LMFDB - Embedded newform 4275.2.a.r.1.2

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:	$2$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1425)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+0.618034 q^{2} -1.61803 q^{4} +0.236068 q^{7} -2.23607 q^{8} +0.763932 q^{11} -3.23607 q^{13} +0.145898 q^{14} +1.85410 q^{16} +6.47214 q^{17} +1.00000 q^{19} +0.472136 q^{22} -8.47214 q^{23} -2.00000 q^{26} -0.381966 q^{28} +9.47214 q^{29} -8.00000 q^{31} +5.61803 q^{32} +4.00000 q^{34} -4.76393 q^{37} +0.618034 q^{38} -1.47214 q^{41} +12.9443 q^{43} -1.23607 q^{44} -5.23607 q^{46} -5.23607 q^{47} -6.94427 q^{49} +5.23607 q^{52} +1.00000 q^{53} -0.527864 q^{56} +5.85410 q^{58} +6.70820 q^{59} -7.47214 q^{61} -4.94427 q^{62} -0.236068 q^{64} -3.70820 q^{67} -10.4721 q^{68} +1.29180 q^{71} -14.4164 q^{73} -2.94427 q^{74} -1.61803 q^{76} +0.180340 q^{77} -4.47214 q^{79} -0.909830 q^{82} +7.70820 q^{83} +8.00000 q^{86} -1.70820 q^{88} +5.00000 q^{89} -0.763932 q^{91} +13.7082 q^{92} -3.23607 q^{94} -3.70820 q^{97} -4.29180 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} - 4 q^{7} + 6 q^{11} - 2 q^{13} + 7 q^{14} - 3 q^{16} + 4 q^{17} + 2 q^{19} - 8 q^{22} - 8 q^{23} - 4 q^{26} - 3 q^{28} + 10 q^{29} - 16 q^{31} + 9 q^{32} + 8 q^{34} - 14 q^{37} - q^{38}+ \cdots - 22 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 - 4 * q^7 + 6 * q^11 - 2 * q^13 + 7 * q^14 - 3 * q^16 + 4 * q^17 + 2 * q^19 - 8 * q^22 - 8 * q^23 - 4 * q^26 - 3 * q^28 + 10 * q^29 - 16 * q^31 + 9 * q^32 + 8 * q^34 - 14 * q^37 - q^38 + 6 * q^41 + 8 * q^43 + 2 * q^44 - 6 * q^46 - 6 * q^47 + 4 * q^49 + 6 * q^52 + 2 * q^53 - 10 * q^56 + 5 * q^58 - 6 * q^61 + 8 * q^62 + 4 * q^64 + 6 * q^67 - 12 * q^68 + 16 * q^71 - 2 * q^73 + 12 * q^74 - q^76 - 22 * q^77 - 13 * q^82 + 2 * q^83 + 16 * q^86 + 10 * q^88 + 10 * q^89 - 6 * q^91 + 14 * q^92 - 2 * q^94 + 6 * q^97 - 22 * 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 $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	−2.23607	−0.790569
$9$	0	0
$10$	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
$13$	−3.23607	−0.897524	−0.448762	−	0.893651i	$-0.648135\pi$
$13$	−3.23607	−0.897524	−0.448762	+	0.893651i	$0.648135\pi$
$14$	0.145898	0.0389929
$15$	0	0
$16$	1.85410	0.463525
$17$	6.47214	1.56972	0.784862	−	0.619671i	$-0.212734\pi$
$17$	6.47214	1.56972	0.784862	+	0.619671i	$0.212734\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0.472136	0.100660
$23$	−8.47214	−1.76656	−0.883281	−	0.468844i	$-0.844671\pi$
$23$	−8.47214	−1.76656	−0.883281	+	0.468844i	$0.844671\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	0	0
$28$	−0.381966	−0.0721848
$29$	9.47214	1.75893	0.879466	−	0.475962i	$-0.157900\pi$
$29$	9.47214	1.75893	0.879466	+	0.475962i	$0.157900\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	5.61803	0.993137
$33$	0	0
$34$	4.00000	0.685994
$35$	0	0
$36$	0	0
$37$	−4.76393	−0.783186	−0.391593	−	0.920139i	$-0.628076\pi$
$37$	−4.76393	−0.783186	−0.391593	+	0.920139i	$0.628076\pi$
$38$	0.618034	0.100258
$39$	0	0
$40$	0	0
$41$	−1.47214	−0.229909	−0.114955	−	0.993371i	$-0.536672\pi$
$41$	−1.47214	−0.229909	−0.114955	+	0.993371i	$0.536672\pi$
$42$	0	0
$43$	12.9443	1.97398	0.986991	−	0.160773i	$-0.0513986\pi$
$43$	12.9443	1.97398	0.986991	+	0.160773i	$0.0513986\pi$
$44$	−1.23607	−0.186344
$45$	0	0
$46$	−5.23607	−0.772016
$47$	−5.23607	−0.763759	−0.381880	−	0.924212i	$-0.624723\pi$
$47$	−5.23607	−0.763759	−0.381880	+	0.924212i	$0.624723\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	0	0
$51$	0	0
$52$	5.23607	0.726112
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	0	0
$55$	0	0
$56$	−0.527864	−0.0705388
$57$	0	0
$58$	5.85410	0.768681
$59$	6.70820	0.873334	0.436667	−	0.899623i	$-0.356159\pi$
$59$	6.70820	0.873334	0.436667	+	0.899623i	$0.356159\pi$
$60$	0	0
$61$	−7.47214	−0.956709	−0.478354	−	0.878167i	$-0.658767\pi$
$61$	−7.47214	−0.956709	−0.478354	+	0.878167i	$0.658767\pi$
$62$	−4.94427	−0.627923
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	0	0
$67$	−3.70820	−0.453029	−0.226515	−	0.974008i	$-0.572733\pi$
$67$	−3.70820	−0.453029	−0.226515	+	0.974008i	$0.572733\pi$
$68$	−10.4721	−1.26993
$69$	0	0
$70$	0	0
$71$	1.29180	0.153308	0.0766540	−	0.997058i	$-0.475576\pi$
$71$	1.29180	0.153308	0.0766540	+	0.997058i	$0.475576\pi$
$72$	0	0
$73$	−14.4164	−1.68731	−0.843656	−	0.536883i	$-0.819602\pi$
$73$	−14.4164	−1.68731	−0.843656	+	0.536883i	$0.819602\pi$
$74$	−2.94427	−0.342265
$75$	0	0
$76$	−1.61803	−0.185601
$77$	0.180340	0.0205516
$78$	0	0
$79$	−4.47214	−0.503155	−0.251577	−	0.967837i	$-0.580949\pi$
