LMFDB - Embedded newform 4275.2.a.bs.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, 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("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4275 = 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4275.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:	$34.1360468641$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.16717036.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} + 26x^{2} - 19 $ x^6 - 10x^4 + 26x^2 - 19
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.53035$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.53035 q^{2} +4.40268 q^{4} -2.62981 q^{7} +6.07962 q^{8} -4.12400 q^{11} -3.54573 q^{13} -6.65435 q^{14} +6.57822 q^{16} -3.91124 q^{17} -1.00000 q^{19} -10.4352 q^{22} -0.936703 q^{23} -8.97195 q^{26} -11.5782 q^{28} +1.01892 q^{29} -3.17554 q^{31} +4.48597 q^{32} -9.89682 q^{34} -7.54573 q^{37} -2.53035 q^{38} +1.95562 q^{41} +6.35109 q^{43} -18.1566 q^{44} -2.37019 q^{46} +8.97195 q^{47} -0.0840822 q^{49} -15.6107 q^{52} -11.1403 q^{53} -15.9883 q^{56} +2.57822 q^{58} -7.01632 q^{59} +12.6107 q^{61} -8.03524 q^{62} -1.80536 q^{64} +3.98090 q^{67} -17.2199 q^{68} +7.01632 q^{71} -7.16816 q^{73} -19.0934 q^{74} -4.40268 q^{76} +10.8454 q^{77} -11.5266 q^{79} +4.94841 q^{82} -14.2454 q^{83} +16.0705 q^{86} -25.0724 q^{88} +13.1782 q^{89} +9.32461 q^{91} -4.12400 q^{92} +22.7022 q^{94} -3.54573 q^{97} -0.212757 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{4} - 16 q^{13} - 6 q^{19} - 10 q^{22} - 30 q^{28} + 2 q^{31} - 12 q^{34} - 40 q^{37} - 4 q^{43} - 30 q^{46} + 10 q^{49} - 20 q^{52} - 24 q^{58} + 2 q^{61} + 26 q^{64} - 34 q^{67} - 22 q^{73}+ \cdots - 16 q^{97}+O(q^{100}) $ 6 * q + 8 * q^4 - 16 * q^13 - 6 * q^19 - 10 * q^22 - 30 * q^28 + 2 * q^31 - 12 * q^34 - 40 * q^37 - 4 * q^43 - 30 * q^46 + 10 * q^49 - 20 * q^52 - 24 * q^58 + 2 * q^61 + 26 * q^64 - 34 * q^67 - 22 * q^73 - 8 * q^76 - 6 * q^79 + 6 * q^82 - 82 * q^88 - 44 * q^91 + 52 * q^94 - 16 * q^97

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 $

$2$	2.53035	1.78923	0.894614	−	0.446839i	$-0.147450\pi$
$2$	2.53035	1.78923	0.894614	+	0.446839i	$0.147450\pi$
$3$	0	0
$3$	0	0
$4$	4.40268	2.20134
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.62981	−0.993976	−0.496988	−	0.867757i	$-0.665561\pi$
$7$	−2.62981	−0.993976	−0.496988	+	0.867757i	$0.665561\pi$
$8$	6.07962	2.14947
$9$	0	0
$10$	0	0
$11$	−4.12400	−1.24343	−0.621716	−	0.783242i	$-0.713564\pi$
$11$	−4.12400	−1.24343	−0.621716	+	0.783242i	$0.713564\pi$
$12$	0	0
$13$	−3.54573	−0.983409	−0.491704	−	0.870762i	$-0.663626\pi$
$13$	−3.54573	−0.983409	−0.491704	+	0.870762i	$0.663626\pi$
$14$	−6.65435	−1.77845
$15$	0	0
$16$	6.57822	1.64456
$17$	−3.91124	−0.948616	−0.474308	−	0.880359i	$-0.657302\pi$
$17$	−3.91124	−0.948616	−0.474308	+	0.880359i	$0.657302\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−10.4352	−2.22479
$23$	−0.936703	−0.195316	−0.0976580	−	0.995220i	$-0.531135\pi$
$23$	−0.936703	−0.195316	−0.0976580	+	0.995220i	$0.531135\pi$
$24$	0	0
$25$	0	0
$26$	−8.97195	−1.75954
$27$	0	0
$28$	−11.5782	−2.18808
$29$	1.01892	0.189209	0.0946043	−	0.995515i	$-0.469841\pi$
$29$	1.01892	0.189209	0.0946043	+	0.995515i	$0.469841\pi$
$30$	0	0
$31$	−3.17554	−0.570345	−0.285172	−	0.958476i	$-0.592051\pi$
$31$	−3.17554	−0.570345	−0.285172	+	0.958476i	$0.592051\pi$
$32$	4.48597	0.793015
$33$	0	0
$34$	−9.89682	−1.69729
$35$	0	0
$36$	0	0
$37$	−7.54573	−1.24051	−0.620255	−	0.784400i	$-0.712971\pi$
$37$	−7.54573	−1.24051	−0.620255	+	0.784400i	$0.712971\pi$
$38$	−2.53035	−0.410477
$39$	0	0
$40$	0	0
$41$	1.95562	0.305417	0.152708	−	0.988271i	$-0.451200\pi$
$41$	1.95562	0.305417	0.152708	+	0.988271i	$0.451200\pi$
$42$	0	0
$43$	6.35109	0.968532	0.484266	−	0.874921i	$-0.339087\pi$
$43$	6.35109	0.968532	0.484266	+	0.874921i	$0.339087\pi$
$44$	−18.1566	−2.73722
$45$	0	0
$46$	−2.37019	−0.349465
$47$	8.97195	1.30869	0.654346	−	0.756195i	$-0.272944\pi$
$47$	8.97195	1.30869	0.654346	+	0.756195i	$0.272944\pi$
$48$	0	0
$49$	−0.0840822	−0.0120117
$50$	0	0
$51$	0	0
$52$	−15.6107	−2.16482
$53$	−11.1403	−1.53024	−0.765121	−	0.643887i	$-0.777321\pi$
$53$	−11.1403	−1.53024	−0.765121	+	0.643887i	$0.777321\pi$
$54$	0	0
$55$	0	0
$56$	−15.9883	−2.13652
$57$	0	0
$58$	2.57822	0.338537
$59$	−7.01632	−0.913448	−0.456724	−	0.889609i	$-0.650977\pi$
$59$	−7.01632	−0.913448	−0.456724	+	0.889609i	$0.650977\pi$
$60$	0	0
$61$	12.6107	1.61464	0.807318	−	0.590116i	$-0.200918\pi$
$61$	12.6107	1.61464	0.807318	+	0.590116i	$0.200918\pi$
$62$	−8.03524	−1.02048
$63$	0	0
$64$	−1.80536	−0.225670
$65$	0	0
$66$	0	0
$67$	3.98090	0.486345	0.243172	−	0.969983i	$-0.421812\pi$
