LMFDB - Embedded newform 6025.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, 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("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$1$
Dimension:	$46$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67602 q^{2} -0.199289 q^{3} +5.16108 q^{4} +0.533301 q^{6} +2.87374 q^{7} -8.45912 q^{8} -2.96028 q^{9} +O(q^{10})$	\(q-2.67602 q^{2} -0.199289 q^{3} +5.16108 q^{4} +0.533301 q^{6} +2.87374 q^{7} -8.45912 q^{8} -2.96028 q^{9} -0.806904 q^{11} -1.02855 q^{12} +5.09752 q^{13} -7.69018 q^{14} +12.3146 q^{16} +0.390647 q^{17} +7.92178 q^{18} +0.489734 q^{19} -0.572704 q^{21} +2.15929 q^{22} +1.83958 q^{23} +1.68581 q^{24} -13.6411 q^{26} +1.18782 q^{27} +14.8316 q^{28} -8.05368 q^{29} -5.55127 q^{31} -16.0359 q^{32} +0.160807 q^{33} -1.04538 q^{34} -15.2783 q^{36} +2.20153 q^{37} -1.31054 q^{38} -1.01588 q^{39} -3.59719 q^{41} +1.53257 q^{42} +9.27459 q^{43} -4.16450 q^{44} -4.92275 q^{46} -10.9779 q^{47} -2.45416 q^{48} +1.25837 q^{49} -0.0778515 q^{51} +26.3087 q^{52} +1.68955 q^{53} -3.17863 q^{54} -24.3093 q^{56} -0.0975985 q^{57} +21.5518 q^{58} -5.02546 q^{59} +3.39923 q^{61} +14.8553 q^{62} -8.50708 q^{63} +18.2832 q^{64} -0.430323 q^{66} -12.9814 q^{67} +2.01616 q^{68} -0.366608 q^{69} +8.48671 q^{71} +25.0414 q^{72} -13.1818 q^{73} -5.89135 q^{74} +2.52756 q^{76} -2.31883 q^{77} +2.71851 q^{78} +8.08502 q^{79} +8.64413 q^{81} +9.62615 q^{82} -0.366678 q^{83} -2.95577 q^{84} -24.8190 q^{86} +1.60501 q^{87} +6.82569 q^{88} -2.24259 q^{89} +14.6489 q^{91} +9.49422 q^{92} +1.10631 q^{93} +29.3772 q^{94} +3.19578 q^{96} -7.32434 q^{97} -3.36741 q^{98} +2.38866 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * 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 $

$2$	−2.67602	−1.89223	−0.946116	−	0.323828i	$-0.895030\pi$
$2$	−2.67602	−1.89223	−0.946116	+	0.323828i	$0.895030\pi$
$3$	−0.199289	−0.115060	−0.0575298	−	0.998344i	$-0.518322\pi$
$3$	−0.199289	−0.115060	−0.0575298	+	0.998344i	$0.518322\pi$
$4$	5.16108	2.58054
$5$	0	0
$5$	0	0
$6$	0.533301	0.217719
$7$	2.87374	1.08617	0.543085	−	0.839678i	$-0.317256\pi$
$7$	2.87374	1.08617	0.543085	+	0.839678i	$0.317256\pi$
$8$	−8.45912	−2.99075
$9$	−2.96028	−0.986761
$10$	0	0
$11$	−0.806904	−0.243291	−0.121645	−	0.992574i	$-0.538817\pi$
$11$	−0.806904	−0.243291	−0.121645	+	0.992574i	$0.538817\pi$
$12$	−1.02855	−0.296916
$13$	5.09752	1.41380	0.706899	−	0.707315i	$-0.250094\pi$
$13$	5.09752	1.41380	0.706899	+	0.707315i	$0.250094\pi$
$14$	−7.69018	−2.05529
$15$	0	0
$16$	12.3146	3.07865
$17$	0.390647	0.0947457	0.0473729	−	0.998877i	$-0.484915\pi$
$17$	0.390647	0.0947457	0.0473729	+	0.998877i	$0.484915\pi$
$18$	7.92178	1.86718
$19$	0.489734	0.112353	0.0561763	−	0.998421i	$-0.482109\pi$
$19$	0.489734	0.112353	0.0561763	+	0.998421i	$0.482109\pi$
$20$	0	0
$21$	−0.572704	−0.124974
$22$	2.15929	0.460362
$23$	1.83958	0.383579	0.191789	−	0.981436i	$-0.438571\pi$
$23$	1.83958	0.383579	0.191789	+	0.981436i	$0.438571\pi$
$24$	1.68581	0.344114
$25$	0	0
$26$	−13.6411	−2.67523
$27$	1.18782	0.228596
$28$	14.8316	2.80291
$29$	−8.05368	−1.49553	−0.747765	−	0.663963i	$-0.768873\pi$
$29$	−8.05368	−1.49553	−0.747765	+	0.663963i	$0.768873\pi$
$30$	0	0
$31$	−5.55127	−0.997038	−0.498519	−	0.866879i	$-0.666123\pi$
$31$	−5.55127	−0.997038	−0.498519	+	0.866879i	$0.666123\pi$
$32$	−16.0359	−2.83477
$33$	0.160807	0.0279929
$34$	−1.04538	−0.179281
$35$	0	0
$36$	−15.2783	−2.54638
$37$	2.20153	0.361930	0.180965	−	0.983490i	$-0.442078\pi$
$37$	2.20153	0.361930	0.180965	+	0.983490i	$0.442078\pi$
$38$	−1.31054	−0.212597
$39$	−1.01588	−0.162671
$40$	0	0
$41$	−3.59719	−0.561787	−0.280893	−	0.959739i	$-0.590631\pi$
$41$	−3.59719	−0.561787	−0.280893	+	0.959739i	$0.590631\pi$
$42$	1.53257	0.236480
$43$	9.27459	1.41436	0.707181	−	0.707033i	$-0.249967\pi$
$43$	9.27459	1.41436	0.707181	+	0.707033i	$0.249967\pi$
$44$	−4.16450	−0.627821
$45$	0	0
$46$	−4.92275	−0.725820
$47$	−10.9779	−1.60130	−0.800649	−	0.599134i	$-0.795512\pi$
$47$	−10.9779	−1.60130	−0.800649	+	0.599134i	$0.795512\pi$
$48$	−2.45416	−0.354228
$49$	1.25837	0.179767
$50$	0	0
$51$	−0.0778515	−0.0109014
$52$	26.3087	3.64836
$53$	1.68955	0.232077	0.116039	−	0.993245i	$-0.462980\pi$
$53$	1.68955	0.232077	0.116039	+	0.993245i	$0.462980\pi$
$54$	−3.17863	−0.432556
$55$	0	0
$56$	−24.3093	−3.24847
$57$	−0.0975985	−0.0129272
$58$	21.5518	2.82989
$59$	−5.02546	−0.654259	−0.327130	−	0.944979i	$-0.606081\pi$
$59$	−5.02546	−0.654259	−0.327130	+	0.944979i	$0.606081\pi$
$60$	0	0
$61$	3.39923	0.435227	0.217613	−	0.976035i	$-0.430173\pi$
$61$	3.39923	0.435227	0.217613	+	0.976035i	$0.430173\pi$
$62$	14.8553	1.88663
$63$	−8.50708	−1.07179
$64$	18.2832	2.28539
$65$	0	0
$66$	−0.430323	−0.0529690
$67$	−12.9814	−1.58594	−0.792968	−	0.609263i	$-0.791465\pi$
$67$	−12.9814	−1.58594	−0.792968	+	0.609263i	$0.791465\pi$
$68$	2.01616	0.244495
$69$	−0.366608	−0.0441344
$70$	0	0
$71$	8.48671	1.00719	0.503594	−	0.863941i	$-0.332011\pi$
$71$	8.48671	1.00719	0.503594	+	0.863941i	$0.332011\pi$
$72$	25.0414	2.95116
$73$	−13.1818	−1.54281	−0.771404	−	0.636346i	$-0.780445\pi$
