LMFDB - Embedded newform 7616.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7616,2,Mod(1,7616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7616, 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("7616.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7616 = 2^{6} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7616.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:	$60.8140661794$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 238)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	7616.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23607 q^{3} -3.23607 q^{5} +1.00000 q^{7} -1.47214 q^{9} +0.763932 q^{11} -6.47214 q^{13} +4.00000 q^{15} +1.00000 q^{17} +0.472136 q^{19} -1.23607 q^{21} -8.00000 q^{23} +5.47214 q^{25} +5.52786 q^{27} -7.70820 q^{29} +10.4721 q^{31} -0.944272 q^{33} -3.23607 q^{35} -1.23607 q^{37} +8.00000 q^{39} -12.4721 q^{41} -6.47214 q^{43} +4.76393 q^{45} -6.47214 q^{47} +1.00000 q^{49} -1.23607 q^{51} -4.47214 q^{53} -2.47214 q^{55} -0.583592 q^{57} -6.00000 q^{59} +3.23607 q^{61} -1.47214 q^{63} +20.9443 q^{65} -10.4721 q^{67} +9.88854 q^{69} -6.47214 q^{71} -13.4164 q^{73} -6.76393 q^{75} +0.763932 q^{77} +1.52786 q^{79} -2.41641 q^{81} +2.00000 q^{83} -3.23607 q^{85} +9.52786 q^{87} +2.00000 q^{89} -6.47214 q^{91} -12.9443 q^{93} -1.52786 q^{95} +0.472136 q^{97} -1.12461 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 6 q^{9} + 6 q^{11} - 4 q^{13} + 8 q^{15} + 2 q^{17} - 8 q^{19} + 2 q^{21} - 16 q^{23} + 2 q^{25} + 20 q^{27} - 2 q^{29} + 12 q^{31} + 16 q^{33} - 2 q^{35} + 2 q^{37}+ \cdots + 38 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 6 * q^9 + 6 * q^11 - 4 * q^13 + 8 * q^15 + 2 * q^17 - 8 * q^19 + 2 * q^21 - 16 * q^23 + 2 * q^25 + 20 * q^27 - 2 * q^29 + 12 * q^31 + 16 * q^33 - 2 * q^35 + 2 * q^37 + 16 * q^39 - 16 * q^41 - 4 * q^43 + 14 * q^45 - 4 * q^47 + 2 * q^49 + 2 * q^51 + 4 * q^55 - 28 * q^57 - 12 * q^59 + 2 * q^61 + 6 * q^63 + 24 * q^65 - 12 * q^67 - 16 * q^69 - 4 * q^71 - 18 * q^75 + 6 * q^77 + 12 * q^79 + 22 * q^81 + 4 * q^83 - 2 * q^85 + 28 * q^87 + 4 * q^89 - 4 * q^91 - 8 * q^93 - 12 * q^95 - 8 * q^97 + 38 * q^99

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
$3$	−1.23607	−0.713644	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	−1.23607	−0.713644	−0.356822	+	0.934172i	$0.616140\pi$
$4$	0	0
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.47214	−0.490712
$10$	0	0
$11$	0.763932	0.230334	0.115167	−	0.993346i	$-0.463260\pi$
$11$	0.763932	0.230334	0.115167	+	0.993346i	$0.463260\pi$
$12$	0	0
$13$	−6.47214	−1.79505	−0.897524	−	0.440966i	$-0.854636\pi$
$13$	−6.47214	−1.79505	−0.897524	+	0.440966i	$0.854636\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	0.472136	0.108315	0.0541577	−	0.998532i	$-0.482753\pi$
$19$	0.472136	0.108315	0.0541577	+	0.998532i	$0.482753\pi$
$20$	0	0
$21$	−1.23607	−0.269732
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	5.52786	1.06384
$28$	0	0
$29$	−7.70820	−1.43138	−0.715689	−	0.698419i	$-0.753887\pi$
$29$	−7.70820	−1.43138	−0.715689	+	0.698419i	$0.753887\pi$
$30$	0	0
$31$	10.4721	1.88085	0.940426	−	0.340000i	$-0.110427\pi$
$31$	10.4721	1.88085	0.940426	+	0.340000i	$0.110427\pi$
$32$	0	0
$33$	−0.944272	−0.164377
$34$	0	0
$35$	−3.23607	−0.546995
$36$	0	0
$37$	−1.23607	−0.203208	−0.101604	−	0.994825i	$-0.532398\pi$
$37$	−1.23607	−0.203208	−0.101604	+	0.994825i	$0.532398\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	0	0
$41$	−12.4721	−1.94782	−0.973910	−	0.226934i	$-0.927130\pi$
$41$	−12.4721	−1.94782	−0.973910	+	0.226934i	$0.927130\pi$
$42$	0	0
$43$	−6.47214	−0.986991	−0.493496	−	0.869748i	$-0.664281\pi$
$43$	−6.47214	−0.986991	−0.493496	+	0.869748i	$0.664281\pi$
$44$	0	0
$45$	4.76393	0.710165
$46$	0	0
$47$	−6.47214	−0.944058	−0.472029	−	0.881583i	$-0.656478\pi$
$47$	−6.47214	−0.944058	−0.472029	+	0.881583i	$0.656478\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.23607	−0.173084
$52$	0	0
$53$	−4.47214	−0.614295	−0.307148	−	0.951662i	$-0.599375\pi$
$53$	−4.47214	−0.614295	−0.307148	+	0.951662i	$0.599375\pi$
$54$	0	0
$55$	−2.47214	−0.333343
$56$	0	0
$57$	−0.583592	−0.0772987
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	3.23607	0.414336	0.207168	−	0.978305i	$-0.433575\pi$
$61$	3.23607	0.414336	0.207168	+	0.978305i	$0.433575\pi$
$62$	0	0
$63$	−1.47214	−0.185472
$64$	0	0
$65$	20.9443	2.59782
$66$	0	0
$67$	−10.4721	−1.27938	−0.639688	−	0.768635i	$-0.720936\pi$
$67$	−10.4721	−1.27938	−0.639688	+	0.768635i	$0.720936\pi$
$68$	0	0