$67$	3.98090	0.486345	0.243172	+	0.969983i	$0.421812\pi$
$68$	−17.2199	−2.08823
$69$	0	0
$70$	0	0
$71$	7.01632	0.832685	0.416342	−	0.909208i	$-0.363312\pi$
$71$	7.01632	0.832685	0.416342	+	0.909208i	$0.363312\pi$
$72$	0	0
$73$	−7.16816	−0.838970	−0.419485	−	0.907762i	$-0.637789\pi$
$73$	−7.16816	−0.838970	−0.419485	+	0.907762i	$0.637789\pi$
$74$	−19.0934	−2.21956
$75$	0	0
$76$	−4.40268	−0.505022
$77$	10.8454	1.23594
$78$	0	0
$79$	−11.5266	−1.29685	−0.648424	−	0.761280i	$-0.724571\pi$
$79$	−11.5266	−1.29685	−0.648424	+	0.761280i	$0.724571\pi$
$80$	0	0
$81$	0	0
$82$	4.94841	0.546460
$83$	−14.2454	−1.56364	−0.781818	−	0.623506i	$-0.785708\pi$
$83$	−14.2454	−1.56364	−0.781818	+	0.623506i	$0.785708\pi$
$84$	0	0
$85$	0	0
$86$	16.0705	1.73293
$87$	0	0
$88$	−25.0724	−2.67272
$89$	13.1782	1.39688	0.698441	−	0.715667i	$-0.253877\pi$
$89$	13.1782	1.39688	0.698441	+	0.715667i	$0.253877\pi$
$90$	0	0
$91$	9.32461	0.977485
$92$	−4.12400	−0.429957
$93$	0	0
$94$	22.7022	2.34155
$95$	0	0
$96$	0	0
$97$	−3.54573	−0.360014	−0.180007	−	0.983665i	$-0.557612\pi$
$97$	−3.54573	−0.360014	−0.180007	+	0.983665i	$0.557612\pi$
$98$	−0.212757	−0.0214917
$99$	0	0
$100$	0	0
$101$	2.76178	0.274808	0.137404	−	0.990515i	$-0.456124\pi$
$101$	2.76178	0.274808	0.137404	+	0.990515i	$0.456124\pi$
$102$	0	0
$103$	7.98090	0.786382	0.393191	−	0.919457i	$-0.371371\pi$
$103$	7.98090	0.786382	0.393191	+	0.919457i	$0.371371\pi$
$104$	−21.5567	−2.11381
$105$	0	0
$106$	−28.1889	−2.73795
$107$	11.3531	1.09754	0.548772	−	0.835972i	$-0.315096\pi$
$107$	11.3531	1.09754	0.548772	+	0.835972i	$0.315096\pi$
$108$	0	0
$109$	12.8054	1.22653	0.613265	−	0.789877i	$-0.289856\pi$
$109$	12.8054	1.22653	0.613265	+	0.789877i	$0.289856\pi$
$110$	0	0
$111$	0	0
$112$	−17.2995	−1.63465
$113$	−1.36222	−0.128147	−0.0640733	−	0.997945i	$-0.520409\pi$
$113$	−1.36222	−0.128147	−0.0640733	+	0.997945i	$0.520409\pi$
$114$	0	0
$115$	0	0
$116$	4.48597	0.416512
$117$	0	0
$118$	−17.7538	−1.63437
$119$	10.2858	0.942901
$120$	0	0
$121$	6.00738	0.546125
$122$	31.9095	2.88895
$123$	0	0
$124$	−13.9809	−1.25552
$125$	0	0
$126$	0	0
$127$	−19.8777	−1.76386	−0.881931	−	0.471378i	$-0.843757\pi$
$127$	−19.8777	−1.76386	−0.881931	+	0.471378i	$0.843757\pi$
$128$	−13.5401	−1.19679
$129$	0	0
$130$	0	0
$131$	1.36222	0.119017	0.0595087	−	0.998228i	$-0.481047\pi$
$131$	1.36222	0.119017	0.0595087	+	0.998228i	$0.481047\pi$
$132$	0	0
$133$	2.62981	0.228034
$134$	10.0731	0.870182
$135$	0	0
$136$	−23.7789	−2.03902
$137$	0.425515	0.0363542	0.0181771	−	0.999835i	$-0.494214\pi$
$137$	0.425515	0.0363542	0.0181771	+	0.999835i	$0.494214\pi$
$138$	0	0
$139$	0.629813	0.0534201	0.0267100	−	0.999643i	$-0.491497\pi$
$139$	0.629813	0.0534201	0.0267100	+	0.999643i	$0.491497\pi$
$140$	0	0
$141$	0	0
$142$	17.7538	1.48986
$143$	14.6226	1.22280
$144$	0	0
$145$	0	0
$146$	−18.1380	−1.50111
$147$	0	0
$148$	−33.2214	−2.73078
$149$	0.723945	0.0593079	0.0296539	−	0.999560i	$-0.490559\pi$
$149$	0.723945	0.0593079	0.0296539	+	0.999560i	$0.490559\pi$
$150$	0	0
$151$	2.16816	0.176443	0.0882214	−	0.996101i	$-0.471882\pi$
$151$	2.16816	0.176443	0.0882214	+	0.996101i	$0.471882\pi$
$152$	−6.07962	−0.493122
$153$	0	0
$154$	27.4426	2.21138
$155$	0	0
$156$	0	0
$157$	20.1373	1.60714	0.803568	−	0.595213i	$-0.202932\pi$
$157$	20.1373	1.60714	0.803568	+	0.595213i	$0.202932\pi$
$158$	−29.1664	−2.32036
$159$	0	0
$160$	0	0
$161$	2.46335	0.194139
$162$	0	0
$163$	−10.6298	−0.832591	−0.416296	−	0.909229i	$-0.636672\pi$
$163$	−10.6298	−0.832591	−0.416296	+	0.909229i	$0.636672\pi$
$164$	8.60997	0.672326
$165$	0	0
$166$	−36.0459	−2.79770
$167$	21.0490	1.62882	0.814409	−	0.580291i	$-0.197061\pi$
$167$	21.0490	1.62882	0.814409	+	0.580291i	$0.197061\pi$
$168$	0	0
$169$	−0.427791	−0.0329070
$170$	0	0
$171$	0	0
$172$	27.9618	2.13207
$173$	−17.3505	−1.31913	−0.659567	−	0.751646i	$-0.729260\pi$
$173$	−17.3505	−1.31913	−0.659567	+	0.751646i	$0.729260\pi$
$174$	0	0
$175$	0	0
$176$	−27.1286	−2.04489
$177$	0	0
$178$	33.3454	2.49934
$179$	−19.1756	−1.43325	−0.716625	−	0.697459i	$-0.754314\pi$
$179$	−19.1756	−1.43325	−0.716625	+	0.697459i	$0.754314\pi$
$180$	0	0
$181$	−13.6107	−1.01168	−0.505838	−	0.862628i	$-0.668817\pi$
$181$	−13.6107	−1.01168	−0.505838	+	0.862628i	$0.668817\pi$
$182$	23.5945	1.74894
$183$	0	0
$184$	−5.69480	−0.419826
$185$	0	0
$186$	0	0
$187$	16.1300	1.17954
$188$	39.5006	2.88088
$189$	0	0
$190$	0	0
$191$	9.18470	0.664582	0.332291	−	0.943177i	$-0.392178\pi$
$191$	9.18470	0.664582	0.332291	+	0.943177i	$0.392178\pi$
$192$	0	0
$193$	−24.3511	−1.75283	−0.876415	−	0.481557i	$-0.840071\pi$
$193$	−24.3511	−1.75283	−0.876415	+	0.481557i	$0.840071\pi$
$194$	−8.97195	−0.644148
