LMFDB - Embedded newform 7191.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7191,2,Mod(1,7191)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7191, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7191.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7191 = 3^{2} \cdot 17 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7191.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:	$57.4204240935$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 24 x^{18} + 108 x^{17} + 221 x^{16} - 1200 x^{15} - 931 x^{14} + 7128 x^{13} + \cdots + 63 $ x^20 - 4x^19 - 24x^18 + 108x^17 + 221x^16 - 1200x^15 - 931x^14 + 7128x^13 + 1280x^12 - 24559x^11 + 3411x^10 + 49829x^9 - 15132x^8 - 58070x^7 + 20659x^6 + 36738x^5 - 11129x^4 - 11299x^3 + 1589x^2 + 1289*x + 63
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 799)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64443$ of defining polynomial
Character	$\chi$	$=$	7191.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64443 q^{2} +4.99303 q^{4} +3.13323 q^{5} -1.81022 q^{7} -7.91488 q^{8} +O(q^{10})$	\(q-2.64443 q^{2} +4.99303 q^{4} +3.13323 q^{5} -1.81022 q^{7} -7.91488 q^{8} -8.28563 q^{10} +0.553008 q^{11} +0.601061 q^{13} +4.78700 q^{14} +10.9443 q^{16} +1.00000 q^{17} +4.50744 q^{19} +15.6443 q^{20} -1.46239 q^{22} +7.28680 q^{23} +4.81714 q^{25} -1.58947 q^{26} -9.03847 q^{28} -2.50389 q^{29} -9.60133 q^{31} -13.1118 q^{32} -2.64443 q^{34} -5.67183 q^{35} -4.56133 q^{37} -11.9196 q^{38} -24.7992 q^{40} -10.8389 q^{41} -4.59190 q^{43} +2.76119 q^{44} -19.2695 q^{46} -1.00000 q^{47} -3.72312 q^{49} -12.7386 q^{50} +3.00112 q^{52} -0.0565505 q^{53} +1.73270 q^{55} +14.3276 q^{56} +6.62137 q^{58} -11.9547 q^{59} +8.36297 q^{61} +25.3901 q^{62} +12.7846 q^{64} +1.88326 q^{65} -15.7774 q^{67} +4.99303 q^{68} +14.9988 q^{70} -4.84854 q^{71} +3.41250 q^{73} +12.0621 q^{74} +22.5058 q^{76} -1.00106 q^{77} +3.47942 q^{79} +34.2911 q^{80} +28.6628 q^{82} +6.02539 q^{83} +3.13323 q^{85} +12.1430 q^{86} -4.37700 q^{88} -16.2853 q^{89} -1.08805 q^{91} +36.3833 q^{92} +2.64443 q^{94} +14.1228 q^{95} +2.05567 q^{97} +9.84554 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{2} + 24 q^{4} - 13 q^{5} - 3 q^{7} - 12 q^{8}+O(q^{10}) $ 20 * q - 4 * q^2 + 24 * q^4 - 13 * q^5 - 3 * q^7 - 12 * q^8	$ 20 q - 4 q^{2} + 24 q^{4} - 13 q^{5} - 3 q^{7} - 12 q^{8} - 2 q^{10} - 10 q^{11} + q^{13} - 13 q^{14} + 28 q^{16} + 20 q^{17} + 5 q^{19} - 37 q^{20} - 19 q^{22} - 4 q^{23} + 41 q^{25} + 9 q^{26} - 25 q^{28} - 26 q^{29} + 6 q^{31} - 28 q^{32} - 4 q^{34} - 21 q^{35} - 8 q^{37} + 21 q^{38} - 25 q^{40} - 69 q^{41} - 7 q^{43} - 16 q^{44} - 24 q^{46} - 20 q^{47} + 53 q^{49} - 16 q^{50} + 18 q^{52} - 29 q^{53} + 5 q^{55} + 22 q^{56} - q^{58} - 55 q^{59} - 17 q^{61} + 7 q^{62} + 58 q^{64} - 40 q^{65} - 6 q^{67} + 24 q^{68} - 15 q^{70} - 47 q^{71} - 32 q^{73} + 67 q^{74} - 5 q^{76} - 4 q^{77} - 26 q^{79} - 108 q^{80} + 25 q^{82} - 3 q^{83} - 13 q^{85} - 8 q^{86} - 47 q^{88} - 119 q^{89} - 35 q^{91} + 4 q^{94} + 48 q^{95} - 13 q^{97} - q^{98}+O(q^{100}) $ 20 * q - 4 * q^2 + 24 * q^4 - 13 * q^5 - 3 * q^7 - 12 * q^8 - 2 * q^10 - 10 * q^11 + q^13 - 13 * q^14 + 28 * q^16 + 20 * q^17 + 5 * q^19 - 37 * q^20 - 19 * q^22 - 4 * q^23 + 41 * q^25 + 9 * q^26 - 25 * q^28 - 26 * q^29 + 6 * q^31 - 28 * q^32 - 4 * q^34 - 21 * q^35 - 8 * q^37 + 21 * q^38 - 25 * q^40 - 69 * q^41 - 7 * q^43 - 16 * q^44 - 24 * q^46 - 20 * q^47 + 53 * q^49 - 16 * q^50 + 18 * q^52 - 29 * q^53 + 5 * q^55 + 22 * q^56 - q^58 - 55 * q^59 - 17 * q^61 + 7 * q^62 + 58 * q^64 - 40 * q^65 - 6 * q^67 + 24 * q^68 - 15 * q^70 - 47 * q^71 - 32 * q^73 + 67 * q^74 - 5 * q^76 - 4 * q^77 - 26 * q^79 - 108 * q^80 + 25 * q^82 - 3 * q^83 - 13 * q^85 - 8 * q^86 - 47 * q^88 - 119 * q^89 - 35 * q^91 + 4 * q^94 + 48 * q^95 - 13 * q^97 - q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.64443	−1.86990	−0.934949	−	0.354783i	$-0.884555\pi$
$2$	−2.64443	−1.86990	−0.934949	+	0.354783i	$0.884555\pi$
$3$	0	0
$3$	0	0
$4$	4.99303	2.49652
$5$	3.13323	1.40122	0.700612	−	0.713543i	$-0.252910\pi$
$5$	3.13323	1.40122	0.700612	+	0.713543i	$0.252910\pi$
$6$	0	0
$7$	−1.81022	−0.684197	−0.342099	−	0.939664i	$-0.611138\pi$
$7$	−1.81022	−0.684197	−0.342099	+	0.939664i	$0.611138\pi$
$8$	−7.91488	−2.79833
$9$	0	0
$10$	−8.28563	−2.62015
$11$	0.553008	0.166738	0.0833691	−	0.996519i	$-0.473432\pi$
$11$	0.553008	0.166738	0.0833691	+	0.996519i	$0.473432\pi$
$12$	0	0
$13$	0.601061	0.166704	0.0833522	−	0.996520i	$-0.473437\pi$
$13$	0.601061	0.166704	0.0833522	+	0.996520i	$0.473437\pi$
$14$	4.78700	1.27938
$15$	0	0
$16$	10.9443	2.73608
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	4.50744	1.03408	0.517038	−	0.855962i	$-0.327034\pi$
$19$	4.50744	1.03408	0.517038	+	0.855962i	$0.327034\pi$
$20$	15.6443	3.49818
$21$	0	0
$22$	−1.46239	−0.311783
$23$	7.28680	1.51940	0.759702	−	0.650272i	$-0.225345\pi$
$23$	7.28680	1.51940	0.759702	+	0.650272i	$0.225345\pi$
$24$	0	0
$25$	4.81714	0.963428
$26$	−1.58947	−0.311720
$27$	0	0
$28$	−9.03847	−1.70811
$29$	−2.50389	−0.464960	−0.232480	−	0.972601i	$-0.574684\pi$
$29$	−2.50389	−0.464960	−0.232480	+	0.972601i	$0.574684\pi$
$30$	0	0
$31$	−9.60133	−1.72445	−0.862225	−	0.506526i	$-0.830930\pi$
$31$	−9.60133	−1.72445	−0.862225	+	0.506526i	$0.830930\pi$
