LMFDB - Embedded newform 6025.2.a.p.1.6

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			1.6
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)

$f(q)$	$=$	$q-1.95888 q^{2} +0.350465 q^{3} +1.83721 q^{4} -0.686519 q^{6} +1.14869 q^{7} +0.318879 q^{8} -2.87717 q^{9} +O(q^{10})$	\(q-1.95888 q^{2} +0.350465 q^{3} +1.83721 q^{4} -0.686519 q^{6} +1.14869 q^{7} +0.318879 q^{8} -2.87717 q^{9} +0.395421 q^{11} +0.643879 q^{12} -0.595326 q^{13} -2.25014 q^{14} -4.29907 q^{16} -7.79008 q^{17} +5.63604 q^{18} +3.25574 q^{19} +0.402575 q^{21} -0.774582 q^{22} +3.49975 q^{23} +0.111756 q^{24} +1.16617 q^{26} -2.05974 q^{27} +2.11038 q^{28} +6.23795 q^{29} -9.01707 q^{31} +7.78361 q^{32} +0.138581 q^{33} +15.2598 q^{34} -5.28598 q^{36} +0.667693 q^{37} -6.37760 q^{38} -0.208641 q^{39} +6.05766 q^{41} -0.788596 q^{42} +9.26190 q^{43} +0.726472 q^{44} -6.85559 q^{46} +13.6965 q^{47} -1.50667 q^{48} -5.68052 q^{49} -2.73015 q^{51} -1.09374 q^{52} +0.719285 q^{53} +4.03479 q^{54} +0.366293 q^{56} +1.14102 q^{57} -12.2194 q^{58} -10.8402 q^{59} +2.84725 q^{61} +17.6634 q^{62} -3.30498 q^{63} -6.64902 q^{64} -0.271464 q^{66} +1.17890 q^{67} -14.3120 q^{68} +1.22654 q^{69} -14.0182 q^{71} -0.917471 q^{72} -4.89629 q^{73} -1.30793 q^{74} +5.98148 q^{76} +0.454215 q^{77} +0.408703 q^{78} +13.1124 q^{79} +7.90966 q^{81} -11.8662 q^{82} +5.86352 q^{83} +0.739616 q^{84} -18.1429 q^{86} +2.18618 q^{87} +0.126092 q^{88} -11.3538 q^{89} -0.683844 q^{91} +6.42979 q^{92} -3.16017 q^{93} -26.8298 q^{94} +2.72788 q^{96} +0.175389 q^{97} +11.1275 q^{98} -1.13769 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.95888	−1.38514	−0.692569	−	0.721352i	$-0.743521\pi$
$2$	−1.95888	−1.38514	−0.692569	+	0.721352i	$0.743521\pi$
$3$	0.350465	0.202341	0.101171	−	0.994869i	$-0.467741\pi$
$3$	0.350465	0.202341	0.101171	+	0.994869i	$0.467741\pi$
$4$	1.83721	0.918607
$5$	0	0
$5$	0	0
$6$	−0.686519	−0.280270
$7$	1.14869	0.434163	0.217082	−	0.976153i	$-0.430346\pi$
$7$	1.14869	0.434163	0.217082	+	0.976153i	$0.430346\pi$
$8$	0.318879	0.112741
$9$	−2.87717	−0.959058
$10$	0	0
$11$	0.395421	0.119224	0.0596119	−	0.998222i	$-0.481014\pi$
$11$	0.395421	0.119224	0.0596119	+	0.998222i	$0.481014\pi$
$12$	0.643879	0.185872
$13$	−0.595326	−0.165114	−0.0825569	−	0.996586i	$-0.526309\pi$
$13$	−0.595326	−0.165114	−0.0825569	+	0.996586i	$0.526309\pi$
$14$	−2.25014	−0.601376
$15$	0	0
$16$	−4.29907	−1.07477
$17$	−7.79008	−1.88937	−0.944686	−	0.327977i	$-0.893633\pi$
$17$	−7.79008	−1.88937	−0.944686	+	0.327977i	$0.893633\pi$
$18$	5.63604	1.32843
$19$	3.25574	0.746917	0.373459	−	0.927647i	$-0.378172\pi$
$19$	3.25574	0.746917	0.373459	+	0.927647i	$0.378172\pi$
$20$	0	0
$21$	0.402575	0.0878490
$22$	−0.774582	−0.165141
$23$	3.49975	0.729748	0.364874	−	0.931057i	$-0.381112\pi$
$23$	3.49975	0.729748	0.364874	+	0.931057i	$0.381112\pi$
$24$	0.111756	0.0228121
$25$	0	0
$26$	1.16617	0.228705
$27$	−2.05974	−0.396398
$28$	2.11038	0.398825
$29$	6.23795	1.15836	0.579179	−	0.815200i	$-0.303373\pi$
$29$	6.23795	1.15836	0.579179	+	0.815200i	$0.303373\pi$
$30$	0	0
$31$	−9.01707	−1.61951	−0.809757	−	0.586766i	$-0.800401\pi$
$31$	−9.01707	−1.61951	−0.809757	+	0.586766i	$0.800401\pi$
$32$	7.78361	1.37596
$33$	0.138581	0.0241239
$34$	15.2598	2.61704
$35$	0	0
$36$	−5.28598	−0.880997
$37$	0.667693	0.109768	0.0548840	−	0.998493i	$-0.482521\pi$
$37$	0.667693	0.109768	0.0548840	+	0.998493i	$0.482521\pi$
$38$	−6.37760	−1.03458
$39$	−0.208641	−0.0334093
$40$	0	0
$41$	6.05766	0.946048	0.473024	−	0.881050i	$-0.343162\pi$
$41$	6.05766	0.946048	0.473024	+	0.881050i	$0.343162\pi$
$42$	−0.788596	−0.121683
$43$	9.26190	1.41243	0.706213	−	0.707999i	$-0.250402\pi$
$43$	9.26190	1.41243	0.706213	+	0.707999i	$0.250402\pi$
$44$	0.726472	0.109520
$45$	0	0
$46$	−6.85559	−1.01080
$47$	13.6965	1.99784	0.998921	−	0.0464433i	$-0.0147887\pi$
$47$	13.6965	1.99784	0.998921	+	0.0464433i	$0.0147887\pi$
$48$	−1.50667	−0.217470
$49$	−5.68052	−0.811502
$50$	0	0
$51$	−2.73015	−0.382297
$52$	−1.09374	−0.151675
$53$	0.719285	0.0988013	0.0494007	−	0.998779i	$-0.484269\pi$
$53$	0.719285	0.0988013	0.0494007	+	0.998779i	$0.484269\pi$
$54$	4.03479	0.549066
$55$	0	0
$56$	0.366293	0.0489479
$57$	1.14102	0.151132
$58$	−12.2194	−1.60449
$59$	−10.8402	−1.41127	−0.705635	−	0.708576i	$-0.749338\pi$
$59$	−10.8402	−1.41127	−0.705635	+	0.708576i	$0.749338\pi$
$60$	0	0
$61$	2.84725	0.364553	0.182276	−	0.983247i	$-0.441653\pi$
$61$	2.84725	0.364553	0.182276	+	0.983247i	$0.441653\pi$
$62$	17.6634	2.24325
$63$	−3.30498	−0.416388
$64$	−6.64902	−0.831128
$65$	0	0
$66$	−0.271464	−0.0334149
$67$	1.17890	0.144026	0.0720129	−	0.997404i	$-0.477058\pi$
$67$	1.17890	0.144026	0.0720129	+	0.997404i	$0.477058\pi$
$68$	−14.3120	−1.73559
$69$	1.22654	0.147658
$70$	0	0
$71$	−14.0182	−1.66366	−0.831828	−	0.555033i	$-0.812706\pi$
$71$	−14.0182	−1.66366	−0.831828	+	0.555033i	$0.812706\pi$
$72$	−0.917471	−0.108125
$73$	−4.89629	−0.573067	−0.286534	−	0.958070i	$-0.592503\pi$
$73$	−4.89629	−0.573067	−0.286534	+	0.958070i	$0.592503\pi$
$74$	−1.30793	−0.152044
$75$	0	0
$76$	5.98148	0.686123
$77$	0.454215	0.0517626
$78$	0.408703	0.0462765