$195$	0	0
$196$	−0.370187	−0.0264419
$197$	1.14946	0.0818956	0.0409478	−	0.999161i	$-0.486962\pi$
$197$	1.14946	0.0818956	0.0409478	+	0.999161i	$0.486962\pi$
$198$	0	0
$199$	8.62981	0.611751	0.305876	−	0.952071i	$-0.401051\pi$
$199$	8.62981	0.611751	0.305876	+	0.952071i	$0.401051\pi$
$200$	0	0
$201$	0	0
$202$	6.98828	0.491694
$203$	−2.67957	−0.188069
$204$	0	0
$205$	0	0
$206$	20.1945	1.40702
$207$	0	0
$208$	−23.3246	−1.61727
$209$	4.12400	0.285263
$210$	0	0
$211$	13.5266	0.931212	0.465606	−	0.884992i	$-0.345836\pi$
$211$	13.5266	0.931212	0.465606	+	0.884992i	$0.345836\pi$
$212$	−49.0473	−3.36858
$213$	0	0
$214$	28.7273	1.96376
$215$	0	0
$216$	0	0
$217$	8.35109	0.566909
$218$	32.4021	2.19454
$219$	0	0
$220$	0	0
$221$	13.8682	0.932877
$222$	0	0
$223$	−5.91592	−0.396159	−0.198080	−	0.980186i	$-0.563470\pi$
$223$	−5.91592	−0.396159	−0.198080	+	0.980186i	$0.563470\pi$
$224$	−11.7973	−0.788238
$225$	0	0
$226$	−3.44689	−0.229284
$227$	−13.3909	−0.888787	−0.444393	−	0.895832i	$-0.646581\pi$
$227$	−13.3909	−0.888787	−0.444393	+	0.895832i	$0.646581\pi$
$228$	0	0
$229$	−27.8777	−1.84221	−0.921106	−	0.389312i	$-0.872713\pi$
$229$	−27.8777	−1.84221	−0.921106	+	0.389312i	$0.872713\pi$
$230$	0	0
$231$	0	0
$232$	6.19464	0.406698
$233$	23.0046	1.50708	0.753541	−	0.657401i	$-0.228344\pi$
$233$	23.0046	1.50708	0.753541	+	0.657401i	$0.228344\pi$
$234$	0	0
$235$	0	0
$236$	−30.8906	−2.01081
$237$	0	0
$238$	26.0268	1.68707
$239$	−27.0803	−1.75168	−0.875839	−	0.482604i	$-0.839691\pi$
$239$	−27.0803	−1.75168	−0.875839	+	0.482604i	$0.839691\pi$
$240$	0	0
$241$	15.9618	1.02819	0.514095	−	0.857733i	$-0.328128\pi$
$241$	15.9618	1.02819	0.514095	+	0.857733i	$0.328128\pi$
$242$	15.2008	0.977143
$243$	0	0
$244$	55.5209	3.55436
$245$	0	0
$246$	0	0
$247$	3.54573	0.225609
$248$	−19.3061	−1.22594
$249$	0	0
$250$	0	0
$251$	18.3694	1.15947	0.579733	−	0.814806i	$-0.303157\pi$
$251$	18.3694	1.15947	0.579733	+	0.814806i	$0.303157\pi$
$252$	0	0
$253$	3.86296	0.242862
$254$	−50.2976	−3.15595
$255$	0	0
$256$	−30.6506	−1.91566
$257$	0.854486	0.0533014	0.0266507	−	0.999645i	$-0.491516\pi$
$257$	0.854486	0.0533014	0.0266507	+	0.999645i	$0.491516\pi$
$258$	0	0
$259$	19.8439	1.23304
$260$	0	0
$261$	0	0
$262$	3.44689	0.212950
$263$	4.12400	0.254297	0.127148	−	0.991884i	$-0.459418\pi$
$263$	4.12400	0.254297	0.127148	+	0.991884i	$0.459418\pi$
$264$	0	0
$265$	0	0
$266$	6.65435	0.408005
$267$	0	0
$268$	17.5266	1.07061
$269$	30.5287	1.86136	0.930682	−	0.365828i	$-0.119214\pi$
$269$	30.5287	1.86136	0.930682	+	0.365828i	$0.119214\pi$
$270$	0	0
$271$	−8.46165	−0.514009	−0.257004	−	0.966410i	$-0.582735\pi$
$271$	−8.46165	−0.514009	−0.257004	+	0.966410i	$0.582735\pi$
$272$	−25.7290	−1.56005
$273$	0	0
$274$	1.07670	0.0650460
$275$	0	0
$276$	0	0
$277$	12.9618	0.778799	0.389400	−	0.921069i	$-0.372683\pi$
$277$	12.9618	0.778799	0.389400	+	0.921069i	$0.372683\pi$
$278$	1.59365	0.0955807
$279$	0	0
$280$	0	0
$281$	−22.9563	−1.36946	−0.684728	−	0.728799i	$-0.740079\pi$
$281$	−22.9563	−1.36946	−0.684728	+	0.728799i	$0.740079\pi$
$282$	0	0
$283$	23.9618	1.42438	0.712191	−	0.701986i	$-0.247703\pi$
$283$	23.9618	1.42438	0.712191	+	0.701986i	$0.247703\pi$
$284$	30.8906	1.83302
$285$	0	0
$286$	37.0003	2.18787
$287$	−5.14292	−0.303577
$288$	0	0
$289$	−1.70218	−0.100128
$290$	0	0
$291$	0	0
$292$	−31.5591	−1.84686
$293$	−17.0072	−0.993571	−0.496785	−	0.867873i	$-0.665486\pi$
$293$	−17.0072	−0.993571	−0.496785	+	0.867873i	$0.665486\pi$
$294$	0	0
$295$	0	0
$296$	−45.8752	−2.66644
$297$	0	0
$298$	1.83184	0.106115
$299$	3.32130	0.192075
$300$	0	0
$301$	−16.7022	−0.962698
$302$	5.48622	0.315696
$303$	0	0
$304$	−6.57822	−0.377287
$305$	0	0
$306$	0	0
$307$	−26.8512	−1.53248	−0.766241	−	0.642554i	$-0.777875\pi$
$307$	−26.8512	−1.53248	−0.766241	+	0.642554i	$0.777875\pi$
$308$	47.7486	2.72073
$309$	0	0
$310$	0	0
$311$	18.2050	1.03231	0.516155	−	0.856495i	$-0.327363\pi$
$311$	18.2050	1.03231	0.516155	+	0.856495i	$0.327363\pi$
$312$	0	0
$313$	−0.824456	−0.0466010	−0.0233005	−	0.999729i	$-0.507417\pi$
$313$	−0.824456	−0.0466010	−0.0233005	+	0.999729i	$0.507417\pi$
$314$	50.9546	2.87553
$315$	0	0
$316$	−50.7481	−2.85480
$317$	−10.9759	−0.616468	−0.308234	−	0.951311i	$-0.599738\pi$
$317$	−10.9759	−0.616468	−0.308234	+	0.951311i	$0.599738\pi$
$318$	0	0
$319$	−4.20202	−0.235268
$320$	0	0
$321$	0	0
$322$	6.23315	0.347360
$323$	3.91124	0.217627
$324$	0	0
$325$	0	0
$326$	−26.8972	−1.48970
$327$	0	0
$328$	11.8894	0.656484
$329$	−23.5945	−1.30081
$330$	0	0
$331$	−10.6181	−0.583623	−0.291812	−	0.956476i	$-0.594258\pi$
$331$	−10.6181	−0.583623	−0.291812	+	0.956476i	$0.594258\pi$
$332$	−62.7179	−3.44209
$333$	0	0