$73$	−13.1818	−1.54281	−0.771404	+	0.636346i	$0.780445\pi$
$74$	−5.89135	−0.684855
$75$	0	0
$76$	2.52756	0.289931
$77$	−2.31883	−0.264255
$78$	2.71851	0.307811
$79$	8.08502	0.909636	0.454818	−	0.890584i	$-0.349704\pi$
$79$	8.08502	0.909636	0.454818	+	0.890584i	$0.349704\pi$
$80$	0	0
$81$	8.64413	0.960459
$82$	9.62615	1.06303
$83$	−0.366678	−0.0402481	−0.0201240	−	0.999797i	$-0.506406\pi$
$83$	−0.366678	−0.0402481	−0.0201240	+	0.999797i	$0.506406\pi$
$84$	−2.95577	−0.322501
$85$	0	0
$86$	−24.8190	−2.67630
$87$	1.60501	0.172075
$88$	6.82569	0.727621
$89$	−2.24259	−0.237715	−0.118857	−	0.992911i	$-0.537923\pi$
$89$	−2.24259	−0.237715	−0.118857	+	0.992911i	$0.537923\pi$
$90$	0	0
$91$	14.6489	1.53563
$92$	9.49422	0.989841
$93$	1.10631	0.114719
$94$	29.3772	3.03003
$95$	0	0
$96$	3.19578	0.326168
$97$	−7.32434	−0.743674	−0.371837	−	0.928298i	$-0.621272\pi$
$97$	−7.32434	−0.743674	−0.371837	+	0.928298i	$0.621272\pi$
$98$	−3.36741	−0.340160
$99$	2.38866	0.240070
$100$	0	0
$101$	3.65598	0.363783	0.181892	−	0.983319i	$-0.441778\pi$
$101$	3.65598	0.363783	0.181892	+	0.983319i	$0.441778\pi$
$102$	0.208332	0.0206280
$103$	19.1218	1.88413	0.942064	−	0.335434i	$-0.108883\pi$
$103$	19.1218	1.88413	0.942064	+	0.335434i	$0.108883\pi$
$104$	−43.1205	−4.22832
$105$	0	0
$106$	−4.52127	−0.439144
$107$	−6.56240	−0.634411	−0.317205	−	0.948357i	$-0.602744\pi$
$107$	−6.56240	−0.634411	−0.317205	+	0.948357i	$0.602744\pi$
$108$	6.13043	0.589901
$109$	−15.2722	−1.46282	−0.731408	−	0.681940i	$-0.761136\pi$
$109$	−15.2722	−1.46282	−0.731408	+	0.681940i	$0.761136\pi$
$110$	0	0
$111$	−0.438741	−0.0416435
$112$	35.3889	3.34394
$113$	−0.183036	−0.0172186	−0.00860929	−	0.999963i	$-0.502740\pi$
$113$	−0.183036	−0.0172186	−0.00860929	+	0.999963i	$0.502740\pi$
$114$	0.261176	0.0244613
$115$	0	0
$116$	−41.5657	−3.85928
$117$	−15.0901	−1.39508
$118$	13.4482	1.23801
$119$	1.12262	0.102910
$120$	0	0
$121$	−10.3489	−0.940810
$122$	−9.09640	−0.823550
$123$	0.716880	0.0646389
$124$	−28.6506	−2.57290
$125$	0	0
$126$	22.7651	2.02808
$127$	−15.1585	−1.34510	−0.672551	−	0.740051i	$-0.734801\pi$
$127$	−15.1585	−1.34510	−0.672551	+	0.740051i	$0.734801\pi$
$128$	−16.8543	−1.48972
$129$	−1.84832	−0.162736
$130$	0	0
$131$	7.10114	0.620430	0.310215	−	0.950667i	$-0.399599\pi$
$131$	7.10114	0.620430	0.310215	+	0.950667i	$0.399599\pi$
$132$	0.829938	0.0722368
$133$	1.40737	0.122034
$134$	34.7386	3.00096
$135$	0	0
$136$	−3.30453	−0.283361
$137$	7.60403	0.649656	0.324828	−	0.945773i	$-0.394694\pi$
$137$	7.60403	0.649656	0.324828	+	0.945773i	$0.394694\pi$
$138$	0.981050	0.0835125
$139$	−21.2414	−1.80167	−0.900835	−	0.434161i	$-0.857045\pi$
$139$	−21.2414	−1.80167	−0.900835	+	0.434161i	$0.857045\pi$
$140$	0	0
$141$	2.18778	0.184245
$142$	−22.7106	−1.90583
$143$	−4.11321	−0.343964
$144$	−36.4547	−3.03789
$145$	0	0
$146$	35.2746	2.91935
$147$	−0.250778	−0.0206839
$148$	11.3623	0.933975
$149$	−16.0916	−1.31827	−0.659137	−	0.752023i	$-0.729078\pi$
$149$	−16.0916	−1.31827	−0.659137	+	0.752023i	$0.729078\pi$
$150$	0	0
$151$	−10.5518	−0.858695	−0.429348	−	0.903139i	$-0.641256\pi$
$151$	−10.5518	−0.858695	−0.429348	+	0.903139i	$0.641256\pi$
$152$	−4.14272	−0.336019
$153$	−1.15642	−0.0934914
$154$	6.20523	0.500032
$155$	0	0
$156$	−5.24304	−0.419779
$157$	−0.254366	−0.0203006	−0.0101503	−	0.999948i	$-0.503231\pi$
$157$	−0.254366	−0.0203006	−0.0101503	+	0.999948i	$0.503231\pi$
$158$	−21.6357	−1.72124
$159$	−0.336708	−0.0267027
$160$	0	0
$161$	5.28647	0.416632
$162$	−23.1319	−1.81741
$163$	−8.82845	−0.691497	−0.345749	−	0.938327i	$-0.612375\pi$
$163$	−8.82845	−0.691497	−0.345749	+	0.938327i	$0.612375\pi$
$164$	−18.5654	−1.44971
$165$	0	0
$166$	0.981236	0.0761587
$167$	3.04162	0.235367	0.117684	−	0.993051i	$-0.462453\pi$
$167$	3.04162	0.235367	0.117684	+	0.993051i	$0.462453\pi$
$168$	4.84457	0.373767
$169$	12.9847	0.998824
$170$	0	0
$171$	−1.44975	−0.110865
$172$	47.8669	3.64982
$173$	−6.02560	−0.458118	−0.229059	−	0.973413i	$-0.573565\pi$
$173$	−6.02560	−0.458118	−0.229059	+	0.973413i	$0.573565\pi$
$174$	−4.29504	−0.325606
$175$	0	0
$176$	−9.93670	−0.749007
$177$	1.00152	0.0752787
$178$	6.00123	0.449811
$179$	7.72419	0.577333	0.288667	−	0.957430i	$-0.406788\pi$
$179$	7.72419	0.577333	0.288667	+	0.957430i	$0.406788\pi$
$180$	0	0
$181$	−10.3144	−0.766662	−0.383331	−	0.923611i	$-0.625223\pi$
$181$	−10.3144	−0.766662	−0.383331	+	0.923611i	$0.625223\pi$
$182$	−39.2008	−2.90576
$183$	−0.677429	−0.0500770
$184$	−15.5612	−1.14719
$185$	0	0
$186$	−2.96050	−0.217074
$187$	−0.315214	−0.0230507
$188$	−56.6581	−4.13221
$189$	3.41348	0.248294
$190$	0	0
$191$	−16.6263	−1.20304	−0.601519	−	0.798858i	$-0.705438\pi$
$191$	−16.6263	−1.20304	−0.601519	+	0.798858i	$0.705438\pi$
$192$	−3.64363	−0.262956
$193$	26.5699	1.91254	0.956271	−	0.292481i	$-0.0944810\pi$
$193$	26.5699	1.91254	0.956271	+	0.292481i	$0.0944810\pi$
$194$	19.6001	1.40720
$195$	0	0
$196$	6.49453	0.463895
$197$	4.42875	0.315536	0.157768	−	0.987476i	$-0.449570\pi$
$197$	4.42875	0.315536	0.157768	+	0.987476i	$0.449570\pi$
$198$	−6.39211	−0.454268
$199$	3.44630	0.244302	0.122151	−	0.992512i	$-0.461021\pi$
$199$	3.44630	0.244302	0.122151	+	0.992512i	$0.461021\pi$
