.900173	+	0.435532i	$0.143440\pi$
$440$	0	0
$441$	37.8264	1.80126
$442$	12.7195	0.605006
$443$	13.1144	0.623086	0.311543	−	0.950232i	$-0.399154\pi$
$443$	13.1144	0.623086	0.311543	+	0.950232i	$0.399154\pi$
$444$	−1.30928	−0.0621358
$445$	0	0
$446$	−11.5626	−0.547504
$447$	−4.21562	−0.199392
$448$	26.8049	1.26641
$449$	11.6905	0.551711	0.275855	−	0.961199i	$-0.411039\pi$
$449$	11.6905	0.551711	0.275855	+	0.961199i	$0.411039\pi$
$450$	0	0
$451$	−4.67346	−0.220064
$452$	2.43489	0.114528
$453$	18.3558	0.862430
$454$	−24.9597	−1.17142
$455$	0	0
$456$	0	0
$457$	−21.4290	−1.00240	−0.501202	−	0.865330i	$-0.667109\pi$
$457$	−21.4290	−1.00240	−0.501202	+	0.865330i	$0.667109\pi$
$458$	0.372350	0.0173988
$459$	10.8044	0.504307
$460$	0	0
$461$	−9.65840	−0.449837	−0.224918	−	0.974378i	$-0.572212\pi$
$461$	−9.65840	−0.449837	−0.224918	+	0.974378i	$0.572212\pi$
$462$	89.6117	4.16911
$463$	1.24981	0.0580838	0.0290419	−	0.999578i	$-0.490754\pi$
$463$	1.24981	0.0580838	0.0290419	+	0.999578i	$0.490754\pi$
$464$	−7.24017	−0.336116
$465$	0	0
$466$	−34.3680	−1.59206
$467$	−31.3476	−1.45059	−0.725296	−	0.688437i	$-0.758297\pi$
$467$	−31.3476	−1.45059	−0.725296	+	0.688437i	$0.758297\pi$
$468$	−4.91092	−0.227007
$469$	−2.95794	−0.136585
$470$	0	0
$471$	63.6736	2.93393
$472$	−1.09296	−0.0503077
$473$	−5.99365	−0.275588
$474$	52.9037	2.42995
$475$	0	0
$476$	1.28487	0.0588918
$477$	35.0782	1.60612
$478$	11.2390	0.514059
$479$	4.75460	0.217243	0.108622	−	0.994083i	$-0.465356\pi$
$479$	4.75460	0.217243	0.108622	+	0.994083i	$0.465356\pi$
$480$	0	0
$481$	11.5952	0.528693
$482$	8.17052	0.372157
$483$	−6.28870	−0.286146
$484$	3.93683	0.178947
$485$	0	0
$486$	18.3230	0.831150
$487$	17.0491	0.772569	0.386285	−	0.922380i	$-0.373758\pi$
$487$	17.0491	0.772569	0.386285	+	0.922380i	$0.373758\pi$
$488$	37.3457	1.69056
$489$	−42.9428	−1.94194
$490$	0	0
$491$	−4.28063	−0.193182	−0.0965910	−	0.995324i	$-0.530794\pi$
$491$	−4.28063	−0.193182	−0.0965910	+	0.995324i	$0.530794\pi$
$492$	0.462806	0.0208649
$493$	2.91285	0.131188
$494$	0	0
$495$	0	0
$496$	−30.5249	−1.37061
$497$	24.1448	1.08304
$498$	−43.9652	−1.97013
$499$	−4.82567	−0.216027	−0.108013	−	0.994149i	$-0.534449\pi$
$499$	−4.82567	−0.216027	−0.108013	+	0.994149i	$0.534449\pi$
$500$	0	0
$501$	27.4206	1.22506
$502$	27.6371	1.23350
$503$	−24.2736	−1.08231	−0.541153	−	0.840924i	$-0.682012\pi$
$503$	−24.2736	−1.08231	−0.541153	+	0.840924i	$0.682012\pi$
$504$	52.2623	2.32795
$505$	0	0
$506$	−4.81901	−0.214231
$507$	31.6352	1.40497
$508$	1.58235	0.0702056
$509$	−10.1138	−0.448286	−0.224143	−	0.974556i	$-0.571958\pi$
$509$	−10.1138	−0.448286	−0.224143	+	0.974556i	$0.571958\pi$
$510$	0	0
$511$	−10.9366	−0.483808
$512$	18.5188	0.818423
$513$	0	0
$514$	−1.02603	−0.0452562
$515$	0	0
$516$	0.593543	0.0261293
$517$	21.6375	0.951615
$518$	13.2504	0.582188
$519$	−25.2314	−1.10753
$520$	0	0