$334$	53.2613	2.91433
$335$	0	0
$336$	0	0
$337$	−14.5193	−0.790914	−0.395457	−	0.918484i	$-0.629414\pi$
$337$	−14.5193	−0.790914	−0.395457	+	0.918484i	$0.629414\pi$
$338$	−1.08246	−0.0588781
$339$	0	0
$340$	0	0
$341$	13.0959	0.709185
$342$	0	0
$343$	18.6298	1.00592
$344$	38.6122	2.08183
$345$	0	0
$346$	−43.9028	−2.36023
$347$	−19.0934	−1.02498	−0.512492	−	0.858692i	$-0.671278\pi$
$347$	−19.0934	−1.02498	−0.512492	+	0.858692i	$0.671278\pi$
$348$	0	0
$349$	−28.7789	−1.54050	−0.770249	−	0.637743i	$-0.779868\pi$
$349$	−28.7789	−1.54050	−0.770249	+	0.637743i	$0.779868\pi$
$350$	0	0
$351$	0	0
$352$	−18.5002	−0.986062
$353$	−18.5338	−0.986457	−0.493228	−	0.869900i	$-0.664183\pi$
$353$	−18.5338	−0.986457	−0.493228	+	0.869900i	$0.664183\pi$
$354$	0	0
$355$	0	0
$356$	58.0192	3.07501
$357$	0	0
$358$	−48.5209	−2.56441
$359$	19.0934	1.00771	0.503854	−	0.863789i	$-0.331915\pi$
$359$	19.0934	1.00771	0.503854	+	0.863789i	$0.331915\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−34.4399	−1.81012
$363$	0	0
$364$	41.0533	2.15178
$365$	0	0
$366$	0	0
$367$	8.53401	0.445472	0.222736	−	0.974879i	$-0.428501\pi$
$367$	8.53401	0.445472	0.222736	+	0.974879i	$0.428501\pi$
$368$	−6.16184	−0.321208
$369$	0	0
$370$	0	0
$371$	29.2970	1.52102
$372$	0	0
$373$	−27.2596	−1.41145	−0.705725	−	0.708486i	$-0.749379\pi$
$373$	−27.2596	−1.41145	−0.705725	+	0.708486i	$0.749379\pi$
$374$	40.8145	2.11047
$375$	0	0
$376$	54.5460	2.81300
$377$	−3.61281	−0.186069
$378$	0	0
$379$	13.3246	0.684439	0.342220	−	0.939620i	$-0.388821\pi$
$379$	13.3246	0.684439	0.342220	+	0.939620i	$0.388821\pi$
$380$	0	0
$381$	0	0
$382$	23.2405	1.18909
$383$	−23.5123	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$383$	−23.5123	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$384$	0	0
$385$	0	0
$386$	−61.6168	−3.13621
$387$	0	0
$388$	−15.6107	−0.792514
$389$	−29.3792	−1.48958	−0.744792	−	0.667297i	$-0.767451\pi$
$389$	−29.3792	−1.48958	−0.744792	+	0.667297i	$0.767451\pi$
$390$	0	0
$391$	3.66367	0.185280
$392$	−0.511188	−0.0258189
$393$	0	0
$394$	2.90854	0.146530
$395$	0	0
$396$	0	0
$397$	−21.6948	−1.08883	−0.544415	−	0.838816i	$-0.683248\pi$
$397$	−21.6948	−1.08883	−0.544415	+	0.838816i	$0.683248\pi$
$398$	21.8365	1.09456
$399$	0	0
$400$	0	0
$401$	33.1743	1.65665	0.828323	−	0.560250i	$-0.189295\pi$
$401$	33.1743	1.65665	0.828323	+	0.560250i	$0.189295\pi$
$402$	0	0
$403$	11.2596	0.560882
$404$	12.1592	0.604945
$405$	0	0
$406$	−6.78025	−0.336498
$407$	31.1186	1.54249
$408$	0	0
$409$	−0.271344	−0.0134171	−0.00670856	−	0.999977i	$-0.502135\pi$
$409$	−0.271344	−0.0134171	−0.00670856	+	0.999977i	$0.502135\pi$
$410$	0	0
$411$	0	0
$412$	35.1373	1.73109
$413$	18.4516	0.907945
$414$	0	0
$415$	0	0
$416$	−15.9061	−0.779858
$417$	0	0
$418$	10.4352	0.510401
$419$	27.6024	1.34847	0.674234	−	0.738518i	$-0.264474\pi$
$419$	27.6024	1.34847	0.674234	+	0.738518i	$0.264474\pi$
$420$	0	0
$421$	−27.1564	−1.32352	−0.661762	−	0.749714i	$-0.730191\pi$
$421$	−27.1564	−1.32352	−0.661762	+	0.749714i	$0.730191\pi$
$422$	34.2271	1.66615
$423$	0	0
$424$	−67.7290	−3.28921
$425$	0	0
$426$	0	0
$427$	−33.1638	−1.60491
$428$	49.9840	2.41607
$429$	0	0
$430$	0	0
$431$	16.8766	0.812920	0.406460	−	0.913669i	$-0.366763\pi$
$431$	16.8766	0.812920	0.406460	+	0.913669i	$0.366763\pi$
$432$	0	0
$433$	−3.19464	−0.153525	−0.0767624	−	0.997049i	$-0.524458\pi$
$433$	−3.19464	−0.153525	−0.0767624	+	0.997049i	$0.524458\pi$
$434$	21.1312	1.01433
$435$	0	0
$436$	56.3779	2.70001
$437$	0.936703	0.0448086
$438$	0	0
$439$	−21.6948	−1.03544	−0.517718	−	0.855551i	$-0.673218\pi$
$439$	−21.6948	−1.03544	−0.517718	+	0.855551i	$0.673218\pi$
$440$	0	0
$441$	0	0
$442$	35.0915	1.66913
$443$	10.8937	0.517574	0.258787	−	0.965934i	$-0.416677\pi$
$443$	10.8937	0.517574	0.258787	+	0.965934i	$0.416677\pi$
$444$	0	0
$445$	0	0
$446$	−14.9694	−0.708819
$447$	0	0
$448$	4.74775	0.224310
$449$	40.0567	1.89039	0.945195	−	0.326506i	$-0.105871\pi$
$449$	40.0567	1.89039	0.945195	+	0.326506i	$0.105871\pi$
$450$	0	0
$451$	−8.06498	−0.379765
$452$	−5.99741	−0.282094
$453$	0	0
$454$	−33.8837	−1.59024
$455$	0	0
$456$	0	0
$457$	−15.4499	−0.722717	−0.361359	−	0.932427i	$-0.617687\pi$
$457$	−15.4499	−0.722717	−0.361359	+	0.932427i	$0.617687\pi$
$458$	−70.5404	−3.29614
$459$	0	0
$460$	0	0
$461$	6.83746	0.318452	0.159226	−	0.987242i	$-0.449100\pi$
$461$	6.83746	0.318452	0.159226	+	0.987242i	$0.449100\pi$
$462$	0	0
$463$	−29.2362	−1.35872	−0.679361	−	0.733805i	$-0.737743\pi$
$463$	−29.2362	−1.35872	−0.679361	+	0.733805i	$0.737743\pi$
$464$	6.70268	0.311164
$465$	0	0
$466$	58.2097	2.69651
$467$	24.1057	1.11548	0.557740	−	0.830016i	$-0.311669\pi$
$467$	24.1057	1.11548	0.557740	+	0.830016i	$0.311669\pi$
