LMFDB - Embedded newform 4275.2.a.bh.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
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			$-0.523976$ 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)

$f(q)$	$=$	$q-0.523976 q^{2} -1.72545 q^{4} -3.20147 q^{7} +1.95205 q^{8} -2.20147 q^{11} +3.04795 q^{13} +1.67750 q^{14} +2.42807 q^{16} +1.00000 q^{19} +1.15352 q^{22} -3.24943 q^{23} -1.59706 q^{26} +5.52398 q^{28} -3.00000 q^{29} -1.24943 q^{31} -5.17635 q^{32} -3.45090 q^{37} -0.523976 q^{38} -8.55646 q^{41} +8.00000 q^{43} +3.79853 q^{44} +1.70262 q^{46} -3.35499 q^{47} +3.24943 q^{49} -5.25909 q^{52} -0.904094 q^{53} -6.24943 q^{56} +1.57193 q^{58} -11.2015 q^{59} +3.40294 q^{61} +0.654669 q^{62} -2.14386 q^{64} -2.29738 q^{67} -7.29738 q^{71} +1.09591 q^{73} +1.80819 q^{74} -1.72545 q^{76} +7.04795 q^{77} +13.5565 q^{79} +4.48339 q^{82} +8.20147 q^{83} -4.19181 q^{86} -4.29738 q^{88} +7.40294 q^{89} -9.75794 q^{91} +5.60672 q^{92} +1.75794 q^{94} -7.14386 q^{97} -1.70262 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} + 9 q^{8} + 3 q^{11} + 6 q^{13} - 3 q^{14} + 12 q^{16} + 3 q^{19} - 3 q^{22} + 3 q^{23} - 24 q^{26} + 15 q^{28} - 9 q^{29} + 9 q^{31} + 18 q^{32} + 12 q^{37} + 24 q^{43} + 21 q^{44} + 21 q^{46}+ \cdots - 21 q^{98}+O(q^{100}) $ 3 * q + 6 * q^4 + 9 * q^8 + 3 * q^11 + 6 * q^13 - 3 * q^14 + 12 * q^16 + 3 * q^19 - 3 * q^22 + 3 * q^23 - 24 * q^26 + 15 * q^28 - 9 * q^29 + 9 * q^31 + 18 * q^32 + 12 * q^37 + 24 * q^43 + 21 * q^44 + 21 * q^46 + 6 * q^47 - 3 * q^49 - 6 * q^52 - 9 * q^53 - 6 * q^56 - 24 * q^59 - 9 * q^61 + 21 * q^62 + 3 * q^64 + 9 * q^67 - 6 * q^71 - 3 * q^73 + 18 * q^74 + 6 * q^76 + 18 * q^77 + 15 * q^79 - 33 * q^82 + 15 * q^83 + 3 * q^88 + 3 * q^89 + 6 * q^91 + 39 * q^92 - 30 * q^94 - 12 * q^97 - 21 * 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 $

$2$	−0.523976	−0.370507	−0.185254	−	0.982691i	$-0.559311\pi$
$2$	−0.523976	−0.370507	−0.185254	+	0.982691i	$0.559311\pi$
$3$	0	0
$3$	0	0
$4$	−1.72545	−0.862724
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.20147	−1.21004	−0.605021	−	0.796209i	$-0.706835\pi$
$7$	−3.20147	−1.21004	−0.605021	+	0.796209i	$0.706835\pi$
$8$	1.95205	0.690153
$9$	0	0
$10$	0	0
$11$	−2.20147	−0.663769	−0.331884	−	0.943320i	$-0.607684\pi$
$11$	−2.20147	−0.663769	−0.331884	+	0.943320i	$0.607684\pi$
$12$	0	0
$13$	3.04795	0.845350	0.422675	−	0.906281i	$-0.361091\pi$
$13$	3.04795	0.845350	0.422675	+	0.906281i	$0.361091\pi$
$14$	1.67750	0.448330
$15$	0	0
$16$	2.42807	0.607018
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	1.15352	0.245931
$23$	−3.24943	−0.677552	−0.338776	−	0.940867i	$-0.610013\pi$
$23$	−3.24943	−0.677552	−0.338776	+	0.940867i	$0.610013\pi$
$24$	0	0
$25$	0	0
$26$	−1.59706	−0.313208
$27$	0	0
$28$	5.52398	1.04393
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−1.24943	−0.224403	−0.112202	−	0.993685i	$-0.535790\pi$
$31$	−1.24943	−0.224403	−0.112202	+	0.993685i	$0.535790\pi$
$32$	−5.17635	−0.915057
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.45090	−0.567324	−0.283662	−	0.958924i	$-0.591549\pi$
$37$	−3.45090	−0.567324	−0.283662	+	0.958924i	$0.591549\pi$
$38$	−0.523976	−0.0850002
$39$	0	0
$40$	0	0
$41$	−8.55646	−1.33630	−0.668148	−	0.744029i	$-0.732913\pi$
$41$	−8.55646	−1.33630	−0.668148	+	0.744029i	$0.732913\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	3.79853	0.572650
$45$	0	0
$46$	1.70262	0.251038
$47$	−3.35499	−0.489376	−0.244688	−	0.969602i	$-0.578686\pi$
$47$	−3.35499	−0.489376	−0.244688	+	0.969602i	$0.578686\pi$
$48$	0	0
$49$	3.24943	0.464204
$50$	0	0
$51$	0	0
$52$	−5.25909	−0.729304
$53$	−0.904094	−0.124187	−0.0620935	−	0.998070i	$-0.519778\pi$
$53$	−0.904094	−0.124187	−0.0620935	+	0.998070i	$0.519778\pi$
$54$	0	0
$55$	0	0
$56$	−6.24943	−0.835115
$57$	0	0
$58$	1.57193	0.206404
$59$	−11.2015	−1.45831	−0.729154	−	0.684350i	$-0.760086\pi$
$59$	−11.2015	−1.45831	−0.729154	+	0.684350i	$0.760086\pi$
$60$	0	0
$61$	3.40294	0.435702	0.217851	−	0.975982i	$-0.430095\pi$
$61$	3.40294	0.435702	0.217851	+	0.975982i	$0.430095\pi$
$62$	0.654669	0.0831431
$63$	0	0
$64$	−2.14386	−0.267982
$65$	0	0
$66$	0	0
$67$	−2.29738	−0.280669	−0.140335	−	0.990104i	$-0.544818\pi$
$67$	−2.29738	−0.280669	−0.140335	+	0.990104i	$0.544818\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.29738	−0.866039	−0.433020	−	0.901384i	$-0.642552\pi$
$71$	−7.29738	−0.866039	−0.433020	+	0.901384i	$0.642552\pi$
$72$	0	0
$73$	1.09591	0.128266	0.0641330	−	0.997941i	$-0.479572\pi$
$73$	1.09591	0.128266	0.0641330	+	0.997941i	$0.479572\pi$
$74$	1.80819	0.210198
$75$	0	0
$76$	−1.72545	−0.197923