$521$	29.3729	1.28685	0.643426	−	0.765508i	$-0.277513\pi$
$521$	29.3729	1.28685	0.643426	+	0.765508i	$0.277513\pi$
$522$	−12.7226	−0.556852
$523$	−11.2000	−0.489740	−0.244870	−	0.969556i	$-0.578745\pi$
$523$	−11.2000	−0.489740	−0.244870	+	0.969556i	$0.578745\pi$
$524$	3.73324	0.163087
$525$	0	0
$526$	−17.2215	−0.750892
$527$	12.2807	0.534957
$528$	69.5272	3.02578
$529$	−22.6618	−0.985296
$530$	0	0
$531$	−2.10858	−0.0915046
$532$	0	0
$533$	−4.09866	−0.177533
$534$	−75.4969	−3.26707
$535$	0	0
$536$	−2.09036	−0.0902899
$537$	30.9037	1.33360
$538$	34.9732	1.50780
$539$	−41.0047	−1.76620
$540$	0	0
$541$	5.69382	0.244797	0.122398	−	0.992481i	$-0.460941\pi$
$541$	5.69382	0.244797	0.122398	+	0.992481i	$0.460941\pi$
$542$	11.3162	0.486072
$543$	30.7019	1.31754
$544$	1.91352	0.0820415
$545$	0	0
$546$	78.5903	3.36335
$547$	11.9818	0.512303	0.256151	−	0.966637i	$-0.417545\pi$
$547$	11.9818	0.512303	0.256151	+	0.966637i	$0.417545\pi$
$548$	0.304851	0.0130226
$549$	72.0486	3.07496
$550$	0	0
$551$	0	0
$552$	−4.44419	−0.189157
$553$	−47.3278	−2.01258
$554$	27.6298	1.17388
$555$	0	0
$556$	−0.397411	−0.0168540
$557$	−34.5573	−1.46424	−0.732119	−	0.681177i	$-0.761469\pi$
$557$	−34.5573	−1.46424	−0.732119	+	0.681177i	$0.761469\pi$
$558$	−53.6390	−2.27072
$559$	−5.25649	−0.222326
$560$	0	0
$561$	−27.9721	−1.18098
$562$	−20.7216	−0.874087
$563$	24.8817	1.04864	0.524318	−	0.851522i	$-0.324320\pi$
$563$	24.8817	1.04864	0.524318	+	0.851522i	$0.324320\pi$
$564$	−2.14273	−0.0902252
$565$	0	0
$566$	36.5363	1.53573
$567$	8.14845	0.342203
$568$	17.0630	0.715947
$569$	−38.7345	−1.62384	−0.811918	−	0.583771i	$-0.801577\pi$
$569$	−38.7345	−1.62384	−0.811918	+	0.583771i	$0.801577\pi$
$570$	0	0
$571$	−4.53514	−0.189790	−0.0948948	−	0.995487i	$-0.530251\pi$
$571$	−4.53514	−0.189790	−0.0948948	+	0.995487i	$0.530251\pi$
$572$	5.32356	0.222589
$573$	−17.9507	−0.749900
$574$	−4.68376	−0.195496
$575$	0	0
$576$	36.5453	1.52272
$577$	21.9514	0.913847	0.456923	−	0.889506i	$-0.348951\pi$
$577$	21.9514	0.913847	0.456923	+	0.889506i	$0.348951\pi$
$578$	20.6432	0.858642
$579$	50.9333	2.11672
$580$	0	0
$581$	39.3314	1.63174
$582$	46.5448	1.92934
$583$	−38.0256	−1.57486
$584$	−7.72886	−0.319823
$585$	0	0
$586$	−15.6686	−0.647264
$587$	41.0464	1.69417	0.847084	−	0.531460i	$-0.178356\pi$
$587$	41.0464	1.69417	0.847084	+	0.531460i	$0.178356\pi$
$588$	4.06064	0.167458
$589$	0	0
$590$	0	0
$591$	−48.5430	−1.99679
$592$	10.2806	0.422530
$593$	9.71964	0.399138	0.199569	−	0.979884i	$-0.436046\pi$
$593$	9.71964	0.399138	0.199569	+	0.979884i	$0.436046\pi$
$594$	51.1561	2.09896
$595$	0	0
$596$	−0.286188	−0.0117227
$597$	46.9769	1.92264
$598$	−4.22632	−0.172827
$599$	−30.4494	−1.24413	−0.622065	−	0.782966i	$-0.713706\pi$
$599$	−30.4494	−1.24413	−0.622065	+	0.782966i	$0.713706\pi$
$600$	0	0
$601$	24.5483	1.00135	0.500673	−	0.865637i	$-0.333086\pi$
