LMFDB - Embedded newform 675.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(1,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.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:	$5.38990213644$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30278 q^{2} +3.30278 q^{4} +2.60555 q^{7} -3.00000 q^{8} +O(q^{10})$	$q-2.30278 q^{2} +3.30278 q^{4} +2.60555 q^{7} -3.00000 q^{8} -4.60555 q^{11} -6.60555 q^{13} -6.00000 q^{14} +0.302776 q^{16} +1.60555 q^{17} +3.60555 q^{19} +10.6056 q^{22} -3.00000 q^{23} +15.2111 q^{26} +8.60555 q^{28} -1.39445 q^{29} -5.60555 q^{31} +5.30278 q^{32} -3.69722 q^{34} -2.00000 q^{37} -8.30278 q^{38} +4.60555 q^{41} -0.605551 q^{43} -15.2111 q^{44} +6.90833 q^{46} -9.21110 q^{47} -0.211103 q^{49} -21.8167 q^{52} +1.60555 q^{53} -7.81665 q^{56} +3.21110 q^{58} +1.39445 q^{59} -4.21110 q^{61} +12.9083 q^{62} -12.8167 q^{64} +0.788897 q^{67} +5.30278 q^{68} -7.39445 q^{71} -12.6056 q^{73} +4.60555 q^{74} +11.9083 q^{76} -12.0000 q^{77} -11.6056 q^{79} -10.6056 q^{82} +3.00000 q^{83} +1.39445 q^{86} +13.8167 q^{88} +13.8167 q^{89} -17.2111 q^{91} -9.90833 q^{92} +21.2111 q^{94} -8.00000 q^{97} +0.486122 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 3 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - q^2 + 3 * q^4 - 2 * q^7 - 6 * q^8	$ 2 q - q^{2} + 3 q^{4} - 2 q^{7} - 6 q^{8} - 2 q^{11} - 6 q^{13} - 12 q^{14} - 3 q^{16} - 4 q^{17} + 14 q^{22} - 6 q^{23} + 16 q^{26} + 10 q^{28} - 10 q^{29} - 4 q^{31} + 7 q^{32} - 11 q^{34} - 4 q^{37} - 13 q^{38} + 2 q^{41} + 6 q^{43} - 16 q^{44} + 3 q^{46} - 4 q^{47} + 14 q^{49} - 22 q^{52} - 4 q^{53} + 6 q^{56} - 8 q^{58} + 10 q^{59} + 6 q^{61} + 15 q^{62} - 4 q^{64} + 16 q^{67} + 7 q^{68} - 22 q^{71} - 18 q^{73} + 2 q^{74} + 13 q^{76} - 24 q^{77} - 16 q^{79} - 14 q^{82} + 6 q^{83} + 10 q^{86} + 6 q^{88} + 6 q^{89} - 20 q^{91} - 9 q^{92} + 28 q^{94} - 16 q^{97} + 19 q^{98}+O(q^{100}) $ 2 * q - q^2 + 3 * q^4 - 2 * q^7 - 6 * q^8 - 2 * q^11 - 6 * q^13 - 12 * q^14 - 3 * q^16 - 4 * q^17 + 14 * q^22 - 6 * q^23 + 16 * q^26 + 10 * q^28 - 10 * q^29 - 4 * q^31 + 7 * q^32 - 11 * q^34 - 4 * q^37 - 13 * q^38 + 2 * q^41 + 6 * q^43 - 16 * q^44 + 3 * q^46 - 4 * q^47 + 14 * q^49 - 22 * q^52 - 4 * q^53 + 6 * q^56 - 8 * q^58 + 10 * q^59 + 6 * q^61 + 15 * q^62 - 4 * q^64 + 16 * q^67 + 7 * q^68 - 22 * q^71 - 18 * q^73 + 2 * q^74 + 13 * q^76 - 24 * q^77 - 16 * q^79 - 14 * q^82 + 6 * q^83 + 10 * q^86 + 6 * q^88 + 6 * q^89 - 20 * q^91 - 9 * q^92 + 28 * q^94 - 16 * q^97 + 19 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.30278	−1.62831	−0.814154	−	0.580649i	$-0.802799\pi$
$2$	−2.30278	−1.62831	−0.814154	+	0.580649i	$0.802799\pi$
$3$	0	0
$3$	0	0
$4$	3.30278	1.65139
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.60555	0.984806	0.492403	−	0.870367i	$-0.336119\pi$
$7$	2.60555	0.984806	0.492403	+	0.870367i	$0.336119\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	0	0
$11$	−4.60555	−1.38863	−0.694313	−	0.719673i	$-0.744292\pi$
$11$	−4.60555	−1.38863	−0.694313	+	0.719673i	$0.744292\pi$
$12$	0	0
$13$	−6.60555	−1.83205	−0.916025	−	0.401121i	$-0.868621\pi$
$13$	−6.60555	−1.83205	−0.916025	+	0.401121i	$0.868621\pi$
$14$	−6.00000	−1.60357
$15$	0	0
$16$	0.302776	0.0756939
$17$	1.60555	0.389403	0.194702	−	0.980863i	$-0.437626\pi$
$17$	1.60555	0.389403	0.194702	+	0.980863i	$0.437626\pi$
$18$	0	0
$19$	3.60555	0.827170	0.413585	−	0.910465i	$-0.364276\pi$
$19$	3.60555	0.827170	0.413585	+	0.910465i	$0.364276\pi$
$20$	0	0
$21$	0	0
$22$	10.6056	2.26111
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	0	0
$26$	15.2111	2.98314
$27$	0	0
$28$	8.60555	1.62630
$29$	−1.39445	−0.258943	−0.129471	−	0.991583i	$-0.541328\pi$
$29$	−1.39445	−0.258943	−0.129471	+	0.991583i	$0.541328\pi$
$30$	0	0
$31$	−5.60555	−1.00679	−0.503393	−	0.864057i	$-0.667915\pi$
$31$	−5.60555	−1.00679	−0.503393	+	0.864057i	$0.667915\pi$
$32$	5.30278	0.937407
$33$	0	0
$34$	−3.69722	−0.634069
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−8.30278	−1.34689
$39$	0	0
$40$	0	0
$41$	4.60555	0.719266	0.359633	−	0.933094i	$-0.382902\pi$
$41$	4.60555	0.719266	0.359633	+	0.933094i	$0.382902\pi$
$42$	0	0
$43$	−0.605551	−0.0923457	−0.0461729	−	0.998933i	$-0.514703\pi$
$43$	−0.605551	−0.0923457	−0.0461729	+	0.998933i	$0.514703\pi$
$44$	−15.2111	−2.29316
$45$	0	0
$46$	6.90833	1.01858
$47$	−9.21110	−1.34358	−0.671789	−	0.740743i	$-0.734474\pi$
$47$	−9.21110	−1.34358	−0.671789	+	0.740743i	$0.734474\pi$
$48$	0	0
$49$	−0.211103	−0.0301575
$50$	0	0
$51$	0	0
$52$	−21.8167	−3.02543
$53$	1.60555	0.220539	0.110270	−	0.993902i	$-0.464829\pi$
$53$	1.60555	0.220539	0.110270	+	0.993902i	$0.464829\pi$
$54$	0	0
$55$	0	0
$56$	−7.81665	−1.04454
$57$	0	0
$58$	3.21110	0.421638
