LMFDB - Embedded newform 4275.2.a.bw.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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 8x^{5} + 26x^{4} + 11x^{3} - 51x^{2} + 12x + 4 $ x^7 - 3x^6 - 8x^5 + 26x^4 + 11x^3 - 51x^2 + 12x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.70383$ 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)

$f(q)$	$=$	$q-1.70383 q^{2} +0.903045 q^{4} +0.338398 q^{7} +1.86903 q^{8} -1.26153 q^{11} -6.57224 q^{13} -0.576574 q^{14} -4.99060 q^{16} +3.65980 q^{17} +1.00000 q^{19} +2.14943 q^{22} -5.14881 q^{23} +11.1980 q^{26} +0.305589 q^{28} +4.73743 q^{29} -3.25083 q^{31} +4.76509 q^{32} -6.23569 q^{34} -4.44536 q^{37} -1.70383 q^{38} +4.93134 q^{41} -9.32202 q^{43} -1.13922 q^{44} +8.77271 q^{46} +8.42466 q^{47} -6.88549 q^{49} -5.93503 q^{52} -6.98052 q^{53} +0.632476 q^{56} -8.07179 q^{58} -2.34081 q^{59} +14.2007 q^{61} +5.53887 q^{62} +1.86228 q^{64} +0.160860 q^{67} +3.30497 q^{68} -0.160860 q^{71} +5.07893 q^{73} +7.57416 q^{74} +0.903045 q^{76} -0.426899 q^{77} +16.6039 q^{79} -8.40218 q^{82} +6.81966 q^{83} +15.8832 q^{86} -2.35783 q^{88} +16.3639 q^{89} -2.22403 q^{91} -4.64961 q^{92} -14.3542 q^{94} -13.7180 q^{97} +11.7317 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{2} + 11 q^{4} - 8 q^{7} + 9 q^{8} + 4 q^{11} - 8 q^{13} + 4 q^{14} + 19 q^{16} + 4 q^{17} + 7 q^{19} - 12 q^{22} + 10 q^{23} + 20 q^{26} - 14 q^{28} + 6 q^{29} + 4 q^{31} + 31 q^{32} + 2 q^{34}+ \cdots + 5 q^{98}+O(q^{100}) $ 7 * q + 3 * q^2 + 11 * q^4 - 8 * q^7 + 9 * q^8 + 4 * q^11 - 8 * q^13 + 4 * q^14 + 19 * q^16 + 4 * q^17 + 7 * q^19 - 12 * q^22 + 10 * q^23 + 20 * q^26 - 14 * q^28 + 6 * q^29 + 4 * q^31 + 31 * q^32 + 2 * q^34 - 14 * q^37 + 3 * q^38 - 2 * q^41 + 2 * q^43 + 32 * q^44 - 4 * q^46 + 30 * q^47 + 17 * q^49 - 18 * q^52 + 22 * q^56 + 40 * q^58 + 18 * q^59 + 12 * q^61 + 18 * q^62 + 11 * q^64 - 18 * q^67 - 12 * q^68 + 18 * q^71 - 10 * q^73 - 6 * q^74 + 11 * q^76 - 18 * q^77 - 4 * q^79 + 16 * q^82 + 18 * q^83 + 46 * q^86 + 18 * q^88 + 8 * q^89 + 12 * q^91 + 34 * q^92 - 20 * q^94 - 20 * q^97 + 5 * 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 $

$2$	−1.70383	−1.20479	−0.602396	−	0.798198i	$-0.705787\pi$
$2$	−1.70383	−1.20479	−0.602396	+	0.798198i	$0.705787\pi$
$3$	0	0
$3$	0	0
$4$	0.903045	0.451523
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.338398	0.127903	0.0639513	−	0.997953i	$-0.479630\pi$
$7$	0.338398	0.127903	0.0639513	+	0.997953i	$0.479630\pi$
$8$	1.86903	0.660801
$9$	0	0
$10$	0	0
$11$	−1.26153	−0.380365	−0.190183	−	0.981749i	$-0.560908\pi$
$11$	−1.26153	−0.380365	−0.190183	+	0.981749i	$0.560908\pi$
$12$	0	0
$13$	−6.57224	−1.82281	−0.911406	−	0.411509i	$-0.865002\pi$
$13$	−6.57224	−1.82281	−0.911406	+	0.411509i	$0.865002\pi$
$14$	−0.576574	−0.154096
$15$	0	0
$16$	−4.99060	−1.24765
$17$	3.65980	0.887633	0.443816	−	0.896118i	$-0.353624\pi$
$17$	3.65980	0.887633	0.443816	+	0.896118i	$0.353624\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	2.14943	0.458261
$23$	−5.14881	−1.07360	−0.536801	−	0.843709i	$-0.680367\pi$
$23$	−5.14881	−1.07360	−0.536801	+	0.843709i	$0.680367\pi$
$24$	0	0
$25$	0	0
$26$	11.1980	2.19611
$27$	0	0
$28$	0.305589	0.0577509
$29$	4.73743	0.879719	0.439860	−	0.898067i	$-0.355028\pi$
$29$	4.73743	0.879719	0.439860	+	0.898067i	$0.355028\pi$
$30$	0	0
$31$	−3.25083	−0.583867	−0.291933	−	0.956439i	$-0.594299\pi$
$31$	−3.25083	−0.583867	−0.291933	+	0.956439i	$0.594299\pi$
$32$	4.76509	0.842357
$33$	0	0
$34$	−6.23569	−1.06941
$35$	0	0
$36$	0	0
$37$	−4.44536	−0.730813	−0.365407	−	0.930848i	$-0.619070\pi$
$37$	−4.44536	−0.730813	−0.365407	+	0.930848i	$0.619070\pi$
$38$	−1.70383	−0.276398
$39$	0	0
$40$	0	0
$41$	4.93134	0.770147	0.385073	−	0.922886i	$-0.374176\pi$
$41$	4.93134	0.770147	0.385073	+	0.922886i	$0.374176\pi$
$42$	0	0
$43$	−9.32202	−1.42159	−0.710797	−	0.703397i	$-0.751666\pi$
$43$	−9.32202	−1.42159	−0.710797	+	0.703397i	$0.751666\pi$
$44$	−1.13922	−0.171743
$45$	0	0
$46$	8.77271	1.29347
$47$	8.42466	1.22886	0.614431	−	0.788970i	$-0.289386\pi$
$47$	8.42466	1.22886	0.614431	+	0.788970i	$0.289386\pi$
$48$	0	0
$49$	−6.88549	−0.983641
$50$	0	0
$51$	0	0
$52$	−5.93503	−0.823040
$53$	−6.98052	−0.958849	−0.479424	−	0.877583i	$-0.659155\pi$
$53$	−6.98052	−0.958849	−0.479424	+	0.877583i	$0.659155\pi$
$54$	0	0
$55$	0	0
$56$	0.632476	0.0845181
$57$	0	0
$58$	−8.07179	−1.05988
$59$	−2.34081	−0.304747	−0.152374	−	0.988323i	$-0.548692\pi$
$59$	−2.34081	−0.304747	−0.152374	+	0.988323i	$0.548692\pi$
$60$	0	0
$61$	14.2007	1.81821	0.909105	−	0.416567i	$-0.136767\pi$
$61$	14.2007	1.81821	0.909105	+	0.416567i	$0.136767\pi$
$62$	5.53887	0.703438
$63$	0	0
$64$	1.86228	0.232785
$65$	0	0
$66$	0	0
$67$	0.160860	0.0196522	0.00982608	−	0.999952i	$-0.496872\pi$
