LMFDB - Embedded newform 6975.2.a.bf.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6975,2,Mod(1,6975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6975, 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("6975.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.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:	$55.6956554098$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.51414$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.51414 q^{2} +4.32088 q^{4} -0.485863 q^{7} +5.83502 q^{8} -5.02827 q^{11} -3.51414 q^{13} -1.22153 q^{14} +6.02827 q^{16} -1.32088 q^{17} -6.64177 q^{19} -12.6418 q^{22} +0.292611 q^{23} -8.83502 q^{26} -2.09936 q^{28} +9.86330 q^{29} +1.00000 q^{31} +3.48586 q^{32} -3.32088 q^{34} -5.51414 q^{37} -16.6983 q^{38} +7.02827 q^{41} +1.02827 q^{43} -21.7266 q^{44} +0.735663 q^{46} -6.93438 q^{47} -6.76394 q^{49} -15.1842 q^{52} -1.70739 q^{53} -2.83502 q^{56} +24.7977 q^{58} +2.19325 q^{59} -2.00000 q^{61} +2.51414 q^{62} -3.29261 q^{64} -9.12763 q^{67} -5.70739 q^{68} -13.4768 q^{71} -12.5424 q^{73} -13.8633 q^{74} -28.6983 q^{76} +2.44305 q^{77} -0.349158 q^{79} +17.6700 q^{82} +10.9344 q^{83} +2.58522 q^{86} -29.3401 q^{88} -5.03374 q^{89} +1.70739 q^{91} +1.26434 q^{92} -17.4340 q^{94} -10.4431 q^{97} -17.0055 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 5 q^{4} - 8 q^{7} + 3 q^{8} - 2 q^{11} - 4 q^{13} + 8 q^{14} + 5 q^{16} + 4 q^{17} - 4 q^{19} - 22 q^{22} + 6 q^{23} - 12 q^{26} - 10 q^{28} + 2 q^{29} + 3 q^{31} + 17 q^{32} - 2 q^{34} - 10 q^{37}+ \cdots - 57 q^{98}+O(q^{100}) $ 3 * q + q^2 + 5 * q^4 - 8 * q^7 + 3 * q^8 - 2 * q^11 - 4 * q^13 + 8 * q^14 + 5 * q^16 + 4 * q^17 - 4 * q^19 - 22 * q^22 + 6 * q^23 - 12 * q^26 - 10 * q^28 + 2 * q^29 + 3 * q^31 + 17 * q^32 - 2 * q^34 - 10 * q^37 - 8 * q^38 + 8 * q^41 - 10 * q^43 - 10 * q^44 - 16 * q^46 - 10 * q^47 + 11 * q^49 - 10 * q^52 + 6 * q^56 + 36 * q^58 + 8 * q^59 - 6 * q^61 + q^62 - 15 * q^64 - 18 * q^67 - 12 * q^68 - 10 * q^71 - 18 * q^73 - 14 * q^74 - 44 * q^76 - 16 * q^77 + 20 * q^79 + 24 * q^82 + 22 * q^83 + 18 * q^86 - 30 * q^88 - 8 * q^89 + 22 * q^92 + 8 * q^94 - 8 * q^97 - 57 * q^98

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 $

$2$	2.51414	1.77776	0.888882	−	0.458137i	$-0.151483\pi$
$2$	2.51414	1.77776	0.888882	+	0.458137i	$0.151483\pi$
$3$	0	0
$3$	0	0
$4$	4.32088	2.16044
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.485863	−0.183639	−0.0918195	−	0.995776i	$-0.529268\pi$
$7$	−0.485863	−0.183639	−0.0918195	+	0.995776i	$0.529268\pi$
$8$	5.83502	2.06299
$9$	0	0
$10$	0	0
$11$	−5.02827	−1.51608	−0.758041	−	0.652207i	$-0.773843\pi$
$11$	−5.02827	−1.51608	−0.758041	+	0.652207i	$0.773843\pi$
$12$	0	0
$13$	−3.51414	−0.974646	−0.487323	−	0.873222i	$-0.662027\pi$
$13$	−3.51414	−0.974646	−0.487323	+	0.873222i	$0.662027\pi$
$14$	−1.22153	−0.326467
$15$	0	0
$16$	6.02827	1.50707
$17$	−1.32088	−0.320362	−0.160181	−	0.987088i	$-0.551208\pi$
$17$	−1.32088	−0.320362	−0.160181	+	0.987088i	$0.551208\pi$
$18$	0	0
$19$	−6.64177	−1.52373	−0.761863	−	0.647738i	$-0.775715\pi$
$19$	−6.64177	−1.52373	−0.761863	+	0.647738i	$0.775715\pi$
$20$	0	0
$21$	0	0
$22$	−12.6418	−2.69523
$23$	0.292611	0.0610135	0.0305068	−	0.999535i	$-0.490288\pi$
$23$	0.292611	0.0610135	0.0305068	+	0.999535i	$0.490288\pi$
$24$	0	0
$25$	0	0
$26$	−8.83502	−1.73269
$27$	0	0
$28$	−2.09936	−0.396741
$29$	9.86330	1.83157	0.915784	−	0.401671i	$-0.131571\pi$
$29$	9.86330	1.83157	0.915784	+	0.401671i	$0.131571\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	3.48586	0.616219
$33$	0	0
$34$	−3.32088	−0.569527
$35$	0	0
$36$	0	0
$37$	−5.51414	−0.906519	−0.453259	−	0.891379i	$-0.649739\pi$
$37$	−5.51414	−0.906519	−0.453259	+	0.891379i	$0.649739\pi$
$38$	−16.6983	−2.70882
$39$	0	0
$40$	0	0
$41$	7.02827	1.09763	0.548816	−	0.835943i	$-0.315079\pi$
$41$	7.02827	1.09763	0.548816	+	0.835943i	$0.315079\pi$
$42$	0	0
$43$	1.02827	0.156810	0.0784051	−	0.996922i	$-0.475017\pi$
$43$	1.02827	0.156810	0.0784051	+	0.996922i	$0.475017\pi$
$44$	−21.7266	−3.27541
$45$	0	0
$46$	0.735663	0.108468
$47$	−6.93438	−1.01148	−0.505742	−	0.862685i	$-0.668781\pi$
$47$	−6.93438	−1.01148	−0.505742	+	0.862685i	$0.668781\pi$
$48$	0	0
$49$	−6.76394	−0.966277
$50$	0	0
$51$	0	0
$52$	−15.1842	−2.10567
$53$	−1.70739	−0.234528	−0.117264	−	0.993101i	$-0.537412\pi$
$53$	−1.70739	−0.234528	−0.117264	+	0.993101i	$0.537412\pi$
$54$	0	0
$55$	0	0
$56$	−2.83502	−0.378846
$57$	0	0
$58$	24.7977	3.25609
$59$	2.19325	0.285537	0.142769	−	0.989756i	$-0.454400\pi$
$59$	2.19325	0.285537	0.142769	+	0.989756i	$0.454400\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	2.51414	0.319296
$63$	0	0
$64$	−3.29261	−0.411576
$65$	0	0
$66$	0	0
$67$	−9.12763	−1.11512	−0.557559	−	0.830137i	$-0.688262\pi$