$601$	24.5483	1.00135	0.500673	+	0.865637i	$0.333086\pi$
$602$	−6.00686	−0.244821
$603$	−4.03280	−0.164228
$604$	1.24613	0.0507042
$605$	0	0
$606$	−21.8292	−0.886751
$607$	−39.0162	−1.58362	−0.791810	−	0.610767i	$-0.790861\pi$
$607$	−39.0162	−1.58362	−0.791810	+	0.610767i	$0.790861\pi$
$608$	0	0
$609$	17.9977	0.729303
$610$	0	0
$611$	18.9763	0.767697
$612$	1.75176	0.0708108
$613$	−4.97416	−0.200904	−0.100452	−	0.994942i	$-0.532029\pi$
$613$	−4.97416	−0.200904	−0.100452	+	0.994942i	$0.532029\pi$
$614$	18.9955	0.766596
$615$	0	0
$616$	−56.6536	−2.28264
$617$	38.6420	1.55567	0.777834	−	0.628470i	$-0.216318\pi$
$617$	38.6420	1.55567	0.777834	+	0.628470i	$0.216318\pi$
$618$	−32.8293	−1.32059
$619$	−27.9053	−1.12161	−0.560804	−	0.827949i	$-0.689508\pi$
$619$	−27.9053	−1.12161	−0.560804	+	0.827949i	$0.689508\pi$
$620$	0	0
$621$	−3.58999	−0.144061
$622$	−5.07615	−0.203535
$623$	67.5398	2.70592
$624$	60.9760	2.44099
$625$	0	0
$626$	−4.21322	−0.168394
$627$	0	0
$628$	4.32264	0.172492
$629$	−4.13608	−0.164916
$630$	0	0
$631$	−13.2402	−0.527085	−0.263543	−	0.964648i	$-0.584891\pi$
$631$	−13.2402	−0.527085	−0.263543	+	0.964648i	$0.584891\pi$
$632$	−33.4463	−1.33042
$633$	44.5441	1.77047
$634$	−6.90451	−0.274213
$635$	0	0
$636$	3.76563	0.149317
$637$	−35.9615	−1.42485
$638$	13.7916	0.546014
$639$	32.9185	1.30224
$640$	0	0
$641$	37.5109	1.48159	0.740796	−	0.671730i	$-0.234448\pi$
$641$	37.5109	1.48159	0.740796	+	0.671730i	$0.234448\pi$
$642$	−53.5360	−2.11290
$643$	24.3939	0.962000	0.481000	−	0.876721i	$-0.340274\pi$
$643$	24.3939	0.962000	0.481000	+	0.876721i	$0.340274\pi$
$644$	−0.426923	−0.0168231
$645$	0	0
$646$	0	0
$647$	−8.05266	−0.316583	−0.158291	−	0.987392i	$-0.550599\pi$
$647$	−8.05266	−0.316583	−0.158291	+	0.987392i	$0.550599\pi$
$648$	5.75847	0.226214
$649$	2.28575	0.0897237
$650$	0	0
$651$	75.8791	2.97394
$652$	−2.91528	−0.114171
$653$	19.9890	0.782228	0.391114	−	0.920342i	$-0.372090\pi$
$653$	19.9890	0.782228	0.391114	+	0.920342i	$0.372090\pi$
$654$	34.6717	1.35577
$655$	0	0
$656$	−3.63400	−0.141884
$657$	−14.9108	−0.581725
$658$	21.6852	0.845375
$659$	−32.4393	−1.26366	−0.631829	−	0.775108i	$-0.717695\pi$
$659$	−32.4393	−1.26366	−0.631829	+	0.775108i	$0.717695\pi$
$660$	0	0
$661$	−35.5542	−1.38290	−0.691448	−	0.722426i	$-0.743027\pi$
$661$	−35.5542	−1.38290	−0.691448	+	0.722426i	$0.743027\pi$
$662$	10.3722	0.403127
$663$	−24.5318	−0.952735
$664$	27.7953	1.07867
$665$	0	0
$666$	18.0653	0.700017
$667$	−0.967855	−0.0374755
$668$	1.86151	0.0720241
$669$	22.3004	0.862185
$670$	0	0
$671$	−78.1024	−3.01511
$672$	11.8231	0.456086
$673$	−27.5963	−1.06376	−0.531880	−	0.846820i	$-0.678514\pi$
$673$	−27.5963	−1.06376	−0.531880	+	0.846820i	$0.678514\pi$
$674$	−20.0518	−0.772368
$675$	0	0
$676$	2.14763	0.0826013
$677$	−21.2114	−0.815221	−0.407610	−	0.913156i	$-0.633638\pi$
$677$	−21.2114	−0.815221	−0.407610	+	0.913156i	$0.633638\pi$