$67$	0.160860	0.0196522	0.00982608	+	0.999952i	$0.496872\pi$
$68$	3.30497	0.400786
$69$	0	0
$70$	0	0
$71$	−0.160860	−0.0190906	−0.00954528	−	0.999954i	$-0.503038\pi$
$71$	−0.160860	−0.0190906	−0.00954528	+	0.999954i	$0.503038\pi$
$72$	0	0
$73$	5.07893	0.594443	0.297222	−	0.954809i	$-0.403940\pi$
$73$	5.07893	0.594443	0.297222	+	0.954809i	$0.403940\pi$
$74$	7.57416	0.880478
$75$	0	0
$76$	0.903045	0.103586
$77$	−0.426899	−0.0486497
$78$	0	0
$79$	16.6039	1.86808	0.934040	−	0.357168i	$-0.116258\pi$
$79$	16.6039	1.86808	0.934040	+	0.357168i	$0.116258\pi$
$80$	0	0
$81$	0	0
$82$	−8.40218	−0.927866
$83$	6.81966	0.748555	0.374278	−	0.927317i	$-0.377891\pi$
$83$	6.81966	0.748555	0.374278	+	0.927317i	$0.377891\pi$
$84$	0	0
$85$	0	0
$86$	15.8832	1.71272
$87$	0	0
$88$	−2.35783	−0.251346
$89$	16.3639	1.73457	0.867283	−	0.497815i	$-0.165864\pi$
$89$	16.3639	1.73457	0.867283	+	0.497815i	$0.165864\pi$
$90$	0	0
$91$	−2.22403	−0.233142
$92$	−4.64961	−0.484755
$93$	0	0
$94$	−14.3542	−1.48052
$95$	0	0
$96$	0	0
$97$	−13.7180	−1.39285	−0.696424	−	0.717631i	$-0.745227\pi$
$97$	−13.7180	−1.39285	−0.696424	+	0.717631i	$0.745227\pi$
$98$	11.7317	1.18508
$99$	0	0
$100$	0	0
$101$	3.01514	0.300018	0.150009	−	0.988685i	$-0.452070\pi$
$101$	3.01514	0.300018	0.150009	+	0.988685i	$0.452070\pi$
$102$	0	0
$103$	−6.03399	−0.594546	−0.297273	−	0.954792i	$-0.596077\pi$
$103$	−6.03399	−0.594546	−0.297273	+	0.954792i	$0.596077\pi$
$104$	−12.2837	−1.20452
$105$	0	0
$106$	11.8936	1.15521
$107$	4.67680	0.452123	0.226062	−	0.974113i	$-0.427415\pi$
$107$	4.67680	0.452123	0.226062	+	0.974113i	$0.427415\pi$
$108$	0	0
$109$	2.19391	0.210138	0.105069	−	0.994465i	$-0.466494\pi$
$109$	2.19391	0.210138	0.105069	+	0.994465i	$0.466494\pi$
$110$	0	0
$111$	0	0
$112$	−1.68881	−0.159578
$113$	−2.54559	−0.239469	−0.119734	−	0.992806i	$-0.538204\pi$
$113$	−2.54559	−0.239469	−0.119734	+	0.992806i	$0.538204\pi$
$114$	0	0
$115$	0	0
$116$	4.27812	0.397213
$117$	0	0
$118$	3.98834	0.367157
$119$	1.23847	0.113531
$120$	0	0
$121$	−9.40854	−0.855322
$122$	−24.1956	−2.19056
$123$	0	0
$124$	−2.93565	−0.263629
$125$	0	0
$126$	0	0
$127$	−6.77093	−0.600823	−0.300411	−	0.953810i	$-0.597124\pi$
$127$	−6.77093	−0.600823	−0.300411	+	0.953810i	$0.597124\pi$
$128$	−12.7032	−1.12281
$129$	0	0
$130$	0	0
$131$	−6.23896	−0.545101	−0.272550	−	0.962142i	$-0.587867\pi$
$131$	−6.23896	−0.545101	−0.272550	+	0.962142i	$0.587867\pi$
$132$	0	0
$133$	0.338398	0.0293429
$134$	−0.274078	−0.0236768
$135$	0	0
$136$	6.84027	0.586549
$137$	−8.88502	−0.759098	−0.379549	−	0.925172i	$-0.623921\pi$
$137$	−8.88502	−0.759098	−0.379549	+	0.925172i	$0.623921\pi$
$138$	0	0
$139$	15.9529	1.35311	0.676553	−	0.736394i	$-0.263473\pi$
$139$	15.9529	1.35311	0.676553	+	0.736394i	$0.263473\pi$
$140$	0	0
$141$	0	0
$142$	0.274078	0.0230001
$143$	8.29107	0.693334
$144$	0	0
$145$	0	0
$146$	−8.65364	−0.716180
$147$	0	0
$148$	−4.01436	−0.329979
$149$	−13.4315	−1.10035	−0.550175	−	0.835049i	$-0.685439\pi$
$149$	−13.4315	−1.10035	−0.550175	+	0.835049i	$0.685439\pi$
$150$	0	0
$151$	−8.59789	−0.699686	−0.349843	−	0.936808i	$-0.613765\pi$
$151$	−8.59789	−0.699686	−0.349843	+	0.936808i	$0.613765\pi$
$152$	1.86903	0.151598
$153$	0	0
$154$	0.727365	0.0586127
$155$	0	0
$156$	0	0
$157$	23.5887	1.88258	0.941292	−	0.337592i	$-0.109612\pi$
$157$	23.5887	1.88258	0.941292	+	0.337592i	$0.109612\pi$
$158$	−28.2902	−2.25065
$159$	0	0
$160$	0	0
$161$	−1.74235	−0.137316
$162$	0	0
$163$	−4.57348	−0.358223	−0.179111	−	0.983829i	$-0.557322\pi$
$163$	−4.57348	−0.358223	−0.179111	+	0.983829i	$0.557322\pi$
$164$	4.45323	0.347738
$165$	0	0
$166$	−11.6196	−0.901853
$167$	−4.77725	−0.369675	−0.184837	−	0.982769i	$-0.559176\pi$
$167$	−4.77725	−0.369675	−0.184837	+	0.982769i	$0.559176\pi$
$168$	0	0
$169$	30.1943	2.32264
$170$	0	0
$171$	0	0
$172$	−8.41820	−0.641882
$173$	12.5915	0.957311	0.478656	−	0.878003i	$-0.341124\pi$
$173$	12.5915	0.957311	0.478656	+	0.878003i	$0.341124\pi$
$174$	0	0
$175$	0	0
$176$	6.29579	0.474563
$177$	0	0
$178$	−27.8813	−2.08979
$179$	−13.5976	−1.01633	−0.508167	−	0.861259i	$-0.669677\pi$
$179$	−13.5976	−1.01633	−0.508167	+	0.861259i	$0.669677\pi$
$180$	0	0
$181$	6.38782	0.474803	0.237401	−	0.971412i	$-0.423704\pi$
$181$	6.38782	0.474803	0.237401	+	0.971412i	$0.423704\pi$
$182$	3.78938	0.280888
$183$	0	0
$184$	−9.62327	−0.709437
$185$	0	0
$186$	0	0
$187$	−4.61695	−0.337625
$188$	7.60784	0.554859
$189$	0	0
$190$	0	0
$191$	−10.9816	−0.794597	−0.397299	−	0.917689i	$-0.630052\pi$
$191$	−10.9816	−0.794597	−0.397299	+	0.917689i	$0.630052\pi$
$192$	0	0
$193$	4.01436	0.288960	0.144480	−	0.989508i	$-0.453849\pi$
$193$	4.01436	0.288960	0.144480	+	0.989508i	$0.453849\pi$
$194$	23.3731	1.67809
$195$	0	0
$196$	−6.21790	−0.444136
$197$	8.84545	0.630213	0.315106	−	0.949056i	$-0.397960\pi$
