LMFDB - Embedded newform 1175.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1175,2,Mod(1,1175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1175, 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("1175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1175 = 5^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1175.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:	$9.38242223750$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	1175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.53209 q^{2} -0.347296 q^{3} +0.347296 q^{4} +0.532089 q^{6} +0.347296 q^{7} +2.53209 q^{8} -2.87939 q^{9} +O(q^{10})$	\(q-1.53209 q^{2} -0.347296 q^{3} +0.347296 q^{4} +0.532089 q^{6} +0.347296 q^{7} +2.53209 q^{8} -2.87939 q^{9} -1.34730 q^{11} -0.120615 q^{12} +3.41147 q^{13} -0.532089 q^{14} -4.57398 q^{16} +0.879385 q^{17} +4.41147 q^{18} -3.53209 q^{19} -0.120615 q^{21} +2.06418 q^{22} +1.53209 q^{23} -0.879385 q^{24} -5.22668 q^{26} +2.04189 q^{27} +0.120615 q^{28} -0.120615 q^{29} +3.82295 q^{31} +1.94356 q^{32} +0.467911 q^{33} -1.34730 q^{34} -1.00000 q^{36} -8.17024 q^{37} +5.41147 q^{38} -1.18479 q^{39} +1.98545 q^{41} +0.184793 q^{42} +3.41147 q^{43} -0.467911 q^{44} -2.34730 q^{46} -1.00000 q^{47} +1.58853 q^{48} -6.87939 q^{49} -0.305407 q^{51} +1.18479 q^{52} +2.81521 q^{53} -3.12836 q^{54} +0.879385 q^{56} +1.22668 q^{57} +0.184793 q^{58} +2.63816 q^{59} -6.69459 q^{61} -5.85710 q^{62} -1.00000 q^{63} +6.17024 q^{64} -0.716881 q^{66} -6.33275 q^{67} +0.305407 q^{68} -0.532089 q^{69} -9.66044 q^{71} -7.29086 q^{72} +2.04189 q^{73} +12.5175 q^{74} -1.22668 q^{76} -0.467911 q^{77} +1.81521 q^{78} -9.94356 q^{79} +7.92902 q^{81} -3.04189 q^{82} -4.87939 q^{83} -0.0418891 q^{84} -5.22668 q^{86} +0.0418891 q^{87} -3.41147 q^{88} -8.55943 q^{89} +1.18479 q^{91} +0.532089 q^{92} -1.32770 q^{93} +1.53209 q^{94} -0.674992 q^{96} -13.9145 q^{97} +10.5398 q^{98} +3.87939 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{6} + 3 q^{8} - 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^6 + 3 * q^8 - 3 * q^9	$ 3 q - 3 q^{6} + 3 q^{8} - 3 q^{9} - 3 q^{11} - 6 q^{12} + 3 q^{14} - 6 q^{16} - 3 q^{17} + 3 q^{18} - 6 q^{19} - 6 q^{21} - 3 q^{22} + 3 q^{24} - 9 q^{26} + 3 q^{27} + 6 q^{28} - 6 q^{29} - 9 q^{31} - 9 q^{32} + 6 q^{33} - 3 q^{34} - 3 q^{36} - 3 q^{37} + 6 q^{38} - 12 q^{41} - 3 q^{42} - 6 q^{44} - 6 q^{46} - 3 q^{47} + 15 q^{48} - 15 q^{49} - 3 q^{51} + 12 q^{53} + 9 q^{54} - 3 q^{56} - 3 q^{57} - 3 q^{58} - 9 q^{59} - 18 q^{61} - 18 q^{62} - 3 q^{63} - 3 q^{64} + 6 q^{66} + 3 q^{68} + 3 q^{69} - 6 q^{71} - 6 q^{72} + 3 q^{73} + 15 q^{74} + 3 q^{76} - 6 q^{77} + 9 q^{78} - 15 q^{79} - 9 q^{81} - 6 q^{82} - 9 q^{83} + 3 q^{84} - 9 q^{86} - 3 q^{87} - 3 q^{92} + 3 q^{96} + 9 q^{97} + 3 q^{98} + 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^6 + 3 * q^8 - 3 * q^9 - 3 * q^11 - 6 * q^12 + 3 * q^14 - 6 * q^16 - 3 * q^17 + 3 * q^18 - 6 * q^19 - 6 * q^21 - 3 * q^22 + 3 * q^24 - 9 * q^26 + 3 * q^27 + 6 * q^28 - 6 * q^29 - 9 * q^31 - 9 * q^32 + 6 * q^33 - 3 * q^34 - 3 * q^36 - 3 * q^37 + 6 * q^38 - 12 * q^41 - 3 * q^42 - 6 * q^44 - 6 * q^46 - 3 * q^47 + 15 * q^48 - 15 * q^49 - 3 * q^51 + 12 * q^53 + 9 * q^54 - 3 * q^56 - 3 * q^57 - 3 * q^58 - 9 * q^59 - 18 * q^61 - 18 * q^62 - 3 * q^63 - 3 * q^64 + 6 * q^66 + 3 * q^68 + 3 * q^69 - 6 * q^71 - 6 * q^72 + 3 * q^73 + 15 * q^74 + 3 * q^76 - 6 * q^77 + 9 * q^78 - 15 * q^79 - 9 * q^81 - 6 * q^82 - 9 * q^83 + 3 * q^84 - 9 * q^86 - 3 * q^87 - 3 * q^92 + 3 * q^96 + 9 * q^97 + 3 * q^98 + 6 * 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 $

$2$	−1.53209	−1.08335	−0.541675	−	0.840588i	$-0.682210\pi$
$2$	−1.53209	−1.08335	−0.541675	+	0.840588i	$0.682210\pi$
$3$	−0.347296	−0.200512	−0.100256	−	0.994962i	$-0.531966\pi$
$3$	−0.347296	−0.200512	−0.100256	+	0.994962i	$0.531966\pi$
$4$	0.347296	0.173648
$5$	0	0
$5$	0	0
$6$	0.532089	0.217224
$7$	0.347296	0.131266	0.0656328	−	0.997844i	$-0.479093\pi$
$7$	0.347296	0.131266	0.0656328	+	0.997844i	$0.479093\pi$
$8$	2.53209	0.895229
$9$	−2.87939	−0.959795
$10$	0	0
$11$	−1.34730	−0.406225	−0.203113	−	0.979155i	$-0.565106\pi$
$11$	−1.34730	−0.406225	−0.203113	+	0.979155i	$0.565106\pi$
$12$	−0.120615	−0.0348185
$13$	3.41147	0.946173	0.473086	−	0.881016i	$-0.343140\pi$
$13$	3.41147	0.946173	0.473086	+	0.881016i	$0.343140\pi$
$14$	−0.532089	−0.142207
$15$	0	0
$16$	−4.57398	−1.14349
$17$	0.879385	0.213282	0.106641	−	0.994298i	$-0.465990\pi$
$17$	0.879385	0.213282	0.106641	+	0.994298i	$0.465990\pi$
$18$	4.41147	1.03979
$19$	−3.53209	−0.810317	−0.405158	−	0.914247i	$-0.632784\pi$
$19$	−3.53209	−0.810317	−0.405158	+	0.914247i	$0.632784\pi$
$20$	0	0
$21$	−0.120615	−0.0263203
$22$	2.06418	0.440084
$23$	1.53209	0.319463	0.159731	−	0.987161i	$-0.448937\pi$
$23$	1.53209	0.319463	0.159731	+	0.987161i	$0.448937\pi$
$24$	−0.879385	−0.179504
$25$	0	0
$26$	−5.22668	−1.02504
$27$	2.04189	0.392962
$28$	0.120615	0.0227940
$29$	−0.120615	−0.0223976	−0.0111988	−	0.999937i	$-0.503565\pi$
$29$	−0.120615	−0.0223976	−0.0111988	+	0.999937i	$0.503565\pi$
$30$	0	0
$31$	3.82295	0.686622	0.343311	−	0.939222i	$-0.388452\pi$
$31$	3.82295	0.686622	0.343311	+	0.939222i	$0.388452\pi$
$32$	1.94356	0.343577
$33$	0.467911	0.0814529
$34$	−1.34730	−0.231059
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−8.17024	−1.34318	−0.671590	−	0.740923i	$-0.734388\pi$
$37$	−8.17024	−1.34318	−0.671590	+	0.740923i	$0.734388\pi$
$38$	5.41147	0.877857
$39$	−1.18479	−0.189719
$40$	0	0
$41$	1.98545	0.310076	0.155038	−	0.987909i	$-0.450450\pi$
$41$	1.98545	0.310076	0.155038	+	0.987909i	$0.450450\pi$