$79$	−4.47214	−0.503155	−0.251577	+	0.967837i	$0.580949\pi$
$80$	0	0
$81$	0	0
$82$	−0.909830	−0.100474
$83$	7.70820	0.846085	0.423043	−	0.906110i	$-0.360962\pi$
$83$	7.70820	0.846085	0.423043	+	0.906110i	$0.360962\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	0	0
$88$	−1.70820	−0.182095
$89$	5.00000	0.529999	0.264999	−	0.964249i	$-0.414628\pi$
$89$	5.00000	0.529999	0.264999	+	0.964249i	$0.414628\pi$
$90$	0	0
$91$	−0.763932	−0.0800818
$92$	13.7082	1.42918
$93$	0	0
$94$	−3.23607	−0.333775
$95$	0	0
$96$	0	0
$97$	−3.70820	−0.376511	−0.188256	−	0.982120i	$-0.560283\pi$
$97$	−3.70820	−0.376511	−0.188256	+	0.982120i	$0.560283\pi$
$98$	−4.29180	−0.433537
$99$	0	0
$100$	0	0
$101$	4.18034	0.415959	0.207980	−	0.978133i	$-0.433311\pi$
$101$	4.18034	0.415959	0.207980	+	0.978133i	$0.433311\pi$
$102$	0	0
$103$	−12.1803	−1.20016	−0.600082	−	0.799938i	$-0.704866\pi$
$103$	−12.1803	−1.20016	−0.600082	+	0.799938i	$0.704866\pi$
$104$	7.23607	0.709555
$105$	0	0
$106$	0.618034	0.0600288
$107$	−4.70820	−0.455159	−0.227580	−	0.973759i	$-0.573081\pi$
$107$	−4.70820	−0.455159	−0.227580	+	0.973759i	$0.573081\pi$
$108$	0	0
$109$	−12.7639	−1.22256	−0.611281	−	0.791413i	$-0.709346\pi$
$109$	−12.7639	−1.22256	−0.611281	+	0.791413i	$0.709346\pi$
$110$	0	0
$111$	0	0
$112$	0.437694	0.0413582
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	−15.3262	−1.42301
$117$	0	0
$118$	4.14590	0.381661
$119$	1.52786	0.140059
$120$	0	0
$121$	−10.4164	−0.946946
$122$	−4.61803	−0.418097
$123$	0	0
$124$	12.9443	1.16243
$125$	0	0
$126$	0	0
$127$	11.4164	1.01304	0.506521	−	0.862228i	$-0.330931\pi$
$127$	11.4164	1.01304	0.506521	+	0.862228i	$0.330931\pi$
$128$	−11.3820	−1.00603
$129$	0	0
$130$	0	0
$131$	−5.41641	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$131$	−5.41641	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$132$	0	0
$133$	0.236068	0.0204697
$134$	−2.29180	−0.197981
$135$	0	0
$136$	−14.4721	−1.24098
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	12.2361	1.03785	0.518925	−	0.854820i	$-0.326332\pi$
$139$	12.2361	1.03785	0.518925	+	0.854820i	$0.326332\pi$
$140$	0	0
$141$	0	0
$142$	0.798374	0.0669980
$143$	−2.47214	−0.206730
$144$	0	0
$145$	0	0
$146$	−8.90983	−0.737383
$147$	0	0
$148$	7.70820	0.633610
$149$	−16.1803	−1.32555	−0.662773	−	0.748821i	$-0.730620\pi$
$149$	−16.1803	−1.32555	−0.662773	+	0.748821i	$0.730620\pi$
$150$	0	0
$151$	−16.9443	−1.37891	−0.689453	−	0.724331i	$-0.742149\pi$
$151$	−16.9443	−1.37891	−0.689453	+	0.724331i	$0.742149\pi$
$152$	−2.23607	−0.181369
$153$	0	0
$154$	0.111456	0.00898139
$155$	0	0
$156$	0	0
$157$	−2.52786	−0.201746	−0.100873	−	0.994899i	$-0.532163\pi$
$157$	−2.52786	−0.201746	−0.100873	+	0.994899i	$0.532163\pi$
$158$	−2.76393	−0.219887
$159$	0	0
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	−18.2361	−1.42836	−0.714180	−	0.699963i	$-0.753200\pi$
$163$	−18.2361	−1.42836	−0.714180	+	0.699963i	$0.753200\pi$
$164$	2.38197	0.186000
$165$	0	0
$166$	4.76393	0.369753
$167$	−1.29180	−0.0999622	−0.0499811	−	0.998750i	$-0.515916\pi$
$167$	−1.29180	−0.0999622	−0.0499811	+	0.998750i	$0.515916\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	0	0
$171$	0	0
$172$	−20.9443	−1.59699
$173$	−14.5279	−1.10453	−0.552267	−	0.833668i	$-0.686237\pi$
$173$	−14.5279	−1.10453	−0.552267	+	0.833668i	$0.686237\pi$
$174$	0	0
$175$	0	0
$176$	1.41641	0.106766
$177$	0	0
$178$	3.09017	0.231618
$179$	7.76393	0.580304	0.290152	−	0.956981i	$-0.406294\pi$
$179$	7.76393	0.580304	0.290152	+	0.956981i	$0.406294\pi$
$180$	0	0
$181$	6.47214	0.481070	0.240535	−	0.970640i	$-0.422677\pi$
$181$	6.47214	0.481070	0.240535	+	0.970640i	$0.422677\pi$
$182$	−0.472136	−0.0349970
$183$	0	0
$184$	18.9443	1.39659
$185$	0	0
$186$	0	0
$187$	4.94427	0.361561
$188$	8.47214	0.617894
$189$	0	0
$190$	0	0
$191$	25.8885	1.87323	0.936615	−	0.350361i	$-0.113941\pi$
$191$	25.8885	1.87323	0.936615	+	0.350361i	$0.113941\pi$
$192$	0	0
$193$	−11.5279	−0.829794	−0.414897	−	0.909868i	$-0.636182\pi$
$193$	−11.5279	−0.829794	−0.414897	+	0.909868i	$0.636182\pi$
$194$	−2.29180	−0.164541
$195$	0	0
$196$	11.2361	0.802576
$197$	−14.1803	−1.01031	−0.505154	−	0.863029i	$-0.668564\pi$
$197$	−14.1803	−1.01031	−0.505154	+	0.863029i	$0.668564\pi$
$198$	0	0
$199$	−13.2918	−0.942230	−0.471115	−	0.882072i	$-0.656148\pi$
$199$	−13.2918	−0.942230	−0.471115	+	0.882072i	$0.656148\pi$
$200$	0	0
$201$	0	0
$202$	2.58359	0.181781
$203$	2.23607	0.156941
$204$	0	0
$205$	0	0
$206$	−7.52786	−0.524491
$207$	0	0
$208$	−6.00000	−0.416025
$209$	0.763932	0.0528423
