LMFDB - Embedded newform 6025.2.a.p.1.7

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			1.7
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)

$f(q)$	$=$	$q-1.92263 q^{2} +0.557567 q^{3} +1.69651 q^{4} -1.07200 q^{6} -0.976687 q^{7} +0.583491 q^{8} -2.68912 q^{9} +O(q^{10})$	\(q-1.92263 q^{2} +0.557567 q^{3} +1.69651 q^{4} -1.07200 q^{6} -0.976687 q^{7} +0.583491 q^{8} -2.68912 q^{9} +5.68399 q^{11} +0.945921 q^{12} -0.519007 q^{13} +1.87781 q^{14} -4.51487 q^{16} +2.78999 q^{17} +5.17019 q^{18} -2.82159 q^{19} -0.544569 q^{21} -10.9282 q^{22} +7.25155 q^{23} +0.325336 q^{24} +0.997860 q^{26} -3.17207 q^{27} -1.65696 q^{28} -2.98708 q^{29} -4.59370 q^{31} +7.51345 q^{32} +3.16920 q^{33} -5.36412 q^{34} -4.56213 q^{36} -9.40071 q^{37} +5.42487 q^{38} -0.289381 q^{39} +1.99776 q^{41} +1.04700 q^{42} -5.68341 q^{43} +9.64296 q^{44} -13.9421 q^{46} +1.33825 q^{47} -2.51734 q^{48} -6.04608 q^{49} +1.55561 q^{51} -0.880503 q^{52} +2.40587 q^{53} +6.09872 q^{54} -0.569888 q^{56} -1.57322 q^{57} +5.74306 q^{58} +5.64524 q^{59} -11.4529 q^{61} +8.83199 q^{62} +2.62643 q^{63} -5.41586 q^{64} -6.09321 q^{66} +5.66108 q^{67} +4.73325 q^{68} +4.04323 q^{69} -10.9432 q^{71} -1.56908 q^{72} +15.5123 q^{73} +18.0741 q^{74} -4.78686 q^{76} -5.55147 q^{77} +0.556374 q^{78} -15.9163 q^{79} +6.29872 q^{81} -3.84096 q^{82} -6.63802 q^{83} -0.923868 q^{84} +10.9271 q^{86} -1.66550 q^{87} +3.31656 q^{88} +13.2676 q^{89} +0.506907 q^{91} +12.3024 q^{92} -2.56129 q^{93} -2.57295 q^{94} +4.18925 q^{96} +1.00517 q^{97} +11.6244 q^{98} -15.2849 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.92263	−1.35951	−0.679753	−	0.733441i	$-0.737913\pi$
$2$	−1.92263	−1.35951	−0.679753	+	0.733441i	$0.737913\pi$
$3$	0.557567	0.321912	0.160956	−	0.986962i	$-0.448542\pi$
$3$	0.557567	0.321912	0.160956	+	0.986962i	$0.448542\pi$
$4$	1.69651	0.848257
$5$	0	0
$5$	0	0
$6$	−1.07200	−0.437641
$7$	−0.976687	−0.369153	−0.184576	−	0.982818i	$-0.559091\pi$
$7$	−0.976687	−0.369153	−0.184576	+	0.982818i	$0.559091\pi$
$8$	0.583491	0.206295
$9$	−2.68912	−0.896373
$10$	0	0
$11$	5.68399	1.71379	0.856893	−	0.515494i	$-0.172392\pi$
$11$	5.68399	1.71379	0.856893	+	0.515494i	$0.172392\pi$
$12$	0.945921	0.273064
$13$	−0.519007	−0.143947	−0.0719733	−	0.997407i	$-0.522930\pi$
$13$	−0.519007	−0.143947	−0.0719733	+	0.997407i	$0.522930\pi$
$14$	1.87781	0.501866
$15$	0	0
$16$	−4.51487	−1.12872
$17$	2.78999	0.676671	0.338336	−	0.941025i	$-0.390136\pi$
$17$	2.78999	0.676671	0.338336	+	0.941025i	$0.390136\pi$
$18$	5.17019	1.21862
$19$	−2.82159	−0.647316	−0.323658	−	0.946174i	$-0.604913\pi$
$19$	−2.82159	−0.647316	−0.323658	+	0.946174i	$0.604913\pi$
$20$	0	0
$21$	−0.544569	−0.118835
$22$	−10.9282	−2.32990
$23$	7.25155	1.51205	0.756027	−	0.654541i	$-0.227138\pi$
$23$	7.25155	1.51205	0.756027	+	0.654541i	$0.227138\pi$
$24$	0.325336	0.0664089
$25$	0	0
$26$	0.997860	0.195696
$27$	−3.17207	−0.610464
$28$	−1.65696	−0.313137
$29$	−2.98708	−0.554687	−0.277344	−	0.960771i	$-0.589454\pi$
$29$	−2.98708	−0.554687	−0.277344	+	0.960771i	$0.589454\pi$
$30$	0	0
$31$	−4.59370	−0.825052	−0.412526	−	0.910946i	$-0.635353\pi$
$31$	−4.59370	−0.825052	−0.412526	+	0.910946i	$0.635353\pi$
$32$	7.51345	1.32820
$33$	3.16920	0.551688
$34$	−5.36412	−0.919939
$35$	0	0
$36$	−4.56213	−0.760355
$37$	−9.40071	−1.54547	−0.772734	−	0.634731i	$-0.781111\pi$
$37$	−9.40071	−1.54547	−0.772734	+	0.634731i	$0.781111\pi$
$38$	5.42487	0.880030
$39$	−0.289381	−0.0463381
$40$	0	0
$41$	1.99776	0.311998	0.155999	−	0.987757i	$-0.450140\pi$
$41$	1.99776	0.311998	0.155999	+	0.987757i	$0.450140\pi$
$42$	1.04700	0.161556
$43$	−5.68341	−0.866712	−0.433356	−	0.901223i	$-0.642671\pi$
$43$	−5.68341	−0.866712	−0.433356	+	0.901223i	$0.642671\pi$
$44$	9.64296	1.45373
$45$	0	0
$46$	−13.9421	−2.05565
$47$	1.33825	0.195203	0.0976016	−	0.995226i	$-0.468883\pi$
$47$	1.33825	0.195203	0.0976016	+	0.995226i	$0.468883\pi$
$48$	−2.51734	−0.363347
$49$	−6.04608	−0.863726
$50$	0	0
$51$	1.55561	0.217828
$52$	−0.880503	−0.122104
$53$	2.40587	0.330471	0.165236	−	0.986254i	$-0.447162\pi$
$53$	2.40587	0.330471	0.165236	+	0.986254i	$0.447162\pi$
$54$	6.09872	0.829930
$55$	0	0
$56$	−0.569888	−0.0761545
$57$	−1.57322	−0.208379
$58$	5.74306	0.754101
$59$	5.64524	0.734947	0.367474	−	0.930034i	$-0.380223\pi$
$59$	5.64524	0.734947	0.367474	+	0.930034i	$0.380223\pi$
$60$	0	0
$61$	−11.4529	−1.46639	−0.733196	−	0.680017i	$-0.761972\pi$
$61$	−11.4529	−1.46639	−0.733196	+	0.680017i	$0.761972\pi$
$62$	8.83199	1.12166
$63$	2.62643	0.330899
$64$	−5.41586	−0.676982
$65$	0	0
$66$	−6.09321	−0.750023
$67$	5.66108	0.691611	0.345805	−	0.938306i	$-0.387606\pi$
$67$	5.66108	0.691611	0.345805	+	0.938306i	$0.387606\pi$
$68$	4.73325	0.573991
$69$	4.04323	0.486748
$70$	0	0
$71$	−10.9432	−1.29872	−0.649360	−	0.760481i	$-0.724963\pi$
$71$	−10.9432	−1.29872	−0.649360	+	0.760481i	$0.724963\pi$
$72$	−1.56908	−0.184918
$73$	15.5123	1.81558	0.907789	−	0.419427i	$-0.137769\pi$
$73$	15.5123	1.81558	0.907789	+	0.419427i	$0.137769\pi$
$74$	18.0741	2.10107
$75$	0	0
$76$	−4.78686	−0.549091
$77$	−5.55147	−0.632649
$78$	0.556374	0.0629969