$468$	0	0
$469$	−10.4690	−0.483415
$470$	0	0
$471$	0	0
$472$	−42.6566	−1.96343
$473$	−26.1919	−1.20430
$474$	0	0
$475$	0	0
$476$	45.2852	2.07565
$477$	0	0
$478$	−68.5226	−3.13415
$479$	34.2271	1.56388	0.781939	−	0.623355i	$-0.214231\pi$
$479$	34.2271	1.56388	0.781939	+	0.623355i	$0.214231\pi$
$480$	0	0
$481$	26.7551	1.21993
$482$	40.3890	1.83967
$483$	0	0
$484$	26.4486	1.20221
$485$	0	0
$486$	0	0
$487$	−40.0650	−1.81552	−0.907759	−	0.419492i	$-0.862208\pi$
$487$	−40.0650	−1.81552	−0.907759	+	0.419492i	$0.862208\pi$
$488$	76.6684	3.47061
$489$	0	0
$490$	0	0
$491$	34.7010	1.56603	0.783017	−	0.622001i	$-0.213680\pi$
$491$	34.7010	1.56603	0.783017	+	0.622001i	$0.213680\pi$
$492$	0	0
$493$	−3.98524	−0.179486
$494$	8.97195	0.403667
$495$	0	0
$496$	−20.8894	−0.937964
$497$	−18.4516	−0.827668
$498$	0	0
$499$	−38.9279	−1.74265	−0.871327	−	0.490703i	$-0.836740\pi$
$499$	−38.9279	−1.74265	−0.871327	+	0.490703i	$0.836740\pi$
$500$	0	0
$501$	0	0
$502$	46.4811	2.07455
$503$	−2.92622	−0.130474	−0.0652368	−	0.997870i	$-0.520780\pi$
$503$	−2.92622	−0.130474	−0.0652368	+	0.997870i	$0.520780\pi$
$504$	0	0
$505$	0	0
$506$	9.77465	0.434536
$507$	0	0
$508$	−87.5152	−3.88286
$509$	−4.66908	−0.206953	−0.103477	−	0.994632i	$-0.532997\pi$
$509$	−4.66908	−0.206953	−0.103477	+	0.994632i	$0.532997\pi$
$510$	0	0
$511$	18.8509	0.833916
$512$	−50.4765	−2.23077
$513$	0	0
$514$	2.16215	0.0953684
$515$	0	0
$516$	0	0
$517$	−37.0003	−1.62727
$518$	50.2120	2.20619
$519$	0	0
$520$	0	0
$521$	−15.6415	−0.685267	−0.342634	−	0.939469i	$-0.611319\pi$
$521$	−15.6415	−0.685267	−0.342634	+	0.939469i	$0.611319\pi$
$522$	0	0
$523$	27.1564	1.18747	0.593734	−	0.804661i	$-0.297653\pi$
$523$	27.1564	1.18747	0.593734	+	0.804661i	$0.297653\pi$
$524$	5.99741	0.261998
$525$	0	0
$526$	10.4352	0.454995
$527$	12.4203	0.541038
$528$	0	0
$529$	−22.1226	−0.961852
$530$	0	0
$531$	0	0
$532$	11.5782	0.501980
$533$	−6.93411	−0.300350
$534$	0	0
$535$	0	0
$536$	24.2024	1.04538
$537$	0	0
$538$	77.2482	3.33041
$539$	0.346755	0.0149358
$540$	0	0
$541$	21.6566	0.931090	0.465545	−	0.885024i	$-0.345858\pi$
$541$	21.6566	0.931090	0.465545	+	0.885024i	$0.345858\pi$
$542$	−21.4109	−0.919679
$543$	0	0
$544$	−17.5457	−0.752267
$545$	0	0
$546$	0	0
$547$	−4.35543	−0.186225	−0.0931123	−	0.995656i	$-0.529682\pi$
$547$	−4.35543	−0.186225	−0.0931123	+	0.995656i	$0.529682\pi$
$548$	1.87341	0.0800279
$549$	0	0
$550$	0	0
$551$	−1.01892	−0.0434074
$552$	0	0
$553$	30.3129	1.28904
$554$	32.7979	1.39345
$555$	0	0
$556$	2.77287	0.117596
$557$	42.4268	1.79768	0.898841	−	0.438275i	$-0.144410\pi$
$557$	42.4268	1.79768	0.898841	+	0.438275i	$0.144410\pi$
$558$	0	0
$559$	−22.5193	−0.952463
$560$	0	0
$561$	0	0
$562$	−58.0874	−2.45027
$563$	25.2891	1.06581	0.532904	−	0.846176i	$-0.321101\pi$
$563$	25.2891	1.06581	0.532904	+	0.846176i	$0.321101\pi$
$564$	0	0
$565$	0	0
$566$	60.6318	2.54854
$567$	0	0
$568$	42.6566	1.78983
$569$	12.2415	0.513189	0.256594	−	0.966519i	$-0.417400\pi$
$569$	12.2415	0.513189	0.256594	+	0.966519i	$0.417400\pi$
$570$	0	0
$571$	28.5769	1.19590	0.597952	−	0.801532i	$-0.295981\pi$
$571$	28.5769	1.19590	0.597952	+	0.801532i	$0.295981\pi$
$572$	64.3786	2.69180
$573$	0	0
$574$	−13.0134	−0.543169
$575$	0	0
$576$	0	0
$577$	15.3055	0.637177	0.318588	−	0.947893i	$-0.396791\pi$
$577$	15.3055	0.637177	0.318588	+	0.947893i	$0.396791\pi$
$578$	−4.30711	−0.179152
$579$	0	0
$580$	0	0
$581$	37.4628	1.55422
$582$	0	0
$583$	45.9427	1.90275
$584$	−43.5797	−1.80334
$585$	0	0
$586$	−43.0342	−1.77773
$587$	−36.0632	−1.48849	−0.744244	−	0.667908i	$-0.767190\pi$
$587$	−36.0632	−1.48849	−0.744244	+	0.667908i	$0.767190\pi$
$588$	0	0
$589$	3.17554	0.130846
$590$	0	0
$591$	0	0
$592$	−49.6375	−2.04009
$593$	−17.2199	−0.707138	−0.353569	−	0.935408i	$-0.615032\pi$
$593$	−17.2199	−0.707138	−0.353569	+	0.935408i	$0.615032\pi$
$594$	0	0
$595$	0	0
$596$	3.18730	0.130557
$597$	0	0
$598$	8.40404	0.343667
$599$	18.3694	0.750554	0.375277	−	0.926913i	$-0.377548\pi$
$599$	18.3694	0.750554	0.375277	+	0.926913i	$0.377548\pi$
$600$	0	0
$601$	−2.23315	−0.0910920	−0.0455460	−	0.998962i	$-0.514503\pi$
$601$	−2.23315	−0.0910920	−0.0455460	+	0.998962i	$0.514503\pi$
$602$	−42.2624	−1.72249
$603$	0	0
$604$	9.54573	0.388410
$605$	0	0
$606$	0	0
$607$	0.331991	0.0134751	0.00673755	−	0.999977i	$-0.497855\pi$
$607$	0.331991	0.0134751	0.00673755	+	0.999977i	$0.497855\pi$
$608$	−4.48597	−0.181930
$609$	0	0
$610$	0	0
$611$	−31.8121	−1.28698
$612$	0	0
$613$	−37.3970	−1.51045	−0.755225	−	0.655466i	$-0.772472\pi$
$613$	−37.3970	−1.51045	−0.755225	+	0.655466i	$0.772472\pi$
$614$	−67.9431	−2.74196
$615$	0	0
$616$	65.9356	2.65662