$42$	0.184793	0.0285141
$43$	3.41147	0.520245	0.260122	−	0.965576i	$-0.416237\pi$
$43$	3.41147	0.520245	0.260122	+	0.965576i	$0.416237\pi$
$44$	−0.467911	−0.0705403
$45$	0	0
$46$	−2.34730	−0.346090
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	1.58853	0.229284
$49$	−6.87939	−0.982769
$50$	0	0
$51$	−0.305407	−0.0427656
$52$	1.18479	0.164301
$53$	2.81521	0.386698	0.193349	−	0.981130i	$-0.438065\pi$
$53$	2.81521	0.386698	0.193349	+	0.981130i	$0.438065\pi$
$54$	−3.12836	−0.425715
$55$	0	0
$56$	0.879385	0.117513
$57$	1.22668	0.162478
$58$	0.184793	0.0242644
$59$	2.63816	0.343459	0.171729	−	0.985144i	$-0.445065\pi$
$59$	2.63816	0.343459	0.171729	+	0.985144i	$0.445065\pi$
$60$	0	0
$61$	−6.69459	−0.857155	−0.428577	−	0.903505i	$-0.640985\pi$
$61$	−6.69459	−0.857155	−0.428577	+	0.903505i	$0.640985\pi$
$62$	−5.85710	−0.743852
$63$	−1.00000	−0.125988
$64$	6.17024	0.771281
$65$	0	0
$66$	−0.716881	−0.0882420
$67$	−6.33275	−0.773668	−0.386834	−	0.922149i	$-0.626431\pi$
$67$	−6.33275	−0.773668	−0.386834	+	0.922149i	$0.626431\pi$
$68$	0.305407	0.0370361
$69$	−0.532089	−0.0640560
$70$	0	0
$71$	−9.66044	−1.14648	−0.573242	−	0.819386i	$-0.694314\pi$
$71$	−9.66044	−1.14648	−0.573242	+	0.819386i	$0.694314\pi$
$72$	−7.29086	−0.859236
$73$	2.04189	0.238985	0.119493	−	0.992835i	$-0.461873\pi$
$73$	2.04189	0.238985	0.119493	+	0.992835i	$0.461873\pi$
$74$	12.5175	1.45513
$75$	0	0
$76$	−1.22668	−0.140710
$77$	−0.467911	−0.0533234
$78$	1.81521	0.205532
$79$	−9.94356	−1.11874	−0.559369	−	0.828919i	$-0.688957\pi$
$79$	−9.94356	−1.11874	−0.559369	+	0.828919i	$0.688957\pi$
$80$	0	0
$81$	7.92902	0.881002
$82$	−3.04189	−0.335920
$83$	−4.87939	−0.535582	−0.267791	−	0.963477i	$-0.586294\pi$
$83$	−4.87939	−0.535582	−0.267791	+	0.963477i	$0.586294\pi$
$84$	−0.0418891	−0.00457047
$85$	0	0
$86$	−5.22668	−0.563608
$87$	0.0418891	0.00449098
$88$	−3.41147	−0.363664
$89$	−8.55943	−0.907298	−0.453649	−	0.891181i	$-0.649878\pi$
$89$	−8.55943	−0.907298	−0.453649	+	0.891181i	$0.649878\pi$
$90$	0	0
$91$	1.18479	0.124200
$92$	0.532089	0.0554741
$93$	−1.32770	−0.137676
$94$	1.53209	0.158023
$95$	0	0
$96$	−0.674992	−0.0688911
$97$	−13.9145	−1.41280	−0.706400	−	0.707813i	$-0.749682\pi$
$97$	−13.9145	−1.41280	−0.706400	+	0.707813i	$0.749682\pi$
$98$	10.5398	1.06468
$99$	3.87939	0.389893
$100$	0	0
$101$	−12.2567	−1.21959	−0.609794	−	0.792560i	$-0.708748\pi$
$101$	−12.2567	−1.21959	−0.609794	+	0.792560i	$0.708748\pi$
$102$	0.467911	0.0463301
$103$	−9.74422	−0.960127	−0.480063	−	0.877234i	$-0.659386\pi$
$103$	−9.74422	−0.960127	−0.480063	+	0.877234i	$0.659386\pi$
$104$	8.63816	0.847041
$105$	0	0
$106$	−4.31315	−0.418930
$107$	7.14796	0.691019	0.345509	−	0.938415i	$-0.387706\pi$
$107$	7.14796	0.691019	0.345509	+	0.938415i	$0.387706\pi$
$108$	0.709141	0.0682371
$109$	−2.21894	−0.212536	−0.106268	−	0.994338i	$-0.533890\pi$
$109$	−2.21894	−0.212536	−0.106268	+	0.994338i	$0.533890\pi$
$110$	0	0
$111$	2.83750	0.269323
$112$	−1.58853	−0.150102
$113$	−13.0273	−1.22551	−0.612755	−	0.790273i	$-0.709939\pi$
$113$	−13.0273	−1.22551	−0.612755	+	0.790273i	$0.709939\pi$
$114$	−1.87939	−0.176021
$115$	0	0
$116$	−0.0418891	−0.00388930
$117$	−9.82295	−0.908132
$118$	−4.04189	−0.372086
$119$	0.305407	0.0279966
$120$	0	0
$121$	−9.18479	−0.834981
$122$	10.2567	0.928599
$123$	−0.689540	−0.0621738
$124$	1.32770	0.119231
$125$	0	0
$126$	1.53209	0.136489
$127$	12.2686	1.08866	0.544330	−	0.838871i	$-0.316784\pi$
$127$	12.2686	1.08866	0.544330	+	0.838871i	$0.316784\pi$
$128$	−13.3405	−1.17914
$129$	−1.18479	−0.104315
$130$	0	0
$131$	3.09833	0.270702	0.135351	−	0.990798i	$-0.456784\pi$
$131$	3.09833	0.270702	0.135351	+	0.990798i	$0.456784\pi$
$132$	0.162504	0.0141441
$133$	−1.22668	−0.106367
$134$	9.70233	0.838154
$135$	0	0
$136$	2.22668	0.190936
$137$	−4.11381	−0.351466	−0.175733	−	0.984438i	$-0.556230\pi$
$137$	−4.11381	−0.351466	−0.175733	+	0.984438i	$0.556230\pi$
$138$	0.815207	0.0693951
$139$	−19.2199	−1.63021	−0.815104	−	0.579314i	$-0.803320\pi$
$139$	−19.2199	−1.63021	−0.815104	+	0.579314i	$0.803320\pi$
$140$	0	0
$141$	0.347296	0.0292476
$142$	14.8007	1.24204
$143$	−4.59627	−0.384359
$144$	13.1702	1.09752
$145$	0	0
$146$	−3.12836	−0.258905
$147$	2.38919	0.197057
$148$	−2.83750	−0.233241
$149$	6.12836	0.502054	0.251027	−	0.967980i	$-0.419232\pi$
$149$	6.12836	0.502054	0.251027	+	0.967980i	$0.419232\pi$
$150$	0	0
$151$	−6.47296	−0.526762	−0.263381	−	0.964692i	$-0.584838\pi$
$151$	−6.47296	−0.526762	−0.263381	+	0.964692i	$0.584838\pi$
$152$	−8.94356	−0.725419
$153$	−2.53209	−0.204707
$154$	0.716881	0.0577680
$155$	0	0
$156$	−0.411474	−0.0329443
$157$	3.92127	0.312952	0.156476	−	0.987682i	$-0.449987\pi$
$157$	3.92127	0.312952	0.156476	+	0.987682i	$0.449987\pi$
$158$	15.2344	1.21199
$159$	−0.977711	−0.0775375
$160$	0	0
$161$	0.532089	0.0419345
$162$	−12.1480	−0.954434
$163$	−15.2713	−1.19614	−0.598069	−	0.801445i	$-0.704065\pi$
$163$	−15.2713	−1.19614	−0.598069	+	0.801445i	$0.704065\pi$
$164$	0.689540	0.0538440
$165$	0	0
$166$	7.47565	0.580223
$167$	12.1061	0.936796	0.468398	−	0.883518i	$-0.344831\pi$
$167$	12.1061	0.936796	0.468398	+	0.883518i	$0.344831\pi$
$168$	−0.305407	−0.0235627
$169$	−1.36184	−0.104757
$170$	0	0
$171$	10.1702	0.777738
$172$	1.18479	0.0903396
$173$	−6.07192	−0.461639	−0.230820	−	0.972997i	$-0.574141\pi$
$173$	−6.07192	−0.461639	−0.230820	+	0.972997i	$0.574141\pi$
$174$	−0.0641778	−0.00486530
$175$	0	0
$176$	6.16250	0.464516
$177$	−0.916222	−0.0688675
$178$	13.1138	0.982921
$179$	16.6236	1.24251	0.621253	−	0.783610i	$-0.286624\pi$
$179$	16.6236	1.24251	0.621253	+	0.783610i	$0.286624\pi$
$180$	0	0