$79$	−15.9163	−1.79073	−0.895363	−	0.445337i	$-0.853084\pi$
$79$	−15.9163	−1.79073	−0.895363	+	0.445337i	$0.853084\pi$
$80$	0	0
$81$	6.29872	0.699857
$82$	−3.84096	−0.424163
$83$	−6.63802	−0.728618	−0.364309	−	0.931278i	$-0.618695\pi$
$83$	−6.63802	−0.728618	−0.364309	+	0.931278i	$0.618695\pi$
$84$	−0.923868	−0.100802
$85$	0	0
$86$	10.9271	1.17830
$87$	−1.66550	−0.178560
$88$	3.31656	0.353546
$89$	13.2676	1.40636	0.703182	−	0.711009i	$-0.251762\pi$
$89$	13.2676	1.40636	0.703182	+	0.711009i	$0.251762\pi$
$90$	0	0
$91$	0.506907	0.0531383
$92$	12.3024	1.28261
$93$	−2.56129	−0.265594
$94$	−2.57295	−0.265380
$95$	0	0
$96$	4.18925	0.427564
$97$	1.00517	0.102060	0.0510298	−	0.998697i	$-0.483750\pi$
$97$	1.00517	0.102060	0.0510298	+	0.998697i	$0.483750\pi$
$98$	11.6244	1.17424
$99$	−15.2849	−1.53619
$100$	0	0
$101$	−7.10243	−0.706718	−0.353359	−	0.935488i	$-0.614961\pi$
$101$	−7.10243	−0.706718	−0.353359	+	0.935488i	$0.614961\pi$
$102$	−2.99086	−0.296139
$103$	5.52638	0.544531	0.272265	−	0.962222i	$-0.412227\pi$
$103$	5.52638	0.544531	0.272265	+	0.962222i	$0.412227\pi$
$104$	−0.302836	−0.0296955
$105$	0	0
$106$	−4.62559	−0.449277
$107$	7.01208	0.677884	0.338942	−	0.940807i	$-0.389931\pi$
$107$	7.01208	0.677884	0.338942	+	0.940807i	$0.389931\pi$
$108$	−5.38146	−0.517831
$109$	−7.33345	−0.702417	−0.351209	−	0.936297i	$-0.614229\pi$
$109$	−7.33345	−0.702417	−0.351209	+	0.936297i	$0.614229\pi$
$110$	0	0
$111$	−5.24153	−0.497504
$112$	4.40961	0.416669
$113$	0.394108	0.0370746	0.0185373	−	0.999828i	$-0.494099\pi$
$113$	0.394108	0.0370746	0.0185373	+	0.999828i	$0.494099\pi$
$114$	3.02473	0.283292
$115$	0	0
$116$	−5.06763	−0.470517
$117$	1.39567	0.129030
$118$	−10.8537	−0.999165
$119$	−2.72494	−0.249795
$120$	0	0
$121$	21.3077	1.93706
$122$	22.0197	1.99357
$123$	1.11389	0.100436
$124$	−7.79327	−0.699857
$125$	0	0
$126$	−5.04965	−0.449859
$127$	−9.43489	−0.837211	−0.418605	−	0.908168i	$-0.637481\pi$
$127$	−9.43489	−0.837211	−0.418605	+	0.908168i	$0.637481\pi$
$128$	−4.61419	−0.407841
$129$	−3.16888	−0.279005
$130$	0	0
$131$	9.36627	0.818335	0.409167	−	0.912459i	$-0.365819\pi$
$131$	9.36627	0.818335	0.409167	+	0.912459i	$0.365819\pi$
$132$	5.37660	0.467973
$133$	2.75581	0.238959
$134$	−10.8842	−0.940249
$135$	0	0
$136$	1.62793	0.139594
$137$	−5.32866	−0.455258	−0.227629	−	0.973748i	$-0.573097\pi$
$137$	−5.32866	−0.455258	−0.227629	+	0.973748i	$0.573097\pi$
$138$	−7.77364	−0.661736
$139$	7.74702	0.657093	0.328547	−	0.944488i	$-0.393441\pi$
$139$	7.74702	0.657093	0.328547	+	0.944488i	$0.393441\pi$
$140$	0	0
$141$	0.746162	0.0628382
$142$	21.0398	1.76562
$143$	−2.95003	−0.246694
$144$	12.1410	1.01175
$145$	0	0
$146$	−29.8245	−2.46829
$147$	−3.37110	−0.278043
$148$	−15.9484	−1.31095
$149$	−19.8084	−1.62277	−0.811383	−	0.584515i	$-0.801285\pi$
$149$	−19.8084	−1.62277	−0.811383	+	0.584515i	$0.801285\pi$
$150$	0	0
$151$	19.6869	1.60209	0.801047	−	0.598601i	$-0.204276\pi$
$151$	19.6869	1.60209	0.801047	+	0.598601i	$0.204276\pi$
$152$	−1.64637	−0.133538
$153$	−7.50261	−0.606550
$154$	10.6734	0.860090
$155$	0	0
$156$	−0.490940	−0.0393066
$157$	−21.5455	−1.71952	−0.859758	−	0.510701i	$-0.829386\pi$
$157$	−21.5455	−1.71952	−0.859758	+	0.510701i	$0.829386\pi$
$158$	30.6012	2.43450
$159$	1.34143	0.106382
$160$	0	0
$161$	−7.08250	−0.558179
$162$	−12.1101	−0.951460
$163$	12.5163	0.980349	0.490175	−	0.871624i	$-0.336933\pi$
$163$	12.5163	0.980349	0.490175	+	0.871624i	$0.336933\pi$
$164$	3.38923	0.264654
$165$	0	0
$166$	12.7625	0.990560
$167$	7.38516	0.571481	0.285740	−	0.958307i	$-0.407761\pi$
$167$	7.38516	0.571481	0.285740	+	0.958307i	$0.407761\pi$
$168$	−0.317751	−0.0245150
$169$	−12.7306	−0.979279
$170$	0	0
$171$	7.58758	0.580237
$172$	−9.64199	−0.735195
$173$	17.6215	1.33974	0.669868	−	0.742480i	$-0.266351\pi$
$173$	17.6215	1.33974	0.669868	+	0.742480i	$0.266351\pi$
$174$	3.20214	0.242754
$175$	0	0
$176$	−25.6624	−1.93438
$177$	3.14760	0.236588
$178$	−25.5088	−1.91196
$179$	−2.22554	−0.166345	−0.0831723	−	0.996535i	$-0.526505\pi$
$179$	−2.22554	−0.166345	−0.0831723	+	0.996535i	$0.526505\pi$
$180$	0	0
$181$	−18.6828	−1.38868	−0.694341	−	0.719646i	$-0.744304\pi$
$181$	−18.6828	−1.38868	−0.694341	+	0.719646i	$0.744304\pi$
$182$	−0.974596	−0.0722419
$183$	−6.38576	−0.472049
$184$	4.23122	0.311930
$185$	0	0
$186$	4.92443	0.361077
$187$	15.8583	1.15967
$188$	2.27035	0.165583
$189$	3.09812	0.225355
$190$	0	0
$191$	−13.4719	−0.974793	−0.487396	−	0.873181i	$-0.662053\pi$
$191$	−13.4719	−0.974793	−0.487396	+	0.873181i	$0.662053\pi$
$192$	−3.01971	−0.217928
$193$	−9.23670	−0.664872	−0.332436	−	0.943126i	$-0.607871\pi$
$193$	−9.23670	−0.664872	−0.332436	+	0.943126i	$0.607871\pi$
$194$	−1.93257	−0.138751
$195$	0	0
$196$	−10.2573	−0.732662
$197$	25.8626	1.84263	0.921315	−	0.388816i	$-0.127116\pi$
$197$	25.8626	1.84263	0.921315	+	0.388816i	$0.127116\pi$
$198$	29.3873	2.08846
$199$	4.26338	0.302223	0.151112	−	0.988517i	$-0.451715\pi$
$199$	4.26338	0.302223	0.151112	+	0.988517i	$0.451715\pi$
$200$	0	0
$201$	3.15643	0.222638
$202$	13.6554	0.960788
$203$	2.91744	0.204764
$204$	2.63911	0.184774
$205$	0	0
$206$	−10.6252	−0.740293