$67$	−9.12763	−1.11512	−0.557559	+	0.830137i	$0.688262\pi$
$68$	−5.70739	−0.692123
$69$	0	0
$70$	0	0
$71$	−13.4768	−1.59940	−0.799700	−	0.600399i	$-0.795008\pi$
$71$	−13.4768	−1.59940	−0.799700	+	0.600399i	$0.795008\pi$
$72$	0	0
$73$	−12.5424	−1.46798	−0.733989	−	0.679161i	$-0.762344\pi$
$73$	−12.5424	−1.46798	−0.733989	+	0.679161i	$0.762344\pi$
$74$	−13.8633	−1.61158
$75$	0	0
$76$	−28.6983	−3.29192
$77$	2.44305	0.278412
$78$	0	0
$79$	−0.349158	−0.0392834	−0.0196417	−	0.999807i	$-0.506253\pi$
$79$	−0.349158	−0.0392834	−0.0196417	+	0.999807i	$0.506253\pi$
$80$	0	0
$81$	0	0
$82$	17.6700	1.95133
$83$	10.9344	1.20020	0.600102	−	0.799923i	$-0.295127\pi$
$83$	10.9344	1.20020	0.600102	+	0.799923i	$0.295127\pi$
$84$	0	0
$85$	0	0
$86$	2.58522	0.278772
$87$	0	0
$88$	−29.3401	−3.12766
$89$	−5.03374	−0.533575	−0.266788	−	0.963755i	$-0.585962\pi$
$89$	−5.03374	−0.533575	−0.266788	+	0.963755i	$0.585962\pi$
$90$	0	0
$91$	1.70739	0.178983
$92$	1.26434	0.131816
$93$	0	0
$94$	−17.4340	−1.79818
$95$	0	0
$96$	0	0
$97$	−10.4431	−1.06033	−0.530166	−	0.847894i	$-0.677870\pi$
$97$	−10.4431	−1.06033	−0.530166	+	0.847894i	$0.677870\pi$
$98$	−17.0055	−1.71781
$99$	0	0
$100$	0	0
$101$	11.6135	1.15559	0.577793	−	0.816183i	$-0.303914\pi$
$101$	11.6135	1.15559	0.577793	+	0.816183i	$0.303914\pi$
$102$	0	0
$103$	3.76940	0.371410	0.185705	−	0.982606i	$-0.440543\pi$
$103$	3.76940	0.371410	0.185705	+	0.982606i	$0.440543\pi$
$104$	−20.5051	−2.01069
$105$	0	0
$106$	−4.29261	−0.416935
$107$	−6.73566	−0.651161	−0.325581	−	0.945514i	$-0.605560\pi$
$107$	−6.73566	−0.651161	−0.325581	+	0.945514i	$0.605560\pi$
$108$	0	0
$109$	−3.90611	−0.374137	−0.187069	−	0.982347i	$-0.559899\pi$
$109$	−3.90611	−0.374137	−0.187069	+	0.982347i	$0.559899\pi$
$110$	0	0
$111$	0	0
$112$	−2.92892	−0.276757
$113$	−15.0848	−1.41906	−0.709530	−	0.704675i	$-0.751093\pi$
$113$	−15.0848	−1.41906	−0.709530	+	0.704675i	$0.751093\pi$
$114$	0	0
$115$	0	0
$116$	42.6182	3.95700
$117$	0	0
$118$	5.51414	0.507617
$119$	0.641769	0.0588309
$120$	0	0
$121$	14.2835	1.29850
$122$	−5.02827	−0.455239
$123$	0	0
$124$	4.32088	0.388027
$125$	0	0
$126$	0	0
$127$	6.25526	0.555065	0.277532	−	0.960716i	$-0.410483\pi$
$127$	6.25526	0.555065	0.277532	+	0.960716i	$0.410483\pi$
$128$	−15.2498	−1.34790
$129$	0	0
$130$	0	0
$131$	22.1186	1.93251	0.966254	−	0.257592i	$-0.0829291\pi$
$131$	22.1186	1.93251	0.966254	+	0.257592i	$0.0829291\pi$
$132$	0	0
$133$	3.22699	0.279816
$134$	−22.9481	−1.98242
$135$	0	0
$136$	−7.70739	−0.660903
$137$	−14.6044	−1.24774	−0.623870	−	0.781528i	$-0.714441\pi$
$137$	−14.6044	−1.24774	−0.623870	+	0.781528i	$0.714441\pi$
$138$	0	0
$139$	12.8970	1.09391	0.546956	−	0.837161i	$-0.315786\pi$
$139$	12.8970	1.09391	0.546956	+	0.837161i	$0.315786\pi$
$140$	0	0
$141$	0	0
$142$	−33.8825	−2.84336
$143$	17.6700	1.47764
$144$	0	0
$145$	0	0
$146$	−31.5333	−2.60972
$147$	0	0
$148$	−23.8259	−1.95848
$149$	−16.3118	−1.33632	−0.668158	−	0.744020i	$-0.732917\pi$
$149$	−16.3118	−1.33632	−0.668158	+	0.744020i	$0.732917\pi$
$150$	0	0
$151$	19.3774	1.57691	0.788457	−	0.615090i	$-0.210881\pi$
$151$	19.3774	1.57691	0.788457	+	0.615090i	$0.210881\pi$
$152$	−38.7549	−3.14343
$153$	0	0
$154$	6.14217	0.494950
$155$	0	0
$156$	0	0
$157$	3.80128	0.303375	0.151688	−	0.988428i	$-0.451529\pi$
$157$	3.80128	0.303375	0.151688	+	0.988428i	$0.451529\pi$
$158$	−0.877832	−0.0698366
$159$	0	0
$160$	0	0
$161$	−0.142169	−0.0112045
$162$	0	0
$163$	−12.5424	−0.982397	−0.491199	−	0.871048i	$-0.663441\pi$
$163$	−12.5424	−0.982397	−0.491199	+	0.871048i	$0.663441\pi$
$164$	30.3684	2.37137
$165$	0	0
$166$	27.4905	2.13368
$167$	22.2553	1.72216	0.861082	−	0.508466i	$-0.169787\pi$
$167$	22.2553	1.72216	0.861082	+	0.508466i	$0.169787\pi$
$168$	0	0
$169$	−0.650842	−0.0500647
$170$	0	0
$171$	0	0
$172$	4.44305	0.338780
$173$	−10.9717	−0.834165	−0.417082	−	0.908869i	$-0.636947\pi$
$173$	−10.9717	−0.834165	−0.417082	+	0.908869i	$0.636947\pi$
$174$	0	0
$175$	0	0
$176$	−30.3118	−2.28484
$177$	0	0
$178$	−12.6555	−0.948570
$179$	15.9253	1.19031	0.595157	−	0.803610i	$-0.297090\pi$
$179$	15.9253	1.19031	0.595157	+	0.803610i	$0.297090\pi$
$180$	0	0
$181$	−8.05655	−0.598838	−0.299419	−	0.954122i	$-0.596793\pi$
$181$	−8.05655	−0.598838	−0.299419	+	0.954122i	$0.596793\pi$
$182$	4.29261	0.318189
$183$	0	0
$184$	1.70739	0.125870
$185$	0	0
$186$	0	0
$187$	6.64177	0.485694
$188$	−29.9627	−2.18525
$189$	0	0
$190$	0	0
$191$	−6.19325	−0.448128	−0.224064	−	0.974574i	$-0.571932\pi$
$191$	−6.19325	−0.448128	−0.224064	+	0.974574i	$0.571932\pi$
$192$	0	0
$193$	2.25526	0.162337	0.0811687	−	0.996700i	$-0.474135\pi$
$193$	2.25526	0.162337	0.0811687	+	0.996700i	$0.474135\pi$
$194$	−26.2553	−1.88502
$195$	0	0
$196$	−29.2262	−2.08759