$210$	0	0
$211$	24.3607	1.67706	0.838529	−	0.544857i	$-0.183416\pi$
$211$	24.3607	1.67706	0.838529	+	0.544857i	$0.183416\pi$
$212$	−1.61803	−0.111127
$213$	0	0
$214$	−2.90983	−0.198912
$215$	0	0
$216$	0	0
$217$	−1.88854	−0.128203
$218$	−7.88854	−0.534280
$219$	0	0
$220$	0	0
$221$	−20.9443	−1.40886
$222$	0	0
$223$	−10.4721	−0.701266	−0.350633	−	0.936513i	$-0.614034\pi$
$223$	−10.4721	−0.701266	−0.350633	+	0.936513i	$0.614034\pi$
$224$	1.32624	0.0886130
$225$	0	0
$226$	−8.65248	−0.575554
$227$	−1.29180	−0.0857395	−0.0428698	−	0.999081i	$-0.513650\pi$
$227$	−1.29180	−0.0857395	−0.0428698	+	0.999081i	$0.513650\pi$
$228$	0	0
$229$	1.05573	0.0697645	0.0348822	−	0.999391i	$-0.488894\pi$
$229$	1.05573	0.0697645	0.0348822	+	0.999391i	$0.488894\pi$
$230$	0	0
$231$	0	0
$232$	−21.1803	−1.39056
$233$	−10.1803	−0.666936	−0.333468	−	0.942761i	$-0.608219\pi$
$233$	−10.1803	−0.666936	−0.333468	+	0.942761i	$0.608219\pi$
$234$	0	0
$235$	0	0
$236$	−10.8541	−0.706542
$237$	0	0
$238$	0.944272	0.0612081
$239$	−19.5967	−1.26761	−0.633804	−	0.773494i	$-0.718507\pi$
$239$	−19.5967	−1.26761	−0.633804	+	0.773494i	$0.718507\pi$
$240$	0	0
$241$	28.8328	1.85728	0.928642	−	0.370976i	$-0.120977\pi$
$241$	28.8328	1.85728	0.928642	+	0.370976i	$0.120977\pi$
$242$	−6.43769	−0.413831
$243$	0	0
$244$	12.0902	0.773994
$245$	0	0
$246$	0	0
$247$	−3.23607	−0.205906
$248$	17.8885	1.13592
$249$	0	0
$250$	0	0
$251$	2.47214	0.156040	0.0780199	−	0.996952i	$-0.475140\pi$
$251$	2.47214	0.156040	0.0780199	+	0.996952i	$0.475140\pi$
$252$	0	0
$253$	−6.47214	−0.406900
$254$	7.05573	0.442716
$255$	0	0
$256$	−6.56231	−0.410144
$257$	2.52786	0.157684	0.0788419	−	0.996887i	$-0.474878\pi$
$257$	2.52786	0.157684	0.0788419	+	0.996887i	$0.474878\pi$
$258$	0	0
$259$	−1.12461	−0.0698800
$260$	0	0
$261$	0	0
$262$	−3.34752	−0.206811
$263$	−24.6525	−1.52014	−0.760068	−	0.649843i	$-0.774835\pi$
$263$	−24.6525	−1.52014	−0.760068	+	0.649843i	$0.774835\pi$
$264$	0	0
$265$	0	0
$266$	0.145898	0.00894558
$267$	0	0
$268$	6.00000	0.366508
$269$	27.8885	1.70039	0.850197	−	0.526464i	$-0.176483\pi$
$269$	27.8885	1.70039	0.850197	+	0.526464i	$0.176483\pi$
$270$	0	0
$271$	17.6525	1.07231	0.536156	−	0.844119i	$-0.319876\pi$
$271$	17.6525	1.07231	0.536156	+	0.844119i	$0.319876\pi$
$272$	12.0000	0.727607
$273$	0	0
$274$	−4.94427	−0.298694
$275$	0	0
$276$	0	0
$277$	16.4164	0.986366	0.493183	−	0.869925i	$-0.335833\pi$
$277$	16.4164	0.986366	0.493183	+	0.869925i	$0.335833\pi$
$278$	7.56231	0.453557
$279$	0	0
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−2.09017	−0.124029
$285$	0	0
$286$	−1.52786	−0.0903445
$287$	−0.347524	−0.0205137
$288$	0	0
$289$	24.8885	1.46403
$290$	0	0
$291$	0	0
$292$	23.3262	1.36506
$293$	14.9443	0.873054	0.436527	−	0.899691i	$-0.356208\pi$
$293$	14.9443	0.873054	0.436527	+	0.899691i	$0.356208\pi$
$294$	0	0
$295$	0	0
$296$	10.6525	0.619163
$297$	0	0
$298$	−10.0000	−0.579284
$299$	27.4164	1.58553
$300$	0	0
$301$	3.05573	0.176129
$302$	−10.4721	−0.602604
$303$	0	0
$304$	1.85410	0.106340
$305$	0	0
$306$	0	0
$307$	−8.18034	−0.466877	−0.233438	−	0.972372i	$-0.574998\pi$
$307$	−8.18034	−0.466877	−0.233438	+	0.972372i	$0.574998\pi$
$308$	−0.291796	−0.0166266
$309$	0	0
$310$	0	0
$311$	−15.4164	−0.874184	−0.437092	−	0.899417i	$-0.643992\pi$
$311$	−15.4164	−0.874184	−0.437092	+	0.899417i	$0.643992\pi$
$312$	0	0
$313$	−32.8328	−1.85582	−0.927910	−	0.372804i	$-0.878396\pi$
$313$	−32.8328	−1.85582	−0.927910	+	0.372804i	$0.878396\pi$
$314$	−1.56231	−0.0881660
$315$	0	0
$316$	7.23607	0.407061
$317$	−25.3607	−1.42440	−0.712199	−	0.701978i	$-0.752301\pi$
$317$	−25.3607	−1.42440	−0.712199	+	0.701978i	$0.752301\pi$
$318$	0	0
$319$	7.23607	0.405142
$320$	0	0
$321$	0	0
$322$	−1.23607	−0.0688834
$323$	6.47214	0.360119
$324$	0	0
$325$	0	0
$326$	−11.2705	−0.624216
$327$	0	0
$328$	3.29180	0.181759
$329$	−1.23607	−0.0681466
$330$	0	0
$331$	14.3607	0.789334	0.394667	−	0.918824i	$-0.370860\pi$
$331$	14.3607	0.789334	0.394667	+	0.918824i	$0.370860\pi$
$332$	−12.4721	−0.684497
$333$	0	0
$334$	−0.798374	−0.0436851
$335$	0	0
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	−1.56231	−0.0849782
$339$	0	0
$340$	0	0
$341$	−6.11146	−0.330954
$342$	0	0
$343$	−3.29180	−0.177740
$344$	−28.9443	−1.56057
$345$	0	0
$346$	−8.97871	−0.482699
$347$	−19.7082	−1.05799	−0.528996	−	0.848624i	$-0.677431\pi$
$347$	−19.7082	−1.05799	−0.528996	+	0.848624i	$0.677431\pi$
$348$	0	0
$349$	−23.9443	−1.28171	−0.640854	−	0.767663i	$-0.721420\pi$
$349$	−23.9443	−1.28171	−0.640854	+	0.767663i	$0.721420\pi$
$350$	0	0
$351$	0	0
