LMFDB - Embedded newform 4275.2.a.bn.1.3

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:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 475)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.80194 q^{2} +5.85086 q^{4} +3.04892 q^{7} +10.7899 q^{8} +O(q^{10})$	$q+2.80194 q^{2} +5.85086 q^{4} +3.04892 q^{7} +10.7899 q^{8} +2.93900 q^{11} -3.24698 q^{13} +8.54288 q^{14} +18.5308 q^{16} -2.15883 q^{17} -1.00000 q^{19} +8.23490 q^{22} +1.19806 q^{23} -9.09783 q^{26} +17.8388 q^{28} +1.77479 q^{29} -9.34481 q^{31} +30.3424 q^{32} -6.04892 q^{34} +1.15883 q^{37} -2.80194 q^{38} -8.57002 q^{41} -5.27413 q^{43} +17.1957 q^{44} +3.35690 q^{46} +2.35690 q^{47} +2.29590 q^{49} -18.9976 q^{52} +8.82371 q^{53} +32.8974 q^{56} +4.97285 q^{58} +5.70171 q^{59} -9.96077 q^{61} -26.1836 q^{62} +47.9560 q^{64} -4.98254 q^{67} -12.6310 q^{68} -2.70171 q^{71} -13.7778 q^{73} +3.24698 q^{74} -5.85086 q^{76} +8.96077 q^{77} +5.66487 q^{79} -24.0127 q^{82} -3.00969 q^{83} -14.7778 q^{86} +31.7114 q^{88} +10.2838 q^{89} -9.89977 q^{91} +7.00969 q^{92} +6.60388 q^{94} -3.24698 q^{97} +6.43296 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{2} + 4 q^{4} + 9 q^{8}+O(q^{10}) $ 3 * q + 4 * q^2 + 4 * q^4 + 9 * q^8	$ 3 q + 4 q^{2} + 4 q^{4} + 9 q^{8} - q^{11} - 5 q^{13} + 7 q^{14} + 18 q^{16} + 2 q^{17} - 3 q^{19} + q^{22} + 8 q^{23} - 9 q^{26} + 21 q^{28} + 7 q^{29} - 5 q^{31} + 27 q^{32} - 9 q^{34} - 5 q^{37} - 4 q^{38} - q^{41} - 5 q^{43} + 15 q^{44} + 6 q^{46} + 3 q^{47} - 7 q^{49} - 16 q^{52} + 19 q^{53} + 35 q^{56} + 21 q^{58} - 10 q^{59} - 17 q^{61} - 23 q^{62} + 49 q^{64} + q^{67} - 23 q^{68} + 19 q^{71} + q^{73} + 5 q^{74} - 4 q^{76} + 14 q^{77} + 18 q^{79} - 6 q^{82} + 13 q^{83} - 2 q^{86} + 46 q^{88} - 2 q^{89} - 7 q^{91} - q^{92} + 11 q^{94} - 5 q^{97}+O(q^{100}) $ 3 * q + 4 * q^2 + 4 * q^4 + 9 * q^8 - q^11 - 5 * q^13 + 7 * q^14 + 18 * q^16 + 2 * q^17 - 3 * q^19 + q^22 + 8 * q^23 - 9 * q^26 + 21 * q^28 + 7 * q^29 - 5 * q^31 + 27 * q^32 - 9 * q^34 - 5 * q^37 - 4 * q^38 - q^41 - 5 * q^43 + 15 * q^44 + 6 * q^46 + 3 * q^47 - 7 * q^49 - 16 * q^52 + 19 * q^53 + 35 * q^56 + 21 * q^58 - 10 * q^59 - 17 * q^61 - 23 * q^62 + 49 * q^64 + q^67 - 23 * q^68 + 19 * q^71 + q^73 + 5 * q^74 - 4 * q^76 + 14 * q^77 + 18 * q^79 - 6 * q^82 + 13 * q^83 - 2 * q^86 + 46 * q^88 - 2 * q^89 - 7 * q^91 - q^92 + 11 * q^94 - 5 * q^97