$197$	9.32088	0.664086	0.332043	−	0.943264i	$-0.392262\pi$
$197$	9.32088	0.664086	0.332043	+	0.943264i	$0.392262\pi$
$198$	0	0
$199$	16.5479	1.17305	0.586524	−	0.809932i	$-0.300496\pi$
$199$	16.5479	1.17305	0.586524	+	0.809932i	$0.300496\pi$
$200$	0	0
$201$	0	0
$202$	29.1979	2.05436
$203$	−4.79221	−0.336347
$204$	0	0
$205$	0	0
$206$	9.47679	0.660279
$207$	0	0
$208$	−21.1842	−1.46886
$209$	33.3966	2.31009
$210$	0	0
$211$	25.0101	1.72177	0.860884	−	0.508801i	$-0.169911\pi$
$211$	25.0101	1.72177	0.860884	+	0.508801i	$0.169911\pi$
$212$	−7.37743	−0.506684
$213$	0	0
$214$	−16.9344	−1.15761
$215$	0	0
$216$	0	0
$217$	−0.485863	−0.0329825
$218$	−9.82048	−0.665127
$219$	0	0
$220$	0	0
$221$	4.64177	0.312239
$222$	0	0
$223$	−19.2835	−1.29132	−0.645661	−	0.763625i	$-0.723418\pi$
$223$	−19.2835	−1.29132	−0.645661	+	0.763625i	$0.723418\pi$
$224$	−1.69365	−0.113162
$225$	0	0
$226$	−37.9253	−2.52275
$227$	23.4340	1.55537	0.777684	−	0.628655i	$-0.216394\pi$
$227$	23.4340	1.55537	0.777684	+	0.628655i	$0.216394\pi$
$228$	0	0
$229$	−25.0283	−1.65391	−0.826957	−	0.562265i	$-0.809930\pi$
$229$	−25.0283	−1.65391	−0.826957	+	0.562265i	$0.809930\pi$
$230$	0	0
$231$	0	0
$232$	57.5525	3.77851
$233$	18.7175	1.22623	0.613113	−	0.789995i	$-0.289917\pi$
$233$	18.7175	1.22623	0.613113	+	0.789995i	$0.289917\pi$
$234$	0	0
$235$	0	0
$236$	9.47679	0.616887
$237$	0	0
$238$	1.61350	0.104587
$239$	3.86876	0.250249	0.125125	−	0.992141i	$-0.460067\pi$
$239$	3.86876	0.250249	0.125125	+	0.992141i	$0.460067\pi$
$240$	0	0
$241$	−1.61350	−0.103934	−0.0519672	−	0.998649i	$-0.516549\pi$
$241$	−1.61350	−0.103934	−0.0519672	+	0.998649i	$0.516549\pi$
$242$	35.9108	2.30843
$243$	0	0
$244$	−8.64177	−0.553233
$245$	0	0
$246$	0	0
$247$	23.3401	1.48509
$248$	5.83502	0.370524
$249$	0	0
$250$	0	0
$251$	25.4713	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$251$	25.4713	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$252$	0	0
$253$	−1.47133	−0.0925015
$254$	15.7266	0.986774
$255$	0	0
$256$	−31.7549	−1.98468
$257$	18.0192	1.12401	0.562003	−	0.827135i	$-0.310031\pi$
$257$	18.0192	1.12401	0.562003	+	0.827135i	$0.310031\pi$
$258$	0	0
$259$	2.67912	0.166472
$260$	0	0
$261$	0	0
$262$	55.6091	3.43554
$263$	1.15951	0.0714987	0.0357494	−	0.999361i	$-0.488618\pi$
$263$	1.15951	0.0714987	0.0357494	+	0.999361i	$0.488618\pi$
$264$	0	0
$265$	0	0
$266$	8.11310	0.497446
$267$	0	0
$268$	−39.4394	−2.40915
$269$	7.80675	0.475986	0.237993	−	0.971267i	$-0.423511\pi$
$269$	7.80675	0.475986	0.237993	+	0.971267i	$0.423511\pi$
$270$	0	0
$271$	17.7831	1.08025	0.540124	−	0.841585i	$-0.318377\pi$
$271$	17.7831	1.08025	0.540124	+	0.841585i	$0.318377\pi$
$272$	−7.96265	−0.482807
$273$	0	0
$274$	−36.7175	−2.21819
$275$	0	0
$276$	0	0
$277$	−25.9390	−1.55853	−0.779263	−	0.626697i	$-0.784406\pi$
$277$	−25.9390	−1.55853	−0.779263	+	0.626697i	$0.784406\pi$
$278$	32.4249	1.94472
$279$	0	0
$280$	0	0
$281$	13.0283	0.777202	0.388601	−	0.921406i	$-0.372959\pi$
$281$	13.0283	0.777202	0.388601	+	0.921406i	$0.372959\pi$
$282$	0	0
$283$	−15.9006	−0.945195	−0.472598	−	0.881278i	$-0.656684\pi$
$283$	−15.9006	−0.945195	−0.472598	+	0.881278i	$0.656684\pi$
$284$	−58.2317	−3.45541
$285$	0	0
$286$	44.4249	2.62690
$287$	−3.41478	−0.201568
$288$	0	0
$289$	−15.2553	−0.897368
$290$	0	0
$291$	0	0
$292$	−54.1943	−3.17148
$293$	−29.4340	−1.71955	−0.859776	−	0.510672i	$-0.829397\pi$
$293$	−29.4340	−1.71955	−0.859776	+	0.510672i	$0.829397\pi$
$294$	0	0
$295$	0	0
$296$	−32.1751	−1.87014
$297$	0	0
$298$	−41.0101	−2.37565
$299$	−1.02827	−0.0594666
$300$	0	0
$301$	−0.499600	−0.0287965
$302$	48.7175	2.80338
$303$	0	0
$304$	−40.0384	−2.29636
$305$	0	0
$306$	0	0
$307$	−9.96812	−0.568911	−0.284455	−	0.958689i	$-0.591813\pi$
$307$	−9.96812	−0.568911	−0.284455	+	0.958689i	$0.591813\pi$
$308$	10.5561	0.601492
$309$	0	0
$310$	0	0
$311$	−15.9945	−0.906967	−0.453483	−	0.891265i	$-0.649819\pi$
$311$	−15.9945	−0.906967	−0.453483	+	0.891265i	$0.649819\pi$
$312$	0	0
$313$	22.8542	1.29180	0.645899	−	0.763423i	$-0.276483\pi$
$313$	22.8542	1.29180	0.645899	+	0.763423i	$0.276483\pi$
$314$	9.55695	0.539330
$315$	0	0
$316$	−1.50867	−0.0848695
$317$	14.6044	0.820266	0.410133	−	0.912026i	$-0.365482\pi$
$317$	14.6044	0.820266	0.410133	+	0.912026i	$0.365482\pi$
$318$	0	0
$319$	−49.5953	−2.77681
$320$	0	0
$321$	0	0
$322$	−0.357432	−0.0199189
$323$	8.77301	0.488143
$324$	0	0
$325$	0	0
$326$	−31.5333	−1.74647
$327$	0	0
$328$	41.0101	2.26441
$329$	3.36916	0.185748
$330$	0	0
$331$	−8.16137	−0.448589	−0.224295	−	0.974521i	$-0.572008\pi$
$331$	−8.16137	−0.448589	−0.224295	+	0.974521i	$0.572008\pi$
$332$	47.2462	2.59297
$333$	0	0
$334$	55.9528	3.06160
$335$	0	0
$336$	0	0
$337$	11.8825	0.647281	0.323640	−	0.946180i	$-0.395093\pi$