$200$	0	0
$201$	2.58706	0.182477
$202$	−9.78347	−0.688362
$203$	−23.1442	−1.62440
$204$	−0.401798	−0.0281315
$205$	0	0
$206$	−51.1703	−3.56521
$207$	−5.44568	−0.378501
$208$	62.7740	4.35259
$209$	−0.395168	−0.0273343
$210$	0	0
$211$	−0.758471	−0.0522153	−0.0261076	−	0.999659i	$-0.508311\pi$
$211$	−0.758471	−0.0522153	−0.0261076	+	0.999659i	$0.508311\pi$
$212$	8.71990	0.598885
$213$	−1.69131	−0.115886
$214$	17.5611	1.20045
$215$	0	0
$216$	−10.0479	−0.683673
$217$	−15.9529	−1.08295
$218$	40.8688	2.76799
$219$	2.62698	0.177515
$220$	0	0
$221$	1.99133	0.133951
$222$	1.17408	0.0787991
$223$	0.994567	0.0666011	0.0333006	−	0.999445i	$-0.489398\pi$
$223$	0.994567	0.0666011	0.0333006	+	0.999445i	$0.489398\pi$
$224$	−46.0830	−3.07905
$225$	0	0
$226$	0.489808	0.0325815
$227$	9.15012	0.607315	0.303657	−	0.952781i	$-0.401792\pi$
$227$	9.15012	0.607315	0.303657	+	0.952781i	$0.401792\pi$
$228$	−0.503714	−0.0333593
$229$	24.8061	1.63924	0.819618	−	0.572910i	$-0.194186\pi$
$229$	24.8061	1.63924	0.819618	+	0.572910i	$0.194186\pi$
$230$	0	0
$231$	0.462117	0.0304051
$232$	68.1270	4.47276
$233$	−20.8307	−1.36466	−0.682332	−	0.731042i	$-0.739034\pi$
$233$	−20.8307	−1.36466	−0.682332	+	0.731042i	$0.739034\pi$
$234$	40.3814	2.63982
$235$	0	0
$236$	−25.9368	−1.68834
$237$	−1.61126	−0.104662
$238$	−3.00414	−0.194730
$239$	−27.2186	−1.76062	−0.880311	−	0.474397i	$-0.842666\pi$
$239$	−27.2186	−1.76062	−0.880311	+	0.474397i	$0.842666\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	27.6939	1.78023
$243$	−5.28614	−0.339106
$244$	17.5437	1.12312
$245$	0	0
$246$	−1.91839	−0.122312
$247$	2.49643	0.158844
$248$	46.9589	2.98189
$249$	0.0730748	0.00463093
$250$	0	0
$251$	17.6839	1.11620	0.558100	−	0.829774i	$-0.311531\pi$
$251$	17.6839	1.11620	0.558100	+	0.829774i	$0.311531\pi$
$252$	−43.9057	−2.76580
$253$	−1.48436	−0.0933211
$254$	40.5645	2.54524
$255$	0	0
$256$	8.53615	0.533509
$257$	24.6433	1.53721	0.768604	−	0.639725i	$-0.220952\pi$
$257$	24.6433	1.53721	0.768604	+	0.639725i	$0.220952\pi$
$258$	4.94615	0.307934
$259$	6.32663	0.393118
$260$	0	0
$261$	23.8412	1.47573
$262$	−19.0028	−1.17400
$263$	21.3217	1.31475	0.657376	−	0.753563i	$-0.271666\pi$
$263$	21.3217	1.31475	0.657376	+	0.753563i	$0.271666\pi$
$264$	−1.36029	−0.0837198
$265$	0	0
$266$	−3.76614	−0.230917
$267$	0.446924	0.0273513
$268$	−66.9983	−4.09257
$269$	21.8448	1.33190	0.665951	−	0.745996i	$-0.268026\pi$
$269$	21.8448	1.33190	0.665951	+	0.745996i	$0.268026\pi$
$270$	0	0
$271$	1.65986	0.100830	0.0504148	−	0.998728i	$-0.483946\pi$
$271$	1.65986	0.100830	0.0504148	+	0.998728i	$0.483946\pi$
$272$	4.81066	0.291689
$273$	−2.91937	−0.176688
$274$	−20.3485	−1.22930
$275$	0	0
$276$	−1.89209	−0.113891
$277$	−17.1120	−1.02816	−0.514080	−	0.857742i	$-0.671866\pi$
$277$	−17.1120	−1.02816	−0.514080	+	0.857742i	$0.671866\pi$
$278$	56.8424	3.40918
$279$	16.4333	0.983839
$280$	0	0
$281$	26.8613	1.60241	0.801205	−	0.598390i	$-0.204193\pi$
$281$	26.8613	1.60241	0.801205	+	0.598390i	$0.204193\pi$
$282$	−5.85455	−0.348633
$283$	4.41536	0.262466	0.131233	−	0.991352i	$-0.458106\pi$
$283$	4.41536	0.262466	0.131233	+	0.991352i	$0.458106\pi$
$284$	43.8006	2.59909
$285$	0	0
$286$	11.0070	0.650859
$287$	−10.3374	−0.610196
$288$	47.4708	2.79724
$289$	−16.8474	−0.991023
$290$	0	0
$291$	1.45966	0.0855668
$292$	−68.0321	−3.98128
$293$	−27.4208	−1.60194	−0.800971	−	0.598703i	$-0.795683\pi$
$293$	−27.4208	−1.60194	−0.800971	+	0.598703i	$0.795683\pi$
$294$	0.671088	0.0391387
$295$	0	0
$296$	−18.6230	−1.08244
$297$	−0.958455	−0.0556152
$298$	43.0614	2.49448
$299$	9.37730	0.542303
$300$	0	0
$301$	26.6527	1.53624
$302$	28.2369	1.62485
$303$	−0.728596	−0.0418567
$304$	6.03088	0.345895
$305$	0	0
$306$	3.09462	0.176907
$307$	−22.4558	−1.28162	−0.640809	−	0.767700i	$-0.721401\pi$
$307$	−22.4558	−1.28162	−0.640809	+	0.767700i	$0.721401\pi$
$308$	−11.9677	−0.681921
$309$	−3.81076	−0.216787
$310$	0	0
$311$	−17.0158	−0.964880	−0.482440	−	0.875929i	$-0.660249\pi$
$311$	−17.0158	−0.964880	−0.482440	+	0.875929i	$0.660249\pi$
$312$	8.59344	0.486508
$313$	30.6076	1.73004	0.865022	−	0.501734i	$-0.167304\pi$
$313$	30.6076	1.73004	0.865022	+	0.501734i	$0.167304\pi$
$314$	0.680689	0.0384135
$315$	0	0
$316$	41.7275	2.34735
$317$	13.1196	0.736869	0.368434	−	0.929654i	$-0.379894\pi$
$317$	13.1196	0.736869	0.368434	+	0.929654i	$0.379894\pi$
$318$	0.901039	0.0505277
$319$	6.49854	0.363848
$320$	0	0
$321$	1.30781	0.0729950
$322$	−14.1467	−0.788365
$323$	0.191313	0.0106449
$324$	44.6131	2.47850
$325$	0	0
$326$	23.6251	1.30847
$327$	3.04359	0.168311
$328$	30.4291	1.68016
$329$	−31.5477	−1.73928
$330$	0	0
$331$	20.2977	1.11566	0.557832	−	0.829954i	$-0.311633\pi$
$331$	20.2977	1.11566	0.557832	+	0.829954i	$0.311633\pi$
$332$	−1.89245	−0.103862
$333$	−6.51716	−0.357138
$334$	−8.13942	−0.445369
$335$	0	0
$336$	−7.05262	−0.384752
$337$	10.6212	0.578575	0.289287	−	0.957242i	$-0.406582\pi$
$337$	10.6212	0.578575	0.289287	+	0.957242i	$0.406582\pi$
$338$	−34.7473	−1.89001
$339$	0.0364771	0.00198116
$340$	0	0
$341$	4.47934	0.242570
$342$	3.87956	0.209783
$343$	−16.4999	−0.890913
$344$	−78.4549	−4.23000
$345$	0	0