$197$	8.84545	0.630213	0.315106	+	0.949056i	$0.397960\pi$
$198$	0	0
$199$	−15.2576	−1.08158	−0.540791	−	0.841157i	$-0.681875\pi$
$199$	−15.2576	−1.08158	−0.540791	+	0.841157i	$0.681875\pi$
$200$	0	0
$201$	0	0
$202$	−5.13729	−0.361459
$203$	1.60314	0.112518
$204$	0	0
$205$	0	0
$206$	10.2809	0.716304
$207$	0	0
$208$	32.7994	2.27423
$209$	−1.26153	−0.0872618
$210$	0	0
$211$	15.9598	1.09872	0.549359	−	0.835586i	$-0.314872\pi$
$211$	15.9598	1.09872	0.549359	+	0.835586i	$0.314872\pi$
$212$	−6.30373	−0.432942
$213$	0	0
$214$	−7.96848	−0.544714
$215$	0	0
$216$	0	0
$217$	−1.10008	−0.0746781
$218$	−3.73806	−0.253173
$219$	0	0
$220$	0	0
$221$	−24.0531	−1.61799
$222$	0	0
$223$	−14.2079	−0.951430	−0.475715	−	0.879600i	$-0.657811\pi$
$223$	−14.2079	−0.951430	−0.475715	+	0.879600i	$0.657811\pi$
$224$	1.61250	0.107740
$225$	0	0
$226$	4.33725	0.288510
$227$	4.98748	0.331031	0.165515	−	0.986207i	$-0.447071\pi$
$227$	4.98748	0.331031	0.165515	+	0.986207i	$0.447071\pi$
$228$	0	0
$229$	17.0988	1.12992	0.564959	−	0.825119i	$-0.308892\pi$
$229$	17.0988	1.12992	0.564959	+	0.825119i	$0.308892\pi$
$230$	0	0
$231$	0	0
$232$	8.85439	0.581319
$233$	−20.3833	−1.33536	−0.667678	−	0.744450i	$-0.732712\pi$
$233$	−20.3833	−1.33536	−0.667678	+	0.744450i	$0.732712\pi$
$234$	0	0
$235$	0	0
$236$	−2.11385	−0.137600
$237$	0	0
$238$	−2.11015	−0.136781
$239$	13.7888	0.891924	0.445962	−	0.895052i	$-0.352862\pi$
$239$	13.7888	0.891924	0.445962	+	0.895052i	$0.352862\pi$
$240$	0	0
$241$	19.2835	1.24216	0.621081	−	0.783746i	$-0.286694\pi$
$241$	19.2835	1.24216	0.621081	+	0.783746i	$0.286694\pi$
$242$	16.0306	1.03048
$243$	0	0
$244$	12.8238	0.820963
$245$	0	0
$246$	0	0
$247$	−6.57224	−0.418182
$248$	−6.07590	−0.385820
$249$	0	0
$250$	0	0
$251$	28.4001	1.79260	0.896299	−	0.443449i	$-0.146245\pi$
$251$	28.4001	1.79260	0.896299	+	0.443449i	$0.146245\pi$
$252$	0	0
$253$	6.49538	0.408361
$254$	11.5365	0.723866
$255$	0	0
$256$	17.9196	1.11997
$257$	−21.6561	−1.35087	−0.675435	−	0.737420i	$-0.736044\pi$
$257$	−21.6561	−1.35087	−0.675435	+	0.737420i	$0.736044\pi$
$258$	0	0
$259$	−1.50430	−0.0934729
$260$	0	0
$261$	0	0
$262$	10.6301	0.656732
$263$	29.9857	1.84900	0.924498	−	0.381186i	$-0.124484\pi$
$263$	29.9857	1.84900	0.924498	+	0.381186i	$0.124484\pi$
$264$	0	0
$265$	0	0
$266$	−0.576574	−0.0353520
$267$	0	0
$268$	0.145264	0.00887339
$269$	−9.71487	−0.592326	−0.296163	−	0.955137i	$-0.595707\pi$
$269$	−9.71487	−0.592326	−0.296163	+	0.955137i	$0.595707\pi$
$270$	0	0
$271$	−17.5794	−1.06787	−0.533937	−	0.845524i	$-0.679288\pi$
$271$	−17.5794	−1.06787	−0.533937	+	0.845524i	$0.679288\pi$
$272$	−18.2646	−1.10746
$273$	0	0
$274$	15.1386	0.914555
$275$	0	0
$276$	0	0
$277$	21.6333	1.29982	0.649910	−	0.760011i	$-0.274807\pi$
$277$	21.6333	1.29982	0.649910	+	0.760011i	$0.274807\pi$
$278$	−27.1811	−1.63021
$279$	0	0
$280$	0	0
$281$	24.3262	1.45118	0.725588	−	0.688129i	$-0.241568\pi$
$281$	24.3262	1.45118	0.725588	+	0.688129i	$0.241568\pi$
$282$	0	0
$283$	26.0867	1.55070	0.775348	−	0.631534i	$-0.217575\pi$
$283$	26.0867	1.55070	0.775348	+	0.631534i	$0.217575\pi$
$284$	−0.145264	−0.00861981
$285$	0	0
$286$	−14.1266	−0.835323
$287$	1.66876	0.0985037
$288$	0	0
$289$	−3.60583	−0.212108
$290$	0	0
$291$	0	0
$292$	4.58650	0.268405
$293$	25.8399	1.50958	0.754790	−	0.655966i	$-0.227739\pi$
$293$	25.8399	1.50958	0.754790	+	0.655966i	$0.227739\pi$
$294$	0	0
$295$	0	0
$296$	−8.30851	−0.482922
$297$	0	0
$298$	22.8850	1.32569
$299$	33.8392	1.95697
$300$	0	0
$301$	−3.15456	−0.181826
$302$	14.6494	0.842976
$303$	0	0
$304$	−4.99060	−0.286231
$305$	0	0
$306$	0	0
$307$	13.4520	0.767744	0.383872	−	0.923386i	$-0.374590\pi$
$307$	13.4520	0.767744	0.383872	+	0.923386i	$0.374590\pi$
$308$	−0.385509	−0.0219664
$309$	0	0
$310$	0	0
$311$	18.3714	1.04175	0.520874	−	0.853633i	$-0.325606\pi$
$311$	18.3714	1.04175	0.520874	+	0.853633i	$0.325606\pi$
$312$	0	0
$313$	5.36525	0.303262	0.151631	−	0.988437i	$-0.451547\pi$
$313$	5.36525	0.303262	0.151631	+	0.988437i	$0.451547\pi$
$314$	−40.1912	−2.26812
$315$	0	0
$316$	14.9940	0.843480
$317$	19.8770	1.11640	0.558201	−	0.829706i	$-0.311492\pi$
$317$	19.8770	1.11640	0.558201	+	0.829706i	$0.311492\pi$
$318$	0	0
$319$	−5.97641	−0.334615
$320$	0	0
$321$	0	0
$322$	2.96867	0.165438
$323$	3.65980	0.203637
$324$	0	0
$325$	0	0
$326$	7.79245	0.431584
$327$	0	0
$328$	9.21682	0.508914
$329$	2.85089	0.157175
$330$	0	0
$331$	−14.9005	−0.819007	−0.409504	−	0.912308i	$-0.634298\pi$
$331$	−14.9005	−0.819007	−0.409504	+	0.912308i	$0.634298\pi$
$332$	6.15846	0.337990
$333$	0	0
$334$	8.13964	0.445381
$335$	0	0
$336$	0	0
$337$	−1.38458	−0.0754229	−0.0377115	−	0.999289i	$-0.512007\pi$
$337$	−1.38458	−0.0754229	−0.0377115	+	0.999289i	$0.512007\pi$
$338$	−51.4461	−2.79830
$339$	0	0