$337$	11.8825	0.647281	0.323640	+	0.946180i	$0.395093\pi$
$338$	−1.63631	−0.0890033
$339$	0	0
$340$	0	0
$341$	−5.02827	−0.272296
$342$	0	0
$343$	6.68739	0.361085
$344$	6.00000	0.323498
$345$	0	0
$346$	−27.5844	−1.48295
$347$	−18.3684	−0.986065	−0.493033	−	0.870011i	$-0.664112\pi$
$347$	−18.3684	−0.986065	−0.493033	+	0.870011i	$0.664112\pi$
$348$	0	0
$349$	11.2462	0.601995	0.300997	−	0.953625i	$-0.402680\pi$
$349$	11.2462	0.601995	0.300997	+	0.953625i	$0.402680\pi$
$350$	0	0
$351$	0	0
$352$	−17.5279	−0.934239
$353$	25.3593	1.34974	0.674869	−	0.737937i	$-0.264200\pi$
$353$	25.3593	1.34974	0.674869	+	0.737937i	$0.264200\pi$
$354$	0	0
$355$	0	0
$356$	−21.7502	−1.15276
$357$	0	0
$358$	40.0384	2.11610
$359$	−15.2890	−0.806923	−0.403461	−	0.914997i	$-0.632193\pi$
$359$	−15.2890	−0.806923	−0.403461	+	0.914997i	$0.632193\pi$
$360$	0	0
$361$	25.1131	1.32174
$362$	−20.2553	−1.06459
$363$	0	0
$364$	7.37743	0.386683
$365$	0	0
$366$	0	0
$367$	−6.05655	−0.316149	−0.158075	−	0.987427i	$-0.550529\pi$
$367$	−6.05655	−0.316149	−0.158075	+	0.987427i	$0.550529\pi$
$368$	1.76394	0.0919516
$369$	0	0
$370$	0	0
$371$	0.829557	0.0430685
$372$	0	0
$373$	−8.06748	−0.417718	−0.208859	−	0.977946i	$-0.566975\pi$
$373$	−8.06748	−0.417718	−0.208859	+	0.977946i	$0.566975\pi$
$374$	16.6983	0.863449
$375$	0	0
$376$	−40.4623	−2.08668
$377$	−34.6610	−1.78513
$378$	0	0
$379$	−36.8114	−1.89088	−0.945438	−	0.325803i	$-0.894365\pi$
$379$	−36.8114	−1.89088	−0.945438	+	0.325803i	$0.894365\pi$
$380$	0	0
$381$	0	0
$382$	−15.5707	−0.796666
$383$	−3.63270	−0.185622	−0.0928111	−	0.995684i	$-0.529585\pi$
$383$	−3.63270	−0.185622	−0.0928111	+	0.995684i	$0.529585\pi$
$384$	0	0
$385$	0	0
$386$	5.67004	0.288598
$387$	0	0
$388$	−45.1232	−2.29078
$389$	−15.2781	−0.774629	−0.387315	−	0.921948i	$-0.626597\pi$
$389$	−15.2781	−0.774629	−0.387315	+	0.921948i	$0.626597\pi$
$390$	0	0
$391$	−0.386505	−0.0195464
$392$	−39.4677	−1.99342
$393$	0	0
$394$	23.4340	1.18059
$395$	0	0
$396$	0	0
$397$	18.5852	0.932766	0.466383	−	0.884583i	$-0.345557\pi$
$397$	18.5852	0.932766	0.466383	+	0.884583i	$0.345557\pi$
$398$	41.6036	2.08540
$399$	0	0
$400$	0	0
$401$	−6.19325	−0.309276	−0.154638	−	0.987971i	$-0.549421\pi$
$401$	−6.19325	−0.309276	−0.154638	+	0.987971i	$0.549421\pi$
$402$	0	0
$403$	−3.51414	−0.175052
$404$	50.1806	2.49658
$405$	0	0
$406$	−12.0483	−0.597946
$407$	27.7266	1.37436
$408$	0	0
$409$	−32.0565	−1.58509	−0.792547	−	0.609811i	$-0.791245\pi$
$409$	−32.0565	−1.58509	−0.792547	+	0.609811i	$0.791245\pi$
$410$	0	0
$411$	0	0
$412$	16.2871	0.802410
$413$	−1.06562	−0.0524357
$414$	0	0
$415$	0	0
$416$	−12.2498	−0.600596
$417$	0	0
$418$	83.9637	4.10680
$419$	−9.78860	−0.478205	−0.239102	−	0.970994i	$-0.576853\pi$
$419$	−9.78860	−0.478205	−0.239102	+	0.970994i	$0.576853\pi$
$420$	0	0
$421$	−14.2745	−0.695695	−0.347847	−	0.937551i	$-0.613087\pi$
$421$	−14.2745	−0.695695	−0.347847	+	0.937551i	$0.613087\pi$
$422$	62.8789	3.06090
$423$	0	0
$424$	−9.96265	−0.483829
$425$	0	0
$426$	0	0
$427$	0.971726	0.0470251
$428$	−29.1040	−1.40680
$429$	0	0
$430$	0	0
$431$	22.8350	1.09992	0.549962	−	0.835190i	$-0.314642\pi$
$431$	22.8350	1.09992	0.549962	+	0.835190i	$0.314642\pi$
$432$	0	0
$433$	−13.8259	−0.664433	−0.332216	−	0.943203i	$-0.607796\pi$
$433$	−13.8259	−0.664433	−0.332216	+	0.943203i	$0.607796\pi$
$434$	−1.22153	−0.0586351
$435$	0	0
$436$	−16.8778	−0.808302
$437$	−1.94345	−0.0929679
$438$	0	0
$439$	39.8506	1.90197	0.950983	−	0.309243i	$-0.100076\pi$
$439$	39.8506	1.90197	0.950983	+	0.309243i	$0.100076\pi$
$440$	0	0
$441$	0	0
$442$	11.6700	0.555087
$443$	−10.1504	−0.482262	−0.241131	−	0.970493i	$-0.577518\pi$
$443$	−10.1504	−0.482262	−0.241131	+	0.970493i	$0.577518\pi$
$444$	0	0
$445$	0	0
$446$	−48.4815	−2.29566
$447$	0	0
$448$	1.59976	0.0755815
$449$	3.75020	0.176983	0.0884914	−	0.996077i	$-0.471795\pi$
$449$	3.75020	0.176983	0.0884914	+	0.996077i	$0.471795\pi$
$450$	0	0
$451$	−35.3401	−1.66410
$452$	−65.1798	−3.06580
$453$	0	0
$454$	58.9162	2.76508
$455$	0	0
$456$	0	0
$457$	1.70193	0.0796127	0.0398064	−	0.999207i	$-0.487326\pi$
$457$	1.70193	0.0796127	0.0398064	+	0.999207i	$0.487326\pi$
$458$	−62.9245	−2.94027
$459$	0	0
$460$	0	0
$461$	−7.40931	−0.345086	−0.172543	−	0.985002i	$-0.555198\pi$
$461$	−7.40931	−0.345086	−0.172543	+	0.985002i	$0.555198\pi$
$462$	0	0
$463$	30.3865	1.41218	0.706090	−	0.708122i	$-0.250457\pi$
$463$	30.3865	1.41218	0.706090	+	0.708122i	$0.250457\pi$
$464$	59.4586	2.76030
$465$	0	0
$466$	47.0584	2.17994
$467$	17.1040	0.791480	0.395740	−	0.918363i	$-0.370488\pi$
$467$	17.1040	0.791480	0.395740	+	0.918363i	$0.370488\pi$
$468$	0	0
$469$	4.43478	0.204779
$470$	0	0
$471$	0	0
$472$	12.7977	0.589061
$473$	−5.17044	−0.237737
$474$	0	0
$475$	0	0
$476$	2.77301	0.127101
$477$	0	0