$32$	−13.1118	−2.31786
$33$	0	0
$34$	−2.64443	−0.453517
$35$	−5.67183	−0.958714
$36$	0	0
$37$	−4.56133	−0.749878	−0.374939	−	0.927049i	$-0.622336\pi$
$37$	−4.56133	−0.749878	−0.374939	+	0.927049i	$0.622336\pi$
$38$	−11.9196	−1.93362
$39$	0	0
$40$	−24.7992	−3.92109
$41$	−10.8389	−1.69276	−0.846378	−	0.532583i	$-0.821222\pi$
$41$	−10.8389	−1.69276	−0.846378	+	0.532583i	$0.821222\pi$
$42$	0	0
$43$	−4.59190	−0.700259	−0.350129	−	0.936701i	$-0.613862\pi$
$43$	−4.59190	−0.700259	−0.350129	+	0.936701i	$0.613862\pi$
$44$	2.76119	0.416265
$45$	0	0
$46$	−19.2695	−2.84113
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−3.72312	−0.531874
$50$	−12.7386	−1.80151
$51$	0	0
$52$	3.00112	0.416180
$53$	−0.0565505	−0.00776781	−0.00388390	−	0.999992i	$-0.501236\pi$
$53$	−0.0565505	−0.00776781	−0.00388390	+	0.999992i	$0.501236\pi$
$54$	0	0
$55$	1.73270	0.233638
$56$	14.3276	1.91461
$57$	0	0
$58$	6.62137	0.869428
$59$	−11.9547	−1.55637	−0.778184	−	0.628036i	$-0.783859\pi$
$59$	−11.9547	−1.55637	−0.778184	+	0.628036i	$0.783859\pi$
$60$	0	0
$61$	8.36297	1.07077	0.535384	−	0.844608i	$-0.320167\pi$
$61$	8.36297	1.07077	0.535384	+	0.844608i	$0.320167\pi$
$62$	25.3901	3.22454
$63$	0	0
$64$	12.7846	1.59807
$65$	1.88326	0.233590
$66$	0	0
$67$	−15.7774	−1.92752	−0.963760	−	0.266772i	$-0.914043\pi$
$67$	−15.7774	−1.92752	−0.963760	+	0.266772i	$0.914043\pi$
$68$	4.99303	0.605494
$69$	0	0
$70$	14.9988	1.79270
$71$	−4.84854	−0.575416	−0.287708	−	0.957718i	$-0.592893\pi$
$71$	−4.84854	−0.575416	−0.287708	+	0.957718i	$0.592893\pi$
$72$	0	0
$73$	3.41250	0.399403	0.199702	−	0.979857i	$-0.436003\pi$
$73$	3.41250	0.399403	0.199702	+	0.979857i	$0.436003\pi$
$74$	12.0621	1.40220
$75$	0	0
$76$	22.5058	2.58159
$77$	−1.00106	−0.114082
$78$	0	0
$79$	3.47942	0.391465	0.195733	−	0.980657i	$-0.437292\pi$
$79$	3.47942	0.391465	0.195733	+	0.980657i	$0.437292\pi$
$80$	34.2911	3.83386
$81$	0	0
$82$	28.6628	3.16528
$83$	6.02539	0.661373	0.330686	−	0.943741i	$-0.392720\pi$
$83$	6.02539	0.661373	0.330686	+	0.943741i	$0.392720\pi$
$84$	0	0
$85$	3.13323	0.339847
$86$	12.1430	1.30941
$87$	0	0
$88$	−4.37700	−0.466589
$89$	−16.2853	−1.72623	−0.863117	−	0.505004i	$-0.831491\pi$
$89$	−16.2853	−1.72623	−0.863117	+	0.505004i	$0.831491\pi$
$90$	0	0
$91$	−1.08805	−0.114059
$92$	36.3833	3.79322
$93$	0	0
$94$	2.64443	0.272753
$95$	14.1228	1.44897
$96$	0	0
$97$	2.05567	0.208722	0.104361	−	0.994539i	$-0.466720\pi$
$97$	2.05567	0.208722	0.104361	+	0.994539i	$0.466720\pi$
$98$	9.84554	0.994550
$99$	0	0
$100$	24.0522	2.40522
$101$	−10.9779	−1.09235	−0.546173	−	0.837672i	$-0.683916\pi$
$101$	−10.9779	−1.09235	−0.546173	+	0.837672i	$0.683916\pi$
$102$	0	0
$103$	−9.96147	−0.981533	−0.490766	−	0.871291i	$-0.663283\pi$
$103$	−9.96147	−0.981533	−0.490766	+	0.871291i	$0.663283\pi$
$104$	−4.75733	−0.466495
$105$	0	0
$106$	0.149544	0.0145250
$107$	−13.6857	−1.32304	−0.661522	−	0.749926i	$-0.730089\pi$
$107$	−13.6857	−1.32304	−0.661522	+	0.749926i	$0.730089\pi$
$108$	0	0
$109$	−0.302525	−0.0289766	−0.0144883	−	0.999895i	$-0.504612\pi$
$109$	−0.302525	−0.0289766	−0.0144883	+	0.999895i	$0.504612\pi$
$110$	−4.58202	−0.436878
$111$	0	0
$112$	−19.8116	−1.87202
$113$	16.2192	1.52577	0.762885	−	0.646534i	$-0.223782\pi$
$113$	16.2192	1.52577	0.762885	+	0.646534i	$0.223782\pi$
$114$	0	0
$115$	22.8312	2.12902
$116$	−12.5020	−1.16078
$117$	0	0
$118$	31.6134	2.91025
$119$	−1.81022	−0.165942
$120$	0	0
$121$	−10.6942	−0.972198
$122$	−22.1153	−2.00223
$123$	0	0
$124$	−47.9398	−4.30512
$125$	−0.572939	−0.0512452
$126$	0	0
$127$	8.81457	0.782166	0.391083	−	0.920355i	$-0.372100\pi$
$127$	8.81457	0.782166	0.391083	+	0.920355i	$0.372100\pi$
$128$	−7.58447	−0.670379
$129$	0	0
$130$	−4.98017	−0.436790
$131$	5.13376	0.448539	0.224269	−	0.974527i	$-0.428000\pi$
$131$	5.13376	0.448539	0.224269	+	0.974527i	$0.428000\pi$
$132$	0	0
$133$	−8.15943	−0.707513
$134$	41.7224	3.60426
$135$	0	0
$136$	−7.91488	−0.678696
$137$	−18.2819	−1.56193	−0.780963	−	0.624578i	$-0.785271\pi$
$137$	−18.2819	−1.56193	−0.780963	+	0.624578i	$0.785271\pi$
$138$	0	0
$139$	12.3582	1.04821	0.524105	−	0.851654i	$-0.324400\pi$
$139$	12.3582	1.04821	0.524105	+	0.851654i	$0.324400\pi$
$140$	−28.3196	−2.39344
$141$	0	0
$142$	12.8217	1.07597
$143$	0.332392	0.0277960
$144$	0	0
$145$	−7.84526	−0.651514
$146$	−9.02414	−0.746843
$147$	0	0
$148$	−22.7749	−1.87208
$149$	−18.1494	−1.48686	−0.743430	−	0.668814i	$-0.766802\pi$
$149$	−18.1494	−1.48686	−0.743430	+	0.668814i	$0.766802\pi$
$150$	0	0
$151$	13.5427	1.10209	0.551046	−	0.834475i	$-0.314229\pi$
$151$	13.5427	1.10209	0.551046	+	0.834475i	$0.314229\pi$
$152$	−35.6758	−2.89369
$153$	0	0
$154$	2.64725	0.213321
$155$	−30.0832	−2.41634
$156$	0	0
$157$	−5.66275	−0.451937	−0.225968	−	0.974135i	$-0.572555\pi$
$157$	−5.66275	−0.451937	−0.225968	+	0.974135i	$0.572555\pi$
$158$	−9.20109	−0.732000
$159$	0	0
$160$	−41.0822	−3.24784
$161$	−13.1907	−1.03957
$162$	0	0
$163$	22.7374	1.78093	0.890467	−	0.455049i	$-0.150378\pi$
$163$	22.7374	1.78093	0.890467	+	0.455049i	$0.150378\pi$
$164$	−54.1191	−4.22599
$165$	0	0
$166$	−15.9338	−1.23670
$167$	−2.77813	−0.214978	−0.107489	−	0.994206i	$-0.534281\pi$
$167$	−2.77813	−0.214978	−0.107489	+	0.994206i	$0.534281\pi$
$168$	0	0
$169$	−12.6387	−0.972210
$170$	−8.28563	−0.635479
$171$	0	0
$172$	−22.9275	−1.74821