$69$	9.88854	1.19044
$70$	0	0
$71$	−6.47214	−0.768101	−0.384051	−	0.923312i	$-0.625471\pi$
$71$	−6.47214	−0.768101	−0.384051	+	0.923312i	$0.625471\pi$
$72$	0	0
$73$	−13.4164	−1.57027	−0.785136	−	0.619324i	$-0.787407\pi$
$73$	−13.4164	−1.57027	−0.785136	+	0.619324i	$0.787407\pi$
$74$	0	0
$75$	−6.76393	−0.781032
$76$	0	0
$77$	0.763932	0.0870581
$78$	0	0
$79$	1.52786	0.171898	0.0859491	−	0.996300i	$-0.472608\pi$
$79$	1.52786	0.171898	0.0859491	+	0.996300i	$0.472608\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	−3.23607	−0.351001
$86$	0	0
$87$	9.52786	1.02149
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−6.47214	−0.678464
$92$	0	0
$93$	−12.9443	−1.34226
$94$	0	0
$95$	−1.52786	−0.156756
$96$	0	0
$97$	0.472136	0.0479381	0.0239691	−	0.999713i	$-0.492370\pi$
$97$	0.472136	0.0479381	0.0239691	+	0.999713i	$0.492370\pi$
$98$	0	0
$99$	−1.12461	−0.113028
$100$	0	0
$101$	−2.47214	−0.245987	−0.122993	−	0.992407i	$-0.539249\pi$
$101$	−2.47214	−0.245987	−0.122993	+	0.992407i	$0.539249\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	3.23607	0.312842	0.156421	−	0.987690i	$-0.450004\pi$
$107$	3.23607	0.312842	0.156421	+	0.987690i	$0.450004\pi$
$108$	0	0
$109$	10.1803	0.975100	0.487550	−	0.873095i	$-0.337891\pi$
$109$	10.1803	0.975100	0.487550	+	0.873095i	$0.337891\pi$
$110$	0	0
$111$	1.52786	0.145018
$112$	0	0
$113$	−13.4164	−1.26211	−0.631055	−	0.775738i	$-0.717378\pi$
$113$	−13.4164	−1.26211	−0.631055	+	0.775738i	$0.717378\pi$
$114$	0	0
$115$	25.8885	2.41412
$116$	0	0
$117$	9.52786	0.880851
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−10.4164	−0.946946
$122$	0	0
$123$	15.4164	1.39005
$124$	0	0
$125$	−1.52786	−0.136656
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	−11.7082	−1.02295	−0.511475	−	0.859298i	$-0.670901\pi$
$131$	−11.7082	−1.02295	−0.511475	+	0.859298i	$0.670901\pi$
$132$	0	0
$133$	0.472136	0.0409394
$134$	0	0
$135$	−17.8885	−1.53960
$136$	0	0
$137$	12.4721	1.06557	0.532783	−	0.846252i	$-0.321146\pi$
$137$	12.4721	1.06557	0.532783	+	0.846252i	$0.321146\pi$
$138$	0	0
$139$	10.1803	0.863485	0.431743	−	0.901997i	$-0.357899\pi$
$139$	10.1803	0.863485	0.431743	+	0.901997i	$0.357899\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	−4.94427	−0.413461
$144$	0	0
$145$	24.9443	2.07151
$146$	0	0
$147$	−1.23607	−0.101949
$148$	0	0
$149$	−1.05573	−0.0864886	−0.0432443	−	0.999065i	$-0.513769\pi$
$149$	−1.05573	−0.0864886	−0.0432443	+	0.999065i	$0.513769\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	−1.47214	−0.119015
$154$	0	0
$155$	−33.8885	−2.72199
$156$	0	0
$157$	−3.05573	−0.243874	−0.121937	−	0.992538i	$-0.538911\pi$
$157$	−3.05573	−0.243874	−0.121937	+	0.992538i	$0.538911\pi$
$158$	0	0
$159$	5.52786	0.438388
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	−13.7082	−1.07371	−0.536855	−	0.843675i	$-0.680388\pi$
$163$	−13.7082	−1.07371	−0.536855	+	0.843675i	$0.680388\pi$
$164$	0	0
$165$	3.05573	0.237888
$166$	0	0
$167$	5.52786	0.427759	0.213879	−	0.976860i	$-0.431390\pi$
$167$	5.52786	0.427759	0.213879	+	0.976860i	$0.431390\pi$
$168$	0	0
$169$	28.8885	2.22220
$170$	0	0
$171$	−0.695048	−0.0531517
$172$	0	0
$173$	10.6525	0.809893	0.404946	−	0.914340i	$-0.367290\pi$
$173$	10.6525	0.809893	0.404946	+	0.914340i	$0.367290\pi$
$174$	0	0
$175$	5.47214	0.413655
$176$	0	0
$177$	7.41641	0.557451
$178$	0	0
$179$	8.94427	0.668526	0.334263	−	0.942480i	$-0.391513\pi$
$179$	8.94427	0.668526	0.334263	+	0.942480i	$0.391513\pi$
$180$	0	0
$181$	7.23607	0.537853	0.268926	−	0.963161i	$-0.413331\pi$
$181$	7.23607	0.537853	0.268926	+	0.963161i	$0.413331\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	0.763932	0.0558642
$188$	0	0
$189$	5.52786	0.402093
$190$	0	0
$191$	4.94427	0.357755	0.178877	−	0.983871i	$-0.442753\pi$
$191$	4.94427	0.357755	0.178877	+	0.983871i	$0.442753\pi$
$192$	0	0
$193$	23.8885	1.71954	0.859768	−	0.510686i	$-0.170608\pi$
$193$	23.8885	1.71954	0.859768	+	0.510686i	$0.170608\pi$
$194$	0	0
$195$	−25.8885	−1.85392
$196$	0	0
$197$	−8.29180	−0.590766	−0.295383	−	0.955379i	$-0.595447\pi$
$197$	−8.29180	−0.590766	−0.295383	+	0.955379i	$0.595447\pi$
$198$	0	0
$199$	−5.52786	−0.391860	−0.195930	−	0.980618i	$-0.562773\pi$
$199$	−5.52786	−0.391860	−0.195930	+	0.980618i	$0.562773\pi$
$200$	0	0
$201$	12.9443	0.913019
$202$	0	0
$203$	−7.70820	−0.541010
$204$	0	0
$205$	40.3607	2.81891
$206$	0	0
$207$	11.7771	0.818564
$208$	0	0
$209$	0.360680	0.0249487
$210$	0	0
$211$	−14.2918	−0.983888	−0.491944	−	0.870627i	$-0.663713\pi$
$211$	−14.2918	−0.983888	−0.491944	+	0.870627i	$0.663713\pi$