$346$	16.1246	0.866865
$347$	−1.73240	−0.0930001	−0.0465001	−	0.998918i	$-0.514807\pi$
$347$	−1.73240	−0.0930001	−0.0465001	+	0.998918i	$0.514807\pi$
$348$	8.28358	0.444047
$349$	−31.0836	−1.66387	−0.831933	−	0.554876i	$-0.812766\pi$
$349$	−31.0836	−1.66387	−0.831933	+	0.554876i	$0.812766\pi$
$350$	0	0
$351$	6.05493	0.323188
$352$	12.9394	0.689673
$353$	−33.0003	−1.75643	−0.878214	−	0.478268i	$-0.841265\pi$
$353$	−33.0003	−1.75643	−0.878214	+	0.478268i	$0.841265\pi$
$354$	−2.68008	−0.142445
$355$	0	0
$356$	−11.5742	−0.613432
$357$	−0.223725	−0.0118408
$358$	−20.6701	−1.09245
$359$	32.1190	1.69518	0.847589	−	0.530654i	$-0.178054\pi$
$359$	32.1190	1.69518	0.847589	+	0.530654i	$0.178054\pi$
$360$	0	0
$361$	−18.7602	−0.987377
$362$	27.6015	1.45070
$363$	2.06242	0.108249
$364$	75.6044	3.96275
$365$	0	0
$366$	1.81281	0.0947572
$367$	−1.93764	−0.101144	−0.0505719	−	0.998720i	$-0.516104\pi$
$367$	−1.93764	−0.101144	−0.0505719	+	0.998720i	$0.516104\pi$
$368$	22.6537	1.18091
$369$	10.6487	0.554349
$370$	0	0
$371$	4.85532	0.252076
$372$	5.70974	0.296036
$373$	8.78738	0.454993	0.227497	−	0.973779i	$-0.426946\pi$
$373$	8.78738	0.454993	0.227497	+	0.973779i	$0.426946\pi$
$374$	0.843519	0.0436173
$375$	0	0
$376$	92.8637	4.78908
$377$	−41.0538	−2.11438
$378$	−9.13454	−0.469830
$379$	15.6012	0.801378	0.400689	−	0.916214i	$-0.368771\pi$
$379$	15.6012	0.801378	0.400689	+	0.916214i	$0.368771\pi$
$380$	0	0
$381$	3.02093	0.154767
$382$	44.4924	2.27643
$383$	3.43332	0.175435	0.0877173	−	0.996145i	$-0.472043\pi$
$383$	3.43332	0.175435	0.0877173	+	0.996145i	$0.472043\pi$
$384$	3.35888	0.171407
$385$	0	0
$386$	−71.1016	−3.61897
$387$	−27.4554	−1.39564
$388$	−37.8015	−1.91908
$389$	−16.1302	−0.817836	−0.408918	−	0.912571i	$-0.634094\pi$
$389$	−16.1302	−0.817836	−0.408918	+	0.912571i	$0.634094\pi$
$390$	0	0
$391$	0.718626	0.0363425
$392$	−10.6447	−0.537637
$393$	−1.41518	−0.0713863
$394$	−11.8514	−0.597067
$395$	0	0
$396$	12.3281	0.619510
$397$	4.27325	0.214468	0.107234	−	0.994234i	$-0.465801\pi$
$397$	4.27325	0.214468	0.107234	+	0.994234i	$0.465801\pi$
$398$	−9.22236	−0.462275
$399$	−0.280472	−0.0140412
$400$	0	0
$401$	−0.0337541	−0.00168560	−0.000842800	−	1.00000i	$-0.500268\pi$
$401$	−0.0337541	−0.00168560	−0.000842800		1.00000i	$0.500268\pi$
$402$	−6.92302	−0.345289
$403$	−28.2977	−1.40961
$404$	18.8688	0.938758
$405$	0	0
$406$	61.9342	3.07374
$407$	−1.77642	−0.0880541
$408$	0.658555	0.0326034
$409$	12.7748	0.631672	0.315836	−	0.948814i	$-0.397715\pi$
$409$	12.7748	0.631672	0.315836	+	0.948814i	$0.397715\pi$
$410$	0	0
$411$	−1.51540	−0.0747491
$412$	98.6892	4.86207
$413$	−14.4419	−0.710637
$414$	14.5727	0.716211
$415$	0	0
$416$	−81.7433	−4.00780
$417$	4.23317	0.207299
$418$	1.05748	0.0517229
$419$	−37.5266	−1.83329	−0.916646	−	0.399699i	$-0.869115\pi$
$419$	−37.5266	−1.83329	−0.916646	+	0.399699i	$0.869115\pi$
$420$	0	0
$421$	−9.23835	−0.450250	−0.225125	−	0.974330i	$-0.572279\pi$
$421$	−9.23835	−0.450250	−0.225125	+	0.974330i	$0.572279\pi$
$422$	2.02968	0.0988034
$423$	32.4978	1.58010
$424$	−14.2921	−0.694086
$425$	0	0
$426$	4.52597	0.219284
$427$	9.76849	0.472730
$428$	−33.8691	−1.63712
$429$	0.819717	0.0395763
$430$	0	0
$431$	−28.2694	−1.36169	−0.680845	−	0.732427i	$-0.738387\pi$
$431$	−28.2694	−1.36169	−0.680845	+	0.732427i	$0.738387\pi$
$432$	14.6275	0.703767
$433$	2.19007	0.105248	0.0526239	−	0.998614i	$-0.483242\pi$
$433$	2.19007	0.105248	0.0526239	+	0.998614i	$0.483242\pi$
$434$	42.6903	2.04920
$435$	0	0
$436$	−78.8213	−3.77486
$437$	0.900904	0.0430961
$438$	−7.02984	−0.335899
$439$	16.6956	0.796839	0.398419	−	0.917203i	$-0.369559\pi$
$439$	16.6956	0.796839	0.398419	+	0.917203i	$0.369559\pi$
$440$	0	0
$441$	−3.72512	−0.177387
$442$	−5.32884	−0.253467
$443$	10.7664	0.511529	0.255764	−	0.966739i	$-0.417673\pi$
$443$	10.7664	0.511529	0.255764	+	0.966739i	$0.417673\pi$
$444$	−2.26438	−0.107463
$445$	0	0
$446$	−2.66148	−0.126025
$447$	3.20688	0.151680
$448$	52.5410	2.48233
$449$	−32.2748	−1.52314	−0.761571	−	0.648082i	$-0.775571\pi$
$449$	−32.2748	−1.52314	−0.761571	+	0.648082i	$0.775571\pi$
$450$	0	0
$451$	2.90259	0.136677
$452$	−0.944664	−0.0444333
$453$	2.10286	0.0988011
$454$	−24.4859	−1.14918
$455$	0	0
$456$	0.825597	0.0386621
$457$	−19.5040	−0.912358	−0.456179	−	0.889888i	$-0.650782\pi$
$457$	−19.5040	−0.912358	−0.456179	+	0.889888i	$0.650782\pi$
$458$	−66.3817	−3.10181
$459$	0.464017	0.0216585
$460$	0	0
$461$	−7.59585	−0.353774	−0.176887	−	0.984231i	$-0.556603\pi$
$461$	−7.59585	−0.353774	−0.176887	+	0.984231i	$0.556603\pi$
$462$	−1.23663	−0.0575334
$463$	−12.5155	−0.581644	−0.290822	−	0.956777i	$-0.593929\pi$
$463$	−12.5155	−0.581644	−0.290822	+	0.956777i	$0.593929\pi$
$464$	−99.1779	−4.60422
$465$	0	0
$466$	55.7433	2.58226
$467$	−11.4282	−0.528833	−0.264416	−	0.964409i	$-0.585179\pi$
$467$	−11.4282	−0.528833	−0.264416	+	0.964409i	$0.585179\pi$
$468$	−77.8813	−3.60006
$469$	−37.3053	−1.72260
$470$	0	0
$471$	0.0506924	0.00233578
$472$	42.5110	1.95673
$473$	−7.48370	−0.344101
$474$	4.31175	0.198045
$475$	0	0
$476$	5.79391	0.265564
$477$	−5.00155	−0.229005
$478$	72.8374	3.33150