$207$	−19.5003	−1.35536
$208$	2.34325	0.162475
$209$	−16.0379	−1.10936
$210$	0	0
$211$	6.03303	0.415331	0.207665	−	0.978200i	$-0.433413\pi$
$211$	6.03303	0.415331	0.207665	+	0.978200i	$0.433413\pi$
$212$	4.08159	0.280324
$213$	−6.10158	−0.418073
$214$	−13.4817	−0.921587
$215$	0	0
$216$	−1.85087	−0.125936
$217$	4.48660	0.304570
$218$	14.0995	0.954941
$219$	8.64915	0.584456
$220$	0	0
$221$	−1.44802	−0.0974046
$222$	10.0775	0.676360
$223$	−21.1766	−1.41809	−0.709046	−	0.705162i	$-0.750874\pi$
$223$	−21.1766	−1.41809	−0.709046	+	0.705162i	$0.750874\pi$
$224$	−7.33828	−0.490310
$225$	0	0
$226$	−0.757725	−0.0504031
$227$	3.50318	0.232514	0.116257	−	0.993219i	$-0.462910\pi$
$227$	3.50318	0.232514	0.116257	+	0.993219i	$0.462910\pi$
$228$	−2.66900	−0.176759
$229$	−19.3330	−1.27756	−0.638781	−	0.769389i	$-0.720561\pi$
$229$	−19.3330	−1.27756	−0.638781	+	0.769389i	$0.720561\pi$
$230$	0	0
$231$	−3.09532	−0.203657
$232$	−1.74294	−0.114429
$233$	−23.0684	−1.51126	−0.755632	−	0.654997i	$-0.772670\pi$
$233$	−23.0684	−1.51126	−0.755632	+	0.654997i	$0.772670\pi$
$234$	−2.68336	−0.175417
$235$	0	0
$236$	9.57722	0.623424
$237$	−8.87442	−0.576456
$238$	5.23906	0.339598
$239$	−3.13382	−0.202710	−0.101355	−	0.994850i	$-0.532318\pi$
$239$	−3.13382	−0.202710	−0.101355	+	0.994850i	$0.532318\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−40.9669	−2.63345
$243$	13.0282	0.835757
$244$	−19.4300	−1.24388
$245$	0	0
$246$	−2.14159	−0.136543
$247$	1.46442	0.0931790
$248$	−2.68038	−0.170204
$249$	−3.70114	−0.234551
$250$	0	0
$251$	8.51378	0.537385	0.268693	−	0.963226i	$-0.413408\pi$
$251$	8.51378	0.537385	0.268693	+	0.963226i	$0.413408\pi$
$252$	4.45577	0.280687
$253$	41.2177	2.59134
$254$	18.1398	1.13819
$255$	0	0
$256$	19.7031	1.23144
$257$	0.300229	0.0187278	0.00936389	−	0.999956i	$-0.497019\pi$
$257$	0.300229	0.0187278	0.00936389	+	0.999956i	$0.497019\pi$
$258$	6.09260	0.379309
$259$	9.18155	0.570514
$260$	0	0
$261$	8.03262	0.497207
$262$	−18.0079	−1.11253
$263$	−15.5157	−0.956740	−0.478370	−	0.878158i	$-0.658772\pi$
$263$	−15.5157	−0.956740	−0.478370	+	0.878158i	$0.658772\pi$
$264$	1.84920	0.113811
$265$	0	0
$266$	−5.29840	−0.324866
$267$	7.39759	0.452725
$268$	9.60410	0.586664
$269$	−28.6731	−1.74823	−0.874115	−	0.485718i	$-0.838558\pi$
$269$	−28.6731	−1.74823	−0.874115	+	0.485718i	$0.838558\pi$
$270$	0	0
$271$	−6.01685	−0.365498	−0.182749	−	0.983160i	$-0.558500\pi$
$271$	−6.01685	−0.365498	−0.182749	+	0.983160i	$0.558500\pi$
$272$	−12.5964	−0.763771
$273$	0.282635	0.0171058
$274$	10.2451	0.618927
$275$	0	0
$276$	6.85940	0.412887
$277$	22.1758	1.33241	0.666207	−	0.745767i	$-0.267917\pi$
$277$	22.1758	1.33241	0.666207	+	0.745767i	$0.267917\pi$
$278$	−14.8947	−0.893322
$279$	12.3530	0.739555
$280$	0	0
$281$	−17.6552	−1.05322	−0.526610	−	0.850107i	$-0.676537\pi$
$281$	−17.6552	−1.05322	−0.526610	+	0.850107i	$0.676537\pi$
$282$	−1.43460	−0.0854289
$283$	3.73383	0.221953	0.110977	−	0.993823i	$-0.464602\pi$
$283$	3.73383	0.221953	0.110977	+	0.993823i	$0.464602\pi$
$284$	−18.5653	−1.10165
$285$	0	0
$286$	5.67182	0.335382
$287$	−1.95119	−0.115175
$288$	−20.2046	−1.19056
$289$	−9.21597	−0.542116
$290$	0	0
$291$	0.560451	0.0328542
$292$	26.3168	1.54008
$293$	−30.3941	−1.77564	−0.887821	−	0.460190i	$-0.847781\pi$
$293$	−30.3941	−1.77564	−0.887821	+	0.460190i	$0.847781\pi$
$294$	6.48138	0.378002
$295$	0	0
$296$	−5.48523	−0.318823
$297$	−18.0300	−1.04621
$298$	38.0842	2.20616
$299$	−3.76361	−0.217655
$300$	0	0
$301$	5.55091	0.319949
$302$	−37.8506	−2.17806
$303$	−3.96008	−0.227501
$304$	12.7391	0.730637
$305$	0	0
$306$	14.4248	0.824608
$307$	5.88448	0.335845	0.167923	−	0.985800i	$-0.446294\pi$
$307$	5.88448	0.335845	0.167923	+	0.985800i	$0.446294\pi$
$308$	−9.41815	−0.536649
$309$	3.08133	0.175291
$310$	0	0
$311$	−8.56865	−0.485883	−0.242942	−	0.970041i	$-0.578112\pi$
$311$	−8.56865	−0.485883	−0.242942	+	0.970041i	$0.578112\pi$
$312$	−0.168852	−0.00955933
$313$	−6.68268	−0.377728	−0.188864	−	0.982003i	$-0.560480\pi$
$313$	−6.68268	−0.377728	−0.188864	+	0.982003i	$0.560480\pi$
$314$	41.4240	2.33769
$315$	0	0
$316$	−27.0023	−1.51900
$317$	18.7500	1.05311	0.526553	−	0.850142i	$-0.323484\pi$
$317$	18.7500	1.05311	0.526553	+	0.850142i	$0.323484\pi$
$318$	−2.57908	−0.144628
$319$	−16.9785	−0.950615
$320$	0	0
$321$	3.90971	0.218219
$322$	13.6170	0.758848
$323$	−7.87219	−0.438020
$324$	10.6859	0.593659
$325$	0	0
$326$	−24.0642	−1.33279
$327$	−4.08889	−0.226116
$328$	1.16568	0.0643637
$329$	−1.30705	−0.0720598
$330$	0	0
$331$	−10.1395	−0.557317	−0.278659	−	0.960390i	$-0.589890\pi$
$331$	−10.1395	−0.557317	−0.278659	+	0.960390i	$0.589890\pi$
$332$	−11.2615	−0.618055
$333$	25.2796	1.38531
$334$	−14.1989	−0.776932
$335$	0	0
$336$	2.45865	0.134131
$337$	−22.7996	−1.24197	−0.620986	−	0.783822i	$-0.713268\pi$
$337$	−22.7996	−1.24197	−0.620986	+	0.783822i	$0.713268\pi$
$338$	24.4763	1.33134
$339$	0.219742	0.0119347
$340$	0	0
$341$	−26.1105	−1.41396
$342$	−14.5881	−0.788835
$343$	12.7419	0.688000
$344$	−3.31622	−0.178799
$345$	0	0
$346$	−33.8796	−1.82138
$347$	33.0431	1.77385	0.886924	−	0.461916i	$-0.152838\pi$