$678$	−53.1254	−2.04027
$679$	−41.6391	−1.59796
$680$	0	0
$681$	48.1391	1.84470
$682$	58.1460	2.22653
$683$	−19.3626	−0.740891	−0.370446	−	0.928854i	$-0.620795\pi$
$683$	−19.3626	−0.740891	−0.370446	+	0.928854i	$0.620795\pi$
$684$	0	0
$685$	0	0
$686$	−1.84656	−0.0705021
$687$	−0.718142	−0.0273988
$688$	−4.66056	−0.177682
$689$	−33.3488	−1.27049
$690$	0	0
$691$	−1.28081	−0.0487241	−0.0243621	−	0.999703i	$-0.507755\pi$
$691$	−1.28081	−0.0487241	−0.0243621	+	0.999703i	$0.507755\pi$
$692$	−1.71289	−0.0651144
$693$	−109.298	−4.15189
$694$	32.5320	1.23490
$695$	0	0
$696$	12.7189	0.482108
$697$	1.46202	0.0553781
$698$	32.8721	1.24423
$699$	66.2845	2.50711
$700$	0	0
$701$	−37.1297	−1.40237	−0.701185	−	0.712979i	$-0.747345\pi$
$701$	−37.1297	−1.40237	−0.701185	+	0.712979i	$0.747345\pi$
$702$	44.8644	1.69330
$703$	0	0
$704$	−39.6159	−1.49308
$705$	0	0
$706$	−1.00575	−0.0378519
$707$	19.5285	0.734445
$708$	−0.226355	−0.00850695
$709$	13.1577	0.494149	0.247074	−	0.968997i	$-0.420531\pi$
$709$	13.1577	0.494149	0.247074	+	0.968997i	$0.420531\pi$
$710$	0	0
$711$	−64.5258	−2.41991
$712$	47.7300	1.78876
$713$	−4.08052	−0.152817
$714$	−28.0337	−1.04914
$715$	0	0
$716$	2.09798	0.0784050
$717$	−21.6763	−0.809516
$718$	37.5120	1.39993
$719$	14.5571	0.542886	0.271443	−	0.962454i	$-0.412499\pi$
$719$	14.5571	0.542886	0.271443	+	0.962454i	$0.412499\pi$
$720$	0	0
$721$	29.3692	1.09376
$722$	0	0
$723$	−15.7583	−0.586056
$724$	2.08427	0.0774613
$725$	0	0
$726$	−85.8952	−3.18787
$727$	−8.97163	−0.332739	−0.166370	−	0.986063i	$-0.553204\pi$
$727$	−8.97163	−0.332739	−0.166370	+	0.986063i	$0.553204\pi$
$728$	−49.6857	−1.84147
$729$	−41.7969	−1.54803
$730$	0	0
$731$	1.87503	0.0693504
$732$	7.73438	0.285871
$733$	−24.7325	−0.913516	−0.456758	−	0.889591i	$-0.650990\pi$
$733$	−24.7325	−0.913516	−0.456758	+	0.889591i	$0.650990\pi$
$734$	17.2753	0.637642
$735$	0	0
$736$	−0.635806	−0.0234361
$737$	4.37165	0.161032
$738$	−6.38574	−0.235062
$739$	−1.56918	−0.0577230	−0.0288615	−	0.999583i	$-0.509188\pi$
$739$	−1.56918	−0.0577230	−0.0288615	+	0.999583i	$0.509188\pi$
$740$	0	0
$741$	0	0
$742$	−38.1094	−1.39904
$743$	24.2926	0.891208	0.445604	−	0.895230i	$-0.352989\pi$
$743$	24.2926	0.891208	0.445604	+	0.895230i	$0.352989\pi$
$744$	53.6234	1.96593
$745$	0	0
$746$	−19.3908	−0.709947
$747$	53.6237	1.96199
$748$	−1.89895	−0.0694326
$749$	47.8935	1.74999
$750$	0	0
$751$	34.9542	1.27550	0.637748	−	0.770245i	$-0.279866\pi$
$751$	34.9542	1.27550	0.637748	+	0.770245i	$0.279866\pi$
$752$	16.8249	0.613541
$753$	−53.3029	−1.94247
$754$	12.0954	0.440487
$755$	0	0
$756$	4.53199	0.164827
$757$	3.22086	0.117064	0.0585321	−	0.998286i	$-0.481358\pi$
$757$	3.22086	0.117064	0.0585321	+	0.998286i	$0.481358\pi$
$758$	48.5189	1.76229
$759$	9.29430	0.337362
$760$	0	0
$761$	3.72402	0.134996	0.0674979	−	0.997719i	$-0.478498\pi$