$181$	−7.68004	−0.570853	−0.285427	−	0.958401i	$-0.592135\pi$
$181$	−7.68004	−0.570853	−0.285427	+	0.958401i	$0.592135\pi$
$182$	−1.81521	−0.134552
$183$	2.32501	0.171870
$184$	3.87939	0.285992
$185$	0	0
$186$	2.03415	0.149151
$187$	−1.18479	−0.0866406
$188$	−0.347296	−0.0253292
$189$	0.709141	0.0515824
$190$	0	0
$191$	1.54664	0.111911	0.0559554	−	0.998433i	$-0.482180\pi$
$191$	1.54664	0.111911	0.0559554	+	0.998433i	$0.482180\pi$
$192$	−2.14290	−0.154651
$193$	−4.63041	−0.333305	−0.166652	−	0.986016i	$-0.553296\pi$
$193$	−4.63041	−0.333305	−0.166652	+	0.986016i	$0.553296\pi$
$194$	21.3182	1.53056
$195$	0	0
$196$	−2.38919	−0.170656
$197$	−0.460170	−0.0327858	−0.0163929	−	0.999866i	$-0.505218\pi$
$197$	−0.460170	−0.0327858	−0.0163929	+	0.999866i	$0.505218\pi$
$198$	−5.94356	−0.422391
$199$	−11.2148	−0.794998	−0.397499	−	0.917603i	$-0.630122\pi$
$199$	−11.2148	−0.794998	−0.397499	+	0.917603i	$0.630122\pi$
$200$	0	0
$201$	2.19934	0.155130
$202$	18.7784	1.32124
$203$	−0.0418891	−0.00294004
$204$	−0.106067	−0.00742616
$205$	0	0
$206$	14.9290	1.04015
$207$	−4.41147	−0.306619
$208$	−15.6040	−1.08194
$209$	4.75877	0.329171
$210$	0	0
$211$	15.3773	1.05862	0.529309	−	0.848429i	$-0.322451\pi$
$211$	15.3773	1.05862	0.529309	+	0.848429i	$0.322451\pi$
$212$	0.977711	0.0671495
$213$	3.35504	0.229883
$214$	−10.9513	−0.748616
$215$	0	0
$216$	5.17024	0.351791
$217$	1.32770	0.0901299
$218$	3.39961	0.230251
$219$	−0.709141	−0.0479193
$220$	0	0
$221$	3.00000	0.201802
$222$	−4.34730	−0.291771
$223$	−18.0419	−1.20817	−0.604087	−	0.796918i	$-0.706462\pi$
$223$	−18.0419	−1.20817	−0.604087	+	0.796918i	$0.706462\pi$
$224$	0.674992	0.0450998
$225$	0	0
$226$	19.9590	1.32766
$227$	12.1206	0.804473	0.402237	−	0.915536i	$-0.368233\pi$
$227$	12.1206	0.804473	0.402237	+	0.915536i	$0.368233\pi$
$228$	0.426022	0.0282140
$229$	−9.41416	−0.622105	−0.311053	−	0.950393i	$-0.600682\pi$
$229$	−9.41416	−0.622105	−0.311053	+	0.950393i	$0.600682\pi$
$230$	0	0
$231$	0.162504	0.0106920
$232$	−0.305407	−0.0200510
$233$	23.0847	1.51233	0.756165	−	0.654381i	$-0.227071\pi$
$233$	23.0847	1.51233	0.756165	+	0.654381i	$0.227071\pi$
$234$	15.0496	0.983825
$235$	0	0
$236$	0.916222	0.0596410
$237$	3.45336	0.224320
$238$	−0.467911	−0.0303302
$239$	3.57903	0.231508	0.115754	−	0.993278i	$-0.463072\pi$
$239$	3.57903	0.231508	0.115754	+	0.993278i	$0.463072\pi$
$240$	0	0
$241$	17.6955	1.13987	0.569935	−	0.821690i	$-0.306969\pi$
$241$	17.6955	1.13987	0.569935	+	0.821690i	$0.306969\pi$
$242$	14.0719	0.904577
$243$	−8.87939	−0.569613
$244$	−2.32501	−0.148843
$245$	0	0
$246$	1.05644	0.0673560
$247$	−12.0496	−0.766700
$248$	9.68004	0.614683
$249$	1.69459	0.107390
$250$	0	0
$251$	−2.74422	−0.173214	−0.0866069	−	0.996243i	$-0.527602\pi$
$251$	−2.74422	−0.173214	−0.0866069	+	0.996243i	$0.527602\pi$
$252$	−0.347296	−0.0218776
$253$	−2.06418	−0.129774
$254$	−18.7965	−1.17940
$255$	0	0
$256$	8.09833	0.506145
$257$	22.6263	1.41139	0.705695	−	0.708516i	$-0.250635\pi$
$257$	22.6263	1.41139	0.705695	+	0.708516i	$0.250635\pi$
$258$	1.81521	0.113010
$259$	−2.83750	−0.176313
$260$	0	0
$261$	0.347296	0.0214971
$262$	−4.74691	−0.293265
$263$	−11.9632	−0.737680	−0.368840	−	0.929493i	$-0.620245\pi$
$263$	−11.9632	−0.737680	−0.368840	+	0.929493i	$0.620245\pi$
$264$	1.18479	0.0729189
$265$	0	0
$266$	1.87939	0.115233
$267$	2.97266	0.181924
$268$	−2.19934	−0.134346
$269$	−2.04694	−0.124804	−0.0624021	−	0.998051i	$-0.519876\pi$
$269$	−2.04694	−0.124804	−0.0624021	+	0.998051i	$0.519876\pi$
$270$	0	0
$271$	−18.1489	−1.10247	−0.551233	−	0.834351i	$-0.685843\pi$
$271$	−18.1489	−1.10247	−0.551233	+	0.834351i	$0.685843\pi$
$272$	−4.02229	−0.243887
$273$	−0.411474	−0.0249035
$274$	6.30272	0.380761
$275$	0	0
$276$	−0.184793	−0.0111232
$277$	0.745977	0.0448214	0.0224107	−	0.999749i	$-0.492866\pi$
$277$	0.745977	0.0448214	0.0224107	+	0.999749i	$0.492866\pi$
$278$	29.4466	1.76609
$279$	−11.0077	−0.659016
$280$	0	0
$281$	9.96316	0.594352	0.297176	−	0.954823i	$-0.403955\pi$
$281$	9.96316	0.594352	0.297176	+	0.954823i	$0.403955\pi$
$282$	−0.532089	−0.0316854
$283$	8.19253	0.486996	0.243498	−	0.969901i	$-0.421705\pi$
$283$	8.19253	0.486996	0.243498	+	0.969901i	$0.421705\pi$
$284$	−3.35504	−0.199085
$285$	0	0
$286$	7.04189	0.416396
$287$	0.689540	0.0407023
$288$	−5.59627	−0.329763
$289$	−16.2267	−0.954511
$290$	0	0
$291$	4.83244	0.283283
$292$	0.709141	0.0414993
$293$	−9.62092	−0.562060	−0.281030	−	0.959699i	$-0.590676\pi$
$293$	−9.62092	−0.562060	−0.281030	+	0.959699i	$0.590676\pi$
$294$	−3.66044	−0.213481
$295$	0	0
$296$	−20.6878	−1.20245
$297$	−2.75103	−0.159631
$298$	−9.38919	−0.543901
$299$	5.22668	0.302267
$300$	0	0
$301$	1.18479	0.0682903
$302$	9.91716	0.570668
$303$	4.25671	0.244542
$304$	16.1557	0.926593
$305$	0	0
$306$	3.87939	0.221770
$307$	9.15476	0.522490	0.261245	−	0.965273i	$-0.415867\pi$
$307$	9.15476	0.522490	0.261245	+	0.965273i	$0.415867\pi$
$308$	−0.162504	−0.00925951
$309$	3.38413	0.192517
$310$	0	0
$311$	23.6236	1.33957	0.669786	−	0.742554i	$-0.266386\pi$
$311$	23.6236	1.33957	0.669786	+	0.742554i	$0.266386\pi$
$312$	−3.00000	−0.169842
$313$	16.2395	0.917909	0.458955	−	0.888460i	$-0.348224\pi$
$313$	16.2395	0.917909	0.458955	+	0.888460i	$0.348224\pi$
$314$	−6.00774	−0.339036
$315$	0	0
$316$	−3.45336	−0.194267
$317$	10.8256	0.608028	0.304014	−	0.952668i	$-0.401673\pi$
$317$	10.8256	0.608028	0.304014	+	0.952668i	$0.401673\pi$
$318$	1.49794	0.0840003
$319$	0.162504	0.00909847
$320$	0	0
$321$	−2.48246	−0.138557
$322$	−0.815207	−0.0454297
$323$	−3.10607	−0.172826
$324$	2.75372	0.152984
$325$	0	0
$326$	23.3969	1.29584
$327$	0.770630	0.0426159
$328$	5.02734	0.277588