$59$	1.39445	0.181542	0.0907709	−	0.995872i	$-0.471067\pi$
$59$	1.39445	0.181542	0.0907709	+	0.995872i	$0.471067\pi$
$60$	0	0
$61$	−4.21110	−0.539176	−0.269588	−	0.962976i	$-0.586888\pi$
$61$	−4.21110	−0.539176	−0.269588	+	0.962976i	$0.586888\pi$
$62$	12.9083	1.63936
$63$	0	0
$64$	−12.8167	−1.60208
$65$	0	0
$66$	0	0
$67$	0.788897	0.0963792	0.0481896	−	0.998838i	$-0.484655\pi$
$67$	0.788897	0.0963792	0.0481896	+	0.998838i	$0.484655\pi$
$68$	5.30278	0.643056
$69$	0	0
$70$	0	0
$71$	−7.39445	−0.877560	−0.438780	−	0.898595i	$-0.644589\pi$
$71$	−7.39445	−0.877560	−0.438780	+	0.898595i	$0.644589\pi$
$72$	0	0
$73$	−12.6056	−1.47537	−0.737684	−	0.675146i	$-0.764081\pi$
$73$	−12.6056	−1.47537	−0.737684	+	0.675146i	$0.764081\pi$
$74$	4.60555	0.535384
$75$	0	0
$76$	11.9083	1.36598
$77$	−12.0000	−1.36753
$78$	0	0
$79$	−11.6056	−1.30573	−0.652863	−	0.757476i	$-0.726432\pi$
$79$	−11.6056	−1.30573	−0.652863	+	0.757476i	$0.726432\pi$
$80$	0	0
$81$	0	0
$82$	−10.6056	−1.17119
$83$	3.00000	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$84$	0	0
$85$	0	0
$86$	1.39445	0.150367
$87$	0	0
$88$	13.8167	1.47286
$89$	13.8167	1.46456	0.732281	−	0.681002i	$-0.238456\pi$
$89$	13.8167	1.46456	0.732281	+	0.681002i	$0.238456\pi$
$90$	0	0
$91$	−17.2111	−1.80421
$92$	−9.90833	−1.03301
$93$	0	0
$94$	21.2111	2.18776
$95$	0	0
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0.486122	0.0491057
$99$	0	0
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	19.8167	1.94318
$105$	0	0
$106$	−3.69722	−0.359106
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	0	0
$111$	0	0
$112$	0.788897	0.0745438
$113$	−15.2111	−1.43094	−0.715470	−	0.698643i	$-0.753787\pi$
$113$	−15.2111	−1.43094	−0.715470	+	0.698643i	$0.753787\pi$
$114$	0	0
$115$	0	0
$116$	−4.60555	−0.427615
$117$	0	0
$118$	−3.21110	−0.295606
$119$	4.18335	0.383487
$120$	0	0
$121$	10.2111	0.928282
$122$	9.69722	0.877945
$123$	0	0
$124$	−18.5139	−1.66260
$125$	0	0
$126$	0	0
$127$	19.2111	1.70471	0.852355	−	0.522964i	$-0.175174\pi$
$127$	19.2111	1.70471	0.852355	+	0.522964i	$0.175174\pi$
$128$	18.9083	1.67128
$129$	0	0
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	9.39445	0.814602
$134$	−1.81665	−0.156935
$135$	0	0
$136$	−4.81665	−0.413025
$137$	16.8167	1.43674	0.718372	−	0.695659i	$-0.244888\pi$
$137$	16.8167	1.43674	0.718372	+	0.695659i	$0.244888\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	17.0278	1.42894
$143$	30.4222	2.54403
$144$	0	0
$145$	0	0
$146$	29.0278	2.40235
$147$	0	0
$148$	−6.60555	−0.542973
$149$	−23.0278	−1.88651	−0.943254	−	0.332073i	$-0.892252\pi$
$149$	−23.0278	−1.88651	−0.943254	+	0.332073i	$0.892252\pi$
$150$	0	0
$151$	14.4222	1.17366	0.586831	−	0.809709i	$-0.300375\pi$
$151$	14.4222	1.17366	0.586831	+	0.809709i	$0.300375\pi$
$152$	−10.8167	−0.877346
$153$	0	0
$154$	27.6333	2.22676
$155$	0	0
$156$	0	0
$157$	17.8167	1.42192	0.710962	−	0.703231i	$-0.248260\pi$
$157$	17.8167	1.42192	0.710962	+	0.703231i	$0.248260\pi$
$158$	26.7250	2.12613
$159$	0	0
$160$	0	0
$161$	−7.81665	−0.616039
$162$	0	0
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	15.2111	1.18779
$165$	0	0
$166$	−6.90833	−0.536190
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	0	0
$169$	30.6333	2.35641
$170$	0	0
$171$	0	0
$172$	−2.00000	−0.152499
$173$	−10.8167	−0.822375	−0.411187	−	0.911551i	$-0.634886\pi$
$173$	−10.8167	−0.822375	−0.411187	+	0.911551i	$0.634886\pi$
$174$	0	0
$175$	0	0
$176$	−1.39445	−0.105111
$177$	0	0
$178$	−31.8167	−2.38476
$179$	21.2111	1.58539	0.792696	−	0.609617i	$-0.208677\pi$
$179$	21.2111	1.58539	0.792696	+	0.609617i	$0.208677\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	39.6333	2.93782
$183$	0	0
$184$	9.00000	0.663489
$185$	0	0
$186$	0	0
$187$	−7.39445	−0.540736
$188$	−30.4222	−2.21877
$189$	0	0
$190$	0	0
$191$	12.4222	0.898839	0.449420	−	0.893321i	$-0.351631\pi$
$191$	12.4222	0.898839	0.449420	+	0.893321i	$0.351631\pi$
$192$	0	0
$193$	−0.183346	−0.0131975	−0.00659877	−	0.999978i	$-0.502100\pi$
$193$	−0.183346	−0.0131975	−0.00659877	+	0.999978i	$0.502100\pi$
$194$	18.4222	1.32264
$195$	0	0
$196$	−0.697224	−0.0498017
$197$	−22.8167	−1.62562	−0.812810	−	0.582529i	$-0.802063\pi$
$197$	−22.8167	−1.62562	−0.812810	+	0.582529i	$0.802063\pi$
$198$	0	0
$199$	−1.21110	−0.0858528	−0.0429264	−	0.999078i	$-0.513668\pi$
$199$	−1.21110	−0.0858528	−0.0429264	+	0.999078i	$0.513668\pi$
$200$	0	0
$201$	0	0
$202$	27.6333	1.94427
$203$	−3.63331	−0.255008
$204$	0	0
$205$	0	0
$206$	−9.21110	−0.641768
$207$	0	0
$208$	−2.00000	−0.138675
$209$	−16.6056	−1.14863
$210$	0	0
$211$	−8.81665	−0.606963	−0.303482	−	0.952837i	$-0.598149\pi$
$211$	−8.81665	−0.606963	−0.303482	+	0.952837i	$0.598149\pi$
$212$	5.30278	0.364196
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−14.6056	−0.991489