$617$	16.9589	0.682738	0.341369	−	0.939929i	$-0.389109\pi$
$617$	16.9589	0.682738	0.341369	+	0.939929i	$0.389109\pi$
$618$	0	0
$619$	18.8127	0.756148	0.378074	−	0.925775i	$-0.376587\pi$
$619$	18.8127	0.756148	0.378074	+	0.925775i	$0.376587\pi$
$620$	0	0
$621$	0	0
$622$	46.0650	1.84704
$623$	−34.6561	−1.38847
$624$	0	0
$625$	0	0
$626$	−2.08616	−0.0833798
$627$	0	0
$628$	88.6583	3.53785
$629$	29.5132	1.17677
$630$	0	0
$631$	5.27439	0.209970	0.104985	−	0.994474i	$-0.466521\pi$
$631$	5.27439	0.209970	0.104985	+	0.994474i	$0.466521\pi$
$632$	−70.0776	−2.78754
$633$	0	0
$634$	−27.7729	−1.10300
$635$	0	0
$636$	0	0
$637$	0.298133	0.0118124
$638$	−10.6326	−0.420948
$639$	0	0
$640$	0	0
$641$	8.50908	0.336089	0.168044	−	0.985779i	$-0.446255\pi$
$641$	8.50908	0.336089	0.168044	+	0.985779i	$0.446255\pi$
$642$	0	0
$643$	−27.8512	−1.09835	−0.549173	−	0.835709i	$-0.685057\pi$
$643$	−27.8512	−1.09835	−0.549173	+	0.835709i	$0.685057\pi$
$644$	10.8454	0.427367
$645$	0	0
$646$	9.89682	0.389385
$647$	−2.54903	−0.100213	−0.0501063	−	0.998744i	$-0.515956\pi$
$647$	−2.54903	−0.100213	−0.0501063	+	0.998744i	$0.515956\pi$
$648$	0	0
$649$	28.9353	1.13581
$650$	0	0
$651$	0	0
$652$	−46.7997	−1.83282
$653$	−33.7159	−1.31941	−0.659704	−	0.751526i	$-0.729318\pi$
$653$	−33.7159	−1.31941	−0.659704	+	0.751526i	$0.729318\pi$
$654$	0	0
$655$	0	0
$656$	12.8645	0.502275
$657$	0	0
$658$	−59.7025	−2.32745
$659$	−10.1214	−0.394274	−0.197137	−	0.980376i	$-0.563164\pi$
$659$	−10.1214	−0.394274	−0.197137	+	0.980376i	$0.563164\pi$
$660$	0	0
$661$	−10.3129	−0.401125	−0.200563	−	0.979681i	$-0.564277\pi$
$661$	−10.3129	−0.401125	−0.200563	+	0.979681i	$0.564277\pi$
$662$	−26.8675	−1.04424
$663$	0	0
$664$	−86.6067	−3.36099
$665$	0	0
$666$	0	0
$667$	−0.954424	−0.0369554
$668$	92.6719	3.58558
$669$	0	0
$670$	0	0
$671$	−52.0066	−2.00769
$672$	0	0
$673$	−30.3779	−1.17098	−0.585491	−	0.810679i	$-0.699098\pi$
$673$	−30.3779	−1.17098	−0.585491	+	0.810679i	$0.699098\pi$
$674$	−36.7388	−1.41513
$675$	0	0
$676$	−1.88343	−0.0724395
$677$	28.9198	1.11148	0.555739	−	0.831357i	$-0.312435\pi$
$677$	28.9198	1.11148	0.555739	+	0.831357i	$0.312435\pi$
$678$	0	0
$679$	9.32461	0.357846
$680$	0	0
$681$	0	0
$682$	33.1373	1.26889
$683$	−31.3348	−1.19899	−0.599497	−	0.800377i	$-0.704633\pi$
$683$	−31.3348	−1.19899	−0.599497	+	0.800377i	$0.704633\pi$
$684$	0	0
$685$	0	0
$686$	47.1400	1.79981
$687$	0	0
$688$	41.7789	1.59280
$689$	39.5006	1.50485
$690$	0	0
$691$	38.3511	1.45894	0.729472	−	0.684011i	$-0.239766\pi$
$691$	38.3511	1.45894	0.729472	+	0.684011i	$0.239766\pi$
$692$	−76.3886	−2.90386
$693$	0	0
$694$	−48.3129	−1.83393
$695$	0	0
$696$	0	0
$697$	−7.64891	−0.289723
$698$	−72.8207	−2.75630
$699$	0	0
$700$	0	0
$701$	7.39697	0.279380	0.139690	−	0.990195i	$-0.455389\pi$
$701$	7.39697	0.279380	0.139690	+	0.990195i	$0.455389\pi$
$702$	0	0
$703$	7.54573	0.284593
$704$	7.44530	0.280605
$705$	0	0
$706$	−46.8971	−1.76500
$707$	−7.26297	−0.273152
$708$	0	0
$709$	−24.4278	−0.917405	−0.458702	−	0.888590i	$-0.651686\pi$
$709$	−24.4278	−0.917405	−0.458702	+	0.888590i	$0.651686\pi$
$710$	0	0
$711$	0	0
$712$	80.1183	3.00256
$713$	2.97454	0.111397
$714$	0	0
$715$	0	0
$716$	−84.4239	−3.15507
$717$	0	0
$718$	48.3129	1.80302
$719$	−24.6652	−0.919858	−0.459929	−	0.887956i	$-0.652125\pi$
$719$	−24.6652	−0.919858	−0.459929	+	0.887956i	$0.652125\pi$
$720$	0	0
$721$	−20.9883	−0.781644
$722$	2.53035	0.0941699
$723$	0	0
$724$	−59.9236	−2.22704
$725$	0	0
$726$	0	0
$727$	−16.6064	−0.615896	−0.307948	−	0.951403i	$-0.599642\pi$
$727$	−16.6064	−0.615896	−0.307948	+	0.951403i	$0.599642\pi$
$728$	56.6901	2.10108
$729$	0	0
$730$	0	0
$731$	−24.8407	−0.918765
$732$	0	0
$733$	−38.4044	−1.41850	−0.709249	−	0.704958i	$-0.750966\pi$
$733$	−38.4044	−1.41850	−0.709249	+	0.704958i	$0.750966\pi$
$734$	21.5941	0.797051
$735$	0	0
$736$	−4.20202	−0.154889
$737$	−16.4172	−0.604737
$738$	0	0
$739$	34.7216	1.27725	0.638627	−	0.769516i	$-0.279503\pi$
$739$	34.7216	1.27725	0.638627	+	0.769516i	$0.279503\pi$
$740$	0	0
$741$	0	0
$742$	74.1316	2.72146
$743$	26.7370	0.980884	0.490442	−	0.871474i	$-0.336835\pi$
$743$	26.7370	0.980884	0.490442	+	0.871474i	$0.336835\pi$
$744$	0	0
$745$	0	0
$746$	−68.9764	−2.52541
$747$	0	0
$748$	71.0151	2.59657
$749$	−29.8565	−1.09093
$750$	0	0
$751$	37.8821	1.38234	0.691168	−	0.722694i	$-0.257096\pi$
$751$	37.8821	1.38234	0.691168	+	0.722694i	$0.257096\pi$
$752$	59.0195	2.15222
$753$	0	0
$754$	−9.14169	−0.332921
$755$	0	0
$756$	0	0
$757$	−36.0533	−1.31038	−0.655189	−	0.755465i	$-0.727411\pi$
$757$	−36.0533	−1.31038	−0.655189	+	0.755465i	$0.727411\pi$
$758$	33.7159	1.22462
$759$	0	0
$760$	0	0
$761$	21.6534	0.784933	0.392467	−	0.919766i	$-0.371622\pi$