$77$	7.04795	0.803189
$78$	0	0
$79$	13.5565	1.52522	0.762611	−	0.646858i	$-0.223917\pi$
$79$	13.5565	1.52522	0.762611	+	0.646858i	$0.223917\pi$
$80$	0	0
$81$	0	0
$82$	4.48339	0.495107
$83$	8.20147	0.900229	0.450114	−	0.892971i	$-0.351383\pi$
$83$	8.20147	0.900229	0.450114	+	0.892971i	$0.351383\pi$
$84$	0	0
$85$	0	0
$86$	−4.19181	−0.452015
$87$	0	0
$88$	−4.29738	−0.458102
$89$	7.40294	0.784711	0.392355	−	0.919814i	$-0.371660\pi$
$89$	7.40294	0.784711	0.392355	+	0.919814i	$0.371660\pi$
$90$	0	0
$91$	−9.75794	−1.02291
$92$	5.60672	0.584541
$93$	0	0
$94$	1.75794	0.181317
$95$	0	0
$96$	0	0
$97$	−7.14386	−0.725349	−0.362674	−	0.931916i	$-0.618136\pi$
$97$	−7.14386	−0.725349	−0.362674	+	0.931916i	$0.618136\pi$
$98$	−1.70262	−0.171991
$99$	0	0
$100$	0	0
$101$	−13.7579	−1.36897	−0.684483	−	0.729029i	$-0.739972\pi$
$101$	−13.7579	−1.36897	−0.684483	+	0.729029i	$0.739972\pi$
$102$	0	0
$103$	−6.20147	−0.611049	−0.305525	−	0.952184i	$-0.598832\pi$
$103$	−6.20147	−0.611049	−0.305525	+	0.952184i	$0.598832\pi$
$104$	5.94975	0.583421
$105$	0	0
$106$	0.473724	0.0460122
$107$	15.6044	1.50854	0.754268	−	0.656567i	$-0.227992\pi$
$107$	15.6044	1.50854	0.754268	+	0.656567i	$0.227992\pi$
$108$	0	0
$109$	7.45090	0.713667	0.356833	−	0.934168i	$-0.383856\pi$
$109$	7.45090	0.713667	0.356833	+	0.934168i	$0.383856\pi$
$110$	0	0
$111$	0	0
$112$	−7.77340	−0.734517
$113$	15.2494	1.43455	0.717273	−	0.696793i	$-0.245390\pi$
$113$	15.2494	1.43455	0.717273	+	0.696793i	$0.245390\pi$
$114$	0	0
$115$	0	0
$116$	5.17635	0.480612
$117$	0	0
$118$	5.86931	0.540314
$119$	0	0
$120$	0	0
$121$	−6.15352	−0.559411
$122$	−1.78306	−0.161431
$123$	0	0
$124$	2.15582	0.193598
$125$	0	0
$126$	0	0
$127$	14.0553	1.24721	0.623604	−	0.781741i	$-0.285668\pi$
$127$	14.0553	1.24721	0.623604	+	0.781741i	$0.285668\pi$
$128$	11.4760	1.01435
$129$	0	0
$130$	0	0
$131$	−4.34763	−0.379854	−0.189927	−	0.981798i	$-0.560825\pi$
$131$	−4.34763	−0.379854	−0.189927	+	0.981798i	$0.560825\pi$
$132$	0	0
$133$	−3.20147	−0.277603
$134$	1.20377	0.103990
$135$	0	0
$136$	0	0
$137$	19.7077	1.68374	0.841871	−	0.539680i	$-0.181455\pi$
$137$	19.7077	1.68374	0.841871	+	0.539680i	$0.181455\pi$
$138$	0	0
$139$	−15.9115	−1.34959	−0.674796	−	0.738004i	$-0.735768\pi$
$139$	−15.9115	−1.34959	−0.674796	+	0.738004i	$0.735768\pi$
$140$	0	0
$141$	0	0
$142$	3.82365	0.320874
$143$	−6.70998	−0.561117
$144$	0	0
$145$	0	0
$146$	−0.574229	−0.0475235
$147$	0	0
$148$	5.95435	0.489444
$149$	15.8538	1.29880	0.649399	−	0.760448i	$-0.275021\pi$
$149$	15.8538	1.29880	0.649399	+	0.760448i	$0.275021\pi$
$150$	0	0
$151$	18.9018	1.53821	0.769103	−	0.639125i	$-0.220703\pi$
$151$	18.9018	1.53821	0.769103	+	0.639125i	$0.220703\pi$
$152$	1.95205	0.158332
$153$	0	0
$154$	−3.69296	−0.297587
$155$	0	0
$156$	0	0
$157$	14.4606	1.15408	0.577039	−	0.816717i	$-0.304208\pi$
$157$	14.4606	1.15408	0.577039	+	0.816717i	$0.304208\pi$
$158$	−7.10327	−0.565106
$159$	0	0
$160$	0	0
$161$	10.4029	0.819867
$162$	0	0
$163$	22.0074	1.72375	0.861875	−	0.507121i	$-0.169290\pi$
$163$	22.0074	1.72375	0.861875	+	0.507121i	$0.169290\pi$
$164$	14.7637	1.15285
$165$	0	0
$166$	−4.29738	−0.333541
$167$	−3.10557	−0.240316	−0.120158	−	0.992755i	$-0.538340\pi$
$167$	−3.10557	−0.240316	−0.120158	+	0.992755i	$0.538340\pi$
$168$	0	0
$169$	−3.70998	−0.285383
$170$	0	0
$171$	0	0
$172$	−13.8036	−1.05251
$173$	15.9977	1.21628	0.608141	−	0.793829i	$-0.291915\pi$
$173$	15.9977	1.21628	0.608141	+	0.793829i	$0.291915\pi$
$174$	0	0
$175$	0	0
$176$	−5.34533	−0.402919
$177$	0	0
$178$	−3.87897	−0.290741
$179$	−2.89443	−0.216340	−0.108170	−	0.994132i	$-0.534499\pi$
$179$	−2.89443	−0.216340	−0.108170	+	0.994132i	$0.534499\pi$
$180$	0	0
$181$	6.90179	0.513006	0.256503	−	0.966543i	$-0.417430\pi$
$181$	6.90179	0.513006	0.256503	+	0.966543i	$0.417430\pi$
$182$	5.11293	0.378995
$183$	0	0
$184$	−6.34303	−0.467614
$185$	0	0
$186$	0	0
$187$	0	0
$188$	5.78887	0.422196
$189$	0	0
$190$	0	0
$191$	9.24943	0.669265	0.334632	−	0.942349i	$-0.391388\pi$
$191$	9.24943	0.669265	0.334632	+	0.942349i	$0.391388\pi$
$192$	0	0
$193$	−1.19411	−0.0859540	−0.0429770	−	0.999076i	$-0.513684\pi$
$193$	−1.19411	−0.0859540	−0.0429770	+	0.999076i	$0.513684\pi$
$194$	3.74321	0.268747
$195$	0	0
$196$	−5.60672	−0.400480
$197$	−17.4509	−1.24332	−0.621662	−	0.783285i	$-0.713542\pi$
$197$	−17.4509	−1.24332	−0.621662	+	0.783285i	$0.713542\pi$
$198$	0	0
$199$	2.79853	0.198382	0.0991912	−	0.995068i	$-0.468374\pi$
$199$	2.79853	0.198382	0.0991912	+	0.995068i	$0.468374\pi$
$200$	0	0
$201$	0	0
$202$	7.20883	0.507212
$203$	9.60442	0.674098
$204$	0	0
$205$	0	0
$206$	3.24943	0.226398
$207$	0	0