$347$	33.0431	1.77385	0.886924	+	0.461916i	$0.152838\pi$
$348$	−2.82554	−0.151465
$349$	−10.7629	−0.576126	−0.288063	−	0.957611i	$-0.593011\pi$
$349$	−10.7629	−0.576126	−0.288063	+	0.957611i	$0.593011\pi$
$350$	0	0
$351$	1.64632	0.0878743
$352$	42.7063	2.27626
$353$	−12.8113	−0.681879	−0.340940	−	0.940085i	$-0.610745\pi$
$353$	−12.8113	−0.681879	−0.340940	+	0.940085i	$0.610745\pi$
$354$	−6.05167	−0.321643
$355$	0	0
$356$	22.5087	1.19296
$357$	−1.51934	−0.0804120
$358$	4.27889	0.226146
$359$	12.7434	0.672570	0.336285	−	0.941760i	$-0.390830\pi$
$359$	12.7434	0.672570	0.336285	+	0.941760i	$0.390830\pi$
$360$	0	0
$361$	−11.0387	−0.580982
$362$	35.9202	1.88792
$363$	11.8805	0.623563
$364$	0.859975	0.0450750
$365$	0	0
$366$	12.2775	0.641753
$367$	−1.03639	−0.0540989	−0.0270495	−	0.999634i	$-0.508611\pi$
$367$	−1.03639	−0.0540989	−0.0270495	+	0.999634i	$0.508611\pi$
$368$	−32.7398	−1.70668
$369$	−5.37222	−0.279666
$370$	0	0
$371$	−2.34978	−0.121994
$372$	−4.34527	−0.225292
$373$	14.7553	0.763999	0.381999	−	0.924163i	$-0.375236\pi$
$373$	14.7553	0.763999	0.381999	+	0.924163i	$0.375236\pi$
$374$	−30.4896	−1.57658
$375$	0	0
$376$	0.780855	0.0402695
$377$	1.55032	0.0798454
$378$	−5.95654	−0.306371
$379$	23.1923	1.19131	0.595653	−	0.803242i	$-0.296893\pi$
$379$	23.1923	1.19131	0.595653	+	0.803242i	$0.296893\pi$
$380$	0	0
$381$	−5.26058	−0.269508
$382$	25.9015	1.32524
$383$	12.7484	0.651415	0.325707	−	0.945471i	$-0.394398\pi$
$383$	12.7484	0.651415	0.325707	+	0.945471i	$0.394398\pi$
$384$	−2.57272	−0.131289
$385$	0	0
$386$	17.7588	0.903898
$387$	15.2834	0.776897
$388$	1.70529	0.0865728
$389$	31.6959	1.60705	0.803523	−	0.595274i	$-0.202957\pi$
$389$	31.6959	1.60705	0.803523	+	0.595274i	$0.202957\pi$
$390$	0	0
$391$	20.2317	1.02316
$392$	−3.52784	−0.178183
$393$	5.22233	0.263432
$394$	−49.7242	−2.50507
$395$	0	0
$396$	−25.9311	−1.30309
$397$	−18.0796	−0.907389	−0.453694	−	0.891157i	$-0.649894\pi$
$397$	−18.0796	−0.907389	−0.453694	+	0.891157i	$0.649894\pi$
$398$	−8.19692	−0.410874
$399$	1.53655	0.0769236
$400$	0	0
$401$	24.8583	1.24136	0.620682	−	0.784063i	$-0.286856\pi$
$401$	24.8583	1.24136	0.620682	+	0.784063i	$0.286856\pi$
$402$	−6.06866	−0.302677
$403$	2.38416	0.118764
$404$	−12.0494	−0.599479
$405$	0	0
$406$	−5.60917	−0.278378
$407$	−53.4335	−2.64860
$408$	0.907683	0.0449370
$409$	4.25506	0.210399	0.105200	−	0.994451i	$-0.466452\pi$
$409$	4.25506	0.210399	0.105200	+	0.994451i	$0.466452\pi$
$410$	0	0
$411$	−2.97109	−0.146553
$412$	9.37559	0.461902
$413$	−5.51363	−0.271308
$414$	37.4919	1.84263
$415$	0	0
$416$	−3.89953	−0.191190
$417$	4.31948	0.211526
$418$	30.8349	1.50818
$419$	16.7521	0.818396	0.409198	−	0.912446i	$-0.365809\pi$
$419$	16.7521	0.818396	0.409198	+	0.912446i	$0.365809\pi$
$420$	0	0
$421$	20.9251	1.01982	0.509912	−	0.860226i	$-0.329678\pi$
$421$	20.9251	1.01982	0.509912	+	0.860226i	$0.329678\pi$
$422$	−11.5993	−0.564645
$423$	−3.59870	−0.174975
$424$	1.40380	0.0681746
$425$	0	0
$426$	11.7311	0.568373
$427$	11.1859	0.541323
$428$	11.8961	0.575020
$429$	−1.64484	−0.0794136
$430$	0	0
$431$	−27.7303	−1.33572	−0.667861	−	0.744286i	$-0.732790\pi$
$431$	−27.7303	−1.33572	−0.667861	+	0.744286i	$0.732790\pi$
$432$	14.3215	0.689042
$433$	−28.0853	−1.34969	−0.674847	−	0.737958i	$-0.735790\pi$
$433$	−28.0853	−1.34969	−0.674847	+	0.737958i	$0.735790\pi$
$434$	−8.62609	−0.414065
$435$	0	0
$436$	−12.4413	−0.595831
$437$	−20.4609	−0.978777
$438$	−16.6291	−0.794571
$439$	−10.0542	−0.479863	−0.239931	−	0.970790i	$-0.577125\pi$
$439$	−10.0542	−0.479863	−0.239931	+	0.970790i	$0.577125\pi$
$440$	0	0
$441$	16.2586	0.774221
$442$	2.78402	0.132422
$443$	−4.38230	−0.208209	−0.104105	−	0.994566i	$-0.533198\pi$
$443$	−4.38230	−0.208209	−0.104105	+	0.994566i	$0.533198\pi$
$444$	−8.89233	−0.422011
$445$	0	0
$446$	40.7149	1.92791
$447$	−11.0445	−0.522387
$448$	5.28960	0.249910
$449$	7.51495	0.354652	0.177326	−	0.984152i	$-0.443255\pi$
$449$	7.51495	0.354652	0.177326	+	0.984152i	$0.443255\pi$
$450$	0	0
$451$	11.3552	0.534698
$452$	0.668610	0.0314488
$453$	10.9768	0.515733
$454$	−6.73533	−0.316105
$455$	0	0
$456$	−0.917963	−0.0429875
$457$	−27.0931	−1.26736	−0.633682	−	0.773594i	$-0.718457\pi$
$457$	−27.0931	−1.26736	−0.633682	+	0.773594i	$0.718457\pi$
$458$	37.1703	1.73685
$459$	−8.85003	−0.413084
$460$	0	0
$461$	8.28811	0.386016	0.193008	−	0.981197i	$-0.438176\pi$
$461$	8.28811	0.386016	0.193008	+	0.981197i	$0.438176\pi$
$462$	5.95116	0.276873
$463$	−39.3319	−1.82791	−0.913955	−	0.405815i	$-0.866988\pi$
$463$	−39.3319	−1.82791	−0.913955	+	0.405815i	$0.866988\pi$
$464$	13.4863	0.626085
$465$	0	0
$466$	44.3521	2.05457
$467$	−1.62543	−0.0752160	−0.0376080	−	0.999293i	$-0.511974\pi$
$467$	−1.62543	−0.0752160	−0.0376080	+	0.999293i	$0.511974\pi$
$468$	2.36778	0.109451
$469$	−5.52910	−0.255310
$470$	0	0
$471$	−12.0131	−0.553532
$472$	3.29395	0.151616
$473$	−32.3044	−1.48536
$474$	17.0622	0.783695
$475$	0	0
$476$	−4.62291	−0.211891
$477$	−6.46966	−0.296225
$478$	6.02518	0.275585
$479$	27.2783	1.24638	0.623189	−	0.782071i	$-0.285837\pi$
$479$	27.2783	1.24638	0.623189	+	0.782071i	$0.285837\pi$
$480$	0	0
$481$	4.87903	0.222465