$218$	16.1194	1.09175
$219$	0	0
$220$	0	0
$221$	−10.6056	−0.713407
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	13.8167	0.923164
$225$	0	0
$226$	35.0278	2.33001
$227$	−11.7889	−0.782457	−0.391228	−	0.920294i	$-0.627950\pi$
$227$	−11.7889	−0.782457	−0.391228	+	0.920294i	$0.627950\pi$
$228$	0	0
$229$	8.21110	0.542605	0.271302	−	0.962494i	$-0.412546\pi$
$229$	8.21110	0.542605	0.271302	+	0.962494i	$0.412546\pi$
$230$	0	0
$231$	0	0
$232$	4.18335	0.274650
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	0	0
$236$	4.60555	0.299796
$237$	0	0
$238$	−9.63331	−0.624435
$239$	15.2111	0.983924	0.491962	−	0.870617i	$-0.336280\pi$
$239$	15.2111	0.983924	0.491962	+	0.870617i	$0.336280\pi$
$240$	0	0
$241$	13.7889	0.888221	0.444110	−	0.895972i	$-0.353520\pi$
$241$	13.7889	0.888221	0.444110	+	0.895972i	$0.353520\pi$
$242$	−23.5139	−1.51153
$243$	0	0
$244$	−13.9083	−0.890389
$245$	0	0
$246$	0	0
$247$	−23.8167	−1.51542
$248$	16.8167	1.06786
$249$	0	0
$250$	0	0
$251$	−27.6333	−1.74420	−0.872099	−	0.489329i	$-0.837242\pi$
$251$	−27.6333	−1.74420	−0.872099	+	0.489329i	$0.837242\pi$
$252$	0	0
$253$	13.8167	0.868646
$254$	−44.2389	−2.77579
$255$	0	0
$256$	−17.9083	−1.11927
$257$	1.18335	0.0738151	0.0369076	−	0.999319i	$-0.488249\pi$
$257$	1.18335	0.0738151	0.0369076	+	0.999319i	$0.488249\pi$
$258$	0	0
$259$	−5.21110	−0.323802
$260$	0	0
$261$	0	0
$262$	13.8167	0.853596
$263$	−2.78890	−0.171971	−0.0859854	−	0.996296i	$-0.527404\pi$
$263$	−2.78890	−0.171971	−0.0859854	+	0.996296i	$0.527404\pi$
$264$	0	0
$265$	0	0
$266$	−21.6333	−1.32642
$267$	0	0
$268$	2.60555	0.159159
$269$	3.21110	0.195784	0.0978922	−	0.995197i	$-0.468790\pi$
$269$	3.21110	0.195784	0.0978922	+	0.995197i	$0.468790\pi$
$270$	0	0
$271$	31.2389	1.89763	0.948813	−	0.315839i	$-0.102286\pi$
$271$	31.2389	1.89763	0.948813	+	0.315839i	$0.102286\pi$
$272$	0.486122	0.0294755
$273$	0	0
$274$	−38.7250	−2.33946
$275$	0	0
$276$	0	0
$277$	−7.02776	−0.422257	−0.211128	−	0.977458i	$-0.567714\pi$
$277$	−7.02776	−0.422257	−0.211128	+	0.977458i	$0.567714\pi$
$278$	9.21110	0.552445
$279$	0	0
$280$	0	0
$281$	19.8167	1.18216	0.591081	−	0.806612i	$-0.298701\pi$
$281$	19.8167	1.18216	0.591081	+	0.806612i	$0.298701\pi$
$282$	0	0
$283$	−3.39445	−0.201779	−0.100890	−	0.994898i	$-0.532169\pi$
$283$	−3.39445	−0.201779	−0.100890	+	0.994898i	$0.532169\pi$
$284$	−24.4222	−1.44919
$285$	0	0
$286$	−70.0555	−4.14247
$287$	12.0000	0.708338
$288$	0	0
$289$	−14.4222	−0.848365
$290$	0	0
$291$	0	0
$292$	−41.6333	−2.43641
$293$	−7.18335	−0.419656	−0.209828	−	0.977738i	$-0.567290\pi$
$293$	−7.18335	−0.419656	−0.209828	+	0.977738i	$0.567290\pi$
$294$	0	0
$295$	0	0
$296$	6.00000	0.348743
$297$	0	0
$298$	53.0278	3.07182
$299$	19.8167	1.14603
$300$	0	0
$301$	−1.57779	−0.0909426
$302$	−33.2111	−1.91108
$303$	0	0
$304$	1.09167	0.0626117
$305$	0	0
$306$	0	0
$307$	−8.42221	−0.480681	−0.240340	−	0.970689i	$-0.577259\pi$
$307$	−8.42221	−0.480681	−0.240340	+	0.970689i	$0.577259\pi$
$308$	−39.6333	−2.25832
$309$	0	0
$310$	0	0
$311$	7.81665	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$311$	7.81665	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$312$	0	0
$313$	19.6333	1.10974	0.554870	−	0.831937i	$-0.312768\pi$
$313$	19.6333	1.10974	0.554870	+	0.831937i	$0.312768\pi$
$314$	−41.0278	−2.31533
$315$	0	0
$316$	−38.3305	−2.15626
$317$	−7.60555	−0.427170	−0.213585	−	0.976924i	$-0.568514\pi$
$317$	−7.60555	−0.427170	−0.213585	+	0.976924i	$0.568514\pi$
$318$	0	0
$319$	6.42221	0.359574
$320$	0	0
$321$	0	0
$322$	18.0000	1.00310
$323$	5.78890	0.322103
$324$	0	0
$325$	0	0
$326$	4.60555	0.255078
$327$	0	0
$328$	−13.8167	−0.762897
$329$	−24.0000	−1.32316
$330$	0	0
$331$	29.2111	1.60559	0.802794	−	0.596257i	$-0.203346\pi$
$331$	29.2111	1.60559	0.802794	+	0.596257i	$0.203346\pi$
$332$	9.90833	0.543790
$333$	0	0
$334$	6.90833	0.378007
$335$	0	0
$336$	0	0
$337$	−6.60555	−0.359827	−0.179914	−	0.983682i	$-0.557582\pi$
$337$	−6.60555	−0.359827	−0.179914	+	0.983682i	$0.557582\pi$
$338$	−70.5416	−3.83696
$339$	0	0
$340$	0	0
$341$	25.8167	1.39805
$342$	0	0
$343$	−18.7889	−1.01451
$344$	1.81665	0.0979474
$345$	0	0
$346$	24.9083	1.33908
$347$	30.4222	1.63315	0.816575	−	0.577240i	$-0.195870\pi$
$347$	30.4222	1.63315	0.816575	+	0.577240i	$0.195870\pi$
$348$	0	0
$349$	−31.8444	−1.70459	−0.852296	−	0.523060i	$-0.824791\pi$
$349$	−31.8444	−1.70459	−0.852296	+	0.523060i	$0.824791\pi$
$350$	0	0
$351$	0	0
$352$	−24.4222	−1.30171
$353$	21.6333	1.15142	0.575712	−	0.817652i	$-0.304725\pi$
$353$	21.6333	1.15142	0.575712	+	0.817652i	$0.304725\pi$
$354$	0	0
$355$	0	0
$356$	45.6333	2.41856
$357$	0	0
$358$	−48.8444	−2.58151
$359$	9.63331	0.508427	0.254213	−	0.967148i	$-0.418183\pi$