$208$	7.40065	0.513142
$209$	−2.20147	−0.152279
$210$	0	0
$211$	−26.2471	−1.80693	−0.903463	−	0.428665i	$-0.858984\pi$
$211$	−26.2471	−1.80693	−0.903463	+	0.428665i	$0.858984\pi$
$212$	1.55997	0.107139
$213$	0	0
$214$	−8.17635	−0.558924
$215$	0	0
$216$	0	0
$217$	4.00000	0.271538
$218$	−3.90409	−0.264419
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−13.2494	−0.887247	−0.443624	−	0.896213i	$-0.646307\pi$
$223$	−13.2494	−0.887247	−0.443624	+	0.896213i	$0.646307\pi$
$224$	16.5719	1.10726
$225$	0	0
$226$	−7.99034	−0.531509
$227$	−7.29738	−0.484344	−0.242172	−	0.970233i	$-0.577860\pi$
$227$	−7.29738	−0.484344	−0.242172	+	0.970233i	$0.577860\pi$
$228$	0	0
$229$	15.8635	1.04829	0.524145	−	0.851629i	$-0.324385\pi$
$229$	15.8635	1.04829	0.524145	+	0.851629i	$0.324385\pi$
$230$	0	0
$231$	0	0
$232$	−5.85614	−0.384475
$233$	8.64501	0.566353	0.283177	−	0.959068i	$-0.408612\pi$
$233$	8.64501	0.566353	0.283177	+	0.959068i	$0.408612\pi$
$234$	0	0
$235$	0	0
$236$	19.3276	1.25812
$237$	0	0
$238$	0	0
$239$	−25.7579	−1.66614	−0.833071	−	0.553166i	$-0.813420\pi$
$239$	−25.7579	−1.66614	−0.833071	+	0.553166i	$0.813420\pi$
$240$	0	0
$241$	9.59706	0.618201	0.309100	−	0.951029i	$-0.399972\pi$
$241$	9.59706	0.618201	0.309100	+	0.951029i	$0.399972\pi$
$242$	3.22430	0.207266
$243$	0	0
$244$	−5.87161	−0.375891
$245$	0	0
$246$	0	0
$247$	3.04795	0.193937
$248$	−2.43894	−0.154873
$249$	0	0
$250$	0	0
$251$	−17.6118	−1.11165	−0.555823	−	0.831301i	$-0.687597\pi$
$251$	−17.6118	−1.11165	−0.555823	+	0.831301i	$0.687597\pi$
$252$	0	0
$253$	7.15352	0.449738
$254$	−7.36465	−0.462099
$255$	0	0
$256$	−1.72545	−0.107841
$257$	−0.692961	−0.0432257	−0.0216129	−	0.999766i	$-0.506880\pi$
$257$	−0.692961	−0.0432257	−0.0216129	+	0.999766i	$0.506880\pi$
$258$	0	0
$259$	11.0480	0.686486
$260$	0	0
$261$	0	0
$262$	2.27806	0.140739
$263$	23.5062	1.44946	0.724728	−	0.689036i	$-0.241966\pi$
$263$	23.5062	1.44946	0.724728	+	0.689036i	$0.241966\pi$
$264$	0	0
$265$	0	0
$266$	1.67750	0.102854
$267$	0	0
$268$	3.96401	0.242140
$269$	10.6141	0.647152	0.323576	−	0.946202i	$-0.395115\pi$
$269$	10.6141	0.647152	0.323576	+	0.946202i	$0.395115\pi$
$270$	0	0
$271$	15.2974	0.929250	0.464625	−	0.885508i	$-0.346189\pi$
$271$	15.2974	0.929250	0.464625	+	0.885508i	$0.346189\pi$
$272$	0	0
$273$	0	0
$274$	−10.3264	−0.623838
$275$	0	0
$276$	0	0
$277$	19.5948	1.17733	0.588667	−	0.808375i	$-0.299653\pi$
$277$	19.5948	1.17733	0.588667	+	0.808375i	$0.299653\pi$
$278$	8.33723	0.500034
$279$	0	0
$280$	0	0
$281$	30.5542	1.82271	0.911354	−	0.411623i	$-0.135038\pi$
$281$	30.5542	1.82271	0.911354	+	0.411623i	$0.135038\pi$
$282$	0	0
$283$	−8.61408	−0.512054	−0.256027	−	0.966670i	$-0.582414\pi$
$283$	−8.61408	−0.512054	−0.256027	+	0.966670i	$0.582414\pi$
$284$	12.5913	0.747153
$285$	0	0
$286$	3.51587	0.207898
$287$	27.3933	1.61697
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	−1.89093	−0.110658
$293$	29.5565	1.72671	0.863354	−	0.504600i	$-0.168360\pi$
$293$	29.5565	1.72671	0.863354	+	0.504600i	$0.168360\pi$
$294$	0	0
$295$	0	0
$296$	−6.73631	−0.391540
$297$	0	0
$298$	−8.30704	−0.481214
$299$	−9.90409	−0.572769
$300$	0	0
$301$	−25.6118	−1.47624
$302$	−9.90409	−0.569917
$303$	0	0
$304$	2.42807	0.139259
$305$	0	0
$306$	0	0
$307$	16.9115	0.965188	0.482594	−	0.875844i	$-0.339695\pi$
$307$	16.9115	0.965188	0.482594	+	0.875844i	$0.339695\pi$
$308$	−12.1609	−0.692931
$309$	0	0
$310$	0	0
$311$	−14.0959	−0.799305	−0.399653	−	0.916667i	$-0.630869\pi$
$311$	−14.0959	−0.799305	−0.399653	+	0.916667i	$0.630869\pi$
$312$	0	0
$313$	−4.05531	−0.229220	−0.114610	−	0.993411i	$-0.536562\pi$
$313$	−4.05531	−0.229220	−0.114610	+	0.993411i	$0.536562\pi$
$314$	−7.57699	−0.427594
$315$	0	0
$316$	−23.3910	−1.31585
$317$	−21.9977	−1.23551	−0.617757	−	0.786369i	$-0.711958\pi$
$317$	−21.9977	−1.23551	−0.617757	+	0.786369i	$0.711958\pi$
$318$	0	0
$319$	6.60442	0.369776
$320$	0	0
$321$	0	0
$322$	−5.45090	−0.303767
$323$	0	0
$324$	0	0
$325$	0	0
$326$	−11.5313	−0.638662
$327$	0	0
$328$	−16.7026	−0.922248
$329$	10.7409	0.592166
$330$	0	0
$331$	12.4583	0.684768	0.342384	−	0.939560i	$-0.388766\pi$
$331$	12.4583	0.684768	0.342384	+	0.939560i	$0.388766\pi$
$332$	−14.1512	−0.776649
$333$	0	0
$334$	1.62724	0.0890388
$335$	0	0
$336$	0	0
$337$	20.4989	1.11664	0.558322	−	0.829624i	$-0.311445\pi$
$337$	20.4989	1.11664	0.558322	+	0.829624i	$0.311445\pi$
$338$	1.94394	0.105737
$339$	0	0
$340$	0	0
$341$	2.75057	0.148952
$342$	0	0
$343$	12.0074	0.648337
$344$	15.6164	0.841979
$345$	0	0
$346$	−8.38242	−0.450642
$347$	4.45320	0.239060	0.119530	−	0.992831i	$-0.461861\pi$
$347$	4.45320	0.239060	0.119530	+	0.992831i	$0.461861\pi$