$482$	−1.92263	−0.0875735
$483$	−3.94897	−0.179684
$484$	36.1488	1.64313
$485$	0	0
$486$	−25.0484	−1.13622
$487$	16.1457	0.731631	0.365815	−	0.930687i	$-0.380790\pi$
$487$	16.1457	0.731631	0.365815	+	0.930687i	$0.380790\pi$
$488$	−6.68266	−0.302510
$489$	6.97866	0.315586
$490$	0	0
$491$	−13.9774	−0.630791	−0.315395	−	0.948960i	$-0.602137\pi$
$491$	−13.9774	−0.630791	−0.315395	+	0.948960i	$0.602137\pi$
$492$	1.88972	0.0851953
$493$	−8.33392	−0.375341
$494$	−2.81555	−0.126677
$495$	0	0
$496$	20.7399	0.931251
$497$	10.6881	0.479427
$498$	7.11594	0.318873
$499$	13.5580	0.606937	0.303469	−	0.952841i	$-0.401855\pi$
$499$	13.5580	0.606937	0.303469	+	0.952841i	$0.401855\pi$
$500$	0	0
$501$	4.11772	0.183966
$502$	−16.3689	−0.730578
$503$	−28.0739	−1.25175	−0.625876	−	0.779923i	$-0.715258\pi$
$503$	−28.0739	−1.25175	−0.625876	+	0.779923i	$0.715258\pi$
$504$	1.53250	0.0682629
$505$	0	0
$506$	−79.2465	−3.52294
$507$	−7.09818	−0.315241
$508$	−16.0064	−0.710170
$509$	−39.5857	−1.75460	−0.877302	−	0.479939i	$-0.840659\pi$
$509$	−39.5857	−1.75460	−0.877302	+	0.479939i	$0.840659\pi$
$510$	0	0
$511$	−15.1507	−0.670226
$512$	−28.6534	−1.26632
$513$	8.95026	0.395164
$514$	−0.577230	−0.0254605
$515$	0	0
$516$	−5.37606	−0.236668
$517$	7.60657	0.334537
$518$	−17.6527	−0.775617
$519$	9.82516	0.431277
$520$	0	0
$521$	13.3893	0.586597	0.293299	−	0.956021i	$-0.405247\pi$
$521$	13.3893	0.586597	0.293299	+	0.956021i	$0.405247\pi$
$522$	−15.4438	−0.675955
$523$	−11.5484	−0.504976	−0.252488	−	0.967600i	$-0.581249\pi$
$523$	−11.5484	−0.504976	−0.252488	+	0.967600i	$0.581249\pi$
$524$	15.8900	0.694158
$525$	0	0
$526$	29.8310	1.30069
$527$	−12.8164	−0.558289
$528$	−14.3085	−0.622699
$529$	29.5850	1.28631
$530$	0	0
$531$	−15.1807	−0.658787
$532$	4.67526	0.202698
$533$	−1.03685	−0.0449110
$534$	−14.2228	−0.615483
$535$	0	0
$536$	3.30319	0.142676
$537$	−1.24089	−0.0535482
$538$	55.1279	2.37673
$539$	−34.3659	−1.48024
$540$	0	0
$541$	−24.7679	−1.06486	−0.532428	−	0.846476i	$-0.678720\pi$
$541$	−24.7679	−1.06486	−0.532428	+	0.846476i	$0.678720\pi$
$542$	11.5682	0.496897
$543$	−10.4169	−0.447033
$544$	20.9624	0.898757
$545$	0	0
$546$	−0.543403	−0.0232555
$547$	−34.4650	−1.47362	−0.736808	−	0.676102i	$-0.763668\pi$
$547$	−34.4650	−1.47362	−0.736808	+	0.676102i	$0.763668\pi$
$548$	−9.04015	−0.386176
$549$	30.7982	1.31443
$550$	0	0
$551$	8.42831	0.359058
$552$	2.35919	0.100414
$553$	15.5453	0.661052
$554$	−42.6359	−1.81142
$555$	0	0
$556$	13.1429	0.557384
$557$	3.13055	0.132646	0.0663228	−	0.997798i	$-0.478873\pi$
$557$	3.13055	0.132646	0.0663228	+	0.997798i	$0.478873\pi$
$558$	−23.7503	−1.00543
$559$	2.94973	0.124760
$560$	0	0
$561$	8.84204	0.373311
$562$	33.9444	1.43186
$563$	−41.5012	−1.74907	−0.874533	−	0.484966i	$-0.838832\pi$
$563$	−41.5012	−1.74907	−0.874533	+	0.484966i	$0.838832\pi$
$564$	1.26587	0.0533029
$565$	0	0
$566$	−7.17878	−0.301747
$567$	−6.15187	−0.258354
$568$	−6.38527	−0.267920
$569$	9.52061	0.399125	0.199562	−	0.979885i	$-0.436048\pi$
$569$	9.52061	0.399125	0.199562	+	0.979885i	$0.436048\pi$
$570$	0	0
$571$	−12.5830	−0.526583	−0.263292	−	0.964716i	$-0.584808\pi$
$571$	−12.5830	−0.526583	−0.263292	+	0.964716i	$0.584808\pi$
$572$	−5.00477	−0.209260
$573$	−7.51149	−0.313797
$574$	3.75141	0.156581
$575$	0	0
$576$	14.5639	0.606829
$577$	4.59818	0.191425	0.0957123	−	0.995409i	$-0.469487\pi$
$577$	4.59818	0.191425	0.0957123	+	0.995409i	$0.469487\pi$
$578$	17.7189	0.737010
$579$	−5.15008	−0.214030
$580$	0	0
$581$	6.48327	0.268971
$582$	−1.07754	−0.0446655
$583$	13.6749	0.566357
$584$	9.05130	0.374545
$585$	0	0
$586$	58.4366	2.41400
$587$	11.2149	0.462887	0.231443	−	0.972848i	$-0.425655\pi$
$587$	11.2149	0.462887	0.231443	+	0.972848i	$0.425655\pi$
$588$	−5.71912	−0.235852
$589$	12.9615	0.534070
$590$	0	0
$591$	14.4201	0.593164
$592$	42.4430	1.74439
$593$	−15.2963	−0.628144	−0.314072	−	0.949399i	$-0.601693\pi$
$593$	−15.2963	−0.628144	−0.314072	+	0.949399i	$0.601693\pi$
$594$	34.6650	1.42232
$595$	0	0
$596$	−33.6052	−1.37652
$597$	2.37712	0.0972892
$598$	7.23603	0.295903
$599$	26.5049	1.08296	0.541480	−	0.840713i	$-0.317864\pi$
$599$	26.5049	1.08296	0.541480	+	0.840713i	$0.317864\pi$
$600$	0	0
$601$	−4.30453	−0.175586	−0.0877928	−	0.996139i	$-0.527981\pi$
$601$	−4.30453	−0.175586	−0.0877928	+	0.996139i	$0.527981\pi$
$602$	−10.6724	−0.434973
$603$	−15.2233	−0.619941
$604$	33.3991	1.35899
$605$	0	0
$606$	7.61378	0.309289
$607$	5.46958	0.222004	0.111002	−	0.993820i	$-0.464594\pi$
$607$	5.46958	0.222004	0.111002	+	0.993820i	$0.464594\pi$
$608$	−21.1998	−0.859767
$609$	1.62667	0.0659160
$610$	0	0
$611$	−0.694559	−0.0280989
$612$	−12.7283	−0.514510
$613$	−34.9065	−1.40986	−0.704930	−	0.709277i	$-0.749021\pi$
$613$	−34.9065	−1.40986	−0.704930	+	0.709277i	$0.749021\pi$
$614$	−11.3137	−0.456584
$615$	0	0
$616$	−3.23924	−0.130513
$617$	24.0459	0.968052	0.484026	−	0.875054i	$-0.339174\pi$
$617$	24.0459	0.968052	0.484026	+	0.875054i	$0.339174\pi$
$618$	−5.92427	−0.238309
$619$	−36.3148	−1.45961	−0.729807	−	0.683653i	$-0.760390\pi$
$619$	−36.3148	−1.45961	−0.729807	+	0.683653i	$0.760390\pi$
$620$	0	0