$173$	−17.9802	−1.36701	−0.683506	−	0.729945i	$-0.739546\pi$
$173$	−17.9802	−1.36701	−0.683506	+	0.729945i	$0.739546\pi$
$174$	0	0
$175$	−8.72007	−0.659175
$176$	6.05230	0.456209
$177$	0	0
$178$	43.0653	3.22788
$179$	4.10372	0.306726	0.153363	−	0.988170i	$-0.450990\pi$
$179$	4.10372	0.306726	0.153363	+	0.988170i	$0.450990\pi$
$180$	0	0
$181$	10.0366	0.746014	0.373007	−	0.927828i	$-0.378327\pi$
$181$	10.0366	0.746014	0.373007	+	0.927828i	$0.378327\pi$
$182$	2.87728	0.213278
$183$	0	0
$184$	−57.6742	−4.25180
$185$	−14.2917	−1.05075
$186$	0	0
$187$	0.553008	0.0404400
$188$	−4.99303	−0.364154
$189$	0	0
$190$	−37.3469	−2.70943
$191$	14.1847	1.02637	0.513183	−	0.858279i	$-0.328466\pi$
$191$	14.1847	1.02637	0.513183	+	0.858279i	$0.328466\pi$
$192$	0	0
$193$	−11.4410	−0.823542	−0.411771	−	0.911287i	$-0.635090\pi$
$193$	−11.4410	−0.823542	−0.411771	+	0.911287i	$0.635090\pi$
$194$	−5.43609	−0.390288
$195$	0	0
$196$	−18.5897	−1.32783
$197$	21.2895	1.51681	0.758406	−	0.651782i	$-0.225978\pi$
$197$	21.2895	1.51681	0.758406	+	0.651782i	$0.225978\pi$
$198$	0	0
$199$	2.39273	0.169616	0.0848079	−	0.996397i	$-0.472972\pi$
$199$	2.39273	0.169616	0.0848079	+	0.996397i	$0.472972\pi$
$200$	−38.1271	−2.69599
$201$	0	0
$202$	29.0305	2.04258
$203$	4.53258	0.318125
$204$	0	0
$205$	−33.9609	−2.37193
$206$	26.3425	1.83537
$207$	0	0
$208$	6.57821	0.456117
$209$	2.49265	0.172420
$210$	0	0
$211$	9.04792	0.622885	0.311442	−	0.950265i	$-0.399188\pi$
$211$	9.04792	0.622885	0.311442	+	0.950265i	$0.399188\pi$
$212$	−0.282358	−0.0193925
$213$	0	0
$214$	36.1909	2.47396
$215$	−14.3875	−0.981219
$216$	0	0
$217$	17.3805	1.17986
$218$	0.800006	0.0541833
$219$	0	0
$220$	8.65145	0.583280
$221$	0.601061	0.0404318
$222$	0	0
$223$	−21.2470	−1.42281	−0.711403	−	0.702785i	$-0.751940\pi$
$223$	−21.2470	−1.42281	−0.711403	+	0.702785i	$0.751940\pi$
$224$	23.7352	1.58587
$225$	0	0
$226$	−42.8905	−2.85303
$227$	6.39796	0.424648	0.212324	−	0.977199i	$-0.431897\pi$
$227$	6.39796	0.424648	0.212324	+	0.977199i	$0.431897\pi$
$228$	0	0
$229$	−21.7587	−1.43786	−0.718929	−	0.695083i	$-0.755368\pi$
$229$	−21.7587	−1.43786	−0.718929	+	0.695083i	$0.755368\pi$
$230$	−60.3757	−3.98106
$231$	0	0
$232$	19.8180	1.30111
$233$	23.4628	1.53710	0.768548	−	0.639792i	$-0.220980\pi$
$233$	23.4628	1.53710	0.768548	+	0.639792i	$0.220980\pi$
$234$	0	0
$235$	−3.13323	−0.204390
$236$	−59.6902	−3.88550
$237$	0	0
$238$	4.78700	0.310295
$239$	−22.3236	−1.44399	−0.721996	−	0.691897i	$-0.756775\pi$
$239$	−22.3236	−1.44399	−0.721996	+	0.691897i	$0.756775\pi$
$240$	0	0
$241$	13.5938	0.875655	0.437828	−	0.899059i	$-0.355748\pi$
$241$	13.5938	0.875655	0.437828	+	0.899059i	$0.355748\pi$
$242$	28.2801	1.81791
$243$	0	0
$244$	41.7566	2.67319
$245$	−11.6654	−0.745275
$246$	0	0
$247$	2.70925	0.172385
$248$	75.9934	4.82559
$249$	0	0
$250$	1.51510	0.0958232
$251$	3.09155	0.195137	0.0975684	−	0.995229i	$-0.468894\pi$
$251$	3.09155	0.195137	0.0975684	+	0.995229i	$0.468894\pi$
$252$	0	0
$253$	4.02966	0.253343
$254$	−23.3095	−1.46257
$255$	0	0
$256$	−5.51256	−0.344535
$257$	1.48432	0.0925894	0.0462947	−	0.998928i	$-0.485259\pi$
$257$	1.48432	0.0925894	0.0462947	+	0.998928i	$0.485259\pi$
$258$	0	0
$259$	8.25700	0.513065
$260$	9.40320	0.583162
$261$	0	0
$262$	−13.5759	−0.838722
$263$	31.1594	1.92137	0.960685	−	0.277641i	$-0.0895526\pi$
$263$	31.1594	1.92137	0.960685	+	0.277641i	$0.0895526\pi$
$264$	0	0
$265$	−0.177186	−0.0108844
$266$	21.5771	1.32298
$267$	0	0
$268$	−78.7772	−4.81208
$269$	6.20598	0.378386	0.189193	−	0.981940i	$-0.439413\pi$
$269$	6.20598	0.378386	0.189193	+	0.981940i	$0.439413\pi$
$270$	0	0
$271$	−13.8384	−0.840624	−0.420312	−	0.907380i	$-0.638079\pi$
$271$	−13.8384	−0.840624	−0.420312	+	0.907380i	$0.638079\pi$
$272$	10.9443	0.663597
$273$	0	0
$274$	48.3452	2.92064
$275$	2.66392	0.160640
$276$	0	0
$277$	−22.3386	−1.34220	−0.671098	−	0.741369i	$-0.734177\pi$
$277$	−22.3386	−1.34220	−0.671098	+	0.741369i	$0.734177\pi$
$278$	−32.6805	−1.96005
$279$	0	0
$280$	44.8918	2.68280
$281$	1.61186	0.0961555	0.0480777	−	0.998844i	$-0.484690\pi$
$281$	1.61186	0.0961555	0.0480777	+	0.998844i	$0.484690\pi$
$282$	0	0
$283$	1.90278	0.113109	0.0565544	−	0.998400i	$-0.481989\pi$
$283$	1.90278	0.113109	0.0565544	+	0.998400i	$0.481989\pi$
$284$	−24.2090	−1.43654
$285$	0	0
$286$	−0.878988	−0.0519757
$287$	19.6208	1.15818
$288$	0	0
$289$	1.00000	0.0588235
$290$	20.7463	1.21826
$291$	0	0
$292$	17.0387	0.997117
$293$	15.8029	0.923218	0.461609	−	0.887083i	$-0.347272\pi$
$293$	15.8029	0.923218	0.461609	+	0.887083i	$0.347272\pi$
$294$	0	0
$295$	−37.4568	−2.18082
$296$	36.1024	2.09841
$297$	0	0
$298$	47.9950	2.78028
$299$	4.37982	0.253291
$300$	0	0
$301$	8.31233	0.479115
$302$	−35.8129	−2.06080
$303$	0	0
$304$	49.3308	2.82932
$305$	26.2031	1.50039
$306$	0	0
$307$	4.93239	0.281507	0.140753	−	0.990045i	$-0.455048\pi$
$307$	4.93239	0.281507	0.140753	+	0.990045i	$0.455048\pi$
$308$	−4.99835	−0.284807
$309$	0	0
$310$	79.5530	4.51831
$311$	−31.9393	−1.81111	−0.905554	−	0.424230i	$-0.860545\pi$
$311$	−31.9393	−1.81111	−0.905554	+	0.424230i	$0.860545\pi$
$312$	0	0
$313$	6.11717	0.345763	0.172882	−	0.984943i	$-0.444692\pi$
$313$	6.11717	0.345763	0.172882	+	0.984943i	$0.444692\pi$
$314$	14.9748	0.845076
$315$	0	0
$316$	17.3729	0.977299