$79$	13.1124	1.47526	0.737632	−	0.675203i	$-0.235944\pi$
$79$	13.1124	1.47526	0.737632	+	0.675203i	$0.235944\pi$
$80$	0	0
$81$	7.90966	0.878851
$82$	−11.8662	−1.31041
$83$	5.86352	0.643604	0.321802	−	0.946807i	$-0.395711\pi$
$83$	5.86352	0.643604	0.321802	+	0.946807i	$0.395711\pi$
$84$	0.739616	0.0806987
$85$	0	0
$86$	−18.1429	−1.95640
$87$	2.18618	0.234383
$88$	0.126092	0.0134414
$89$	−11.3538	−1.20350	−0.601751	−	0.798683i	$-0.705530\pi$
$89$	−11.3538	−1.20350	−0.601751	+	0.798683i	$0.705530\pi$
$90$	0	0
$91$	−0.683844	−0.0716863
$92$	6.42979	0.670352
$93$	−3.16017	−0.327694
$94$	−26.8298	−2.76729
$95$	0	0
$96$	2.72788	0.278413
$97$	0.175389	0.0178081	0.00890404	−	0.999960i	$-0.497166\pi$
$97$	0.175389	0.0178081	0.00890404	+	0.999960i	$0.497166\pi$
$98$	11.1275	1.12404
$99$	−1.13769	−0.114343
$100$	0	0
$101$	−11.2535	−1.11976	−0.559882	−	0.828572i	$-0.689154\pi$
$101$	−11.2535	−1.11976	−0.559882	+	0.828572i	$0.689154\pi$
$102$	5.34804	0.529534
$103$	9.35760	0.922032	0.461016	−	0.887392i	$-0.347485\pi$
$103$	9.35760	0.922032	0.461016	+	0.887392i	$0.347485\pi$
$104$	−0.189837	−0.0186151
$105$	0	0
$106$	−1.40899	−0.136853
$107$	−11.7603	−1.13691	−0.568456	−	0.822714i	$-0.692459\pi$
$107$	−11.7603	−1.13691	−0.568456	+	0.822714i	$0.692459\pi$
$108$	−3.78419	−0.364134
$109$	−4.89496	−0.468852	−0.234426	−	0.972134i	$-0.575321\pi$
$109$	−4.89496	−0.468852	−0.234426	+	0.972134i	$0.575321\pi$
$110$	0	0
$111$	0.234003	0.0222106
$112$	−4.93829	−0.466625
$113$	7.26082	0.683041	0.341520	−	0.939874i	$-0.389058\pi$
$113$	7.26082	0.683041	0.341520	+	0.939874i	$0.389058\pi$
$114$	−2.23512	−0.209339
$115$	0	0
$116$	11.4604	1.06408
$117$	1.71286	0.158354
$118$	21.2346	1.95480
$119$	−8.94837	−0.820296
$120$	0	0
$121$	−10.8436	−0.985786
$122$	−5.57742	−0.504956
$123$	2.12300	0.191424
$124$	−16.5663	−1.48770
$125$	0	0
$126$	6.47405	0.576754
$127$	12.5876	1.11697	0.558483	−	0.829516i	$-0.311384\pi$
$127$	12.5876	1.11697	0.558483	+	0.829516i	$0.311384\pi$
$128$	−2.54258	−0.224735
$129$	3.24597	0.285792
$130$	0	0
$131$	−14.3195	−1.25110	−0.625552	−	0.780182i	$-0.715126\pi$
$131$	−14.3195	−1.25110	−0.625552	+	0.780182i	$0.715126\pi$
$132$	0.254603	0.0221604
$133$	3.73982	0.324284
$134$	−2.30933	−0.199495
$135$	0	0
$136$	−2.48409	−0.213009
$137$	−5.34099	−0.456312	−0.228156	−	0.973625i	$-0.573270\pi$
$137$	−5.34099	−0.456312	−0.228156	+	0.973625i	$0.573270\pi$
$138$	−2.40264	−0.204527
$139$	−13.1396	−1.11448	−0.557241	−	0.830351i	$-0.688140\pi$
$139$	−13.1396	−1.11448	−0.557241	+	0.830351i	$0.688140\pi$
$140$	0	0
$141$	4.80015	0.404245
$142$	27.4600	2.30439
$143$	−0.235404	−0.0196855
$144$	12.3692	1.03077
$145$	0	0
$146$	9.59125	0.793777
$147$	−1.99082	−0.164200
$148$	1.22669	0.100834
$149$	−6.09589	−0.499395	−0.249697	−	0.968324i	$-0.580331\pi$
$149$	−6.09589	−0.499395	−0.249697	+	0.968324i	$0.580331\pi$
$150$	0	0
$151$	−2.45933	−0.200137	−0.100069	−	0.994981i	$-0.531906\pi$
$151$	−2.45933	−0.200137	−0.100069	+	0.994981i	$0.531906\pi$
$152$	1.03819	0.0842081
$153$	22.4134	1.81202
$154$	−0.889753	−0.0716983
$155$	0	0
$156$	−0.383318	−0.0306900
$157$	−2.93713	−0.234409	−0.117204	−	0.993108i	$-0.537393\pi$
$157$	−2.93713	−0.234409	−0.117204	+	0.993108i	$0.537393\pi$
$158$	−25.6857	−2.04344
$159$	0.252084	0.0199916
$160$	0	0
$161$	4.02012	0.316830
$162$	−15.4941	−1.21733
$163$	23.5480	1.84443	0.922213	−	0.386683i	$-0.126379\pi$
$163$	23.5480	1.84443	0.922213	+	0.386683i	$0.126379\pi$
$164$	11.1292	0.869046
$165$	0	0
$166$	−11.4859	−0.891481
$167$	−15.3378	−1.18688	−0.593439	−	0.804879i	$-0.702230\pi$
$167$	−15.3378	−1.18688	−0.593439	+	0.804879i	$0.702230\pi$
$168$	0.128373	0.00990418
$169$	−12.6456	−0.972737
$170$	0	0
$171$	−9.36732	−0.716337
$172$	17.0161	1.29746
$173$	−7.62958	−0.580066	−0.290033	−	0.957017i	$-0.593666\pi$
$173$	−7.62958	−0.580066	−0.290033	+	0.957017i	$0.593666\pi$
$174$	−4.28247	−0.324653
$175$	0	0
$176$	−1.69994	−0.128138
$177$	−3.79910	−0.285558
$178$	22.2408	1.66702
$179$	13.4465	1.00504	0.502520	−	0.864566i	$-0.332406\pi$
$179$	13.4465	1.00504	0.502520	+	0.864566i	$0.332406\pi$
$180$	0	0
$181$	24.3051	1.80658	0.903292	−	0.429027i	$-0.141144\pi$
$181$	24.3051	1.80658	0.903292	+	0.429027i	$0.141144\pi$
$182$	1.33957	0.0992954
$183$	0.997861	0.0737640
$184$	1.11600	0.0822725
$185$	0	0
$186$	6.19039	0.453901
$187$	−3.08036	−0.225258
$188$	25.1634	1.83523
$189$	−2.36600	−0.172101
$190$	0	0
$191$	16.5298	1.19605	0.598026	−	0.801476i	$-0.295952\pi$
$191$	16.5298	1.19605	0.598026	+	0.801476i	$0.295952\pi$
$192$	−2.33025	−0.168171
$193$	−14.0074	−1.00828	−0.504138	−	0.863623i	$-0.668190\pi$
$193$	−14.0074	−1.00828	−0.504138	+	0.863623i	$0.668190\pi$
$194$	−0.343566	−0.0246666
$195$	0	0
$196$	−10.4363	−0.745451
$197$	−25.8379	−1.84087	−0.920437	−	0.390892i	$-0.872167\pi$
$197$	−25.8379	−1.84087	−0.920437	+	0.390892i	$0.872167\pi$
$198$	2.22861	0.158380
$199$	−5.06385	−0.358967	−0.179483	−	0.983761i	$-0.557443\pi$
$199$	−5.06385	−0.358967	−0.179483	+	0.983761i	$0.557443\pi$
$200$	0	0
$201$	0.413164	0.0291423
$202$	22.0443	1.55103
$203$	7.16546	0.502916
$204$	−5.01587	−0.351181
$205$	0	0
$206$	−18.3304	−1.27714
