LMFDB - Embedded newform 8024.2.a.y.1.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.21
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.49483 q^{3} +1.95871 q^{5} -4.15658 q^{7} +3.22420 q^{9} +O(q^{10})$	$q+2.49483 q^{3} +1.95871 q^{5} -4.15658 q^{7} +3.22420 q^{9} +5.07386 q^{11} -3.39523 q^{13} +4.88666 q^{15} -1.00000 q^{17} -7.96131 q^{19} -10.3700 q^{21} -1.52248 q^{23} -1.16344 q^{25} +0.559335 q^{27} +2.94482 q^{29} -9.57103 q^{31} +12.6584 q^{33} -8.14155 q^{35} -3.43470 q^{37} -8.47053 q^{39} +1.52328 q^{41} -0.867593 q^{43} +6.31528 q^{45} -1.96425 q^{47} +10.2772 q^{49} -2.49483 q^{51} +3.60603 q^{53} +9.93824 q^{55} -19.8622 q^{57} -1.00000 q^{59} -2.74860 q^{61} -13.4016 q^{63} -6.65028 q^{65} +16.0573 q^{67} -3.79833 q^{69} +1.96161 q^{71} -10.9901 q^{73} -2.90260 q^{75} -21.0899 q^{77} -9.09546 q^{79} -8.27714 q^{81} -17.0316 q^{83} -1.95871 q^{85} +7.34684 q^{87} +2.26374 q^{89} +14.1125 q^{91} -23.8781 q^{93} -15.5939 q^{95} +4.97767 q^{97} +16.3591 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * 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 $

$2$	0	0
$2$	0	0
$3$	2.49483	1.44039	0.720197	−	0.693770i	$-0.244052\pi$
$3$	2.49483	1.44039	0.720197	+	0.693770i	$0.244052\pi$
$4$	0	0
$5$	1.95871	0.875963	0.437981	−	0.898984i	$-0.355694\pi$
$5$	1.95871	0.875963	0.437981	+	0.898984i	$0.355694\pi$
$6$	0	0
$7$	−4.15658	−1.57104	−0.785520	−	0.618836i	$-0.787605\pi$
$7$	−4.15658	−1.57104	−0.785520	+	0.618836i	$0.787605\pi$
$8$	0	0
$9$	3.22420	1.07473
$10$	0	0
$11$	5.07386	1.52983	0.764914	−	0.644133i	$-0.222782\pi$
$11$	5.07386	1.52983	0.764914	+	0.644133i	$0.222782\pi$
$12$	0	0
$13$	−3.39523	−0.941667	−0.470833	−	0.882222i	$-0.656047\pi$
$13$	−3.39523	−0.941667	−0.470833	+	0.882222i	$0.656047\pi$
$14$	0	0
$15$	4.88666	1.26173
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−7.96131	−1.82645	−0.913225	−	0.407455i	$-0.866416\pi$
$19$	−7.96131	−1.82645	−0.913225	+	0.407455i	$0.866416\pi$
$20$	0	0
$21$	−10.3700	−2.26292
$22$	0	0
$23$	−1.52248	−0.317459	−0.158729	−	0.987322i	$-0.550740\pi$
$23$	−1.52248	−0.317459	−0.158729	+	0.987322i	$0.550740\pi$
$24$	0	0
$25$	−1.16344	−0.232689
$26$	0	0
$27$	0.559335	0.107644
$28$	0	0
$29$	2.94482	0.546840	0.273420	−	0.961895i	$-0.411845\pi$
$29$	2.94482	0.546840	0.273420	+	0.961895i	$0.411845\pi$
$30$	0	0
$31$	−9.57103	−1.71901	−0.859504	−	0.511129i	$-0.829228\pi$
$31$	−9.57103	−1.71901	−0.859504	+	0.511129i	$0.829228\pi$
$32$	0	0
$33$	12.6584	2.20355
$34$	0	0
$35$	−8.14155	−1.37617
$36$	0	0
$37$	−3.43470	−0.564661	−0.282330	−	0.959317i	$-0.591107\pi$
$37$	−3.43470	−0.564661	−0.282330	+	0.959317i	$0.591107\pi$
$38$	0	0
$39$	−8.47053	−1.35637
$40$	0	0
$41$	1.52328	0.237896	0.118948	−	0.992901i	$-0.462048\pi$
$41$	1.52328	0.237896	0.118948	+	0.992901i	$0.462048\pi$
$42$	0	0
$43$	−0.867593	−0.132307	−0.0661534	−	0.997809i	$-0.521073\pi$
$43$	−0.867593	−0.132307	−0.0661534	+	0.997809i	$0.521073\pi$
$44$	0	0
$45$	6.31528	0.941426
$46$	0	0
$47$	−1.96425	−0.286515	−0.143258	−	0.989685i	$-0.545758\pi$
$47$	−1.96425	−0.286515	−0.143258	+	0.989685i	$0.545758\pi$
$48$	0	0
$49$	10.2772	1.46817
$50$	0	0
$51$	−2.49483	−0.349347
$52$	0	0
$53$	3.60603	0.495327	0.247663	−	0.968846i	$-0.420337\pi$
$53$	3.60603	0.495327	0.247663	+	0.968846i	$0.420337\pi$
$54$	0	0
$55$	9.93824	1.34007
$56$	0	0
$57$	−19.8622	−2.63081
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−2.74860	−0.351922	−0.175961	−	0.984397i	$-0.556303\pi$
$61$	−2.74860	−0.351922	−0.175961	+	0.984397i	$0.556303\pi$
$62$	0	0
$63$	−13.4016	−1.68845
$64$	0	0
$65$	−6.65028	−0.824865
$66$	0	0
$67$	16.0573	1.96172	0.980859	−	0.194721i	$-0.0623800\pi$
$67$	16.0573	1.96172	0.980859	+	0.194721i	$0.0623800\pi$
$68$	0	0
$69$	−3.79833	−0.457266
$70$	0	0
$71$	1.96161	0.232800	0.116400	−	0.993202i	$-0.462865\pi$
$71$	1.96161	0.232800	0.116400	+	0.993202i	$0.462865\pi$
$72$	0	0
$73$	−10.9901	−1.28630	−0.643149	−	0.765741i	$-0.722373\pi$
$73$	−10.9901	−1.28630	−0.643149	+	0.765741i	$0.722373\pi$
$74$	0	0
$75$	−2.90260	−0.335163
$76$	0	0
$77$	−21.0899	−2.40342
$78$	0	0
$79$	−9.09546	−1.02332	−0.511659	−	0.859188i	$-0.670969\pi$
$79$	−9.09546	−1.02332	−0.511659	+	0.859188i	$0.670969\pi$
$80$	0	0
$81$	−8.27714	−0.919683
$82$	0	0
$83$	−17.0316	−1.86946	−0.934728	−	0.355363i	$-0.884357\pi$
$83$	−17.0316	−1.86946	−0.934728	+	0.355363i	$0.884357\pi$
$84$	0	0
$85$	−1.95871	−0.212452
