LMFDB - Embedded newform 9025.2.a.bs.1.5

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:	$1$
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			1.5
Root			$2.85674$ 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	$-0.324558\pi$
$2$	1.48119	1.04736	0.523681	+	0.851914i	$0.324558\pi$
$3$	−2.85674	−1.64934	−0.824669	−	0.565615i	$-0.808639\pi$
$3$	−2.85674	−1.64934	−0.824669	+	0.565615i	$0.808639\pi$
$4$	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	$-0.261801\pi$
$13$	4.90652	1.36082	0.680411	+	0.732830i	$0.261801\pi$
$14$	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	$-0.549385\pi$
$29$	−1.66431	−0.309054	−0.154527	+	0.987989i	$0.549385\pi$
$30$	0	0
$31$	−7.01680	−1.26025	−0.630127	−	0.776492i	$-0.716997\pi$
$31$	−7.01680	−1.26025	−0.630127	+	0.776492i	$0.716997\pi$
$32$	−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	$-0.437771\pi$
$37$	2.36322	0.388510	0.194255	+	0.980951i	$0.437771\pi$
$38$	0	0
$39$	−14.0166	−2.24446
$40$	0	0
$41$	−0.835351	−0.130460	−0.0652300	−	0.997870i	$-0.520778\pi$
$41$	−0.835351	−0.130460	−0.0652300	+	0.997870i	$0.520778\pi$
$42$	−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	$-0.654596\pi$
$53$	−6.79684	−0.933618	−0.466809	+	0.884358i	$0.654596\pi$
$54$	−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	$-0.491533\pi$
$59$	0.408564	0.0531905	0.0265953	+	0.999646i	$0.491533\pi$
$60$	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	$-0.484801\pi$
$67$	0.781406	0.0954639	0.0477319	+	0.998860i	$0.484801\pi$
$68$	0.339426	0.0411614
$69$	1.66130	0.199997
$70$	0	0
$71$	−6.37838	−0.756974	−0.378487	−	0.925607i	$-0.623556\pi$
$71$	−6.37838	−0.756974	−0.378487	+	0.925607i	$0.623556\pi$
$72$	−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	$-0.251695\pi$
$79$	12.5027	1.40666	0.703331	+	0.710862i	$0.251695\pi$
$80$	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	$-0.894554\pi$
$89$	−17.8421	−1.89126	−0.945631	+	0.325242i	$0.894554\pi$
$90$	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	$-0.311402\pi$
$97$	10.9999	1.11687	0.558435	+	0.829548i	$0.311402\pi$
$98$	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	$-0.624845\pi$
$103$	−7.75851	−0.764469	−0.382235	+	0.924065i	$0.624845\pi$
$104$	−13.1256	−1.28707
$105$	0	0
$106$	−10.0674	−0.977837
$107$	−12.6521	−1.22313	−0.611564	−	0.791195i	$-0.709459\pi$
$107$	−12.6521	−1.22313	−0.611564	+	0.791195i	$0.709459\pi$
$108$	−1.19723	−0.115203
$109$	8.19394	0.784837	0.392418	−	0.919787i	$-0.371639\pi$
$109$	8.19394	0.784837	0.392418	+	0.919787i	$0.371639\pi$
$110$	0	0
$111$	−6.75109	−0.640785
$112$	−16.4675	−1.55604
$113$	−12.5551	−1.18108	−0.590541	−	0.807007i	$-0.701086\pi$
$113$	−12.5551	−1.18108	−0.590541	+	0.807007i	$0.701086\pi$
$114$	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	$-0.617907\pi$
$127$	−8.15913	−0.724006	−0.362003	+	0.932177i	$0.617907\pi$
$128$	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	$-0.584200\pi$
$151$	−6.42543	−0.522894	−0.261447	+	0.965218i	$0.584200\pi$
$152$	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	$-0.621115\pi$
$167$	−9.59857	−0.742759	−0.371380	+	0.928481i	$0.621115\pi$
$168$	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	$-0.391010\pi$
$173$	8.83223	0.671502	0.335751	+	0.941951i	$0.391010\pi$
$174$	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	$-0.632478\pi$
$179$	−10.8178	−0.808564	−0.404282	+	0.914634i	$0.632478\pi$
$180$	0	0
$181$	−10.7472	−0.798831	−0.399416	−	0.916770i	$-0.630787\pi$
$181$	−10.7472	−0.798831	−0.399416	+	0.916770i	$0.630787\pi$
$182$	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	$-0.721765\pi$
$193$	−17.8292	−1.28337	−0.641686	+	0.766967i	$0.721765\pi$
$194$	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	$-0.680337\pi$
$211$	−15.5926	−1.07344	−0.536721	+	0.843760i	$0.680337\pi$
$212$	−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	$-0.584175\pi$
$223$	−7.80626	−0.522746	−0.261373	+	0.965238i	$0.584175\pi$
$224$	−4.13867	−0.276527
$225$	0	0
$226$	−18.5965	−1.23702
$227$	−16.8511	−1.11845	−0.559223	−	0.829017i	$-0.688900\pi$
$227$	−16.8511	−1.11845	−0.559223	+	0.829017i	$0.688900\pi$
$228$	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	$-0.443146\pi$
$241$	5.51617	0.355328	0.177664	+	0.984091i	$0.443146\pi$
$242$	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	$-0.506878\pi$