$352$	4.29180	0.228753
$353$	−1.88854	−0.100517	−0.0502585	−	0.998736i	$-0.516005\pi$
$353$	−1.88854	−0.100517	−0.0502585	+	0.998736i	$0.516005\pi$
$354$	0	0
$355$	0	0
$356$	−8.09017	−0.428778
$357$	0	0
$358$	4.79837	0.253602
$359$	−3.81966	−0.201594	−0.100797	−	0.994907i	$-0.532139\pi$
$359$	−3.81966	−0.201594	−0.100797	+	0.994907i	$0.532139\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	4.00000	0.210235
$363$	0	0
$364$	1.23607	0.0647876
$365$	0	0
$366$	0	0
$367$	−3.05573	−0.159508	−0.0797539	−	0.996815i	$-0.525413\pi$
$367$	−3.05573	−0.159508	−0.0797539	+	0.996815i	$0.525413\pi$
$368$	−15.7082	−0.818847
$369$	0	0
$370$	0	0
$371$	0.236068	0.0122560
$372$	0	0
$373$	2.94427	0.152449	0.0762243	−	0.997091i	$-0.475713\pi$
$373$	2.94427	0.152449	0.0762243	+	0.997091i	$0.475713\pi$
$374$	3.05573	0.158008
$375$	0	0
$376$	11.7082	0.603805
$377$	−30.6525	−1.57868
$378$	0	0
$379$	10.6525	0.547181	0.273590	−	0.961846i	$-0.411789\pi$
$379$	10.6525	0.547181	0.273590	+	0.961846i	$0.411789\pi$
$380$	0	0
$381$	0	0
$382$	16.0000	0.818631
$383$	−0.708204	−0.0361875	−0.0180938	−	0.999836i	$-0.505760\pi$
$383$	−0.708204	−0.0361875	−0.0180938	+	0.999836i	$0.505760\pi$
$384$	0	0
$385$	0	0
$386$	−7.12461	−0.362633
$387$	0	0
$388$	6.00000	0.304604
$389$	−18.2918	−0.927431	−0.463715	−	0.885984i	$-0.653484\pi$
$389$	−18.2918	−0.927431	−0.463715	+	0.885984i	$0.653484\pi$
$390$	0	0
$391$	−54.8328	−2.77301
$392$	15.5279	0.784276
$393$	0	0
$394$	−8.76393	−0.441521
$395$	0	0
$396$	0	0
$397$	−7.52786	−0.377813	−0.188906	−	0.981995i	$-0.560494\pi$
$397$	−7.52786	−0.377813	−0.188906	+	0.981995i	$0.560494\pi$
$398$	−8.21478	−0.411770
$399$	0	0
$400$	0	0
$401$	0.111456	0.00556586	0.00278293	−	0.999996i	$-0.499114\pi$
$401$	0.111456	0.00556586	0.00278293	+	0.999996i	$0.499114\pi$
$402$	0	0
$403$	25.8885	1.28960
$404$	−6.76393	−0.336518
$405$	0	0
$406$	1.38197	0.0685858
$407$	−3.63932	−0.180394
$408$	0	0
$409$	−8.29180	−0.410003	−0.205001	−	0.978762i	$-0.565720\pi$
$409$	−8.29180	−0.410003	−0.205001	+	0.978762i	$0.565720\pi$
$410$	0	0
$411$	0	0
$412$	19.7082	0.970954
$413$	1.58359	0.0779235
$414$	0	0
$415$	0	0
$416$	−18.1803	−0.891364
$417$	0	0
$418$	0.472136	0.0230929
$419$	20.6525	1.00894	0.504470	−	0.863429i	$-0.331688\pi$
$419$	20.6525	1.00894	0.504470	+	0.863429i	$0.331688\pi$
$420$	0	0
$421$	18.1803	0.886056	0.443028	−	0.896508i	$-0.353904\pi$
$421$	18.1803	0.886056	0.443028	+	0.896508i	$0.353904\pi$
$422$	15.0557	0.732901
$423$	0	0
$424$	−2.23607	−0.108593
$425$	0	0
$426$	0	0
$427$	−1.76393	−0.0853627
$428$	7.61803	0.368232
$429$	0	0
$430$	0	0
$431$	38.1246	1.83640	0.918199	−	0.396120i	$-0.129643\pi$
$431$	38.1246	1.83640	0.918199	+	0.396120i	$0.129643\pi$
$432$	0	0
$433$	−13.2361	−0.636085	−0.318042	−	0.948076i	$-0.603025\pi$
$433$	−13.2361	−0.636085	−0.318042	+	0.948076i	$0.603025\pi$
$434$	−1.16718	−0.0560266
$435$	0	0
$436$	20.6525	0.989074
$437$	−8.47214	−0.405277
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	0	0
$442$	−12.9443	−0.615696
$443$	−5.05573	−0.240205	−0.120102	−	0.992762i	$-0.538322\pi$
$443$	−5.05573	−0.240205	−0.120102	+	0.992762i	$0.538322\pi$
$444$	0	0
$445$	0	0
$446$	−6.47214	−0.306465
$447$	0	0
$448$	−0.0557281	−0.00263290
$449$	36.3050	1.71334	0.856668	−	0.515868i	$-0.172530\pi$
$449$	36.3050	1.71334	0.856668	+	0.515868i	$0.172530\pi$
$450$	0	0
$451$	−1.12461	−0.0529559
$452$	22.6525	1.06548
$453$	0	0
$454$	−0.798374	−0.0374695
$455$	0	0
$456$	0	0
$457$	8.52786	0.398917	0.199458	−	0.979906i	$-0.436082\pi$
$457$	8.52786	0.398917	0.199458	+	0.979906i	$0.436082\pi$
$458$	0.652476	0.0304882
$459$	0	0
$460$	0	0
$461$	4.18034	0.194698	0.0973489	−	0.995250i	$-0.468964\pi$
$461$	4.18034	0.194698	0.0973489	+	0.995250i	$0.468964\pi$
$462$	0	0
$463$	−1.52786	−0.0710059	−0.0355029	−	0.999370i	$-0.511303\pi$
$463$	−1.52786	−0.0710059	−0.0355029	+	0.999370i	$0.511303\pi$
$464$	17.5623	0.815310
$465$	0	0
$466$	−6.29180	−0.291462
$467$	4.76393	0.220448	0.110224	−	0.993907i	$-0.464843\pi$
$467$	4.76393	0.220448	0.110224	+	0.993907i	$0.464843\pi$
$468$	0	0
$469$	−0.875388	−0.0404217
$470$	0	0
$471$	0	0
$472$	−15.0000	−0.690431
$473$	9.88854	0.454676
$474$	0	0
$475$	0	0
$476$	−2.47214	−0.113310
$477$	0	0
$478$	−12.1115	−0.553965
$479$	−21.7082	−0.991873	−0.495937	−	0.868359i	$-0.665175\pi$
$479$	−21.7082	−0.991873	−0.495937	+	0.868359i	$0.665175\pi$
$480$	0	0
$481$	15.4164	0.702928
$482$	17.8197	0.811663
$483$	0	0
$484$	16.8541	0.766096
$485$	0	0
$486$	0	0
$487$	−26.0689	−1.18129	−0.590647	−	0.806930i	$-0.701127\pi$