$621$	−23.0024	−0.923055
$622$	16.4744	0.660562
$623$	−12.9583	−0.519164
$624$	1.30652	0.0523026
$625$	0	0
$626$	12.8483	0.513523
$627$	−8.94218	−0.357116
$628$	−36.5522	−1.45859
$629$	−26.2279	−1.04577
$630$	0	0
$631$	−19.2957	−0.768150	−0.384075	−	0.923302i	$-0.625480\pi$
$631$	−19.2957	−0.768150	−0.384075	+	0.923302i	$0.625480\pi$
$632$	−9.28704	−0.369419
$633$	3.36382	0.133700
$634$	−36.0494	−1.43171
$635$	0	0
$636$	2.27576	0.0902397
$637$	3.13796	0.124330
$638$	32.6435	1.29237
$639$	29.4276	1.16414
$640$	0	0
$641$	−29.0190	−1.14618	−0.573091	−	0.819492i	$-0.694256\pi$
$641$	−29.0190	−1.14618	−0.573091	+	0.819492i	$0.694256\pi$
$642$	−7.51693	−0.296670
$643$	−29.2586	−1.15385	−0.576924	−	0.816798i	$-0.695747\pi$
$643$	−29.2586	−1.15385	−0.576924	+	0.816798i	$0.695747\pi$
$644$	−12.0156	−0.473479
$645$	0	0
$646$	15.1353	0.595491
$647$	−32.7993	−1.28947	−0.644736	−	0.764405i	$-0.723033\pi$
$647$	−32.7993	−1.28947	−0.644736	+	0.764405i	$0.723033\pi$
$648$	3.67525	0.144377
$649$	32.0874	1.25954
$650$	0	0
$651$	2.50158	0.0980448
$652$	21.2340	0.831588
$653$	−15.9479	−0.624092	−0.312046	−	0.950067i	$-0.601014\pi$
$653$	−15.9479	−0.624092	−0.312046	+	0.950067i	$0.601014\pi$
$654$	7.86144	0.307407
$655$	0	0
$656$	−9.01963	−0.352157
$657$	−41.7144	−1.62743
$658$	2.51297	0.0979658
$659$	−19.1492	−0.745947	−0.372973	−	0.927842i	$-0.621662\pi$
$659$	−19.1492	−0.745947	−0.372973	+	0.927842i	$0.621662\pi$
$660$	0	0
$661$	−2.28059	−0.0887045	−0.0443523	−	0.999016i	$-0.514122\pi$
$661$	−2.28059	−0.0887045	−0.0443523	+	0.999016i	$0.514122\pi$
$662$	19.4945	0.757676
$663$	−0.807370	−0.0313557
$664$	−3.87323	−0.150310
$665$	0	0
$666$	−48.6034	−1.88334
$667$	−21.6610	−0.838717
$668$	12.5290	0.484763
$669$	−11.8074	−0.456500
$670$	0	0
$671$	−65.0981	−2.51308
$672$	−4.09159	−0.157836
$673$	−24.0503	−0.927073	−0.463536	−	0.886078i	$-0.653420\pi$
$673$	−24.0503	−0.927073	−0.463536	+	0.886078i	$0.653420\pi$
$674$	43.8352	1.68847
$675$	0	0
$676$	−21.5977	−0.830681
$677$	−47.6572	−1.83161	−0.915807	−	0.401619i	$-0.868448\pi$
$677$	−47.6572	−1.83161	−0.915807	+	0.401619i	$0.868448\pi$
$678$	−0.422483	−0.0162254
$679$	−0.981737	−0.0376756
$680$	0	0
$681$	1.95326	0.0748491
$682$	50.2009	1.92229
$683$	−30.1166	−1.15238	−0.576190	−	0.817316i	$-0.695461\pi$
$683$	−30.1166	−1.15238	−0.576190	+	0.817316i	$0.695461\pi$
$684$	12.8724	0.492190
$685$	0	0
$686$	−24.4981	−0.935340
$687$	−10.7795	−0.411262
$688$	25.6599	0.978273
$689$	−1.24866	−0.0475702
$690$	0	0
$691$	6.81543	0.259271	0.129636	−	0.991562i	$-0.458619\pi$
$691$	6.81543	0.259271	0.129636	+	0.991562i	$0.458619\pi$
$692$	29.8951	1.13644
$693$	14.9286	0.567090
$694$	−63.5297	−2.41156
$695$	0	0
$696$	−0.971804	−0.0368361
$697$	5.57373	0.211120
$698$	20.6931	0.783247
$699$	−12.8622	−0.486493
$700$	0	0
$701$	38.7899	1.46507	0.732537	−	0.680727i	$-0.238336\pi$
$701$	38.7899	1.46507	0.732537	+	0.680727i	$0.238336\pi$
$702$	−3.16528	−0.119466
$703$	26.5249	1.00041
$704$	−30.7837	−1.16020
$705$	0	0
$706$	24.6315	0.927019
$707$	6.93685	0.260887
$708$	5.33995	0.200687
$709$	−8.72623	−0.327720	−0.163860	−	0.986484i	$-0.552395\pi$
$709$	−8.72623	−0.327720	−0.163860	+	0.986484i	$0.552395\pi$
$710$	0	0
$711$	42.8009	1.60516
$712$	7.74154	0.290127
$713$	−33.3114	−1.24752
$714$	2.92113	0.109321
$715$	0	0
$716$	−3.77566	−0.141103
$717$	−1.74732	−0.0652547
$718$	−24.5008	−0.914362
$719$	−22.6803	−0.845832	−0.422916	−	0.906169i	$-0.638994\pi$
$719$	−22.6803	−0.845832	−0.422916	+	0.906169i	$0.638994\pi$
$720$	0	0
$721$	−5.39755	−0.201015
$722$	21.2233	0.789848
$723$	0.557567	0.0207361
$724$	−31.6956	−1.17796
$725$	0	0
$726$	−22.8418	−0.847738
$727$	−7.41044	−0.274838	−0.137419	−	0.990513i	$-0.543881\pi$
$727$	−7.41044	−0.274838	−0.137419	+	0.990513i	$0.543881\pi$
$728$	0.295776	0.0109622
$729$	−11.6321	−0.430818
$730$	0	0
$731$	−15.8567	−0.586479
$732$	−10.8335	−0.400419
$733$	−9.84342	−0.363575	−0.181787	−	0.983338i	$-0.558188\pi$
$733$	−9.84342	−0.363575	−0.181787	+	0.983338i	$0.558188\pi$
$734$	1.99259	0.0735478
$735$	0	0
$736$	54.4842	2.00831
$737$	32.1775	1.18527
$738$	10.3288	0.380208
$739$	−25.0992	−0.923289	−0.461645	−	0.887065i	$-0.652740\pi$
$739$	−25.0992	−0.923289	−0.461645	+	0.887065i	$0.652740\pi$
$740$	0	0
$741$	0.816514	0.0299954
$742$	4.51776	0.165852
$743$	45.4541	1.66755	0.833774	−	0.552106i	$-0.186176\pi$
$743$	45.4541	1.66755	0.833774	+	0.552106i	$0.186176\pi$
$744$	−1.49449	−0.0547908
$745$	0	0
$746$	−28.3690	−1.03866
$747$	17.8504	0.653113
$748$	26.9038	0.983699
$749$	−6.84861	−0.250243
$750$	0	0
$751$	0.291173	0.0106251	0.00531253	−	0.999986i	$-0.498309\pi$
$751$	0.291173	0.0106251	0.00531253	+	0.999986i	$0.498309\pi$
$752$	−6.04200	−0.220329
$753$	4.74701	0.172991
$754$	−2.98069	−0.108550
$755$	0	0
$756$	5.25600	0.191159
$757$	8.45373	0.307256	0.153628	−	0.988129i	$-0.450904\pi$
$757$	8.45373	0.307256	0.153628	+	0.988129i	$0.450904\pi$
$758$	−44.5902	−1.61959
$759$	22.9817	0.834181
$760$	0	0
$761$	−18.3586	−0.665498	−0.332749	−	0.943015i	$-0.607976\pi$
$761$	−18.3586	−0.665498	−0.332749	+	0.943015i	$0.607976\pi$
$762$	10.1142	0.366398