$86$	0	0
$87$	7.34684	0.787664
$88$	0	0
$89$	2.26374	0.239956	0.119978	−	0.992777i	$-0.461718\pi$
$89$	2.26374	0.239956	0.119978	+	0.992777i	$0.461718\pi$
$90$	0	0
$91$	14.1125	1.47940
$92$	0	0
$93$	−23.8781	−2.47605
$94$	0	0
$95$	−15.5939	−1.59990
$96$	0	0
$97$	4.97767	0.505406	0.252703	−	0.967544i	$-0.418681\pi$
$97$	4.97767	0.505406	0.252703	+	0.967544i	$0.418681\pi$
$98$	0	0
$99$	16.3591	1.64416
$100$	0	0
$101$	−16.8370	−1.67534	−0.837672	−	0.546173i	$-0.816084\pi$
$101$	−16.8370	−1.67534	−0.837672	+	0.546173i	$0.816084\pi$
$102$	0	0
$103$	−2.70143	−0.266180	−0.133090	−	0.991104i	$-0.542490\pi$
$103$	−2.70143	−0.266180	−0.133090	+	0.991104i	$0.542490\pi$
$104$	0	0
$105$	−20.3118	−1.98223
$106$	0	0
$107$	−6.95144	−0.672021	−0.336010	−	0.941858i	$-0.609078\pi$
$107$	−6.95144	−0.672021	−0.336010	+	0.941858i	$0.609078\pi$
$108$	0	0
$109$	−1.12492	−0.107748	−0.0538741	−	0.998548i	$-0.517157\pi$
$109$	−1.12492	−0.107748	−0.0538741	+	0.998548i	$0.517157\pi$
$110$	0	0
$111$	−8.56900	−0.813333
$112$	0	0
$113$	19.0005	1.78741	0.893707	−	0.448651i	$-0.148095\pi$
$113$	19.0005	1.78741	0.893707	+	0.448651i	$0.148095\pi$
$114$	0	0
$115$	−2.98210	−0.278082
$116$	0	0
$117$	−10.9469	−1.01204
$118$	0	0
$119$	4.15658	0.381033
$120$	0	0
$121$	14.7441	1.34037
$122$	0	0
$123$	3.80032	0.342663
$124$	0	0
$125$	−12.0724	−1.07979
$126$	0	0
$127$	1.75929	0.156111	0.0780557	−	0.996949i	$-0.475129\pi$
$127$	1.75929	0.156111	0.0780557	+	0.996949i	$0.475129\pi$
$128$	0	0
$129$	−2.16450	−0.190574
$130$	0	0
$131$	18.3019	1.59904	0.799520	−	0.600639i	$-0.205087\pi$
$131$	18.3019	1.59904	0.799520	+	0.600639i	$0.205087\pi$
$132$	0	0
$133$	33.0918	2.86943
$134$	0	0
$135$	1.09558	0.0942922
$136$	0	0
$137$	2.73848	0.233964	0.116982	−	0.993134i	$-0.462678\pi$
$137$	2.73848	0.233964	0.116982	+	0.993134i	$0.462678\pi$
$138$	0	0
$139$	−22.1496	−1.87870	−0.939352	−	0.342953i	$-0.888573\pi$
$139$	−22.1496	−1.87870	−0.939352	+	0.342953i	$0.888573\pi$
$140$	0	0
$141$	−4.90048	−0.412695
$142$	0	0
$143$	−17.2269	−1.44059
$144$	0	0
$145$	5.76806	0.479011
$146$	0	0
$147$	25.6398	2.11474
$148$	0	0
$149$	−9.17745	−0.751846	−0.375923	−	0.926651i	$-0.622674\pi$
$149$	−9.17745	−0.751846	−0.375923	+	0.926651i	$0.622674\pi$
$150$	0	0
$151$	14.4278	1.17412	0.587060	−	0.809543i	$-0.300285\pi$
$151$	14.4278	1.17412	0.587060	+	0.809543i	$0.300285\pi$
$152$	0	0
$153$	−3.22420	−0.260661
$154$	0	0
$155$	−18.7469	−1.50579
$156$	0	0
$157$	2.42017	0.193151	0.0965753	−	0.995326i	$-0.469211\pi$
$157$	2.42017	0.193151	0.0965753	+	0.995326i	$0.469211\pi$
$158$	0	0
$159$	8.99646	0.713465
$160$	0	0
$161$	6.32831	0.498741
$162$	0	0
$163$	−2.49849	−0.195697	−0.0978485	−	0.995201i	$-0.531196\pi$
$163$	−2.49849	−0.195697	−0.0978485	+	0.995201i	$0.531196\pi$
$164$	0	0
$165$	24.7943	1.93023
$166$	0	0
$167$	2.38461	0.184526	0.0922632	−	0.995735i	$-0.470590\pi$
$167$	2.38461	0.184526	0.0922632	+	0.995735i	$0.470590\pi$
$168$	0	0
$169$	−1.47243	−0.113264
$170$	0	0
$171$	−25.6688	−1.96295
$172$	0	0
$173$	20.3344	1.54600	0.773000	−	0.634407i	$-0.218755\pi$
$173$	20.3344	1.54600	0.773000	+	0.634407i	$0.218755\pi$
$174$	0	0
$175$	4.83595	0.365564
$176$	0	0
$177$	−2.49483	−0.187523
$178$	0	0
$179$	−22.1784	−1.65769	−0.828847	−	0.559476i	$-0.811003\pi$
$179$	−22.1784	−1.65769	−0.828847	+	0.559476i	$0.811003\pi$
$180$	0	0
$181$	−2.16570	−0.160975	−0.0804876	−	0.996756i	$-0.525648\pi$
$181$	−2.16570	−0.160975	−0.0804876	+	0.996756i	$0.525648\pi$
$182$	0	0
$183$	−6.85730	−0.506906
$184$	0	0
$185$	−6.72758	−0.494622
$186$	0	0
$187$	−5.07386	−0.371038
$188$	0	0
$189$	−2.32492	−0.169113
$190$	0	0
$191$	−16.1452	−1.16822	−0.584111	−	0.811673i	$-0.698557\pi$
$191$	−16.1452	−1.16822	−0.584111	+	0.811673i	$0.698557\pi$
$192$	0	0
$193$	10.7483	0.773677	0.386839	−	0.922147i	$-0.373567\pi$
$193$	10.7483	0.773677	0.386839	+	0.922147i	$0.373567\pi$
$194$	0	0
$195$	−16.5913	−1.18813
$196$	0	0
$197$	4.34131	0.309306	0.154653	−	0.987969i	$-0.450574\pi$
$197$	4.34131	0.309306	0.154653	+	0.987969i	$0.450574\pi$
$198$	0	0
$199$	18.6832	1.32442	0.662210	−	0.749319i	$-0.269619\pi$
$199$	18.6832	1.32442	0.662210	+	0.749319i	$0.269619\pi$
$200$	0	0
$201$	40.0604	2.82564
$202$	0	0
$203$	−12.2404	−0.859107
$204$	0	0
$205$	2.98366	0.208388
$206$	0	0
$207$	−4.90877	−0.341183
$208$	0	0
$209$	−40.3946	−2.79415
$210$	0	0
$211$	16.2894	1.12141	0.560705	−	0.828015i	$-0.310530\pi$