$329$	−0.347296	−0.0191471
$330$	0	0
$331$	31.2550	1.71793	0.858964	−	0.512036i	$-0.171109\pi$
$331$	31.2550	1.71793	0.858964	+	0.512036i	$0.171109\pi$
$332$	−1.69459	−0.0930029
$333$	23.5253	1.28918
$334$	−18.5476	−1.01488
$335$	0	0
$336$	0.551689	0.0300971
$337$	29.9668	1.63239	0.816197	−	0.577773i	$-0.196078\pi$
$337$	29.9668	1.63239	0.816197	+	0.577773i	$0.196078\pi$
$338$	2.08647	0.113489
$339$	4.52435	0.245729
$340$	0	0
$341$	−5.15064	−0.278923
$342$	−15.5817	−0.842563
$343$	−4.82026	−0.260270
$344$	8.63816	0.465738
$345$	0	0
$346$	9.30272	0.500117
$347$	18.7033	1.00404	0.502022	−	0.864855i	$-0.332590\pi$
$347$	18.7033	1.00404	0.502022	+	0.864855i	$0.332590\pi$
$348$	0.0145479	0.000779850 0
$349$	−0.639910	−0.0342536	−0.0171268	−	0.999853i	$-0.505452\pi$
$349$	−0.639910	−0.0342536	−0.0171268	+	0.999853i	$0.505452\pi$
$350$	0	0
$351$	6.96585	0.371810
$352$	−2.61856	−0.139569
$353$	−0.288171	−0.0153378	−0.00766890	−	0.999971i	$-0.502441\pi$
$353$	−0.288171	−0.0153378	−0.00766890	+	0.999971i	$0.502441\pi$
$354$	1.40373	0.0746076
$355$	0	0
$356$	−2.97266	−0.157551
$357$	−0.106067	−0.00561365
$358$	−25.4688	−1.34607
$359$	19.6486	1.03701	0.518506	−	0.855074i	$-0.326488\pi$
$359$	19.6486	1.03701	0.518506	+	0.855074i	$0.326488\pi$
$360$	0	0
$361$	−6.52435	−0.343387
$362$	11.7665	0.618434
$363$	3.18984	0.167423
$364$	0.411474	0.0215671
$365$	0	0
$366$	−3.56212	−0.186195
$367$	13.8402	0.722452	0.361226	−	0.932478i	$-0.382358\pi$
$367$	13.8402	0.722452	0.361226	+	0.932478i	$0.382358\pi$
$368$	−7.00774	−0.365304
$369$	−5.71688	−0.297609
$370$	0	0
$371$	0.977711	0.0507602
$372$	−0.461104	−0.0239071
$373$	−11.5098	−0.595955	−0.297977	−	0.954573i	$-0.596312\pi$
$373$	−11.5098	−0.595955	−0.297977	+	0.954573i	$0.596312\pi$
$374$	1.81521	0.0938621
$375$	0	0
$376$	−2.53209	−0.130583
$377$	−0.411474	−0.0211920
$378$	−1.08647	−0.0558818
$379$	15.9564	0.819623	0.409811	−	0.912170i	$-0.365594\pi$
$379$	15.9564	0.819623	0.409811	+	0.912170i	$0.365594\pi$
$380$	0	0
$381$	−4.26083	−0.218289
$382$	−2.36959	−0.121239
$383$	29.5381	1.50933	0.754663	−	0.656113i	$-0.227801\pi$
$383$	29.5381	1.50933	0.754663	+	0.656113i	$0.227801\pi$
$384$	4.63310	0.236432
$385$	0	0
$386$	7.09421	0.361086
$387$	−9.82295	−0.499329
$388$	−4.83244	−0.245330
$389$	−27.4712	−1.39285	−0.696423	−	0.717632i	$-0.745226\pi$
$389$	−27.4712	−1.39285	−0.696423	+	0.717632i	$0.745226\pi$
$390$	0	0
$391$	1.34730	0.0681357
$392$	−17.4192	−0.879803
$393$	−1.07604	−0.0542789
$394$	0.705022	0.0355185
$395$	0	0
$396$	1.34730	0.0677042
$397$	−30.3628	−1.52386	−0.761932	−	0.647657i	$-0.775749\pi$
$397$	−30.3628	−1.52386	−0.761932	+	0.647657i	$0.775749\pi$
$398$	17.1821	0.861261
$399$	0.426022	0.0213278
$400$	0	0
$401$	−32.6955	−1.63274	−0.816368	−	0.577532i	$-0.804016\pi$
$401$	−32.6955	−1.63274	−0.816368	+	0.577532i	$0.804016\pi$
$402$	−3.36959	−0.168060
$403$	13.0419	0.649663
$404$	−4.25671	−0.211779
$405$	0	0
$406$	0.0641778	0.00318509
$407$	11.0077	0.545633
$408$	−0.773318	−0.0382850
$409$	−6.93313	−0.342821	−0.171411	−	0.985200i	$-0.554833\pi$
$409$	−6.93313	−0.342821	−0.171411	+	0.985200i	$0.554833\pi$
$410$	0	0
$411$	1.42871	0.0704731
$412$	−3.38413	−0.166724
$413$	0.916222	0.0450843
$414$	6.75877	0.332175
$415$	0	0
$416$	6.63041	0.325083
$417$	6.67499	0.326876
$418$	−7.29086	−0.356608
$419$	−8.09926	−0.395675	−0.197837	−	0.980235i	$-0.563392\pi$
$419$	−8.09926	−0.395675	−0.197837	+	0.980235i	$0.563392\pi$
$420$	0	0
$421$	−29.4296	−1.43431	−0.717157	−	0.696912i	$-0.754557\pi$
$421$	−29.4296	−1.43431	−0.717157	+	0.696912i	$0.754557\pi$
$422$	−23.5594	−1.14686
$423$	2.87939	0.140001
$424$	7.12836	0.346184
$425$	0	0
$426$	−5.14022	−0.249044
$427$	−2.32501	−0.112515
$428$	2.48246	0.119994
$429$	1.59627	0.0770685
$430$	0	0
$431$	8.44150	0.406613	0.203306	−	0.979115i	$-0.434831\pi$
$431$	8.44150	0.406613	0.203306	+	0.979115i	$0.434831\pi$
$432$	−9.33956	−0.449350
$433$	−26.4739	−1.27225	−0.636127	−	0.771584i	$-0.719465\pi$
$433$	−26.4739	−1.27225	−0.636127	+	0.771584i	$0.719465\pi$
$434$	−2.03415	−0.0976422
$435$	0	0
$436$	−0.770630	−0.0369065
$437$	−5.41147	−0.258866
$438$	1.08647	0.0519134
$439$	−3.98545	−0.190215	−0.0951076	−	0.995467i	$-0.530320\pi$
$439$	−3.98545	−0.190215	−0.0951076	+	0.995467i	$0.530320\pi$
$440$	0	0
$441$	19.8084	0.943257
$442$	−4.59627	−0.218622
$443$	−9.81016	−0.466095	−0.233047	−	0.972465i	$-0.574870\pi$
$443$	−9.81016	−0.466095	−0.233047	+	0.972465i	$0.574870\pi$
$444$	0.985452	0.0467675
$445$	0	0
$446$	27.6418	1.30888
$447$	−2.12836	−0.100668
$448$	2.14290	0.101243
$449$	−8.43376	−0.398014	−0.199007	−	0.979998i	$-0.563772\pi$
$449$	−8.43376	−0.398014	−0.199007	+	0.979998i	$0.563772\pi$
$450$	0	0
$451$	−2.67499	−0.125960
$452$	−4.52435	−0.212807
$453$	2.24804	0.105622
$454$	−18.5699	−0.871527
$455$	0	0
$456$	3.10607	0.145455
$457$	25.8553	1.20946	0.604731	−	0.796430i	$-0.293281\pi$
$457$	25.8553	1.20946	0.604731	+	0.796430i	$0.293281\pi$
$458$	14.4233	0.673958
$459$	1.79561	0.0838118
$460$	0	0
$461$	6.88981	0.320891	0.160445	−	0.987045i	$-0.448707\pi$
$461$	6.88981	0.320891	0.160445	+	0.987045i	$0.448707\pi$
$462$	−0.248970	−0.0115831
$463$	1.96854	0.0914858	0.0457429	−	0.998953i	$-0.485434\pi$
$463$	1.96854	0.0914858	0.0457429	+	0.998953i	$0.485434\pi$
$464$	0.551689	0.0256115
$465$	0	0
$466$	−35.3678	−1.63838
$467$	−29.2645	−1.35420	−0.677099	−	0.735892i	$-0.736763\pi$
$467$	−29.2645	−1.35420	−0.677099	+	0.735892i	$0.736763\pi$
$468$	−3.41147	−0.157695
$469$	−2.19934	−0.101556
$470$	0	0
$471$	−1.36184	−0.0627505
$472$	6.68004	0.307474
$473$	−4.59627	−0.211337
$474$	−5.29086	−0.243017
$475$	0	0