$478$	9.72659	0.444884
$479$	−21.2890	−0.972719	−0.486360	−	0.873759i	$-0.661676\pi$
$479$	−21.2890	−0.972719	−0.486360	+	0.873759i	$0.661676\pi$
$480$	0	0
$481$	19.3774	0.883535
$482$	−4.05655	−0.184771
$483$	0	0
$484$	61.7175	2.80534
$485$	0	0
$486$	0	0
$487$	−20.4996	−0.928926	−0.464463	−	0.885593i	$-0.653753\pi$
$487$	−20.4996	−0.928926	−0.464463	+	0.885593i	$0.653753\pi$
$488$	−11.6700	−0.528278
$489$	0	0
$490$	0	0
$491$	31.0667	1.40202	0.701010	−	0.713152i	$-0.252733\pi$
$491$	31.0667	1.40202	0.701010	+	0.713152i	$0.252733\pi$
$492$	0	0
$493$	−13.0283	−0.586764
$494$	58.6802	2.64015
$495$	0	0
$496$	6.02827	0.270677
$497$	6.54787	0.293712
$498$	0	0
$499$	15.3401	0.686717	0.343358	−	0.939205i	$-0.388435\pi$
$499$	15.3401	0.686717	0.343358	+	0.939205i	$0.388435\pi$
$500$	0	0
$501$	0	0
$502$	64.0384	2.85817
$503$	−14.9344	−0.665891	−0.332946	−	0.942946i	$-0.608043\pi$
$503$	−14.9344	−0.665891	−0.332946	+	0.942946i	$0.608043\pi$
$504$	0	0
$505$	0	0
$506$	−3.69912	−0.164446
$507$	0	0
$508$	27.0283	1.19919
$509$	1.22153	0.0541432	0.0270716	−	0.999633i	$-0.491382\pi$
$509$	1.22153	0.0541432	0.0270716	+	0.999633i	$0.491382\pi$
$510$	0	0
$511$	6.09389	0.269578
$512$	−49.3365	−2.18038
$513$	0	0
$514$	45.3027	1.99822
$515$	0	0
$516$	0	0
$517$	34.8680	1.53349
$518$	6.73566	0.295948
$519$	0	0
$520$	0	0
$521$	−4.57429	−0.200403	−0.100202	−	0.994967i	$-0.531949\pi$
$521$	−4.57429	−0.200403	−0.100202	+	0.994967i	$0.531949\pi$
$522$	0	0
$523$	29.7831	1.30233	0.651163	−	0.758938i	$-0.274281\pi$
$523$	29.7831	1.30233	0.651163	+	0.758938i	$0.274281\pi$
$524$	95.5717	4.17507
$525$	0	0
$526$	2.91518	0.127108
$527$	−1.32088	−0.0575386
$528$	0	0
$529$	−22.9144	−0.996277
$530$	0	0
$531$	0	0
$532$	13.9435	0.604525
$533$	−24.6983	−1.06980
$534$	0	0
$535$	0	0
$536$	−53.2599	−2.30048
$537$	0	0
$538$	19.6272	0.846190
$539$	34.0109	1.46495
$540$	0	0
$541$	−3.04748	−0.131021	−0.0655106	−	0.997852i	$-0.520868\pi$
$541$	−3.04748	−0.131021	−0.0655106	+	0.997852i	$0.520868\pi$
$542$	44.7092	1.92043
$543$	0	0
$544$	−4.60442	−0.197413
$545$	0	0
$546$	0	0
$547$	−9.82595	−0.420127	−0.210064	−	0.977688i	$-0.567367\pi$
$547$	−9.82595	−0.420127	−0.210064	+	0.977688i	$0.567367\pi$
$548$	−63.1040	−2.69567
$549$	0	0
$550$	0	0
$551$	−65.5097	−2.79081
$552$	0	0
$553$	0.169643	0.00721396
$554$	−65.2143	−2.77069
$555$	0	0
$556$	55.7266	2.36333
$557$	−38.7922	−1.64368	−0.821839	−	0.569719i	$-0.807052\pi$
$557$	−38.7922	−1.64368	−0.821839	+	0.569719i	$0.807052\pi$
$558$	0	0
$559$	−3.61350	−0.152835
$560$	0	0
$561$	0	0
$562$	32.7549	1.38168
$563$	2.73566	0.115294	0.0576472	−	0.998337i	$-0.481640\pi$
$563$	2.73566	0.115294	0.0576472	+	0.998337i	$0.481640\pi$
$564$	0	0
$565$	0	0
$566$	−39.9764	−1.68033
$567$	0	0
$568$	−78.6374	−3.29955
$569$	−4.39197	−0.184121	−0.0920605	−	0.995753i	$-0.529345\pi$
$569$	−4.39197	−0.184121	−0.0920605	+	0.995753i	$0.529345\pi$
$570$	0	0
$571$	9.67004	0.404679	0.202339	−	0.979315i	$-0.435146\pi$
$571$	9.67004	0.404679	0.202339	+	0.979315i	$0.435146\pi$
$572$	76.3502	3.19236
$573$	0	0
$574$	−8.58522	−0.358340
$575$	0	0
$576$	0	0
$577$	−31.4148	−1.30781	−0.653907	−	0.756575i	$-0.726871\pi$
$577$	−31.4148	−1.30781	−0.653907	+	0.756575i	$0.726871\pi$
$578$	−38.3538	−1.59531
$579$	0	0
$580$	0	0
$581$	−5.31261	−0.220404
$582$	0	0
$583$	8.58522	0.355564
$584$	−73.1852	−3.02843
$585$	0	0
$586$	−74.0011	−3.05696
$587$	−27.2161	−1.12333	−0.561664	−	0.827366i	$-0.689838\pi$
$587$	−27.2161	−1.12333	−0.561664	+	0.827366i	$0.689838\pi$
$588$	0	0
$589$	−6.64177	−0.273669
$590$	0	0
$591$	0	0
$592$	−33.2407	−1.36619
$593$	32.5561	1.33692	0.668460	−	0.743748i	$-0.266954\pi$
$593$	32.5561	1.33692	0.668460	+	0.743748i	$0.266954\pi$
$594$	0	0
$595$	0	0
$596$	−70.4815	−2.88703
$597$	0	0
$598$	−2.58522	−0.105718
$599$	−27.8067	−1.13615	−0.568076	−	0.822976i	$-0.692312\pi$
$599$	−27.8067	−1.13615	−0.568076	+	0.822976i	$0.692312\pi$
$600$	0	0
$601$	17.6519	0.720036	0.360018	−	0.932945i	$-0.382771\pi$
$601$	17.6519	0.720036	0.360018	+	0.932945i	$0.382771\pi$
$602$	−1.25606	−0.0511933
$603$	0	0
$604$	83.7276	3.40683
$605$	0	0
$606$	0	0
$607$	−44.9673	−1.82517	−0.912584	−	0.408890i	$-0.865916\pi$
$607$	−44.9673	−1.82517	−0.912584	+	0.408890i	$0.865916\pi$
$608$	−23.1523	−0.938950
$609$	0	0
$610$	0	0
$611$	24.3684	0.985838
$612$	0	0
$613$	−33.7694	−1.36393	−0.681967	−	0.731383i	$-0.738875\pi$
$613$	−33.7694	−1.36393	−0.681967	+	0.731383i	$0.738875\pi$
$614$	−25.0612	−1.01139
$615$	0	0
$616$	14.2553	0.574361
$617$	−26.1131	−1.05127	−0.525637	−	0.850709i	$-0.676173\pi$
$617$	−26.1131	−1.05127	−0.525637	+	0.850709i	$0.676173\pi$
$618$	0	0
$619$	−20.8597	−0.838422	−0.419211	−	0.907889i	$-0.637693\pi$
$619$	−20.8597	−0.838422	−0.419211	+	0.907889i	$0.637693\pi$