$359$	9.63331	0.508427	0.254213	+	0.967148i	$0.418183\pi$
$360$	0	0
$361$	−6.00000	−0.315789
$362$	16.1194	0.847218
$363$	0	0
$364$	−56.8444	−2.97946
$365$	0	0
$366$	0	0
$367$	2.60555	0.136009	0.0680043	−	0.997685i	$-0.478337\pi$
$367$	2.60555	0.136009	0.0680043	+	0.997685i	$0.478337\pi$
$368$	−0.908327	−0.0473498
$369$	0	0
$370$	0	0
$371$	4.18335	0.217189
$372$	0	0
$373$	25.2111	1.30538	0.652691	−	0.757624i	$-0.273640\pi$
$373$	25.2111	1.30538	0.652691	+	0.757624i	$0.273640\pi$
$374$	17.0278	0.880484
$375$	0	0
$376$	27.6333	1.42508
$377$	9.21110	0.474396
$378$	0	0
$379$	21.6056	1.10980	0.554901	−	0.831916i	$-0.312756\pi$
$379$	21.6056	1.10980	0.554901	+	0.831916i	$0.312756\pi$
$380$	0	0
$381$	0	0
$382$	−28.6056	−1.46359
$383$	24.6333	1.25870	0.629352	−	0.777121i	$-0.283321\pi$
$383$	24.6333	1.25870	0.629352	+	0.777121i	$0.283321\pi$
$384$	0	0
$385$	0	0
$386$	0.422205	0.0214897
$387$	0	0
$388$	−26.4222	−1.34138
$389$	−25.8167	−1.30896	−0.654478	−	0.756081i	$-0.727112\pi$
$389$	−25.8167	−1.30896	−0.654478	+	0.756081i	$0.727112\pi$
$390$	0	0
$391$	−4.81665	−0.243589
$392$	0.633308	0.0319869
$393$	0	0
$394$	52.5416	2.64701
$395$	0	0
$396$	0	0
$397$	5.39445	0.270740	0.135370	−	0.990795i	$-0.456778\pi$
$397$	5.39445	0.270740	0.135370	+	0.990795i	$0.456778\pi$
$398$	2.78890	0.139795
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	37.0278	1.84448
$404$	−39.6333	−1.97183
$405$	0	0
$406$	8.36669	0.415232
$407$	9.21110	0.456577
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	0	0
$411$	0	0
$412$	13.2111	0.650864
$413$	3.63331	0.178783
$414$	0	0
$415$	0	0
$416$	−35.0278	−1.71738
$417$	0	0
$418$	38.2389	1.87032
$419$	−23.0278	−1.12498	−0.562490	−	0.826804i	$-0.690156\pi$
$419$	−23.0278	−1.12498	−0.562490	+	0.826804i	$0.690156\pi$
$420$	0	0
$421$	5.42221	0.264262	0.132131	−	0.991232i	$-0.457818\pi$
$421$	5.42221	0.264262	0.132131	+	0.991232i	$0.457818\pi$
$422$	20.3028	0.988324
$423$	0	0
$424$	−4.81665	−0.233917
$425$	0	0
$426$	0	0
$427$	−10.9722	−0.530984
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.18335	−0.201505	−0.100752	−	0.994912i	$-0.532125\pi$
$431$	−4.18335	−0.201505	−0.100752	+	0.994912i	$0.532125\pi$
$432$	0	0
$433$	−22.2389	−1.06873	−0.534366	−	0.845253i	$-0.679449\pi$
$433$	−22.2389	−1.06873	−0.534366	+	0.845253i	$0.679449\pi$
$434$	33.6333	1.61445
$435$	0	0
$436$	−23.1194	−1.10722
$437$	−10.8167	−0.517431
$438$	0	0
$439$	27.6056	1.31754	0.658771	−	0.752344i	$-0.271077\pi$
$439$	27.6056	1.31754	0.658771	+	0.752344i	$0.271077\pi$
$440$	0	0
$441$	0	0
$442$	24.4222	1.16165
$443$	−24.6333	−1.17036	−0.585182	−	0.810902i	$-0.698977\pi$
$443$	−24.6333	−1.17036	−0.585182	+	0.810902i	$0.698977\pi$
$444$	0	0
$445$	0	0
$446$	−23.0278	−1.09040
$447$	0	0
$448$	−33.3944	−1.57774
$449$	−38.2389	−1.80460	−0.902302	−	0.431105i	$-0.858124\pi$
$449$	−38.2389	−1.80460	−0.902302	+	0.431105i	$0.858124\pi$
$450$	0	0
$451$	−21.2111	−0.998792
$452$	−50.2389	−2.36304
$453$	0	0
$454$	27.1472	1.27408
$455$	0	0
$456$	0	0
$457$	13.2111	0.617989	0.308995	−	0.951064i	$-0.400007\pi$
$457$	13.2111	0.617989	0.308995	+	0.951064i	$0.400007\pi$
$458$	−18.9083	−0.883528
$459$	0	0
$460$	0	0
$461$	−21.6333	−1.00756	−0.503782	−	0.863831i	$-0.668058\pi$
$461$	−21.6333	−1.00756	−0.503782	+	0.863831i	$0.668058\pi$
$462$	0	0
$463$	0.788897	0.0366632	0.0183316	−	0.999832i	$-0.494165\pi$
$463$	0.788897	0.0366632	0.0183316	+	0.999832i	$0.494165\pi$
$464$	−0.422205	−0.0196004
$465$	0	0
$466$	41.4500	1.92013
$467$	12.2111	0.565062	0.282531	−	0.959258i	$-0.408826\pi$
$467$	12.2111	0.565062	0.282531	+	0.959258i	$0.408826\pi$
$468$	0	0
$469$	2.05551	0.0949148
$470$	0	0
$471$	0	0
$472$	−4.18335	−0.192554
$473$	2.78890	0.128234
$474$	0	0
$475$	0	0
$476$	13.8167	0.633285
$477$	0	0
$478$	−35.0278	−1.60213
$479$	37.8167	1.72789	0.863944	−	0.503589i	$-0.167987\pi$
$479$	37.8167	1.72789	0.863944	+	0.503589i	$0.167987\pi$
$480$	0	0
$481$	13.2111	0.602374
$482$	−31.7527	−1.44630
$483$	0	0
$484$	33.7250	1.53295
$485$	0	0
$486$	0	0
$487$	29.8167	1.35112	0.675561	−	0.737304i	$-0.263902\pi$
$487$	29.8167	1.35112	0.675561	+	0.737304i	$0.263902\pi$
$488$	12.6333	0.571883
$489$	0	0
$490$	0	0
$491$	−27.2111	−1.22802	−0.614010	−	0.789298i	$-0.710445\pi$
$491$	−27.2111	−1.22802	−0.614010	+	0.789298i	$0.710445\pi$
$492$	0	0
$493$	−2.23886	−0.100833
$494$	54.8444	2.46757
$495$	0	0
$496$	−1.69722	−0.0762076
$497$	−19.2666	−0.864226
$498$	0	0
$499$	−20.3944	−0.912981	−0.456490	−	0.889728i	$-0.650894\pi$
$499$	−20.3944	−0.912981	−0.456490	+	0.889728i	$0.650894\pi$
$500$	0	0
$501$	0	0
$502$	63.6333	2.84009
$503$	2.57779	0.114938	0.0574691	−	0.998347i	$-0.481697\pi$
$503$	2.57779	0.114938	0.0574691	+	0.998347i	$0.481697\pi$
$504$	0	0
$505$	0	0
$506$	−31.8167	−1.41442