$479$	24.3966	1.11471	0.557354	−	0.830275i	$-0.311817\pi$
$479$	24.3966	1.11471	0.557354	+	0.830275i	$0.311817\pi$
$480$	0	0
$481$	11.2224	0.511696
$482$	−2.67602	−0.121889
$483$	−1.05353	−0.0479375
$484$	−53.4116	−2.42780
$485$	0	0
$486$	14.1458	0.641667
$487$	32.5012	1.47277	0.736385	−	0.676563i	$-0.236531\pi$
$487$	32.5012	1.47277	0.736385	+	0.676563i	$0.236531\pi$
$488$	−28.7545	−1.30165
$489$	1.75941	0.0795633
$490$	0	0
$491$	15.3890	0.694498	0.347249	−	0.937773i	$-0.387116\pi$
$491$	15.3890	0.694498	0.347249	+	0.937773i	$0.387116\pi$
$492$	3.69988	0.166803
$493$	−3.14614	−0.141695
$494$	−6.68049	−0.300569
$495$	0	0
$496$	−68.3618	−3.06953
$497$	24.3886	1.09398
$498$	−0.195550	−0.00876278
$499$	−33.4862	−1.49905	−0.749525	−	0.661976i	$-0.769718\pi$
$499$	−33.4862	−1.49905	−0.749525	+	0.661976i	$0.769718\pi$
$500$	0	0
$501$	−0.606160	−0.0270812
$502$	−47.3225	−2.11211
$503$	38.0060	1.69460	0.847302	−	0.531111i	$-0.178225\pi$
$503$	38.0060	1.69460	0.847302	+	0.531111i	$0.178225\pi$
$504$	71.9624	3.20546
$505$	0	0
$506$	3.97219	0.176585
$507$	−2.58771	−0.114924
$508$	−78.2344	−3.47109
$509$	28.9990	1.28536	0.642679	−	0.766136i	$-0.277823\pi$
$509$	28.9990	1.28536	0.642679	+	0.766136i	$0.277823\pi$
$510$	0	0
$511$	−37.8809	−1.67575
$512$	10.8657	0.480202
$513$	0.581715	0.0256833
$514$	−65.9460	−2.90875
$515$	0	0
$516$	−9.53935	−0.419946
$517$	8.85814	0.389581
$518$	−16.9302	−0.743870
$519$	1.20083	0.0527108
$520$	0	0
$521$	−8.52198	−0.373355	−0.186677	−	0.982421i	$-0.559772\pi$
$521$	−8.52198	−0.373355	−0.186677	+	0.982421i	$0.559772\pi$
$522$	−63.7995	−2.79243
$523$	6.18981	0.270662	0.135331	−	0.990800i	$-0.456790\pi$
$523$	6.18981	0.270662	0.135331	+	0.990800i	$0.456790\pi$
$524$	36.6496	1.60104
$525$	0	0
$526$	−57.0573	−2.48782
$527$	−2.16859	−0.0944651
$528$	1.98027	0.0861804
$529$	−19.6159	−0.852867
$530$	0	0
$531$	14.8768	0.645598
$532$	7.26353	0.314914
$533$	−18.3367	−0.794253
$534$	−1.19598	−0.0517550
$535$	0	0
$536$	109.812	4.74314
$537$	−1.53935	−0.0664277
$538$	−58.4572	−2.52027
$539$	−1.01538	−0.0437355
$540$	0	0
$541$	23.0092	0.989244	0.494622	−	0.869108i	$-0.335306\pi$
$541$	23.0092	0.989244	0.494622	+	0.869108i	$0.335306\pi$
$542$	−4.44183	−0.190793
$543$	2.05554	0.0882117
$544$	−6.26437	−0.268583
$545$	0	0
$546$	7.81229	0.334335
$547$	34.2241	1.46332	0.731658	−	0.681672i	$-0.238747\pi$
$547$	34.2241	1.46332	0.731658	+	0.681672i	$0.238747\pi$
$548$	39.2450	1.67646
$549$	−10.0627	−0.429465
$550$	0	0
$551$	−3.94416	−0.168027
$552$	3.10118	0.131995
$553$	23.2342	0.988020
$554$	45.7920	1.94552
$555$	0	0
$556$	−109.629	−4.64929
$557$	−8.34480	−0.353580	−0.176790	−	0.984249i	$-0.556571\pi$
$557$	−8.34480	−0.353580	−0.176790	+	0.984249i	$0.556571\pi$
$558$	−43.9760	−1.86165
$559$	47.2774	1.99962
$560$	0	0
$561$	0.0628187	0.00265221
$562$	−71.8813	−3.03213
$563$	−21.1210	−0.890146	−0.445073	−	0.895494i	$-0.646822\pi$
$563$	−21.1210	−0.890146	−0.445073	+	0.895494i	$0.646822\pi$
$564$	11.2913	0.475451
$565$	0	0
$566$	−11.8156	−0.496647
$567$	24.8410	1.04322
$568$	−71.7901	−3.01225
$569$	38.2880	1.60512	0.802559	−	0.596573i	$-0.203471\pi$
$569$	38.2880	1.60512	0.802559	+	0.596573i	$0.203471\pi$
$570$	0	0
$571$	4.36826	0.182806	0.0914031	−	0.995814i	$-0.470865\pi$
$571$	4.36826	0.182806	0.0914031	+	0.995814i	$0.470865\pi$
$572$	−21.2286	−0.887612
$573$	3.31344	0.138421
$574$	27.6630	1.15463
$575$	0	0
$576$	−54.1233	−2.25514
$577$	9.27981	0.386324	0.193162	−	0.981167i	$-0.438126\pi$
$577$	9.27981	0.386324	0.193162	+	0.981167i	$0.438126\pi$
$578$	45.0840	1.87525
$579$	−5.29508	−0.220056
$580$	0	0
$581$	−1.05373	−0.0437163
$582$	−3.90608	−0.161912
$583$	−1.36330	−0.0564623
$584$	111.506	4.61415
$585$	0	0
$586$	73.3787	3.03125
$587$	−12.4373	−0.513342	−0.256671	−	0.966499i	$-0.582626\pi$
$587$	−12.4373	−0.513342	−0.256671	+	0.966499i	$0.582626\pi$
$588$	−1.29429	−0.0533755
$589$	−2.71865	−0.112020
$590$	0	0
$591$	−0.882602	−0.0363054
$592$	27.1110	1.11426
$593$	−37.5611	−1.54245	−0.771225	−	0.636563i	$-0.780356\pi$
$593$	−37.5611	−1.54245	−0.771225	+	0.636563i	$0.780356\pi$
$594$	2.56484	0.105237
$595$	0	0
$596$	−83.0500	−3.40186
$597$	−0.686809	−0.0281092
$598$	−25.0938	−1.02616
$599$	−22.2278	−0.908204	−0.454102	−	0.890950i	$-0.650040\pi$
$599$	−22.2278	−0.908204	−0.454102	+	0.890950i	$0.650040\pi$
$600$	0	0
$601$	−46.9863	−1.91661	−0.958305	−	0.285747i	$-0.907758\pi$
$601$	−46.9863	−1.91661	−0.958305	+	0.285747i	$0.907758\pi$
$602$	−71.3233	−2.90692
$603$	38.4288	1.56494
$604$	−54.4588	−2.21590
$605$	0	0
$606$	1.94974	0.0792026
$607$	0.527341	0.0214041	0.0107021	−	0.999943i	$-0.496593\pi$
$607$	0.527341	0.0214041	0.0107021	+	0.999943i	$0.496593\pi$
$608$	−7.85332	−0.318494
$609$	4.61237	0.186903
$610$	0	0
$611$	−55.9603	−2.26391
$612$	−5.96840	−0.241258
$613$	9.71818	0.392514	0.196257	−	0.980553i	$-0.437121\pi$
$613$	9.71818	0.392514	0.196257	+	0.980553i	$0.437121\pi$
$614$	60.0921	2.42512
$615$	0	0
$616$	19.6153	0.790321
$617$	−2.81867	−0.113476	−0.0567378	−	0.998389i	$-0.518070\pi$
$617$	−2.81867	−0.113476	−0.0567378	+	0.998389i	$0.518070\pi$
$618$	10.1977	0.410211
$619$	−2.75560	−0.110757	−0.0553785	−	0.998465i	$-0.517637\pi$