$317$	−4.30598	−0.241848	−0.120924	−	0.992662i	$-0.538586\pi$
$317$	−4.30598	−0.241848	−0.120924	+	0.992662i	$0.538586\pi$
$318$	0	0
$319$	−1.38467	−0.0775267
$320$	40.0571	2.23926
$321$	0	0
$322$	34.8819	1.94389
$323$	4.50744	0.250800
$324$	0	0
$325$	2.89540	0.160608
$326$	−60.1276	−3.33016
$327$	0	0
$328$	85.7888	4.73690
$329$	1.81022	0.0998004
$330$	0	0
$331$	−3.63561	−0.199831	−0.0999157	−	0.994996i	$-0.531857\pi$
$331$	−3.63561	−0.199831	−0.0999157	+	0.994996i	$0.531857\pi$
$332$	30.0850	1.65113
$333$	0	0
$334$	7.34659	0.401987
$335$	−49.4343	−2.70089
$336$	0	0
$337$	−28.5515	−1.55530	−0.777648	−	0.628700i	$-0.783588\pi$
$337$	−28.5515	−1.55530	−0.777648	+	0.628700i	$0.783588\pi$
$338$	33.4223	1.81793
$339$	0	0
$340$	15.6443	0.848433
$341$	−5.30961	−0.287532
$342$	0	0
$343$	19.4112	1.04810
$344$	36.3444	1.95956
$345$	0	0
$346$	47.5476	2.55617
$347$	11.4415	0.614211	0.307106	−	0.951675i	$-0.400639\pi$
$347$	11.4415	0.614211	0.307106	+	0.951675i	$0.400639\pi$
$348$	0	0
$349$	1.80988	0.0968809	0.0484405	−	0.998826i	$-0.484575\pi$
$349$	1.80988	0.0968809	0.0484405	+	0.998826i	$0.484575\pi$
$350$	23.0596	1.23259
$351$	0	0
$352$	−7.25092	−0.386475
$353$	6.62644	0.352690	0.176345	−	0.984328i	$-0.443573\pi$
$353$	6.62644	0.352690	0.176345	+	0.984328i	$0.443573\pi$
$354$	0	0
$355$	−15.1916	−0.806287
$356$	−81.3129	−4.30957
$357$	0	0
$358$	−10.8520	−0.573547
$359$	−0.741280	−0.0391233	−0.0195616	−	0.999809i	$-0.506227\pi$
$359$	−0.741280	−0.0391233	−0.0195616	+	0.999809i	$0.506227\pi$
$360$	0	0
$361$	1.31698	0.0693146
$362$	−26.5411	−1.39497
$363$	0	0
$364$	−5.43267	−0.284750
$365$	10.6922	0.559653
$366$	0	0
$367$	−7.02173	−0.366531	−0.183266	−	0.983063i	$-0.558667\pi$
$367$	−7.02173	−0.366531	−0.183266	+	0.983063i	$0.558667\pi$
$368$	79.7491	4.15721
$369$	0	0
$370$	37.7935	1.96479
$371$	0.102369	0.00531471
$372$	0	0
$373$	14.0870	0.729398	0.364699	−	0.931125i	$-0.381172\pi$
$373$	14.0870	0.729398	0.364699	+	0.931125i	$0.381172\pi$
$374$	−1.46239	−0.0756186
$375$	0	0
$376$	7.91488	0.408179
$377$	−1.50499	−0.0775110
$378$	0	0
$379$	−8.26406	−0.424496	−0.212248	−	0.977216i	$-0.568079\pi$
$379$	−8.26406	−0.424496	−0.212248	+	0.977216i	$0.568079\pi$
$380$	70.5158	3.61739
$381$	0	0
$382$	−37.5104	−1.91920
$383$	5.36991	0.274389	0.137195	−	0.990544i	$-0.456191\pi$
$383$	5.36991	0.274389	0.137195	+	0.990544i	$0.456191\pi$
$384$	0	0
$385$	−3.13657	−0.159854
$386$	30.2550	1.53994
$387$	0	0
$388$	10.2640	0.521078
$389$	−5.39200	−0.273385	−0.136693	−	0.990614i	$-0.543647\pi$
$389$	−5.39200	−0.273385	−0.136693	+	0.990614i	$0.543647\pi$
$390$	0	0
$391$	7.28680	0.368510
$392$	29.4680	1.48836
$393$	0	0
$394$	−56.2986	−2.83628
$395$	10.9018	0.548530
$396$	0	0
$397$	−14.7214	−0.738848	−0.369424	−	0.929261i	$-0.620445\pi$
$397$	−14.7214	−0.738848	−0.369424	+	0.929261i	$0.620445\pi$
$398$	−6.32741	−0.317164
$399$	0	0
$400$	52.7203	2.63602
$401$	−16.3151	−0.814738	−0.407369	−	0.913264i	$-0.633554\pi$
$401$	−16.3151	−0.814738	−0.407369	+	0.913264i	$0.633554\pi$
$402$	0	0
$403$	−5.77099	−0.287473
$404$	−54.8133	−2.72706
$405$	0	0
$406$	−11.9861	−0.594861
$407$	−2.52245	−0.125033
$408$	0	0
$409$	−28.2570	−1.39722	−0.698609	−	0.715504i	$-0.746197\pi$
$409$	−28.2570	−1.39722	−0.698609	+	0.715504i	$0.746197\pi$
$410$	89.8073	4.43527
$411$	0	0
$412$	−49.7380	−2.45041
$413$	21.6406	1.06486
$414$	0	0
$415$	18.8790	0.926731
$416$	−7.88098	−0.386397
$417$	0	0
$418$	−6.59165	−0.322408
$419$	14.9348	0.729611	0.364805	−	0.931084i	$-0.381136\pi$
$419$	14.9348	0.729611	0.364805	+	0.931084i	$0.381136\pi$
$420$	0	0
$421$	18.1123	0.882738	0.441369	−	0.897326i	$-0.354493\pi$
$421$	18.1123	0.882738	0.441369	+	0.897326i	$0.354493\pi$
$422$	−23.9266	−1.16473
$423$	0	0
$424$	0.447590	0.0217369
$425$	4.81714	0.233666
$426$	0	0
$427$	−15.1388	−0.732617
$428$	−68.3330	−3.30300
$429$	0	0
$430$	38.0468	1.83478
$431$	−19.4432	−0.936547	−0.468273	−	0.883584i	$-0.655124\pi$
$431$	−19.4432	−0.936547	−0.468273	+	0.883584i	$0.655124\pi$
$432$	0	0
$433$	3.50895	0.168629	0.0843147	−	0.996439i	$-0.473130\pi$
$433$	3.50895	0.168629	0.0843147	+	0.996439i	$0.473130\pi$
$434$	−45.9615	−2.20622
$435$	0	0
$436$	−1.51052	−0.0723406
$437$	32.8448	1.57118
$438$	0	0
$439$	−25.6234	−1.22294	−0.611470	−	0.791267i	$-0.709422\pi$
$439$	−25.6234	−1.22294	−0.611470	+	0.791267i	$0.709422\pi$
$440$	−13.7141	−0.653796
$441$	0	0
$442$	−1.58947	−0.0756033
$443$	11.9297	0.566795	0.283398	−	0.959003i	$-0.408538\pi$
$443$	11.9297	0.566795	0.283398	+	0.959003i	$0.408538\pi$
$444$	0	0
$445$	−51.0255	−2.41884
$446$	56.1863	2.66050
$447$	0	0
$448$	−23.1429	−1.09340
$449$	28.2323	1.33236	0.666182	−	0.745789i	$-0.267927\pi$
$449$	28.2323	1.33236	0.666182	+	0.745789i	$0.267927\pi$
$450$	0	0
$451$	−5.99402	−0.282247
$452$	80.9828	3.80911
$453$	0	0
$454$	−16.9190	−0.794048
$455$	−3.40912	−0.159822
$456$	0	0
$457$	36.4110	1.70323	0.851617	−	0.524165i	$-0.175623\pi$
$457$	36.4110	1.70323	0.851617	+	0.524165i	$0.175623\pi$
$458$	57.5396	2.68865
$459$	0	0
$460$	113.997	5.31515
$461$	4.53770	0.211342	0.105671	−	0.994401i	$-0.466301\pi$
$461$	4.53770	0.211342	0.105671	+	0.994401i	$0.466301\pi$
$462$	0	0
$463$	39.3894	1.83058	0.915290	−	0.402795i	$-0.131961\pi$
$463$	39.3894	1.83058	0.915290	+	0.402795i	$0.131961\pi$
$464$	−27.4034	−1.27217
$465$	0	0