$212$	0	0
$213$	8.00000	0.548151
$214$	0	0
$215$	20.9443	1.42839
$216$	0	0
$217$	10.4721	0.710895
$218$	0	0
$219$	16.5836	1.12062
$220$	0	0
$221$	−6.47214	−0.435363
$222$	0	0
$223$	−10.4721	−0.701266	−0.350633	−	0.936513i	$-0.614034\pi$
$223$	−10.4721	−0.701266	−0.350633	+	0.936513i	$0.614034\pi$
$224$	0	0
$225$	−8.05573	−0.537049
$226$	0	0
$227$	−21.5967	−1.43343	−0.716713	−	0.697368i	$-0.754354\pi$
$227$	−21.5967	−1.43343	−0.716713	+	0.697368i	$0.754354\pi$
$228$	0	0
$229$	−12.9443	−0.855382	−0.427691	−	0.903925i	$-0.640673\pi$
$229$	−12.9443	−0.855382	−0.427691	+	0.903925i	$0.640673\pi$
$230$	0	0
$231$	−0.944272	−0.0621285
$232$	0	0
$233$	−15.8885	−1.04089	−0.520447	−	0.853894i	$-0.674235\pi$
$233$	−15.8885	−1.04089	−0.520447	+	0.853894i	$0.674235\pi$
$234$	0	0
$235$	20.9443	1.36625
$236$	0	0
$237$	−1.88854	−0.122674
$238$	0	0
$239$	−12.9443	−0.837295	−0.418648	−	0.908149i	$-0.637496\pi$
$239$	−12.9443	−0.837295	−0.418648	+	0.908149i	$0.637496\pi$
$240$	0	0
$241$	−23.8885	−1.53880	−0.769398	−	0.638769i	$-0.779444\pi$
$241$	−23.8885	−1.53880	−0.769398	+	0.638769i	$0.779444\pi$
$242$	0	0
$243$	−13.5967	−0.872232
$244$	0	0
$245$	−3.23607	−0.206745
$246$	0	0
$247$	−3.05573	−0.194431
$248$	0	0
$249$	−2.47214	−0.156665
$250$	0	0
$251$	14.0000	0.883672	0.441836	−	0.897096i	$-0.354327\pi$
$251$	14.0000	0.883672	0.441836	+	0.897096i	$0.354327\pi$
$252$	0	0
$253$	−6.11146	−0.384224
$254$	0	0
$255$	4.00000	0.250490
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	−1.23607	−0.0768055
$260$	0	0
$261$	11.3475	0.702394
$262$	0	0
$263$	−24.9443	−1.53813	−0.769065	−	0.639171i	$-0.779278\pi$
$263$	−24.9443	−1.53813	−0.769065	+	0.639171i	$0.779278\pi$
$264$	0	0
$265$	14.4721	0.889016
$266$	0	0
$267$	−2.47214	−0.151292
$268$	0	0
$269$	13.7082	0.835804	0.417902	−	0.908492i	$-0.362766\pi$
$269$	13.7082	0.835804	0.417902	+	0.908492i	$0.362766\pi$
$270$	0	0
$271$	7.41641	0.450515	0.225257	−	0.974299i	$-0.427678\pi$
$271$	7.41641	0.450515	0.225257	+	0.974299i	$0.427678\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	0	0
$275$	4.18034	0.252084
$276$	0	0
$277$	−8.65248	−0.519877	−0.259938	−	0.965625i	$-0.583702\pi$
$277$	−8.65248	−0.519877	−0.259938	+	0.965625i	$0.583702\pi$
$278$	0	0
$279$	−15.4164	−0.922956
$280$	0	0
$281$	12.4721	0.744025	0.372013	−	0.928228i	$-0.378668\pi$
$281$	12.4721	0.744025	0.372013	+	0.928228i	$0.378668\pi$
$282$	0	0
$283$	23.7082	1.40931	0.704653	−	0.709552i	$-0.251103\pi$
$283$	23.7082	1.40931	0.704653	+	0.709552i	$0.251103\pi$
$284$	0	0
$285$	1.88854	0.111868
$286$	0	0
$287$	−12.4721	−0.736207
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−0.583592	−0.0342108
$292$	0	0
$293$	28.0000	1.63578	0.817889	−	0.575376i	$-0.195144\pi$
$293$	28.0000	1.63578	0.817889	+	0.575376i	$0.195144\pi$
$294$	0	0
$295$	19.4164	1.13047
$296$	0	0
$297$	4.22291	0.245038
$298$	0	0
$299$	51.7771	2.99435
$300$	0	0
$301$	−6.47214	−0.373048
$302$	0	0
$303$	3.05573	0.175547
$304$	0	0
$305$	−10.4721	−0.599633
$306$	0	0
$307$	−12.4721	−0.711822	−0.355911	−	0.934520i	$-0.615829\pi$
$307$	−12.4721	−0.711822	−0.355911	+	0.934520i	$0.615829\pi$
$308$	0	0
$309$	−4.94427	−0.281270
$310$	0	0
$311$	3.05573	0.173274	0.0866372	−	0.996240i	$-0.472388\pi$
$311$	3.05573	0.173274	0.0866372	+	0.996240i	$0.472388\pi$
$312$	0	0
$313$	21.4164	1.21053	0.605263	−	0.796025i	$-0.293068\pi$
$313$	21.4164	1.21053	0.605263	+	0.796025i	$0.293068\pi$
$314$	0	0
$315$	4.76393	0.268417
$316$	0	0
$317$	5.81966	0.326865	0.163432	−	0.986555i	$-0.447743\pi$
$317$	5.81966	0.326865	0.163432	+	0.986555i	$0.447743\pi$
$318$	0	0
$319$	−5.88854	−0.329695
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	0.472136	0.0262703
$324$	0	0
$325$	−35.4164	−1.96455
$326$	0	0
$327$	−12.5836	−0.695874
$328$	0	0
$329$	−6.47214	−0.356820
$330$	0	0
$331$	24.3607	1.33898	0.669492	−	0.742819i	$-0.266512\pi$
$331$	24.3607	1.33898	0.669492	+	0.742819i	$0.266512\pi$
$332$	0	0
$333$	1.81966	0.0997168
$334$	0	0
$335$	33.8885	1.85153
$336$	0	0
$337$	−11.5279	−0.627963	−0.313981	−	0.949429i	$-0.601663\pi$
$337$	−11.5279	−0.627963	−0.313981	+	0.949429i	$0.601663\pi$
$338$	0	0
$339$	16.5836	0.900697
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−32.0000	−1.72282
$346$	0	0
$347$	−26.6525	−1.43078	−0.715390	−	0.698725i	$-0.753751\pi$
$347$	−26.6525	−1.43078	−0.715390	+	0.698725i	$0.753751\pi$
$348$	0	0
$349$	30.4721	1.63114	0.815568	−	0.578661i	$-0.196425\pi$
$349$	30.4721	1.63114	0.815568	+	0.578661i	$0.196425\pi$