$207$	−10.0694	−0.699871
$208$	2.55935	0.177459
$209$	1.28739	0.0890503
$210$	0	0
$211$	−12.2747	−0.845026	−0.422513	−	0.906357i	$-0.638852\pi$
$211$	−12.2747	−0.845026	−0.422513	+	0.906357i	$0.638852\pi$
$212$	1.32148	0.0907596
$213$	−4.91289	−0.336626
$214$	23.0370	1.57478
$215$	0	0
$216$	−0.656810	−0.0446902
$217$	−10.3578	−0.703133
$218$	9.58864	0.649425
$219$	−1.71598	−0.115955
$220$	0	0
$221$	4.63764	0.311961
$222$	−0.458384	−0.0307647
$223$	20.5388	1.37538	0.687689	−	0.726005i	$-0.258625\pi$
$223$	20.5388	1.37538	0.687689	+	0.726005i	$0.258625\pi$
$224$	8.94094	0.597392
$225$	0	0
$226$	−14.2231	−0.946106
$227$	15.6294	1.03736	0.518679	−	0.854969i	$-0.326424\pi$
$227$	15.6294	1.03736	0.518679	+	0.854969i	$0.326424\pi$
$228$	2.09630	0.138831
$229$	−7.68533	−0.507861	−0.253930	−	0.967222i	$-0.581723\pi$
$229$	−7.68533	−0.507861	−0.253930	+	0.967222i	$0.581723\pi$
$230$	0	0
$231$	0.159186	0.0104737
$232$	1.98915	0.130594
$233$	−28.4122	−1.86135	−0.930673	−	0.365851i	$-0.880778\pi$
$233$	−28.4122	−1.86135	−0.930673	+	0.365851i	$0.880778\pi$
$234$	−3.35528	−0.219342
$235$	0	0
$236$	−19.9157	−1.29640
$237$	4.59545	0.298507
$238$	17.5288	1.13622
$239$	−2.95254	−0.190984	−0.0954918	−	0.995430i	$-0.530442\pi$
$239$	−2.95254	−0.190984	−0.0954918	+	0.995430i	$0.530442\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	21.2414	1.36545
$243$	8.95129	0.574225
$244$	5.23100	0.334881
$245$	0	0
$246$	−4.15870	−0.265149
$247$	−1.93822	−0.123326
$248$	−2.87536	−0.182585
$249$	2.05496	0.130228
$250$	0	0
$251$	12.1180	0.764879	0.382439	−	0.923981i	$-0.375084\pi$
$251$	12.1180	0.764879	0.382439	+	0.923981i	$0.375084\pi$
$252$	−6.07195	−0.382497
$253$	1.38387	0.0870034
$254$	−24.6575	−1.54715
$255$	0	0
$256$	18.2787	1.14242
$257$	−28.0054	−1.74693	−0.873464	−	0.486888i	$-0.838132\pi$
$257$	−28.0054	−1.74693	−0.873464	+	0.486888i	$0.838132\pi$
$258$	−6.35847	−0.395861
$259$	0.766971	0.0476572
$260$	0	0
$261$	−17.9477	−1.11093
$262$	28.0503	1.73295
$263$	−9.93706	−0.612746	−0.306373	−	0.951912i	$-0.599115\pi$
$263$	−9.93706	−0.612746	−0.306373	+	0.951912i	$0.599115\pi$
$264$	0.0441907	0.00271975
$265$	0	0
$266$	−7.32587	−0.449178
$267$	−3.97912	−0.243518
$268$	2.16589	0.132303
$269$	−19.8870	−1.21253	−0.606266	−	0.795262i	$-0.707333\pi$
$269$	−19.8870	−1.21253	−0.606266	+	0.795262i	$0.707333\pi$
$270$	0	0
$271$	7.60072	0.461711	0.230855	−	0.972988i	$-0.425848\pi$
$271$	7.60072	0.461711	0.230855	+	0.972988i	$0.425848\pi$
$272$	33.4901	2.03064
$273$	−0.239663	−0.0145051
$274$	10.4624	0.632055
$275$	0	0
$276$	2.25342	0.135640
$277$	−2.57322	−0.154610	−0.0773048	−	0.997008i	$-0.524631\pi$
$277$	−2.57322	−0.154610	−0.0773048	+	0.997008i	$0.524631\pi$
$278$	25.7388	1.54371
$279$	25.9437	1.55321
$280$	0	0
$281$	−4.36381	−0.260323	−0.130161	−	0.991493i	$-0.541550\pi$
$281$	−4.36381	−0.260323	−0.130161	+	0.991493i	$0.541550\pi$
$282$	−9.40292	−0.559936
$283$	−9.97677	−0.593058	−0.296529	−	0.955024i	$-0.595829\pi$
$283$	−9.97677	−0.593058	−0.296529	+	0.955024i	$0.595829\pi$
$284$	−25.7545	−1.52825
$285$	0	0
$286$	0.461129	0.0272671
$287$	6.95836	0.410739
$288$	−22.3948	−1.31963
$289$	43.6853	2.56972
$290$	0	0
$291$	0.0614678	0.00360330
$292$	−8.99553	−0.526424
$293$	8.13620	0.475322	0.237661	−	0.971348i	$-0.423619\pi$
$293$	8.13620	0.475322	0.237661	+	0.971348i	$0.423619\pi$
$294$	3.89978	0.227440
$295$	0	0
$296$	0.212913	0.0123753
$297$	−0.814465	−0.0472601
$298$	11.9411	0.691731
$299$	−2.08349	−0.120491
$300$	0	0
$301$	10.6390	0.613223
$302$	4.81753	0.277217
$303$	−3.94396	−0.226574
$304$	−13.9967	−0.802763
$305$	0	0
$306$	−43.9052	−2.50989
$307$	−16.9541	−0.967624	−0.483812	−	0.875172i	$-0.660748\pi$
$307$	−16.9541	−0.967624	−0.483812	+	0.875172i	$0.660748\pi$
$308$	0.834490	0.0475495
$309$	3.27951	0.186565
$310$	0	0
$311$	17.6072	0.998412	0.499206	−	0.866483i	$-0.333625\pi$
$311$	17.6072	0.998412	0.499206	+	0.866483i	$0.333625\pi$
$312$	−0.0665313	−0.00376659
$313$	0.775970	0.0438604	0.0219302	−	0.999760i	$-0.493019\pi$
$313$	0.775970	0.0438604	0.0219302	+	0.999760i	$0.493019\pi$
$314$	5.75349	0.324688
$315$	0	0
$316$	24.0903	1.35519
$317$	−20.2080	−1.13499	−0.567496	−	0.823376i	$-0.692088\pi$
$317$	−20.2080	−1.13499	−0.567496	+	0.823376i	$0.692088\pi$
$318$	−0.493803	−0.0276911
$319$	2.46661	0.138104
$320$	0	0
$321$	−4.12157	−0.230044
$322$	−7.87494	−0.438853
$323$	−25.3624	−1.41120
$324$	14.5317	0.807318
$325$	0	0
$326$	−46.1278	−2.55478
$327$	−1.71551	−0.0948680
$328$	1.93166	0.106658
$329$	15.7330	0.867389
$330$	0	0
$331$	3.47783	0.191159	0.0955794	−	0.995422i	$-0.469530\pi$
$331$	3.47783	0.191159	0.0955794	+	0.995422i	$0.469530\pi$
$332$	10.7725	0.591219
$333$	−1.92107	−0.105274
$334$	30.0450	1.64399
$335$	0	0
$336$	−1.73070	−0.0944174
$337$	−28.8161	−1.56971	−0.784857	−	0.619677i	$-0.787263\pi$
$337$	−28.8161	−1.56971	−0.784857	+	0.619677i	$0.787263\pi$
$338$	24.7712	1.34738
$339$	2.54466	0.138207
$340$	0	0
$341$	−3.56554	−0.193085
$342$	18.3495	0.992225
$343$	−14.5660	−0.786488
$344$	2.95343	0.159238
$345$	0	0
$346$	14.9454	0.803472
$347$	−2.81946	−0.151356	−0.0756782	−	0.997132i	$-0.524112\pi$