$507$	0	0
$508$	63.4500	2.81514
$509$	24.8444	1.10121	0.550605	−	0.834766i	$-0.314397\pi$
$509$	24.8444	1.10121	0.550605	+	0.834766i	$0.314397\pi$
$510$	0	0
$511$	−32.8444	−1.45295
$512$	3.42221	0.151242
$513$	0	0
$514$	−2.72498	−0.120194
$515$	0	0
$516$	0	0
$517$	42.4222	1.86573
$518$	12.0000	0.527250
$519$	0	0
$520$	0	0
$521$	28.6056	1.25323	0.626616	−	0.779328i	$-0.284439\pi$
$521$	28.6056	1.25323	0.626616	+	0.779328i	$0.284439\pi$
$522$	0	0
$523$	−30.6056	−1.33829	−0.669144	−	0.743133i	$-0.733339\pi$
$523$	−30.6056	−1.33829	−0.669144	+	0.743133i	$0.733339\pi$
$524$	−19.8167	−0.865695
$525$	0	0
$526$	6.42221	0.280021
$527$	−9.00000	−0.392046
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	0	0
$532$	31.0278	1.34522
$533$	−30.4222	−1.31773
$534$	0	0
$535$	0	0
$536$	−2.36669	−0.102226
$537$	0	0
$538$	−7.39445	−0.318797
$539$	0.972244	0.0418775
$540$	0	0
$541$	−40.4222	−1.73789	−0.868943	−	0.494912i	$-0.835200\pi$
$541$	−40.4222	−1.73789	−0.868943	+	0.494912i	$0.835200\pi$
$542$	−71.9361	−3.08992
$543$	0	0
$544$	8.51388	0.365030
$545$	0	0
$546$	0	0
$547$	−0.605551	−0.0258915	−0.0129458	−	0.999916i	$-0.504121\pi$
$547$	−0.605551	−0.0258915	−0.0129458	+	0.999916i	$0.504121\pi$
$548$	55.5416	2.37262
$549$	0	0
$550$	0	0
$551$	−5.02776	−0.214190
$552$	0	0
$553$	−30.2389	−1.28589
$554$	16.1833	0.687564
$555$	0	0
$556$	−13.2111	−0.560276
$557$	9.63331	0.408176	0.204088	−	0.978953i	$-0.434577\pi$
$557$	9.63331	0.408176	0.204088	+	0.978953i	$0.434577\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	−45.6333	−1.92492
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	0	0
$566$	7.81665	0.328558
$567$	0	0
$568$	22.1833	0.930793
$569$	−16.1833	−0.678441	−0.339221	−	0.940707i	$-0.610163\pi$
$569$	−16.1833	−0.678441	−0.339221	+	0.940707i	$0.610163\pi$
$570$	0	0
$571$	28.4500	1.19059	0.595297	−	0.803506i	$-0.297034\pi$
$571$	28.4500	1.19059	0.595297	+	0.803506i	$0.297034\pi$
$572$	100.478	4.20118
$573$	0	0
$574$	−27.6333	−1.15339
$575$	0	0
$576$	0	0
$577$	−6.18335	−0.257416	−0.128708	−	0.991683i	$-0.541083\pi$
$577$	−6.18335	−0.257416	−0.128708	+	0.991683i	$0.541083\pi$
$578$	33.2111	1.38140
$579$	0	0
$580$	0	0
$581$	7.81665	0.324289
$582$	0	0
$583$	−7.39445	−0.306247
$584$	37.8167	1.56486
$585$	0	0
$586$	16.5416	0.683329
$587$	21.0000	0.866763	0.433381	−	0.901211i	$-0.357320\pi$
$587$	21.0000	0.866763	0.433381	+	0.901211i	$0.357320\pi$
$588$	0	0
$589$	−20.2111	−0.832784
$590$	0	0
$591$	0	0
$592$	−0.605551	−0.0248880
$593$	−29.2389	−1.20070	−0.600348	−	0.799739i	$-0.704971\pi$
$593$	−29.2389	−1.20070	−0.600348	+	0.799739i	$0.704971\pi$
$594$	0	0
$595$	0	0
$596$	−76.0555	−3.11536
$597$	0	0
$598$	−45.6333	−1.86608
$599$	22.6056	0.923638	0.461819	−	0.886974i	$-0.347197\pi$
$599$	22.6056	0.923638	0.461819	+	0.886974i	$0.347197\pi$
$600$	0	0
$601$	−10.6333	−0.433742	−0.216871	−	0.976200i	$-0.569585\pi$
$601$	−10.6333	−0.433742	−0.216871	+	0.976200i	$0.569585\pi$
$602$	3.63331	0.148083
$603$	0	0
$604$	47.6333	1.93817
$605$	0	0
$606$	0	0
$607$	−24.6056	−0.998709	−0.499354	−	0.866398i	$-0.666429\pi$
$607$	−24.6056	−0.998709	−0.499354	+	0.866398i	$0.666429\pi$
$608$	19.1194	0.775395
$609$	0	0
$610$	0	0
$611$	60.8444	2.46150
$612$	0	0
$613$	28.8444	1.16501	0.582507	−	0.812825i	$-0.302072\pi$
$613$	28.8444	1.16501	0.582507	+	0.812825i	$0.302072\pi$
$614$	19.3944	0.782696
$615$	0	0
$616$	36.0000	1.45048
$617$	38.4500	1.54794	0.773969	−	0.633224i	$-0.218269\pi$
$617$	38.4500	1.54794	0.773969	+	0.633224i	$0.218269\pi$
$618$	0	0
$619$	35.6333	1.43222	0.716112	−	0.697986i	$-0.245920\pi$
$619$	35.6333	1.43222	0.716112	+	0.697986i	$0.245920\pi$
$620$	0	0
$621$	0	0
$622$	−18.0000	−0.721734
$623$	36.0000	1.44231
$624$	0	0
$625$	0	0
$626$	−45.2111	−1.80700
$627$	0	0
$628$	58.8444	2.34815
$629$	−3.21110	−0.128035
$630$	0	0
$631$	−6.02776	−0.239961	−0.119981	−	0.992776i	$-0.538283\pi$
$631$	−6.02776	−0.239961	−0.119981	+	0.992776i	$0.538283\pi$
$632$	34.8167	1.38493
$633$	0	0
$634$	17.5139	0.695565
$635$	0	0
$636$	0	0
$637$	1.39445	0.0552501
$638$	−14.7889	−0.585498
$639$	0	0
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	−13.0278	−0.513765	−0.256882	−	0.966443i	$-0.582695\pi$
$643$	−13.0278	−0.513765	−0.256882	+	0.966443i	$0.582695\pi$
$644$	−25.8167	−1.01732
$645$	0	0
$646$	−13.3305	−0.524483
$647$	23.7889	0.935238	0.467619	−	0.883930i	$-0.345112\pi$
$647$	23.7889	0.935238	0.467619	+	0.883930i	$0.345112\pi$
$648$	0	0
$649$	−6.42221	−0.252094
$650$	0	0
$651$	0	0
$652$	−6.60555	−0.258693
$653$	−23.2389	−0.909407	−0.454703	−	0.890643i	$-0.650255\pi$
$653$	−23.2389	−0.909407	−0.454703	+	0.890643i	$0.650255\pi$
$654$	0	0
$655$	0	0
$656$	1.39445	0.0544441
$657$	0	0
$658$	55.2666	2.15452