$487$	−26.0689	−1.18129	−0.590647	+	0.806930i	$0.701127\pi$
$488$	16.7082	0.756345
$489$	0	0
$490$	0	0
$491$	−20.9443	−0.945202	−0.472601	−	0.881277i	$-0.656685\pi$
$491$	−20.9443	−0.945202	−0.472601	+	0.881277i	$0.656685\pi$
$492$	0	0
$493$	61.3050	2.76104
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	−14.8328	−0.666013
$497$	0.304952	0.0136790
$498$	0	0
$499$	35.6525	1.59602	0.798012	−	0.602642i	$-0.205885\pi$
$499$	35.6525	1.59602	0.798012	+	0.602642i	$0.205885\pi$
$500$	0	0
$501$	0	0
$502$	1.52786	0.0681919
$503$	12.1803	0.543095	0.271547	−	0.962425i	$-0.412465\pi$
$503$	12.1803	0.543095	0.271547	+	0.962425i	$0.412465\pi$
$504$	0	0
$505$	0	0
$506$	−4.00000	−0.177822
$507$	0	0
$508$	−18.4721	−0.819569
$509$	−43.9443	−1.94780	−0.973898	−	0.226987i	$-0.927113\pi$
$509$	−43.9443	−1.94780	−0.973898	+	0.226987i	$0.927113\pi$
$510$	0	0
$511$	−3.40325	−0.150551
$512$	18.7082	0.826794
$513$	0	0
$514$	1.56231	0.0689104
$515$	0	0
$516$	0	0
$517$	−4.00000	−0.175920
$518$	−0.695048	−0.0305387
$519$	0	0
$520$	0	0
$521$	25.3607	1.11107	0.555536	−	0.831493i	$-0.312513\pi$
$521$	25.3607	1.11107	0.555536	+	0.831493i	$0.312513\pi$
$522$	0	0
$523$	26.7639	1.17031	0.585153	−	0.810923i	$-0.301035\pi$
$523$	26.7639	1.17031	0.585153	+	0.810923i	$0.301035\pi$
$524$	8.76393	0.382854
$525$	0	0
$526$	−15.2361	−0.664324
$527$	−51.7771	−2.25545
$528$	0	0
$529$	48.7771	2.12074
$530$	0	0
$531$	0	0
$532$	−0.381966	−0.0165603
$533$	4.76393	0.206349
$534$	0	0
$535$	0	0
$536$	8.29180	0.358151
$537$	0	0
$538$	17.2361	0.743100
$539$	−5.30495	−0.228500
$540$	0	0
$541$	19.8885	0.855075	0.427538	−	0.903998i	$-0.359381\pi$
$541$	19.8885	0.855075	0.427538	+	0.903998i	$0.359381\pi$
$542$	10.9098	0.468617
$543$	0	0
$544$	36.3607	1.55895
$545$	0	0
$546$	0	0
$547$	−4.76393	−0.203691	−0.101846	−	0.994800i	$-0.532475\pi$
$547$	−4.76393	−0.203691	−0.101846	+	0.994800i	$0.532475\pi$
$548$	12.9443	0.552952
$549$	0	0
$550$	0	0
$551$	9.47214	0.403527
$552$	0	0
$553$	−1.05573	−0.0448941
$554$	10.1459	0.431058
$555$	0	0
$556$	−19.7984	−0.839638
$557$	35.4164	1.50064	0.750321	−	0.661074i	$-0.229899\pi$
$557$	35.4164	1.50064	0.750321	+	0.661074i	$0.229899\pi$
$558$	0	0
$559$	−41.8885	−1.77170
$560$	0	0
$561$	0	0
$562$	−13.5967	−0.573544
$563$	−19.6525	−0.828253	−0.414127	−	0.910219i	$-0.635913\pi$
$563$	−19.6525	−0.828253	−0.414127	+	0.910219i	$0.635913\pi$
$564$	0	0
$565$	0	0
$566$	2.47214	0.103912
$567$	0	0
$568$	−2.88854	−0.121201
$569$	41.8328	1.75372	0.876861	−	0.480743i	$-0.159633\pi$
$569$	41.8328	1.75372	0.876861	+	0.480743i	$0.159633\pi$
$570$	0	0
$571$	−20.2361	−0.846853	−0.423427	−	0.905930i	$-0.639173\pi$
$571$	−20.2361	−0.846853	−0.423427	+	0.905930i	$0.639173\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	−0.214782	−0.00896482
$575$	0	0
$576$	0	0
$577$	19.3050	0.803676	0.401838	−	0.915711i	$-0.368372\pi$
$577$	19.3050	0.803676	0.401838	+	0.915711i	$0.368372\pi$
$578$	15.3820	0.639805
$579$	0	0
$580$	0	0
$581$	1.81966	0.0754922
$582$	0	0
$583$	0.763932	0.0316388
$584$	32.2361	1.33394
$585$	0	0
$586$	9.23607	0.381538
$587$	20.9443	0.864463	0.432231	−	0.901763i	$-0.357726\pi$
$587$	20.9443	0.864463	0.432231	+	0.901763i	$0.357726\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	0	0
$592$	−8.83282	−0.363026
$593$	−36.7639	−1.50971	−0.754857	−	0.655890i	$-0.772294\pi$
$593$	−36.7639	−1.50971	−0.754857	+	0.655890i	$0.772294\pi$
$594$	0	0
$595$	0	0
$596$	26.1803	1.07239
$597$	0	0
$598$	16.9443	0.692903
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−4.18034	−0.170520	−0.0852598	−	0.996359i	$-0.527172\pi$
$601$	−4.18034	−0.170520	−0.0852598	+	0.996359i	$0.527172\pi$
$602$	1.88854	0.0769713
$603$	0	0
$604$	27.4164	1.11556
$605$	0	0
$606$	0	0
$607$	18.6525	0.757081	0.378540	−	0.925585i	$-0.376426\pi$
$607$	18.6525	0.757081	0.378540	+	0.925585i	$0.376426\pi$
$608$	5.61803	0.227841
$609$	0	0
$610$	0	0
$611$	16.9443	0.685492
$612$	0	0
$613$	−29.9443	−1.20944	−0.604719	−	0.796439i	$-0.706715\pi$
$613$	−29.9443	−1.20944	−0.604719	+	0.796439i	$0.706715\pi$
$614$	−5.05573	−0.204033
$615$	0	0
$616$	−0.403252	−0.0162475
$617$	16.0689	0.646909	0.323454	−	0.946244i	$-0.395156\pi$
$617$	16.0689	0.646909	0.323454	+	0.946244i	$0.395156\pi$
$618$	0	0
$619$	−27.7639	−1.11593	−0.557963	−	0.829866i	$-0.688417\pi$
$619$	−27.7639	−1.11593	−0.557963	+	0.829866i	$0.688417\pi$
$620$	0	0
$621$	0	0
$622$	−9.52786	−0.382033
$623$	1.18034	0.0472893
$624$	0	0
$625$	0	0
$626$	−20.2918	−0.811023
$627$	0	0
$628$	4.09017	0.163216
$629$	−30.8328	−1.22938
$630$	0	0