$340$	0	0
$341$	4.10102	0.222083
$342$	0	0
$343$	−4.69883	−0.253713
$344$	−17.4231	−0.939391
$345$	0	0
$346$	−21.4537	−1.15336
$347$	6.07341	0.326038	0.163019	−	0.986623i	$-0.447877\pi$
$347$	6.07341	0.326038	0.163019	+	0.986623i	$0.447877\pi$
$348$	0	0
$349$	−24.3054	−1.30104	−0.650518	−	0.759491i	$-0.725448\pi$
$349$	−24.3054	−1.30104	−0.650518	+	0.759491i	$0.725448\pi$
$350$	0	0
$351$	0	0
$352$	−6.01130	−0.320403
$353$	−1.64440	−0.0875225	−0.0437612	−	0.999042i	$-0.513934\pi$
$353$	−1.64440	−0.0875225	−0.0437612	+	0.999042i	$0.513934\pi$
$354$	0	0
$355$	0	0
$356$	14.7773	0.783196
$357$	0	0
$358$	23.1680	1.22447
$359$	−7.55763	−0.398877	−0.199438	−	0.979910i	$-0.563912\pi$
$359$	−7.55763	−0.398877	−0.199438	+	0.979910i	$0.563912\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−10.8838	−0.572039
$363$	0	0
$364$	−2.00840	−0.105269
$365$	0	0
$366$	0	0
$367$	27.8973	1.45623	0.728114	−	0.685457i	$-0.240397\pi$
$367$	27.8973	1.45623	0.728114	+	0.685457i	$0.240397\pi$
$368$	25.6957	1.33948
$369$	0	0
$370$	0	0
$371$	−2.36220	−0.122639
$372$	0	0
$373$	4.48495	0.232222	0.116111	−	0.993236i	$-0.462957\pi$
$373$	4.48495	0.232222	0.116111	+	0.993236i	$0.462957\pi$
$374$	7.86651	0.406767
$375$	0	0
$376$	15.7459	0.812034
$377$	−31.1355	−1.60356
$378$	0	0
$379$	−0.870441	−0.0447116	−0.0223558	−	0.999750i	$-0.507117\pi$
$379$	−0.870441	−0.0447116	−0.0223558	+	0.999750i	$0.507117\pi$
$380$	0	0
$381$	0	0
$382$	18.7107	0.957324
$383$	24.1047	1.23169	0.615845	−	0.787867i	$-0.288815\pi$
$383$	24.1047	1.23169	0.615845	+	0.787867i	$0.288815\pi$
$384$	0	0
$385$	0	0
$386$	−6.83980	−0.348137
$387$	0	0
$388$	−12.3879	−0.628902
$389$	9.42190	0.477709	0.238855	−	0.971055i	$-0.423228\pi$
$389$	9.42190	0.477709	0.238855	+	0.971055i	$0.423228\pi$
$390$	0	0
$391$	−18.8436	−0.952964
$392$	−12.8692	−0.649991
$393$	0	0
$394$	−15.0712	−0.759275
$395$	0	0
$396$	0	0
$397$	22.4127	1.12486	0.562429	−	0.826845i	$-0.309867\pi$
$397$	22.4127	1.12486	0.562429	+	0.826845i	$0.309867\pi$
$398$	25.9964	1.30308
$399$	0	0
$400$	0	0
$401$	19.1341	0.955513	0.477757	−	0.878492i	$-0.341450\pi$
$401$	19.1341	0.955513	0.477757	+	0.878492i	$0.341450\pi$
$402$	0	0
$403$	21.3653	1.06428
$404$	2.72281	0.135465
$405$	0	0
$406$	−2.73148	−0.135561
$407$	5.60796	0.277976
$408$	0	0
$409$	−29.9491	−1.48089	−0.740445	−	0.672117i	$-0.765385\pi$
$409$	−29.9491	−1.48089	−0.740445	+	0.672117i	$0.765385\pi$
$410$	0	0
$411$	0	0
$412$	−5.44896	−0.268451
$413$	−0.792125	−0.0389779
$414$	0	0
$415$	0	0
$416$	−31.3173	−1.53546
$417$	0	0
$418$	2.14943	0.105132
$419$	9.25091	0.451936	0.225968	−	0.974135i	$-0.427445\pi$
$419$	9.25091	0.451936	0.225968	+	0.974135i	$0.427445\pi$
$420$	0	0
$421$	−1.37526	−0.0670263	−0.0335131	−	0.999438i	$-0.510670\pi$
$421$	−1.37526	−0.0670263	−0.0335131	+	0.999438i	$0.510670\pi$
$422$	−27.1928	−1.32373
$423$	0	0
$424$	−13.0468	−0.633608
$425$	0	0
$426$	0	0
$427$	4.80548	0.232554
$428$	4.22336	0.204144
$429$	0	0
$430$	0	0
$431$	37.8011	1.82082	0.910408	−	0.413713i	$-0.135768\pi$
$431$	37.8011	1.82082	0.910408	+	0.413713i	$0.135768\pi$
$432$	0	0
$433$	−11.8369	−0.568844	−0.284422	−	0.958699i	$-0.591802\pi$
$433$	−11.8369	−0.568844	−0.284422	+	0.958699i	$0.591802\pi$
$434$	1.87435	0.0899715
$435$	0	0
$436$	1.98120	0.0948822
$437$	−5.14881	−0.246301
$438$	0	0
$439$	1.26164	0.0602148	0.0301074	−	0.999547i	$-0.490415\pi$
$439$	1.26164	0.0602148	0.0301074	+	0.999547i	$0.490415\pi$
$440$	0	0
$441$	0	0
$442$	40.9825	1.94934
$443$	12.3103	0.584880	0.292440	−	0.956284i	$-0.405533\pi$
$443$	12.3103	0.584880	0.292440	+	0.956284i	$0.405533\pi$
$444$	0	0
$445$	0	0
$446$	24.2078	1.14627
$447$	0	0
$448$	0.630194	0.0297738
$449$	−20.5974	−0.972049	−0.486025	−	0.873945i	$-0.661554\pi$
$449$	−20.5974	−0.972049	−0.486025	+	0.873945i	$0.661554\pi$
$450$	0	0
$451$	−6.22103	−0.292937
$452$	−2.29878	−0.108125
$453$	0	0
$454$	−8.49783	−0.398823
$455$	0	0
$456$	0	0
$457$	5.44702	0.254801	0.127401	−	0.991851i	$-0.459337\pi$
$457$	5.44702	0.254801	0.127401	+	0.991851i	$0.459337\pi$
$458$	−29.1334	−1.36132
$459$	0	0
$460$	0	0
$461$	20.2718	0.944152	0.472076	−	0.881558i	$-0.343505\pi$
$461$	20.2718	0.944152	0.472076	+	0.881558i	$0.343505\pi$
$462$	0	0
$463$	−5.96485	−0.277210	−0.138605	−	0.990348i	$-0.544262\pi$
$463$	−5.96485	−0.277210	−0.138605	+	0.990348i	$0.544262\pi$
$464$	−23.6426	−1.09758
$465$	0	0
$466$	34.7298	1.60883
$467$	23.9168	1.10674	0.553368	−	0.832937i	$-0.313342\pi$
$467$	23.9168	1.10674	0.553368	+	0.832937i	$0.313342\pi$
$468$	0	0
$469$	0.0544347	0.00251356
$470$	0	0
$471$	0	0
$472$	−4.37503	−0.201377
$473$	11.7600	0.540725
$474$	0	0
$475$	0	0
$476$	1.11840	0.0512616
$477$	0	0
$478$	−23.4938	−1.07458
$479$	22.8932	1.04602	0.523009	−	0.852327i	$-0.324809\pi$