$761$	3.72402	0.134996	0.0674979	+	0.997719i	$0.478498\pi$
$762$	−34.5244	−1.25069
$763$	−31.0174	−1.12291
$764$	−1.21862	−0.0440883
$765$	0	0
$766$	−37.9859	−1.37249
$767$	2.00463	0.0723829
$768$	13.1756	0.475435
$769$	−32.2064	−1.16139	−0.580695	−	0.814121i	$-0.697219\pi$
$769$	−32.2064	−1.16139	−0.580695	+	0.814121i	$0.697219\pi$
$770$	0	0
$771$	1.97887	0.0712674
$772$	3.45773	0.124446
$773$	−20.6833	−0.743926	−0.371963	−	0.928248i	$-0.621315\pi$
$773$	−20.6833	−0.743926	−0.371963	+	0.928248i	$0.621315\pi$
$774$	−8.18963	−0.294370
$775$	0	0
$776$	−29.4262	−1.05634
$777$	−25.5556	−0.916804
$778$	27.4711	0.984886
$779$	0	0
$780$	0	0
$781$	−35.6845	−1.27689
$782$	1.50756	0.0539102
$783$	10.2742	0.367171
$784$	−31.8845	−1.13873
$785$	0	0
$786$	−81.4534	−2.90535
$787$	46.0227	1.64053	0.820266	−	0.571982i	$-0.193825\pi$
$787$	46.0227	1.64053	0.820266	+	0.571982i	$0.193825\pi$
$788$	−3.29546	−0.117396
$789$	33.2146	1.18247
$790$	0	0
$791$	47.5261	1.68983
$792$	−77.2403	−2.74462
$793$	−68.4965	−2.43238
$794$	22.8047	0.809307
$795$	0	0
$796$	3.18914	0.113036
$797$	25.0847	0.888546	0.444273	−	0.895891i	$-0.353462\pi$
$797$	25.0847	0.888546	0.444273	+	0.895891i	$0.353462\pi$
$798$	0	0
$799$	−6.76897	−0.239469
$800$	0	0
$801$	92.0824	3.25357
$802$	30.4997	1.07698
$803$	16.1637	0.570403
$804$	−0.432919	−0.0152679
$805$	0	0
$806$	50.9946	1.79621
$807$	−67.4518	−2.37442
$808$	13.8007	0.485507
$809$	27.6272	0.971319	0.485660	−	0.874148i	$-0.338579\pi$
$809$	27.6272	0.971319	0.485660	+	0.874148i	$0.338579\pi$
$810$	0	0
$811$	−38.4507	−1.35019	−0.675094	−	0.737732i	$-0.735897\pi$
$811$	−38.4507	−1.35019	−0.675094	+	0.737732i	$0.735897\pi$
$812$	1.22182	0.0428774
$813$	−21.8252	−0.765443
$814$	−19.5832	−0.686392
$815$	0	0
$816$	−21.7506	−0.761423
$817$	0	0
$818$	2.40522	0.0840966
$819$	−95.8553	−3.34946
$820$	0	0
$821$	52.6678	1.83812	0.919060	−	0.394117i	$-0.128950\pi$
$821$	52.6678	1.83812	0.919060	+	0.394117i	$0.128950\pi$
$822$	−6.65136	−0.231993
$823$	4.67194	0.162854	0.0814269	−	0.996679i	$-0.474052\pi$
$823$	4.67194	0.162854	0.0814269	+	0.996679i	$0.474052\pi$
$824$	20.7550	0.723036
$825$	0	0
$826$	2.29079	0.0797068
$827$	27.4248	0.953653	0.476827	−	0.878997i	$-0.341787\pi$
$827$	27.4248	0.953653	0.476827	+	0.878997i	$0.341787\pi$
$828$	−0.582059	−0.0202279
$829$	0.156181	0.00542438	0.00271219	−	0.999996i	$-0.499137\pi$
$829$	0.156181	0.00542438	0.00271219	+	0.999996i	$0.499137\pi$
$830$	0	0
$831$	−53.2888	−1.84857
$832$	−34.7435	−1.20452
$833$	12.8277	0.444455
$834$	8.67088	0.300248
$835$	0	0
$836$	0	0
$837$	43.3167	1.49724
$838$	−24.8970	−0.860053
$839$	−29.1892	−1.00772	−0.503862	−	0.863784i	$-0.668088\pi$
$839$	−29.1892	−1.00772	−0.503862	+	0.863784i	$0.668088\pi$
$840$	0	0
$841$	−26.2301	−0.904486
$842$	−31.0216	−1.06907
$843$	39.9651	1.37647
$844$	3.02398	0.104090
$845$	0	0
$846$	29.5651	1.01647
$847$	76.8421	2.64033