$211$	16.2894	1.12141	0.560705	+	0.828015i	$0.310530\pi$
$212$	0	0
$213$	4.89389	0.335324
$214$	0	0
$215$	−1.69937	−0.115896
$216$	0	0
$217$	39.7828	2.70063
$218$	0	0
$219$	−27.4186	−1.85278
$220$	0	0
$221$	3.39523	0.228388
$222$	0	0
$223$	9.34606	0.625858	0.312929	−	0.949776i	$-0.398690\pi$
$223$	9.34606	0.625858	0.312929	+	0.949776i	$0.398690\pi$
$224$	0	0
$225$	−3.75117	−0.250078
$226$	0	0
$227$	14.8574	0.986118	0.493059	−	0.869996i	$-0.335879\pi$
$227$	14.8574	0.986118	0.493059	+	0.869996i	$0.335879\pi$
$228$	0	0
$229$	−6.24150	−0.412450	−0.206225	−	0.978505i	$-0.566118\pi$
$229$	−6.24150	−0.412450	−0.206225	+	0.978505i	$0.566118\pi$
$230$	0	0
$231$	−52.6159	−3.46187
$232$	0	0
$233$	21.2458	1.39186	0.695930	−	0.718109i	$-0.254992\pi$
$233$	21.2458	1.39186	0.695930	+	0.718109i	$0.254992\pi$
$234$	0	0
$235$	−3.84740	−0.250977
$236$	0	0
$237$	−22.6917	−1.47398
$238$	0	0
$239$	−17.2626	−1.11663	−0.558313	−	0.829631i	$-0.688551\pi$
$239$	−17.2626	−1.11663	−0.558313	+	0.829631i	$0.688551\pi$
$240$	0	0
$241$	5.96786	0.384424	0.192212	−	0.981353i	$-0.438434\pi$
$241$	5.96786	0.384424	0.192212	+	0.981353i	$0.438434\pi$
$242$	0	0
$243$	−22.3281	−1.43235
$244$	0	0
$245$	20.1300	1.28606
$246$	0	0
$247$	27.0305	1.71991
$248$	0	0
$249$	−42.4909	−2.69275
$250$	0	0
$251$	23.7652	1.50005	0.750023	−	0.661411i	$-0.230042\pi$
$251$	23.7652	1.50005	0.750023	+	0.661411i	$0.230042\pi$
$252$	0	0
$253$	−7.72485	−0.485657
$254$	0	0
$255$	−4.88666	−0.306015
$256$	0	0
$257$	3.71800	0.231922	0.115961	−	0.993254i	$-0.463005\pi$
$257$	3.71800	0.231922	0.115961	+	0.993254i	$0.463005\pi$
$258$	0	0
$259$	14.2766	0.887105
$260$	0	0
$261$	9.49469	0.587706
$262$	0	0
$263$	−24.1595	−1.48974	−0.744870	−	0.667209i	$-0.767489\pi$
$263$	−24.1595	−1.48974	−0.744870	+	0.667209i	$0.767489\pi$
$264$	0	0
$265$	7.06318	0.433888
$266$	0	0
$267$	5.64765	0.345631
$268$	0	0
$269$	−12.7537	−0.777606	−0.388803	−	0.921321i	$-0.627111\pi$
$269$	−12.7537	−0.777606	−0.388803	+	0.921321i	$0.627111\pi$
$270$	0	0
$271$	−12.7180	−0.772566	−0.386283	−	0.922380i	$-0.626241\pi$
$271$	−12.7180	−0.772566	−0.386283	+	0.922380i	$0.626241\pi$
$272$	0	0
$273$	35.2085	2.13091
$274$	0	0
$275$	−5.90316	−0.355974
$276$	0	0
$277$	−0.236744	−0.0142246	−0.00711229	−	0.999975i	$-0.502264\pi$
$277$	−0.236744	−0.0142246	−0.00711229	+	0.999975i	$0.502264\pi$
$278$	0	0
$279$	−30.8589	−1.84747
$280$	0	0
$281$	25.9046	1.54534	0.772669	−	0.634809i	$-0.218921\pi$
$281$	25.9046	1.54534	0.772669	+	0.634809i	$0.218921\pi$
$282$	0	0
$283$	−4.05950	−0.241312	−0.120656	−	0.992694i	$-0.538500\pi$
$283$	−4.05950	−0.241312	−0.120656	+	0.992694i	$0.538500\pi$
$284$	0	0
$285$	−38.9043	−2.30449
$286$	0	0
$287$	−6.33162	−0.373744
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	12.4185	0.727983
$292$	0	0
$293$	−1.75208	−0.102357	−0.0511787	−	0.998690i	$-0.516298\pi$
$293$	−1.75208	−0.102357	−0.0511787	+	0.998690i	$0.516298\pi$
$294$	0	0
$295$	−1.95871	−0.114041
$296$	0	0
$297$	2.83799	0.164677
$298$	0	0
$299$	5.16916	0.298940
$300$	0	0
$301$	3.60622	0.207859
$302$	0	0
$303$	−42.0055	−2.41316
$304$	0	0
$305$	−5.38372	−0.308271
$306$	0	0
$307$	6.44245	0.367690	0.183845	−	0.982955i	$-0.441145\pi$
$307$	6.44245	0.367690	0.183845	+	0.982955i	$0.441145\pi$
$308$	0	0
$309$	−6.73962	−0.383404
$310$	0	0
$311$	2.20637	0.125112	0.0625560	−	0.998041i	$-0.480075\pi$
$311$	2.20637	0.125112	0.0625560	+	0.998041i	$0.480075\pi$
$312$	0	0
$313$	−22.4719	−1.27019	−0.635095	−	0.772434i	$-0.719039\pi$
$313$	−22.4719	−1.27019	−0.635095	+	0.772434i	$0.719039\pi$
$314$	0	0
$315$	−26.2500	−1.47902
$316$	0	0
$317$	−11.7292	−0.658778	−0.329389	−	0.944194i	$-0.606843\pi$
$317$	−11.7292	−0.658778	−0.329389	+	0.944194i	$0.606843\pi$
$318$	0	0
$319$	14.9416	0.836571
$320$	0	0
$321$	−17.3427	−0.967974
$322$	0	0
$323$	7.96131	0.442979
$324$	0	0
$325$	3.95016	0.219115
$326$	0	0
$327$	−2.80650	−0.155200
$328$	0	0
$329$	8.16457	0.450127
$330$	0	0
$331$	35.4802	1.95017	0.975085	−	0.221830i	$-0.0712030\pi$
$331$	35.4802	1.95017	0.975085	+	0.221830i	$0.0712030\pi$
$332$	0	0
$333$	−11.0741	−0.606859
$334$	0	0
$335$	31.4517	1.71839
$336$	0	0
$337$	−10.1240	−0.551488	−0.275744	−	0.961231i	$-0.588924\pi$
$337$	−10.1240	−0.551488	−0.275744	+	0.961231i	$0.588924\pi$
$338$	0	0
$339$	47.4030	2.57458
$340$	0	0
$341$	−48.5621	−2.62979
$342$	0	0
$343$	−13.6218	−0.735511
$344$	0	0
$345$	−7.43984	−0.400548
$346$	0	0