$476$	0.106067	0.00486157
$477$	−8.10607	−0.371151
$478$	−5.48339	−0.250805
$479$	7.95399	0.363427	0.181714	−	0.983351i	$-0.441836\pi$
$479$	7.95399	0.363427	0.181714	+	0.983351i	$0.441836\pi$
$480$	0	0
$481$	−27.8726	−1.27088
$482$	−27.1111	−1.23488
$483$	−0.184793	−0.00840835
$484$	−3.18984	−0.144993
$485$	0	0
$486$	13.6040	0.617090
$487$	−15.8007	−0.715996	−0.357998	−	0.933722i	$-0.616541\pi$
$487$	−15.8007	−0.715996	−0.357998	+	0.933722i	$0.616541\pi$
$488$	−16.9513	−0.767349
$489$	5.30365	0.239839
$490$	0	0
$491$	−34.2799	−1.54703	−0.773516	−	0.633777i	$-0.781504\pi$
$491$	−34.2799	−1.54703	−0.773516	+	0.633777i	$0.781504\pi$
$492$	−0.239475	−0.0107964
$493$	−0.106067	−0.00477701
$494$	18.4611	0.830604
$495$	0	0
$496$	−17.4861	−0.785148
$497$	−3.35504	−0.150494
$498$	−2.59627	−0.116341
$499$	−18.3277	−0.820460	−0.410230	−	0.911982i	$-0.634552\pi$
$499$	−18.3277	−0.820460	−0.410230	+	0.911982i	$0.634552\pi$
$500$	0	0
$501$	−4.20439	−0.187838
$502$	4.20439	0.187651
$503$	−37.1421	−1.65608	−0.828042	−	0.560666i	$-0.810545\pi$
$503$	−37.1421	−1.65608	−0.828042	+	0.560666i	$0.810545\pi$
$504$	−2.53209	−0.112788
$505$	0	0
$506$	3.16250	0.140590
$507$	0.472964	0.0210050
$508$	4.26083	0.189044
$509$	41.9786	1.86067	0.930335	−	0.366710i	$-0.119516\pi$
$509$	41.9786	1.86067	0.930335	+	0.366710i	$0.119516\pi$
$510$	0	0
$511$	0.709141	0.0313705
$512$	14.2736	0.630811
$513$	−7.21213	−0.318423
$514$	−34.6655	−1.52903
$515$	0	0
$516$	−0.411474	−0.0181141
$517$	1.34730	0.0592540
$518$	4.34730	0.191009
$519$	2.10876	0.0925641
$520$	0	0
$521$	−36.9326	−1.61805	−0.809024	−	0.587775i	$-0.800004\pi$
$521$	−36.9326	−1.61805	−0.809024	+	0.587775i	$0.800004\pi$
$522$	−0.532089	−0.0232889
$523$	−4.74186	−0.207347	−0.103673	−	0.994611i	$-0.533060\pi$
$523$	−4.74186	−0.207347	−0.103673	+	0.994611i	$0.533060\pi$
$524$	1.07604	0.0470069
$525$	0	0
$526$	18.3286	0.799166
$527$	3.36184	0.146444
$528$	−2.14022	−0.0931409
$529$	−20.6527	−0.897944
$530$	0	0
$531$	−7.59627	−0.329650
$532$	−0.426022	−0.0184704
$533$	6.77332	0.293385
$534$	−4.55438	−0.197087
$535$	0	0
$536$	−16.0351	−0.692610
$537$	−5.77332	−0.249137
$538$	3.13610	0.135207
$539$	9.26857	0.399226
$540$	0	0
$541$	−31.8111	−1.36767	−0.683833	−	0.729639i	$-0.739688\pi$
$541$	−31.8111	−1.36767	−0.683833	+	0.729639i	$0.739688\pi$
$542$	27.8057	1.19436
$543$	2.66725	0.114463
$544$	1.70914	0.0732788
$545$	0	0
$546$	0.630415	0.0269793
$547$	42.6150	1.82209	0.911044	−	0.412309i	$-0.135278\pi$
$547$	42.6150	1.82209	0.911044	+	0.412309i	$0.135278\pi$
$548$	−1.42871	−0.0610315
$549$	19.2763	0.822693
$550$	0	0
$551$	0.426022	0.0181491
$552$	−1.34730	−0.0573447
$553$	−3.45336	−0.146852
$554$	−1.14290	−0.0485573
$555$	0	0
$556$	−6.67499	−0.283083
$557$	−29.6578	−1.25664	−0.628320	−	0.777955i	$-0.716257\pi$
$557$	−29.6578	−1.25664	−0.628320	+	0.777955i	$0.716257\pi$
$558$	16.8648	0.713945
$559$	11.6382	0.492242
$560$	0	0
$561$	0.411474	0.0173725
$562$	−15.2645	−0.643892
$563$	15.1162	0.637071	0.318535	−	0.947911i	$-0.396809\pi$
$563$	15.1162	0.637071	0.318535	+	0.947911i	$0.396809\pi$
$564$	0.120615	0.00507880
$565$	0	0
$566$	−12.5517	−0.527587
$567$	2.75372	0.115645
$568$	−24.4611	−1.02637
$569$	15.2986	0.641351	0.320675	−	0.947189i	$-0.396090\pi$
$569$	15.2986	0.641351	0.320675	+	0.947189i	$0.396090\pi$
$570$	0	0
$571$	−31.4151	−1.31468	−0.657340	−	0.753594i	$-0.728319\pi$
$571$	−31.4151	−1.31468	−0.657340	+	0.753594i	$0.728319\pi$
$572$	−1.59627	−0.0667433
$573$	−0.537141	−0.0224394
$574$	−1.05644	−0.0440948
$575$	0	0
$576$	−17.7665	−0.740271
$577$	−26.3131	−1.09543	−0.547715	−	0.836665i	$-0.684502\pi$
$577$	−26.3131	−1.09543	−0.547715	+	0.836665i	$0.684502\pi$
$578$	24.8607	1.03407
$579$	1.60813	0.0668314
$580$	0	0
$581$	−1.69459	−0.0703036
$582$	−7.40373	−0.306895
$583$	−3.79292	−0.157087
$584$	5.17024	0.213946
$585$	0	0
$586$	14.7401	0.608908
$587$	8.03239	0.331532	0.165766	−	0.986165i	$-0.446990\pi$
$587$	8.03239	0.331532	0.165766	+	0.986165i	$0.446990\pi$
$588$	0.829755	0.0342185
$589$	−13.5030	−0.556381
$590$	0	0
$591$	0.159815	0.00657393
$592$	37.3705	1.53592
$593$	4.70470	0.193199	0.0965994	−	0.995323i	$-0.469203\pi$
$593$	4.70470	0.193199	0.0965994	+	0.995323i	$0.469203\pi$
$594$	4.21482	0.172936
$595$	0	0
$596$	2.12836	0.0871808
$597$	3.89487	0.159406
$598$	−8.00774	−0.327461
$599$	42.7134	1.74522	0.872611	−	0.488417i	$-0.162425\pi$
$599$	42.7134	1.74522	0.872611	+	0.488417i	$0.162425\pi$
$600$	0	0
$601$	−36.5425	−1.49060	−0.745300	−	0.666729i	$-0.767694\pi$
$601$	−36.5425	−1.49060	−0.745300	+	0.666729i	$0.767694\pi$
$602$	−1.81521	−0.0739823
$603$	18.2344	0.742563
$604$	−2.24804	−0.0914713
$605$	0	0
$606$	−6.52166	−0.264924
$607$	9.67230	0.392587	0.196293	−	0.980545i	$-0.437110\pi$
$607$	9.67230	0.392587	0.196293	+	0.980545i	$0.437110\pi$
$608$	−6.86484	−0.278406
$609$	0.0145479	0.000589511 0
$610$	0	0
$611$	−3.41147	−0.138013
$612$	−0.879385	−0.0355470
$613$	10.9040	0.440410	0.220205	−	0.975454i	$-0.429327\pi$
$613$	10.9040	0.440410	0.220205	+	0.975454i	$0.429327\pi$
$614$	−14.0259	−0.566040
$615$	0	0
$616$	−1.18479	−0.0477367
$617$	12.1652	0.489752	0.244876	−	0.969554i	$-0.421253\pi$
$617$	12.1652	0.489752	0.244876	+	0.969554i	$0.421253\pi$
$618$	−5.18479	−0.208563
$619$	4.94532	0.198769	0.0993846	−	0.995049i	$-0.468313\pi$
$619$	4.94532	0.198769	0.0993846	+	0.995049i	$0.468313\pi$
$620$	0	0
$621$	3.12836	0.125537
$622$	−36.1935	−1.45123
$623$	−2.97266	−0.119097
$624$	5.41921	0.216942
$625$	0	0
$626$	−24.8803	−0.994418
$627$	−1.65270	−0.0660026
$628$	1.36184	0.0543435
$629$	−7.18479	−0.286476
$630$	0	0