$761$	21.6534	0.784933	0.392467	+	0.919766i	$0.371622\pi$
$762$	0	0
$763$	−33.6757	−1.21914
$764$	40.4373	1.46297
$765$	0	0
$766$	−59.4945	−2.14962
$767$	24.8780	0.898293
$768$	0	0
$769$	22.2596	0.802703	0.401351	−	0.915924i	$-0.368541\pi$
$769$	22.2596	0.802703	0.401351	+	0.915924i	$0.368541\pi$
$770$	0	0
$771$	0	0
$772$	−107.210	−3.85857
$773$	32.4843	1.16838	0.584189	−	0.811618i	$-0.301413\pi$
$773$	32.4843	1.16838	0.584189	+	0.811618i	$0.301413\pi$
$774$	0	0
$775$	0	0
$776$	−21.5567	−0.773841
$777$	0	0
$778$	−74.3397	−2.66521
$779$	−1.95562	−0.0700674
$780$	0	0
$781$	−28.9353	−1.03539
$782$	9.27038	0.331508
$783$	0	0
$784$	−0.553111	−0.0197540
$785$	0	0
$786$	0	0
$787$	4.70652	0.167769	0.0838846	−	0.996475i	$-0.473267\pi$
$787$	4.70652	0.167769	0.0838846	+	0.996475i	$0.473267\pi$
$788$	5.06070	0.180280
$789$	0	0
$790$	0	0
$791$	3.58238	0.127375
$792$	0	0
$793$	−44.7142	−1.58785
$794$	−54.8955	−1.94817
$795$	0	0
$796$	37.9943	1.34667
$797$	−13.8538	−0.490726	−0.245363	−	0.969431i	$-0.578907\pi$
$797$	−13.8538	−0.490726	−0.245363	+	0.969431i	$0.578907\pi$
$798$	0	0
$799$	−35.0915	−1.24145
$800$	0	0
$801$	0	0
$802$	83.9427	2.96412
$803$	29.5615	1.04320
$804$	0	0
$805$	0	0
$806$	28.4908	1.00355
$807$	0	0
$808$	16.7906	0.590691
$809$	−49.3609	−1.73544	−0.867719	−	0.497055i	$-0.834415\pi$
$809$	−49.3609	−1.73544	−0.867719	+	0.497055i	$0.834415\pi$
$810$	0	0
$811$	16.8747	0.592550	0.296275	−	0.955103i	$-0.404255\pi$
$811$	16.8747	0.592550	0.296275	+	0.955103i	$0.404255\pi$
$812$	−11.7973	−0.414003
$813$	0	0
$814$	78.7410	2.75987
$815$	0	0
$816$	0	0
$817$	−6.35109	−0.222196
$818$	−0.686597	−0.0240063
$819$	0	0
$820$	0	0
$821$	−52.6449	−1.83732	−0.918659	−	0.395051i	$-0.870727\pi$
$821$	−52.6449	−1.83732	−0.918659	+	0.395051i	$0.870727\pi$
$822$	0	0
$823$	46.7598	1.62994	0.814972	−	0.579501i	$-0.196752\pi$
$823$	46.7598	1.62994	0.814972	+	0.579501i	$0.196752\pi$
$824$	48.5209	1.69030
$825$	0	0
$826$	46.6891	1.62452
$827$	−20.1462	−0.700551	−0.350275	−	0.936647i	$-0.613912\pi$
$827$	−20.1462	−0.700551	−0.350275	+	0.936647i	$0.613912\pi$
$828$	0	0
$829$	−9.49278	−0.329698	−0.164849	−	0.986319i	$-0.552714\pi$
$829$	−9.49278	−0.329698	−0.164849	+	0.986319i	$0.552714\pi$
$830$	0	0
$831$	0	0
$832$	6.40131	0.221926
$833$	0.328866	0.0113945
$834$	0	0
$835$	0	0
$836$	18.1566	0.627961
$837$	0	0
$838$	69.8439	2.41272
$839$	−19.2722	−0.665351	−0.332675	−	0.943041i	$-0.607951\pi$
$839$	−19.2722	−0.665351	−0.332675	+	0.943041i	$0.607951\pi$
$840$	0	0
$841$	−27.9618	−0.964200
$842$	−68.7154	−2.36809
$843$	0	0
$844$	59.5534	2.04991
$845$	0	0
$846$	0	0
$847$	−15.7983	−0.542836
$848$	−73.2835	−2.51657
$849$	0	0
$850$	0	0
$851$	7.06811	0.242292
$852$	0	0
$853$	−1.61810	−0.0554025	−0.0277013	−	0.999616i	$-0.508819\pi$
$853$	−1.61810	−0.0554025	−0.0277013	+	0.999616i	$0.508819\pi$
$854$	−83.9161	−2.87155
$855$	0	0
$856$	69.0225	2.35914
$857$	20.5861	0.703208	0.351604	−	0.936149i	$-0.385636\pi$
$857$	20.5861	0.703208	0.351604	+	0.936149i	$0.385636\pi$
$858$	0	0
$859$	11.6831	0.398622	0.199311	−	0.979936i	$-0.436130\pi$
$859$	11.6831	0.398622	0.199311	+	0.979936i	$0.436130\pi$
$860$	0	0
$861$	0	0
$862$	42.7039	1.45450
$863$	10.5021	0.357494	0.178747	−	0.983895i	$-0.442796\pi$
$863$	10.5021	0.357494	0.178747	+	0.983895i	$0.442796\pi$
$864$	0	0
$865$	0	0
$866$	−8.08357	−0.274691
$867$	0	0
$868$	36.7672	1.24796
$869$	47.5358	1.61254
$870$	0	0
$871$	−14.1152	−0.478276
$872$	77.8517	2.63639
$873$	0	0
$874$	2.37019	0.0801728
$875$	0	0
$876$	0	0
$877$	21.4693	0.724968	0.362484	−	0.931990i	$-0.381929\pi$
$877$	21.4693	0.724968	0.362484	+	0.931990i	$0.381929\pi$
$878$	−54.8955	−1.85263
$879$	0	0
$880$	0	0
$881$	−44.0018	−1.48246	−0.741229	−	0.671252i	$-0.765757\pi$
$881$	−44.0018	−1.48246	−0.741229	+	0.671252i	$0.765757\pi$
$882$	0	0
$883$	−27.1109	−0.912353	−0.456177	−	0.889889i	$-0.650782\pi$
$883$	−27.1109	−0.912353	−0.456177	+	0.889889i	$0.650782\pi$
$884$	61.0573	2.05358
$885$	0	0
$886$	27.5648	0.926059
$887$	−13.8682	−0.465649	−0.232825	−	0.972519i	$-0.574797\pi$
$887$	−13.8682	−0.465649	−0.232825	+	0.972519i	$0.574797\pi$
$888$	0	0
$889$	52.2747	1.75324
$890$	0	0
$891$	0	0
$892$	−26.0459	−0.872081
$893$	−8.97195	−0.300235
$894$	0	0
$895$	0	0
$896$	35.6080	1.18958
$897$	0	0
$898$	101.357	3.38234
$899$	−3.23562	−0.107914
$900$	0	0
$901$	43.5725	1.45161
$902$	−20.4072	−0.679487
$903$	0	0
$904$	−8.28177	−0.275447
$905$	0	0
$906$	0	0
$907$	−0.908538	−0.0301675	−0.0150838	−	0.999886i	$-0.504801\pi$
$907$	−0.908538	−0.0301675	−0.0150838	+	0.999886i	$0.504801\pi$
$908$	−58.9559	−1.95652
$909$	0	0
$910$	0	0
$911$	−1.91827	−0.0635552	−0.0317776	−	0.999495i	$-0.510117\pi$