$631$	37.5279	1.49396	0.746980	−	0.664846i	$-0.231503\pi$
$631$	37.5279	1.49396	0.746980	+	0.664846i	$0.231503\pi$
$632$	10.0000	0.397779
$633$	0	0
$634$	−15.6738	−0.622485
$635$	0	0
$636$	0	0
$637$	22.4721	0.890378
$638$	4.47214	0.177054
$639$	0	0
$640$	0	0
$641$	15.8885	0.627560	0.313780	−	0.949496i	$-0.398405\pi$
$641$	15.8885	0.627560	0.313780	+	0.949496i	$0.398405\pi$
$642$	0	0
$643$	26.2361	1.03465	0.517325	−	0.855789i	$-0.326928\pi$
$643$	26.2361	1.03465	0.517325	+	0.855789i	$0.326928\pi$
$644$	3.23607	0.127519
$645$	0	0
$646$	4.00000	0.157378
$647$	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
$649$	5.12461	0.201159
$650$	0	0
$651$	0	0
$652$	29.5066	1.15557
$653$	41.1246	1.60933	0.804665	−	0.593729i	$-0.202345\pi$
$653$	41.1246	1.60933	0.804665	+	0.593729i	$0.202345\pi$
$654$	0	0
$655$	0	0
$656$	−2.72949	−0.106569
$657$	0	0
$658$	−0.763932	−0.0297812
$659$	−40.0000	−1.55818	−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000	−1.55818	−0.779089	+	0.626913i	$0.784318\pi$
$660$	0	0
$661$	−34.8328	−1.35484	−0.677420	−	0.735597i	$-0.736902\pi$
$661$	−34.8328	−1.35484	−0.677420	+	0.735597i	$0.736902\pi$
$662$	8.87539	0.344952
$663$	0	0
$664$	−17.2361	−0.668889
$665$	0	0
$666$	0	0
$667$	−80.2492	−3.10726
$668$	2.09017	0.0808711
$669$	0	0
$670$	0	0
$671$	−5.70820	−0.220363
$672$	0	0
$673$	−6.65248	−0.256434	−0.128217	−	0.991746i	$-0.540925\pi$
$673$	−6.65248	−0.256434	−0.128217	+	0.991746i	$0.540925\pi$
$674$	11.1246	0.428504
$675$	0	0
$676$	4.09017	0.157314
$677$	−9.83282	−0.377906	−0.188953	−	0.981986i	$-0.560509\pi$
$677$	−9.83282	−0.377906	−0.188953	+	0.981986i	$0.560509\pi$
$678$	0	0
$679$	−0.875388	−0.0335943
$680$	0	0
$681$	0	0
$682$	−3.77709	−0.144632
$683$	8.23607	0.315144	0.157572	−	0.987507i	$-0.449633\pi$
$683$	8.23607	0.315144	0.157572	+	0.987507i	$0.449633\pi$
$684$	0	0
$685$	0	0
$686$	−2.03444	−0.0776754
$687$	0	0
$688$	24.0000	0.914991
$689$	−3.23607	−0.123284
$690$	0	0
$691$	−20.3607	−0.774557	−0.387278	−	0.921963i	$-0.626585\pi$
$691$	−20.3607	−0.774557	−0.387278	+	0.921963i	$0.626585\pi$
$692$	23.5066	0.893586
$693$	0	0
$694$	−12.1803	−0.462359
$695$	0	0
$696$	0	0
$697$	−9.52786	−0.360894
$698$	−14.7984	−0.560127
$699$	0	0
$700$	0	0
$701$	9.05573	0.342030	0.171015	−	0.985268i	$-0.445295\pi$
$701$	9.05573	0.342030	0.171015	+	0.985268i	$0.445295\pi$
$702$	0	0
$703$	−4.76393	−0.179675
$704$	−0.180340	−0.00679682
$705$	0	0
$706$	−1.16718	−0.0439276
$707$	0.986844	0.0371141
$708$	0	0
$709$	−2.88854	−0.108482	−0.0542408	−	0.998528i	$-0.517274\pi$
$709$	−2.88854	−0.108482	−0.0542408	+	0.998528i	$0.517274\pi$
$710$	0	0
$711$	0	0
$712$	−11.1803	−0.419001
$713$	67.7771	2.53827
$714$	0	0
$715$	0	0
$716$	−12.5623	−0.469475
$717$	0	0
$718$	−2.36068	−0.0880998
$719$	−14.4721	−0.539720	−0.269860	−	0.962900i	$-0.586977\pi$
$719$	−14.4721	−0.539720	−0.269860	+	0.962900i	$0.586977\pi$
$720$	0	0
$721$	−2.87539	−0.107085
$722$	0.618034	0.0230008
$723$	0	0
$724$	−10.4721	−0.389194
$725$	0	0
$726$	0	0
$727$	12.5967	0.467188	0.233594	−	0.972334i	$-0.424951\pi$
$727$	12.5967	0.467188	0.233594	+	0.972334i	$0.424951\pi$
$728$	1.70820	0.0633102
$729$	0	0
$730$	0	0
$731$	83.7771	3.09861
$732$	0	0
$733$	−9.94427	−0.367300	−0.183650	−	0.982992i	$-0.558791\pi$
$733$	−9.94427	−0.367300	−0.183650	+	0.982992i	$0.558791\pi$
$734$	−1.88854	−0.0697074
$735$	0	0
$736$	−47.5967	−1.75444
$737$	−2.83282	−0.104348
$738$	0	0
$739$	−39.0689	−1.43717	−0.718586	−	0.695438i	$-0.755210\pi$
$739$	−39.0689	−1.43717	−0.718586	+	0.695438i	$0.755210\pi$
$740$	0	0
$741$	0	0
$742$	0.145898	0.00535609
$743$	25.0689	0.919688	0.459844	−	0.888000i	$-0.347905\pi$
$743$	25.0689	0.919688	0.459844	+	0.888000i	$0.347905\pi$
$744$	0	0
$745$	0	0
$746$	1.81966	0.0666225
$747$	0	0
$748$	−8.00000	−0.292509
$749$	−1.11146	−0.0406117
$750$	0	0
$751$	−32.0689	−1.17021	−0.585105	−	0.810957i	$-0.698947\pi$
$751$	−32.0689	−1.17021	−0.585105	+	0.810957i	$0.698947\pi$
$752$	−9.70820	−0.354022
$753$	0	0
$754$	−18.9443	−0.689910
$755$	0	0
$756$	0	0
$757$	−51.4721	−1.87079	−0.935393	−	0.353609i	$-0.884954\pi$
$757$	−51.4721	−1.87079	−0.935393	+	0.353609i	$0.884954\pi$
$758$	6.58359	0.239127
$759$	0	0
$760$	0	0
$761$	−12.6525	−0.458652	−0.229326	−	0.973350i	$-0.573652\pi$
$761$	−12.6525	−0.458652	−0.229326	+	0.973350i	$0.573652\pi$
$762$	0	0
$763$	−3.01316	−0.109084
$764$	−41.8885	−1.51547
$765$	0	0
$766$	−0.437694	−0.0158145
$767$	−21.7082	−0.783838
$768$	0	0
$769$	−10.5279	−0.379644	−0.189822	−	0.981818i	$-0.560791\pi$
$769$	−10.5279	−0.379644	−0.189822	+	0.981818i	$0.560791\pi$
$770$	0	0