$348$	0	0
$349$	−9.30704	−0.498194	−0.249097	−	0.968478i	$-0.580134\pi$
$349$	−9.30704	−0.498194	−0.249097	+	0.968478i	$0.580134\pi$
$350$	0	0
$351$	0	0
$352$	11.3956	0.607387
$353$	16.6141	0.884278	0.442139	−	0.896947i	$-0.354220\pi$
$353$	16.6141	0.884278	0.442139	+	0.896947i	$0.354220\pi$
$354$	0	0
$355$	0	0
$356$	−12.7734	−0.676989
$357$	0	0
$358$	1.51661	0.0801556
$359$	−29.9497	−1.58069	−0.790344	−	0.612664i	$-0.790098\pi$
$359$	−29.9497	−1.58069	−0.790344	+	0.612664i	$0.790098\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−3.61638	−0.190073
$363$	0	0
$364$	16.8368	0.882489
$365$	0	0
$366$	0	0
$367$	−28.1106	−1.46736	−0.733681	−	0.679494i	$-0.762200\pi$
$367$	−28.1106	−1.46736	−0.733681	+	0.679494i	$0.762200\pi$
$368$	−7.88983	−0.411286
$369$	0	0
$370$	0	0
$371$	2.89443	0.150271
$372$	0	0
$373$	10.5948	0.548576	0.274288	−	0.961648i	$-0.411558\pi$
$373$	10.5948	0.548576	0.274288	+	0.961648i	$0.411558\pi$
$374$	0	0
$375$	0	0
$376$	−6.54910	−0.337744
$377$	−9.14386	−0.470933
$378$	0	0
$379$	3.04795	0.156563	0.0782814	−	0.996931i	$-0.475057\pi$
$379$	3.04795	0.156563	0.0782814	+	0.996931i	$0.475057\pi$
$380$	0	0
$381$	0	0
$382$	−4.84648	−0.247968
$383$	16.6021	0.848329	0.424164	−	0.905585i	$-0.360568\pi$
$383$	16.6021	0.848329	0.424164	+	0.905585i	$0.360568\pi$
$384$	0	0
$385$	0	0
$386$	0.625686	0.0318466
$387$	0	0
$388$	12.3264	0.625776
$389$	−7.25909	−0.368050	−0.184025	−	0.982922i	$-0.558913\pi$
$389$	−7.25909	−0.368050	−0.184025	+	0.982922i	$0.558913\pi$
$390$	0	0
$391$	0	0
$392$	6.34303	0.320371
$393$	0	0
$394$	9.14386	0.460661
$395$	0	0
$396$	0	0
$397$	20.2664	1.01714	0.508572	−	0.861020i	$-0.330174\pi$
$397$	20.2664	1.01714	0.508572	+	0.861020i	$0.330174\pi$
$398$	−1.46636	−0.0735021
$399$	0	0
$400$	0	0
$401$	3.24943	0.162269	0.0811343	−	0.996703i	$-0.474146\pi$
$401$	3.24943	0.162269	0.0811343	+	0.996703i	$0.474146\pi$
$402$	0	0
$403$	−3.80819	−0.189699
$404$	23.7386	1.18104
$405$	0	0
$406$	−5.03249	−0.249758
$407$	7.59706	0.376572
$408$	0	0
$409$	34.0457	1.68345	0.841725	−	0.539907i	$-0.181541\pi$
$409$	34.0457	1.68345	0.841725	+	0.539907i	$0.181541\pi$
$410$	0	0
$411$	0	0
$412$	10.7003	0.527167
$413$	35.8612	1.76461
$414$	0	0
$415$	0	0
$416$	−15.7773	−0.773544
$417$	0	0
$418$	1.15352	0.0564205
$419$	−26.2568	−1.28273	−0.641364	−	0.767237i	$-0.721631\pi$
$419$	−26.2568	−1.28273	−0.641364	+	0.767237i	$0.721631\pi$
$420$	0	0
$421$	−11.7579	−0.573047	−0.286523	−	0.958073i	$-0.592500\pi$
$421$	−11.7579	−0.573047	−0.286523	+	0.958073i	$0.592500\pi$
$422$	13.7529	0.669479
$423$	0	0
$424$	−1.76483	−0.0857080
$425$	0	0
$426$	0	0
$427$	−10.8944	−0.527219
$428$	−26.9246	−1.30145
$429$	0	0
$430$	0	0
$431$	27.0051	1.30079	0.650394	−	0.759597i	$-0.274604\pi$
$431$	27.0051	1.30079	0.650394	+	0.759597i	$0.274604\pi$
$432$	0	0
$433$	14.6597	0.704502	0.352251	−	0.935906i	$-0.385416\pi$
$433$	14.6597	0.704502	0.352251	+	0.935906i	$0.385416\pi$
$434$	−2.09591	−0.100607
$435$	0	0
$436$	−12.8561	−0.615698
$437$	−3.24943	−0.155441
$438$	0	0
$439$	−22.7653	−1.08653	−0.543264	−	0.839562i	$-0.682812\pi$
$439$	−22.7653	−1.08653	−0.543264	+	0.839562i	$0.682812\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−39.3601	−1.87005	−0.935026	−	0.354578i	$-0.884624\pi$
$443$	−39.3601	−1.87005	−0.935026	+	0.354578i	$0.884624\pi$
$444$	0	0
$445$	0	0
$446$	6.94239	0.328732
$447$	0	0
$448$	6.86350	0.324270
$449$	12.3047	0.580697	0.290348	−	0.956921i	$-0.406229\pi$
$449$	12.3047	0.580697	0.290348	+	0.956921i	$0.406229\pi$
$450$	0	0
$451$	18.8368	0.886991
$452$	−26.3121	−1.23762
$453$	0	0
$454$	3.82365	0.179453
$455$	0	0
$456$	0	0
$457$	−3.15122	−0.147408	−0.0737039	−	0.997280i	$-0.523482\pi$
$457$	−3.15122	−0.147408	−0.0737039	+	0.997280i	$0.523482\pi$
$458$	−8.31210	−0.388399
$459$	0	0
$460$	0	0
$461$	32.0457	1.49251	0.746257	−	0.665657i	$-0.231849\pi$
$461$	32.0457	1.49251	0.746257	+	0.665657i	$0.231849\pi$
$462$	0	0
$463$	−19.0936	−0.887355	−0.443678	−	0.896186i	$-0.646326\pi$
$463$	−19.0936	−0.887355	−0.443678	+	0.896186i	$0.646326\pi$
$464$	−7.28421	−0.338161
$465$	0	0
$466$	−4.52978	−0.209838
$467$	0.105567	0.00488505	0.00244252	−	0.999997i	$-0.499223\pi$
$467$	0.105567	0.00488505	0.00244252	+	0.999997i	$0.499223\pi$
$468$	0	0
$469$	7.35499	0.339622
$470$	0	0
$471$	0	0
$472$	−21.8658	−1.00646
$473$	−17.6118	−0.809790
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	13.4966	0.617318
$479$	29.2951	1.33853	0.669263	−	0.743025i	$-0.266610\pi$
$479$	29.2951	1.33853	0.669263	+	0.743025i	$0.266610\pi$
$480$	0	0
$481$	−10.5182	−0.479587
$482$	−5.02863	−0.229048
$483$	0	0