$619$	−2.75560	−0.110757	−0.0553785	+	0.998465i	$0.517637\pi$
$620$	0	0
$621$	2.18509	0.0876845
$622$	45.5347	1.82578
$623$	−6.44463	−0.258199
$624$	−12.5102	−0.500807
$625$	0	0
$626$	−81.9065	−3.27364
$627$	0.0787526	0.00314507
$628$	−1.31281	−0.0523866
$629$	0.860022	0.0342913
$630$	0	0
$631$	−21.5819	−0.859161	−0.429580	−	0.903029i	$-0.641338\pi$
$631$	−21.5819	−0.859161	−0.429580	+	0.903029i	$0.641338\pi$
$632$	−68.3922	−2.72050
$633$	0.151155	0.00600787
$634$	−35.1083	−1.39433
$635$	0	0
$636$	−1.73778	−0.0689075
$637$	6.41455	0.254154
$638$	−17.3902	−0.688486
$639$	−25.1231	−0.993853
$640$	0	0
$641$	−21.4224	−0.846135	−0.423068	−	0.906098i	$-0.639047\pi$
$641$	−21.4224	−0.846135	−0.423068	+	0.906098i	$0.639047\pi$
$642$	−3.49973	−0.138123
$643$	−29.1001	−1.14760	−0.573798	−	0.818997i	$-0.694531\pi$
$643$	−29.1001	−1.14760	−0.573798	+	0.818997i	$0.694531\pi$
$644$	27.2839	1.07514
$645$	0	0
$646$	−0.511957	−0.0201427
$647$	8.60753	0.338397	0.169198	−	0.985582i	$-0.445882\pi$
$647$	8.60753	0.338397	0.169198	+	0.985582i	$0.445882\pi$
$648$	−73.1218	−2.87249
$649$	4.05506	0.159175
$650$	0	0
$651$	3.17924	0.124604
$652$	−45.5643	−1.78444
$653$	10.3765	0.406062	0.203031	−	0.979172i	$-0.434921\pi$
$653$	10.3765	0.406062	0.203031	+	0.979172i	$0.434921\pi$
$654$	−8.14470	−0.318483
$655$	0	0
$656$	−44.2980	−1.72955
$657$	39.0217	1.52238
$658$	84.4223	3.29113
$659$	−39.6868	−1.54598	−0.772990	−	0.634418i	$-0.781240\pi$
$659$	−39.6868	−1.54598	−0.772990	+	0.634418i	$0.781240\pi$
$660$	0	0
$661$	−37.6874	−1.46587	−0.732934	−	0.680300i	$-0.761850\pi$
$661$	−37.6874	−1.46587	−0.732934	+	0.680300i	$0.761850\pi$
$662$	−54.3171	−2.11109
$663$	−0.396850	−0.0154124
$664$	3.10177	0.120372
$665$	0	0
$666$	17.4401	0.675789
$667$	−14.8154	−0.573654
$668$	15.6980	0.607375
$669$	−0.198206	−0.00766309
$670$	0	0
$671$	−2.74285	−0.105887
$672$	9.18382	0.354274
$673$	−31.7217	−1.22278	−0.611390	−	0.791329i	$-0.709390\pi$
$673$	−31.7217	−1.22278	−0.611390	+	0.791329i	$0.709390\pi$
$674$	−28.4226	−1.09480
$675$	0	0
$676$	67.0152	2.57751
$677$	−45.3409	−1.74259	−0.871297	−	0.490757i	$-0.836720\pi$
$677$	−45.3409	−1.74259	−0.871297	+	0.490757i	$0.836720\pi$
$678$	−0.0976133	−0.00374882
$679$	−21.0482	−0.807757
$680$	0	0
$681$	−1.82352	−0.0698773
$682$	−11.9868	−0.458999
$683$	−7.82593	−0.299451	−0.149726	−	0.988728i	$-0.547839\pi$
$683$	−7.82593	−0.299451	−0.149726	+	0.988728i	$0.547839\pi$
$684$	−7.48228	−0.286092
$685$	0	0
$686$	44.1542	1.68581
$687$	−4.94359	−0.188610
$688$	114.213	4.35433
$689$	8.61251	0.328111
$690$	0	0
$691$	−35.0023	−1.33155	−0.665775	−	0.746152i	$-0.731899\pi$
$691$	−35.0023	−1.33155	−0.665775	+	0.746152i	$0.731899\pi$
$692$	−31.0986	−1.18219
$693$	6.86439	0.260757
$694$	4.63594	0.175978
$695$	0	0
$696$	−13.5770	−0.514633
$697$	−1.40523	−0.0532269
$698$	83.1803	3.14842
$699$	4.15133	0.157018
$700$	0	0
$701$	−8.28122	−0.312778	−0.156389	−	0.987696i	$-0.549985\pi$
$701$	−8.28122	−0.312778	−0.156389	+	0.987696i	$0.549985\pi$
$702$	−16.2031	−0.611547
$703$	1.07816	0.0406638
$704$	−14.7527	−0.556015
$705$	0	0
$706$	88.3094	3.32357
$707$	10.5063	0.395131
$708$	5.16892	0.194260
$709$	−23.7901	−0.893458	−0.446729	−	0.894669i	$-0.647411\pi$
$709$	−23.7901	−0.893458	−0.446729	+	0.894669i	$0.647411\pi$
$710$	0	0
$711$	−23.9340	−0.897594
$712$	18.9704	0.710945
$713$	−10.2120	−0.382443
$714$	0.598692	0.0224055
$715$	0	0
$716$	39.8652	1.48983
$717$	5.42436	0.202576
$718$	−85.9512	−3.20767
$719$	33.8950	1.26407	0.632035	−	0.774940i	$-0.282220\pi$
$719$	33.8950	1.26407	0.632035	+	0.774940i	$0.282220\pi$
$720$	0	0
$721$	54.9510	2.04648
$722$	50.2026	1.86835
$723$	−0.199289	−0.00741163
$724$	−53.2334	−1.97840
$725$	0	0
$726$	−5.51908	−0.204832
$727$	−26.5707	−0.985451	−0.492725	−	0.870185i	$-0.663999\pi$
$727$	−26.5707	−0.985451	−0.492725	+	0.870185i	$0.663999\pi$
$728$	−123.917	−4.59267
$729$	−24.8789	−0.921442
$730$	0	0
$731$	3.62309	0.134005
$732$	−3.49627	−0.129226
$733$	−50.9266	−1.88102	−0.940509	−	0.339768i	$-0.889651\pi$
$733$	−50.9266	−1.88102	−0.940509	+	0.339768i	$0.889651\pi$
$734$	5.18516	0.191388
$735$	0	0
$736$	−29.4993	−1.08736
$737$	10.4748	0.385843
$738$	−28.4961	−1.04896
$739$	37.8448	1.39214	0.696071	−	0.717973i	$-0.254930\pi$
$739$	37.8448	1.39214	0.696071	+	0.717973i	$0.254930\pi$
$740$	0	0
$741$	−0.497510	−0.0182765
$742$	−12.9929	−0.476986
$743$	−18.8351	−0.690992	−0.345496	−	0.938420i	$-0.612289\pi$
$743$	−18.8351	−0.690992	−0.345496	+	0.938420i	$0.612289\pi$
$744$	−9.35839	−0.343095
$745$	0	0
$746$	−23.5152	−0.860953
$747$	1.08547	0.0397153
$748$	−1.62685	−0.0594834
$749$	−18.8586	−0.689079
$750$	0	0
$751$	40.1498	1.46509	0.732544	−	0.680720i	$-0.238333\pi$
$751$	40.1498	1.46509	0.732544	+	0.680720i	$0.238333\pi$
$752$	−135.189	−4.92984
$753$	−3.52421	−0.128429
$754$	109.861	4.00089
$755$	0	0
$756$	17.6172	0.640733
$757$	−11.3231	−0.411546	−0.205773	−	0.978600i	$-0.565971\pi$
$757$	−11.3231	−0.411546	−0.205773	+	0.978600i	$0.565971\pi$
$758$	−41.7490	−1.51639
$759$	0.295817	0.0107375
$760$	0	0
$761$	27.9931	1.01475	0.507374	−	0.861726i	$-0.330616\pi$