$763$	7.16249	0.259299
$764$	−22.8553	−0.826875
$765$	0	0
$766$	−24.5106	−0.885602
$767$	−2.92992	−0.105793
$768$	10.9858	0.396416
$769$	−45.6484	−1.64612	−0.823061	−	0.567953i	$-0.807736\pi$
$769$	−45.6484	−1.64612	−0.823061	+	0.567953i	$0.807736\pi$
$770$	0	0
$771$	0.167398	0.00602869
$772$	−15.6702	−0.563983
$773$	47.5774	1.71124	0.855620	−	0.517604i	$-0.173176\pi$
$773$	47.5774	1.71124	0.855620	+	0.517604i	$0.173176\pi$
$774$	−29.3843	−1.05620
$775$	0	0
$776$	0.586509	0.0210544
$777$	5.11933	0.183655
$778$	−60.9395	−2.18479
$779$	−5.63685	−0.201961
$780$	0	0
$781$	−62.2011	−2.22573
$782$	−38.8982	−1.39100
$783$	9.47522	0.338617
$784$	27.2973	0.974902
$785$	0	0
$786$	−10.0406	−0.358137
$787$	−48.3487	−1.72345	−0.861723	−	0.507378i	$-0.830615\pi$
$787$	−48.3487	−1.72345	−0.861723	+	0.507378i	$0.830615\pi$
$788$	43.8762	1.56302
$789$	−8.65106	−0.307986
$790$	0	0
$791$	−0.384920	−0.0136862
$792$	−8.91862	−0.316909
$793$	5.94413	0.211082
$794$	34.7604	1.23360
$795$	0	0
$796$	7.23289	0.256363
$797$	26.2387	0.929422	0.464711	−	0.885463i	$-0.346158\pi$
$797$	26.2387	0.929422	0.464711	+	0.885463i	$0.346158\pi$
$798$	−2.95421	−0.104578
$799$	3.73369	0.132088
$800$	0	0
$801$	−35.6782	−1.26063
$802$	−47.7933	−1.68764
$803$	88.1717	3.11151
$804$	5.35493	0.188854
$805$	0	0
$806$	−4.58386	−0.161460
$807$	−15.9872	−0.562776
$808$	−4.14421	−0.145793
$809$	8.43180	0.296446	0.148223	−	0.988954i	$-0.452645\pi$
$809$	8.43180	0.296446	0.148223	+	0.988954i	$0.452645\pi$
$810$	0	0
$811$	30.9938	1.08834	0.544170	−	0.838975i	$-0.316845\pi$
$811$	30.9938	1.08834	0.544170	+	0.838975i	$0.316845\pi$
$812$	4.94948	0.173693
$813$	−3.35480	−0.117658
$814$	102.733	3.60079
$815$	0	0
$816$	−7.02335	−0.245867
$817$	16.0362	0.561037
$818$	−8.18092	−0.286039
$819$	−1.36313	−0.0476317
$820$	0	0
$821$	30.6989	1.07140	0.535699	−	0.844409i	$-0.320048\pi$
$821$	30.6989	1.07140	0.535699	+	0.844409i	$0.320048\pi$
$822$	5.71231	0.199240
$823$	20.4776	0.713804	0.356902	−	0.934142i	$-0.383833\pi$
$823$	20.4776	0.713804	0.356902	+	0.934142i	$0.383833\pi$
$824$	3.22460	0.112334
$825$	0	0
$826$	10.6007	0.368845
$827$	−27.5776	−0.958967	−0.479484	−	0.877551i	$-0.659176\pi$
$827$	−27.5776	−0.958967	−0.479484	+	0.877551i	$0.659176\pi$
$828$	−33.0825	−1.14970
$829$	−13.2760	−0.461094	−0.230547	−	0.973061i	$-0.574052\pi$
$829$	−13.2760	−0.461094	−0.230547	+	0.973061i	$0.574052\pi$
$830$	0	0
$831$	12.3645	0.428919
$832$	2.81087	0.0974493
$833$	−16.8685	−0.584459
$834$	−8.30478	−0.287571
$835$	0	0
$836$	−27.2084	−0.941024
$837$	14.5715	0.503665
$838$	−32.2082	−1.11261
$839$	10.9056	0.376504	0.188252	−	0.982121i	$-0.439718\pi$
$839$	10.9056	0.376504	0.188252	+	0.982121i	$0.439718\pi$
$840$	0	0
$841$	−20.0773	−0.692322
$842$	−40.2312	−1.38646
$843$	−9.84395	−0.339043
$844$	10.2351	0.352307
$845$	0	0
$846$	6.91898	0.237879
$847$	−20.8109	−0.715073
$848$	−10.8622	−0.373008
$849$	2.08186	0.0714493
$850$	0	0
$851$	−68.1697	−2.33683
$852$	−10.3514	−0.354634
$853$	−8.12490	−0.278191	−0.139096	−	0.990279i	$-0.544420\pi$
$853$	−8.12490	−0.278191	−0.139096	+	0.990279i	$0.544420\pi$
$854$	−21.5063	−0.735932
$855$	0	0
$856$	4.09149	0.139844
$857$	−34.5296	−1.17951	−0.589755	−	0.807582i	$-0.700776\pi$
$857$	−34.5296	−1.17951	−0.589755	+	0.807582i	$0.700776\pi$
$858$	3.16242	0.107963
$859$	−29.8705	−1.01917	−0.509584	−	0.860421i	$-0.670201\pi$
$859$	−29.8705	−1.01917	−0.509584	+	0.860421i	$0.670201\pi$
$860$	0	0
$861$	−1.08792	−0.0370761
$862$	53.3152	1.81592
$863$	13.2136	0.449797	0.224899	−	0.974382i	$-0.427795\pi$
$863$	13.2136	0.449797	0.224899	+	0.974382i	$0.427795\pi$
$864$	−23.8332	−0.810820
$865$	0	0
$866$	53.9977	1.83492
$867$	−5.13852	−0.174513
$868$	7.61159	0.258354
$869$	−90.4682	−3.06892
$870$	0	0
$871$	−2.93814	−0.0995550
$872$	−4.27901	−0.144905
$873$	−2.70302	−0.0914835
$874$	39.3387	1.33065
$875$	0	0
$876$	14.6734	0.495769
$877$	30.5723	1.03235	0.516176	−	0.856482i	$-0.327355\pi$
$877$	30.5723	1.03235	0.516176	+	0.856482i	$0.327355\pi$
$878$	19.3306	0.652377
$879$	−16.9467	−0.571600
$880$	0	0
$881$	23.4110	0.788736	0.394368	−	0.918953i	$-0.370964\pi$
$881$	23.4110	0.788736	0.394368	+	0.918953i	$0.370964\pi$
$882$	−31.2594	−1.05256
$883$	−25.5119	−0.858545	−0.429273	−	0.903175i	$-0.641230\pi$
$883$	−25.5119	−0.858545	−0.429273	+	0.903175i	$0.641230\pi$
$884$	−2.45659	−0.0826241
$885$	0	0
$886$	8.42555	0.283062
$887$	6.77343	0.227430	0.113715	−	0.993513i	$-0.463725\pi$
$887$	6.77343	0.227430	0.113715	+	0.993513i	$0.463725\pi$
$888$	−3.05839	−0.102633
$889$	9.21493	0.309059
$890$	0	0
$891$	35.8018	1.19941
$892$	−35.9265	−1.20291
$893$	−3.77598	−0.126358
$894$	21.2345	0.710188
$895$	0	0
$896$	4.50662	0.150556
$897$	−2.09846	−0.0700657
$898$	−14.4485	−0.482152
$899$	13.7217	0.457646
$900$	0	0
$901$	6.71234	0.223620
$902$	−21.8320	−0.726925
$903$	3.09501	0.102995
$904$	0.229959	0.00764831
$905$	0	0
$906$	−21.1043	−0.701142
$907$	55.9625	1.85821	0.929103	−	0.369822i	$-0.120581\pi$
$907$	55.9625	1.85821	0.929103	+	0.369822i	$0.120581\pi$
$908$	5.94320	0.197232
$909$	19.0993	0.633483
$910$	0	0
$911$	−13.0286	−0.431656	−0.215828	−	0.976431i	$-0.569245\pi$