$484$	10.6176	0.482617
$485$	0	0
$486$	0	0
$487$	6.24206	0.282855	0.141427	−	0.989949i	$-0.454831\pi$
$487$	6.24206	0.282855	0.141427	+	0.989949i	$0.454831\pi$
$488$	6.64271	0.300701
$489$	0	0
$490$	0	0
$491$	14.5182	0.655196	0.327598	−	0.944817i	$-0.393761\pi$
$491$	14.5182	0.655196	0.327598	+	0.944817i	$0.393761\pi$
$492$	0	0
$493$	0	0
$494$	−1.59706	−0.0718549
$495$	0	0
$496$	−3.03369	−0.136217
$497$	23.3624	1.04794
$498$	0	0
$499$	0.913756	0.0409053	0.0204527	−	0.999791i	$-0.493489\pi$
$499$	0.913756	0.0409053	0.0204527	+	0.999791i	$0.493489\pi$
$500$	0	0
$501$	0	0
$502$	9.22816	0.411873
$503$	27.8538	1.24194	0.620971	−	0.783834i	$-0.286739\pi$
$503$	27.8538	1.24194	0.620971	+	0.783834i	$0.286739\pi$
$504$	0	0
$505$	0	0
$506$	−3.74828	−0.166631
$507$	0	0
$508$	−24.2517	−1.07600
$509$	31.5136	1.39681	0.698407	−	0.715701i	$-0.253892\pi$
$509$	31.5136	1.39681	0.698407	+	0.715701i	$0.253892\pi$
$510$	0	0
$511$	−3.50851	−0.155207
$512$	−22.0480	−0.974391
$513$	0	0
$514$	0.363095	0.0160154
$515$	0	0
$516$	0	0
$517$	7.38592	0.324832
$518$	−5.78887	−0.254348
$519$	0	0
$520$	0	0
$521$	9.49885	0.416152	0.208076	−	0.978113i	$-0.433280\pi$
$521$	9.49885	0.416152	0.208076	+	0.978113i	$0.433280\pi$
$522$	0	0
$523$	−23.9691	−1.04809	−0.524047	−	0.851689i	$-0.675578\pi$
$523$	−23.9691	−1.04809	−0.524047	+	0.851689i	$0.675578\pi$
$524$	7.50161	0.327709
$525$	0	0
$526$	−12.3167	−0.537034
$527$	0	0
$528$	0	0
$529$	−12.4412	−0.540923
$530$	0	0
$531$	0	0
$532$	5.52398	0.239495
$533$	−26.0797	−1.12964
$534$	0	0
$535$	0	0
$536$	−4.48459	−0.193705
$537$	0	0
$538$	−5.56153	−0.239774
$539$	−7.15352	−0.308124
$540$	0	0
$541$	21.9401	0.943278	0.471639	−	0.881792i	$-0.343663\pi$
$541$	21.9401	0.943278	0.471639	+	0.881792i	$0.343663\pi$
$542$	−8.01546	−0.344294
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−39.0074	−1.66783	−0.833917	−	0.551890i	$-0.813907\pi$
$547$	−39.0074	−1.66783	−0.833917	+	0.551890i	$0.813907\pi$
$548$	−34.0046	−1.45260
$549$	0	0
$550$	0	0
$551$	−3.00000	−0.127804
$552$	0	0
$553$	−43.4006	−1.84558
$554$	−10.2672	−0.436211
$555$	0	0
$556$	27.4544	1.16433
$557$	26.0959	1.10572	0.552860	−	0.833274i	$-0.313537\pi$
$557$	26.0959	1.10572	0.552860	+	0.833274i	$0.313537\pi$
$558$	0	0
$559$	24.3836	1.03132
$560$	0	0
$561$	0	0
$562$	−16.0097	−0.675327
$563$	−20.8944	−0.880595	−0.440298	−	0.897852i	$-0.645127\pi$
$563$	−20.8944	−0.880595	−0.440298	+	0.897852i	$0.645127\pi$
$564$	0	0
$565$	0	0
$566$	4.51357	0.189720
$567$	0	0
$568$	−14.2448	−0.597700
$569$	39.8419	1.67026	0.835129	−	0.550054i	$-0.185393\pi$
$569$	39.8419	1.67026	0.835129	+	0.550054i	$0.185393\pi$
$570$	0	0
$571$	16.7173	0.699599	0.349800	−	0.936825i	$-0.386250\pi$
$571$	16.7173	0.699599	0.349800	+	0.936825i	$0.386250\pi$
$572$	11.5777	0.484089
$573$	0	0
$574$	−14.3534	−0.599101
$575$	0	0
$576$	0	0
$577$	36.0360	1.50020	0.750099	−	0.661326i	$-0.230006\pi$
$577$	36.0360	1.50020	0.750099	+	0.661326i	$0.230006\pi$
$578$	8.90760	0.370507
$579$	0	0
$580$	0	0
$581$	−26.2568	−1.08932
$582$	0	0
$583$	1.99034	0.0824314
$584$	2.13926	0.0885232
$585$	0	0
$586$	−15.4869	−0.639758
$587$	−3.95941	−0.163422	−0.0817111	−	0.996656i	$-0.526038\pi$
$587$	−3.95941	−0.163422	−0.0817111	+	0.996656i	$0.526038\pi$
$588$	0	0
$589$	−1.24943	−0.0514817
$590$	0	0
$591$	0	0
$592$	−8.37902	−0.344376
$593$	−21.1439	−0.868274	−0.434137	−	0.900847i	$-0.642947\pi$
$593$	−21.1439	−0.868274	−0.434137	+	0.900847i	$0.642947\pi$
$594$	0	0
$595$	0	0
$596$	−27.3550	−1.12050
$597$	0	0
$598$	5.18951	0.212215
$599$	−4.19181	−0.171273	−0.0856364	−	0.996326i	$-0.527292\pi$
$599$	−4.19181	−0.171273	−0.0856364	+	0.996326i	$0.527292\pi$
$600$	0	0
$601$	−4.54910	−0.185562	−0.0927809	−	0.995687i	$-0.529576\pi$
$601$	−4.54910	−0.185562	−0.0927809	+	0.995687i	$0.529576\pi$
$602$	13.4200	0.546957
$603$	0	0
$604$	−32.6141	−1.32705
$605$	0	0
$606$	0	0
$607$	−34.6044	−1.40455	−0.702275	−	0.711906i	$-0.747832\pi$
$607$	−34.6044	−1.40455	−0.702275	+	0.711906i	$0.747832\pi$
$608$	−5.17635	−0.209929
$609$	0	0
$610$	0	0
$611$	−10.2259	−0.413694
$612$	0	0
$613$	4.13420	0.166979	0.0834893	−	0.996509i	$-0.473394\pi$
$613$	4.13420	0.166979	0.0834893	+	0.996509i	$0.473394\pi$
$614$	−8.86120	−0.357609
$615$	0	0
$616$	13.7579	0.554323
$617$	17.2398	0.694047	0.347023	−	0.937856i	$-0.387192\pi$
$617$	17.2398	0.694047	0.347023	+	0.937856i	$0.387192\pi$
$618$	0	0
$619$	−39.0244	−1.56852	−0.784261	−	0.620431i	$-0.786958\pi$
$619$	−39.0244	−1.56852	−0.784261	+	0.620431i	$0.786958\pi$
$620$	0	0
$621$	0	0
$622$	7.38592	0.296148
$623$	−23.7003	−0.949533
$624$	0	0
$625$	0	0
$626$	2.12489	0.0849276
$627$	0	0