$761$	27.9931	1.01475	0.507374	+	0.861726i	$0.330616\pi$
$762$	−8.08406	−0.292855
$763$	−43.8884	−1.58887
$764$	−85.8098	−3.10449
$765$	0	0
$766$	−9.18764	−0.331963
$767$	−25.6174	−0.924990
$768$	−1.70116	−0.0613853
$769$	−8.32041	−0.300042	−0.150021	−	0.988683i	$-0.547934\pi$
$769$	−8.32041	−0.300042	−0.150021	+	0.988683i	$0.547934\pi$
$770$	0	0
$771$	−4.91114	−0.176870
$772$	137.129	4.93539
$773$	11.4321	0.411183	0.205591	−	0.978638i	$-0.434088\pi$
$773$	11.4321	0.411183	0.205591	+	0.978638i	$0.434088\pi$
$774$	73.4713	2.64087
$775$	0	0
$776$	61.9575	2.22414
$777$	−1.26083	−0.0452319
$778$	43.1649	1.54754
$779$	−1.76167	−0.0631182
$780$	0	0
$781$	−6.84796	−0.245039
$782$	−1.92306	−0.0687684
$783$	−9.56631	−0.341872
$784$	15.4963	0.553439
$785$	0	0
$786$	3.78705	0.135079
$787$	−11.5127	−0.410384	−0.205192	−	0.978722i	$-0.565782\pi$
$787$	−11.5127	−0.410384	−0.205192	+	0.978722i	$0.565782\pi$
$788$	22.8572	0.814253
$789$	−4.24918	−0.151275
$790$	0	0
$791$	−0.525997	−0.0187023
$792$	−20.2060	−0.717989
$793$	17.3276	0.615323
$794$	−11.4353	−0.405824
$795$	0	0
$796$	17.7866	0.630430
$797$	−8.19431	−0.290257	−0.145129	−	0.989413i	$-0.546360\pi$
$797$	−8.19431	−0.290257	−0.145129	+	0.989413i	$0.546360\pi$
$798$	0.750550	0.0265692
$799$	−4.28850	−0.151716
$800$	0	0
$801$	6.63872	0.234568
$802$	0.0903266	0.00318954
$803$	10.6364	0.375351
$804$	13.3520	0.470889
$805$	0	0
$806$	75.7253	2.66731
$807$	−4.35343	−0.153248
$808$	−30.9263	−1.08799
$809$	9.19094	0.323136	0.161568	−	0.986862i	$-0.448345\pi$
$809$	9.19094	0.323136	0.161568	+	0.986862i	$0.448345\pi$
$810$	0	0
$811$	−23.9711	−0.841740	−0.420870	−	0.907121i	$-0.638275\pi$
$811$	−23.9711	−0.841740	−0.420870	+	0.907121i	$0.638275\pi$
$812$	−119.449	−4.19183
$813$	−0.330793	−0.0116014
$814$	4.75375	0.166619
$815$	0	0
$816$	−0.958711	−0.0335616
$817$	4.54208	0.158907
$818$	−34.1856	−1.19527
$819$	−43.3650	−1.51530
$820$	0	0
$821$	−5.58377	−0.194875	−0.0974375	−	0.995242i	$-0.531065\pi$
$821$	−5.58377	−0.194875	−0.0974375	+	0.995242i	$0.531065\pi$
$822$	4.05524	0.141443
$823$	−2.02001	−0.0704131	−0.0352066	−	0.999380i	$-0.511209\pi$
$823$	−2.02001	−0.0704131	−0.0352066	+	0.999380i	$0.511209\pi$
$824$	−161.754	−5.63495
$825$	0	0
$826$	38.6467	1.34469
$827$	−24.2638	−0.843735	−0.421867	−	0.906658i	$-0.638625\pi$
$827$	−24.2638	−0.843735	−0.421867	+	0.906658i	$0.638625\pi$
$828$	−28.1056	−0.976737
$829$	0.464596	0.0161361	0.00806805	−	0.999967i	$-0.497432\pi$
$829$	0.464596	0.0161361	0.00806805	+	0.999967i	$0.497432\pi$
$830$	0	0
$831$	3.41023	0.118299
$832$	93.1988	3.23109
$833$	0.491577	0.0170321
$834$	−11.3281	−0.392258
$835$	0	0
$836$	−2.03949	−0.0705374
$837$	−6.59391	−0.227919
$838$	100.422	3.46901
$839$	11.0915	0.382921	0.191461	−	0.981500i	$-0.438678\pi$
$839$	11.0915	0.382921	0.191461	+	0.981500i	$0.438678\pi$
$840$	0	0
$841$	35.8617	1.23661
$842$	24.7220	0.851977
$843$	−5.35316	−0.184372
$844$	−3.91453	−0.134744
$845$	0	0
$846$	−86.9648	−2.98991
$847$	−29.7400	−1.02188
$848$	20.8061	0.714486
$849$	−0.879933	−0.0301992
$850$	0	0
$851$	4.04990	0.138829
$852$	−8.72897	−0.299050
$853$	38.1172	1.30511	0.652553	−	0.757743i	$-0.273698\pi$
$853$	38.1172	1.30511	0.652553	+	0.757743i	$0.273698\pi$
$854$	−26.1407	−0.894516
$855$	0	0
$856$	55.5121	1.89736
$857$	17.6418	0.602634	0.301317	−	0.953524i	$-0.402574\pi$
$857$	17.6418	0.602634	0.301317	+	0.953524i	$0.402574\pi$
$858$	−2.19358	−0.0748875
$859$	−27.4875	−0.937862	−0.468931	−	0.883235i	$-0.655361\pi$
$859$	−27.4875	−0.937862	−0.468931	+	0.883235i	$0.655361\pi$
$860$	0	0
$861$	2.06013	0.0702089
$862$	75.6495	2.57663
$863$	19.2496	0.655263	0.327632	−	0.944806i	$-0.393750\pi$
$863$	19.2496	0.655263	0.327632	+	0.944806i	$0.393750\pi$
$864$	−19.0477	−0.648017
$865$	0	0
$866$	−5.86066	−0.199153
$867$	3.35750	0.114027
$868$	−82.3342	−2.79461
$869$	−6.52383	−0.221306
$870$	0	0
$871$	−66.1732	−2.24219
$872$	129.190	4.37492
$873$	21.6821	0.733829
$874$	−2.41084	−0.0815478
$875$	0	0
$876$	13.5580	0.458084
$877$	41.2611	1.39329	0.696644	−	0.717417i	$-0.254676\pi$
$877$	41.2611	1.39329	0.696644	+	0.717417i	$0.254676\pi$
$878$	−44.6778	−1.50780
$879$	5.46467	0.184319
$880$	0	0
$881$	32.1377	1.08275	0.541373	−	0.840783i	$-0.317905\pi$
$881$	32.1377	1.08275	0.541373	+	0.840783i	$0.317905\pi$
$882$	9.96850	0.335657
$883$	−9.13498	−0.307416	−0.153708	−	0.988116i	$-0.549122\pi$
$883$	−9.13498	−0.307416	−0.153708	+	0.988116i	$0.549122\pi$
$884$	10.2774	0.345667
$885$	0	0
$886$	−28.8112	−0.967931
$887$	−13.8765	−0.465928	−0.232964	−	0.972485i	$-0.574842\pi$
$887$	−13.8765	−0.465928	−0.232964	+	0.972485i	$0.574842\pi$
$888$	3.71136	0.124545
$889$	−43.5616	−1.46101
$890$	0	0
$891$	−6.97498	−0.233671
$892$	5.13304	0.171867
$893$	−5.37627	−0.179910
$894$	−8.58167	−0.287014
$895$	0	0
$896$	−48.4349	−1.61809
$897$	−1.86879	−0.0623971
$898$	86.3680	2.88214
$899$	44.7082	1.49110
$900$	0	0
$901$	0.660017	0.0219883
$902$	−7.76738	−0.258625
$903$	−5.31160	−0.176759
$904$	1.54832	0.0514965
$905$	0	0
$906$	−5.62730	−0.186954
$907$	20.5930	0.683778	0.341889	−	0.939740i	$-0.388933\pi$
$907$	20.5930	0.683778	0.341889	+	0.939740i	$0.388933\pi$