$466$	−62.0457	−2.87421
$467$	5.87990	0.272089	0.136045	−	0.990703i	$-0.456561\pi$
$467$	5.87990	0.272089	0.136045	+	0.990703i	$0.456561\pi$
$468$	0	0
$469$	28.5605	1.31880
$470$	8.28563	0.382187
$471$	0	0
$472$	94.6200	4.35524
$473$	−2.53936	−0.116760
$474$	0	0
$475$	21.7130	0.996259
$476$	−9.03847	−0.414278
$477$	0	0
$478$	59.0332	2.70012
$479$	16.0596	0.733780	0.366890	−	0.930264i	$-0.380423\pi$
$479$	16.0596	0.733780	0.366890	+	0.930264i	$0.380423\pi$
$480$	0	0
$481$	−2.74164	−0.125008
$482$	−35.9480	−1.63739
$483$	0	0
$484$	−53.3964	−2.42711
$485$	6.44089	0.292466
$486$	0	0
$487$	−31.7392	−1.43824	−0.719121	−	0.694885i	$-0.755455\pi$
$487$	−31.7392	−1.43824	−0.719121	+	0.694885i	$0.755455\pi$
$488$	−66.1920	−2.99637
$489$	0	0
$490$	30.8484	1.39359
$491$	17.6360	0.795903	0.397952	−	0.917406i	$-0.369721\pi$
$491$	17.6360	0.795903	0.397952	+	0.917406i	$0.369721\pi$
$492$	0	0
$493$	−2.50389	−0.112769
$494$	−7.16442	−0.322343
$495$	0	0
$496$	−105.080	−4.71823
$497$	8.77691	0.393698
$498$	0	0
$499$	−19.2620	−0.862284	−0.431142	−	0.902284i	$-0.641889\pi$
$499$	−19.2620	−0.862284	−0.431142	+	0.902284i	$0.641889\pi$
$500$	−2.86070	−0.127934
$501$	0	0
$502$	−8.17540	−0.364886
$503$	1.16135	0.0517820	0.0258910	−	0.999665i	$-0.491758\pi$
$503$	1.16135	0.0517820	0.0258910	+	0.999665i	$0.491758\pi$
$504$	0	0
$505$	−34.3965	−1.53062
$506$	−10.6562	−0.473725
$507$	0	0
$508$	44.0114	1.95269
$509$	−36.4195	−1.61427	−0.807134	−	0.590368i	$-0.798982\pi$
$509$	−36.4195	−1.61427	−0.807134	+	0.590368i	$0.798982\pi$
$510$	0	0
$511$	−6.17737	−0.273271
$512$	29.7465	1.31462
$513$	0	0
$514$	−3.92519	−0.173133
$515$	−31.2116	−1.37535
$516$	0	0
$517$	−0.553008	−0.0243213
$518$	−21.8351	−0.959379
$519$	0	0
$520$	−14.9058	−0.653663
$521$	17.6202	0.771955	0.385977	−	0.922508i	$-0.373864\pi$
$521$	17.6202	0.771955	0.385977	+	0.922508i	$0.373864\pi$
$522$	0	0
$523$	24.0011	1.04950	0.524748	−	0.851258i	$-0.324160\pi$
$523$	24.0011	1.04950	0.524748	+	0.851258i	$0.324160\pi$
$524$	25.6330	1.11978
$525$	0	0
$526$	−82.3989	−3.59276
$527$	−9.60133	−0.418240
$528$	0	0
$529$	30.0975	1.30859
$530$	0.468556	0.0203528
$531$	0	0
$532$	−40.7403	−1.76632
$533$	−6.51486	−0.282190
$534$	0	0
$535$	−42.8804	−1.85388
$536$	124.876	5.39384
$537$	0	0
$538$	−16.4113	−0.707542
$539$	−2.05891	−0.0886837
$540$	0	0
$541$	29.8116	1.28170	0.640850	−	0.767666i	$-0.278582\pi$
$541$	29.8116	1.28170	0.640850	+	0.767666i	$0.278582\pi$
$542$	36.5948	1.57188
$543$	0	0
$544$	−13.1118	−0.562163
$545$	−0.947880	−0.0406027
$546$	0	0
$547$	−2.89547	−0.123801	−0.0619006	−	0.998082i	$-0.519716\pi$
$547$	−2.89547	−0.123801	−0.0619006	+	0.998082i	$0.519716\pi$
$548$	−91.2820	−3.89937
$549$	0	0
$550$	−7.04456	−0.300381
$551$	−11.2861	−0.480805
$552$	0	0
$553$	−6.29850	−0.267839
$554$	59.0729	2.50977
$555$	0	0
$556$	61.7050	2.61687
$557$	−28.2886	−1.19863	−0.599314	−	0.800514i	$-0.704560\pi$
$557$	−28.2886	−1.19863	−0.599314	+	0.800514i	$0.704560\pi$
$558$	0	0
$559$	−2.76001	−0.116736
$560$	−62.0743	−2.62312
$561$	0	0
$562$	−4.26246	−0.179801
$563$	−12.9456	−0.545592	−0.272796	−	0.962072i	$-0.587948\pi$
$563$	−12.9456	−0.545592	−0.272796	+	0.962072i	$0.587948\pi$
$564$	0	0
$565$	50.8184	2.13795
$566$	−5.03179	−0.211502
$567$	0	0
$568$	38.3757	1.61021
$569$	−32.7516	−1.37302	−0.686509	−	0.727121i	$-0.740858\pi$
$569$	−32.7516	−1.37302	−0.686509	+	0.727121i	$0.740858\pi$
$570$	0	0
$571$	−3.80076	−0.159057	−0.0795284	−	0.996833i	$-0.525341\pi$
$571$	−3.80076	−0.159057	−0.0795284	+	0.996833i	$0.525341\pi$
$572$	1.65964	0.0693932
$573$	0	0
$574$	−51.8859	−2.16568
$575$	35.1016	1.46384
$576$	0	0
$577$	35.8041	1.49054	0.745272	−	0.666761i	$-0.232320\pi$
$577$	35.8041	1.49054	0.745272	+	0.666761i	$0.232320\pi$
$578$	−2.64443	−0.109994
$579$	0	0
$580$	−39.1717	−1.62652
$581$	−10.9073	−0.452510
$582$	0	0
$583$	−0.0312729	−0.00129519
$584$	−27.0096	−1.11766
$585$	0	0
$586$	−41.7899	−1.72632
$587$	11.7990	0.486997	0.243498	−	0.969901i	$-0.421705\pi$
$587$	11.7990	0.486997	0.243498	+	0.969901i	$0.421705\pi$
$588$	0	0
$589$	−43.2774	−1.78321
$590$	99.0522	4.07791
$591$	0	0
$592$	−49.9207	−2.05173
$593$	−25.3555	−1.04123	−0.520613	−	0.853793i	$-0.674296\pi$
$593$	−25.3555	−1.04123	−0.520613	+	0.853793i	$0.674296\pi$
$594$	0	0
$595$	−5.67183	−0.232522
$596$	−90.6207	−3.71197
$597$	0	0
$598$	−11.5821	−0.473629
$599$	−6.12955	−0.250446	−0.125223	−	0.992129i	$-0.539965\pi$
$599$	−6.12955	−0.250446	−0.125223	+	0.992129i	$0.539965\pi$
$600$	0	0
$601$	30.6272	1.24931	0.624654	−	0.780902i	$-0.285240\pi$
$601$	30.6272	1.24931	0.624654	+	0.780902i	$0.285240\pi$
$602$	−21.9814	−0.895896
$603$	0	0
$604$	67.6193	2.75139
$605$	−33.5074	−1.36227
$606$	0	0
$607$	−3.16682	−0.128537	−0.0642686	−	0.997933i	$-0.520471\pi$
$607$	−3.16682	−0.128537	−0.0642686	+	0.997933i	$0.520471\pi$
$608$	−59.1005	−2.39684
$609$	0	0
$610$	−69.2925	−2.80557
$611$	−0.601061	−0.0243163
$612$	0	0
$613$	−13.1859	−0.532574	−0.266287	−	0.963894i	$-0.585797\pi$
$613$	−13.1859	−0.532574	−0.266287	+	0.963894i	$0.585797\pi$
$614$	−13.0434	−0.526388
$615$	0	0
$616$	7.92331	0.319239
$617$	−17.0538	−0.686560	−0.343280	−	0.939233i	$-0.611538\pi$
$617$	−17.0538	−0.686560	−0.343280	+	0.939233i	$0.611538\pi$
$618$	0	0
$619$	−26.9188	−1.08196	−0.540979	−	0.841036i	$-0.681946\pi$