$628$	−24.9510	−0.995651
$629$	0	0
$630$	0	0
$631$	34.0959	1.35734	0.678668	−	0.734445i	$-0.262557\pi$
$631$	34.0959	1.35734	0.678668	+	0.734445i	$0.262557\pi$
$632$	26.4629	1.05264
$633$	0	0
$634$	11.5263	0.457767
$635$	0	0
$636$	0	0
$637$	9.90409	0.392415
$638$	−3.46056	−0.137005
$639$	0	0
$640$	0	0
$641$	−27.8036	−1.09818	−0.549088	−	0.835765i	$-0.685025\pi$
$641$	−27.8036	−1.09818	−0.549088	+	0.835765i	$0.685025\pi$
$642$	0	0
$643$	40.2185	1.58606	0.793031	−	0.609181i	$-0.208502\pi$
$643$	40.2185	1.58606	0.793031	+	0.609181i	$0.208502\pi$
$644$	−17.9497	−0.707319
$645$	0	0
$646$	0	0
$647$	−20.1512	−0.792226	−0.396113	−	0.918202i	$-0.629641\pi$
$647$	−20.1512	−0.792226	−0.396113	+	0.918202i	$0.629641\pi$
$648$	0	0
$649$	24.6597	0.967979
$650$	0	0
$651$	0	0
$652$	−37.9726	−1.48712
$653$	−26.7556	−1.04703	−0.523514	−	0.852017i	$-0.675379\pi$
$653$	−26.7556	−1.04703	−0.523514	+	0.852017i	$0.675379\pi$
$654$	0	0
$655$	0	0
$656$	−20.7757	−0.811155
$657$	0	0
$658$	−5.62799	−0.219402
$659$	−17.6118	−0.686057	−0.343029	−	0.939325i	$-0.611453\pi$
$659$	−17.6118	−0.686057	−0.343029	+	0.939325i	$0.611453\pi$
$660$	0	0
$661$	11.5159	0.447916	0.223958	−	0.974599i	$-0.428102\pi$
$661$	11.5159	0.447916	0.223958	+	0.974599i	$0.428102\pi$
$662$	−6.52783	−0.253711
$663$	0	0
$664$	16.0097	0.621295
$665$	0	0
$666$	0	0
$667$	9.74828	0.377455
$668$	5.35850	0.207326
$669$	0	0
$670$	0	0
$671$	−7.49149	−0.289206
$672$	0	0
$673$	−7.85384	−0.302743	−0.151372	−	0.988477i	$-0.548369\pi$
$673$	−7.85384	−0.302743	−0.151372	+	0.988477i	$0.548369\pi$
$674$	−10.7409	−0.413725
$675$	0	0
$676$	6.40139	0.246207
$677$	−43.5136	−1.67236	−0.836181	−	0.548453i	$-0.815217\pi$
$677$	−43.5136	−1.67236	−0.836181	+	0.548453i	$0.815217\pi$
$678$	0	0
$679$	22.8709	0.877703
$680$	0	0
$681$	0	0
$682$	−1.44124	−0.0551878
$683$	4.60212	0.176095	0.0880476	−	0.996116i	$-0.471937\pi$
$683$	4.60212	0.176095	0.0880476	+	0.996116i	$0.471937\pi$
$684$	0	0
$685$	0	0
$686$	−6.29157	−0.240213
$687$	0	0
$688$	19.4246	0.740555
$689$	−2.75564	−0.104981
$690$	0	0
$691$	28.5948	1.08780	0.543898	−	0.839151i	$-0.316948\pi$
$691$	28.5948	1.08780	0.543898	+	0.839151i	$0.316948\pi$
$692$	−27.6032	−1.04932
$693$	0	0
$694$	−2.33337	−0.0885735
$695$	0	0
$696$	0	0
$697$	0	0
$698$	4.87667	0.184585
$699$	0	0
$700$	0	0
$701$	18.2111	0.687825	0.343913	−	0.939002i	$-0.388248\pi$
$701$	18.2111	0.687825	0.343913	+	0.939002i	$0.388248\pi$
$702$	0	0
$703$	−3.45090	−0.130153
$704$	4.71964	0.177878
$705$	0	0
$706$	−8.70538	−0.327631
$707$	44.0457	1.65651
$708$	0	0
$709$	−10.6930	−0.401583	−0.200791	−	0.979634i	$-0.564351\pi$
$709$	−10.6930	−0.401583	−0.200791	+	0.979634i	$0.564351\pi$
$710$	0	0
$711$	0	0
$712$	14.4509	0.541570
$713$	4.05991	0.152045
$714$	0	0
$715$	0	0
$716$	4.99420	0.186642
$717$	0	0
$718$	15.6930	0.585656
$719$	41.9548	1.56465	0.782325	−	0.622870i	$-0.214034\pi$
$719$	41.9548	1.56465	0.782325	+	0.622870i	$0.214034\pi$
$720$	0	0
$721$	19.8538	0.739396
$722$	−0.523976	−0.0195004
$723$	0	0
$724$	−11.9087	−0.442583
$725$	0	0
$726$	0	0
$727$	−28.4103	−1.05368	−0.526840	−	0.849964i	$-0.676623\pi$
$727$	−28.4103	−1.05368	−0.526840	+	0.849964i	$0.676623\pi$
$728$	−19.0480	−0.705964
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	14.6930	0.542697	0.271348	−	0.962481i	$-0.412530\pi$
$733$	14.6930	0.542697	0.271348	+	0.962481i	$0.412530\pi$
$734$	14.7293	0.543669
$735$	0	0
$736$	16.8201	0.619999
$737$	5.05761	0.186300
$738$	0	0
$739$	14.4103	0.530092	0.265046	−	0.964236i	$-0.414613\pi$
$739$	14.4103	0.530092	0.265046	+	0.964236i	$0.414613\pi$
$740$	0	0
$741$	0	0
$742$	−1.51661	−0.0556767
$743$	−31.5851	−1.15874	−0.579372	−	0.815063i	$-0.696702\pi$
$743$	−31.5851	−1.15874	−0.579372	+	0.815063i	$0.696702\pi$
$744$	0	0
$745$	0	0
$746$	−5.55140	−0.203251
$747$	0	0
$748$	0	0
$749$	−49.9571	−1.82539
$750$	0	0
$751$	25.4509	0.928716	0.464358	−	0.885647i	$-0.346285\pi$
$751$	25.4509	0.928716	0.464358	+	0.885647i	$0.346285\pi$
$752$	−8.14616	−0.297060
$753$	0	0
$754$	4.79117	0.174484
$755$	0	0
$756$	0	0
$757$	14.0170	0.509457	0.254729	−	0.967013i	$-0.418014\pi$
$757$	14.0170	0.509457	0.254729	+	0.967013i	$0.418014\pi$
$758$	−1.59706	−0.0580077
$759$	0	0
$760$	0	0
$761$	47.2738	1.71367	0.856837	−	0.515587i	$-0.172426\pi$
$761$	47.2738	1.71367	0.856837	+	0.515587i	$0.172426\pi$
$762$	0	0
$763$	−23.8538	−0.863567
$764$	−15.9594	−0.577391
$765$	0	0
$766$	−8.69912	−0.314312
$767$	−34.1416	−1.23278
$768$	0	0
$769$	26.0936	0.940960	0.470480	−	0.882411i	$-0.344081\pi$
$769$	26.0936	0.940960	0.470480	+	0.882411i	$0.344081\pi$
$770$	0	0
$771$	0	0
$772$	2.06038	0.0741546