$631$	10.6604	0.424386	0.212193	−	0.977228i	$-0.431940\pi$
$631$	10.6604	0.424386	0.212193	+	0.977228i	$0.431940\pi$
$632$	−25.1780	−1.00153
$633$	−5.34049	−0.212265
$634$	−16.5858	−0.658708
$635$	0	0
$636$	−0.339556	−0.0134643
$637$	−23.4688	−0.929869
$638$	−0.248970	−0.00985683
$639$	27.8161	1.10039
$640$	0	0
$641$	11.6186	0.458905	0.229453	−	0.973320i	$-0.426306\pi$
$641$	11.6186	0.458905	0.229453	+	0.973320i	$0.426306\pi$
$642$	3.80335	0.150106
$643$	1.82026	0.0717840	0.0358920	−	0.999356i	$-0.488573\pi$
$643$	1.82026	0.0717840	0.0358920	+	0.999356i	$0.488573\pi$
$644$	0.184793	0.00728185
$645$	0	0
$646$	4.75877	0.187231
$647$	−27.8794	−1.09605	−0.548026	−	0.836461i	$-0.684620\pi$
$647$	−27.8794	−1.09605	−0.548026	+	0.836461i	$0.684620\pi$
$648$	20.0770	0.788698
$649$	−3.55438	−0.139522
$650$	0	0
$651$	−0.461104	−0.0180721
$652$	−5.30365	−0.207707
$653$	36.5918	1.43195	0.715974	−	0.698127i	$-0.245983\pi$
$653$	36.5918	1.43195	0.715974	+	0.698127i	$0.245983\pi$
$654$	−1.18067	−0.0461680
$655$	0	0
$656$	−9.08141	−0.354570
$657$	−5.87939	−0.229377
$658$	0.532089	0.0207430
$659$	39.2867	1.53039	0.765197	−	0.643796i	$-0.222642\pi$
$659$	39.2867	1.53039	0.765197	+	0.643796i	$0.222642\pi$
$660$	0	0
$661$	−10.6646	−0.414803	−0.207402	−	0.978256i	$-0.566501\pi$
$661$	−10.6646	−0.414803	−0.207402	+	0.978256i	$0.566501\pi$
$662$	−47.8854	−1.86112
$663$	−1.04189	−0.0404636
$664$	−12.3550	−0.479468
$665$	0	0
$666$	−36.0428	−1.39663
$667$	−0.184793	−0.00715520
$668$	4.20439	0.162673
$669$	6.26588	0.242253
$670$	0	0
$671$	9.01960	0.348198
$672$	−0.234422	−0.00904304
$673$	21.5503	0.830701	0.415351	−	0.909661i	$-0.363659\pi$
$673$	21.5503	0.830701	0.415351	+	0.909661i	$0.363659\pi$
$674$	−45.9118	−1.76846
$675$	0	0
$676$	−0.472964	−0.0181909
$677$	−35.4561	−1.36269	−0.681343	−	0.731964i	$-0.738604\pi$
$677$	−35.4561	−1.36269	−0.681343	+	0.731964i	$0.738604\pi$
$678$	−6.93170	−0.266210
$679$	−4.83244	−0.185452
$680$	0	0
$681$	−4.20945	−0.161306
$682$	7.89124	0.302171
$683$	−0.616801	−0.0236012	−0.0118006	−	0.999930i	$-0.503756\pi$
$683$	−0.616801	−0.0236012	−0.0118006	+	0.999930i	$0.503756\pi$
$684$	3.53209	0.135053
$685$	0	0
$686$	7.38507	0.281963
$687$	3.26950	0.124739
$688$	−15.6040	−0.594897
$689$	9.60401	0.365884
$690$	0	0
$691$	50.6255	1.92588	0.962942	−	0.269709i	$-0.0869275\pi$
$691$	50.6255	1.92588	0.962942	+	0.269709i	$0.0869275\pi$
$692$	−2.10876	−0.0801628
$693$	1.34730	0.0511796
$694$	−28.6551	−1.08773
$695$	0	0
$696$	0.106067	0.00402045
$697$	1.74598	0.0661336
$698$	0.980400	0.0371087
$699$	−8.01724	−0.303240
$700$	0	0
$701$	−47.2671	−1.78526	−0.892628	−	0.450795i	$-0.851141\pi$
$701$	−47.2671	−1.78526	−0.892628	+	0.450795i	$0.851141\pi$
$702$	−10.6723	−0.402800
$703$	28.8580	1.08840
$704$	−8.31315	−0.313314
$705$	0	0
$706$	0.441504	0.0166162
$707$	−4.25671	−0.160090
$708$	−0.318201	−0.0119587
$709$	−21.3158	−0.800533	−0.400267	−	0.916399i	$-0.631082\pi$
$709$	−21.3158	−0.800533	−0.400267	+	0.916399i	$0.631082\pi$
$710$	0	0
$711$	28.6313	1.07376
$712$	−21.6732	−0.812239
$713$	5.85710	0.219350
$714$	0.162504	0.00608155
$715$	0	0
$716$	5.77332	0.215759
$717$	−1.24298	−0.0464201
$718$	−30.1034	−1.12345
$719$	−4.90766	−0.183025	−0.0915124	−	0.995804i	$-0.529170\pi$
$719$	−4.90766	−0.183025	−0.0915124	+	0.995804i	$0.529170\pi$
$720$	0	0
$721$	−3.38413	−0.126032
$722$	9.99588	0.372008
$723$	−6.14559	−0.228557
$724$	−2.66725	−0.0991276
$725$	0	0
$726$	−4.88713	−0.181378
$727$	18.0550	0.669623	0.334812	−	0.942285i	$-0.391327\pi$
$727$	18.0550	0.669623	0.334812	+	0.942285i	$0.391327\pi$
$728$	3.00000	0.111187
$729$	−20.7033	−0.766788
$730$	0	0
$731$	3.00000	0.110959
$732$	0.807467	0.0298448
$733$	12.4183	0.458680	0.229340	−	0.973346i	$-0.426343\pi$
$733$	12.4183	0.458680	0.229340	+	0.973346i	$0.426343\pi$
$734$	−21.2044	−0.782668
$735$	0	0
$736$	2.97771	0.109760
$737$	8.53209	0.314284
$738$	8.75877	0.322415
$739$	4.38682	0.161372	0.0806859	−	0.996740i	$-0.474289\pi$
$739$	4.38682	0.161372	0.0806859	+	0.996740i	$0.474289\pi$
$740$	0	0
$741$	4.18479	0.153732
$742$	−1.49794	−0.0549911
$743$	−40.6468	−1.49119	−0.745594	−	0.666401i	$-0.767834\pi$
$743$	−40.6468	−1.49119	−0.745594	+	0.666401i	$0.767834\pi$
$744$	−3.36184	−0.123251
$745$	0	0
$746$	17.6340	0.645628
$747$	14.0496	0.514049
$748$	−0.411474	−0.0150450
$749$	2.48246	0.0907071
$750$	0	0
$751$	31.0597	1.13339	0.566693	−	0.823929i	$-0.308223\pi$
$751$	31.0597	1.13339	0.566693	+	0.823929i	$0.308223\pi$
$752$	4.57398	0.166796
$753$	0.953058	0.0347314
$754$	0.630415	0.0229584
$755$	0	0
$756$	0.246282	0.00895719
$757$	41.2995	1.50106	0.750529	−	0.660838i	$-0.229799\pi$
$757$	41.2995	1.50106	0.750529	+	0.660838i	$0.229799\pi$
$758$	−24.4466	−0.887939
$759$	0.716881	0.0260211
$760$	0	0
$761$	28.0104	1.01538	0.507689	−	0.861541i	$-0.330500\pi$
$761$	28.0104	1.01538	0.507689	+	0.861541i	$0.330500\pi$
$762$	6.52797	0.236483
$763$	−0.770630	−0.0278987
$764$	0.537141	0.0194331
$765$	0	0
$766$	−45.2550	−1.63513
$767$	9.00000	0.324971
$768$	−2.81252	−0.101488
$769$	−6.44057	−0.232253	−0.116126	−	0.993234i	$-0.537048\pi$
$769$	−6.44057	−0.232253	−0.116126	+	0.993234i	$0.537048\pi$
$770$	0	0
$771$	−7.85803	−0.283000
$772$	−1.60813	−0.0578777
$773$	13.3105	0.478744	0.239372	−	0.970928i	$-0.423058\pi$
$773$	13.3105	0.478744	0.239372	+	0.970928i	$0.423058\pi$
$774$	15.0496	0.540948
$775$	0	0
$776$	−35.2327	−1.26478
$777$	0.985452	0.0353529
$778$	42.0883	1.50894
$779$	−7.01279	−0.251259
$780$	0	0
$781$	13.0155	0.465731
$782$	−2.06418	−0.0738148
$783$	−0.246282	−0.00880140
$784$	31.4662	1.12379
$785$	0	0
$786$	1.64858	0.0588031