$619$	−26.9188	−1.08196	−0.540979	+	0.841036i	$0.681946\pi$
$620$	−150.206	−6.03243
$621$	0	0
$622$	84.4613	3.38659
$623$	29.4798	1.18108
$624$	0	0
$625$	−25.8809	−1.03523
$626$	−16.1765	−0.646542
$627$	0	0
$628$	−28.2743	−1.12827
$629$	−4.56133	−0.181872
$630$	0	0
$631$	12.3723	0.492534	0.246267	−	0.969202i	$-0.420796\pi$
$631$	12.3723	0.492534	0.246267	+	0.969202i	$0.420796\pi$
$632$	−27.5392	−1.09545
$633$	0	0
$634$	11.3869	0.452231
$635$	27.6181	1.09599
$636$	0	0
$637$	−2.23782	−0.0886657
$638$	3.66167	0.144967
$639$	0	0
$640$	−23.7639	−0.939351
$641$	6.86240	0.271049	0.135524	−	0.990774i	$-0.456728\pi$
$641$	6.86240	0.271049	0.135524	+	0.990774i	$0.456728\pi$
$642$	0	0
$643$	22.8976	0.902995	0.451498	−	0.892272i	$-0.350890\pi$
$643$	22.8976	0.902995	0.451498	+	0.892272i	$0.350890\pi$
$644$	−65.8616	−2.59531
$645$	0	0
$646$	−11.9196	−0.468971
$647$	49.5515	1.94807	0.974035	−	0.226397i	$-0.0726946\pi$
$647$	49.5515	1.94807	0.974035	+	0.226397i	$0.0726946\pi$
$648$	0	0
$649$	−6.61105	−0.259506
$650$	−7.65669	−0.300320
$651$	0	0
$652$	113.529	4.44613
$653$	−44.0053	−1.72206	−0.861029	−	0.508555i	$-0.830180\pi$
$653$	−44.0053	−1.72206	−0.861029	+	0.508555i	$0.830180\pi$
$654$	0	0
$655$	16.0853	0.628503
$656$	−118.625	−4.63152
$657$	0	0
$658$	−4.78700	−0.186617
$659$	39.6406	1.54418	0.772089	−	0.635515i	$-0.219212\pi$
$659$	39.6406	1.54418	0.772089	+	0.635515i	$0.219212\pi$
$660$	0	0
$661$	−8.97888	−0.349238	−0.174619	−	0.984636i	$-0.555869\pi$
$661$	−8.97888	−0.349238	−0.174619	+	0.984636i	$0.555869\pi$
$662$	9.61415	0.373664
$663$	0	0
$664$	−47.6903	−1.85074
$665$	−25.5654	−0.991383
$666$	0	0
$667$	−18.2453	−0.706463
$668$	−13.8713	−0.536697
$669$	0	0
$670$	130.726	5.05038
$671$	4.62479	0.178538
$672$	0	0
$673$	11.7692	0.453669	0.226834	−	0.973933i	$-0.427162\pi$
$673$	11.7692	0.453669	0.226834	+	0.973933i	$0.427162\pi$
$674$	75.5024	2.90825
$675$	0	0
$676$	−63.1056	−2.42714
$677$	−22.1849	−0.852634	−0.426317	−	0.904574i	$-0.640189\pi$
$677$	−22.1849	−0.852634	−0.426317	+	0.904574i	$0.640189\pi$
$678$	0	0
$679$	−3.72121	−0.142807
$680$	−24.7992	−0.951005
$681$	0	0
$682$	14.0409	0.537655
$683$	1.17426	0.0449318	0.0224659	−	0.999748i	$-0.492848\pi$
$683$	1.17426	0.0449318	0.0224659	+	0.999748i	$0.492848\pi$
$684$	0	0
$685$	−57.2813	−2.18861
$686$	−51.3315	−1.95985
$687$	0	0
$688$	−50.2553	−1.91596
$689$	−0.0339903	−0.00129493
$690$	0	0
$691$	−37.1299	−1.41249	−0.706244	−	0.707969i	$-0.749612\pi$
$691$	−37.1299	−1.41249	−0.706244	+	0.707969i	$0.749612\pi$
$692$	−89.7759	−3.41277
$693$	0	0
$694$	−30.2563	−1.14851
$695$	38.7212	1.46878
$696$	0	0
$697$	−10.8389	−0.410554
$698$	−4.78612	−0.181157
$699$	0	0
$700$	−43.5396	−1.64564
$701$	−27.3371	−1.03251	−0.516255	−	0.856435i	$-0.672674\pi$
$701$	−27.3371	−1.03251	−0.516255	+	0.856435i	$0.672674\pi$
$702$	0	0
$703$	−20.5599	−0.775432
$704$	7.06999	0.266460
$705$	0	0
$706$	−17.5232	−0.659494
$707$	19.8725	0.747381
$708$	0	0
$709$	−13.0898	−0.491597	−0.245799	−	0.969321i	$-0.579050\pi$
$709$	−13.0898	−0.491597	−0.245799	+	0.969321i	$0.579050\pi$
$710$	40.1732	1.50767
$711$	0	0
$712$	128.896	4.83058
$713$	−69.9630	−2.62014
$714$	0	0
$715$	1.04146	0.0389484
$716$	20.4900	0.765748
$717$	0	0
$718$	1.96027	0.0731565
$719$	−2.48890	−0.0928204	−0.0464102	−	0.998922i	$-0.514778\pi$
$719$	−2.48890	−0.0928204	−0.0464102	+	0.998922i	$0.514778\pi$
$720$	0	0
$721$	18.0324	0.671562
$722$	−3.48266	−0.129611
$723$	0	0
$724$	50.1131	1.86244
$725$	−12.0616	−0.447956
$726$	0	0
$727$	10.9052	0.404452	0.202226	−	0.979339i	$-0.435182\pi$
$727$	10.9052	0.404452	0.202226	+	0.979339i	$0.435182\pi$
$728$	8.61180	0.319174
$729$	0	0
$730$	−28.2747	−1.04649
$731$	−4.59190	−0.169838
$732$	0	0
$733$	8.86060	0.327274	0.163637	−	0.986521i	$-0.447677\pi$
$733$	8.86060	0.327274	0.163637	+	0.986521i	$0.447677\pi$
$734$	18.5685	0.685376
$735$	0	0
$736$	−95.5430	−3.52176
$737$	−8.72505	−0.321391
$738$	0	0
$739$	−45.8965	−1.68833	−0.844165	−	0.536084i	$-0.819903\pi$
$739$	−45.8965	−1.68833	−0.844165	+	0.536084i	$0.819903\pi$
$740$	−71.3590	−2.62321
$741$	0	0
$742$	−0.270707	−0.00993797
$743$	26.0962	0.957377	0.478688	−	0.877985i	$-0.341112\pi$
$743$	26.0962	0.957377	0.478688	+	0.877985i	$0.341112\pi$
$744$	0	0
$745$	−56.8664	−2.08342
$746$	−37.2522	−1.36390
$747$	0	0
$748$	2.76119	0.100959
$749$	24.7740	0.905223
$750$	0	0
$751$	29.9043	1.09122	0.545611	−	0.838038i	$-0.316298\pi$
$751$	29.9043	1.09122	0.545611	+	0.838038i	$0.316298\pi$
$752$	−10.9443	−0.399098
$753$	0	0
$754$	3.97985	0.144938
$755$	42.4325	1.54428
$756$	0	0
$757$	−12.5101	−0.454687	−0.227344	−	0.973815i	$-0.573004\pi$
$757$	−12.5101	−0.454687	−0.227344	+	0.973815i	$0.573004\pi$
$758$	21.8538	0.793765
$759$	0	0
$760$	−111.781	−4.05471
$761$	−11.5782	−0.419708	−0.209854	−	0.977733i	$-0.567299\pi$
$761$	−11.5782	−0.419708	−0.209854	+	0.977733i	$0.567299\pi$
$762$	0	0
$763$	0.547635	0.0198257
$764$	70.8246	2.56234
$765$	0	0
$766$	−14.2004	−0.513080
$767$	−7.18551	−0.259454
$768$	0	0
$769$	4.10679	0.148095	0.0740473	−	0.997255i	$-0.476408\pi$
$769$	4.10679	0.148095	0.0740473	+	0.997255i	$0.476408\pi$
$770$	8.29445	0.298911
$771$	0	0
$772$	−57.1253	−2.05599
$773$	2.29506	0.0825477	0.0412738	−	0.999148i	$-0.486858\pi$
$773$	2.29506	0.0825477	0.0412738	+	0.999148i	$0.486858\pi$