$479$	22.8932	1.04602	0.523009	+	0.852327i	$0.324809\pi$
$480$	0	0
$481$	29.2160	1.33213
$482$	−32.8559	−1.49655
$483$	0	0
$484$	−8.49634	−0.386197
$485$	0	0
$486$	0	0
$487$	−29.2434	−1.32515	−0.662573	−	0.748997i	$-0.730536\pi$
$487$	−29.2434	−1.32515	−0.662573	+	0.748997i	$0.730536\pi$
$488$	26.5414	1.20147
$489$	0	0
$490$	0	0
$491$	−38.4307	−1.73435	−0.867177	−	0.498001i	$-0.834068\pi$
$491$	−38.4307	−1.73435	−0.867177	+	0.498001i	$0.834068\pi$
$492$	0	0
$493$	17.3381	0.780868
$494$	11.1980	0.503822
$495$	0	0
$496$	16.2236	0.728461
$497$	−0.0544347	−0.00244173
$498$	0	0
$499$	−18.0065	−0.806081	−0.403041	−	0.915182i	$-0.632047\pi$
$499$	−18.0065	−0.806081	−0.403041	+	0.915182i	$0.632047\pi$
$500$	0	0
$501$	0	0
$502$	−48.3890	−2.15971
$503$	21.9944	0.980683	0.490341	−	0.871530i	$-0.336872\pi$
$503$	21.9944	0.980683	0.490341	+	0.871530i	$0.336872\pi$
$504$	0	0
$505$	0	0
$506$	−11.0670	−0.491990
$507$	0	0
$508$	−6.11445	−0.271285
$509$	28.2815	1.25356	0.626778	−	0.779198i	$-0.284373\pi$
$509$	28.2815	1.25356	0.626778	+	0.779198i	$0.284373\pi$
$510$	0	0
$511$	1.71870	0.0760308
$512$	−5.12552	−0.226518
$513$	0	0
$514$	36.8983	1.62752
$515$	0	0
$516$	0	0
$517$	−10.6279	−0.467417
$518$	2.56308	0.112615
$519$	0	0
$520$	0	0
$521$	38.9817	1.70782	0.853909	−	0.520422i	$-0.174225\pi$
$521$	38.9817	1.70782	0.853909	+	0.520422i	$0.174225\pi$
$522$	0	0
$523$	−15.9957	−0.699445	−0.349723	−	0.936853i	$-0.613724\pi$
$523$	−15.9957	−0.699445	−0.349723	+	0.936853i	$0.613724\pi$
$524$	−5.63406	−0.246125
$525$	0	0
$526$	−51.0906	−2.22766
$527$	−11.8974	−0.518259
$528$	0	0
$529$	3.51028	0.152621
$530$	0	0
$531$	0	0
$532$	0.305589	0.0132490
$533$	−32.4100	−1.40383
$534$	0	0
$535$	0	0
$536$	0.300652	0.0129862
$537$	0	0
$538$	16.5525	0.713629
$539$	8.68624	0.374143
$540$	0	0
$541$	29.8743	1.28439	0.642197	−	0.766539i	$-0.278023\pi$
$541$	29.8743	1.28439	0.642197	+	0.766539i	$0.278023\pi$
$542$	29.9524	1.28656
$543$	0	0
$544$	17.4393	0.747704
$545$	0	0
$546$	0	0
$547$	−10.7906	−0.461375	−0.230687	−	0.973028i	$-0.574097\pi$
$547$	−10.7906	−0.461375	−0.230687	+	0.973028i	$0.574097\pi$
$548$	−8.02357	−0.342750
$549$	0	0
$550$	0	0
$551$	4.73743	0.201821
$552$	0	0
$553$	5.61872	0.238932
$554$	−36.8595	−1.56601
$555$	0	0
$556$	14.4062	0.610958
$557$	21.5086	0.911349	0.455674	−	0.890147i	$-0.349398\pi$
$557$	21.5086	0.911349	0.455674	+	0.890147i	$0.349398\pi$
$558$	0	0
$559$	61.2665	2.59130
$560$	0	0
$561$	0	0
$562$	−41.4477	−1.74837
$563$	−38.2483	−1.61197	−0.805986	−	0.591934i	$-0.798364\pi$
$563$	−38.2483	−1.61197	−0.805986	+	0.591934i	$0.798364\pi$
$564$	0	0
$565$	0	0
$566$	−44.4474	−1.86826
$567$	0	0
$568$	−0.300652	−0.0126151
$569$	−1.58310	−0.0663672	−0.0331836	−	0.999449i	$-0.510565\pi$
$569$	−1.58310	−0.0663672	−0.0331836	+	0.999449i	$0.510565\pi$
$570$	0	0
$571$	16.7896	0.702625	0.351312	−	0.936258i	$-0.385736\pi$
$571$	16.7896	0.702625	0.351312	+	0.936258i	$0.385736\pi$
$572$	7.48721	0.313056
$573$	0	0
$574$	−2.84329	−0.118676
$575$	0	0
$576$	0	0
$577$	−12.4194	−0.517026	−0.258513	−	0.966008i	$-0.583232\pi$
$577$	−12.4194	−0.517026	−0.258513	+	0.966008i	$0.583232\pi$
$578$	6.14374	0.255546
$579$	0	0
$580$	0	0
$581$	2.30776	0.0957422
$582$	0	0
$583$	8.80613	0.364713
$584$	9.49265	0.392809
$585$	0	0
$586$	−44.0268	−1.81873
$587$	36.8546	1.52115	0.760577	−	0.649248i	$-0.224916\pi$
$587$	36.8546	1.52115	0.760577	+	0.649248i	$0.224916\pi$
$588$	0	0
$589$	−3.25083	−0.133948
$590$	0	0
$591$	0	0
$592$	22.1850	0.911799
$593$	−25.3678	−1.04173	−0.520866	−	0.853638i	$-0.674391\pi$
$593$	−25.3678	−1.04173	−0.520866	+	0.853638i	$0.674391\pi$
$594$	0	0
$595$	0	0
$596$	−12.1292	−0.496833
$597$	0	0
$598$	−57.6564	−2.35774
$599$	−9.06026	−0.370192	−0.185096	−	0.982720i	$-0.559260\pi$
$599$	−9.06026	−0.370192	−0.185096	+	0.982720i	$0.559260\pi$
$600$	0	0
$601$	−20.2910	−0.827688	−0.413844	−	0.910348i	$-0.635814\pi$
$601$	−20.2910	−0.827688	−0.413844	+	0.910348i	$0.635814\pi$
$602$	5.37483	0.219062
$603$	0	0
$604$	−7.76428	−0.315924
$605$	0	0
$606$	0	0
$607$	15.1815	0.616199	0.308099	−	0.951354i	$-0.400307\pi$
$607$	15.1815	0.616199	0.308099	+	0.951354i	$0.400307\pi$
$608$	4.76509	0.193250
$609$	0	0
$610$	0	0
$611$	−55.3689	−2.23998
$612$	0	0
$613$	−0.895386	−0.0361643	−0.0180821	−	0.999837i	$-0.505756\pi$
$613$	−0.895386	−0.0361643	−0.0180821	+	0.999837i	$0.505756\pi$
$614$	−22.9199	−0.924971
$615$	0	0
$616$	−0.797887	−0.0321478
$617$	−8.13119	−0.327349	−0.163675	−	0.986514i	$-0.552335\pi$
$617$	−8.13119	−0.327349	−0.163675	+	0.986514i	$0.552335\pi$
$618$	0	0
$619$	−30.5308	−1.22714	−0.613568	−	0.789642i	$-0.710266\pi$
$619$	−30.5308	−1.22714	−0.613568	+	0.789642i	$0.710266\pi$
$620$	0	0
$621$	0	0
$622$	−31.3018	−1.25509
$623$	5.53750	0.221855
$624$	0	0
$625$	0	0
$626$	−9.14149	−0.365367