$787$	22.7110	0.809560	0.404780	−	0.914414i	$-0.367348\pi$
$787$	22.7110	0.809560	0.404780	+	0.914414i	$0.367348\pi$
$788$	−0.159815	−0.00569319
$789$	4.15476	0.147913
$790$	0	0
$791$	−4.52435	−0.160867
$792$	9.82295	0.349043
$793$	−22.8384	−0.811016
$794$	46.5185	1.65088
$795$	0	0
$796$	−3.89487	−0.138050
$797$	−42.7401	−1.51393	−0.756966	−	0.653454i	$-0.773319\pi$
$797$	−42.7401	−1.51393	−0.756966	+	0.653454i	$0.773319\pi$
$798$	−0.652704	−0.0231055
$799$	−0.879385	−0.0311104
$800$	0	0
$801$	24.6459	0.870820
$802$	50.0925	1.76883
$803$	−2.75103	−0.0970817
$804$	0.763823	0.0269380
$805$	0	0
$806$	−19.9813	−0.703812
$807$	0.710895	0.0250247
$808$	−31.0351	−1.09181
$809$	−19.5986	−0.689051	−0.344526	−	0.938777i	$-0.611960\pi$
$809$	−19.5986	−0.689051	−0.344526	+	0.938777i	$0.611960\pi$
$810$	0	0
$811$	36.1385	1.26899	0.634496	−	0.772926i	$-0.281207\pi$
$811$	36.1385	1.26899	0.634496	+	0.772926i	$0.281207\pi$
$812$	−0.0145479	−0.000510532 0
$813$	6.30304	0.221057
$814$	−16.8648	−0.591112
$815$	0	0
$816$	1.39693	0.0489022
$817$	−12.0496	−0.421563
$818$	10.6222	0.371396
$819$	−3.41147	−0.119207
$820$	0	0
$821$	36.4107	1.27074	0.635370	−	0.772208i	$-0.280847\pi$
$821$	36.4107	1.27074	0.635370	+	0.772208i	$0.280847\pi$
$822$	−2.18891	−0.0763470
$823$	42.9181	1.49603	0.748015	−	0.663682i	$-0.231007\pi$
$823$	42.9181	1.49603	0.748015	+	0.663682i	$0.231007\pi$
$824$	−24.6732	−0.859533
$825$	0	0
$826$	−1.40373	−0.0488421
$827$	25.6827	0.893076	0.446538	−	0.894765i	$-0.352657\pi$
$827$	25.6827	0.893076	0.446538	+	0.894765i	$0.352657\pi$
$828$	−1.53209	−0.0532438
$829$	−18.0033	−0.625280	−0.312640	−	0.949872i	$-0.601213\pi$
$829$	−18.0033	−0.625280	−0.312640	+	0.949872i	$0.601213\pi$
$830$	0	0
$831$	−0.259075	−0.00898722
$832$	21.0496	0.729765
$833$	−6.04963	−0.209607
$834$	−10.2267	−0.354121
$835$	0	0
$836$	1.65270	0.0571600
$837$	7.80604	0.269816
$838$	12.4088	0.428654
$839$	−49.7461	−1.71743	−0.858713	−	0.512457i	$-0.828735\pi$
$839$	−49.7461	−1.71743	−0.858713	+	0.512457i	$0.828735\pi$
$840$	0	0
$841$	−28.9855	−0.999498
$842$	45.0888	1.55386
$843$	−3.46017	−0.119175
$844$	5.34049	0.183827
$845$	0	0
$846$	−4.41147	−0.151670
$847$	−3.18984	−0.109604
$848$	−12.8767	−0.442188
$849$	−2.84524	−0.0976483
$850$	0	0
$851$	−12.5175	−0.429096
$852$	1.16519	0.0399188
$853$	57.4698	1.96773	0.983864	−	0.178919i	$-0.0572599\pi$
$853$	57.4698	1.96773	0.983864	+	0.178919i	$0.0572599\pi$
$854$	3.56212	0.121893
$855$	0	0
$856$	18.0993	0.618620
$857$	−20.9317	−0.715013	−0.357507	−	0.933911i	$-0.616373\pi$
$857$	−20.9317	−0.715013	−0.357507	+	0.933911i	$0.616373\pi$
$858$	−2.44562	−0.0834922
$859$	9.86247	0.336503	0.168252	−	0.985744i	$-0.446188\pi$
$859$	9.86247	0.336503	0.168252	+	0.985744i	$0.446188\pi$
$860$	0	0
$861$	−0.239475	−0.00816128
$862$	−12.9331	−0.440504
$863$	−21.9855	−0.748393	−0.374197	−	0.927349i	$-0.622082\pi$
$863$	−21.9855	−0.748393	−0.374197	+	0.927349i	$0.622082\pi$
$864$	3.96854	0.135012
$865$	0	0
$866$	40.5604	1.37830
$867$	5.63547	0.191391
$868$	0.461104	0.0156509
$869$	13.3969	0.454460
$870$	0	0
$871$	−21.6040	−0.732024
$872$	−5.61856	−0.190268
$873$	40.0651	1.35600
$874$	8.29086	0.280443
$875$	0	0
$876$	−0.246282	−0.00832110
$877$	12.5885	0.425084	0.212542	−	0.977152i	$-0.431826\pi$
$877$	12.5885	0.425084	0.212542	+	0.977152i	$0.431826\pi$
$878$	6.10607	0.206070
$879$	3.34131	0.112700
$880$	0	0
$881$	42.2217	1.42249	0.711243	−	0.702946i	$-0.248132\pi$
$881$	42.2217	1.42249	0.711243	+	0.702946i	$0.248132\pi$
$882$	−30.3482	−1.02188
$883$	20.4989	0.689842	0.344921	−	0.938632i	$-0.387906\pi$
$883$	20.4989	0.689842	0.344921	+	0.938632i	$0.387906\pi$
$884$	1.04189	0.0350425
$885$	0	0
$886$	15.0300	0.504944
$887$	−10.7365	−0.360496	−0.180248	−	0.983621i	$-0.557690\pi$
$887$	−10.7365	−0.360496	−0.180248	+	0.983621i	$0.557690\pi$
$888$	7.18479	0.241106
$889$	4.26083	0.142904
$890$	0	0
$891$	−10.6827	−0.357885
$892$	−6.26588	−0.209797
$893$	3.53209	0.118197
$894$	3.26083	0.109058
$895$	0	0
$896$	−4.63310	−0.154781
$897$	−1.81521	−0.0606080
$898$	12.9213	0.431189
$899$	−0.461104	−0.0153787
$900$	0	0
$901$	2.47565	0.0824759
$902$	4.09833	0.136459
$903$	−0.411474	−0.0136930
$904$	−32.9864	−1.09711
$905$	0	0
$906$	−3.44419	−0.114426
$907$	−11.2627	−0.373972	−0.186986	−	0.982363i	$-0.559872\pi$
$907$	−11.2627	−0.373972	−0.186986	+	0.982363i	$0.559872\pi$
$908$	4.20945	0.139695
$909$	35.2918	1.17055
$910$	0	0
$911$	22.8324	0.756473	0.378236	−	0.925709i	$-0.376531\pi$
$911$	22.8324	0.756473	0.378236	+	0.925709i	$0.376531\pi$
$912$	−5.61081	−0.185793
$913$	6.57398	0.217567
$914$	−39.6127	−1.31027
$915$	0	0
$916$	−3.26950	−0.108027
$917$	1.07604	0.0355339
$918$	−2.75103	−0.0907975
$919$	−0.154763	−0.00510516	−0.00255258	−	0.999997i	$-0.500813\pi$
$919$	−0.154763	−0.00510516	−0.00255258	+	0.999997i	$0.500813\pi$
$920$	0	0
$921$	−3.17942	−0.104765
$922$	−10.5558	−0.347637
$923$	−32.9564	−1.08477
$924$	0.0564370	0.00185664
$925$	0	0
$926$	−3.01598	−0.0991112
$927$	28.0574	0.921525
$928$	−0.234422	−0.00769529
$929$	−43.5850	−1.42998	−0.714989	−	0.699136i	$-0.753568\pi$
$929$	−43.5850	−1.42998	−0.714989	+	0.699136i	$0.753568\pi$
$930$	0	0
$931$	24.2986	0.796354
$932$	8.01724	0.262613
$933$	−8.20439	−0.268600
$934$	44.8357	1.46707
$935$	0	0
$936$	−24.8726	−0.812986
$937$	−27.4328	−0.896191	−0.448096	−	0.893986i	$-0.647898\pi$
$937$	−27.4328	−0.896191	−0.448096	+	0.893986i	$0.647898\pi$
$938$	3.36959	0.110021
$939$	−5.63991	−0.184052
$940$	0	0
$941$	−3.78013	−0.123229	−0.0616143	−	0.998100i	$-0.519625\pi$
$941$	−3.78013	−0.123229	−0.0616143	+	0.998100i	$0.519625\pi$
$942$	2.08647	0.0679808
$943$	3.04189	0.0990575