$774$	0	0
$775$	−46.2510	−1.66138
$776$	−16.2704	−0.584073
$777$	0	0
$778$	14.2588	0.511202
$779$	−48.8558	−1.75044
$780$	0	0
$781$	−2.68129	−0.0959439
$782$	−19.2695	−0.689075
$783$	0	0
$784$	−40.7470	−1.45525
$785$	−17.7427	−0.633265
$786$	0	0
$787$	−14.0524	−0.500913	−0.250457	−	0.968128i	$-0.580581\pi$
$787$	−14.0524	−0.500913	−0.250457	+	0.968128i	$0.580581\pi$
$788$	106.299	3.78675
$789$	0	0
$790$	−28.8292	−1.02570
$791$	−29.3602	−1.04393
$792$	0	0
$793$	5.02666	0.178502
$794$	38.9299	1.38157
$795$	0	0
$796$	11.9470	0.423449
$797$	34.1633	1.21013	0.605063	−	0.796178i	$-0.293148\pi$
$797$	34.1633	1.21013	0.605063	+	0.796178i	$0.293148\pi$
$798$	0	0
$799$	−1.00000	−0.0353775
$800$	−63.1613	−2.23309
$801$	0	0
$802$	43.1443	1.52348
$803$	1.88714	0.0665958
$804$	0	0
$805$	−41.3295	−1.45667
$806$	15.2610	0.537546
$807$	0	0
$808$	86.8892	3.05675
$809$	3.92390	0.137957	0.0689785	−	0.997618i	$-0.478026\pi$
$809$	3.92390	0.137957	0.0689785	+	0.997618i	$0.478026\pi$
$810$	0	0
$811$	9.16119	0.321693	0.160847	−	0.986979i	$-0.448578\pi$
$811$	9.16119	0.321693	0.160847	+	0.986979i	$0.448578\pi$
$812$	22.6313	0.794204
$813$	0	0
$814$	6.67047	0.233800
$815$	71.2416	2.49549
$816$	0	0
$817$	−20.6977	−0.724121
$818$	74.7237	2.61265
$819$	0	0
$820$	−169.568	−5.92156
$821$	34.7544	1.21294	0.606468	−	0.795108i	$-0.292586\pi$
$821$	34.7544	1.21294	0.606468	+	0.795108i	$0.292586\pi$
$822$	0	0
$823$	−38.2623	−1.33374	−0.666870	−	0.745174i	$-0.732366\pi$
$823$	−38.2623	−1.33374	−0.666870	+	0.745174i	$0.732366\pi$
$824$	78.8439	2.74666
$825$	0	0
$826$	−57.2271	−1.99119
$827$	−35.6867	−1.24095	−0.620474	−	0.784227i	$-0.713060\pi$
$827$	−35.6867	−1.24095	−0.620474	+	0.784227i	$0.713060\pi$
$828$	0	0
$829$	−32.7275	−1.13667	−0.568336	−	0.822797i	$-0.692413\pi$
$829$	−32.7275	−1.13667	−0.568336	+	0.822797i	$0.692413\pi$
$830$	−49.9242	−1.73289
$831$	0	0
$832$	7.68433	0.266406
$833$	−3.72312	−0.128998
$834$	0	0
$835$	−8.70453	−0.301233
$836$	12.4459	0.430450
$837$	0	0
$838$	−39.4940	−1.36430
$839$	−8.66003	−0.298977	−0.149489	−	0.988763i	$-0.547763\pi$
$839$	−8.66003	−0.298977	−0.149489	+	0.988763i	$0.547763\pi$
$840$	0	0
$841$	−22.7305	−0.783812
$842$	−47.8967	−1.65063
$843$	0	0
$844$	45.1766	1.55504
$845$	−39.6001	−1.36228
$846$	0	0
$847$	19.3588	0.665176
$848$	−0.618907	−0.0212533
$849$	0	0
$850$	−12.7386	−0.436931
$851$	−33.2375	−1.13937
$852$	0	0
$853$	47.6875	1.63279	0.816394	−	0.577495i	$-0.195970\pi$
$853$	47.6875	1.63279	0.816394	+	0.577495i	$0.195970\pi$
$854$	40.0335	1.36992
$855$	0	0
$856$	108.320	3.70232
$857$	16.0804	0.549296	0.274648	−	0.961545i	$-0.411439\pi$
$857$	16.0804	0.549296	0.274648	+	0.961545i	$0.411439\pi$
$858$	0	0
$859$	−48.2924	−1.64772	−0.823858	−	0.566796i	$-0.808183\pi$
$859$	−48.2924	−1.64772	−0.823858	+	0.566796i	$0.808183\pi$
$860$	−71.8373	−2.44963
$861$	0	0
$862$	51.4163	1.75125
$863$	−26.8559	−0.914185	−0.457092	−	0.889419i	$-0.651109\pi$
$863$	−26.8559	−0.914185	−0.457092	+	0.889419i	$0.651109\pi$
$864$	0	0
$865$	−56.3363	−1.91549
$866$	−9.27919	−0.315320
$867$	0	0
$868$	86.7813	2.94555
$869$	1.92415	0.0652722
$870$	0	0
$871$	−9.48320	−0.321326
$872$	2.39445	0.0810862
$873$	0	0
$874$	−86.8559	−2.93795
$875$	1.03714	0.0350618
$876$	0	0
$877$	14.0899	0.475783	0.237892	−	0.971292i	$-0.423544\pi$
$877$	14.0899	0.475783	0.237892	+	0.971292i	$0.423544\pi$
$878$	67.7595	2.28677
$879$	0	0
$880$	18.9633	0.639251
$881$	−19.8027	−0.667171	−0.333585	−	0.942720i	$-0.608259\pi$
$881$	−19.8027	−0.667171	−0.333585	+	0.942720i	$0.608259\pi$
$882$	0	0
$883$	27.7650	0.934365	0.467183	−	0.884161i	$-0.345269\pi$
$883$	27.7650	0.934365	0.467183	+	0.884161i	$0.345269\pi$
$884$	3.00112	0.100939
$885$	0	0
$886$	−31.5472	−1.05985
$887$	−48.2129	−1.61883	−0.809416	−	0.587236i	$-0.800216\pi$
$887$	−48.2129	−1.61883	−0.809416	+	0.587236i	$0.800216\pi$
$888$	0	0
$889$	−15.9563	−0.535156
$890$	134.934	4.52298
$891$	0	0
$892$	−106.087	−3.55206
$893$	−4.50744	−0.150836
$894$	0	0
$895$	12.8579	0.429792
$896$	13.7295	0.458671
$897$	0	0
$898$	−74.6585	−2.49138
$899$	24.0407	0.801801
$900$	0	0
$901$	−0.0565505	−0.00188397
$902$	15.8508	0.527773
$903$	0	0
$904$	−128.373	−4.26961
$905$	31.4470	1.04533
$906$	0	0
$907$	39.2276	1.30253	0.651266	−	0.758850i	$-0.274238\pi$
$907$	39.2276	1.30253	0.651266	+	0.758850i	$0.274238\pi$
$908$	31.9453	1.06014
$909$	0	0
$910$	9.01518	0.298850
$911$	34.7576	1.15157	0.575785	−	0.817601i	$-0.304697\pi$
$911$	34.7576	1.15157	0.575785	+	0.817601i	$0.304697\pi$
$912$	0	0
$913$	3.33209	0.110276
$914$	−96.2864	−3.18487
$915$	0	0
$916$	−108.642	−3.58964
$917$	−9.29322	−0.306889
$918$	0	0
$919$	−14.9943	−0.494617	−0.247308	−	0.968937i	$-0.579546\pi$
$919$	−14.9943	−0.494617	−0.247308	+	0.968937i	$0.579546\pi$
$920$	−180.707	−5.95772
$921$	0	0
$922$	−11.9997	−0.395188
$923$	−2.91427	−0.0959245
$924$	0	0
$925$	−21.9726	−0.722454
$926$	−104.163	−3.42300
$927$	0	0
$928$	32.8304	1.07771
$929$	−13.0823	−0.429217	−0.214608	−	0.976700i	$-0.568847\pi$
$929$	−13.0823	−0.429217	−0.214608	+	0.976700i	$0.568847\pi$
$930$	0	0
$931$	−16.7817	−0.549999
$932$	117.150	3.83739
$933$	0	0
$934$	−15.5490	−0.508779
$935$	1.73270	0.0566654
$936$	0	0
$937$	−34.2302	−1.11825	−0.559126	−	0.829083i	$-0.688863\pi$
$937$	−34.2302	−1.11825	−0.559126	+	0.829083i	$0.688863\pi$