$620$	0	0
$621$	0	0
$622$	−40.2125	−1.61237
$623$	2.44571	0.0979852
$624$	0	0
$625$	0	0
$626$	57.4586	2.29651
$627$	0	0
$628$	16.4249	0.655425
$629$	7.28354	0.290414
$630$	0	0
$631$	−20.5105	−0.816511	−0.408256	−	0.912868i	$-0.633863\pi$
$631$	−20.5105	−0.816511	−0.408256	+	0.912868i	$0.633863\pi$
$632$	−2.03735	−0.0810413
$633$	0	0
$634$	36.7175	1.45824
$635$	0	0
$636$	0	0
$637$	23.7694	0.941778
$638$	−124.690	−4.93650
$639$	0	0
$640$	0	0
$641$	−23.9945	−0.947727	−0.473864	−	0.880598i	$-0.657141\pi$
$641$	−23.9945	−0.947727	−0.473864	+	0.880598i	$0.657141\pi$
$642$	0	0
$643$	13.6700	0.539094	0.269547	−	0.962987i	$-0.413126\pi$
$643$	13.6700	0.539094	0.269547	+	0.962987i	$0.413126\pi$
$644$	−0.614295	−0.0242066
$645$	0	0
$646$	22.0565	0.867803
$647$	−11.1523	−0.438442	−0.219221	−	0.975675i	$-0.570352\pi$
$647$	−11.1523	−0.438442	−0.219221	+	0.975675i	$0.570352\pi$
$648$	0	0
$649$	−11.0283	−0.432898
$650$	0	0
$651$	0	0
$652$	−54.1943	−2.12241
$653$	−39.1896	−1.53361	−0.766805	−	0.641881i	$-0.778154\pi$
$653$	−39.1896	−1.53361	−0.766805	+	0.641881i	$0.778154\pi$
$654$	0	0
$655$	0	0
$656$	42.3684	1.65421
$657$	0	0
$658$	8.47053	0.330216
$659$	31.6464	1.23277	0.616385	−	0.787445i	$-0.288597\pi$
$659$	31.6464	1.23277	0.616385	+	0.787445i	$0.288597\pi$
$660$	0	0
$661$	−3.59535	−0.139843	−0.0699215	−	0.997553i	$-0.522275\pi$
$661$	−3.59535	−0.139843	−0.0699215	+	0.997553i	$0.522275\pi$
$662$	−20.5188	−0.797486
$663$	0	0
$664$	63.8023	2.47601
$665$	0	0
$666$	0	0
$667$	2.88611	0.111750
$668$	96.1624	3.72064
$669$	0	0
$670$	0	0
$671$	10.0565	0.388229
$672$	0	0
$673$	7.24073	0.279110	0.139555	−	0.990214i	$-0.455433\pi$
$673$	7.24073	0.279110	0.139555	+	0.990214i	$0.455433\pi$
$674$	29.8742	1.15071
$675$	0	0
$676$	−2.81221	−0.108162
$677$	38.5188	1.48040	0.740199	−	0.672388i	$-0.234731\pi$
$677$	38.5188	1.48040	0.740199	+	0.672388i	$0.234731\pi$
$678$	0	0
$679$	5.07389	0.194718
$680$	0	0
$681$	0	0
$682$	−12.6418	−0.484078
$683$	40.0192	1.53129	0.765646	−	0.643262i	$-0.222419\pi$
$683$	40.0192	1.53129	0.765646	+	0.643262i	$0.222419\pi$
$684$	0	0
$685$	0	0
$686$	16.8130	0.641924
$687$	0	0
$688$	6.19872	0.236324
$689$	6.00000	0.228582
$690$	0	0
$691$	−11.8013	−0.448942	−0.224471	−	0.974481i	$-0.572065\pi$
$691$	−11.8013	−0.448942	−0.224471	+	0.974481i	$0.572065\pi$
$692$	−47.4076	−1.80217
$693$	0	0
$694$	−46.1806	−1.75299
$695$	0	0
$696$	0	0
$697$	−9.28354	−0.351639
$698$	28.2745	1.07020
$699$	0	0
$700$	0	0
$701$	−5.17044	−0.195285	−0.0976425	−	0.995222i	$-0.531130\pi$
$701$	−5.17044	−0.195285	−0.0976425	+	0.995222i	$0.531130\pi$
$702$	0	0
$703$	36.6236	1.38129
$704$	16.5561	0.623983
$705$	0	0
$706$	63.7567	2.39952
$707$	−5.64257	−0.212211
$708$	0	0
$709$	47.7722	1.79412	0.897062	−	0.441906i	$-0.145697\pi$
$709$	47.7722	1.79412	0.897062	+	0.441906i	$0.145697\pi$
$710$	0	0
$711$	0	0
$712$	−29.3720	−1.10076
$713$	0.292611	0.0109584
$714$	0	0
$715$	0	0
$716$	68.8114	2.57160
$717$	0	0
$718$	−38.4386	−1.43452
$719$	−23.0848	−0.860919	−0.430459	−	0.902610i	$-0.641648\pi$
$719$	−23.0848	−0.860919	−0.430459	+	0.902610i	$0.641648\pi$
$720$	0	0
$721$	−1.83141	−0.0682054
$722$	63.1378	2.34974
$723$	0	0
$724$	−34.8114	−1.29376
$725$	0	0
$726$	0	0
$727$	22.2125	0.823814	0.411907	−	0.911226i	$-0.364863\pi$
$727$	22.2125	0.823814	0.411907	+	0.911226i	$0.364863\pi$
$728$	9.96265	0.369241
$729$	0	0
$730$	0	0
$731$	−1.35823	−0.0502360
$732$	0	0
$733$	43.9072	1.62175	0.810874	−	0.585221i	$-0.198992\pi$
$733$	43.9072	1.62175	0.810874	+	0.585221i	$0.198992\pi$
$734$	−15.2270	−0.562038
$735$	0	0
$736$	1.02000	0.0375977
$737$	45.8962	1.69061
$738$	0	0
$739$	6.86690	0.252603	0.126302	−	0.991992i	$-0.459689\pi$
$739$	6.86690	0.252603	0.126302	+	0.991992i	$0.459689\pi$
$740$	0	0
$741$	0	0
$742$	2.08562	0.0765656
$743$	29.2726	1.07391	0.536954	−	0.843612i	$-0.319575\pi$
$743$	29.2726	1.07391	0.536954	+	0.843612i	$0.319575\pi$
$744$	0	0
$745$	0	0
$746$	−20.2827	−0.742604
$747$	0	0
$748$	28.6983	1.04931
$749$	3.27261	0.119579
$750$	0	0
$751$	9.67004	0.352865	0.176432	−	0.984313i	$-0.443544\pi$
$751$	9.67004	0.352865	0.176432	+	0.984313i	$0.443544\pi$
$752$	−41.8023	−1.52437
$753$	0	0
$754$	−87.1424	−3.17354
$755$	0	0
$756$	0	0
$757$	49.4104	1.79585	0.897925	−	0.440148i	$-0.145074\pi$
$757$	49.4104	1.79585	0.897925	+	0.440148i	$0.145074\pi$
$758$	−92.5489	−3.36153
$759$	0	0
$760$	0	0
$761$	−47.2599	−1.71317	−0.856586	−	0.516005i	$-0.827419\pi$
$761$	−47.2599	−1.71317	−0.856586	+	0.516005i	$0.827419\pi$
$762$	0	0
$763$	1.89783	0.0687062
$764$	−26.7603	−0.968155
$765$	0	0
$766$	−9.13310	−0.329992
$767$	−7.70739	−0.278298
$768$	0	0
$769$	9.49053	0.342237	0.171119	−	0.985250i	$-0.445262\pi$
$769$	9.49053	0.342237	0.171119	+	0.985250i	$0.445262\pi$