$627$	0	0
$628$	21.3017	0.850029
$629$	−16.2692	−0.648694
$630$	0	0
$631$	−31.1834	−1.24139	−0.620696	−	0.784051i	$-0.713150\pi$
$631$	−31.1834	−1.24139	−0.620696	+	0.784051i	$0.713150\pi$
$632$	31.0331	1.23443
$633$	0	0
$634$	−33.8670	−1.34503
$635$	0	0
$636$	0	0
$637$	45.2531	1.79299
$638$	10.1828	0.403141
$639$	0	0
$640$	0	0
$641$	−41.9990	−1.65886	−0.829430	−	0.558611i	$-0.811335\pi$
$641$	−41.9990	−1.65886	−0.829430	+	0.558611i	$0.811335\pi$
$642$	0	0
$643$	37.0767	1.46216	0.731081	−	0.682291i	$-0.239016\pi$
$643$	37.0767	1.46216	0.731081	+	0.682291i	$0.239016\pi$
$644$	−1.57342	−0.0620014
$645$	0	0
$646$	−6.23569	−0.245340
$647$	11.4455	0.449967	0.224984	−	0.974363i	$-0.427767\pi$
$647$	11.4455	0.449967	0.224984	+	0.974363i	$0.427767\pi$
$648$	0	0
$649$	2.95300	0.115915
$650$	0	0
$651$	0	0
$652$	−4.13006	−0.161746
$653$	10.3303	0.404258	0.202129	−	0.979359i	$-0.435214\pi$
$653$	10.3303	0.404258	0.202129	+	0.979359i	$0.435214\pi$
$654$	0	0
$655$	0	0
$656$	−24.6104	−0.960873
$657$	0	0
$658$	−4.85744	−0.189363
$659$	33.0422	1.28714	0.643571	−	0.765387i	$-0.277452\pi$
$659$	33.0422	1.28714	0.643571	+	0.765387i	$0.277452\pi$
$660$	0	0
$661$	44.3029	1.72318	0.861592	−	0.507602i	$-0.169468\pi$
$661$	44.3029	1.72318	0.861592	+	0.507602i	$0.169468\pi$
$662$	25.3880	0.986733
$663$	0	0
$664$	12.7461	0.494646
$665$	0	0
$666$	0	0
$667$	−24.3922	−0.944468
$668$	−4.31407	−0.166917
$669$	0	0
$670$	0	0
$671$	−17.9146	−0.691584
$672$	0	0
$673$	24.5532	0.946458	0.473229	−	0.880939i	$-0.343088\pi$
$673$	24.5532	0.946458	0.473229	+	0.880939i	$0.343088\pi$
$674$	2.35909	0.0908689
$675$	0	0
$676$	27.2668	1.04872
$677$	31.1408	1.19684	0.598420	−	0.801183i	$-0.295796\pi$
$677$	31.1408	1.19684	0.598420	+	0.801183i	$0.295796\pi$
$678$	0	0
$679$	−4.64213	−0.178149
$680$	0	0
$681$	0	0
$682$	−6.98745	−0.267563
$683$	−1.25288	−0.0479401	−0.0239701	−	0.999713i	$-0.507631\pi$
$683$	−1.25288	−0.0479401	−0.0239701	+	0.999713i	$0.507631\pi$
$684$	0	0
$685$	0	0
$686$	8.00601	0.305671
$687$	0	0
$688$	46.5225	1.77365
$689$	45.8777	1.74780
$690$	0	0
$691$	8.26243	0.314318	0.157159	−	0.987573i	$-0.449767\pi$
$691$	8.26243	0.314318	0.157159	+	0.987573i	$0.449767\pi$
$692$	11.3707	0.432248
$693$	0	0
$694$	−10.3481	−0.392808
$695$	0	0
$696$	0	0
$697$	18.0478	0.683607
$698$	41.4123	1.56748
$699$	0	0
$700$	0	0
$701$	34.3808	1.29855	0.649273	−	0.760555i	$-0.275073\pi$
$701$	34.3808	1.29855	0.649273	+	0.760555i	$0.275073\pi$
$702$	0	0
$703$	−4.44536	−0.167660
$704$	−2.34932	−0.0885435
$705$	0	0
$706$	2.80178	0.105446
$707$	1.02032	0.0383730
$708$	0	0
$709$	−20.5553	−0.771972	−0.385986	−	0.922505i	$-0.626139\pi$
$709$	−20.5553	−0.771972	−0.385986	+	0.922505i	$0.626139\pi$
$710$	0	0
$711$	0	0
$712$	30.5845	1.14620
$713$	16.7379	0.626840
$714$	0	0
$715$	0	0
$716$	−12.2793	−0.458897
$717$	0	0
$718$	12.8769	0.480563
$719$	40.8151	1.52215	0.761074	−	0.648665i	$-0.224672\pi$
$719$	40.8151	1.52215	0.761074	+	0.648665i	$0.224672\pi$
$720$	0	0
$721$	−2.04189	−0.0760440
$722$	−1.70383	−0.0634101
$723$	0	0
$724$	5.76849	0.214384
$725$	0	0
$726$	0	0
$727$	−7.23285	−0.268252	−0.134126	−	0.990964i	$-0.542823\pi$
$727$	−7.23285	−0.268252	−0.134126	+	0.990964i	$0.542823\pi$
$728$	−4.15678	−0.154061
$729$	0	0
$730$	0	0
$731$	−34.1168	−1.26185
$732$	0	0
$733$	−14.8463	−0.548361	−0.274180	−	0.961678i	$-0.588407\pi$
$733$	−14.8463	−0.548361	−0.274180	+	0.961678i	$0.588407\pi$
$734$	−47.5323	−1.75445
$735$	0	0
$736$	−24.5346	−0.904356
$737$	−0.202929	−0.00747500
$738$	0	0
$739$	46.2049	1.69967	0.849837	−	0.527046i	$-0.176701\pi$
$739$	46.2049	1.69967	0.849837	+	0.527046i	$0.176701\pi$
$740$	0	0
$741$	0	0
$742$	4.02479	0.147755
$743$	−27.5405	−1.01036	−0.505181	−	0.863014i	$-0.668574\pi$
$743$	−27.5405	−1.01036	−0.505181	+	0.863014i	$0.668574\pi$
$744$	0	0
$745$	0	0
$746$	−7.64160	−0.279779
$747$	0	0
$748$	−4.16931	−0.152445
$749$	1.58262	0.0578277
$750$	0	0
$751$	4.64254	0.169409	0.0847044	−	0.996406i	$-0.473005\pi$
$751$	4.64254	0.169409	0.0847044	+	0.996406i	$0.473005\pi$
$752$	−42.0441	−1.53319
$753$	0	0
$754$	53.0498	1.93196
$755$	0	0
$756$	0	0
$757$	21.0894	0.766506	0.383253	−	0.923643i	$-0.374804\pi$
$757$	21.0894	0.766506	0.383253	+	0.923643i	$0.374804\pi$
$758$	1.48309	0.0538681
$759$	0	0
$760$	0	0
$761$	−25.7211	−0.932391	−0.466195	−	0.884682i	$-0.654376\pi$
$761$	−25.7211	−0.932391	−0.466195	+	0.884682i	$0.654376\pi$
$762$	0	0
$763$	0.742416	0.0268772
$764$	−9.91683	−0.358778
$765$	0	0
$766$	−41.0703	−1.48393
$767$	15.3843	0.555496
$768$	0	0
$769$	4.09791	0.147774	0.0738872	−	0.997267i	$-0.476460\pi$
$769$	4.09791	0.147774	0.0738872	+	0.997267i	$0.476460\pi$
$770$	0	0
$771$	0	0
$772$	3.62515	0.130472
$773$	−6.10515	−0.219587	−0.109794	−	0.993954i	$-0.535019\pi$