$350$	0	0
$351$	−35.7771	−1.90964
$352$	0	0
$353$	23.8885	1.27146	0.635729	−	0.771912i	$-0.280699\pi$
$353$	23.8885	1.27146	0.635729	+	0.771912i	$0.280699\pi$
$354$	0	0
$355$	20.9443	1.11161
$356$	0	0
$357$	−1.23607	−0.0654197
$358$	0	0
$359$	30.8328	1.62729	0.813647	−	0.581359i	$-0.197479\pi$
$359$	30.8328	1.62729	0.813647	+	0.581359i	$0.197479\pi$
$360$	0	0
$361$	−18.7771	−0.988268
$362$	0	0
$363$	12.8754	0.675783
$364$	0	0
$365$	43.4164	2.27252
$366$	0	0
$367$	11.0557	0.577104	0.288552	−	0.957464i	$-0.406826\pi$
$367$	11.0557	0.577104	0.288552	+	0.957464i	$0.406826\pi$
$368$	0	0
$369$	18.3607	0.955819
$370$	0	0
$371$	−4.47214	−0.232182
$372$	0	0
$373$	13.4164	0.694675	0.347338	−	0.937740i	$-0.387086\pi$
$373$	13.4164	0.694675	0.347338	+	0.937740i	$0.387086\pi$
$374$	0	0
$375$	1.88854	0.0975240
$376$	0	0
$377$	49.8885	2.56939
$378$	0	0
$379$	31.5967	1.62302	0.811508	−	0.584341i	$-0.198647\pi$
$379$	31.5967	1.62302	0.811508	+	0.584341i	$0.198647\pi$
$380$	0	0
$381$	14.8328	0.759908
$382$	0	0
$383$	−8.94427	−0.457031	−0.228515	−	0.973540i	$-0.573387\pi$
$383$	−8.94427	−0.457031	−0.228515	+	0.973540i	$0.573387\pi$
$384$	0	0
$385$	−2.47214	−0.125992
$386$	0	0
$387$	9.52786	0.484329
$388$	0	0
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	0	0
$393$	14.4721	0.730023
$394$	0	0
$395$	−4.94427	−0.248773
$396$	0	0
$397$	4.18034	0.209805	0.104903	−	0.994482i	$-0.466547\pi$
$397$	4.18034	0.209805	0.104903	+	0.994482i	$0.466547\pi$
$398$	0	0
$399$	−0.583592	−0.0292161
$400$	0	0
$401$	−14.9443	−0.746281	−0.373141	−	0.927775i	$-0.621719\pi$
$401$	−14.9443	−0.746281	−0.373141	+	0.927775i	$0.621719\pi$
$402$	0	0
$403$	−67.7771	−3.37622
$404$	0	0
$405$	7.81966	0.388562
$406$	0	0
$407$	−0.944272	−0.0468058
$408$	0	0
$409$	−2.94427	−0.145585	−0.0727924	−	0.997347i	$-0.523191\pi$
$409$	−2.94427	−0.145585	−0.0727924	+	0.997347i	$0.523191\pi$
$410$	0	0
$411$	−15.4164	−0.760435
$412$	0	0
$413$	−6.00000	−0.295241
$414$	0	0
$415$	−6.47214	−0.317705
$416$	0	0
$417$	−12.5836	−0.616221
$418$	0	0
$419$	28.0689	1.37125	0.685627	−	0.727953i	$-0.259528\pi$
$419$	28.0689	1.37125	0.685627	+	0.727953i	$0.259528\pi$
$420$	0	0
$421$	24.4721	1.19270	0.596349	−	0.802725i	$-0.296617\pi$
$421$	24.4721	1.19270	0.596349	+	0.802725i	$0.296617\pi$
$422$	0	0
$423$	9.52786	0.463261
$424$	0	0
$425$	5.47214	0.265438
$426$	0	0
$427$	3.23607	0.156604
$428$	0	0
$429$	6.11146	0.295064
$430$	0	0
$431$	−4.94427	−0.238157	−0.119079	−	0.992885i	$-0.537994\pi$
$431$	−4.94427	−0.238157	−0.119079	+	0.992885i	$0.537994\pi$
$432$	0	0
$433$	−6.94427	−0.333720	−0.166860	−	0.985981i	$-0.553363\pi$
$433$	−6.94427	−0.333720	−0.166860	+	0.985981i	$0.553363\pi$
$434$	0	0
$435$	−30.8328	−1.47832
$436$	0	0
$437$	−3.77709	−0.180683
$438$	0	0
$439$	20.9443	0.999616	0.499808	−	0.866136i	$-0.333404\pi$
$439$	20.9443	0.999616	0.499808	+	0.866136i	$0.333404\pi$
$440$	0	0
$441$	−1.47214	−0.0701017
$442$	0	0
$443$	36.3607	1.72755	0.863774	−	0.503879i	$-0.168094\pi$
$443$	36.3607	1.72755	0.863774	+	0.503879i	$0.168094\pi$
$444$	0	0
$445$	−6.47214	−0.306809
$446$	0	0
$447$	1.30495	0.0617221
$448$	0	0
$449$	−17.4164	−0.821931	−0.410966	−	0.911651i	$-0.634808\pi$
$449$	−17.4164	−0.821931	−0.410966	+	0.911651i	$0.634808\pi$
$450$	0	0
$451$	−9.52786	−0.448650
$452$	0	0
$453$	4.94427	0.232302
$454$	0	0
$455$	20.9443	0.981883
$456$	0	0
$457$	−19.8885	−0.930347	−0.465173	−	0.885220i	$-0.654008\pi$
$457$	−19.8885	−0.930347	−0.465173	+	0.885220i	$0.654008\pi$
$458$	0	0
$459$	5.52786	0.258019
$460$	0	0
$461$	14.4721	0.674035	0.337017	−	0.941498i	$-0.390582\pi$
$461$	14.4721	0.674035	0.337017	+	0.941498i	$0.390582\pi$
$462$	0	0
$463$	−15.0557	−0.699699	−0.349850	−	0.936806i	$-0.613767\pi$
$463$	−15.0557	−0.699699	−0.349850	+	0.936806i	$0.613767\pi$
$464$	0	0
$465$	41.8885	1.94253
$466$	0	0
$467$	34.3607	1.59002	0.795011	−	0.606595i	$-0.207465\pi$
$467$	34.3607	1.59002	0.795011	+	0.606595i	$0.207465\pi$
$468$	0	0
$469$	−10.4721	−0.483558
$470$	0	0
$471$	3.77709	0.174039
$472$	0	0
$473$	−4.94427	−0.227338
$474$	0	0
$475$	2.58359	0.118543
$476$	0	0
$477$	6.58359	0.301442
$478$	0	0
$479$	−7.41641	−0.338864	−0.169432	−	0.985542i	$-0.554193\pi$
$479$	−7.41641	−0.338864	−0.169432	+	0.985542i	$0.554193\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	9.88854	0.449944
$484$	0	0
$485$	−1.52786	−0.0693767
$486$	0	0
$487$	−20.9443	−0.949076	−0.474538	−	0.880235i	$-0.657385\pi$
$487$	−20.9443	−0.949076	−0.474538	+	0.880235i	$0.657385\pi$
$488$	0	0
$489$	16.9443	0.766246
$490$	0	0