$347$	−2.81946	−0.151356	−0.0756782	+	0.997132i	$0.524112\pi$
$348$	4.01648	0.215306
$349$	−2.99211	−0.160164	−0.0800820	−	0.996788i	$-0.525518\pi$
$349$	−2.99211	−0.160164	−0.0800820	+	0.996788i	$0.525518\pi$
$350$	0	0
$351$	1.22622	0.0654507
$352$	3.07780	0.164047
$353$	21.1653	1.12652	0.563259	−	0.826281i	$-0.309547\pi$
$353$	21.1653	1.12652	0.563259	+	0.826281i	$0.309547\pi$
$354$	7.44198	0.395537
$355$	0	0
$356$	−20.8594	−1.10555
$357$	−3.13609	−0.165979
$358$	−26.3401	−1.39212
$359$	−21.1263	−1.11500	−0.557501	−	0.830176i	$-0.688240\pi$
$359$	−21.1263	−1.11500	−0.557501	+	0.830176i	$0.688240\pi$
$360$	0	0
$361$	−8.40018	−0.442115
$362$	−47.6108	−2.50237
$363$	−3.80032	−0.199465
$364$	−1.25637	−0.0658515
$365$	0	0
$366$	−1.95469	−0.102173
$367$	−7.88596	−0.411644	−0.205822	−	0.978589i	$-0.565987\pi$
$367$	−7.88596	−0.411644	−0.205822	+	0.978589i	$0.565987\pi$
$368$	−15.0457	−0.784310
$369$	−17.4290	−0.907315
$370$	0	0
$371$	0.826233	0.0428959
$372$	−5.80590	−0.301022
$373$	37.2140	1.92687	0.963434	−	0.267944i	$-0.0863443\pi$
$373$	37.2140	1.92687	0.963434	+	0.267944i	$0.0863443\pi$
$374$	6.03405	0.312014
$375$	0	0
$376$	4.36754	0.225238
$377$	−3.71361	−0.191261
$378$	4.63472	0.238384
$379$	−2.84659	−0.146219	−0.0731097	−	0.997324i	$-0.523292\pi$
$379$	−2.84659	−0.146219	−0.0731097	+	0.997324i	$0.523292\pi$
$380$	0	0
$381$	4.41150	0.226008
$382$	−32.3799	−1.65670
$383$	25.2530	1.29037	0.645183	−	0.764028i	$-0.276781\pi$
$383$	25.2530	1.29037	0.645183	+	0.764028i	$0.276781\pi$
$384$	−0.891087	−0.0454731
$385$	0	0
$386$	27.4388	1.39660
$387$	−26.6481	−1.35460
$388$	0.322227	0.0163586
$389$	−24.4296	−1.23863	−0.619315	−	0.785143i	$-0.712589\pi$
$389$	−24.4296	−1.23863	−0.619315	+	0.785143i	$0.712589\pi$
$390$	0	0
$391$	−27.2633	−1.37877
$392$	−1.81140	−0.0914895
$393$	−5.01850	−0.253150
$394$	50.6133	2.54986
$395$	0	0
$396$	−2.09019	−0.105036
$397$	38.2694	1.92069	0.960343	−	0.278822i	$-0.0899439\pi$
$397$	38.2694	1.92069	0.960343	+	0.278822i	$0.0899439\pi$
$398$	9.91947	0.497218
$399$	1.31068	0.0656159
$400$	0	0
$401$	−16.9783	−0.847856	−0.423928	−	0.905696i	$-0.639349\pi$
$401$	−16.9783	−0.847856	−0.423928	+	0.905696i	$0.639349\pi$
$402$	−0.809338	−0.0403661
$403$	5.36809	0.267404
$404$	−20.6751	−1.02862
$405$	0	0
$406$	−14.0363	−0.696609
$407$	0.264020	0.0130870
$408$	−0.870588	−0.0431005
$409$	−22.7177	−1.12332	−0.561659	−	0.827369i	$-0.689837\pi$
$409$	−22.7177	−1.12332	−0.561659	+	0.827369i	$0.689837\pi$
$410$	0	0
$411$	−1.87183	−0.0923306
$412$	17.1919	0.846985
$413$	−12.4520	−0.612721
$414$	19.7247	0.969418
$415$	0	0
$416$	−4.63379	−0.227190
$417$	−4.60495	−0.225505
$418$	−2.52183	−0.123347
$419$	−16.7307	−0.817346	−0.408673	−	0.912681i	$-0.634008\pi$
$419$	−16.7307	−0.817346	−0.408673	+	0.912681i	$0.634008\pi$
$420$	0	0
$421$	−24.0746	−1.17333	−0.586663	−	0.809831i	$-0.699559\pi$
$421$	−24.0746	−1.17333	−0.586663	+	0.809831i	$0.699559\pi$
$422$	24.0447	1.17048
$423$	−39.4073	−1.91605
$424$	0.229365	0.0111389
$425$	0	0
$426$	9.62377	0.466273
$427$	3.27060	0.158275
$428$	−21.6062	−1.04437
$429$	−0.0825009	−0.00398318
$430$	0	0
$431$	22.8880	1.10248	0.551239	−	0.834348i	$-0.314155\pi$
$431$	22.8880	1.10248	0.551239	+	0.834348i	$0.314155\pi$
$432$	8.85499	0.426036
$433$	−27.4679	−1.32002	−0.660011	−	0.751256i	$-0.729448\pi$
$433$	−27.4679	−1.32002	−0.660011	+	0.751256i	$0.729448\pi$
$434$	20.2897	0.973936
$435$	0	0
$436$	−8.99309	−0.430691
$437$	11.3943	0.545062
$438$	3.36140	0.160614
$439$	−0.960670	−0.0458503	−0.0229251	−	0.999737i	$-0.507298\pi$
$439$	−0.960670	−0.0458503	−0.0229251	+	0.999737i	$0.507298\pi$
$440$	0	0
$441$	16.3438	0.778278
$442$	−9.08457	−0.432109
$443$	−22.0797	−1.04904	−0.524519	−	0.851399i	$-0.675755\pi$
$443$	−22.0797	−1.04904	−0.524519	+	0.851399i	$0.675755\pi$
$444$	0.429913	0.0204028
$445$	0	0
$446$	−40.2330	−1.90509
$447$	−2.13640	−0.101048
$448$	−7.63765	−0.360845
$449$	20.6202	0.973128	0.486564	−	0.873645i	$-0.338250\pi$
$449$	20.6202	0.973128	0.486564	+	0.873645i	$0.338250\pi$
$450$	0	0
$451$	2.39533	0.112791
$452$	13.3397	0.627446
$453$	−0.861907	−0.0404959
$454$	−30.6161	−1.43689
$455$	0	0
$456$	0.363848	0.0170388
$457$	17.3096	0.809711	0.404855	−	0.914381i	$-0.367322\pi$
$457$	17.3096	0.809711	0.404855	+	0.914381i	$0.367322\pi$
$458$	15.0546	0.703457
$459$	16.0456	0.748943
$460$	0	0
$461$	9.15093	0.426201	0.213101	−	0.977030i	$-0.431644\pi$
$461$	9.15093	0.426201	0.213101	+	0.977030i	$0.431644\pi$
$462$	−0.311827	−0.0145075
$463$	−14.2244	−0.661063	−0.330532	−	0.943795i	$-0.607228\pi$
$463$	−14.2244	−0.661063	−0.330532	+	0.943795i	$0.607228\pi$
$464$	−26.8174	−1.24497
$465$	0	0
$466$	55.6562	2.57822
$467$	−16.2381	−0.751411	−0.375705	−	0.926739i	$-0.622600\pi$
$467$	−16.2381	−0.751411	−0.375705	+	0.926739i	$0.622600\pi$
$468$	3.14688	0.145465
$469$	1.35419	0.0625307
$470$	0	0
$471$	−1.02936	−0.0474305
$472$	−3.45670	−0.159108
$473$	3.66235	0.168395
$474$	−9.00194	−0.413473
$475$	0	0
$476$	−16.4401	−0.753529
$477$	−2.06951	−0.0947562
$478$	5.78367	0.264539
$479$	32.4129	1.48098	0.740492	−	0.672066i	$-0.234593\pi$
$479$	32.4129	1.48098	0.740492	+	0.672066i	$0.234593\pi$