$908$	47.2245	1.56720
$909$	−10.8227	−0.358967
$910$	0	0
$911$	−29.8513	−0.989018	−0.494509	−	0.869173i	$-0.664652\pi$
$911$	−29.8513	−0.989018	−0.494509	+	0.869173i	$0.664652\pi$
$912$	−1.20189	−0.0397985
$913$	0.295873	0.00979198
$914$	52.1931	1.72639
$915$	0	0
$916$	128.027	4.23012
$917$	20.4068	0.673892
$918$	−1.24172	−0.0409828
$919$	−44.8288	−1.47877	−0.739383	−	0.673285i	$-0.764883\pi$
$919$	−44.8288	−1.47877	−0.739383	+	0.673285i	$0.764883\pi$
$920$	0	0
$921$	4.47519	0.147462
$922$	20.3266	0.669422
$923$	43.2612	1.42396
$924$	2.38502	0.0784615
$925$	0	0
$926$	33.4917	1.10061
$927$	−56.6060	−1.85918
$928$	129.148	4.23949
$929$	−53.9902	−1.77136	−0.885681	−	0.464294i	$-0.846308\pi$
$929$	−53.9902	−1.77136	−0.885681	+	0.464294i	$0.846308\pi$
$930$	0	0
$931$	0.616264	0.0201972
$932$	−107.509	−3.52157
$933$	3.39107	0.111019
$934$	30.5820	1.00067
$935$	0	0
$936$	127.649	4.17234
$937$	−41.8090	−1.36584	−0.682920	−	0.730493i	$-0.739290\pi$
$937$	−41.8090	−1.36584	−0.682920	+	0.730493i	$0.739290\pi$
$938$	99.8296	3.25955
$939$	−6.09975	−0.199058
$940$	0	0
$941$	17.5326	0.571548	0.285774	−	0.958297i	$-0.407749\pi$
$941$	17.5326	0.571548	0.285774	+	0.958297i	$0.407749\pi$
$942$	−0.135654	−0.00441984
$943$	−6.61732	−0.215490
$944$	−61.8866	−2.01424
$945$	0	0
$946$	20.0265	0.651119
$947$	−60.8556	−1.97754	−0.988770	−	0.149444i	$-0.952252\pi$
$947$	−60.8556	−1.97754	−0.988770	+	0.149444i	$0.952252\pi$
$948$	−8.31582	−0.270085
$949$	−67.1943	−2.18122
$950$	0	0
$951$	−2.61459	−0.0847838
$952$	−9.49634	−0.307778
$953$	−47.7538	−1.54690	−0.773449	−	0.633859i	$-0.781470\pi$
$953$	−47.7538	−1.54690	−0.773449	+	0.633859i	$0.781470\pi$
$954$	13.3842	0.433331
$955$	0	0
$956$	−140.477	−4.54336
$957$	−1.29509	−0.0418642
$958$	−65.2857	−2.10928
$959$	21.8520	0.705637
$960$	0	0
$961$	−0.183358	−0.00591476
$962$	−30.0313	−0.968247
$963$	19.4266	0.626012
$964$	5.16108	0.166227
$965$	0	0
$966$	2.81928	0.0907088
$967$	−10.5625	−0.339666	−0.169833	−	0.985473i	$-0.554323\pi$
$967$	−10.5625	−0.339666	−0.169833	+	0.985473i	$0.554323\pi$
$968$	87.5426	2.81373
$969$	−0.0381265	−0.00122480
$970$	0	0
$971$	−42.2601	−1.35619	−0.678096	−	0.734973i	$-0.737195\pi$
$971$	−42.2601	−1.35619	−0.678096	+	0.734973i	$0.737195\pi$
$972$	−27.2822	−0.875076
$973$	−61.0422	−1.95692
$974$	−86.9739	−2.78682
$975$	0	0
$976$	41.8602	1.33991
$977$	0.878630	0.0281099	0.0140549	−	0.999901i	$-0.495526\pi$
$977$	0.878630	0.0281099	0.0140549	+	0.999901i	$0.495526\pi$
$978$	−4.70822	−0.150552
$979$	1.80956	0.0578337
$980$	0	0
$981$	45.2102	1.44345
$982$	−41.1814	−1.31415
$983$	20.9666	0.668731	0.334365	−	0.942444i	$-0.391478\pi$
$983$	20.9666	0.668731	0.334365	+	0.942444i	$0.391478\pi$
$984$	−6.06417	−0.193319
$985$	0	0
$986$	8.41914	0.268120
$987$	6.28711	0.200121
$988$	12.8843	0.409903
$989$	17.0614	0.542519
$990$	0	0
$991$	3.56254	0.113168	0.0565840	−	0.998398i	$-0.481979\pi$
$991$	3.56254	0.113168	0.0565840	+	0.998398i	$0.481979\pi$
$992$	89.0196	2.82638
$993$	−4.04511	−0.128368
$994$	−65.2643	−2.07006
$995$	0	0
$996$	0.377145	0.0119503
$997$	5.26746	0.166822	0.0834111	−	0.996515i	$-0.473419\pi$
$997$	5.26746	0.166822	0.0834111	+	0.996515i	$0.473419\pi$
$998$	89.6098	2.83655
$999$	2.61502	0.0827356

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.p.1.2		46
5.2	odd	4		1205.2.b.c.724.2	✓	46
5.3	odd	4		1205.2.b.c.724.45	yes	46
5.4	even	2	inner	6025.2.a.p.1.45		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.c.724.2	✓	46	5.2	odd	4
1205.2.b.c.724.45	yes	46	5.3	odd	4
6025.2.a.p.1.2		46	1.1	even	1	trivial
6025.2.a.p.1.45		46	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.67602 q^{2} -0.199289 q^{3} +5.16108 q^{4} +0.533301 q^{6} +2.87374 q^{7} -8.45912 q^{8} -2.96028 q^{9} +O(q^{10})\)	\(q-2.67602 q^{2} -0.199289 q^{3} +5.16108 q^{4} +0.533301 q^{6} +2.87374 q^{7} -8.45912 q^{8} -2.96028 q^{9} -0.806904 q^{11} -1.02855 q^{12} +5.09752 q^{13} -7.69018 q^{14} +12.3146 q^{16} +0.390647 q^{17} +7.92178 q^{18} +0.489734 q^{19} -0.572704 q^{21} +2.15929 q^{22} +1.83958 q^{23} +1.68581 q^{24} -13.6411 q^{26} +1.18782 q^{27} +14.8316 q^{28} -8.05368 q^{29} -5.55127 q^{31} -16.0359 q^{32} +0.160807 q^{33} -1.04538 q^{34} -15.2783 q^{36} +2.20153 q^{37} -1.31054 q^{38} -1.01588 q^{39} -3.59719 q^{41} +1.53257 q^{42} +9.27459 q^{43} -4.16450 q^{44} -4.92275 q^{46} -10.9779 q^{47} -2.45416 q^{48} +1.25837 q^{49} -0.0778515 q^{51} +26.3087 q^{52} +1.68955 q^{53} -3.17863 q^{54} -24.3093 q^{56} -0.0975985 q^{57} +21.5518 q^{58} -5.02546 q^{59} +3.39923 q^{61} +14.8553 q^{62} -8.50708 q^{63} +18.2832 q^{64} -0.430323 q^{66} -12.9814 q^{67} +2.01616 q^{68} -0.366608 q^{69} +8.48671 q^{71} +25.0414 q^{72} -13.1818 q^{73} -5.89135 q^{74} +2.52756 q^{76} -2.31883 q^{77} +2.71851 q^{78} +8.08502 q^{79} +8.64413 q^{81} +9.62615 q^{82} -0.366678 q^{83} -2.95577 q^{84} -24.8190 q^{86} +1.60501 q^{87} +6.82569 q^{88} -2.24259 q^{89} +14.6489 q^{91} +9.49422 q^{92} +1.10631 q^{93} +29.3772 q^{94} +3.19578 q^{96} -7.32434 q^{97} -3.36741 q^{98} +2.38866 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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