$347$	−17.9611	−0.964201	−0.482100	−	0.876116i	$-0.660126\pi$
$347$	−17.9611	−0.964201	−0.482100	+	0.876116i	$0.660126\pi$
$348$	0	0
$349$	17.0279	0.911482	0.455741	−	0.890113i	$-0.349374\pi$
$349$	17.0279	0.911482	0.455741	+	0.890113i	$0.349374\pi$
$350$	0	0
$351$	−1.89907	−0.101365
$352$	0	0
$353$	−20.2093	−1.07563	−0.537815	−	0.843063i	$-0.680750\pi$
$353$	−20.2093	−1.07563	−0.537815	+	0.843063i	$0.680750\pi$
$354$	0	0
$355$	3.84223	0.203924
$356$	0	0
$357$	10.3700	0.548838
$358$	0	0
$359$	−4.00853	−0.211562	−0.105781	−	0.994389i	$-0.533734\pi$
$359$	−4.00853	−0.211562	−0.105781	+	0.994389i	$0.533734\pi$
$360$	0	0
$361$	44.3825	2.33592
$362$	0	0
$363$	36.7841	1.93066
$364$	0	0
$365$	−21.5265	−1.12675
$366$	0	0
$367$	−27.9747	−1.46027	−0.730135	−	0.683303i	$-0.760543\pi$
$367$	−27.9747	−1.46027	−0.730135	+	0.683303i	$0.760543\pi$
$368$	0	0
$369$	4.91134	0.255674
$370$	0	0
$371$	−14.9888	−0.778178
$372$	0	0
$373$	−18.8210	−0.974514	−0.487257	−	0.873259i	$-0.662003\pi$
$373$	−18.8210	−0.974514	−0.487257	+	0.873259i	$0.662003\pi$
$374$	0	0
$375$	−30.1187	−1.55532
$376$	0	0
$377$	−9.99834	−0.514941
$378$	0	0
$379$	−0.264458	−0.0135843	−0.00679214	−	0.999977i	$-0.502162\pi$
$379$	−0.264458	−0.0135843	−0.00679214	+	0.999977i	$0.502162\pi$
$380$	0	0
$381$	4.38913	0.224862
$382$	0	0
$383$	−13.6906	−0.699559	−0.349779	−	0.936832i	$-0.613743\pi$
$383$	−13.6906	−0.699559	−0.349779	+	0.936832i	$0.613743\pi$
$384$	0	0
$385$	−41.3091	−2.10531
$386$	0	0
$387$	−2.79729	−0.142194
$388$	0	0
$389$	15.8494	0.803596	0.401798	−	0.915728i	$-0.368385\pi$
$389$	15.8494	0.803596	0.401798	+	0.915728i	$0.368385\pi$
$390$	0	0
$391$	1.52248	0.0769951
$392$	0	0
$393$	45.6601	2.30325
$394$	0	0
$395$	−17.8154	−0.896390
$396$	0	0
$397$	1.11324	0.0558720	0.0279360	−	0.999610i	$-0.491107\pi$
$397$	1.11324	0.0558720	0.0279360	+	0.999610i	$0.491107\pi$
$398$	0	0
$399$	82.5587	4.13310
$400$	0	0
$401$	−32.2902	−1.61249	−0.806247	−	0.591579i	$-0.798505\pi$
$401$	−32.2902	−1.61249	−0.806247	+	0.591579i	$0.798505\pi$
$402$	0	0
$403$	32.4958	1.61873
$404$	0	0
$405$	−16.2125	−0.805608
$406$	0	0
$407$	−17.4272	−0.863834
$408$	0	0
$409$	−20.5922	−1.01822	−0.509110	−	0.860701i	$-0.670025\pi$
$409$	−20.5922	−1.01822	−0.509110	+	0.860701i	$0.670025\pi$
$410$	0	0
$411$	6.83206	0.337001
$412$	0	0
$413$	4.15658	0.204532
$414$	0	0
$415$	−33.3599	−1.63757
$416$	0	0
$417$	−55.2596	−2.70607
$418$	0	0
$419$	8.71227	0.425622	0.212811	−	0.977093i	$-0.431738\pi$
$419$	8.71227	0.425622	0.212811	+	0.977093i	$0.431738\pi$
$420$	0	0
$421$	−25.2900	−1.23256	−0.616279	−	0.787528i	$-0.711360\pi$
$421$	−25.2900	−1.23256	−0.616279	+	0.787528i	$0.711360\pi$
$422$	0	0
$423$	−6.33313	−0.307927
$424$	0	0
$425$	1.16344	0.0564353
$426$	0	0
$427$	11.4248	0.552884
$428$	0	0
$429$	−42.9783	−2.07501
$430$	0	0
$431$	−19.1408	−0.921982	−0.460991	−	0.887405i	$-0.652506\pi$
$431$	−19.1408	−0.921982	−0.460991	+	0.887405i	$0.652506\pi$
$432$	0	0
$433$	10.8874	0.523217	0.261609	−	0.965174i	$-0.415747\pi$
$433$	10.8874	0.523217	0.261609	+	0.965174i	$0.415747\pi$
$434$	0	0
$435$	14.3904	0.689965
$436$	0	0
$437$	12.1209	0.579823
$438$	0	0
$439$	−20.8713	−0.996130	−0.498065	−	0.867140i	$-0.665956\pi$
$439$	−20.8713	−0.996130	−0.498065	+	0.867140i	$0.665956\pi$
$440$	0	0
$441$	33.1356	1.57789
$442$	0	0
$443$	−2.97172	−0.141190	−0.0705952	−	0.997505i	$-0.522490\pi$
$443$	−2.97172	−0.141190	−0.0705952	+	0.997505i	$0.522490\pi$
$444$	0	0
$445$	4.43401	0.210192
$446$	0	0
$447$	−22.8962	−1.08295
$448$	0	0
$449$	8.36040	0.394552	0.197276	−	0.980348i	$-0.436791\pi$
$449$	8.36040	0.394552	0.197276	+	0.980348i	$0.436791\pi$
$450$	0	0
$451$	7.72890	0.363939
$452$	0	0
$453$	35.9950	1.69119
$454$	0	0
$455$	27.6424	1.29590
$456$	0	0
$457$	−0.329733	−0.0154242	−0.00771212	−	0.999970i	$-0.502455\pi$
$457$	−0.329733	−0.0154242	−0.00771212	+	0.999970i	$0.502455\pi$
$458$	0	0
$459$	−0.559335	−0.0261075
$460$	0	0
$461$	35.1733	1.63818	0.819092	−	0.573662i	$-0.194478\pi$
$461$	35.1733	1.63818	0.819092	+	0.573662i	$0.194478\pi$
$462$	0	0
$463$	3.14096	0.145973	0.0729864	−	0.997333i	$-0.476747\pi$
$463$	3.14096	0.145973	0.0729864	+	0.997333i	$0.476747\pi$
$464$	0	0
$465$	−46.7704	−2.16893
$466$	0	0
$467$	8.22698	0.380699	0.190350	−	0.981716i	$-0.439038\pi$
$467$	8.22698	0.380699	0.190350	+	0.981716i	$0.439038\pi$
$468$	0	0
$469$	−66.7437	−3.08194
$470$	0	0
$471$	6.03792	0.278213
$472$	0	0