$770$	0	0
$771$	0	0
$772$	9.74474	0.350721
$773$	41.7567	1.50188	0.750942	−	0.660368i	$-0.229600\pi$
$773$	41.7567	1.50188	0.750942	+	0.660368i	$0.229600\pi$
$774$	0	0
$775$	0	0
$776$	−60.9354	−2.18745
$777$	0	0
$778$	−38.4112	−1.37711
$779$	−46.6802	−1.67249
$780$	0	0
$781$	67.7650	2.42482
$782$	−0.971726	−0.0347489
$783$	0	0
$784$	−40.7749	−1.45625
$785$	0	0
$786$	0	0
$787$	40.0950	1.42923	0.714615	−	0.699518i	$-0.246602\pi$
$787$	40.0950	1.42923	0.714615	+	0.699518i	$0.246602\pi$
$788$	40.2745	1.43472
$789$	0	0
$790$	0	0
$791$	7.32916	0.260595
$792$	0	0
$793$	7.02827	0.249581
$794$	46.7258	1.65824
$795$	0	0
$796$	71.5015	2.53430
$797$	−31.1150	−1.10215	−0.551074	−	0.834456i	$-0.685782\pi$
$797$	−31.1150	−1.10215	−0.551074	+	0.834456i	$0.685782\pi$
$798$	0	0
$799$	9.15951	0.324040
$800$	0	0
$801$	0	0
$802$	−15.5707	−0.549820
$803$	63.0667	2.22557
$804$	0	0
$805$	0	0
$806$	−8.83502	−0.311200
$807$	0	0
$808$	67.7650	2.38396
$809$	−46.6291	−1.63939	−0.819696	−	0.572799i	$-0.805857\pi$
$809$	−46.6291	−1.63939	−0.819696	+	0.572799i	$0.805857\pi$
$810$	0	0
$811$	−47.0283	−1.65139	−0.825693	−	0.564120i	$-0.809216\pi$
$811$	−47.0283	−1.65139	−0.825693	+	0.564120i	$0.809216\pi$
$812$	−20.7066	−0.726659
$813$	0	0
$814$	69.7084	2.44328
$815$	0	0
$816$	0	0
$817$	−6.82956	−0.238936
$818$	−80.5946	−2.81792
$819$	0	0
$820$	0	0
$821$	−35.3912	−1.23516	−0.617580	−	0.786508i	$-0.711887\pi$
$821$	−35.3912	−1.23516	−0.617580	+	0.786508i	$0.711887\pi$
$822$	0	0
$823$	17.2161	0.600114	0.300057	−	0.953921i	$-0.402994\pi$
$823$	17.2161	0.600114	0.300057	+	0.953921i	$0.402994\pi$
$824$	21.9945	0.766216
$825$	0	0
$826$	−2.67912	−0.0932184
$827$	16.3310	0.567885	0.283942	−	0.958841i	$-0.408358\pi$
$827$	16.3310	0.567885	0.283942	+	0.958841i	$0.408358\pi$
$828$	0	0
$829$	−29.0667	−1.00953	−0.504764	−	0.863258i	$-0.668420\pi$
$829$	−29.0667	−1.00953	−0.504764	+	0.863258i	$0.668420\pi$
$830$	0	0
$831$	0	0
$832$	11.5707	0.401141
$833$	8.93438	0.309558
$834$	0	0
$835$	0	0
$836$	144.303	4.99082
$837$	0	0
$838$	−24.6099	−0.850134
$839$	−27.6646	−0.955087	−0.477544	−	0.878608i	$-0.658473\pi$
$839$	−27.6646	−0.955087	−0.477544	+	0.878608i	$0.658473\pi$
$840$	0	0
$841$	68.2846	2.35464
$842$	−35.8880	−1.23678
$843$	0	0
$844$	108.066	3.71978
$845$	0	0
$846$	0	0
$847$	−6.93984	−0.238456
$848$	−10.2926	−0.353450
$849$	0	0
$850$	0	0
$851$	−1.61350	−0.0553099
$852$	0	0
$853$	−32.5105	−1.11314	−0.556570	−	0.830801i	$-0.687883\pi$
$853$	−32.5105	−1.11314	−0.556570	+	0.830801i	$0.687883\pi$
$854$	2.44305	0.0835995
$855$	0	0
$856$	−39.3027	−1.34334
$857$	−45.9144	−1.56841	−0.784203	−	0.620505i	$-0.786928\pi$
$857$	−45.9144	−1.56841	−0.784203	+	0.620505i	$0.786928\pi$
$858$	0	0
$859$	−19.6026	−0.668831	−0.334415	−	0.942426i	$-0.608539\pi$
$859$	−19.6026	−0.668831	−0.334415	+	0.942426i	$0.608539\pi$
$860$	0	0
$861$	0	0
$862$	57.4104	1.95540
$863$	46.1696	1.57163	0.785816	−	0.618460i	$-0.212243\pi$
$863$	46.1696	1.57163	0.785816	+	0.618460i	$0.212243\pi$
$864$	0	0
$865$	0	0
$866$	−34.7603	−1.18120
$867$	0	0
$868$	−2.09936	−0.0712569
$869$	1.75566	0.0595568
$870$	0	0
$871$	32.0757	1.08685
$872$	−22.7922	−0.771842
$873$	0	0
$874$	−4.88611	−0.165275
$875$	0	0
$876$	0	0
$877$	−11.3017	−0.381631	−0.190815	−	0.981626i	$-0.561113\pi$
$877$	−11.3017	−0.381631	−0.190815	+	0.981626i	$0.561113\pi$
$878$	100.190	3.38125
$879$	0	0
$880$	0	0
$881$	9.60803	0.323703	0.161851	−	0.986815i	$-0.448253\pi$
$881$	9.60803	0.323703	0.161851	+	0.986815i	$0.448253\pi$
$882$	0	0
$883$	33.5279	1.12830	0.564151	−	0.825671i	$-0.309203\pi$
$883$	33.5279	1.12830	0.564151	+	0.825671i	$0.309203\pi$
$884$	20.0565	0.674575
$885$	0	0
$886$	−25.5196	−0.857348
$887$	−9.76394	−0.327841	−0.163920	−	0.986474i	$-0.552414\pi$
$887$	−9.76394	−0.327841	−0.163920	+	0.986474i	$0.552414\pi$
$888$	0	0
$889$	−3.03920	−0.101932
$890$	0	0
$891$	0	0
$892$	−83.3219	−2.78982
$893$	46.0565	1.54122
$894$	0	0
$895$	0	0
$896$	7.40931	0.247528
$897$	0	0
$898$	9.42852	0.314634
$899$	9.86330	0.328959
$900$	0	0
$901$	2.25526	0.0751337
$902$	−88.8498	−2.95838
$903$	0	0
$904$	−88.0203	−2.92751
$905$	0	0
$906$	0	0
$907$	9.18418	0.304956	0.152478	−	0.988307i	$-0.451275\pi$
$907$	9.18418	0.304956	0.152478	+	0.988307i	$0.451275\pi$
$908$	101.256	3.36028
$909$	0	0
$910$	0	0
$911$	12.2553	0.406035	0.203018	−	0.979175i	$-0.434925\pi$
$911$	12.2553	0.406035	0.203018	+	0.979175i	$0.434925\pi$
$912$	0	0
$913$	−54.9811	−1.81961
$914$	4.27887	0.141533
$915$	0	0
$916$	−108.144	−3.57319
$917$	−10.7466	−0.354884
$918$	0	0
$919$	−47.5663	−1.56907	−0.784533	−	0.620087i	$-0.787097\pi$
$919$	−47.5663	−1.56907	−0.784533	+	0.620087i	$0.787097\pi$
$920$	0	0
$921$	0	0
$922$	−18.6280	−0.613482
$923$	47.3593	1.55885
$924$	0	0
$925$	0	0