$480$	0	0
$481$	−0.397495	−0.0181242
$482$	−1.95888	−0.0892246
$483$	1.40891	0.0641077
$484$	−19.9221	−0.905549
$485$	0	0
$486$	−17.5345	−0.795381
$487$	20.8714	0.945772	0.472886	−	0.881123i	$-0.343212\pi$
$487$	20.8714	0.945772	0.472886	+	0.881123i	$0.343212\pi$
$488$	0.907929	0.0411000
$489$	8.25276	0.373203
$490$	0	0
$491$	−1.44239	−0.0650941	−0.0325471	−	0.999470i	$-0.510362\pi$
$491$	−1.44239	−0.0650941	−0.0325471	+	0.999470i	$0.510362\pi$
$492$	3.90040	0.175844
$493$	−48.5941	−2.18857
$494$	3.79675	0.170824
$495$	0	0
$496$	38.7650	1.74060
$497$	−16.1026	−0.722299
$498$	−4.02541	−0.180383
$499$	12.2754	0.549524	0.274762	−	0.961512i	$-0.411401\pi$
$499$	12.2754	0.549524	0.274762	+	0.961512i	$0.411401\pi$
$500$	0	0
$501$	−5.37537	−0.240154
$502$	−23.7376	−1.05946
$503$	−22.8011	−1.01665	−0.508325	−	0.861165i	$-0.669735\pi$
$503$	−22.8011	−1.01665	−0.508325	+	0.861165i	$0.669735\pi$
$504$	−1.05389	−0.0469439
$505$	0	0
$506$	−2.71084	−0.120512
$507$	−4.43183	−0.196825
$508$	23.1260	1.02605
$509$	−34.5559	−1.53166	−0.765831	−	0.643041i	$-0.777672\pi$
$509$	−34.5559	−1.53166	−0.765831	+	0.643041i	$0.777672\pi$
$510$	0	0
$511$	−5.62431	−0.248805
$512$	−30.7206	−1.35767
$513$	−6.70598	−0.296076
$514$	54.8592	2.41974
$515$	0	0
$516$	5.96354	0.262530
$517$	5.41589	0.238190
$518$	−1.50240	−0.0660118
$519$	−2.67390	−0.117371
$520$	0	0
$521$	−16.4128	−0.719056	−0.359528	−	0.933134i	$-0.617062\pi$
$521$	−16.4128	−0.719056	−0.359528	+	0.933134i	$0.617062\pi$
$522$	35.1573	1.53879
$523$	22.5542	0.986228	0.493114	−	0.869965i	$-0.335859\pi$
$523$	22.5542	0.986228	0.493114	+	0.869965i	$0.335859\pi$
$524$	−26.3081	−1.14927
$525$	0	0
$526$	19.4655	0.848737
$527$	70.2437	3.05986
$528$	−0.595770	−0.0259276
$529$	−10.7517	−0.467467
$530$	0	0
$531$	31.1890	1.35349
$532$	6.87086	0.297889
$533$	−3.60628	−0.156205
$534$	7.79462	0.337306
$535$	0	0
$536$	0.375927	0.0162376
$537$	4.71253	0.203361
$538$	38.9562	1.67952
$539$	−2.24619	−0.0967504
$540$	0	0
$541$	−23.0909	−0.992757	−0.496379	−	0.868106i	$-0.665337\pi$
$541$	−23.0909	−0.992757	−0.496379	+	0.868106i	$0.665337\pi$
$542$	−14.8889	−0.639533
$543$	8.51808	0.365546
$544$	−60.6350	−2.59970
$545$	0	0
$546$	0.469472	0.0200915
$547$	−14.5566	−0.622395	−0.311198	−	0.950345i	$-0.600730\pi$
$547$	−14.5566	−0.622395	−0.311198	+	0.950345i	$0.600730\pi$
$548$	−9.81254	−0.419171
$549$	−8.19203	−0.349627
$550$	0	0
$551$	20.3091	0.865197
$552$	0.391118	0.0166471
$553$	15.0621	0.640506
$554$	5.04063	0.214156
$555$	0	0
$556$	−24.1402	−1.02377
$557$	8.58684	0.363836	0.181918	−	0.983314i	$-0.441769\pi$
$557$	8.58684	0.363836	0.181918	+	0.983314i	$0.441769\pi$
$558$	−50.8206	−2.15141
$559$	−5.51385	−0.233211
$560$	0	0
$561$	−1.07956	−0.0455790
$562$	8.54818	0.360583
$563$	−6.54355	−0.275778	−0.137889	−	0.990448i	$-0.544032\pi$
$563$	−6.54355	−0.275778	−0.137889	+	0.990448i	$0.544032\pi$
$564$	8.81890	0.371343
$565$	0	0
$566$	19.5433	0.821466
$567$	9.08573	0.381565
$568$	−4.47012	−0.187562
$569$	−45.8495	−1.92211	−0.961056	−	0.276355i	$-0.910874\pi$
$569$	−45.8495	−1.92211	−0.961056	+	0.276355i	$0.910874\pi$
$570$	0	0
$571$	−8.61219	−0.360409	−0.180205	−	0.983629i	$-0.557676\pi$
$571$	−8.61219	−0.360409	−0.180205	+	0.983629i	$0.557676\pi$
$572$	−0.432488	−0.0180832
$573$	5.79311	0.242011
$574$	−13.6306	−0.568930
$575$	0	0
$576$	19.1304	0.797100
$577$	−8.25954	−0.343849	−0.171925	−	0.985110i	$-0.554999\pi$
$577$	−8.25954	−0.343849	−0.171925	+	0.985110i	$0.554999\pi$
$578$	−85.5743	−3.55942
$579$	−4.90911	−0.204015
$580$	0	0
$581$	6.73535	0.279429
$582$	−0.120408	−0.00499107
$583$	0.284420	0.0117795
$584$	−1.56133	−0.0646081
$585$	0	0
$586$	−15.9378	−0.658386
$587$	−9.63478	−0.397670	−0.198835	−	0.980033i	$-0.563716\pi$
$587$	−9.63478	−0.397670	−0.198835	+	0.980033i	$0.563716\pi$
$588$	−3.65756	−0.150835
$589$	−29.3572	−1.20964
$590$	0	0
$591$	−9.05527	−0.372484
$592$	−2.87046	−0.117975
$593$	−22.0788	−0.906668	−0.453334	−	0.891341i	$-0.649766\pi$
$593$	−22.0788	−0.906668	−0.453334	+	0.891341i	$0.649766\pi$
$594$	1.59544	0.0654617
$595$	0	0
$596$	−11.1995	−0.458747
$597$	−1.77470	−0.0726337
$598$	4.08131	0.166897
$599$	−26.5132	−1.08330	−0.541650	−	0.840604i	$-0.682200\pi$
$599$	−26.5132	−1.08330	−0.541650	+	0.840604i	$0.682200\pi$
$600$	0	0
$601$	31.0912	1.26823	0.634117	−	0.773237i	$-0.281364\pi$
$601$	31.0912	1.26823	0.634117	+	0.773237i	$0.281364\pi$
$602$	−20.8406	−0.849399
$603$	−3.39190	−0.138129
$604$	−4.51831	−0.183847
$605$	0	0
$606$	7.72574	0.313837
$607$	−5.83591	−0.236872	−0.118436	−	0.992962i	$-0.537788\pi$
$607$	−5.83591	−0.236872	−0.118436	+	0.992962i	$0.537788\pi$
$608$	25.3414	1.02773
$609$	2.51124	0.101761
$610$	0	0
$611$	−8.15389	−0.329871
$612$	41.1782	1.66453
$613$	7.15782	0.289101	0.144551	−	0.989497i	$-0.453826\pi$
$613$	7.15782	0.289101	0.144551	+	0.989497i	$0.453826\pi$
$614$	33.2111	1.34029
$615$	0	0
$616$	0.144840	0.00583576
$617$	7.92936	0.319224	0.159612	−	0.987180i	$-0.448976\pi$
$617$	7.92936	0.319224	0.159612	+	0.987180i	$0.448976\pi$
$618$	−6.42417	−0.258418
$619$	−27.1369	−1.09072	−0.545362	−	0.838200i	$-0.683608\pi$