$773$	−23.6501	−0.850634	−0.425317	−	0.905044i	$-0.639837\pi$
$773$	−23.6501	−0.850634	−0.425317	+	0.905044i	$0.639837\pi$
$774$	0	0
$775$	0	0
$776$	−13.9451	−0.500602
$777$	0	0
$778$	3.80359	0.136365
$779$	−8.55646	−0.306567
$780$	0	0
$781$	16.0650	0.574850
$782$	0	0
$783$	0	0
$784$	7.88983	0.281780
$785$	0	0
$786$	0	0
$787$	−22.8156	−0.813287	−0.406643	−	0.913587i	$-0.633301\pi$
$787$	−22.8156	−0.813287	−0.406643	+	0.913587i	$0.633301\pi$
$788$	30.1106	1.07265
$789$	0	0
$790$	0	0
$791$	−48.8206	−1.73586
$792$	0	0
$793$	10.3720	0.368321
$794$	−10.6191	−0.376859
$795$	0	0
$796$	−4.82872	−0.171149
$797$	−0.748275	−0.0265053	−0.0132526	−	0.999912i	$-0.504219\pi$
$797$	−0.748275	−0.0265053	−0.0132526	+	0.999912i	$0.504219\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	−1.70262	−0.0601217
$803$	−2.41261	−0.0851390
$804$	0	0
$805$	0	0
$806$	1.99540	0.0702850
$807$	0	0
$808$	−26.8561	−0.944796
$809$	−50.1563	−1.76340	−0.881700	−	0.471810i	$-0.843601\pi$
$809$	−50.1563	−1.76340	−0.881700	+	0.471810i	$0.843601\pi$
$810$	0	0
$811$	27.9903	0.982874	0.491437	−	0.870913i	$-0.336472\pi$
$811$	27.9903	0.982874	0.491437	+	0.870913i	$0.336472\pi$
$812$	−16.5719	−0.581561
$813$	0	0
$814$	−3.98068	−0.139523
$815$	0	0
$816$	0	0
$817$	8.00000	0.279885
$818$	−17.8391	−0.623730
$819$	0	0
$820$	0	0
$821$	42.3218	1.47704	0.738520	−	0.674232i	$-0.235525\pi$
$821$	42.3218	1.47704	0.738520	+	0.674232i	$0.235525\pi$
$822$	0	0
$823$	35.6044	1.24109	0.620546	−	0.784170i	$-0.286911\pi$
$823$	35.6044	1.24109	0.620546	+	0.784170i	$0.286911\pi$
$824$	−12.1056	−0.421717
$825$	0	0
$826$	−18.7904	−0.653802
$827$	−51.4154	−1.78789	−0.893944	−	0.448179i	$-0.852073\pi$
$827$	−51.4154	−1.78789	−0.893944	+	0.448179i	$0.852073\pi$
$828$	0	0
$829$	−26.9520	−0.936083	−0.468042	−	0.883706i	$-0.655040\pi$
$829$	−26.9520	−0.936083	−0.468042	+	0.883706i	$0.655040\pi$
$830$	0	0
$831$	0	0
$832$	−6.53438	−0.226539
$833$	0	0
$834$	0	0
$835$	0	0
$836$	3.79853	0.131375
$837$	0	0
$838$	13.7579	0.475260
$839$	−34.7026	−1.19807	−0.599034	−	0.800724i	$-0.704448\pi$
$839$	−34.7026	−1.19807	−0.599034	+	0.800724i	$0.704448\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	6.16088	0.212318
$843$	0	0
$844$	45.2881	1.55888
$845$	0	0
$846$	0	0
$847$	19.7003	0.676911
$848$	−2.19521	−0.0753837
$849$	0	0
$850$	0	0
$851$	11.2134	0.384392
$852$	0	0
$853$	−0.134197	−0.00459483	−0.00229741	−	0.999997i	$-0.500731\pi$
$853$	−0.134197	−0.00459483	−0.00229741	+	0.999997i	$0.500731\pi$
$854$	5.70843	0.195338
$855$	0	0
$856$	30.4606	1.04112
$857$	30.5371	1.04313	0.521564	−	0.853212i	$-0.325349\pi$
$857$	30.5371	1.04313	0.521564	+	0.853212i	$0.325349\pi$
$858$	0	0
$859$	−57.1010	−1.94826	−0.974130	−	0.225989i	$-0.927439\pi$
$859$	−57.1010	−1.94826	−0.974130	+	0.225989i	$0.927439\pi$
$860$	0	0
$861$	0	0
$862$	−14.1500	−0.481951
$863$	−0.376261	−0.0128081	−0.00640403	−	0.999979i	$-0.502038\pi$
$863$	−0.376261	−0.0128081	−0.00640403	+	0.999979i	$0.502038\pi$
$864$	0	0
$865$	0	0
$866$	−7.68135	−0.261023
$867$	0	0
$868$	−6.90179	−0.234262
$869$	−29.8442	−1.01239
$870$	0	0
$871$	−7.00230	−0.237264
$872$	14.5445	0.492539
$873$	0	0
$874$	1.70262	0.0575921
$875$	0	0
$876$	0	0
$877$	−5.54680	−0.187302	−0.0936511	−	0.995605i	$-0.529854\pi$
$877$	−5.54680	−0.187302	−0.0936511	+	0.995605i	$0.529854\pi$
$878$	11.9285	0.402567
$879$	0	0
$880$	0	0
$881$	−13.3357	−0.449290	−0.224645	−	0.974441i	$-0.572122\pi$
$881$	−13.3357	−0.449290	−0.224645	+	0.974441i	$0.572122\pi$
$882$	0	0
$883$	11.6044	0.390520	0.195260	−	0.980752i	$-0.437445\pi$
$883$	11.6044	0.390520	0.195260	+	0.980752i	$0.437445\pi$
$884$	0	0
$885$	0	0
$886$	20.6237	0.692868
$887$	33.2282	1.11569	0.557846	−	0.829944i	$-0.311628\pi$
$887$	33.2282	1.11569	0.557846	+	0.829944i	$0.311628\pi$
$888$	0	0
$889$	−44.9977	−1.50917
$890$	0	0
$891$	0	0
$892$	22.8612	0.765450
$893$	−3.35499	−0.112271
$894$	0	0
$895$	0	0
$896$	−36.7402	−1.22740
$897$	0	0
$898$	−6.44739	−0.215152
$899$	3.74828	0.125012
$900$	0	0
$901$	0	0
$902$	−9.87005	−0.328637
$903$	0	0
$904$	29.7676	0.990056
$905$	0	0
$906$	0	0
$907$	18.9018	0.627624	0.313812	−	0.949485i	$-0.398394\pi$
$907$	18.9018	0.627624	0.313812	+	0.949485i	$0.398394\pi$
$908$	12.5913	0.417855
$909$	0	0
$910$	0	0
$911$	10.3910	0.344269	0.172134	−	0.985073i	$-0.444934\pi$
$911$	10.3910	0.344269	0.172134	+	0.985073i	$0.444934\pi$
$912$	0	0
$913$	−18.0553	−0.597544
$914$	1.65116	0.0546157
$915$	0	0
$916$	−27.3717	−0.904385
$917$	13.9188	0.459640
$918$	0	0
$919$	−0.894433	−0.0295046	−0.0147523	−	0.999891i	$-0.504696\pi$
$919$	−0.894433	−0.0295046	−0.0147523	+	0.999891i	$0.504696\pi$