$257$	−0.692703	−0.0432097	−0.0216048	+	0.999767i	$0.506878\pi$
$258$	−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	$-0.244230\pi$
$269$	23.6115	1.43962	0.719809	+	0.694173i	$0.244230\pi$
$270$	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	$-0.637017\pi$
$281$	−13.9898	−0.834560	−0.417280	+	0.908778i	$0.637017\pi$
$282$	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	$-0.599993\pi$
$293$	−10.5784	−0.617994	−0.308997	+	0.951063i	$0.599993\pi$
$294$	−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	$-0.380739\pi$
$307$	12.8244	0.731930	0.365965	+	0.930629i	$0.380739\pi$
$308$	−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	$-0.541789\pi$
$317$	−4.66145	−0.261813	−0.130907	+	0.991395i	$0.541789\pi$
$318$	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	$-0.438357\pi$
$331$	7.00260	0.384898	0.192449	+	0.981307i	$0.438357\pi$
$332$	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	$-0.620204\pi$
$337$	−13.5376	−0.737441	−0.368721	+	0.929540i	$0.620204\pi$
$338$	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	$-0.610062\pi$
$373$	−13.0913	−0.677843	−0.338921	+	0.940815i	$0.610062\pi$
$374$	−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	$-0.181792\pi$
$379$	32.7566	1.68259	0.841297	+	0.540573i	$0.181792\pi$
$380$	0	0
$381$	23.3085	1.19413
$382$	−9.30727	−0.476201
$383$	−25.6455	−1.31042	−0.655211	−	0.755446i	$-0.727420\pi$
$383$	−25.6455	−1.31042	−0.655211	+	0.755446i	$0.727420\pi$
$384$	−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	$-0.328111\pi$
$401$	20.5913	1.02828	0.514140	+	0.857706i	$0.328111\pi$
$402$	−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	$-0.487217\pi$
$409$	1.62384	0.0802936	0.0401468	+	0.999194i	$0.487217\pi$
$410$	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	$-0.670490\pi$
$421$	−20.9436	−1.02073	−0.510365	+	0.859958i	$0.670490\pi$
$422$	−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	$-0.802495\pi$
$431$	−33.7815	−1.62720	−0.813600	+	0.581425i	$0.802495\pi$
$432$	26.8554	1.29208
$433$	15.0289	0.722241	0.361120	−	0.932519i	$-0.382394\pi$
$433$	15.0289	0.722241	0.361120	+	0.932519i	$0.382394\pi$
$434$	−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	$-0.856560\pi$
$439$	−37.7214	−1.80035	−0.900173	+	0.435532i	$0.856560\pi$
$440$	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	$-0.588961\pi$
$449$	−11.6905	−0.551711	−0.275855	+	0.961199i	$0.588961\pi$
$450$	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	$-0.626242\pi$
$487$	−17.0491	−0.772569	−0.386285	+	0.922380i	$0.626242\pi$
$488$	−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	$-0.428042\pi$
$509$	10.1138	0.448286	0.224143	+	0.974556i	$0.428042\pi$
$510$	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	$-0.722487\pi$
$521$	−29.3729	−1.28685	−0.643426	+	0.765508i	$0.722487\pi$
$522$	−12.7226	−0.556852
$523$	11.2000	0.489740	0.244870	−	0.969556i	$-0.421255\pi$
$523$	11.2000	0.489740	0.244870	+	0.969556i	$0.421255\pi$
$524$	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	$-0.582455\pi$
$547$	−11.9818	−0.512303	−0.256151	+	0.966637i	$0.582455\pi$
$548$	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	$-0.675680\pi$
$563$	−24.8817	−1.04864	−0.524318	+	0.851522i	$0.675680\pi$
$564$	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	$-0.198423\pi$
$569$	38.7345	1.62384	0.811918	+	0.583771i	$0.198423\pi$
$570$	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	$-0.286294\pi$
$599$	30.4494	1.24413	0.622065	+	0.782966i	$0.286294\pi$
$600$	0	0
$601$	−24.5483	−1.00135	−0.500673	−	0.865637i	$-0.666914\pi$
$601$	−24.5483	−1.00135	−0.500673	+	0.865637i	$0.666914\pi$
$602$	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	$-0.209139\pi$
$607$	39.0162	1.58362	0.791810	+	0.610767i	$0.209139\pi$
$608$	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	$-0.765552\pi$
$641$	−37.5109	−1.48159	−0.740796	+	0.671730i	$0.765552\pi$
$642$	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	$-0.282305\pi$
$659$	32.4393	1.26366	0.631829	+	0.775108i	$0.282305\pi$
$660$	0	0
$661$	35.5542	1.38290	0.691448	−	0.722426i	$-0.256973\pi$
$661$	35.5542	1.38290	0.691448	+	0.722426i	$0.256973\pi$
$662$	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	$-0.321486\pi$
$673$	27.5963	1.06376	0.531880	+	0.846820i	$0.321486\pi$
$674$	−20.0518	−0.772368