$619$	−27.1369	−1.09072	−0.545362	+	0.838200i	$0.683608\pi$
$620$	0	0
$621$	−7.20859	−0.289271
$622$	−34.4904	−1.38294
$623$	−13.0420	−0.522517
$624$	0.896962	0.0359072
$625$	0	0
$626$	−1.52003	−0.0607527
$627$	0.451184	0.0180185
$628$	−5.39614	−0.215329
$629$	−5.20138	−0.207393
$630$	0	0
$631$	21.9630	0.874333	0.437167	−	0.899380i	$-0.355982\pi$
$631$	21.9630	0.874333	0.437167	+	0.899380i	$0.355982\pi$
$632$	4.18129	0.166323
$633$	−4.30185	−0.170983
$634$	39.5850	1.57212
$635$	0	0
$636$	0.463132	0.0183644
$637$	3.38176	0.133990
$638$	−4.83180	−0.191293
$639$	40.3329	1.59554
$640$	0	0
$641$	8.90572	0.351755	0.175877	−	0.984412i	$-0.443724\pi$
$641$	8.90572	0.351755	0.175877	+	0.984412i	$0.443724\pi$
$642$	8.07367	0.318642
$643$	−33.1787	−1.30844	−0.654221	−	0.756303i	$-0.727003\pi$
$643$	−33.1787	−1.30844	−0.654221	+	0.756303i	$0.727003\pi$
$644$	7.38582	0.291042
$645$	0	0
$646$	49.6820	1.95471
$647$	45.8517	1.80262	0.901309	−	0.433176i	$-0.142607\pi$
$647$	45.8517	1.80262	0.901309	+	0.433176i	$0.142607\pi$
$648$	2.52223	0.0990824
$649$	−4.28643	−0.168257
$650$	0	0
$651$	−3.63004	−0.142273
$652$	43.2628	1.69430
$653$	−12.5825	−0.492390	−0.246195	−	0.969220i	$-0.579180\pi$
$653$	−12.5825	−0.492390	−0.246195	+	0.969220i	$0.579180\pi$
$654$	3.36048	0.131405
$655$	0	0
$656$	−26.0423	−1.01678
$657$	14.0875	0.549605
$658$	−30.8191	−1.20145
$659$	47.3761	1.84551	0.922756	−	0.385385i	$-0.125931\pi$
$659$	47.3761	1.84551	0.922756	+	0.385385i	$0.125931\pi$
$660$	0	0
$661$	14.2092	0.552675	0.276338	−	0.961061i	$-0.410879\pi$
$661$	14.2092	0.552675	0.276338	+	0.961061i	$0.410879\pi$
$662$	−6.81266	−0.264781
$663$	1.62533	0.0631225
$664$	1.86975	0.0725605
$665$	0	0
$666$	3.76314	0.145819
$667$	21.8313	0.845310
$668$	−28.1789	−1.09027
$669$	7.19812	0.278295
$670$	0	0
$671$	1.12586	0.0434634
$672$	3.13349	0.120877
$673$	−4.54188	−0.175077	−0.0875384	−	0.996161i	$-0.527900\pi$
$673$	−4.54188	−0.175077	−0.0875384	+	0.996161i	$0.527900\pi$
$674$	56.4473	2.17427
$675$	0	0
$676$	−23.2326	−0.893563
$677$	−33.7570	−1.29739	−0.648694	−	0.761050i	$-0.724684\pi$
$677$	−33.7570	−1.29739	−0.648694	+	0.761050i	$0.724684\pi$
$678$	−4.98469	−0.191436
$679$	0.201467	0.00773161
$680$	0	0
$681$	5.47755	0.209900
$682$	6.98446	0.267449
$683$	−17.9641	−0.687376	−0.343688	−	0.939084i	$-0.611676\pi$
$683$	−17.9641	−0.687376	−0.343688	+	0.939084i	$0.611676\pi$
$684$	−17.2098	−0.658032
$685$	0	0
$686$	28.5330	1.08939
$687$	−2.69344	−0.102761
$688$	−39.8176	−1.51803
$689$	−0.428209	−0.0163135
$690$	0	0
$691$	−23.3108	−0.886785	−0.443392	−	0.896328i	$-0.646225\pi$
$691$	−23.3108	−0.886785	−0.443392	+	0.896328i	$0.646225\pi$
$692$	−14.0172	−0.532853
$693$	−1.30686	−0.0496433
$694$	5.52298	0.209650
$695$	0	0
$696$	0.697128	0.0264246
$697$	−47.1897	−1.78744
$698$	5.86119	0.221849
$699$	−9.95749	−0.376627
$700$	0	0
$701$	−31.4253	−1.18692	−0.593458	−	0.804865i	$-0.702238\pi$
$701$	−31.4253	−1.18692	−0.593458	+	0.804865i	$0.702238\pi$
$702$	−2.40202	−0.0906583
$703$	2.17383	0.0819876
$704$	−2.62916	−0.0990902
$705$	0	0
$706$	−41.4604	−1.56038
$707$	−12.9268	−0.486161
$708$	−6.97975	−0.262315
$709$	−33.0144	−1.23988	−0.619941	−	0.784648i	$-0.712844\pi$
$709$	−33.0144	−1.23988	−0.619941	+	0.784648i	$0.712844\pi$
$710$	0	0
$711$	−37.7268	−1.41486
$712$	−3.62050	−0.135684
$713$	−31.5575	−1.18184
$714$	6.14322	0.229904
$715$	0	0
$716$	24.7041	0.923236
$717$	−1.03476	−0.0386438
$718$	41.3839	1.54443
$719$	29.1086	1.08557	0.542785	−	0.839872i	$-0.317370\pi$
$719$	29.1086	1.08557	0.542785	+	0.839872i	$0.317370\pi$
$720$	0	0
$721$	10.7490	0.400312
$722$	16.4550	0.612390
$723$	0.350465	0.0130339
$724$	44.6536	1.65954
$725$	0	0
$726$	7.44437	0.276286
$727$	38.7905	1.43866	0.719330	−	0.694668i	$-0.244449\pi$
$727$	38.7905	1.43866	0.719330	+	0.694668i	$0.244449\pi$
$728$	−0.218064	−0.00808198
$729$	−20.5919	−0.762661
$730$	0	0
$731$	−72.1509	−2.66860
$732$	1.83328	0.0677601
$733$	41.6461	1.53823	0.769117	−	0.639108i	$-0.220696\pi$
$733$	41.6461	1.53823	0.769117	+	0.639108i	$0.220696\pi$
$734$	15.4477	0.570183
$735$	0	0
$736$	27.2407	1.00411
$737$	0.466162	0.0171713
$738$	34.1412	1.25676
$739$	−27.7225	−1.01979	−0.509894	−	0.860237i	$-0.670315\pi$
$739$	−27.7225	−1.01979	−0.509894	+	0.860237i	$0.670315\pi$
$740$	0	0
$741$	−0.679280	−0.0249540
$742$	−1.61849	−0.0594167
$743$	0.633484	0.0232403	0.0116201	−	0.999932i	$-0.496301\pi$
$743$	0.633484	0.0232403	0.0116201	+	0.999932i	$0.496301\pi$
$744$	−1.00771	−0.0369445
$745$	0	0
$746$	−72.8978	−2.66898
$747$	−16.8704	−0.617254
$748$	−5.65928	−0.206924
$749$	−13.5089	−0.493605
$750$	0	0
$751$	−42.8427	−1.56335	−0.781677	−	0.623684i	$-0.785635\pi$
$751$	−42.8427	−1.56335	−0.781677	+	0.623684i	$0.785635\pi$
$752$	−58.8823	−2.14722
$753$	4.24692	0.154766
$754$	7.27452	0.264923
$755$	0	0
$756$	−4.34685	−0.158093
$757$	−52.4715	−1.90711	−0.953554	−	0.301221i	$-0.902606\pi$
$757$	−52.4715	−1.90711	−0.953554	+	0.301221i	$0.902606\pi$
$758$	5.57612	0.202534
$759$	0.484999	0.0176044
$760$	0	0
$761$	18.1523	0.658022	0.329011	−	0.944326i	$-0.393285\pi$
$761$	18.1523	0.658022	0.329011	+	0.944326i	$0.393285\pi$
$762$	−8.64160	−0.313052