$659$	13.3944	0.521774	0.260887	−	0.965369i	$-0.415985\pi$
$659$	13.3944	0.521774	0.260887	+	0.965369i	$0.415985\pi$
$660$	0	0
$661$	−34.8444	−1.35529	−0.677645	−	0.735389i	$-0.736999\pi$
$661$	−34.8444	−1.35529	−0.677645	+	0.735389i	$0.736999\pi$
$662$	−67.2666	−2.61439
$663$	0	0
$664$	−9.00000	−0.349268
$665$	0	0
$666$	0	0
$667$	4.18335	0.161980
$668$	−9.90833	−0.383365
$669$	0	0
$670$	0	0
$671$	19.3944	0.748714
$672$	0	0
$673$	−25.0278	−0.964749	−0.482375	−	0.875965i	$-0.660226\pi$
$673$	−25.0278	−0.964749	−0.482375	+	0.875965i	$0.660226\pi$
$674$	15.2111	0.585910
$675$	0	0
$676$	101.175	3.89134
$677$	21.6333	0.831436	0.415718	−	0.909494i	$-0.363530\pi$
$677$	21.6333	0.831436	0.415718	+	0.909494i	$0.363530\pi$
$678$	0	0
$679$	−20.8444	−0.799935
$680$	0	0
$681$	0	0
$682$	−59.4500	−2.27646
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	0	0
$685$	0	0
$686$	43.2666	1.65193
$687$	0	0
$688$	−0.183346	−0.00699001
$689$	−10.6056	−0.404039
$690$	0	0
$691$	9.60555	0.365412	0.182706	−	0.983168i	$-0.441514\pi$
$691$	9.60555	0.365412	0.182706	+	0.983168i	$0.441514\pi$
$692$	−35.7250	−1.35806
$693$	0	0
$694$	−70.0555	−2.65927
$695$	0	0
$696$	0	0
$697$	7.39445	0.280085
$698$	73.3305	2.77560
$699$	0	0
$700$	0	0
$701$	−11.5778	−0.437287	−0.218644	−	0.975805i	$-0.570163\pi$
$701$	−11.5778	−0.437287	−0.218644	+	0.975805i	$0.570163\pi$
$702$	0	0
$703$	−7.21110	−0.271972
$704$	59.0278	2.22469
$705$	0	0
$706$	−49.8167	−1.87487
$707$	−31.2666	−1.17590
$708$	0	0
$709$	−22.8444	−0.857940	−0.428970	−	0.903319i	$-0.641123\pi$
$709$	−22.8444	−0.857940	−0.428970	+	0.903319i	$0.641123\pi$
$710$	0	0
$711$	0	0
$712$	−41.4500	−1.55340
$713$	16.8167	0.629789
$714$	0	0
$715$	0	0
$716$	70.0555	2.61810
$717$	0	0
$718$	−22.1833	−0.827875
$719$	−37.2666	−1.38981	−0.694905	−	0.719101i	$-0.744554\pi$
$719$	−37.2666	−1.38981	−0.694905	+	0.719101i	$0.744554\pi$
$720$	0	0
$721$	10.4222	0.388143
$722$	13.8167	0.514203
$723$	0	0
$724$	−23.1194	−0.859227
$725$	0	0
$726$	0	0
$727$	−35.6333	−1.32157	−0.660783	−	0.750577i	$-0.729776\pi$
$727$	−35.6333	−1.32157	−0.660783	+	0.750577i	$0.729776\pi$
$728$	51.6333	1.91366
$729$	0	0
$730$	0	0
$731$	−0.972244	−0.0359597
$732$	0	0
$733$	−32.0000	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$733$	−32.0000	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$734$	−6.00000	−0.221464
$735$	0	0
$736$	−15.9083	−0.586389
$737$	−3.63331	−0.133835
$738$	0	0
$739$	−6.02776	−0.221735	−0.110867	−	0.993835i	$-0.535363\pi$
$739$	−6.02776	−0.221735	−0.110867	+	0.993835i	$0.535363\pi$
$740$	0	0
$741$	0	0
$742$	−9.63331	−0.353650
$743$	−5.57779	−0.204629	−0.102315	−	0.994752i	$-0.532625\pi$
$743$	−5.57779	−0.204629	−0.102315	+	0.994752i	$0.532625\pi$
$744$	0	0
$745$	0	0
$746$	−58.0555	−2.12556
$747$	0	0
$748$	−24.4222	−0.892964
$749$	0	0
$750$	0	0
$751$	−30.0278	−1.09573	−0.547864	−	0.836567i	$-0.684559\pi$
$751$	−30.0278	−1.09573	−0.547864	+	0.836567i	$0.684559\pi$
$752$	−2.78890	−0.101701
$753$	0	0
$754$	−21.2111	−0.772463
$755$	0	0
$756$	0	0
$757$	−16.7889	−0.610203	−0.305101	−	0.952320i	$-0.598690\pi$
$757$	−16.7889	−0.610203	−0.305101	+	0.952320i	$0.598690\pi$
$758$	−49.7527	−1.80710
$759$	0	0
$760$	0	0
$761$	−11.4500	−0.415061	−0.207530	−	0.978229i	$-0.566543\pi$
$761$	−11.4500	−0.415061	−0.207530	+	0.978229i	$0.566543\pi$
$762$	0	0
$763$	−18.2389	−0.660291
$764$	41.0278	1.48433
$765$	0	0
$766$	−56.7250	−2.04956
$767$	−9.21110	−0.332594
$768$	0	0
$769$	41.0000	1.47850	0.739249	−	0.673432i	$-0.235181\pi$
$769$	41.0000	1.47850	0.739249	+	0.673432i	$0.235181\pi$
$770$	0	0
$771$	0	0
$772$	−0.605551	−0.0217943
$773$	−4.81665	−0.173243	−0.0866215	−	0.996241i	$-0.527607\pi$
$773$	−4.81665	−0.173243	−0.0866215	+	0.996241i	$0.527607\pi$
$774$	0	0
$775$	0	0
$776$	24.0000	0.861550
$777$	0	0
$778$	59.4500	2.13138
$779$	16.6056	0.594956
$780$	0	0
$781$	34.0555	1.21860
$782$	11.0917	0.396637
$783$	0	0
$784$	−0.0639167	−0.00228274
$785$	0	0
$786$	0	0
$787$	39.0278	1.39119	0.695595	−	0.718434i	$-0.255141\pi$
$787$	39.0278	1.39119	0.695595	+	0.718434i	$0.255141\pi$
$788$	−75.3583	−2.68453
$789$	0	0
$790$	0	0
$791$	−39.6333	−1.40920
$792$	0	0
$793$	27.8167	0.987798
$794$	−12.4222	−0.440848
$795$	0	0
$796$	−4.00000	−0.141776
$797$	41.6611	1.47571	0.737855	−	0.674959i	$-0.235839\pi$
$797$	41.6611	1.47571	0.737855	+	0.674959i	$0.235839\pi$
$798$	0	0
$799$	−14.7889	−0.523194
$800$	0	0
$801$	0	0
$802$	−69.0833	−2.43942
$803$	58.0555	2.04873
$804$	0	0
$805$	0	0
$806$	−85.2666	−3.00339
$807$	0	0
$808$	36.0000	1.26648
$809$	−43.3944	−1.52567	−0.762834	−	0.646595i	$-0.776193\pi$
$809$	−43.3944	−1.52567	−0.762834	+	0.646595i	$0.776193\pi$
$810$	0	0
$811$	13.5778	0.476781	0.238390	−	0.971169i	$-0.423380\pi$
$811$	13.5778	0.476781	0.238390	+	0.971169i	$0.423380\pi$