$491$	−15.0557	−0.679455	−0.339728	−	0.940524i	$-0.610335\pi$
$491$	−15.0557	−0.679455	−0.339728	+	0.940524i	$0.610335\pi$
$492$	0	0
$493$	−7.70820	−0.347160
$494$	0	0
$495$	3.63932	0.163575
$496$	0	0
$497$	−6.47214	−0.290315
$498$	0	0
$499$	−9.34752	−0.418453	−0.209226	−	0.977867i	$-0.567095\pi$
$499$	−9.34752	−0.418453	−0.209226	+	0.977867i	$0.567095\pi$
$500$	0	0
$501$	−6.83282	−0.305268
$502$	0	0
$503$	4.94427	0.220454	0.110227	−	0.993906i	$-0.464842\pi$
$503$	4.94427	0.220454	0.110227	+	0.993906i	$0.464842\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	0	0
$507$	−35.7082	−1.58586
$508$	0	0
$509$	20.9443	0.928339	0.464169	−	0.885747i	$-0.346353\pi$
$509$	20.9443	0.928339	0.464169	+	0.885747i	$0.346353\pi$
$510$	0	0
$511$	−13.4164	−0.593507
$512$	0	0
$513$	2.60990	0.115230
$514$	0	0
$515$	−12.9443	−0.570393
$516$	0	0
$517$	−4.94427	−0.217449
$518$	0	0
$519$	−13.1672	−0.577975
$520$	0	0
$521$	−18.9443	−0.829964	−0.414982	−	0.909830i	$-0.636212\pi$
$521$	−18.9443	−0.829964	−0.414982	+	0.909830i	$0.636212\pi$
$522$	0	0
$523$	−25.0557	−1.09561	−0.547805	−	0.836606i	$-0.684537\pi$
$523$	−25.0557	−1.09561	−0.547805	+	0.836606i	$0.684537\pi$
$524$	0	0
$525$	−6.76393	−0.295202
$526$	0	0
$527$	10.4721	0.456173
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	8.83282	0.383312
$532$	0	0
$533$	80.7214	3.49643
$534$	0	0
$535$	−10.4721	−0.452750
$536$	0	0
$537$	−11.0557	−0.477090
$538$	0	0
$539$	0.763932	0.0329049
$540$	0	0
$541$	18.7639	0.806724	0.403362	−	0.915040i	$-0.367841\pi$
$541$	18.7639	0.806724	0.403362	+	0.915040i	$0.367841\pi$
$542$	0	0
$543$	−8.94427	−0.383835
$544$	0	0
$545$	−32.9443	−1.41118
$546$	0	0
$547$	−7.59675	−0.324813	−0.162407	−	0.986724i	$-0.551926\pi$
$547$	−7.59675	−0.324813	−0.162407	+	0.986724i	$0.551926\pi$
$548$	0	0
$549$	−4.76393	−0.203320
$550$	0	0
$551$	−3.63932	−0.155040
$552$	0	0
$553$	1.52786	0.0649714
$554$	0	0
$555$	−4.94427	−0.209873
$556$	0	0
$557$	10.3607	0.438996	0.219498	−	0.975613i	$-0.429558\pi$
$557$	10.3607	0.438996	0.219498	+	0.975613i	$0.429558\pi$
$558$	0	0
$559$	41.8885	1.77170
$560$	0	0
$561$	−0.944272	−0.0398672
$562$	0	0
$563$	20.8328	0.877999	0.438999	−	0.898487i	$-0.355333\pi$
$563$	20.8328	0.877999	0.438999	+	0.898487i	$0.355333\pi$
$564$	0	0
$565$	43.4164	1.82654
$566$	0	0
$567$	−2.41641	−0.101480
$568$	0	0
$569$	−19.8885	−0.833771	−0.416886	−	0.908959i	$-0.636878\pi$
$569$	−19.8885	−0.833771	−0.416886	+	0.908959i	$0.636878\pi$
$570$	0	0
$571$	−8.76393	−0.366759	−0.183380	−	0.983042i	$-0.558704\pi$
$571$	−8.76393	−0.366759	−0.183380	+	0.983042i	$0.558704\pi$
$572$	0	0
$573$	−6.11146	−0.255310
$574$	0	0
$575$	−43.7771	−1.82563
$576$	0	0
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	0	0
$579$	−29.5279	−1.22714
$580$	0	0
$581$	2.00000	0.0829740
$582$	0	0
$583$	−3.41641	−0.141493
$584$	0	0
$585$	−30.8328	−1.27478
$586$	0	0
$587$	−5.41641	−0.223559	−0.111780	−	0.993733i	$-0.535655\pi$
$587$	−5.41641	−0.223559	−0.111780	+	0.993733i	$0.535655\pi$
$588$	0	0
$589$	4.94427	0.203725
$590$	0	0
$591$	10.2492	0.421597
$592$	0	0
$593$	14.9443	0.613688	0.306844	−	0.951760i	$-0.400727\pi$
$593$	14.9443	0.613688	0.306844	+	0.951760i	$0.400727\pi$
$594$	0	0
$595$	−3.23607	−0.132666
$596$	0	0
$597$	6.83282	0.279649
$598$	0	0
$599$	39.7771	1.62525	0.812624	−	0.582789i	$-0.198038\pi$
$599$	39.7771	1.62525	0.812624	+	0.582789i	$0.198038\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	15.4164	0.627805
$604$	0	0
$605$	33.7082	1.37043
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	9.52786	0.386089
$610$	0	0
$611$	41.8885	1.69463
$612$	0	0
$613$	−8.83282	−0.356754	−0.178377	−	0.983962i	$-0.557085\pi$
$613$	−8.83282	−0.356754	−0.178377	+	0.983962i	$0.557085\pi$
$614$	0	0
$615$	−49.8885	−2.01170
$616$	0	0
$617$	−34.3607	−1.38331	−0.691654	−	0.722229i	$-0.743118\pi$
$617$	−34.3607	−1.38331	−0.691654	+	0.722229i	$0.743118\pi$
$618$	0	0
$619$	12.2918	0.494049	0.247024	−	0.969009i	$-0.420547\pi$
$619$	12.2918	0.494049	0.247024	+	0.969009i	$0.420547\pi$
$620$	0	0
$621$	−44.2229	−1.77460
$622$	0	0
$623$	2.00000	0.0801283
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	−0.445825	−0.0178045
$628$	0	0
$629$	−1.23607	−0.0492853
$630$	0	0
$631$	−20.9443	−0.833778	−0.416889	−	0.908957i	$-0.636880\pi$
$631$	−20.9443	−0.833778	−0.416889	+	0.908957i	$0.636880\pi$
$632$	0	0
$633$	17.6656	0.702146
$634$	0	0
$635$	38.8328	1.54103
$636$	0	0
$637$	−6.47214	−0.256435
$638$	0	0
$639$	9.52786	0.376916
$640$	0	0
$641$	26.9443	1.06423	0.532117	−	0.846671i	$-0.321397\pi$