$911$	−1.91827	−0.0635552	−0.0317776	+	0.999495i	$0.510117\pi$
$912$	0	0
$913$	58.7481	1.94428
$914$	−39.0938	−1.29311
$915$	0	0
$916$	−122.737	−4.05533
$917$	−3.58238	−0.118301
$918$	0	0
$919$	−26.0724	−0.860048	−0.430024	−	0.902817i	$-0.641495\pi$
$919$	−26.0724	−0.860048	−0.430024	+	0.902817i	$0.641495\pi$
$920$	0	0
$921$	0	0
$922$	17.3012	0.569784
$923$	−24.8780	−0.818869
$924$	0	0
$925$	0	0
$926$	−73.9778	−2.43106
$927$	0	0
$928$	4.57084	0.150045
$929$	−20.2428	−0.664145	−0.332073	−	0.943254i	$-0.607748\pi$
$929$	−20.2428	−0.664145	−0.332073	+	0.943254i	$0.607748\pi$
$930$	0	0
$931$	0.0840822	0.00275568
$932$	101.282	3.31760
$933$	0	0
$934$	60.9960	1.99585
$935$	0	0
$936$	0	0
$937$	20.0074	0.653613	0.326806	−	0.945091i	$-0.394028\pi$
$937$	20.0074	0.653613	0.326806	+	0.945091i	$0.394028\pi$
$938$	−26.4903	−0.864940
$939$	0	0
$940$	0	0
$941$	23.4784	0.765375	0.382688	−	0.923878i	$-0.374999\pi$
$941$	23.4784	0.765375	0.382688	+	0.923878i	$0.374999\pi$
$942$	0	0
$943$	−1.83184	−0.0596528
$944$	−46.1549	−1.50222
$945$	0	0
$946$	−66.2747	−2.15478
$947$	4.17233	0.135582	0.0677912	−	0.997700i	$-0.478405\pi$
$947$	4.17233	0.135582	0.0677912	+	0.997700i	$0.478405\pi$
$948$	0	0
$949$	25.4164	0.825051
$950$	0	0
$951$	0	0
$952$	62.5340	2.02674
$953$	36.5709	1.18465	0.592324	−	0.805700i	$-0.298210\pi$
$953$	36.5709	1.18465	0.592324	+	0.805700i	$0.298210\pi$
$954$	0	0
$955$	0	0
$956$	−119.226	−3.85604
$957$	0	0
$958$	86.6067	2.79814
$959$	−1.11902	−0.0361352
$960$	0	0
$961$	−20.9159	−0.674707
$962$	67.6999	2.18273
$963$	0	0
$964$	70.2747	2.26340
$965$	0	0
$966$	0	0
$967$	34.3853	1.10576	0.552878	−	0.833262i	$-0.313530\pi$
$967$	34.3853	1.10576	0.552878	+	0.833262i	$0.313530\pi$
$968$	36.5226	1.17388
$969$	0	0
$970$	0	0
$971$	44.4417	1.42620	0.713102	−	0.701061i	$-0.247290\pi$
$971$	44.4417	1.42620	0.713102	+	0.701061i	$0.247290\pi$
$972$	0	0
$973$	−1.65629	−0.0530983
$974$	−101.378	−3.24838
$975$	0	0
$976$	82.9561	2.65536
$977$	44.3969	1.42038	0.710191	−	0.704009i	$-0.248609\pi$
$977$	44.3969	1.42038	0.710191	+	0.704009i	$0.248609\pi$
$978$	0	0
$979$	−54.3468	−1.73693
$980$	0	0
$981$	0	0
$982$	87.8057	2.80199
$983$	7.72584	0.246416	0.123208	−	0.992381i	$-0.460682\pi$
$983$	7.72584	0.246416	0.123208	+	0.992381i	$0.460682\pi$
$984$	0	0
$985$	0	0
$986$	−10.0841	−0.321142
$987$	0	0
$988$	15.6107	0.496643
$989$	−5.94908	−0.189170
$990$	0	0
$991$	−41.3705	−1.31418	−0.657089	−	0.753813i	$-0.728212\pi$
$991$	−41.3705	−1.31418	−0.657089	+	0.753813i	$0.728212\pi$
$992$	−14.2454	−0.452292
$993$	0	0
$994$	−46.6891	−1.48089
$995$	0	0
$996$	0	0
$997$	−39.2673	−1.24361	−0.621804	−	0.783173i	$-0.713600\pi$
$997$	−39.2673	−1.24361	−0.621804	+	0.783173i	$0.713600\pi$
$998$	−98.5014	−3.11801
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bs.1.6	yes	6
3.2	odd	2	inner	4275.2.a.bs.1.1	✓	6
5.4	even	2		4275.2.a.bt.1.1	yes	6
15.14	odd	2		4275.2.a.bt.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4275.2.a.bs.1.1	✓	6	3.2	odd	2	inner
4275.2.a.bs.1.6	yes	6	1.1	even	1	trivial
4275.2.a.bt.1.1	yes	6	5.4	even	2
4275.2.a.bt.1.6	yes	6	15.14	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q+2.53035 q^{2} +4.40268 q^{4} -2.62981 q^{7} +6.07962 q^{8} -4.12400 q^{11} -3.54573 q^{13} -6.65435 q^{14} +6.57822 q^{16} -3.91124 q^{17} -1.00000 q^{19} -10.4352 q^{22} -0.936703 q^{23} -8.97195 q^{26} -11.5782 q^{28} +1.01892 q^{29} -3.17554 q^{31} +4.48597 q^{32} -9.89682 q^{34} -7.54573 q^{37} -2.53035 q^{38} +1.95562 q^{41} +6.35109 q^{43} -18.1566 q^{44} -2.37019 q^{46} +8.97195 q^{47} -0.0840822 q^{49} -15.6107 q^{52} -11.1403 q^{53} -15.9883 q^{56} +2.57822 q^{58} -7.01632 q^{59} +12.6107 q^{61} -8.03524 q^{62} -1.80536 q^{64} +3.98090 q^{67} -17.2199 q^{68} +7.01632 q^{71} -7.16816 q^{73} -19.0934 q^{74} -4.40268 q^{76} +10.8454 q^{77} -11.5266 q^{79} +4.94841 q^{82} -14.2454 q^{83} +16.0705 q^{86} -25.0724 q^{88} +13.1782 q^{89} +9.32461 q^{91} -4.12400 q^{92} +22.7022 q^{94} -3.54573 q^{97} -0.212757 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{4} - 16 q^{13} - 6 q^{19} - 10 q^{22} - 30 q^{28} + 2 q^{31} - 12 q^{34} - 40 q^{37} - 4 q^{43} - 30 q^{46} + 10 q^{49} - 20 q^{52} - 24 q^{58} + 2 q^{61} + 26 q^{64} - 34 q^{67} - 22 q^{73}+ \cdots - 16 q^{97}+O(q^{100}) \) 6 * q + 8 * q^4 - 16 * q^13 - 6 * q^19 - 10 * q^22 - 30 * q^28 + 2 * q^31 - 12 * q^34 - 40 * q^37 - 4 * q^43 - 30 * q^46 + 10 * q^49 - 20 * q^52 - 24 * q^58 + 2 * q^61 + 26 * q^64 - 34 * q^67 - 22 * q^73 - 8 * q^76 - 6 * q^79 + 6 * q^82 - 82 * q^88 - 44 * q^91 + 52 * q^94 - 16 * q^97

Introduction

L-functions

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