$675$	0	0
$676$	2.14763	0.0826013
$677$	21.2114	0.815221	0.407610	−	0.913156i	$-0.366362\pi$
$677$	21.2114	0.815221	0.407610	+	0.913156i	$0.366362\pi$
$678$	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	$-0.379205\pi$
$683$	19.3626	0.740891	0.370446	+	0.928854i	$0.379205\pi$
$684$	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	$-0.647011\pi$
$743$	−24.2926	−0.891208	−0.445604	+	0.895230i	$0.647011\pi$
$744$	−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	$-0.720134\pi$
$751$	−34.9542	−1.27550	−0.637748	+	0.770245i	$0.720134\pi$
$752$	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	$-0.378685\pi$
$773$	20.6833	0.743926	0.371963	+	0.928248i	$0.378685\pi$
$774$	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	$-0.806175\pi$
$787$	−46.0227	−1.64053	−0.820266	+	0.571982i	$0.806175\pi$
$788$	−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	$-0.646538\pi$
$797$	−25.0847	−0.888546	−0.444273	+	0.895891i	$0.646538\pi$
$798$	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	$-0.264103\pi$
$811$	38.4507	1.35019	0.675094	+	0.737732i	$0.264103\pi$
$812$	−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	$-0.658213\pi$
$827$	−27.4248	−0.953653	−0.476827	+	0.878997i	$0.658213\pi$
$828$	−0.582059	−0.0202279
$829$	−0.156181	−0.00542438	−0.00271219	−	0.999996i	$-0.500863\pi$
$829$	−0.156181	−0.00542438	−0.00271219	+	0.999996i	$0.500863\pi$
$830$	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	$-0.331912\pi$
$839$	29.1892	1.00772	0.503862	+	0.863784i	$0.331912\pi$
$840$	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	$-0.581997\pi$
$857$	−14.9161	−0.509525	−0.254763	+	0.967004i	$0.581997\pi$
$858$	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	$-0.616099\pi$
$863$	−20.9577	−0.713407	−0.356704	+	0.934218i	$0.616099\pi$
$864$	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	$-0.354133\pi$
$877$	26.2017	0.884769	0.442385	+	0.896825i	$0.354133\pi$
$878$	−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	$-0.727328\pi$
$887$	−39.0147	−1.30998	−0.654992	+	0.755636i	$0.727328\pi$
$888$	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	$-0.186748\pi$
$907$	50.1608	1.66556	0.832780	+	0.553603i	$0.186748\pi$
$908$	−3.26804	−0.108454
$909$	26.6248	0.883088
$910$	0	0
$911$	50.4028	1.66992	0.834959	−	0.550312i	$-0.185491\pi$
$911$	50.4028	1.66992	0.834959	+	0.550312i	$0.185491\pi$
$912$	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	$-0.544845\pi$
$941$	−8.61494	−0.280839	−0.140419	+	0.990092i	$0.544845\pi$
$942$	−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	$-0.454005\pi$
$953$	8.89041	0.287988	0.143994	+	0.989579i	$0.454005\pi$
$954$	−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	$-0.529974\pi$
$971$	−5.85997	−0.188055	−0.0940277	+	0.995570i	$0.529974\pi$
$972$	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	$-0.487772\pi$
$977$	2.40082	0.0768089	0.0384044	+	0.999262i	$0.487772\pi$
$978$	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	$-0.594631\pi$
$983$	−18.3685	−0.585864	−0.292932	+	0.956133i	$0.594631\pi$
$984$	−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	$-0.552282\pi$
$991$	−10.2947	−0.327021	−0.163511	+	0.986542i	$0.552282\pi$
$992$	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.bs.1.5		6
5.4	even	2		9025.2.a.by.1.2		6
19.8	odd	6		475.2.e.f.26.5	✓	12
19.12	odd	6		475.2.e.f.201.5	yes	12
19.18	odd	2		9025.2.a.bz.1.2		6
95.8	even	12		475.2.j.d.349.10		24
95.12	even	12		475.2.j.d.49.10		24
95.27	even	12		475.2.j.d.349.3		24
95.69	odd	6		475.2.e.h.201.2	yes	12
95.84	odd	6		475.2.e.h.26.2	yes	12
95.88	even	12		475.2.j.d.49.3		24
95.94	odd	2		9025.2.a.br.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.e.f.26.5	✓	12	19.8	odd	6
475.2.e.f.201.5	yes	12	19.12	odd	6
475.2.e.h.26.2	yes	12	95.84	odd	6
475.2.e.h.201.2	yes	12	95.69	odd	6
475.2.j.d.49.3		24	95.88	even	12
475.2.j.d.49.10		24	95.12	even	12
475.2.j.d.349.3		24	95.27	even	12
475.2.j.d.349.10		24	95.8	even	12
9025.2.a.br.1.5		6	95.94	odd	2
9025.2.a.bs.1.5		6	1.1	even	1	trivial
9025.2.a.by.1.2		6	5.4	even	2
9025.2.a.bz.1.2		6	19.18	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 $\F_{q}$

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

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Database

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