$641$	26.9443	1.06423	0.532117	+	0.846671i	$0.321397\pi$
$642$	0	0
$643$	−27.1246	−1.06969	−0.534845	−	0.844950i	$-0.679630\pi$
$643$	−27.1246	−1.06969	−0.534845	+	0.844950i	$0.679630\pi$
$644$	0	0
$645$	−25.8885	−1.01936
$646$	0	0
$647$	24.3607	0.957717	0.478859	−	0.877892i	$-0.341051\pi$
$647$	24.3607	0.957717	0.478859	+	0.877892i	$0.341051\pi$
$648$	0	0
$649$	−4.58359	−0.179922
$650$	0	0
$651$	−12.9443	−0.507326
$652$	0	0
$653$	−41.5967	−1.62781	−0.813903	−	0.581000i	$-0.802661\pi$
$653$	−41.5967	−1.62781	−0.813903	+	0.581000i	$0.802661\pi$
$654$	0	0
$655$	37.8885	1.48043
$656$	0	0
$657$	19.7508	0.770551
$658$	0	0
$659$	−3.05573	−0.119034	−0.0595171	−	0.998227i	$-0.518956\pi$
$659$	−3.05573	−0.119034	−0.0595171	+	0.998227i	$0.518956\pi$
$660$	0	0
$661$	−23.4164	−0.910793	−0.455396	−	0.890289i	$-0.650502\pi$
$661$	−23.4164	−0.910793	−0.455396	+	0.890289i	$0.650502\pi$
$662$	0	0
$663$	8.00000	0.310694
$664$	0	0
$665$	−1.52786	−0.0592480
$666$	0	0
$667$	61.6656	2.38770
$668$	0	0
$669$	12.9443	0.500454
$670$	0	0
$671$	2.47214	0.0954358
$672$	0	0
$673$	39.3050	1.51509	0.757547	−	0.652780i	$-0.226398\pi$
$673$	39.3050	1.51509	0.757547	+	0.652780i	$0.226398\pi$
$674$	0	0
$675$	30.2492	1.16429
$676$	0	0
$677$	−11.8197	−0.454266	−0.227133	−	0.973864i	$-0.572935\pi$
$677$	−11.8197	−0.454266	−0.227133	+	0.973864i	$0.572935\pi$
$678$	0	0
$679$	0.472136	0.0181189
$680$	0	0
$681$	26.6950	1.02296
$682$	0	0
$683$	23.8197	0.911434	0.455717	−	0.890125i	$-0.349383\pi$
$683$	23.8197	0.911434	0.455717	+	0.890125i	$0.349383\pi$
$684$	0	0
$685$	−40.3607	−1.54210
$686$	0	0
$687$	16.0000	0.610438
$688$	0	0
$689$	28.9443	1.10269
$690$	0	0
$691$	−3.34752	−0.127346	−0.0636729	−	0.997971i	$-0.520281\pi$
$691$	−3.34752	−0.127346	−0.0636729	+	0.997971i	$0.520281\pi$
$692$	0	0
$693$	−1.12461	−0.0427205
$694$	0	0
$695$	−32.9443	−1.24965
$696$	0	0
$697$	−12.4721	−0.472416
$698$	0	0
$699$	19.6393	0.742827
$700$	0	0
$701$	−44.2492	−1.67127	−0.835635	−	0.549285i	$-0.814900\pi$
$701$	−44.2492	−1.67127	−0.835635	+	0.549285i	$0.814900\pi$
$702$	0	0
$703$	−0.583592	−0.0220106
$704$	0	0
$705$	−25.8885	−0.975019
$706$	0	0
$707$	−2.47214	−0.0929742
$708$	0	0
$709$	−25.5967	−0.961306	−0.480653	−	0.876911i	$-0.659600\pi$
$709$	−25.5967	−0.961306	−0.480653	+	0.876911i	$0.659600\pi$
$710$	0	0
$711$	−2.24922	−0.0843525
$712$	0	0
$713$	−83.7771	−3.13748
$714$	0	0
$715$	16.0000	0.598366
$716$	0	0
$717$	16.0000	0.597531
$718$	0	0
$719$	0.583592	0.0217643	0.0108822	−	0.999941i	$-0.496536\pi$
$719$	0.583592	0.0217643	0.0108822	+	0.999941i	$0.496536\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	29.5279	1.09815
$724$	0	0
$725$	−42.1803	−1.56654
$726$	0	0
$727$	46.4721	1.72356	0.861778	−	0.507285i	$-0.169351\pi$
$727$	46.4721	1.72356	0.861778	+	0.507285i	$0.169351\pi$
$728$	0	0
$729$	24.0557	0.890953
$730$	0	0
$731$	−6.47214	−0.239381
$732$	0	0
$733$	31.4164	1.16039	0.580196	−	0.814477i	$-0.302976\pi$
$733$	31.4164	1.16039	0.580196	+	0.814477i	$0.302976\pi$
$734$	0	0
$735$	4.00000	0.147542
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	−6.83282	−0.251349	−0.125675	−	0.992072i	$-0.540110\pi$
$739$	−6.83282	−0.251349	−0.125675	+	0.992072i	$0.540110\pi$
$740$	0	0
$741$	3.77709	0.138755
$742$	0	0
$743$	−38.4721	−1.41141	−0.705703	−	0.708508i	$-0.749369\pi$
$743$	−38.4721	−1.41141	−0.705703	+	0.708508i	$0.749369\pi$
$744$	0	0
$745$	3.41641	0.125167
$746$	0	0
$747$	−2.94427	−0.107725
$748$	0	0
$749$	3.23607	0.118243
$750$	0	0
$751$	−11.0557	−0.403429	−0.201715	−	0.979444i	$-0.564651\pi$
$751$	−11.0557	−0.403429	−0.201715	+	0.979444i	$0.564651\pi$
$752$	0	0
$753$	−17.3050	−0.630627
$754$	0	0
$755$	12.9443	0.471090
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	0	0
$759$	7.55418	0.274199
$760$	0	0
$761$	−31.8885	−1.15596	−0.577979	−	0.816051i	$-0.696159\pi$
$761$	−31.8885	−1.15596	−0.577979	+	0.816051i	$0.696159\pi$
$762$	0	0
$763$	10.1803	0.368553
$764$	0	0
$765$	4.76393	0.172240
$766$	0	0
$767$	38.8328	1.40217
$768$	0	0
$769$	48.8328	1.76096	0.880478	−	0.474087i	$-0.157222\pi$
$769$	48.8328	1.76096	0.880478	+	0.474087i	$0.157222\pi$
$770$	0	0
$771$	2.47214	0.0890318
$772$	0	0
$773$	−4.00000	−0.143870	−0.0719350	−	0.997409i	$-0.522917\pi$
$773$	−4.00000	−0.143870	−0.0719350	+	0.997409i	$0.522917\pi$
$774$	0	0
$775$	57.3050	2.05845
$776$	0	0
$777$	1.52786	0.0548118
$778$	0	0
$779$	−5.88854	−0.210979
$780$	0	0
$781$	−4.94427	−0.176920
$782$	0	0
$783$	−42.6099	−1.52275
$784$	0	0
$785$	9.88854	0.352937