$473$	−4.40205	−0.202407
$474$	0	0
$475$	9.26254	0.424995
$476$	0	0
$477$	11.6266	0.532344
$478$	0	0
$479$	−17.3022	−0.790557	−0.395279	−	0.918561i	$-0.629352\pi$
$479$	−17.3022	−0.790557	−0.395279	+	0.918561i	$0.629352\pi$
$480$	0	0
$481$	11.6616	0.531722
$482$	0	0
$483$	15.7881	0.718383
$484$	0	0
$485$	9.74982	0.442717
$486$	0	0
$487$	21.2261	0.961847	0.480924	−	0.876763i	$-0.340301\pi$
$487$	21.2261	0.961847	0.480924	+	0.876763i	$0.340301\pi$
$488$	0	0
$489$	−6.23332	−0.281881
$490$	0	0
$491$	−9.65183	−0.435581	−0.217791	−	0.975996i	$-0.569885\pi$
$491$	−9.65183	−0.435581	−0.217791	+	0.975996i	$0.569885\pi$
$492$	0	0
$493$	−2.94482	−0.132628
$494$	0	0
$495$	32.0429	1.44022
$496$	0	0
$497$	−8.15359	−0.365739
$498$	0	0
$499$	10.2273	0.457838	0.228919	−	0.973445i	$-0.426481\pi$
$499$	10.2273	0.457838	0.228919	+	0.973445i	$0.426481\pi$
$500$	0	0
$501$	5.94920	0.265791
$502$	0	0
$503$	−33.0575	−1.47396	−0.736981	−	0.675913i	$-0.763749\pi$
$503$	−33.0575	−1.47396	−0.736981	+	0.675913i	$0.763749\pi$
$504$	0	0
$505$	−32.9789	−1.46754
$506$	0	0
$507$	−3.67347	−0.163144
$508$	0	0
$509$	33.4375	1.48209	0.741046	−	0.671454i	$-0.234330\pi$
$509$	33.4375	1.48209	0.741046	+	0.671454i	$0.234330\pi$
$510$	0	0
$511$	45.6814	2.02083
$512$	0	0
$513$	−4.45304	−0.196606
$514$	0	0
$515$	−5.29133	−0.233164
$516$	0	0
$517$	−9.96634	−0.438319
$518$	0	0
$519$	50.7311	2.22685
$520$	0	0
$521$	8.92197	0.390879	0.195439	−	0.980716i	$-0.437387\pi$
$521$	8.92197	0.390879	0.195439	+	0.980716i	$0.437387\pi$
$522$	0	0
$523$	−6.43207	−0.281255	−0.140627	−	0.990063i	$-0.544912\pi$
$523$	−6.43207	−0.281255	−0.140627	+	0.990063i	$0.544912\pi$
$524$	0	0
$525$	12.0649	0.526555
$526$	0	0
$527$	9.57103	0.416921
$528$	0	0
$529$	−20.6821	−0.899220
$530$	0	0
$531$	−3.22420	−0.139918
$532$	0	0
$533$	−5.17187	−0.224018
$534$	0	0
$535$	−13.6159	−0.588665
$536$	0	0
$537$	−55.3315	−2.38773
$538$	0	0
$539$	52.1450	2.24604
$540$	0	0
$541$	36.3185	1.56146	0.780728	−	0.624871i	$-0.214849\pi$
$541$	36.3185	1.56146	0.780728	+	0.624871i	$0.214849\pi$
$542$	0	0
$543$	−5.40306	−0.231868
$544$	0	0
$545$	−2.20340	−0.0943834
$546$	0	0
$547$	−21.8774	−0.935409	−0.467705	−	0.883885i	$-0.654919\pi$
$547$	−21.8774	−0.935409	−0.467705	+	0.883885i	$0.654919\pi$
$548$	0	0
$549$	−8.86202	−0.378222
$550$	0	0
$551$	−23.4446	−0.998776
$552$	0	0
$553$	37.8060	1.60768
$554$	0	0
$555$	−16.7842	−0.712450
$556$	0	0
$557$	4.30285	0.182318	0.0911589	−	0.995836i	$-0.470943\pi$
$557$	4.30285	0.182318	0.0911589	+	0.995836i	$0.470943\pi$
$558$	0	0
$559$	2.94568	0.124589
$560$	0	0
$561$	−12.6584	−0.534440
$562$	0	0
$563$	10.6496	0.448827	0.224414	−	0.974494i	$-0.427953\pi$
$563$	10.6496	0.448827	0.224414	+	0.974494i	$0.427953\pi$
$564$	0	0
$565$	37.2165	1.56571
$566$	0	0
$567$	34.4046	1.44486
$568$	0	0
$569$	33.9934	1.42508	0.712538	−	0.701633i	$-0.247545\pi$
$569$	33.9934	1.42508	0.712538	+	0.701633i	$0.247545\pi$
$570$	0	0
$571$	−12.8186	−0.536442	−0.268221	−	0.963357i	$-0.586436\pi$
$571$	−12.8186	−0.536442	−0.268221	+	0.963357i	$0.586436\pi$
$572$	0	0
$573$	−40.2795	−1.68270
$574$	0	0
$575$	1.77132	0.0738691
$576$	0	0
$577$	−12.7848	−0.532239	−0.266120	−	0.963940i	$-0.585742\pi$
$577$	−12.7848	−0.532239	−0.266120	+	0.963940i	$0.585742\pi$
$578$	0	0
$579$	26.8151	1.11440
$580$	0	0
$581$	70.7931	2.93699
$582$	0	0
$583$	18.2965	0.757765
$584$	0	0
$585$	−21.4418	−0.886509
$586$	0	0
$587$	37.0671	1.52992	0.764961	−	0.644076i	$-0.222758\pi$
$587$	37.0671	1.52992	0.764961	+	0.644076i	$0.222758\pi$
$588$	0	0
$589$	76.1980	3.13968
$590$	0	0
$591$	10.8308	0.445522
$592$	0	0
$593$	33.6190	1.38057	0.690284	−	0.723539i	$-0.257486\pi$
$593$	33.6190	1.38057	0.690284	+	0.723539i	$0.257486\pi$
$594$	0	0
$595$	8.14155	0.333771
$596$	0	0
$597$	46.6116	1.90768
$598$	0	0
$599$	−19.3714	−0.791493	−0.395746	−	0.918360i	$-0.629514\pi$
$599$	−19.3714	−0.791493	−0.395746	+	0.918360i	$0.629514\pi$
$600$	0	0
$601$	1.25408	0.0511548	0.0255774	−	0.999673i	$-0.491858\pi$
$601$	1.25408	0.0511548	0.0255774	+	0.999673i	$0.491858\pi$
$602$	0	0
$603$	51.7721	2.10832
$604$	0	0
$605$	28.8795	1.17412
$606$	0	0
$607$	−34.8340	−1.41387	−0.706935	−	0.707278i	$-0.749923\pi$
$607$	−34.8340	−1.41387	−0.706935	+	0.707278i	$0.749923\pi$
$608$	0	0
$609$	−30.5378	−1.23745
$610$	0	0
$611$	6.66908	0.269802
$612$	0	0
$613$	7.96954	0.321887	0.160943	−	0.986964i	$-0.448546\pi$