$773$	−6.10515	−0.219587	−0.109794	+	0.993954i	$0.535019\pi$
$774$	0	0
$775$	0	0
$776$	−25.6392	−0.920395
$777$	0	0
$778$	−16.0533	−0.575540
$779$	4.93134	0.176684
$780$	0	0
$781$	0.202929	0.00726138
$782$	32.1064	1.14812
$783$	0	0
$784$	34.3627	1.22724
$785$	0	0
$786$	0	0
$787$	5.40729	0.192749	0.0963746	−	0.995345i	$-0.469275\pi$
$787$	5.40729	0.192749	0.0963746	+	0.995345i	$0.469275\pi$
$788$	7.98784	0.284555
$789$	0	0
$790$	0	0
$791$	−0.861422	−0.0306287
$792$	0	0
$793$	−93.3302	−3.31425
$794$	−38.1874	−1.35522
$795$	0	0
$796$	−13.7783	−0.488359
$797$	−15.2892	−0.541572	−0.270786	−	0.962640i	$-0.587284\pi$
$797$	−15.2892	−0.541572	−0.270786	+	0.962640i	$0.587284\pi$
$798$	0	0
$799$	30.8326	1.09078
$800$	0	0
$801$	0	0
$802$	−32.6014	−1.15119
$803$	−6.40721	−0.226106
$804$	0	0
$805$	0	0
$806$	−36.4028	−1.28223
$807$	0	0
$808$	5.63538	0.198252
$809$	14.1943	0.499046	0.249523	−	0.968369i	$-0.419726\pi$
$809$	14.1943	0.499046	0.249523	+	0.968369i	$0.419726\pi$
$810$	0	0
$811$	7.04888	0.247520	0.123760	−	0.992312i	$-0.460505\pi$
$811$	7.04888	0.247520	0.123760	+	0.992312i	$0.460505\pi$
$812$	1.44771	0.0508046
$813$	0	0
$814$	−9.55502	−0.334903
$815$	0	0
$816$	0	0
$817$	−9.32202	−0.326136
$818$	51.0283	1.78416
$819$	0	0
$820$	0	0
$821$	−4.55973	−0.159136	−0.0795678	−	0.996829i	$-0.525354\pi$
$821$	−4.55973	−0.159136	−0.0795678	+	0.996829i	$0.525354\pi$
$822$	0	0
$823$	51.4832	1.79459	0.897296	−	0.441430i	$-0.145529\pi$
$823$	51.4832	1.79459	0.897296	+	0.441430i	$0.145529\pi$
$824$	−11.2777	−0.392877
$825$	0	0
$826$	1.34965	0.0469603
$827$	−5.27138	−0.183304	−0.0916519	−	0.995791i	$-0.529215\pi$
$827$	−5.27138	−0.183304	−0.0916519	+	0.995791i	$0.529215\pi$
$828$	0	0
$829$	−3.33691	−0.115896	−0.0579478	−	0.998320i	$-0.518456\pi$
$829$	−3.33691	−0.115896	−0.0579478	+	0.998320i	$0.518456\pi$
$830$	0	0
$831$	0	0
$832$	−12.2394	−0.424324
$833$	−25.1995	−0.873112
$834$	0	0
$835$	0	0
$836$	−1.13922	−0.0394007
$837$	0	0
$838$	−15.7620	−0.544489
$839$	−46.6426	−1.61028	−0.805141	−	0.593084i	$-0.797910\pi$
$839$	−46.6426	−1.61028	−0.805141	+	0.593084i	$0.797910\pi$
$840$	0	0
$841$	−6.55672	−0.226094
$842$	2.34322	0.0807527
$843$	0	0
$844$	14.4124	0.496096
$845$	0	0
$846$	0	0
$847$	−3.18384	−0.109398
$848$	34.8370	1.19631
$849$	0	0
$850$	0	0
$851$	22.8883	0.784602
$852$	0	0
$853$	−54.9577	−1.88171	−0.940857	−	0.338803i	$-0.889978\pi$
$853$	−54.9577	−1.88171	−0.940857	+	0.338803i	$0.889978\pi$
$854$	−8.18774	−0.280179
$855$	0	0
$856$	8.74106	0.298763
$857$	−20.9465	−0.715520	−0.357760	−	0.933814i	$-0.616459\pi$
$857$	−20.9465	−0.715520	−0.357760	+	0.933814i	$0.616459\pi$
$858$	0	0
$859$	−22.8957	−0.781191	−0.390595	−	0.920562i	$-0.627731\pi$
$859$	−22.8957	−0.781191	−0.390595	+	0.920562i	$0.627731\pi$
$860$	0	0
$861$	0	0
$862$	−64.4067	−2.19370
$863$	−1.14448	−0.0389585	−0.0194792	−	0.999810i	$-0.506201\pi$
$863$	−1.14448	−0.0389585	−0.0194792	+	0.999810i	$0.506201\pi$
$864$	0	0
$865$	0	0
$866$	20.1681	0.685339
$867$	0	0
$868$	−0.993419	−0.0337188
$869$	−20.9463	−0.710553
$870$	0	0
$871$	−1.05721	−0.0358222
$872$	4.10048	0.138860
$873$	0	0
$874$	8.77271	0.296742
$875$	0	0
$876$	0	0
$877$	−24.0552	−0.812285	−0.406143	−	0.913810i	$-0.633126\pi$
$877$	−24.0552	−0.812285	−0.406143	+	0.913810i	$0.633126\pi$
$878$	−2.14962	−0.0725463
$879$	0	0
$880$	0	0
$881$	−6.21009	−0.209223	−0.104612	−	0.994513i	$-0.533360\pi$
$881$	−6.21009	−0.209223	−0.104612	+	0.994513i	$0.533360\pi$
$882$	0	0
$883$	48.6495	1.63719	0.818594	−	0.574373i	$-0.194754\pi$
$883$	48.6495	1.63719	0.818594	+	0.574373i	$0.194754\pi$
$884$	−21.7210	−0.730558
$885$	0	0
$886$	−20.9747	−0.704659
$887$	−21.6544	−0.727085	−0.363542	−	0.931578i	$-0.618433\pi$
$887$	−21.6544	−0.727085	−0.363542	+	0.931578i	$0.618433\pi$
$888$	0	0
$889$	−2.29127	−0.0768468
$890$	0	0
$891$	0	0
$892$	−12.8303	−0.429592
$893$	8.42466	0.281920
$894$	0	0
$895$	0	0
$896$	−4.29874	−0.143611
$897$	0	0
$898$	35.0944	1.17112
$899$	−15.4006	−0.513639
$900$	0	0
$901$	−25.5474	−0.851106
$902$	10.5996	0.352928
$903$	0	0
$904$	−4.75777	−0.158241
$905$	0	0
$906$	0	0
$907$	−34.4404	−1.14357	−0.571787	−	0.820402i	$-0.693750\pi$
$907$	−34.4404	−1.14357	−0.571787	+	0.820402i	$0.693750\pi$
$908$	4.50392	0.149468
$909$	0	0
$910$	0	0
$911$	−0.736061	−0.0243868	−0.0121934	−	0.999926i	$-0.503881\pi$
$911$	−0.736061	−0.0243868	−0.0121934	+	0.999926i	$0.503881\pi$
$912$	0	0
$913$	−8.60321	−0.284725
$914$	−9.28082	−0.306982
$915$	0	0
$916$	15.4410	0.510184
$917$	−2.11125	−0.0697198
$918$	0	0
$919$	−50.4080	−1.66281	−0.831403	−	0.555670i	$-0.812462\pi$
$919$	−50.4080	−1.66281	−0.831403	+	0.555670i	$0.812462\pi$
$920$	0	0
$921$	0	0
$922$	−34.5397	−1.13751
$923$	1.05721	0.0347985
$924$	0	0
$925$	0	0
$926$	10.1631	0.333980
$927$	0	0