$786$	0	0
$787$	−3.70820	−0.132183	−0.0660916	−	0.997814i	$-0.521053\pi$
$787$	−3.70820	−0.132183	−0.0660916	+	0.997814i	$0.521053\pi$
$788$	0	0
$789$	30.8328	1.09768
$790$	0	0
$791$	−13.4164	−0.477033
$792$	0	0
$793$	−20.9443	−0.743753
$794$	0	0
$795$	−17.8885	−0.634441
$796$	0	0
$797$	−12.5836	−0.445734	−0.222867	−	0.974849i	$-0.571542\pi$
$797$	−12.5836	−0.445734	−0.222867	+	0.974849i	$0.571542\pi$
$798$	0	0
$799$	−6.47214	−0.228968
$800$	0	0
$801$	−2.94427	−0.104031
$802$	0	0
$803$	−10.2492	−0.361687
$804$	0	0
$805$	25.8885	0.912451
$806$	0	0
$807$	−16.9443	−0.596467
$808$	0	0
$809$	18.9443	0.666045	0.333023	−	0.942919i	$-0.391931\pi$
$809$	18.9443	0.666045	0.333023	+	0.942919i	$0.391931\pi$
$810$	0	0
$811$	−42.1803	−1.48115	−0.740576	−	0.671973i	$-0.765447\pi$
$811$	−42.1803	−1.48115	−0.740576	+	0.671973i	$0.765447\pi$
$812$	0	0
$813$	−9.16718	−0.321507
$814$	0	0
$815$	44.3607	1.55389
$816$	0	0
$817$	−3.05573	−0.106906
$818$	0	0
$819$	9.52786	0.332931
$820$	0	0
$821$	−33.0132	−1.15217	−0.576084	−	0.817391i	$-0.695420\pi$
$821$	−33.0132	−1.15217	−0.576084	+	0.817391i	$0.695420\pi$
$822$	0	0
$823$	−6.47214	−0.225604	−0.112802	−	0.993617i	$-0.535983\pi$
$823$	−6.47214	−0.225604	−0.112802	+	0.993617i	$0.535983\pi$
$824$	0	0
$825$	−5.16718	−0.179898
$826$	0	0
$827$	11.5967	0.403258	0.201629	−	0.979462i	$-0.435376\pi$
$827$	11.5967	0.403258	0.201629	+	0.979462i	$0.435376\pi$
$828$	0	0
$829$	−23.0557	−0.800759	−0.400379	−	0.916350i	$-0.631122\pi$
$829$	−23.0557	−0.800759	−0.400379	+	0.916350i	$0.631122\pi$
$830$	0	0
$831$	10.6950	0.371007
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−17.8885	−0.619059
$836$	0	0
$837$	57.8885	2.00092
$838$	0	0
$839$	−52.3607	−1.80769	−0.903846	−	0.427859i	$-0.859268\pi$
$839$	−52.3607	−1.80769	−0.903846	+	0.427859i	$0.859268\pi$
$840$	0	0
$841$	30.4164	1.04884
$842$	0	0
$843$	−15.4164	−0.530969
$844$	0	0
$845$	−93.4853	−3.21599
$846$	0	0
$847$	−10.4164	−0.357912
$848$	0	0
$849$	−29.3050	−1.00574
$850$	0	0
$851$	9.88854	0.338975
$852$	0	0
$853$	16.7639	0.573986	0.286993	−	0.957933i	$-0.407344\pi$
$853$	16.7639	0.573986	0.286993	+	0.957933i	$0.407344\pi$
$854$	0	0
$855$	2.24922	0.0769218
$856$	0	0
$857$	−12.4721	−0.426040	−0.213020	−	0.977048i	$-0.568330\pi$
$857$	−12.4721	−0.426040	−0.213020	+	0.977048i	$0.568330\pi$
$858$	0	0
$859$	−0.111456	−0.00380284	−0.00190142	−	0.999998i	$-0.500605\pi$
$859$	−0.111456	−0.00380284	−0.00190142	+	0.999998i	$0.500605\pi$
$860$	0	0
$861$	15.4164	0.525390
$862$	0	0
$863$	0.944272	0.0321434	0.0160717	−	0.999871i	$-0.494884\pi$
$863$	0.944272	0.0321434	0.0160717	+	0.999871i	$0.494884\pi$
$864$	0	0
$865$	−34.4721	−1.17209
$866$	0	0
$867$	−1.23607	−0.0419791
$868$	0	0
$869$	1.16718	0.0395940
$870$	0	0
$871$	67.7771	2.29654
$872$	0	0
$873$	−0.695048	−0.0235238
$874$	0	0
$875$	−1.52786	−0.0516512
$876$	0	0
$877$	31.1246	1.05100	0.525502	−	0.850793i	$-0.323878\pi$
$877$	31.1246	1.05100	0.525502	+	0.850793i	$0.323878\pi$
$878$	0	0
$879$	−34.6099	−1.16736
$880$	0	0
$881$	−29.0557	−0.978912	−0.489456	−	0.872028i	$-0.662805\pi$
$881$	−29.0557	−0.978912	−0.489456	+	0.872028i	$0.662805\pi$
$882$	0	0
$883$	−23.4164	−0.788025	−0.394012	−	0.919105i	$-0.628913\pi$
$883$	−23.4164	−0.788025	−0.394012	+	0.919105i	$0.628913\pi$
$884$	0	0
$885$	−24.0000	−0.806751
$886$	0	0
$887$	−20.3607	−0.683645	−0.341822	−	0.939765i	$-0.611044\pi$
$887$	−20.3607	−0.683645	−0.341822	+	0.939765i	$0.611044\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	0	0
$891$	−1.84597	−0.0618424
$892$	0	0
$893$	−3.05573	−0.102256
$894$	0	0
$895$	−28.9443	−0.967500
$896$	0	0
$897$	−64.0000	−2.13690
$898$	0	0
$899$	−80.7214	−2.69221
$900$	0	0
$901$	−4.47214	−0.148988
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	−23.4164	−0.778388
$906$	0	0
$907$	−11.2361	−0.373088	−0.186544	−	0.982447i	$-0.559729\pi$
$907$	−11.2361	−0.373088	−0.186544	+	0.982447i	$0.559729\pi$
$908$	0	0
$909$	3.63932	0.120709
$910$	0	0
$911$	32.3607	1.07216	0.536079	−	0.844168i	$-0.319905\pi$
$911$	32.3607	1.07216	0.536079	+	0.844168i	$0.319905\pi$
$912$	0	0
$913$	1.52786	0.0505649
$914$	0	0
$915$	12.9443	0.427924
$916$	0	0
$917$	−11.7082	−0.386639
$918$	0	0
$919$	44.9443	1.48257	0.741287	−	0.671188i	$-0.234216\pi$
$919$	44.9443	1.48257	0.741287	+	0.671188i	$0.234216\pi$
$920$	0	0
$921$	15.4164	0.507988
$922$	0	0
$923$	41.8885	1.37878
$924$	0	0
$925$	−6.76393	−0.222397
$926$	0	0
$927$	−5.88854	−0.193405
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	0.472136	0.0154736
$932$	0	0