$613$	7.96954	0.321887	0.160943	+	0.986964i	$0.448546\pi$
$614$	0	0
$615$	7.44374	0.300160
$616$	0	0
$617$	45.2532	1.82182	0.910912	−	0.412600i	$-0.135379\pi$
$617$	45.2532	1.82182	0.910912	+	0.412600i	$0.135379\pi$
$618$	0	0
$619$	−8.33466	−0.334998	−0.167499	−	0.985872i	$-0.553569\pi$
$619$	−8.33466	−0.334998	−0.167499	+	0.985872i	$0.553569\pi$
$620$	0	0
$621$	−0.851575	−0.0341725
$622$	0	0
$623$	−9.40941	−0.376980
$624$	0	0
$625$	−17.8292	−0.713167
$626$	0	0
$627$	−100.778	−4.02468
$628$	0	0
$629$	3.43470	0.136950
$630$	0	0
$631$	−9.03711	−0.359762	−0.179881	−	0.983688i	$-0.557571\pi$
$631$	−9.03711	−0.359762	−0.179881	+	0.983688i	$0.557571\pi$
$632$	0	0
$633$	40.6394	1.61527
$634$	0	0
$635$	3.44594	0.136748
$636$	0	0
$637$	−34.8933	−1.38252
$638$	0	0
$639$	6.32462	0.250198
$640$	0	0
$641$	−19.6647	−0.776709	−0.388355	−	0.921510i	$-0.626956\pi$
$641$	−19.6647	−0.776709	−0.388355	+	0.921510i	$0.626956\pi$
$642$	0	0
$643$	18.7175	0.738147	0.369073	−	0.929400i	$-0.379675\pi$
$643$	18.7175	0.738147	0.369073	+	0.929400i	$0.379675\pi$
$644$	0	0
$645$	−4.23964	−0.166936
$646$	0	0
$647$	−5.56139	−0.218641	−0.109320	−	0.994007i	$-0.534867\pi$
$647$	−5.56139	−0.218641	−0.109320	+	0.994007i	$0.534867\pi$
$648$	0	0
$649$	−5.07386	−0.199167
$650$	0	0
$651$	99.2515	3.88997
$652$	0	0
$653$	14.0390	0.549389	0.274694	−	0.961532i	$-0.411423\pi$
$653$	14.0390	0.549389	0.274694	+	0.961532i	$0.411423\pi$
$654$	0	0
$655$	35.8481	1.40070
$656$	0	0
$657$	−35.4344	−1.38243
$658$	0	0
$659$	−24.3579	−0.948848	−0.474424	−	0.880296i	$-0.657344\pi$
$659$	−24.3579	−0.948848	−0.474424	+	0.880296i	$0.657344\pi$
$660$	0	0
$661$	30.7820	1.19728	0.598640	−	0.801018i	$-0.295708\pi$
$661$	30.7820	1.19728	0.598640	+	0.801018i	$0.295708\pi$
$662$	0	0
$663$	8.47053	0.328968
$664$	0	0
$665$	64.8174	2.51351
$666$	0	0
$667$	−4.48343	−0.173599
$668$	0	0
$669$	23.3169	0.901482
$670$	0	0
$671$	−13.9460	−0.538380
$672$	0	0
$673$	−42.1774	−1.62582	−0.812909	−	0.582391i	$-0.802117\pi$
$673$	−42.1774	−1.62582	−0.812909	+	0.582391i	$0.802117\pi$
$674$	0	0
$675$	−0.650755	−0.0250476
$676$	0	0
$677$	−29.4906	−1.13341	−0.566707	−	0.823919i	$-0.691783\pi$
$677$	−29.4906	−1.13341	−0.566707	+	0.823919i	$0.691783\pi$
$678$	0	0
$679$	−20.6901	−0.794013
$680$	0	0
$681$	37.0667	1.42040
$682$	0	0
$683$	−23.9468	−0.916297	−0.458149	−	0.888876i	$-0.651487\pi$
$683$	−23.9468	−0.916297	−0.458149	+	0.888876i	$0.651487\pi$
$684$	0	0
$685$	5.36390	0.204944
$686$	0	0
$687$	−15.5715	−0.594090
$688$	0	0
$689$	−12.2433	−0.466433
$690$	0	0
$691$	5.64635	0.214797	0.107399	−	0.994216i	$-0.465748\pi$
$691$	5.64635	0.214797	0.107399	+	0.994216i	$0.465748\pi$
$692$	0	0
$693$	−67.9981	−2.58303
$694$	0	0
$695$	−43.3847	−1.64568
$696$	0	0
$697$	−1.52328	−0.0576982
$698$	0	0
$699$	53.0048	2.00483
$700$	0	0
$701$	−3.23244	−0.122087	−0.0610437	−	0.998135i	$-0.519443\pi$
$701$	−3.23244	−0.122087	−0.0610437	+	0.998135i	$0.519443\pi$
$702$	0	0
$703$	27.3447	1.03132
$704$	0	0
$705$	−9.59863	−0.361505
$706$	0	0
$707$	69.9844	2.63203
$708$	0	0
$709$	4.09903	0.153942	0.0769711	−	0.997033i	$-0.475475\pi$
$709$	4.09903	0.153942	0.0769711	+	0.997033i	$0.475475\pi$
$710$	0	0
$711$	−29.3255	−1.09979
$712$	0	0
$713$	14.5717	0.545715
$714$	0	0
$715$	−33.7426	−1.26190
$716$	0	0
$717$	−43.0673	−1.60838
$718$	0	0
$719$	−48.6080	−1.81277	−0.906386	−	0.422451i	$-0.861170\pi$
$719$	−48.6080	−1.81277	−0.906386	+	0.422451i	$0.861170\pi$
$720$	0	0
$721$	11.2287	0.418179
$722$	0	0
$723$	14.8888	0.553721
$724$	0	0
$725$	−3.42614	−0.127244
$726$	0	0
$727$	29.7180	1.10218	0.551089	−	0.834447i	$-0.314213\pi$
$727$	29.7180	1.10218	0.551089	+	0.834447i	$0.314213\pi$
$728$	0	0
$729$	−30.8735	−1.14346
$730$	0	0
$731$	0.867593	0.0320891
$732$	0	0
$733$	24.7866	0.915515	0.457757	−	0.889077i	$-0.348653\pi$
$733$	24.7866	0.915515	0.457757	+	0.889077i	$0.348653\pi$
$734$	0	0
$735$	50.2211	1.85243
$736$	0	0
$737$	81.4728	3.00109
$738$	0	0
$739$	12.0931	0.444850	0.222425	−	0.974950i	$-0.428603\pi$
$739$	12.0931	0.444850	0.222425	+	0.974950i	$0.428603\pi$
$740$	0	0
$741$	67.4365	2.47734
$742$	0	0
$743$	−19.6411	−0.720562	−0.360281	−	0.932844i	$-0.617319\pi$
$743$	−19.6411	−0.720562	−0.360281	+	0.932844i	$0.617319\pi$
$744$	0	0
$745$	−17.9760	−0.658589
$746$	0	0
$747$	−54.9131	−2.00917
$748$	0	0
$749$	28.8942	1.05577
$750$	0	0
$751$	−4.66232	−0.170131	−0.0850653	−	0.996375i	$-0.527110\pi$
$751$	−4.66232	−0.170131	−0.0850653	+	0.996375i	$0.527110\pi$