$938$	−75.5265	−2.46603
$939$	0	0
$940$	−15.6443	−0.510262
$941$	22.2717	0.726037	0.363018	−	0.931782i	$-0.381746\pi$
$941$	22.2717	0.726037	0.363018	+	0.931782i	$0.381746\pi$
$942$	0	0
$943$	−78.9811	−2.57198
$944$	−130.836	−4.25835
$945$	0	0
$946$	6.71517	0.218329
$947$	−35.0927	−1.14036	−0.570180	−	0.821520i	$-0.693127\pi$
$947$	−35.0927	−1.14036	−0.570180	+	0.821520i	$0.693127\pi$
$948$	0	0
$949$	2.05112	0.0665823
$950$	−57.4185	−1.86290
$951$	0	0
$952$	14.3276	0.464362
$953$	−33.2502	−1.07708	−0.538540	−	0.842600i	$-0.681024\pi$
$953$	−33.2502	−1.07708	−0.538540	+	0.842600i	$0.681024\pi$
$954$	0	0
$955$	44.4439	1.43817
$956$	−111.462	−3.60495
$957$	0	0
$958$	−42.4684	−1.37209
$959$	33.0941	1.06867
$960$	0	0
$961$	61.1855	1.97373
$962$	7.25009	0.233752
$963$	0	0
$964$	67.8744	2.18609
$965$	−35.8473	−1.15397
$966$	0	0
$967$	50.2009	1.61435	0.807175	−	0.590312i	$-0.200995\pi$
$967$	50.2009	1.61435	0.807175	+	0.590312i	$0.200995\pi$
$968$	84.6432	2.72054
$969$	0	0
$970$	−17.0325	−0.546881
$971$	31.6070	1.01432	0.507158	−	0.861853i	$-0.330696\pi$
$971$	31.6070	1.01432	0.507158	+	0.861853i	$0.330696\pi$
$972$	0	0
$973$	−22.3710	−0.717183
$974$	83.9323	2.68936
$975$	0	0
$976$	91.5271	2.92971
$977$	−21.8888	−0.700285	−0.350142	−	0.936696i	$-0.613867\pi$
$977$	−21.8888	−0.700285	−0.350142	+	0.936696i	$0.613867\pi$
$978$	0	0
$979$	−9.00588	−0.287829
$980$	−58.2457	−1.86059
$981$	0	0
$982$	−46.6373	−1.48826
$983$	−3.43164	−0.109452	−0.0547262	−	0.998501i	$-0.517429\pi$
$983$	−3.43164	−0.109452	−0.0547262	+	0.998501i	$0.517429\pi$
$984$	0	0
$985$	66.7049	2.12539
$986$	6.62137	0.210867
$987$	0	0
$988$	13.5274	0.430362
$989$	−33.4603	−1.06398
$990$	0	0
$991$	32.7586	1.04061	0.520305	−	0.853981i	$-0.325818\pi$
$991$	32.7586	1.04061	0.520305	+	0.853981i	$0.325818\pi$
$992$	125.891	3.99703
$993$	0	0
$994$	−23.2100	−0.736176
$995$	7.49696	0.237670
$996$	0	0
$997$	−31.7979	−1.00705	−0.503524	−	0.863981i	$-0.667964\pi$
$997$	−31.7979	−1.00705	−0.503524	+	0.863981i	$0.667964\pi$
$998$	50.9370	1.61238
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7191.2.a.bb.1.2		20
3.2	odd	2		799.2.a.g.1.19	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
799.2.a.g.1.19	✓	20	3.2	odd	2
7191.2.a.bb.1.2		20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7191 = 3^{2} \cdot 17 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7191.a (trivial)

\(f(q)\)	\(=\)	\(q-2.64443 q^{2} +4.99303 q^{4} +3.13323 q^{5} -1.81022 q^{7} -7.91488 q^{8} +O(q^{10})\)	\(q-2.64443 q^{2} +4.99303 q^{4} +3.13323 q^{5} -1.81022 q^{7} -7.91488 q^{8} -8.28563 q^{10} +0.553008 q^{11} +0.601061 q^{13} +4.78700 q^{14} +10.9443 q^{16} +1.00000 q^{17} +4.50744 q^{19} +15.6443 q^{20} -1.46239 q^{22} +7.28680 q^{23} +4.81714 q^{25} -1.58947 q^{26} -9.03847 q^{28} -2.50389 q^{29} -9.60133 q^{31} -13.1118 q^{32} -2.64443 q^{34} -5.67183 q^{35} -4.56133 q^{37} -11.9196 q^{38} -24.7992 q^{40} -10.8389 q^{41} -4.59190 q^{43} +2.76119 q^{44} -19.2695 q^{46} -1.00000 q^{47} -3.72312 q^{49} -12.7386 q^{50} +3.00112 q^{52} -0.0565505 q^{53} +1.73270 q^{55} +14.3276 q^{56} +6.62137 q^{58} -11.9547 q^{59} +8.36297 q^{61} +25.3901 q^{62} +12.7846 q^{64} +1.88326 q^{65} -15.7774 q^{67} +4.99303 q^{68} +14.9988 q^{70} -4.84854 q^{71} +3.41250 q^{73} +12.0621 q^{74} +22.5058 q^{76} -1.00106 q^{77} +3.47942 q^{79} +34.2911 q^{80} +28.6628 q^{82} +6.02539 q^{83} +3.13323 q^{85} +12.1430 q^{86} -4.37700 q^{88} -16.2853 q^{89} -1.08805 q^{91} +36.3833 q^{92} +2.64443 q^{94} +14.1228 q^{95} +2.05567 q^{97} +9.84554 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{2} + 24 q^{4} - 13 q^{5} - 3 q^{7} - 12 q^{8}+O(q^{10}) \) 20 * q - 4 * q^2 + 24 * q^4 - 13 * q^5 - 3 * q^7 - 12 * q^8	\( 20 q - 4 q^{2} + 24 q^{4} - 13 q^{5} - 3 q^{7} - 12 q^{8} - 2 q^{10} - 10 q^{11} + q^{13} - 13 q^{14} + 28 q^{16} + 20 q^{17} + 5 q^{19} - 37 q^{20} - 19 q^{22} - 4 q^{23} + 41 q^{25} + 9 q^{26} - 25 q^{28} - 26 q^{29} + 6 q^{31} - 28 q^{32} - 4 q^{34} - 21 q^{35} - 8 q^{37} + 21 q^{38} - 25 q^{40} - 69 q^{41} - 7 q^{43} - 16 q^{44} - 24 q^{46} - 20 q^{47} + 53 q^{49} - 16 q^{50} + 18 q^{52} - 29 q^{53} + 5 q^{55} + 22 q^{56} - q^{58} - 55 q^{59} - 17 q^{61} + 7 q^{62} + 58 q^{64} - 40 q^{65} - 6 q^{67} + 24 q^{68} - 15 q^{70} - 47 q^{71} - 32 q^{73} + 67 q^{74} - 5 q^{76} - 4 q^{77} - 26 q^{79} - 108 q^{80} + 25 q^{82} - 3 q^{83} - 13 q^{85} - 8 q^{86} - 47 q^{88} - 119 q^{89} - 35 q^{91} + 4 q^{94} + 48 q^{95} - 13 q^{97} - q^{98}+O(q^{100}) \) 20 * q - 4 * q^2 + 24 * q^4 - 13 * q^5 - 3 * q^7 - 12 * q^8 - 2 * q^10 - 10 * q^11 + q^13 - 13 * q^14 + 28 * q^16 + 20 * q^17 + 5 * q^19 - 37 * q^20 - 19 * q^22 - 4 * q^23 + 41 * q^25 + 9 * q^26 - 25 * q^28 - 26 * q^29 + 6 * q^31 - 28 * q^32 - 4 * q^34 - 21 * q^35 - 8 * q^37 + 21 * q^38 - 25 * q^40 - 69 * q^41 - 7 * q^43 - 16 * q^44 - 24 * q^46 - 20 * q^47 + 53 * q^49 - 16 * q^50 + 18 * q^52 - 29 * q^53 + 5 * q^55 + 22 * q^56 - q^58 - 55 * q^59 - 17 * q^61 + 7 * q^62 + 58 * q^64 - 40 * q^65 - 6 * q^67 + 24 * q^68 - 15 * q^70 - 47 * q^71 - 32 * q^73 + 67 * q^74 - 5 * q^76 - 4 * q^77 - 26 * q^79 - 108 * q^80 + 25 * q^82 - 3 * q^83 - 13 * q^85 - 8 * q^86 - 47 * q^88 - 119 * q^89 - 35 * q^91 + 4 * q^94 + 48 * q^95 - 13 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more