$771$	0	0
$772$	18.6525	0.671317
$773$	−15.8328	−0.569467	−0.284733	−	0.958607i	$-0.591905\pi$
$773$	−15.8328	−0.569467	−0.284733	+	0.958607i	$0.591905\pi$
$774$	0	0
$775$	0	0
$776$	8.29180	0.297658
$777$	0	0
$778$	−11.3050	−0.405302
$779$	−1.47214	−0.0527447
$780$	0	0
$781$	0.986844	0.0353121
$782$	−33.8885	−1.21185
$783$	0	0
$784$	−12.8754	−0.459835
$785$	0	0
$786$	0	0
$787$	53.1246	1.89369	0.946844	−	0.321693i	$-0.104252\pi$
$787$	53.1246	1.89369	0.946844	+	0.321693i	$0.104252\pi$
$788$	22.9443	0.817356
$789$	0	0
$790$	0	0
$791$	−3.30495	−0.117511
$792$	0	0
$793$	24.1803	0.858669
$794$	−4.65248	−0.165110
$795$	0	0
$796$	21.5066	0.762280
$797$	21.4721	0.760582	0.380291	−	0.924867i	$-0.375824\pi$
$797$	21.4721	0.760582	0.380291	+	0.924867i	$0.375824\pi$
$798$	0	0
$799$	−33.8885	−1.19889
$800$	0	0
$801$	0	0
$802$	0.0688837	0.00243237
$803$	−11.0132	−0.388646
$804$	0	0
$805$	0	0
$806$	16.0000	0.563576
$807$	0	0
$808$	−9.34752	−0.328845
$809$	24.0689	0.846217	0.423108	−	0.906079i	$-0.360939\pi$
$809$	24.0689	0.846217	0.423108	+	0.906079i	$0.360939\pi$
$810$	0	0
$811$	−10.7639	−0.377973	−0.188986	−	0.981980i	$-0.560520\pi$
$811$	−10.7639	−0.377973	−0.188986	+	0.981980i	$0.560520\pi$
$812$	−3.61803	−0.126968
$813$	0	0
$814$	−2.24922	−0.0788352
$815$	0	0
$816$	0	0
$817$	12.9443	0.452863
$818$	−5.12461	−0.179178
$819$	0	0
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	0	0
$823$	37.2918	1.29991	0.649955	−	0.759973i	$-0.274788\pi$
$823$	37.2918	1.29991	0.649955	+	0.759973i	$0.274788\pi$
$824$	27.2361	0.948813
$825$	0	0
$826$	0.978714	0.0340538
$827$	40.9443	1.42377	0.711886	−	0.702295i	$-0.247841\pi$
$827$	40.9443	1.42377	0.711886	+	0.702295i	$0.247841\pi$
$828$	0	0
$829$	35.1246	1.21993	0.609964	−	0.792429i	$-0.291184\pi$
$829$	35.1246	1.21993	0.609964	+	0.792429i	$0.291184\pi$
$830$	0	0
$831$	0	0
$832$	0.763932	0.0264846
$833$	−44.9443	−1.55723
$834$	0	0
$835$	0	0
$836$	−1.23607	−0.0427503
$837$	0	0
$838$	12.7639	0.440923
$839$	34.5967	1.19441	0.597206	−	0.802088i	$-0.296277\pi$
$839$	34.5967	1.19441	0.597206	+	0.802088i	$0.296277\pi$
$840$	0	0
$841$	60.7214	2.09384
$842$	11.2361	0.387220
$843$	0	0
$844$	−39.4164	−1.35677
$845$	0	0
$846$	0	0
$847$	−2.45898	−0.0844916
$848$	1.85410	0.0636701
$849$	0	0
$850$	0	0
$851$	40.3607	1.38355
$852$	0	0
$853$	−51.0000	−1.74621	−0.873103	−	0.487535i	$-0.837896\pi$
$853$	−51.0000	−1.74621	−0.873103	+	0.487535i	$0.837896\pi$
$854$	−1.09017	−0.0373048
$855$	0	0
$856$	10.5279	0.359835
$857$	7.00000	0.239115	0.119558	−	0.992827i	$-0.461852\pi$
$857$	7.00000	0.239115	0.119558	+	0.992827i	$0.461852\pi$
$858$	0	0
$859$	30.1246	1.02784	0.513919	−	0.857839i	$-0.328193\pi$
$859$	30.1246	1.02784	0.513919	+	0.857839i	$0.328193\pi$
$860$	0	0
$861$	0	0
$862$	23.5623	0.802535
$863$	−22.0132	−0.749337	−0.374668	−	0.927159i	$-0.622243\pi$
$863$	−22.0132	−0.749337	−0.374668	+	0.927159i	$0.622243\pi$
$864$	0	0
$865$	0	0
$866$	−8.18034	−0.277979
$867$	0	0
$868$	3.05573	0.103718
$869$	−3.41641	−0.115894
$870$	0	0
$871$	12.0000	0.406604
$872$	28.5410	0.966521
$873$	0	0
$874$	−5.23607	−0.177113
$875$	0	0
$876$	0	0
$877$	−2.65248	−0.0895677	−0.0447839	−	0.998997i	$-0.514260\pi$
$877$	−2.65248	−0.0895677	−0.0447839	+	0.998997i	$0.514260\pi$
$878$	−6.18034	−0.208576
$879$	0	0
$880$	0	0
$881$	16.5410	0.557281	0.278641	−	0.960395i	$-0.410116\pi$
$881$	16.5410	0.557281	0.278641	+	0.960395i	$0.410116\pi$
$882$	0	0
$883$	−20.5967	−0.693136	−0.346568	−	0.938025i	$-0.612653\pi$
$883$	−20.5967	−0.693136	−0.346568	+	0.938025i	$0.612653\pi$
$884$	33.8885	1.13980
$885$	0	0
$886$	−3.12461	−0.104973
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	2.69505	0.0903890
$890$	0	0
$891$	0	0
$892$	16.9443	0.567336
$893$	−5.23607	−0.175218
$894$	0	0
$895$	0	0
$896$	−2.68692	−0.0897636
$897$	0	0
$898$	22.4377	0.748756
$899$	−75.7771	−2.52731
$900$	0	0
$901$	6.47214	0.215618
$902$	−0.695048	−0.0231426
$903$	0	0
$904$	31.3050	1.04119
$905$	0	0
$906$	0	0
$907$	−20.9443	−0.695443	−0.347722	−	0.937598i	$-0.613045\pi$
$907$	−20.9443	−0.695443	−0.347722	+	0.937598i	$0.613045\pi$
$908$	2.09017	0.0693647
$909$	0	0
$910$	0	0
$911$	29.1803	0.966788	0.483394	−	0.875403i	$-0.339404\pi$
$911$	29.1803	0.966788	0.483394	+	0.875403i	$0.339404\pi$
$912$	0	0
$913$	5.88854	0.194882
$914$	5.27051	0.174333
$915$	0	0
$916$	−1.70820	−0.0564406
$917$	−1.27864	−0.0422244
$918$	0	0
$919$	−23.2918	−0.768325	−0.384163	−	0.923265i	$-0.625510\pi$
$919$	−23.2918	−0.768325	−0.384163	+	0.923265i	$0.625510\pi$
$920$	0	0
$921$	0	0
$922$	2.58359	0.0850861