$752$	0	0
$753$	59.2902	2.16066
$754$	0	0
$755$	28.2600	1.02849
$756$	0	0
$757$	13.5388	0.492078	0.246039	−	0.969260i	$-0.420871\pi$
$757$	13.5388	0.492078	0.246039	+	0.969260i	$0.420871\pi$
$758$	0	0
$759$	−19.2722	−0.699538
$760$	0	0
$761$	−53.7250	−1.94753	−0.973764	−	0.227559i	$-0.926926\pi$
$761$	−53.7250	−1.94753	−0.973764	+	0.227559i	$0.926926\pi$
$762$	0	0
$763$	4.67584	0.169277
$764$	0	0
$765$	−6.31528	−0.228329
$766$	0	0
$767$	3.39523	0.122595
$768$	0	0
$769$	−20.7788	−0.749304	−0.374652	−	0.927166i	$-0.622238\pi$
$769$	−20.7788	−0.749304	−0.374652	+	0.927166i	$0.622238\pi$
$770$	0	0
$771$	9.27579	0.334059
$772$	0	0
$773$	−0.555990	−0.0199976	−0.00999878	−	0.999950i	$-0.503183\pi$
$773$	−0.555990	−0.0199976	−0.00999878	+	0.999950i	$0.503183\pi$
$774$	0	0
$775$	11.1354	0.399994
$776$	0	0
$777$	35.6177	1.27778
$778$	0	0
$779$	−12.1273	−0.434505
$780$	0	0
$781$	9.95294	0.356144
$782$	0	0
$783$	1.64714	0.0588640
$784$	0	0
$785$	4.74042	0.169193
$786$	0	0
$787$	13.2699	0.473022	0.236511	−	0.971629i	$-0.423996\pi$
$787$	13.2699	0.473022	0.236511	+	0.971629i	$0.423996\pi$
$788$	0	0
$789$	−60.2740	−2.14581
$790$	0	0
$791$	−78.9770	−2.80810
$792$	0	0
$793$	9.33212	0.331393
$794$	0	0
$795$	17.6215	0.624969
$796$	0	0
$797$	37.3646	1.32352	0.661762	−	0.749714i	$-0.269809\pi$
$797$	37.3646	1.32352	0.661762	+	0.749714i	$0.269809\pi$
$798$	0	0
$799$	1.96425	0.0694902
$800$	0	0
$801$	7.29874	0.257888
$802$	0	0
$803$	−55.7625	−1.96782
$804$	0	0
$805$	12.3953	0.436878
$806$	0	0
$807$	−31.8183	−1.12006
$808$	0	0
$809$	−44.1944	−1.55379	−0.776897	−	0.629628i	$-0.783207\pi$
$809$	−44.1944	−1.55379	−0.776897	+	0.629628i	$0.783207\pi$
$810$	0	0
$811$	30.2166	1.06105	0.530523	−	0.847670i	$-0.321995\pi$
$811$	30.2166	1.06105	0.530523	+	0.847670i	$0.321995\pi$
$812$	0	0
$813$	−31.7294	−1.11280
$814$	0	0
$815$	−4.89383	−0.171423
$816$	0	0
$817$	6.90718	0.241652
$818$	0	0
$819$	45.5016	1.58996
$820$	0	0
$821$	−31.9293	−1.11434	−0.557171	−	0.830398i	$-0.688113\pi$
$821$	−31.9293	−1.11434	−0.557171	+	0.830398i	$0.688113\pi$
$822$	0	0
$823$	−33.8783	−1.18092	−0.590462	−	0.807065i	$-0.701055\pi$
$823$	−33.8783	−1.18092	−0.590462	+	0.807065i	$0.701055\pi$
$824$	0	0
$825$	−14.7274	−0.512742
$826$	0	0
$827$	−10.2833	−0.357585	−0.178792	−	0.983887i	$-0.557219\pi$
$827$	−10.2833	−0.357585	−0.178792	+	0.983887i	$0.557219\pi$
$828$	0	0
$829$	−23.8004	−0.826623	−0.413311	−	0.910590i	$-0.635628\pi$
$829$	−23.8004	−0.826623	−0.413311	+	0.910590i	$0.635628\pi$
$830$	0	0
$831$	−0.590637	−0.0204890
$832$	0	0
$833$	−10.2772	−0.356083
$834$	0	0
$835$	4.67076	0.161638
$836$	0	0
$837$	−5.35341	−0.185041
$838$	0	0
$839$	−44.1011	−1.52254	−0.761270	−	0.648435i	$-0.775424\pi$
$839$	−44.1011	−1.52254	−0.761270	+	0.648435i	$0.775424\pi$
$840$	0	0
$841$	−20.3280	−0.700966
$842$	0	0
$843$	64.6276	2.22589
$844$	0	0
$845$	−2.88407	−0.0992149
$846$	0	0
$847$	−61.2851	−2.10578
$848$	0	0
$849$	−10.1278	−0.347585
$850$	0	0
$851$	5.22925	0.179257
$852$	0	0
$853$	−46.8777	−1.60506	−0.802531	−	0.596610i	$-0.796514\pi$
$853$	−46.8777	−1.60506	−0.802531	+	0.596610i	$0.796514\pi$
$854$	0	0
$855$	−50.2779	−1.71947
$856$	0	0
$857$	10.3430	0.353310	0.176655	−	0.984273i	$-0.443472\pi$
$857$	10.3430	0.353310	0.176655	+	0.984273i	$0.443472\pi$
$858$	0	0
$859$	−46.8356	−1.59801	−0.799006	−	0.601324i	$-0.794640\pi$
$859$	−46.8356	−1.59801	−0.799006	+	0.601324i	$0.794640\pi$
$860$	0	0
$861$	−15.7963	−0.538338
$862$	0	0
$863$	37.5650	1.27873	0.639363	−	0.768905i	$-0.279198\pi$
$863$	37.5650	1.27873	0.639363	+	0.768905i	$0.279198\pi$
$864$	0	0
$865$	39.8293	1.35424
$866$	0	0
$867$	2.49483	0.0847290
$868$	0	0
$869$	−46.1491	−1.56550
$870$	0	0
$871$	−54.5184	−1.84728
$872$	0	0
$873$	16.0490	0.543176
$874$	0	0
$875$	50.1800	1.69639
$876$	0	0
$877$	52.6751	1.77871	0.889356	−	0.457215i	$-0.151153\pi$
$877$	52.6751	1.77871	0.889356	+	0.457215i	$0.151153\pi$
$878$	0	0
$879$	−4.37114	−0.147435
$880$	0	0
$881$	−34.9116	−1.17620	−0.588101	−	0.808787i	$-0.700124\pi$
$881$	−34.9116	−1.17620	−0.588101	+	0.808787i	$0.700124\pi$
$882$	0	0
$883$	−5.98790	−0.201509	−0.100754	−	0.994911i	$-0.532126\pi$
$883$	−5.98790	−0.201509	−0.100754	+	0.994911i	$0.532126\pi$
$884$	0	0
$885$	−4.88666	−0.164263
$886$	0	0
$887$	54.2685	1.82216	0.911079	−	0.412232i	$-0.135251\pi$
$887$	54.2685	1.82216	0.911079	+	0.412232i	$0.135251\pi$
$888$	0	0