$763$	−5.62278	−0.203558
$764$	30.3687	1.09870
$765$	0	0
$766$	−49.4676	−1.78734
$767$	6.45343	0.233020
$768$	6.40603	0.231158
$769$	−1.59167	−0.0573971	−0.0286986	−	0.999588i	$-0.509136\pi$
$769$	−1.59167	−0.0573971	−0.0286986	+	0.999588i	$0.509136\pi$
$770$	0	0
$771$	−9.81491	−0.353475
$772$	−25.7346	−0.926209
$773$	−8.08891	−0.290938	−0.145469	−	0.989363i	$-0.546469\pi$
$773$	−8.08891	−0.290938	−0.145469	+	0.989363i	$0.546469\pi$
$774$	52.2004	1.87631
$775$	0	0
$776$	0.0559280	0.00200770
$777$	0.268796	0.00964302
$778$	47.8547	1.71567
$779$	19.7222	0.706619
$780$	0	0
$781$	−5.54310	−0.198348
$782$	53.4056	1.90978
$783$	−12.8486	−0.459171
$784$	24.4210	0.872177
$785$	0	0
$786$	9.83064	0.350647
$787$	13.2956	0.473936	0.236968	−	0.971518i	$-0.423846\pi$
$787$	13.2956	0.473936	0.236968	+	0.971518i	$0.423846\pi$
$788$	−47.4697	−1.69104
$789$	−3.48259	−0.123984
$790$	0	0
$791$	8.34042	0.296551
$792$	−0.362787	−0.0128911
$793$	−1.69504	−0.0601927
$794$	−74.9652	−2.66041
$795$	0	0
$796$	−9.30337	−0.329749
$797$	15.6546	0.554515	0.277258	−	0.960796i	$-0.410574\pi$
$797$	15.6546	0.554515	0.277258	+	0.960796i	$0.410574\pi$
$798$	−2.56746	−0.0908871
$799$	−106.697	−3.77467
$800$	0	0
$801$	32.6669	1.15423
$802$	33.2585	1.17440
$803$	−1.93609	−0.0683233
$804$	0.759070	0.0267703
$805$	0	0
$806$	−10.5155	−0.370391
$807$	−6.96969	−0.245345
$808$	−3.58851	−0.126243
$809$	−52.3375	−1.84009	−0.920045	−	0.391813i	$-0.871848\pi$
$809$	−52.3375	−1.84009	−0.920045	+	0.391813i	$0.871848\pi$
$810$	0	0
$811$	32.0875	1.12675	0.563373	−	0.826203i	$-0.309504\pi$
$811$	32.0875	1.12675	0.563373	+	0.826203i	$0.309504\pi$
$812$	13.1645	0.461982
$813$	2.66378	0.0934230
$814$	−0.517183	−0.0181273
$815$	0	0
$816$	11.7371	0.410881
$817$	30.1543	1.05497
$818$	44.5013	1.55595
$819$	1.96754	0.0687513
$820$	0	0
$821$	−37.5310	−1.30984	−0.654920	−	0.755698i	$-0.727298\pi$
$821$	−37.5310	−1.30984	−0.654920	+	0.755698i	$0.727298\pi$
$822$	3.66669	0.127891
$823$	−36.0696	−1.25731	−0.628654	−	0.777685i	$-0.716394\pi$
$823$	−36.0696	−1.25731	−0.628654	+	0.777685i	$0.716394\pi$
$824$	2.98395	0.103951
$825$	0	0
$826$	24.3919	0.848703
$827$	−40.8256	−1.41964	−0.709822	−	0.704381i	$-0.751225\pi$
$827$	−40.8256	−1.41964	−0.709822	+	0.704381i	$0.751225\pi$
$828$	−18.4996	−0.642906
$829$	9.33570	0.324242	0.162121	−	0.986771i	$-0.448166\pi$
$829$	9.33570	0.324242	0.162121	+	0.986771i	$0.448166\pi$
$830$	0	0
$831$	−0.901823	−0.0312839
$832$	3.95834	0.137231
$833$	44.2517	1.53323
$834$	9.02055	0.312356
$835$	0	0
$836$	2.36520	0.0818022
$837$	18.5728	0.641971
$838$	32.7733	1.13214
$839$	−38.9297	−1.34400	−0.672002	−	0.740549i	$-0.734565\pi$
$839$	−38.9297	−1.34400	−0.672002	+	0.740549i	$0.734565\pi$
$840$	0	0
$841$	9.91200	0.341793
$842$	47.1593	1.62522
$843$	−1.52936	−0.0526740
$844$	−22.5513	−0.776246
$845$	0	0
$846$	77.1941	2.65399
$847$	−12.4560	−0.427992
$848$	−3.09226	−0.106189
$849$	−3.49651	−0.120000
$850$	0	0
$851$	2.33676	0.0801030
$852$	−9.02604	−0.309227
$853$	−18.9683	−0.649462	−0.324731	−	0.945806i	$-0.605274\pi$
$853$	−18.9683	−0.649462	−0.324731	+	0.945806i	$0.605274\pi$
$854$	−6.40671	−0.219233
$855$	0	0
$856$	−3.75012	−0.128176
$857$	−9.99880	−0.341552	−0.170776	−	0.985310i	$-0.554628\pi$
$857$	−9.99880	−0.341552	−0.170776	+	0.985310i	$0.554628\pi$
$858$	0.161609	0.00551726
$859$	38.2098	1.30370	0.651852	−	0.758347i	$-0.273993\pi$
$859$	38.2098	1.30370	0.651852	+	0.758347i	$0.273993\pi$
$860$	0	0
$861$	2.43866	0.0831094
$862$	−44.8349	−1.52708
$863$	−30.2182	−1.02864	−0.514320	−	0.857598i	$-0.671956\pi$
$863$	−30.2182	−1.02864	−0.514320	+	0.857598i	$0.671956\pi$
$864$	−16.0322	−0.545428
$865$	0	0
$866$	53.8063	1.82841
$867$	15.3102	0.519961
$868$	−19.0295	−0.645903
$869$	5.18493	0.175887
$870$	0	0
$871$	−0.701831	−0.0237806
$872$	−1.56090	−0.0528588
$873$	−0.504625	−0.0170790
$874$	−22.3200	−0.754985
$875$	0	0
$876$	−3.15262	−0.106517
$877$	−51.3849	−1.73515	−0.867573	−	0.497310i	$-0.834321\pi$
$877$	−51.3849	−1.73515	−0.867573	+	0.497310i	$0.834321\pi$
$878$	1.88184	0.0635090
$879$	2.85145	0.0961771
$880$	0	0
$881$	−37.5670	−1.26566	−0.632832	−	0.774289i	$-0.718107\pi$
$881$	−37.5670	−1.26566	−0.632832	+	0.774289i	$0.718107\pi$
$882$	−32.0156	−1.07802
$883$	−33.6934	−1.13387	−0.566937	−	0.823761i	$-0.691872\pi$
$883$	−33.6934	−1.13387	−0.566937	+	0.823761i	$0.691872\pi$
$884$	8.52033	0.286570
$885$	0	0
$886$	43.2515	1.45306
$887$	3.08274	0.103508	0.0517541	−	0.998660i	$-0.483519\pi$
$887$	3.08274	0.103508	0.0517541	+	0.998660i	$0.483519\pi$
$888$	0.0746187	0.00250404
$889$	14.4592	0.484945
$890$	0	0
$891$	3.12764	0.104780
$892$	37.7341	1.26343
$893$	44.5922	1.49222
$894$	4.18494	0.139965
$895$	0	0
$896$	−2.92064	−0.0975716
$897$	−0.730191	−0.0243804
$898$	−40.3926	−1.34792
$899$	−56.2480	−1.87598
$900$	0	0
$901$	−5.60328	−0.186672
$902$	−4.69216	−0.156232
$903$	3.72861	0.124080
$904$	2.31533	0.0770066
$905$	0	0
$906$	1.68837	0.0560925
$907$	−15.3474	−0.509603	−0.254801	−	0.966993i	$-0.582010\pi$
$907$	−15.3474	−0.509603	−0.254801	+	0.966993i	$0.582010\pi$
$908$	28.7145	0.952925
$909$	32.3783	1.07392
$910$	0	0