$812$	−12.0000	−0.421117
$813$	0	0
$814$	−21.2111	−0.743449
$815$	0	0
$816$	0	0
$817$	−2.18335	−0.0763856
$818$	−11.5139	−0.402573
$819$	0	0
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	17.8167	0.621050	0.310525	−	0.950565i	$-0.399495\pi$
$823$	17.8167	0.621050	0.310525	+	0.950565i	$0.399495\pi$
$824$	−12.0000	−0.418040
$825$	0	0
$826$	−8.36669	−0.291114
$827$	−48.2111	−1.67646	−0.838232	−	0.545314i	$-0.816411\pi$
$827$	−48.2111	−1.67646	−0.838232	+	0.545314i	$0.816411\pi$
$828$	0	0
$829$	−12.7889	−0.444177	−0.222088	−	0.975027i	$-0.571287\pi$
$829$	−12.7889	−0.444177	−0.222088	+	0.975027i	$0.571287\pi$
$830$	0	0
$831$	0	0
$832$	84.6611	2.93509
$833$	−0.338936	−0.0117434
$834$	0	0
$835$	0	0
$836$	−54.8444	−1.89683
$837$	0	0
$838$	53.0278	1.83181
$839$	−33.2111	−1.14657	−0.573287	−	0.819354i	$-0.694332\pi$
$839$	−33.2111	−1.14657	−0.573287	+	0.819354i	$0.694332\pi$
$840$	0	0
$841$	−27.0555	−0.932949
$842$	−12.4861	−0.430300
$843$	0	0
$844$	−29.1194	−1.00233
$845$	0	0
$846$	0	0
$847$	26.6056	0.914178
$848$	0.486122	0.0166935
$849$	0	0
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	−29.2111	−1.00017	−0.500085	−	0.865977i	$-0.666698\pi$
$853$	−29.2111	−1.00017	−0.500085	+	0.865977i	$0.666698\pi$
$854$	25.2666	0.864606
$855$	0	0
$856$	0	0
$857$	5.23886	0.178956	0.0894780	−	0.995989i	$-0.471480\pi$
$857$	5.23886	0.178956	0.0894780	+	0.995989i	$0.471480\pi$
$858$	0	0
$859$	−30.0278	−1.02453	−0.512267	−	0.858826i	$-0.671194\pi$
$859$	−30.0278	−1.02453	−0.512267	+	0.858826i	$0.671194\pi$
$860$	0	0
$861$	0	0
$862$	9.63331	0.328112
$863$	30.2111	1.02840	0.514199	−	0.857671i	$-0.328089\pi$
$863$	30.2111	1.02840	0.514199	+	0.857671i	$0.328089\pi$
$864$	0	0
$865$	0	0
$866$	51.2111	1.74022
$867$	0	0
$868$	−48.2389	−1.63733
$869$	53.4500	1.81317
$870$	0	0
$871$	−5.21110	−0.176571
$872$	21.0000	0.711150
$873$	0	0
$874$	24.9083	0.842537
$875$	0	0
$876$	0	0
$877$	−22.6611	−0.765210	−0.382605	−	0.923912i	$-0.624973\pi$
$877$	−22.6611	−0.765210	−0.382605	+	0.923912i	$0.624973\pi$
$878$	−63.5694	−2.14536
$879$	0	0
$880$	0	0
$881$	25.3944	0.855561	0.427780	−	0.903883i	$-0.359296\pi$
$881$	25.3944	0.855561	0.427780	+	0.903883i	$0.359296\pi$
$882$	0	0
$883$	−15.8167	−0.532273	−0.266136	−	0.963935i	$-0.585747\pi$
$883$	−15.8167	−0.532273	−0.266136	+	0.963935i	$0.585747\pi$
$884$	−35.0278	−1.17811
$885$	0	0
$886$	56.7250	1.90571
$887$	−30.6333	−1.02857	−0.514283	−	0.857621i	$-0.671942\pi$
$887$	−30.6333	−1.02857	−0.514283	+	0.857621i	$0.671942\pi$
$888$	0	0
$889$	50.0555	1.67881
$890$	0	0
$891$	0	0
$892$	33.0278	1.10585
$893$	−33.2111	−1.11137
$894$	0	0
$895$	0	0
$896$	49.2666	1.64588
$897$	0	0
$898$	88.0555	2.93845
$899$	7.81665	0.260700
$900$	0	0
$901$	2.57779	0.0858788
$902$	48.8444	1.62634
$903$	0	0
$904$	45.6333	1.51774
$905$	0	0
$906$	0	0
$907$	−1.57779	−0.0523898	−0.0261949	−	0.999657i	$-0.508339\pi$
$907$	−1.57779	−0.0523898	−0.0261949	+	0.999657i	$0.508339\pi$
$908$	−38.9361	−1.29214
$909$	0	0
$910$	0	0
$911$	5.02776	0.166577	0.0832885	−	0.996525i	$-0.473458\pi$
$911$	5.02776	0.166577	0.0832885	+	0.996525i	$0.473458\pi$
$912$	0	0
$913$	−13.8167	−0.457265
$914$	−30.4222	−1.00628
$915$	0	0
$916$	27.1194	0.896051
$917$	−15.6333	−0.516257
$918$	0	0
$919$	26.4222	0.871588	0.435794	−	0.900046i	$-0.356468\pi$
$919$	26.4222	0.871588	0.435794	+	0.900046i	$0.356468\pi$
$920$	0	0
$921$	0	0
$922$	49.8167	1.64062
$923$	48.8444	1.60773
$924$	0	0
$925$	0	0
$926$	−1.81665	−0.0596989
$927$	0	0
$928$	−7.39445	−0.242735
$929$	−19.3944	−0.636311	−0.318156	−	0.948039i	$-0.603063\pi$
$929$	−19.3944	−0.636311	−0.318156	+	0.948039i	$0.603063\pi$
$930$	0	0
$931$	−0.761141	−0.0249454
$932$	−59.4500	−1.94735
$933$	0	0
$934$	−28.1194	−0.920096
$935$	0	0
$936$	0	0
$937$	−41.2111	−1.34631	−0.673154	−	0.739502i	$-0.735061\pi$
$937$	−41.2111	−1.34631	−0.673154	+	0.739502i	$0.735061\pi$
$938$	−4.73338	−0.154550
$939$	0	0
$940$	0	0
$941$	0.422205	0.0137635	0.00688175	−	0.999976i	$-0.497809\pi$
$941$	0.422205	0.0137635	0.00688175	+	0.999976i	$0.497809\pi$
$942$	0	0
$943$	−13.8167	−0.449932
$944$	0.422205	0.0137416
$945$	0	0
$946$	−6.42221	−0.208804
$947$	39.0000	1.26733	0.633665	−	0.773608i	$-0.281550\pi$
$947$	39.0000	1.26733	0.633665	+	0.773608i	$0.281550\pi$
$948$	0	0
$949$	83.2666	2.70295
$950$	0	0
$951$	0	0
$952$	−12.5500	−0.406749
$953$	−30.8444	−0.999148	−0.499574	−	0.866271i	$-0.666510\pi$
$953$	−30.8444	−0.999148	−0.499574	+	0.866271i	$0.666510\pi$
$954$	0	0
$955$	0	0
$956$	50.2389	1.62484
$957$	0	0
$958$	−87.0833	−2.81353
$959$	43.8167	1.41491
$960$	0	0
$961$	0.422205	0.0136195
$962$	−30.4222	−0.980851
$963$	0	0
$964$	45.5416	1.46680
$965$	0	0
$966$	0	0
$967$	−50.0000	−1.60789	−0.803946	−	0.594703i	$-0.797270\pi$