$911$	−13.0286	−0.431656	−0.215828	+	0.976431i	$0.569245\pi$
$912$	7.10290	0.235200
$913$	−37.7304	−1.24870
$914$	52.0901	1.72299
$915$	0	0
$916$	−32.7987	−1.08370
$917$	−9.14791	−0.302091
$918$	17.0153	0.561590
$919$	28.4860	0.939668	0.469834	−	0.882755i	$-0.344314\pi$
$919$	28.4860	0.939668	0.469834	+	0.882755i	$0.344314\pi$
$920$	0	0
$921$	3.28100	0.108112
$922$	−15.9350	−0.524791
$923$	5.67961	0.186947
$924$	−5.25125	−0.172754
$925$	0	0
$926$	75.6209	2.48506
$927$	−14.8611	−0.488103
$928$	−22.4433	−0.736737
$929$	−29.8029	−0.977801	−0.488900	−	0.872340i	$-0.662602\pi$
$929$	−29.8029	−0.977801	−0.488900	+	0.872340i	$0.662602\pi$
$930$	0	0
$931$	17.0595	0.559104
$932$	−39.1359	−1.28194
$933$	−4.77760	−0.156412
$934$	3.12511	0.102257
$935$	0	0
$936$	0.814362	0.0266183
$937$	−8.54890	−0.279281	−0.139640	−	0.990202i	$-0.544595\pi$
$937$	−8.54890	−0.279281	−0.139640	+	0.990202i	$0.544595\pi$
$938$	10.6304	0.347096
$939$	−3.72605	−0.121595
$940$	0	0
$941$	−5.33942	−0.174060	−0.0870300	−	0.996206i	$-0.527738\pi$
$941$	−5.33942	−0.174060	−0.0870300	+	0.996206i	$0.527738\pi$
$942$	23.0967	0.752531
$943$	14.4869	0.471757
$944$	−25.4875	−0.829547
$945$	0	0
$946$	62.1096	2.01936
$947$	−24.4488	−0.794481	−0.397240	−	0.917715i	$-0.630032\pi$
$947$	−24.4488	−0.794481	−0.397240	+	0.917715i	$0.630032\pi$
$948$	−15.0556	−0.488983
$949$	−8.05100	−0.261346
$950$	0	0
$951$	10.4544	0.339007
$952$	−1.58998	−0.0515316
$953$	42.1998	1.36699	0.683493	−	0.729957i	$-0.260460\pi$
$953$	42.1998	1.36699	0.683493	+	0.729957i	$0.260460\pi$
$954$	12.4388	0.402720
$955$	0	0
$956$	−5.31657	−0.171950
$957$	−9.46667	−0.306014
$958$	−52.4462	−1.69446
$959$	5.20443	0.168060
$960$	0	0
$961$	−9.89795	−0.319289
$962$	−9.38059	−0.302442
$963$	−18.8563	−0.607636
$964$	1.69651	0.0546410
$965$	0	0
$966$	7.59241	0.244282
$967$	−12.1031	−0.389211	−0.194605	−	0.980882i	$-0.562343\pi$
$967$	−12.1031	−0.389211	−0.194605	+	0.980882i	$0.562343\pi$
$968$	12.4329	0.399607
$969$	−4.38928	−0.141004
$970$	0	0
$971$	5.26565	0.168983	0.0844913	−	0.996424i	$-0.473073\pi$
$971$	5.26565	0.168983	0.0844913	+	0.996424i	$0.473073\pi$
$972$	22.1025	0.708937
$973$	−7.56641	−0.242568
$974$	−31.0422	−0.994657
$975$	0	0
$976$	51.7083	1.65514
$977$	9.31335	0.297961	0.148980	−	0.988840i	$-0.452401\pi$
$977$	9.31335	0.297961	0.148980	+	0.988840i	$0.452401\pi$
$978$	−13.4174	−0.429041
$979$	75.4130	2.41021
$980$	0	0
$981$	19.7205	0.629628
$982$	26.8734	0.857564
$983$	−27.0262	−0.862003	−0.431002	−	0.902351i	$-0.641840\pi$
$983$	−27.0262	−0.862003	−0.431002	+	0.902351i	$0.641840\pi$
$984$	0.649943	0.0207194
$985$	0	0
$986$	16.0231	0.510278
$987$	−0.728767	−0.0231969
$988$	2.48441	0.0790397
$989$	−41.2136	−1.31052
$990$	0	0
$991$	−22.7776	−0.723554	−0.361777	−	0.932265i	$-0.617830\pi$
$991$	−22.7776	−0.723554	−0.361777	+	0.932265i	$0.617830\pi$
$992$	−34.5145	−1.09584
$993$	−5.65345	−0.179407
$994$	−20.5493	−0.651783
$995$	0	0
$996$	−6.27904	−0.198959
$997$	24.9695	0.790791	0.395395	−	0.918511i	$-0.370608\pi$
$997$	24.9695	0.790791	0.395395	+	0.918511i	$0.370608\pi$
$998$	−26.0670	−0.825135
$999$	29.8197	0.943453

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.p.1.7		46
5.2	odd	4		1205.2.b.c.724.7	✓	46
5.3	odd	4		1205.2.b.c.724.40	yes	46
5.4	even	2	inner	6025.2.a.p.1.40		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.c.724.7	✓	46	5.2	odd	4
1205.2.b.c.724.40	yes	46	5.3	odd	4
6025.2.a.p.1.7		46	1.1	even	1	trivial
6025.2.a.p.1.40		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)

\(f(q)\)	\(=\)	\(q-1.92263 q^{2} +0.557567 q^{3} +1.69651 q^{4} -1.07200 q^{6} -0.976687 q^{7} +0.583491 q^{8} -2.68912 q^{9} +O(q^{10})\)	\(q-1.92263 q^{2} +0.557567 q^{3} +1.69651 q^{4} -1.07200 q^{6} -0.976687 q^{7} +0.583491 q^{8} -2.68912 q^{9} +5.68399 q^{11} +0.945921 q^{12} -0.519007 q^{13} +1.87781 q^{14} -4.51487 q^{16} +2.78999 q^{17} +5.17019 q^{18} -2.82159 q^{19} -0.544569 q^{21} -10.9282 q^{22} +7.25155 q^{23} +0.325336 q^{24} +0.997860 q^{26} -3.17207 q^{27} -1.65696 q^{28} -2.98708 q^{29} -4.59370 q^{31} +7.51345 q^{32} +3.16920 q^{33} -5.36412 q^{34} -4.56213 q^{36} -9.40071 q^{37} +5.42487 q^{38} -0.289381 q^{39} +1.99776 q^{41} +1.04700 q^{42} -5.68341 q^{43} +9.64296 q^{44} -13.9421 q^{46} +1.33825 q^{47} -2.51734 q^{48} -6.04608 q^{49} +1.55561 q^{51} -0.880503 q^{52} +2.40587 q^{53} +6.09872 q^{54} -0.569888 q^{56} -1.57322 q^{57} +5.74306 q^{58} +5.64524 q^{59} -11.4529 q^{61} +8.83199 q^{62} +2.62643 q^{63} -5.41586 q^{64} -6.09321 q^{66} +5.66108 q^{67} +4.73325 q^{68} +4.04323 q^{69} -10.9432 q^{71} -1.56908 q^{72} +15.5123 q^{73} +18.0741 q^{74} -4.78686 q^{76} -5.55147 q^{77} +0.556374 q^{78} -15.9163 q^{79} +6.29872 q^{81} -3.84096 q^{82} -6.63802 q^{83} -0.923868 q^{84} +10.9271 q^{86} -1.66550 q^{87} +3.31656 q^{88} +13.2676 q^{89} +0.506907 q^{91} +12.3024 q^{92} -2.56129 q^{93} -2.57295 q^{94} +4.18925 q^{96} +1.00517 q^{97} +11.6244 q^{98} -15.2849 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) \) 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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