$848$	−29.5680	−1.01537
$849$	−70.4665	−2.41840
$850$	0	0
$851$	1.37430	0.0471103
$852$	3.53379	0.121066
$853$	−45.7663	−1.56701	−0.783505	−	0.621386i	$-0.786570\pi$
$853$	−45.7663	−1.56701	−0.783505	+	0.621386i	$0.786570\pi$
$854$	−78.2745	−2.67850
$855$	0	0
$856$	33.8461	1.15684
$857$	14.9161	0.509525	0.254763	−	0.967004i	$-0.418003\pi$
$857$	14.9161	0.509525	0.254763	+	0.967004i	$0.418003\pi$
$858$	−116.151	−3.96535
$859$	49.2200	1.67936	0.839682	−	0.543078i	$-0.182741\pi$
$859$	49.2200	1.67936	0.839682	+	0.543078i	$0.182741\pi$
$860$	0	0
$861$	9.03343	0.307858
$862$	−50.0370	−1.70427
$863$	20.9577	0.713407	0.356704	−	0.934218i	$-0.383901\pi$
$863$	20.9577	0.713407	0.356704	+	0.934218i	$0.383901\pi$
$864$	6.74938	0.229619
$865$	0	0
$866$	22.2607	0.756448
$867$	−39.8139	−1.35215
$868$	5.15124	0.174844
$869$	69.9476	2.37281
$870$	0	0
$871$	3.83398	0.129909
$872$	−21.9199	−0.742300
$873$	−56.7700	−1.92137
$874$	0	0
$875$	0	0
$876$	−1.60067	−0.0540815
$877$	−26.2017	−0.884769	−0.442385	−	0.896825i	$-0.645867\pi$
$877$	−26.2017	−0.884769	−0.442385	+	0.896825i	$0.645867\pi$
$878$	−55.8728	−1.88562
$879$	30.2196	1.01928
$880$	0	0
$881$	−45.3499	−1.52788	−0.763938	−	0.645289i	$-0.776737\pi$
$881$	−45.3499	−1.52788	−0.763938	+	0.645289i	$0.776737\pi$
$882$	−56.0282	−1.88657
$883$	−43.3170	−1.45773	−0.728867	−	0.684655i	$-0.759953\pi$
$883$	−43.3170	−1.45773	−0.728867	+	0.684655i	$0.759953\pi$
$884$	−1.66540	−0.0560134
$885$	0	0
$886$	−19.4250	−0.652597
$887$	39.0147	1.30998	0.654992	−	0.755636i	$-0.272672\pi$
$887$	39.0147	1.30998	0.654992	+	0.755636i	$0.272672\pi$
$888$	−18.0600	−0.606055
$889$	30.8857	1.03587
$890$	0	0
$891$	−12.0429	−0.403452
$892$	1.51392	0.0506898
$893$	0	0
$894$	6.24416	0.208836
$895$	0	0
$896$	−47.9806	−1.60292
$897$	8.15118	0.272160
$898$	−17.3160	−0.577841
$899$	11.6781	0.389486
$900$	0	0
$901$	11.8958	0.396306
$902$	6.92230	0.230487
$903$	11.5853	0.385533
$904$	33.5865	1.11707
$905$	0	0
$906$	−27.1885	−0.903277
$907$	−50.1608	−1.66556	−0.832780	−	0.553603i	$-0.813252\pi$
$907$	−50.1608	−1.66556	−0.832780	+	0.553603i	$0.813252\pi$
$908$	3.26804	0.108454
$909$	26.6248	0.883088
$910$	0	0
$911$	−50.4028	−1.66992	−0.834959	−	0.550312i	$-0.814509\pi$
$911$	−50.4028	−1.66992	−0.834959	+	0.550312i	$0.814509\pi$
$912$	0	0
$913$	−58.1294	−1.92380
$914$	31.7405	1.04988
$915$	0	0
$916$	−0.0487528	−0.00161084
$917$	72.8685	2.40633
$918$	−16.0035	−0.528193
$919$	48.7246	1.60728	0.803638	−	0.595118i	$-0.202895\pi$
$919$	48.7246	1.60728	0.803638	+	0.595118i	$0.202895\pi$
$920$	0	0
$921$	−36.6361	−1.20720
$922$	14.3060	0.471142
$923$	−31.2956	−1.03011
$924$	−11.7331	−0.385990
$925$	0	0
$926$	−1.85122	−0.0608348
$927$	40.0413	1.31513
$928$	1.81962	0.0597320
$929$	8.03664	0.263673	0.131837	−	0.991271i	$-0.457913\pi$
$929$	8.03664	0.263673	0.131837	+	0.991271i	$0.457913\pi$
$930$	0	0
$931$	0	0