$967$	−50.0000	−1.60789	−0.803946	+	0.594703i	$0.797270\pi$
$968$	−30.6333	−0.984592
$969$	0	0
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	−10.4222	−0.334121
$974$	−68.6611	−2.20004
$975$	0	0
$976$	−1.27502	−0.0408124
$977$	−15.2111	−0.486646	−0.243323	−	0.969945i	$-0.578238\pi$
$977$	−15.2111	−0.486646	−0.243323	+	0.969945i	$0.578238\pi$
$978$	0	0
$979$	−63.6333	−2.03373
$980$	0	0
$981$	0	0
$982$	62.6611	1.99959
$983$	−42.6333	−1.35979	−0.679896	−	0.733309i	$-0.737975\pi$
$983$	−42.6333	−1.35979	−0.679896	+	0.733309i	$0.737975\pi$
$984$	0	0
$985$	0	0
$986$	5.15559	0.164187
$987$	0	0
$988$	−78.6611	−2.50254
$989$	1.81665	0.0577662
$990$	0	0
$991$	−29.1833	−0.927040	−0.463520	−	0.886087i	$-0.653414\pi$
$991$	−29.1833	−0.927040	−0.463520	+	0.886087i	$0.653414\pi$
$992$	−29.7250	−0.943769
$993$	0	0
$994$	44.3667	1.40723
$995$	0	0
$996$	0	0
$997$	39.8722	1.26276	0.631382	−	0.775472i	$-0.282488\pi$
$997$	39.8722	1.26276	0.631382	+	0.775472i	$0.282488\pi$
$998$	46.9638	1.48661
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.2.a.k.1.1		2
3.2	odd	2		675.2.a.p.1.2		2
5.2	odd	4		675.2.b.h.649.1		4
5.3	odd	4		675.2.b.h.649.4		4
5.4	even	2		135.2.a.d.1.2	yes	2
15.2	even	4		675.2.b.i.649.4		4
15.8	even	4		675.2.b.i.649.1		4
15.14	odd	2		135.2.a.c.1.1	✓	2
20.19	odd	2		2160.2.a.y.1.2		2
35.34	odd	2		6615.2.a.v.1.2		2
40.19	odd	2		8640.2.a.cy.1.2		2
40.29	even	2		8640.2.a.df.1.1		2
45.4	even	6		405.2.e.j.136.1		4
45.14	odd	6		405.2.e.k.136.2		4
45.29	odd	6		405.2.e.k.271.2		4
45.34	even	6		405.2.e.j.271.1		4
60.59	even	2		2160.2.a.ba.1.2		2
105.104	even	2		6615.2.a.p.1.1		2
120.29	odd	2		8640.2.a.cr.1.1		2
120.59	even	2		8640.2.a.ck.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.a.c.1.1	✓	2	15.14	odd	2
135.2.a.d.1.2	yes	2	5.4	even	2
405.2.e.j.136.1		4	45.4	even	6
405.2.e.j.271.1		4	45.34	even	6
405.2.e.k.136.2		4	45.14	odd	6
405.2.e.k.271.2		4	45.29	odd	6
675.2.a.k.1.1		2	1.1	even	1	trivial
675.2.a.p.1.2		2	3.2	odd	2
675.2.b.h.649.1		4	5.2	odd	4
675.2.b.h.649.4		4	5.3	odd	4
675.2.b.i.649.1		4	15.8	even	4
675.2.b.i.649.4		4	15.2	even	4
2160.2.a.y.1.2		2	20.19	odd	2
2160.2.a.ba.1.2		2	60.59	even	2
6615.2.a.p.1.1		2	105.104	even	2
6615.2.a.v.1.2		2	35.34	odd	2
8640.2.a.ck.1.2		2	120.59	even	2
8640.2.a.cr.1.1		2	120.29	odd	2
8640.2.a.cy.1.2		2	40.19	odd	2
8640.2.a.df.1.1		2	40.29	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30278 q^{2} +3.30278 q^{4} +2.60555 q^{7} -3.00000 q^{8} +O(q^{10})\)	\(q-2.30278 q^{2} +3.30278 q^{4} +2.60555 q^{7} -3.00000 q^{8} -4.60555 q^{11} -6.60555 q^{13} -6.00000 q^{14} +0.302776 q^{16} +1.60555 q^{17} +3.60555 q^{19} +10.6056 q^{22} -3.00000 q^{23} +15.2111 q^{26} +8.60555 q^{28} -1.39445 q^{29} -5.60555 q^{31} +5.30278 q^{32} -3.69722 q^{34} -2.00000 q^{37} -8.30278 q^{38} +4.60555 q^{41} -0.605551 q^{43} -15.2111 q^{44} +6.90833 q^{46} -9.21110 q^{47} -0.211103 q^{49} -21.8167 q^{52} +1.60555 q^{53} -7.81665 q^{56} +3.21110 q^{58} +1.39445 q^{59} -4.21110 q^{61} +12.9083 q^{62} -12.8167 q^{64} +0.788897 q^{67} +5.30278 q^{68} -7.39445 q^{71} -12.6056 q^{73} +4.60555 q^{74} +11.9083 q^{76} -12.0000 q^{77} -11.6056 q^{79} -10.6056 q^{82} +3.00000 q^{83} +1.39445 q^{86} +13.8167 q^{88} +13.8167 q^{89} -17.2111 q^{91} -9.90833 q^{92} +21.2111 q^{94} -8.00000 q^{97} +0.486122 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 3 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - q^2 + 3 * q^4 - 2 * q^7 - 6 * q^8	\( 2 q - q^{2} + 3 q^{4} - 2 q^{7} - 6 q^{8} - 2 q^{11} - 6 q^{13} - 12 q^{14} - 3 q^{16} - 4 q^{17} + 14 q^{22} - 6 q^{23} + 16 q^{26} + 10 q^{28} - 10 q^{29} - 4 q^{31} + 7 q^{32} - 11 q^{34} - 4 q^{37} - 13 q^{38} + 2 q^{41} + 6 q^{43} - 16 q^{44} + 3 q^{46} - 4 q^{47} + 14 q^{49} - 22 q^{52} - 4 q^{53} + 6 q^{56} - 8 q^{58} + 10 q^{59} + 6 q^{61} + 15 q^{62} - 4 q^{64} + 16 q^{67} + 7 q^{68} - 22 q^{71} - 18 q^{73} + 2 q^{74} + 13 q^{76} - 24 q^{77} - 16 q^{79} - 14 q^{82} + 6 q^{83} + 10 q^{86} + 6 q^{88} + 6 q^{89} - 20 q^{91} - 9 q^{92} + 28 q^{94} - 16 q^{97} + 19 q^{98}+O(q^{100}) \) 2 * q - q^2 + 3 * q^4 - 2 * q^7 - 6 * q^8 - 2 * q^11 - 6 * q^13 - 12 * q^14 - 3 * q^16 - 4 * q^17 + 14 * q^22 - 6 * q^23 + 16 * q^26 + 10 * q^28 - 10 * q^29 - 4 * q^31 + 7 * q^32 - 11 * q^34 - 4 * q^37 - 13 * q^38 + 2 * q^41 + 6 * q^43 - 16 * q^44 + 3 * q^46 - 4 * q^47 + 14 * q^49 - 22 * q^52 - 4 * q^53 + 6 * q^56 - 8 * q^58 + 10 * q^59 + 6 * q^61 + 15 * q^62 - 4 * q^64 + 16 * q^67 + 7 * q^68 - 22 * q^71 - 18 * q^73 + 2 * q^74 + 13 * q^76 - 24 * q^77 - 16 * q^79 - 14 * q^82 + 6 * q^83 + 10 * q^86 + 6 * q^88 + 6 * q^89 - 20 * q^91 - 9 * q^92 + 28 * q^94 - 16 * q^97 + 19 * q^98

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