$920$	0	0
$921$	0	0
$922$	−16.7912	−0.552988
$923$	−22.2421	−0.732106
$924$	0	0
$925$	0	0
$926$	10.0046	0.328772
$927$	0	0
$928$	15.5290	0.509766
$929$	3.69296	0.121162	0.0605811	−	0.998163i	$-0.480705\pi$
$929$	3.69296	0.121162	0.0605811	+	0.998163i	$0.480705\pi$
$930$	0	0
$931$	3.24943	0.106496
$932$	−14.9165	−0.488607
$933$	0	0
$934$	−0.0553145	−0.00180995
$935$	0	0
$936$	0	0
$937$	−16.2494	−0.530846	−0.265423	−	0.964132i	$-0.585512\pi$
$937$	−16.2494	−0.530846	−0.265423	+	0.964132i	$0.585512\pi$
$938$	−3.85384	−0.125832
$939$	0	0
$940$	0	0
$941$	−13.1535	−0.428792	−0.214396	−	0.976747i	$-0.568778\pi$
$941$	−13.1535	−0.428792	−0.214396	+	0.976747i	$0.568778\pi$
$942$	0	0
$943$	27.8036	0.905409
$944$	−27.1980	−0.885218
$945$	0	0
$946$	9.22816	0.300033
$947$	40.5136	1.31651	0.658257	−	0.752793i	$-0.271294\pi$
$947$	40.5136	1.31651	0.658257	+	0.752793i	$0.271294\pi$
$948$	0	0
$949$	3.34027	0.108430
$950$	0	0
$951$	0	0
$952$	0	0
$953$	44.6884	1.44760	0.723799	−	0.690011i	$-0.242394\pi$
$953$	44.6884	1.44760	0.723799	+	0.690011i	$0.242394\pi$
$954$	0	0
$955$	0	0
$956$	44.4440	1.43742
$957$	0	0
$958$	−15.3499	−0.495934
$959$	−63.0936	−2.03740
$960$	0	0
$961$	−29.4389	−0.949643
$962$	5.51127	0.177691
$963$	0	0
$964$	−16.5592	−0.533337
$965$	0	0
$966$	0	0
$967$	−38.3910	−1.23457	−0.617285	−	0.786739i	$-0.711768\pi$
$967$	−38.3910	−1.23457	−0.617285	+	0.786739i	$0.711768\pi$
$968$	−12.0120	−0.386079
$969$	0	0
$970$	0	0
$971$	12.6980	0.407499	0.203749	−	0.979023i	$-0.434687\pi$
$971$	12.6980	0.407499	0.203749	+	0.979023i	$0.434687\pi$
$972$	0	0
$973$	50.9401	1.63306
$974$	−3.27069	−0.104800
$975$	0	0
$976$	8.26259	0.264479
$977$	3.48183	0.111394	0.0556968	−	0.998448i	$-0.482262\pi$
$977$	3.48183	0.111394	0.0556968	+	0.998448i	$0.482262\pi$
$978$	0	0
$979$	−16.2974	−0.520866
$980$	0	0
$981$	0	0
$982$	−7.60718	−0.242755
$983$	−10.1152	−0.322626	−0.161313	−	0.986903i	$-0.551573\pi$
$983$	−10.1152	−0.322626	−0.161313	+	0.986903i	$0.551573\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−5.25909	−0.167314
$989$	−25.9954	−0.826606
$990$	0	0
$991$	23.7985	0.755985	0.377993	−	0.925809i	$-0.376614\pi$
$991$	23.7985	0.755985	0.377993	+	0.925809i	$0.376614\pi$
$992$	6.46746	0.205342
$993$	0	0
$994$	−12.2413	−0.388271
$995$	0	0
$996$	0	0
$997$	−38.9571	−1.23378	−0.616892	−	0.787048i	$-0.711608\pi$
$997$	−38.9571	−1.23378	−0.616892	+	0.787048i	$0.711608\pi$
$998$	−0.478786	−0.0151557
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bh.1.2		3
3.2	odd	2		1425.2.a.v.1.2	yes	3
5.4	even	2		4275.2.a.be.1.2		3
15.2	even	4		1425.2.c.p.799.4		6
15.8	even	4		1425.2.c.p.799.3		6
15.14	odd	2		1425.2.a.u.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.u.1.2	✓	3	15.14	odd	2
1425.2.a.v.1.2	yes	3	3.2	odd	2
1425.2.c.p.799.3		6	15.8	even	4
1425.2.c.p.799.4		6	15.2	even	4
4275.2.a.be.1.2		3	5.4	even	2
4275.2.a.bh.1.2		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.523976 q^{2} -1.72545 q^{4} -3.20147 q^{7} +1.95205 q^{8} -2.20147 q^{11} +3.04795 q^{13} +1.67750 q^{14} +2.42807 q^{16} +1.00000 q^{19} +1.15352 q^{22} -3.24943 q^{23} -1.59706 q^{26} +5.52398 q^{28} -3.00000 q^{29} -1.24943 q^{31} -5.17635 q^{32} -3.45090 q^{37} -0.523976 q^{38} -8.55646 q^{41} +8.00000 q^{43} +3.79853 q^{44} +1.70262 q^{46} -3.35499 q^{47} +3.24943 q^{49} -5.25909 q^{52} -0.904094 q^{53} -6.24943 q^{56} +1.57193 q^{58} -11.2015 q^{59} +3.40294 q^{61} +0.654669 q^{62} -2.14386 q^{64} -2.29738 q^{67} -7.29738 q^{71} +1.09591 q^{73} +1.80819 q^{74} -1.72545 q^{76} +7.04795 q^{77} +13.5565 q^{79} +4.48339 q^{82} +8.20147 q^{83} -4.19181 q^{86} -4.29738 q^{88} +7.40294 q^{89} -9.75794 q^{91} +5.60672 q^{92} +1.75794 q^{94} -7.14386 q^{97} -1.70262 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} + 9 q^{8} + 3 q^{11} + 6 q^{13} - 3 q^{14} + 12 q^{16} + 3 q^{19} - 3 q^{22} + 3 q^{23} - 24 q^{26} + 15 q^{28} - 9 q^{29} + 9 q^{31} + 18 q^{32} + 12 q^{37} + 24 q^{43} + 21 q^{44} + 21 q^{46}+ \cdots - 21 q^{98}+O(q^{100}) \) 3 * q + 6 * q^4 + 9 * q^8 + 3 * q^11 + 6 * q^13 - 3 * q^14 + 12 * q^16 + 3 * q^19 - 3 * q^22 + 3 * q^23 - 24 * q^26 + 15 * q^28 - 9 * q^29 + 9 * q^31 + 18 * q^32 + 12 * q^37 + 24 * q^43 + 21 * q^44 + 21 * q^46 + 6 * q^47 - 3 * q^49 - 6 * q^52 - 9 * q^53 - 6 * q^56 - 24 * q^59 - 9 * q^61 + 21 * q^62 + 3 * q^64 + 9 * q^67 - 6 * q^71 - 3 * q^73 + 18 * q^74 + 6 * q^76 + 18 * q^77 + 15 * q^79 - 33 * q^82 + 15 * q^83 + 3 * q^88 + 3 * q^89 + 6 * q^91 + 39 * q^92 - 30 * q^94 - 12 * q^97 - 21 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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