$933$	−3.77709	−0.123656
$934$	0	0
$935$	−2.47214	−0.0808475
$936$	0	0
$937$	55.8885	1.82580	0.912900	−	0.408184i	$-0.133838\pi$
$937$	55.8885	1.82580	0.912900	+	0.408184i	$0.133838\pi$
$938$	0	0
$939$	−26.4721	−0.863886
$940$	0	0
$941$	−18.8754	−0.615320	−0.307660	−	0.951496i	$-0.599546\pi$
$941$	−18.8754	−0.615320	−0.307660	+	0.951496i	$0.599546\pi$
$942$	0	0
$943$	99.7771	3.24919
$944$	0	0
$945$	−17.8885	−0.581914
$946$	0	0
$947$	−6.87539	−0.223420	−0.111710	−	0.993741i	$-0.535633\pi$
$947$	−6.87539	−0.223420	−0.111710	+	0.993741i	$0.535633\pi$
$948$	0	0
$949$	86.8328	2.81871
$950$	0	0
$951$	−7.19350	−0.233265
$952$	0	0
$953$	22.9443	0.743238	0.371619	−	0.928385i	$-0.378803\pi$
$953$	22.9443	0.743238	0.371619	+	0.928385i	$0.378803\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	0	0
$957$	7.27864	0.235285
$958$	0	0
$959$	12.4721	0.402746
$960$	0	0
$961$	78.6656	2.53760
$962$	0	0
$963$	−4.76393	−0.153516
$964$	0	0
$965$	−77.3050	−2.48853
$966$	0	0
$967$	−27.0557	−0.870054	−0.435027	−	0.900418i	$-0.643261\pi$
$967$	−27.0557	−0.870054	−0.435027	+	0.900418i	$0.643261\pi$
$968$	0	0
$969$	−0.583592	−0.0187477
$970$	0	0
$971$	−28.8328	−0.925289	−0.462645	−	0.886544i	$-0.653099\pi$
$971$	−28.8328	−0.925289	−0.462645	+	0.886544i	$0.653099\pi$
$972$	0	0
$973$	10.1803	0.326367
$974$	0	0
$975$	43.7771	1.40199
$976$	0	0
$977$	−41.0557	−1.31349	−0.656745	−	0.754113i	$-0.728067\pi$
$977$	−41.0557	−1.31349	−0.656745	+	0.754113i	$0.728067\pi$
$978$	0	0
$979$	1.52786	0.0488307
$980$	0	0
$981$	−14.9868	−0.478493
$982$	0	0
$983$	−45.5279	−1.45211	−0.726057	−	0.687635i	$-0.758649\pi$
$983$	−45.5279	−1.45211	−0.726057	+	0.687635i	$0.758649\pi$
$984$	0	0
$985$	26.8328	0.854965
$986$	0	0
$987$	8.00000	0.254643
$988$	0	0
$989$	51.7771	1.64642
$990$	0	0
$991$	−58.2492	−1.85035	−0.925174	−	0.379544i	$-0.876081\pi$
$991$	−58.2492	−1.85035	−0.925174	+	0.379544i	$0.876081\pi$
$992$	0	0
$993$	−30.1115	−0.955558
$994$	0	0
$995$	17.8885	0.567105
$996$	0	0
$997$	−37.4853	−1.18717	−0.593586	−	0.804771i	$-0.702288\pi$
$997$	−37.4853	−1.18717	−0.593586	+	0.804771i	$0.702288\pi$
$998$	0	0
$999$	−6.83282	−0.216181

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.y.1.1		2
4.3	odd	2		7616.2.a.n.1.2		2
8.3	odd	2		238.2.a.f.1.1	✓	2
8.5	even	2		1904.2.a.f.1.2		2
24.11	even	2		2142.2.a.x.1.1		2
40.19	odd	2		5950.2.a.x.1.2		2
56.27	even	2		1666.2.a.o.1.2		2
136.67	odd	2		4046.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
238.2.a.f.1.1	✓	2	8.3	odd	2
1666.2.a.o.1.2		2	56.27	even	2
1904.2.a.f.1.2		2	8.5	even	2
2142.2.a.x.1.1		2	24.11	even	2
4046.2.a.v.1.2		2	136.67	odd	2
5950.2.a.x.1.2		2	40.19	odd	2
7616.2.a.n.1.2		2	4.3	odd	2
7616.2.a.y.1.1		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7616.a (trivial)

\(f(q)\)	\(=\)	\(q-1.23607 q^{3} -3.23607 q^{5} +1.00000 q^{7} -1.47214 q^{9} +0.763932 q^{11} -6.47214 q^{13} +4.00000 q^{15} +1.00000 q^{17} +0.472136 q^{19} -1.23607 q^{21} -8.00000 q^{23} +5.47214 q^{25} +5.52786 q^{27} -7.70820 q^{29} +10.4721 q^{31} -0.944272 q^{33} -3.23607 q^{35} -1.23607 q^{37} +8.00000 q^{39} -12.4721 q^{41} -6.47214 q^{43} +4.76393 q^{45} -6.47214 q^{47} +1.00000 q^{49} -1.23607 q^{51} -4.47214 q^{53} -2.47214 q^{55} -0.583592 q^{57} -6.00000 q^{59} +3.23607 q^{61} -1.47214 q^{63} +20.9443 q^{65} -10.4721 q^{67} +9.88854 q^{69} -6.47214 q^{71} -13.4164 q^{73} -6.76393 q^{75} +0.763932 q^{77} +1.52786 q^{79} -2.41641 q^{81} +2.00000 q^{83} -3.23607 q^{85} +9.52786 q^{87} +2.00000 q^{89} -6.47214 q^{91} -12.9443 q^{93} -1.52786 q^{95} +0.472136 q^{97} -1.12461 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 6 q^{9} + 6 q^{11} - 4 q^{13} + 8 q^{15} + 2 q^{17} - 8 q^{19} + 2 q^{21} - 16 q^{23} + 2 q^{25} + 20 q^{27} - 2 q^{29} + 12 q^{31} + 16 q^{33} - 2 q^{35} + 2 q^{37}+ \cdots + 38 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 6 * q^9 + 6 * q^11 - 4 * q^13 + 8 * q^15 + 2 * q^17 - 8 * q^19 + 2 * q^21 - 16 * q^23 + 2 * q^25 + 20 * q^27 - 2 * q^29 + 12 * q^31 + 16 * q^33 - 2 * q^35 + 2 * q^37 + 16 * q^39 - 16 * q^41 - 4 * q^43 + 14 * q^45 - 4 * q^47 + 2 * q^49 + 2 * q^51 + 4 * q^55 - 28 * q^57 - 12 * q^59 + 2 * q^61 + 6 * q^63 + 24 * q^65 - 12 * q^67 - 16 * q^69 - 4 * q^71 - 18 * q^75 + 6 * q^77 + 12 * q^79 + 22 * q^81 + 4 * q^83 - 2 * q^85 + 28 * q^87 + 4 * q^89 - 4 * q^91 - 8 * q^93 - 12 * q^95 - 8 * q^97 + 38 * q^99

Introduction

L-functions

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Database

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