$932$	4.49989	0.147399
$933$	9.79022	0.320518
$934$	46.4318	1.51930
$935$	0	0
$936$	−67.7405	−2.21417
$937$	−25.5426	−0.834441	−0.417221	−	0.908805i	$-0.636996\pi$
$937$	−25.5426	−0.834441	−0.417221	+	0.908805i	$0.636996\pi$
$938$	4.38129	0.143054
$939$	8.12592	0.265179
$940$	0	0
$941$	8.61494	0.280839	0.140419	−	0.990092i	$-0.455155\pi$
$941$	8.61494	0.280839	0.140419	+	0.990092i	$0.455155\pi$
$942$	−94.3130	−3.07288
$943$	−0.485787	−0.0158194
$944$	1.77736	0.0578482
$945$	0	0
$946$	8.87776	0.288641
$947$	38.4276	1.24873	0.624364	−	0.781133i	$-0.285358\pi$
$947$	38.4276	1.24873	0.624364	+	0.781133i	$0.285358\pi$
$948$	−6.92682	−0.224973
$949$	14.1757	0.460162
$950$	0	0
$951$	13.3165	0.431819
$952$	17.7233	0.574414
$953$	−8.89041	−0.287988	−0.143994	−	0.989579i	$-0.545995\pi$
$953$	−8.89041	−0.287988	−0.143994	+	0.989579i	$0.545995\pi$
$954$	−51.9576	−1.68219
$955$	0	0
$956$	−1.47155	−0.0475933
$957$	−26.5995	−0.859838
$958$	−7.04248	−0.227532
$959$	5.95033	0.192146
$960$	0	0
$961$	18.2354	0.588239
$962$	−17.1747	−0.553734
$963$	65.2971	2.10417
$964$	−1.06979	−0.0344555
$965$	0	0
$966$	9.31478	0.299698
$967$	61.6297	1.98188	0.990939	−	0.134314i	$-0.0428831\pi$
$967$	61.6297	1.98188	0.990939	+	0.134314i	$0.0428831\pi$
$968$	54.3040	1.74540
$969$	0	0
$970$	0	0
$971$	5.85997	0.188055	0.0940277	−	0.995570i	$-0.470026\pi$
$971$	5.85997	0.188055	0.0940277	+	0.995570i	$0.470026\pi$
$972$	−2.39908	−0.0769507
$973$	−7.75700	−0.248678
$974$	−25.2531	−0.809160
$975$	0	0
$976$	−60.7310	−1.94395
$977$	−2.40082	−0.0768089	−0.0384044	−	0.999262i	$-0.512228\pi$
$977$	−2.40082	−0.0768089	−0.0384044	+	0.999262i	$0.512228\pi$
$978$	63.6066	2.03392
$979$	−99.8196	−3.19025
$980$	0	0
$981$	−42.2885	−1.35017
$982$	6.34044	0.202332
$983$	18.3685	0.585864	0.292932	−	0.956133i	$-0.405369\pi$
$983$	18.3685	0.585864	0.292932	+	0.956133i	$0.405369\pi$
$984$	6.38388	0.203511
$985$	0	0
$986$	−4.31450	−0.137402
$987$	−41.8236	−1.33126
$988$	0	0
$989$	−0.623016	−0.0198108
$990$	0	0
$991$	10.2947	0.327021	0.163511	−	0.986542i	$-0.447718\pi$
$991$	10.2947	0.327021	0.163511	+	0.986542i	$0.447718\pi$
$992$	7.67161	0.243574
$993$	−20.0046	−0.634826
$994$	−35.7631	−1.13434
$995$	0	0
$996$	5.75648	0.182401
$997$	2.30230	0.0729146	0.0364573	−	0.999335i	$-0.488393\pi$
$997$	2.30230	0.0729146	0.0364573	+	0.999335i	$0.488393\pi$
$998$	7.14776	0.226258
$999$	−14.5888	−0.461569

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bz.1.2		6
5.4	even	2		9025.2.a.br.1.5		6
19.7	even	3		475.2.e.f.201.5	yes	12
19.11	even	3		475.2.e.f.26.5	✓	12
19.18	odd	2		9025.2.a.bs.1.5		6
95.7	odd	12		475.2.j.d.49.10		24
95.49	even	6		475.2.e.h.26.2	yes	12
95.64	even	6		475.2.e.h.201.2	yes	12
95.68	odd	12		475.2.j.d.349.10		24
95.83	odd	12		475.2.j.d.49.3		24
95.87	odd	12		475.2.j.d.349.3		24
95.94	odd	2		9025.2.a.by.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.e.f.26.5	✓	12	19.11	even	3