$926$	76.3958	2.51052
$927$	0	0
$928$	34.3821	1.12865
$929$	−23.5333	−0.772104	−0.386052	−	0.922477i	$-0.626161\pi$
$929$	−23.5333	−0.772104	−0.386052	+	0.922477i	$0.626161\pi$
$930$	0	0
$931$	44.9245	1.47234
$932$	80.8762	2.64919
$933$	0	0
$934$	43.0019	1.40706
$935$	0	0
$936$	0	0
$937$	56.7258	1.85315	0.926575	−	0.376109i	$-0.122738\pi$
$937$	56.7258	1.85315	0.926575	+	0.376109i	$0.122738\pi$
$938$	11.1496	0.364049
$939$	0	0
$940$	0	0
$941$	33.5333	1.09316	0.546578	−	0.837408i	$-0.315930\pi$
$941$	33.5333	1.09316	0.546578	+	0.837408i	$0.315930\pi$
$942$	0	0
$943$	2.05655	0.0669704
$944$	13.2215	0.430324
$945$	0	0
$946$	−12.9992	−0.422640
$947$	−52.0685	−1.69200	−0.846000	−	0.533183i	$-0.820996\pi$
$947$	−52.0685	−1.69200	−0.846000	+	0.533183i	$0.820996\pi$
$948$	0	0
$949$	44.0757	1.43076
$950$	0	0
$951$	0	0
$952$	3.74474	0.121368
$953$	−11.1896	−0.362468	−0.181234	−	0.983440i	$-0.558009\pi$
$953$	−11.1896	−0.362468	−0.181234	+	0.983440i	$0.558009\pi$
$954$	0	0
$955$	0	0
$956$	16.7165	0.540649
$957$	0	0
$958$	−53.5235	−1.72926
$959$	7.09575	0.229134
$960$	0	0
$961$	1.00000	0.0322581
$962$	48.7175	1.57072
$963$	0	0
$964$	−6.97173	−0.224544
$965$	0	0
$966$	0	0
$967$	−36.8296	−1.18436	−0.592179	−	0.805806i	$-0.701732\pi$
$967$	−36.8296	−1.18436	−0.592179	+	0.805806i	$0.701732\pi$
$968$	83.3448	2.67880
$969$	0	0
$970$	0	0
$971$	−18.7494	−0.601697	−0.300848	−	0.953672i	$-0.597270\pi$
$971$	−18.7494	−0.601697	−0.300848	+	0.953672i	$0.597270\pi$
$972$	0	0
$973$	−6.26619	−0.200885
$974$	−51.5388	−1.65141
$975$	0	0
$976$	−12.0565	−0.385921
$977$	−50.2262	−1.60688	−0.803439	−	0.595387i	$-0.796999\pi$
$977$	−50.2262	−1.60688	−0.803439	+	0.595387i	$0.796999\pi$
$978$	0	0
$979$	25.3110	0.808943
$980$	0	0
$981$	0	0
$982$	78.1059	2.49246
$983$	15.5279	0.495262	0.247631	−	0.968854i	$-0.420348\pi$
$983$	15.5279	0.495262	0.247631	+	0.968854i	$0.420348\pi$
$984$	0	0
$985$	0	0
$986$	−32.7549	−1.04313
$987$	0	0
$988$	100.850	3.20846
$989$	0.300884	0.00956755
$990$	0	0
$991$	−20.1022	−0.638566	−0.319283	−	0.947659i	$-0.603442\pi$
$991$	−20.1022	−0.638566	−0.319283	+	0.947659i	$0.603442\pi$
$992$	3.48586	0.110676
$993$	0	0
$994$	16.4623	0.522151
$995$	0	0
$996$	0	0
$997$	20.8186	0.659333	0.329666	−	0.944098i	$-0.393064\pi$
$997$	20.8186	0.659333	0.329666	+	0.944098i	$0.393064\pi$
$998$	38.5671	1.22082
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bf.1.3		3
3.2	odd	2		2325.2.a.r.1.1		3
5.4	even	2		1395.2.a.j.1.1		3
15.2	even	4		2325.2.c.k.1024.1		6
15.8	even	4		2325.2.c.k.1024.6		6
15.14	odd	2		465.2.a.e.1.3	✓	3
60.59	even	2		7440.2.a.bs.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.e.1.3	✓	3	15.14	odd	2
1395.2.a.j.1.1		3	5.4	even	2
2325.2.a.r.1.1		3	3.2	odd	2
2325.2.c.k.1024.1		6	15.2	even	4
2325.2.c.k.1024.6		6	15.8	even	4
6975.2.a.bf.1.3		3	1.1	even	1	trivial
7440.2.a.bs.1.3		3	60.59	even	2

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:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q+2.51414 q^{2} +4.32088 q^{4} -0.485863 q^{7} +5.83502 q^{8} -5.02827 q^{11} -3.51414 q^{13} -1.22153 q^{14} +6.02827 q^{16} -1.32088 q^{17} -6.64177 q^{19} -12.6418 q^{22} +0.292611 q^{23} -8.83502 q^{26} -2.09936 q^{28} +9.86330 q^{29} +1.00000 q^{31} +3.48586 q^{32} -3.32088 q^{34} -5.51414 q^{37} -16.6983 q^{38} +7.02827 q^{41} +1.02827 q^{43} -21.7266 q^{44} +0.735663 q^{46} -6.93438 q^{47} -6.76394 q^{49} -15.1842 q^{52} -1.70739 q^{53} -2.83502 q^{56} +24.7977 q^{58} +2.19325 q^{59} -2.00000 q^{61} +2.51414 q^{62} -3.29261 q^{64} -9.12763 q^{67} -5.70739 q^{68} -13.4768 q^{71} -12.5424 q^{73} -13.8633 q^{74} -28.6983 q^{76} +2.44305 q^{77} -0.349158 q^{79} +17.6700 q^{82} +10.9344 q^{83} +2.58522 q^{86} -29.3401 q^{88} -5.03374 q^{89} +1.70739 q^{91} +1.26434 q^{92} -17.4340 q^{94} -10.4431 q^{97} -17.0055 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 5 q^{4} - 8 q^{7} + 3 q^{8} - 2 q^{11} - 4 q^{13} + 8 q^{14} + 5 q^{16} + 4 q^{17} - 4 q^{19} - 22 q^{22} + 6 q^{23} - 12 q^{26} - 10 q^{28} + 2 q^{29} + 3 q^{31} + 17 q^{32} - 2 q^{34} - 10 q^{37}+ \cdots - 57 q^{98}+O(q^{100}) \) 3 * q + q^2 + 5 * q^4 - 8 * q^7 + 3 * q^8 - 2 * q^11 - 4 * q^13 + 8 * q^14 + 5 * q^16 + 4 * q^17 - 4 * q^19 - 22 * q^22 + 6 * q^23 - 12 * q^26 - 10 * q^28 + 2 * q^29 + 3 * q^31 + 17 * q^32 - 2 * q^34 - 10 * q^37 - 8 * q^38 + 8 * q^41 - 10 * q^43 - 10 * q^44 - 16 * q^46 - 10 * q^47 + 11 * q^49 - 10 * q^52 + 6 * q^56 + 36 * q^58 + 8 * q^59 - 6 * q^61 + q^62 - 15 * q^64 - 18 * q^67 - 12 * q^68 - 10 * q^71 - 18 * q^73 - 14 * q^74 - 44 * q^76 - 16 * q^77 + 20 * q^79 + 24 * q^82 + 22 * q^83 + 18 * q^86 - 30 * q^88 - 8 * q^89 + 22 * q^92 + 8 * q^94 - 8 * q^97 - 57 * q^98

Introduction

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