$923$	−4.18034	−0.137598
$924$	0	0
$925$	0	0
$926$	−0.944272	−0.0310307
$927$	0	0
$928$	53.2148	1.74686
$929$	33.4164	1.09636	0.548178	−	0.836361i	$-0.315321\pi$
$929$	33.4164	1.09636	0.548178	+	0.836361i	$0.315321\pi$
$930$	0	0
$931$	−6.94427	−0.227589
$932$	16.4721	0.539563
$933$	0	0
$934$	2.94427	0.0963395
$935$	0	0
$936$	0	0
$937$	−32.7771	−1.07078	−0.535390	−	0.844605i	$-0.679836\pi$
$937$	−32.7771	−1.07078	−0.535390	+	0.844605i	$0.679836\pi$
$938$	−0.541020	−0.0176649
$939$	0	0
$940$	0	0
$941$	12.4721	0.406580	0.203290	−	0.979119i	$-0.434837\pi$
$941$	12.4721	0.406580	0.203290	+	0.979119i	$0.434837\pi$
$942$	0	0
$943$	12.4721	0.406149
$944$	12.4377	0.404812
$945$	0	0
$946$	6.11146	0.198701
$947$	53.3050	1.73218	0.866089	−	0.499890i	$-0.166626\pi$
$947$	53.3050	1.73218	0.866089	+	0.499890i	$0.166626\pi$
$948$	0	0
$949$	46.6525	1.51440
$950$	0	0
$951$	0	0
$952$	−3.41641	−0.110726
$953$	26.5279	0.859322	0.429661	−	0.902990i	$-0.358633\pi$
$953$	26.5279	0.859322	0.429661	+	0.902990i	$0.358633\pi$
$954$	0	0
$955$	0	0
$956$	31.7082	1.02552
$957$	0	0
$958$	−13.4164	−0.433464
$959$	−1.88854	−0.0609843
$960$	0	0
$961$	33.0000	1.06452
$962$	9.52786	0.307191
$963$	0	0
$964$	−46.6525	−1.50258
$965$	0	0
$966$	0	0
$967$	17.0689	0.548898	0.274449	−	0.961602i	$-0.411505\pi$
$967$	17.0689	0.548898	0.274449	+	0.961602i	$0.411505\pi$
$968$	23.2918	0.748627
$969$	0	0
$970$	0	0
$971$	2.34752	0.0753356	0.0376678	−	0.999290i	$-0.488007\pi$
$971$	2.34752	0.0753356	0.0376678	+	0.999290i	$0.488007\pi$
$972$	0	0
$973$	2.88854	0.0926025
$974$	−16.1115	−0.516244
$975$	0	0
$976$	−13.8541	−0.443459
$977$	−52.4721	−1.67873	−0.839366	−	0.543566i	$-0.817074\pi$
$977$	−52.4721	−1.67873	−0.839366	+	0.543566i	$0.817074\pi$
$978$	0	0
$979$	3.81966	0.122077
$980$	0	0
$981$	0	0
$982$	−12.9443	−0.413068
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	0	0
$985$	0	0
$986$	37.8885	1.20662
$987$	0	0
$988$	5.23607	0.166582
$989$	−109.666	−3.48716
$990$	0	0
$991$	18.1803	0.577518	0.288759	−	0.957402i	$-0.406757\pi$
$991$	18.1803	0.577518	0.288759	+	0.957402i	$0.406757\pi$
$992$	−44.9443	−1.42698
$993$	0	0
$994$	0.188471	0.00597792
$995$	0	0
$996$	0	0
$997$	28.2492	0.894662	0.447331	−	0.894369i	$-0.352375\pi$
$997$	28.2492	0.894662	0.447331	+	0.894369i	$0.352375\pi$
$998$	22.0344	0.697488
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.r.1.2		2
3.2	odd	2		1425.2.a.r.1.1	yes	2
5.4	even	2		4275.2.a.w.1.1		2
15.2	even	4		1425.2.c.n.799.2		4
15.8	even	4		1425.2.c.n.799.3		4
15.14	odd	2		1425.2.a.m.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.m.1.2	✓	2	15.14	odd	2
1425.2.a.r.1.1	yes	2	3.2	odd	2
1425.2.c.n.799.2		4	15.2	even	4
1425.2.c.n.799.3		4	15.8	even	4
4275.2.a.r.1.2		2	1.1	even	1	trivial
4275.2.a.w.1.1		2	5.4	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -1.61803 q^{4} +0.236068 q^{7} -2.23607 q^{8} +0.763932 q^{11} -3.23607 q^{13} +0.145898 q^{14} +1.85410 q^{16} +6.47214 q^{17} +1.00000 q^{19} +0.472136 q^{22} -8.47214 q^{23} -2.00000 q^{26} -0.381966 q^{28} +9.47214 q^{29} -8.00000 q^{31} +5.61803 q^{32} +4.00000 q^{34} -4.76393 q^{37} +0.618034 q^{38} -1.47214 q^{41} +12.9443 q^{43} -1.23607 q^{44} -5.23607 q^{46} -5.23607 q^{47} -6.94427 q^{49} +5.23607 q^{52} +1.00000 q^{53} -0.527864 q^{56} +5.85410 q^{58} +6.70820 q^{59} -7.47214 q^{61} -4.94427 q^{62} -0.236068 q^{64} -3.70820 q^{67} -10.4721 q^{68} +1.29180 q^{71} -14.4164 q^{73} -2.94427 q^{74} -1.61803 q^{76} +0.180340 q^{77} -4.47214 q^{79} -0.909830 q^{82} +7.70820 q^{83} +8.00000 q^{86} -1.70820 q^{88} +5.00000 q^{89} -0.763932 q^{91} +13.7082 q^{92} -3.23607 q^{94} -3.70820 q^{97} -4.29180 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - 4 q^{7} + 6 q^{11} - 2 q^{13} + 7 q^{14} - 3 q^{16} + 4 q^{17} + 2 q^{19} - 8 q^{22} - 8 q^{23} - 4 q^{26} - 3 q^{28} + 10 q^{29} - 16 q^{31} + 9 q^{32} + 8 q^{34} - 14 q^{37} - q^{38}+ \cdots - 22 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 - 4 * q^7 + 6 * q^11 - 2 * q^13 + 7 * q^14 - 3 * q^16 + 4 * q^17 + 2 * q^19 - 8 * q^22 - 8 * q^23 - 4 * q^26 - 3 * q^28 + 10 * q^29 - 16 * q^31 + 9 * q^32 + 8 * q^34 - 14 * q^37 - q^38 + 6 * q^41 + 8 * q^43 + 2 * q^44 - 6 * q^46 - 6 * q^47 + 4 * q^49 + 6 * q^52 + 2 * q^53 - 10 * q^56 + 5 * q^58 - 6 * q^61 + 8 * q^62 + 4 * q^64 + 6 * q^67 - 12 * q^68 + 16 * q^71 - 2 * q^73 + 12 * q^74 - q^76 - 22 * q^77 - 13 * q^82 + 2 * q^83 + 16 * q^86 + 10 * q^88 + 10 * q^89 - 6 * q^91 + 14 * q^92 - 2 * q^94 + 6 * q^97 - 22 * q^98

Introduction

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