$889$	−7.31262	−0.245257
$890$	0	0
$891$	−41.9971	−1.40696
$892$	0	0
$893$	15.6380	0.523306
$894$	0	0
$895$	−43.4412	−1.45208
$896$	0	0
$897$	12.8962	0.430592
$898$	0	0
$899$	−28.1850	−0.940022
$900$	0	0
$901$	−3.60603	−0.120134
$902$	0	0
$903$	8.99693	0.299399
$904$	0	0
$905$	−4.24198	−0.141008
$906$	0	0
$907$	−7.60874	−0.252644	−0.126322	−	0.991989i	$-0.540317\pi$
$907$	−7.60874	−0.252644	−0.126322	+	0.991989i	$0.540317\pi$
$908$	0	0
$909$	−54.2858	−1.80055
$910$	0	0
$911$	13.5690	0.449563	0.224781	−	0.974409i	$-0.427833\pi$
$911$	13.5690	0.449563	0.224781	+	0.974409i	$0.427833\pi$
$912$	0	0
$913$	−86.4158	−2.85995
$914$	0	0
$915$	−13.4315	−0.444031
$916$	0	0
$917$	−76.0731	−2.51216
$918$	0	0
$919$	43.3564	1.43020	0.715098	−	0.699025i	$-0.246382\pi$
$919$	43.3564	1.43020	0.715098	+	0.699025i	$0.246382\pi$
$920$	0	0
$921$	16.0729	0.529619
$922$	0	0
$923$	−6.66011	−0.219220
$924$	0	0
$925$	3.99608	0.131390
$926$	0	0
$927$	−8.70995	−0.286072
$928$	0	0
$929$	28.4955	0.934908	0.467454	−	0.884017i	$-0.345171\pi$
$929$	28.4955	0.934908	0.467454	+	0.884017i	$0.345171\pi$
$930$	0	0
$931$	−81.8198	−2.68154
$932$	0	0
$933$	5.50453	0.180210
$934$	0	0
$935$	−9.93824	−0.325015
$936$	0	0
$937$	21.5175	0.702947	0.351474	−	0.936198i	$-0.385681\pi$
$937$	21.5175	0.702947	0.351474	+	0.936198i	$0.385681\pi$
$938$	0	0
$939$	−56.0638	−1.82957
$940$	0	0
$941$	−21.9120	−0.714312	−0.357156	−	0.934045i	$-0.616254\pi$
$941$	−21.9120	−0.714312	−0.357156	+	0.934045i	$0.616254\pi$
$942$	0	0
$943$	−2.31916	−0.0755221
$944$	0	0
$945$	−4.55385	−0.148137
$946$	0	0
$947$	−49.8678	−1.62049	−0.810243	−	0.586094i	$-0.800665\pi$
$947$	−49.8678	−1.62049	−0.810243	+	0.586094i	$0.800665\pi$
$948$	0	0
$949$	37.3140	1.21126
$950$	0	0
$951$	−29.2624	−0.948899
$952$	0	0
$953$	28.3647	0.918822	0.459411	−	0.888224i	$-0.348060\pi$
$953$	28.3647	0.918822	0.459411	+	0.888224i	$0.348060\pi$
$954$	0	0
$955$	−31.6237	−1.02332
$956$	0	0
$957$	37.2769	1.20499
$958$	0	0
$959$	−11.3827	−0.367568
$960$	0	0
$961$	60.6047	1.95499
$962$	0	0
$963$	−22.4128	−0.722243
$964$	0	0
$965$	21.0528	0.677712
$966$	0	0
$967$	0.103808	0.00333823	0.00166912	−	0.999999i	$-0.499469\pi$
$967$	0.103808	0.00333823	0.00166912	+	0.999999i	$0.499469\pi$
$968$	0	0
$969$	19.8622	0.638064
$970$	0	0
$971$	17.2099	0.552292	0.276146	−	0.961116i	$-0.410943\pi$
$971$	17.2099	0.552292	0.276146	+	0.961116i	$0.410943\pi$
$972$	0	0
$973$	92.0666	2.95152
$974$	0	0
$975$	9.85499	0.315612
$976$	0	0
$977$	51.7432	1.65541	0.827706	−	0.561162i	$-0.189645\pi$
$977$	51.7432	1.65541	0.827706	+	0.561162i	$0.189645\pi$
$978$	0	0
$979$	11.4859	0.367091
$980$	0	0
$981$	−3.62698	−0.115800
$982$	0	0
$983$	20.0873	0.640684	0.320342	−	0.947302i	$-0.396202\pi$
$983$	20.0873	0.640684	0.320342	+	0.947302i	$0.396202\pi$
$984$	0	0
$985$	8.50338	0.270940
$986$	0	0
$987$	20.3693	0.648360
$988$	0	0
$989$	1.32089	0.0420020
$990$	0	0
$991$	50.0134	1.58873	0.794364	−	0.607442i	$-0.207804\pi$
$991$	50.0134	1.58873	0.794364	+	0.607442i	$0.207804\pi$
$992$	0	0
$993$	88.5173	2.80901
$994$	0	0
$995$	36.5951	1.16014
$996$	0	0
$997$	−7.20554	−0.228202	−0.114101	−	0.993469i	$-0.536399\pi$
$997$	−7.20554	−0.228202	−0.114101	+	0.993469i	$0.536399\pi$
$998$	0	0
$999$	−1.92114	−0.0607823

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.y.1.21	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.y.1.21	✓	23	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q+2.49483 q^{3} +1.95871 q^{5} -4.15658 q^{7} +3.22420 q^{9} +O(q^{10})\)	\(q+2.49483 q^{3} +1.95871 q^{5} -4.15658 q^{7} +3.22420 q^{9} +5.07386 q^{11} -3.39523 q^{13} +4.88666 q^{15} -1.00000 q^{17} -7.96131 q^{19} -10.3700 q^{21} -1.52248 q^{23} -1.16344 q^{25} +0.559335 q^{27} +2.94482 q^{29} -9.57103 q^{31} +12.6584 q^{33} -8.14155 q^{35} -3.43470 q^{37} -8.47053 q^{39} +1.52328 q^{41} -0.867593 q^{43} +6.31528 q^{45} -1.96425 q^{47} +10.2772 q^{49} -2.49483 q^{51} +3.60603 q^{53} +9.93824 q^{55} -19.8622 q^{57} -1.00000 q^{59} -2.74860 q^{61} -13.4016 q^{63} -6.65028 q^{65} +16.0573 q^{67} -3.79833 q^{69} +1.96161 q^{71} -10.9901 q^{73} -2.90260 q^{75} -21.0899 q^{77} -9.09546 q^{79} -8.27714 q^{81} -17.0316 q^{83} -1.95871 q^{85} +7.34684 q^{87} +2.26374 q^{89} +14.1125 q^{91} -23.8781 q^{93} -15.5939 q^{95} +4.97767 q^{97} +16.3591 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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