$928$	22.5743	0.741038
$929$	−48.1908	−1.58109	−0.790544	−	0.612405i	$-0.790202\pi$
$929$	−48.1908	−1.58109	−0.790544	+	0.612405i	$0.790202\pi$
$930$	0	0
$931$	−6.88549	−0.225663
$932$	−18.4071	−0.602943
$933$	0	0
$934$	−40.7502	−1.33339
$935$	0	0
$936$	0	0
$937$	54.6458	1.78520	0.892601	−	0.450848i	$-0.148878\pi$
$937$	54.6458	1.78520	0.892601	+	0.450848i	$0.148878\pi$
$938$	−0.0927476	−0.00302832
$939$	0	0
$940$	0	0
$941$	55.7010	1.81580	0.907901	−	0.419185i	$-0.137684\pi$
$941$	55.7010	1.81580	0.907901	+	0.419185i	$0.137684\pi$
$942$	0	0
$943$	−25.3906	−0.826831
$944$	11.6820	0.380218
$945$	0	0
$946$	−20.0371	−0.651461
$947$	−40.3454	−1.31105	−0.655524	−	0.755174i	$-0.727552\pi$
$947$	−40.3454	−1.31105	−0.655524	+	0.755174i	$0.727552\pi$
$948$	0	0
$949$	−33.3799	−1.08356
$950$	0	0
$951$	0	0
$952$	2.31474	0.0750211
$953$	−38.4655	−1.24602	−0.623009	−	0.782214i	$-0.714090\pi$
$953$	−38.4655	−1.24602	−0.623009	+	0.782214i	$0.714090\pi$
$954$	0	0
$955$	0	0
$956$	12.4519	0.402724
$957$	0	0
$958$	−39.0062	−1.26023
$959$	−3.00667	−0.0970906
$960$	0	0
$961$	−20.4321	−0.659099
$962$	−49.7792	−1.60494
$963$	0	0
$964$	17.4139	0.560864
$965$	0	0
$966$	0	0
$967$	−31.7715	−1.02170	−0.510851	−	0.859669i	$-0.670670\pi$
$967$	−31.7715	−1.02170	−0.510851	+	0.859669i	$0.670670\pi$
$968$	−17.5848	−0.565198
$969$	0	0
$970$	0	0
$971$	−43.7049	−1.40256	−0.701279	−	0.712887i	$-0.747387\pi$
$971$	−43.7049	−1.40256	−0.701279	+	0.712887i	$0.747387\pi$
$972$	0	0
$973$	5.39843	0.173066
$974$	49.8259	1.59652
$975$	0	0
$976$	−70.8699	−2.26849
$977$	−3.98536	−0.127503	−0.0637515	−	0.997966i	$-0.520306\pi$
$977$	−3.98536	−0.127503	−0.0637515	+	0.997966i	$0.520306\pi$
$978$	0	0
$979$	−20.6435	−0.659769
$980$	0	0
$981$	0	0
$982$	65.4795	2.08953
$983$	51.4517	1.64105	0.820527	−	0.571608i	$-0.193680\pi$
$983$	51.4517	1.64105	0.820527	+	0.571608i	$0.193680\pi$
$984$	0	0
$985$	0	0
$986$	−29.5412	−0.940783
$987$	0	0
$988$	−5.93503	−0.188818
$989$	47.9973	1.52623
$990$	0	0
$991$	24.0024	0.762463	0.381231	−	0.924480i	$-0.375500\pi$
$991$	24.0024	0.762463	0.381231	+	0.924480i	$0.375500\pi$
$992$	−15.4905	−0.491824
$993$	0	0
$994$	0.0927476	0.00294178
$995$	0	0
$996$	0	0
$997$	42.9027	1.35874	0.679371	−	0.733795i	$-0.262253\pi$
$997$	42.9027	1.35874	0.679371	+	0.733795i	$0.262253\pi$
$998$	30.6800	0.971160
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bw.1.2		7
3.2	odd	2		1425.2.a.y.1.6		7
5.2	odd	4		855.2.c.g.514.4		14
5.3	odd	4		855.2.c.g.514.11		14
5.4	even	2		4275.2.a.bv.1.6		7
15.2	even	4		285.2.c.b.229.11	yes	14
15.8	even	4		285.2.c.b.229.4	✓	14
15.14	odd	2		1425.2.a.z.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.c.b.229.4	✓	14	15.8	even	4
285.2.c.b.229.11	yes	14	15.2	even	4
855.2.c.g.514.4		14	5.2	odd	4
855.2.c.g.514.11		14	5.3	odd	4
1425.2.a.y.1.6		7	3.2	odd	2
1425.2.a.z.1.2		7	15.14	odd	2
4275.2.a.bv.1.6		7	5.4	even	2
4275.2.a.bw.1.2		7	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)

\(f(q)\)	\(=\)	\(q-1.70383 q^{2} +0.903045 q^{4} +0.338398 q^{7} +1.86903 q^{8} -1.26153 q^{11} -6.57224 q^{13} -0.576574 q^{14} -4.99060 q^{16} +3.65980 q^{17} +1.00000 q^{19} +2.14943 q^{22} -5.14881 q^{23} +11.1980 q^{26} +0.305589 q^{28} +4.73743 q^{29} -3.25083 q^{31} +4.76509 q^{32} -6.23569 q^{34} -4.44536 q^{37} -1.70383 q^{38} +4.93134 q^{41} -9.32202 q^{43} -1.13922 q^{44} +8.77271 q^{46} +8.42466 q^{47} -6.88549 q^{49} -5.93503 q^{52} -6.98052 q^{53} +0.632476 q^{56} -8.07179 q^{58} -2.34081 q^{59} +14.2007 q^{61} +5.53887 q^{62} +1.86228 q^{64} +0.160860 q^{67} +3.30497 q^{68} -0.160860 q^{71} +5.07893 q^{73} +7.57416 q^{74} +0.903045 q^{76} -0.426899 q^{77} +16.6039 q^{79} -8.40218 q^{82} +6.81966 q^{83} +15.8832 q^{86} -2.35783 q^{88} +16.3639 q^{89} -2.22403 q^{91} -4.64961 q^{92} -14.3542 q^{94} -13.7180 q^{97} +11.7317 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{2} + 11 q^{4} - 8 q^{7} + 9 q^{8} + 4 q^{11} - 8 q^{13} + 4 q^{14} + 19 q^{16} + 4 q^{17} + 7 q^{19} - 12 q^{22} + 10 q^{23} + 20 q^{26} - 14 q^{28} + 6 q^{29} + 4 q^{31} + 31 q^{32} + 2 q^{34}+ \cdots + 5 q^{98}+O(q^{100}) \) 7 * q + 3 * q^2 + 11 * q^4 - 8 * q^7 + 9 * q^8 + 4 * q^11 - 8 * q^13 + 4 * q^14 + 19 * q^16 + 4 * q^17 + 7 * q^19 - 12 * q^22 + 10 * q^23 + 20 * q^26 - 14 * q^28 + 6 * q^29 + 4 * q^31 + 31 * q^32 + 2 * q^34 - 14 * q^37 + 3 * q^38 - 2 * q^41 + 2 * q^43 + 32 * q^44 - 4 * q^46 + 30 * q^47 + 17 * q^49 - 18 * q^52 + 22 * q^56 + 40 * q^58 + 18 * q^59 + 12 * q^61 + 18 * q^62 + 11 * q^64 - 18 * q^67 - 12 * q^68 + 18 * q^71 - 10 * q^73 - 6 * q^74 + 11 * q^76 - 18 * q^77 - 4 * q^79 + 16 * q^82 + 18 * q^83 + 46 * q^86 + 18 * q^88 + 8 * q^89 + 12 * q^91 + 34 * q^92 - 20 * q^94 - 20 * q^97 + 5 * q^98

Introduction

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