475.2.e.f.201.5	yes	12	19.7	even	3
475.2.e.h.26.2	yes	12	95.49	even	6
475.2.e.h.201.2	yes	12	95.64	even	6
475.2.j.d.49.3		24	95.83	odd	12
475.2.j.d.49.10		24	95.7	odd	12
475.2.j.d.349.3		24	95.87	odd	12
475.2.j.d.349.10		24	95.68	odd	12
9025.2.a.br.1.5		6	5.4	even	2
9025.2.a.bs.1.5		6	19.18	odd	2
9025.2.a.by.1.2		6	95.94	odd	2
9025.2.a.bz.1.2		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q-1.48119 q^{2} +2.85674 q^{3} +0.193937 q^{4} -4.23138 q^{6} +3.78541 q^{7} +2.67513 q^{8} +5.16096 q^{9} -5.59460 q^{11} +0.554026 q^{12} -4.90652 q^{13} -5.60693 q^{14} -4.35026 q^{16} +1.75019 q^{17} -7.64438 q^{18} +10.8139 q^{21} +8.28669 q^{22} -0.581537 q^{23} +7.64215 q^{24} +7.26750 q^{26} +6.17328 q^{27} +0.734130 q^{28} +1.66431 q^{29} +7.01680 q^{31} +1.09332 q^{32} -15.9823 q^{33} -2.59237 q^{34} +1.00090 q^{36} -2.36322 q^{37} -14.0166 q^{39} +0.835351 q^{41} -16.0175 q^{42} +1.07133 q^{43} -1.08500 q^{44} +0.861369 q^{46} -3.86756 q^{47} -12.4276 q^{48} +7.32934 q^{49} +4.99984 q^{51} -0.951553 q^{52} +6.79684 q^{53} -9.14383 q^{54} +10.1265 q^{56} -2.46516 q^{58} -0.408564 q^{59} +13.9603 q^{61} -10.3932 q^{62} +19.5363 q^{63} +7.08110 q^{64} +23.6729 q^{66} -0.781406 q^{67} +0.339426 q^{68} -1.66130 q^{69} +6.37838 q^{71} +13.8062 q^{72} -2.88915 q^{73} +3.50038 q^{74} -21.1779 q^{77} +20.7614 q^{78} -12.5027 q^{79} +2.15259 q^{81} -1.23732 q^{82} +10.3903 q^{83} +2.09722 q^{84} -1.58684 q^{86} +4.75449 q^{87} -14.9663 q^{88} +17.8421 q^{89} -18.5732 q^{91} -0.112781 q^{92} +20.0452 q^{93} +5.72861 q^{94} +3.12333 q^{96} -10.9999 q^{97} -10.8562 q^{98} -28.8735 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - q^{6} + 2 q^{7} + 6 q^{8} + 7 q^{9} - q^{11} + 7 q^{12} + 5 q^{13} - 6 q^{14} - 6 q^{16} - 3 q^{17} + 7 q^{18} + 3 q^{21} + 9 q^{22} - 6 q^{23} + 11 q^{24} + 19 q^{26}+ \cdots - 20 q^{99}+O(q^{100}) \) 6 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - q^6 + 2 * q^7 + 6 * q^8 + 7 * q^9 - q^11 + 7 * q^12 + 5 * q^13 - 6 * q^14 - 6 * q^16 - 3 * q^17 + 7 * q^18 + 3 * q^21 + 9 * q^22 - 6 * q^23 + 11 * q^24 + 19 * q^26 + 18 * q^27 - 4 * q^28 + 3 * q^29 - 3 * q^31 - 6 * q^32 - 18 * q^33 - q^34 + 13 * q^36 - 6 * q^37 + 8 * q^39 + 11 * q^41 - 11 * q^42 + 13 * q^43 + 21 * q^44 - 12 * q^46 - 6 * q^47 - 19 * q^48 + 4 * q^49 - 17 * q^51 - q^52 + 18 * q^53 + 18 * q^54 + 4 * q^56 + 5 * q^58 + 4 * q^59 + 25 * q^61 - 21 * q^62 + 43 * q^63 - 22 * q^64 + 34 * q^66 + 6 * q^67 - q^68 + 13 * q^69 + 18 * q^71 + 13 * q^72 + q^73 - 6 * q^74 - 11 * q^77 + 72 * q^78 + 3 * q^79 + 2 * q^81 + 31 * q^82 - 23 * q^83 + 37 * q^84 + 9 * q^86 + 11 * q^87 + 11 * q^88 + 12 * q^89 - 11 * q^91 + 28 * q^92 - 13 * q^93 + 8 * q^94 - 13 * q^96 + 3 * q^97 - 22 * q^98 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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