$944$	−12.0669	−0.392743
$945$	0	0
$946$	7.04189	0.228952
$947$	−31.3296	−1.01807	−0.509037	−	0.860745i	$-0.669998\pi$
$947$	−31.3296	−1.01807	−0.509037	+	0.860745i	$0.669998\pi$
$948$	1.19934	0.0389528
$949$	6.96585	0.226121
$950$	0	0
$951$	−3.75970	−0.121917
$952$	0.773318	0.0250634
$953$	5.80747	0.188122	0.0940611	−	0.995566i	$-0.470015\pi$
$953$	5.80747	0.188122	0.0940611	+	0.995566i	$0.470015\pi$
$954$	12.4192	0.402087
$955$	0	0
$956$	1.24298	0.0402010
$957$	−0.0564370	−0.00182435
$958$	−12.1862	−0.393719
$959$	−1.42871	−0.0461355
$960$	0	0
$961$	−16.3851	−0.528551
$962$	42.7033	1.37681
$963$	−20.5817	−0.663237
$964$	6.14559	0.197936
$965$	0	0
$966$	0.283119	0.00910919
$967$	47.3773	1.52355	0.761776	−	0.647840i	$-0.224328\pi$
$967$	47.3773	1.52355	0.761776	+	0.647840i	$0.224328\pi$
$968$	−23.2567	−0.747499
$969$	1.07873	0.0346537
$970$	0	0
$971$	33.7784	1.08400	0.542000	−	0.840379i	$-0.317667\pi$
$971$	33.7784	1.08400	0.542000	+	0.840379i	$0.317667\pi$
$972$	−3.08378	−0.0989122
$973$	−6.67499	−0.213990
$974$	24.2080	0.775675
$975$	0	0
$976$	30.6209	0.980152
$977$	−39.0583	−1.24959	−0.624793	−	0.780790i	$-0.714817\pi$
$977$	−39.0583	−1.24959	−0.624793	+	0.780790i	$0.714817\pi$
$978$	−8.12567	−0.259830
$979$	11.5321	0.368567
$980$	0	0
$981$	6.38919	0.203991
$982$	52.5199	1.67598
$983$	38.9472	1.24222	0.621111	−	0.783722i	$-0.286682\pi$
$983$	38.9472	1.24222	0.621111	+	0.783722i	$0.286682\pi$
$984$	−1.74598	−0.0556597
$985$	0	0
$986$	0.162504	0.00517518
$987$	0.120615	0.00383921
$988$	−4.18479	−0.133136
$989$	5.22668	0.166199
$990$	0	0
$991$	−11.4424	−0.363481	−0.181740	−	0.983347i	$-0.558173\pi$
$991$	−11.4424	−0.363481	−0.181740	+	0.983347i	$0.558173\pi$
$992$	7.43014	0.235907
$993$	−10.8547	−0.344465
$994$	5.14022	0.163038
$995$	0	0
$996$	0.588526	0.0186482
$997$	−8.67230	−0.274655	−0.137327	−	0.990526i	$-0.543851\pi$
$997$	−8.67230	−0.274655	−0.137327	+	0.990526i	$0.543851\pi$
$998$	28.0797	0.888846
$999$	−16.6827	−0.527818

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1175.2.a.e.1.1	yes	3
5.2	odd	4		1175.2.c.d.424.2		6
5.3	odd	4		1175.2.c.d.424.5		6
5.4	even	2		1175.2.a.d.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1175.2.a.d.1.3	✓	3	5.4	even	2
1175.2.a.e.1.1	yes	3	1.1	even	1	trivial
1175.2.c.d.424.2		6	5.2	odd	4
1175.2.c.d.424.5		6	5.3	odd	4

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:	\( N \)	\(=\)	\( 1175 = 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1175.a (trivial)

\(f(q)\)	\(=\)	\(q-1.53209 q^{2} -0.347296 q^{3} +0.347296 q^{4} +0.532089 q^{6} +0.347296 q^{7} +2.53209 q^{8} -2.87939 q^{9} +O(q^{10})\)	\(q-1.53209 q^{2} -0.347296 q^{3} +0.347296 q^{4} +0.532089 q^{6} +0.347296 q^{7} +2.53209 q^{8} -2.87939 q^{9} -1.34730 q^{11} -0.120615 q^{12} +3.41147 q^{13} -0.532089 q^{14} -4.57398 q^{16} +0.879385 q^{17} +4.41147 q^{18} -3.53209 q^{19} -0.120615 q^{21} +2.06418 q^{22} +1.53209 q^{23} -0.879385 q^{24} -5.22668 q^{26} +2.04189 q^{27} +0.120615 q^{28} -0.120615 q^{29} +3.82295 q^{31} +1.94356 q^{32} +0.467911 q^{33} -1.34730 q^{34} -1.00000 q^{36} -8.17024 q^{37} +5.41147 q^{38} -1.18479 q^{39} +1.98545 q^{41} +0.184793 q^{42} +3.41147 q^{43} -0.467911 q^{44} -2.34730 q^{46} -1.00000 q^{47} +1.58853 q^{48} -6.87939 q^{49} -0.305407 q^{51} +1.18479 q^{52} +2.81521 q^{53} -3.12836 q^{54} +0.879385 q^{56} +1.22668 q^{57} +0.184793 q^{58} +2.63816 q^{59} -6.69459 q^{61} -5.85710 q^{62} -1.00000 q^{63} +6.17024 q^{64} -0.716881 q^{66} -6.33275 q^{67} +0.305407 q^{68} -0.532089 q^{69} -9.66044 q^{71} -7.29086 q^{72} +2.04189 q^{73} +12.5175 q^{74} -1.22668 q^{76} -0.467911 q^{77} +1.81521 q^{78} -9.94356 q^{79} +7.92902 q^{81} -3.04189 q^{82} -4.87939 q^{83} -0.0418891 q^{84} -5.22668 q^{86} +0.0418891 q^{87} -3.41147 q^{88} -8.55943 q^{89} +1.18479 q^{91} +0.532089 q^{92} -1.32770 q^{93} +1.53209 q^{94} -0.674992 q^{96} -13.9145 q^{97} +10.5398 q^{98} +3.87939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{6} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^6 + 3 * q^8 - 3 * q^9	\( 3 q - 3 q^{6} + 3 q^{8} - 3 q^{9} - 3 q^{11} - 6 q^{12} + 3 q^{14} - 6 q^{16} - 3 q^{17} + 3 q^{18} - 6 q^{19} - 6 q^{21} - 3 q^{22} + 3 q^{24} - 9 q^{26} + 3 q^{27} + 6 q^{28} - 6 q^{29} - 9 q^{31} - 9 q^{32} + 6 q^{33} - 3 q^{34} - 3 q^{36} - 3 q^{37} + 6 q^{38} - 12 q^{41} - 3 q^{42} - 6 q^{44} - 6 q^{46} - 3 q^{47} + 15 q^{48} - 15 q^{49} - 3 q^{51} + 12 q^{53} + 9 q^{54} - 3 q^{56} - 3 q^{57} - 3 q^{58} - 9 q^{59} - 18 q^{61} - 18 q^{62} - 3 q^{63} - 3 q^{64} + 6 q^{66} + 3 q^{68} + 3 q^{69} - 6 q^{71} - 6 q^{72} + 3 q^{73} + 15 q^{74} + 3 q^{76} - 6 q^{77} + 9 q^{78} - 15 q^{79} - 9 q^{81} - 6 q^{82} - 9 q^{83} + 3 q^{84} - 9 q^{86} - 3 q^{87} - 3 q^{92} + 3 q^{96} + 9 q^{97} + 3 q^{98} + 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^6 + 3 * q^8 - 3 * q^9 - 3 * q^11 - 6 * q^12 + 3 * q^14 - 6 * q^16 - 3 * q^17 + 3 * q^18 - 6 * q^19 - 6 * q^21 - 3 * q^22 + 3 * q^24 - 9 * q^26 + 3 * q^27 + 6 * q^28 - 6 * q^29 - 9 * q^31 - 9 * q^32 + 6 * q^33 - 3 * q^34 - 3 * q^36 - 3 * q^37 + 6 * q^38 - 12 * q^41 - 3 * q^42 - 6 * q^44 - 6 * q^46 - 3 * q^47 + 15 * q^48 - 15 * q^49 - 3 * q^51 + 12 * q^53 + 9 * q^54 - 3 * q^56 - 3 * q^57 - 3 * q^58 - 9 * q^59 - 18 * q^61 - 18 * q^62 - 3 * q^63 - 3 * q^64 + 6 * q^66 + 3 * q^68 + 3 * q^69 - 6 * q^71 - 6 * q^72 + 3 * q^73 + 15 * q^74 + 3 * q^76 - 6 * q^77 + 9 * q^78 - 15 * q^79 - 9 * q^81 - 6 * q^82 - 9 * q^83 + 3 * q^84 - 9 * q^86 - 3 * q^87 - 3 * q^92 + 3 * q^96 + 9 * q^97 + 3 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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