$911$	1.34862	0.0446818	0.0223409	−	0.999750i	$-0.492888\pi$
$911$	1.34862	0.0446818	0.0223409	+	0.999750i	$0.492888\pi$
$912$	−4.90534	−0.162432
$913$	2.31856	0.0767330
$914$	−33.9075	−1.12156
$915$	0	0
$916$	−14.1196	−0.466525
$917$	−16.4487	−0.543184
$918$	−31.4313	−1.03739
$919$	34.5210	1.13874	0.569371	−	0.822081i	$-0.307187\pi$
$919$	34.5210	1.13874	0.569371	+	0.822081i	$0.307187\pi$
$920$	0	0
$921$	−5.94183	−0.195790
$922$	−17.9256	−0.590348
$923$	8.34541	0.274693
$924$	0.292459	0.00962121
$925$	0	0
$926$	27.8639	0.915663
$927$	−26.9234	−0.884282
$928$	48.5538	1.59386
$929$	25.8554	0.848287	0.424144	−	0.905595i	$-0.360575\pi$
$929$	25.8554	0.848287	0.424144	+	0.905595i	$0.360575\pi$
$930$	0	0
$931$	−18.4943	−0.606125
$932$	−52.1993	−1.70985
$933$	6.17070	0.202020
$934$	31.8086	1.04081
$935$	0	0
$936$	0.546195	0.0178529
$937$	7.11017	0.232279	0.116140	−	0.993233i	$-0.462948\pi$
$937$	7.11017	0.232279	0.116140	+	0.993233i	$0.462948\pi$
$938$	−2.65270	−0.0866136
$939$	0.271950	0.00887476
$940$	0	0
$941$	−16.9120	−0.551317	−0.275658	−	0.961256i	$-0.588896\pi$
$941$	−16.9120	−0.551317	−0.275658	+	0.961256i	$0.588896\pi$
$942$	2.01640	0.0656978
$943$	21.2003	0.690377
$944$	46.6027	1.51679
$945$	0	0
$946$	−7.17410	−0.233250
$947$	−44.2056	−1.43649	−0.718244	−	0.695791i	$-0.755054\pi$
$947$	−44.2056	−1.43649	−0.718244	+	0.695791i	$0.755054\pi$
$948$	8.44282	0.274210
$949$	2.91489	0.0946213
$950$	0	0
$951$	−7.08218	−0.229655
$952$	−2.85345	−0.0924808
$953$	−39.1355	−1.26772	−0.633862	−	0.773446i	$-0.718531\pi$
$953$	−39.1355	−1.26772	−0.633862	+	0.773446i	$0.718531\pi$
$954$	4.05392	0.131250
$955$	0	0
$956$	−5.42444	−0.175439
$957$	0.864462	0.0279441
$958$	−63.4930	−2.05137
$959$	−6.13513	−0.198114
$960$	0	0
$961$	50.3075	1.62282
$962$	0.778645	0.0251045
$963$	33.8364	1.09036
$964$	1.83721	0.0591727
$965$	0	0
$966$	−2.75989	−0.0887980
$967$	38.7564	1.24632	0.623161	−	0.782094i	$-0.285848\pi$
$967$	38.7564	1.24632	0.623161	+	0.782094i	$0.285848\pi$
$968$	−3.45781	−0.111138
$969$	−8.88865	−0.285544
$970$	0	0
$971$	−35.9268	−1.15295	−0.576473	−	0.817116i	$-0.695572\pi$
$971$	−35.9268	−1.15295	−0.576473	+	0.817116i	$0.695572\pi$
$972$	16.4454	0.527487
$973$	−15.0932	−0.483867
$974$	−40.8845	−1.31003
$975$	0	0
$976$	−12.2405	−0.391810
$977$	1.39554	0.0446474	0.0223237	−	0.999751i	$-0.492894\pi$
$977$	1.39554	0.0446474	0.0223237	+	0.999751i	$0.492894\pi$
$978$	−16.1662	−0.516937
$979$	−4.48954	−0.143486
$980$	0	0
$981$	14.0837	0.449656
$982$	2.82547	0.0901643
$983$	3.40982	0.108756	0.0543782	−	0.998520i	$-0.482682\pi$
$983$	3.40982	0.108756	0.0543782	+	0.998520i	$0.482682\pi$
$984$	0.676980	0.0215813
$985$	0	0
$986$	95.1900	3.03147
$987$	5.51387	0.175508
$988$	−3.56093	−0.113288
$989$	32.4143	1.03072
$990$	0	0
$991$	−21.1169	−0.670799	−0.335400	−	0.942076i	$-0.608871\pi$
$991$	−21.1169	−0.670799	−0.335400	+	0.942076i	$0.608871\pi$
$992$	−70.1854	−2.22839
$993$	1.21886	0.0386793
$994$	31.5430	1.00048
$995$	0	0
$996$	3.77539	0.119628
$997$	12.2441	0.387775	0.193888	−	0.981024i	$-0.437890\pi$
$997$	12.2441	0.387775	0.193888	+	0.981024i	$0.437890\pi$
$998$	−24.0461	−0.761166
$999$	−1.37528	−0.0435118

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.p.1.6		46
5.2	odd	4		1205.2.b.c.724.6	✓	46
5.3	odd	4		1205.2.b.c.724.41	yes	46
5.4	even	2	inner	6025.2.a.p.1.41		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.c.724.6	✓	46	5.2	odd	4
1205.2.b.c.724.41	yes	46	5.3	odd	4
6025.2.a.p.1.6		46	1.1	even	1	trivial
6025.2.a.p.1.41		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)

\(f(q)\)	\(=\)	\(q-1.95888 q^{2} +0.350465 q^{3} +1.83721 q^{4} -0.686519 q^{6} +1.14869 q^{7} +0.318879 q^{8} -2.87717 q^{9} +O(q^{10})\)	\(q-1.95888 q^{2} +0.350465 q^{3} +1.83721 q^{4} -0.686519 q^{6} +1.14869 q^{7} +0.318879 q^{8} -2.87717 q^{9} +0.395421 q^{11} +0.643879 q^{12} -0.595326 q^{13} -2.25014 q^{14} -4.29907 q^{16} -7.79008 q^{17} +5.63604 q^{18} +3.25574 q^{19} +0.402575 q^{21} -0.774582 q^{22} +3.49975 q^{23} +0.111756 q^{24} +1.16617 q^{26} -2.05974 q^{27} +2.11038 q^{28} +6.23795 q^{29} -9.01707 q^{31} +7.78361 q^{32} +0.138581 q^{33} +15.2598 q^{34} -5.28598 q^{36} +0.667693 q^{37} -6.37760 q^{38} -0.208641 q^{39} +6.05766 q^{41} -0.788596 q^{42} +9.26190 q^{43} +0.726472 q^{44} -6.85559 q^{46} +13.6965 q^{47} -1.50667 q^{48} -5.68052 q^{49} -2.73015 q^{51} -1.09374 q^{52} +0.719285 q^{53} +4.03479 q^{54} +0.366293 q^{56} +1.14102 q^{57} -12.2194 q^{58} -10.8402 q^{59} +2.84725 q^{61} +17.6634 q^{62} -3.30498 q^{63} -6.64902 q^{64} -0.271464 q^{66} +1.17890 q^{67} -14.3120 q^{68} +1.22654 q^{69} -14.0182 q^{71} -0.917471 q^{72} -4.89629 q^{73} -1.30793 q^{74} +5.98148 q^{76} +0.454215 q^{77} +0.408703 q^{78} +13.1124 q^{79} +7.90966 q^{81} -11.8662 q^{82} +5.86352 q^{83} +0.739616 q^{84} -18.1429 q^{86} +2.18618 q^{87} +0.126092 q^{88} -11.3538 q^{89} -0.683844 q^{91} +6.42979 q^{92} -3.16017 q^{93} -26.8298 q^{94} +2.72788 q^{96} +0.175389 q^{97} +11.1275 q^{98} -1.13769 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) \) 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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