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 $

$2$	2.80194	1.98127	0.990635	−	0.136540i	$-0.0435982\pi$
$2$	2.80194	1.98127	0.990635	+	0.136540i	$0.0435982\pi$
$3$	0	0
$3$	0	0
$4$	5.85086	2.92543
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.04892	1.15238	0.576191	−	0.817315i	$-0.304538\pi$
$7$	3.04892	1.15238	0.576191	+	0.817315i	$0.304538\pi$
$8$	10.7899	3.81479
$9$	0	0
$10$	0	0
$11$	2.93900	0.886142	0.443071	−	0.896486i	$-0.353889\pi$
$11$	2.93900	0.886142	0.443071	+	0.896486i	$0.353889\pi$
$12$	0	0
$13$	−3.24698	−0.900550	−0.450275	−	0.892890i	$-0.648674\pi$
$13$	−3.24698	−0.900550	−0.450275	+	0.892890i	$0.648674\pi$
$14$	8.54288	2.28318
$15$	0	0
$16$	18.5308	4.63270
$17$	−2.15883	−0.523594	−0.261797	−	0.965123i	$-0.584315\pi$
$17$	−2.15883	−0.523594	−0.261797	+	0.965123i	$0.584315\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	8.23490	1.75569
$23$	1.19806	0.249813	0.124907	−	0.992169i	$-0.460137\pi$
$23$	1.19806	0.249813	0.124907	+	0.992169i	$0.460137\pi$
$24$	0	0
$25$	0	0
$26$	−9.09783	−1.78423
$27$	0	0
$28$	17.8388	3.37121
$29$	1.77479	0.329570	0.164785	−	0.986329i	$-0.447307\pi$
$29$	1.77479	0.329570	0.164785	+	0.986329i	$0.447307\pi$
$30$	0	0
$31$	−9.34481	−1.67838	−0.839189	−	0.543840i	$-0.816970\pi$
$31$	−9.34481	−1.67838	−0.839189	+	0.543840i	$0.816970\pi$
$32$	30.3424	5.36383
$33$	0	0
$34$	−6.04892	−1.03738
$35$	0	0
$36$	0	0
$37$	1.15883	0.190511	0.0952555	−	0.995453i	$-0.469633\pi$
$37$	1.15883	0.190511	0.0952555	+	0.995453i	$0.469633\pi$
$38$	−2.80194	−0.454534
$39$	0	0
$40$	0	0
$41$	−8.57002	−1.33841	−0.669206	−	0.743077i	$-0.733366\pi$
$41$	−8.57002	−1.33841	−0.669206	+	0.743077i	$0.733366\pi$
$42$	0	0
$43$	−5.27413	−0.804297	−0.402148	−	0.915575i	$-0.631736\pi$
$43$	−5.27413	−0.804297	−0.402148	+	0.915575i	$0.631736\pi$
$44$	17.1957	2.59234
$45$	0	0
$46$	3.35690	0.494947
$47$	2.35690	0.343789	0.171894	−	0.985115i	$-0.445011\pi$
$47$	2.35690	0.343789	0.171894	+	0.985115i	$0.445011\pi$
$48$	0	0
$49$	2.29590	0.327985
$50$	0	0
$51$	0	0
$52$	−18.9976	−2.63449
$53$	8.82371	1.21203	0.606015	−	0.795453i	$-0.292767\pi$
$53$	8.82371	1.21203	0.606015	+	0.795453i	$0.292767\pi$
$54$	0	0
$55$	0	0
$56$	32.8974	4.39610
$57$	0	0
$58$	4.97285	0.652968
$59$	5.70171	0.742299	0.371150	−	0.928573i	$-0.378964\pi$
$59$	5.70171	0.742299	0.371150	+	0.928573i	$0.378964\pi$
$60$	0	0
$61$	−9.96077	−1.27535	−0.637673	−	0.770307i	$-0.720103\pi$
$61$	−9.96077	−1.27535	−0.637673	+	0.770307i	$0.720103\pi$
$62$	−26.1836	−3.32532
$63$	0	0
$64$	47.9560	5.99450
$65$	0	0
$66$	0	0
$67$	−4.98254	−0.608714	−0.304357	−	0.952558i	$-0.598442\pi$
$67$	−4.98254	−0.608714	−0.304357	+	0.952558i	$0.598442\pi$
$68$	−12.6310	−1.53174
$69$	0	0
$70$	0	0
$71$	−2.70171	−0.320634	−0.160317	−	0.987066i	$-0.551252\pi$
$71$	−2.70171	−0.320634	−0.160317	+	0.987066i	$0.551252\pi$
$72$	0	0
$73$	−13.7778	−1.61257	−0.806283	−	0.591530i	$-0.798524\pi$
$73$	−13.7778	−1.61257	−0.806283	+	0.591530i	$0.798524\pi$
$74$	3.24698	0.377454
$75$	0	0
$76$	−5.85086	−0.671139
$77$	8.96077	1.02117
$78$	0	0
$79$	5.66487	0.637348	0.318674	−	0.947864i	$-0.396762\pi$
$79$	5.66487	0.637348	0.318674	+	0.947864i	$0.396762\pi$
$80$	0	0
$81$	0	0
$82$	−24.0127	−2.65176
$83$	−3.00969	−0.330356	−0.165178	−	0.986264i	$-0.552820\pi$
$83$	−3.00969	−0.330356	−0.165178	+	0.986264i	$0.552820\pi$
$84$	0	0
$85$	0	0
$86$	−14.7778	−1.59353
$87$	0	0
$88$	31.7114	3.38045
$89$	10.2838	1.09008	0.545041	−	0.838409i	$-0.316514\pi$
$89$	10.2838	1.09008	0.545041	+	0.838409i	$0.316514\pi$
$90$	0	0
$91$	−9.89977	−1.03778
$92$	7.00969	0.730811
$93$	0	0
$94$	6.60388	0.681138
$95$	0	0
$96$	0	0
$97$	−3.24698	−0.329681	−0.164840	−	0.986320i	$-0.552711\pi$
$97$	−3.24698	−0.329681	−0.164840	+	0.986320i	$0.552711\pi$
$98$	6.43296	0.649827
$99$	0	0
$100$	0	0
$101$	−5.09246	−0.506718	−0.253359	−	0.967372i	$-0.581535\pi$
$101$	−5.09246	−0.506718	−0.253359	+	0.967372i	$0.581535\pi$
$102$	0	0
$103$	14.3110	1.41010	0.705051	−	0.709157i	$-0.250924\pi$
$103$	14.3110	1.41010	0.705051	+	0.709157i	$0.250924\pi$
$104$	−35.0344	−3.43541
$105$	0	0
$106$	24.7235	2.40136
$107$	−8.18598	−0.791369	−0.395684	−	0.918387i	$-0.629493\pi$
$107$	−8.18598	−0.791369	−0.395684	+	0.918387i	$0.629493\pi$
$108$	0	0
$109$	−11.3274	−1.08496	−0.542482	−	0.840067i	$-0.682515\pi$
$109$	−11.3274	−1.08496	−0.542482	+	0.840067i	$0.682515\pi$
$110$	0	0
$111$	0	0
$112$	56.4989	5.33864
$113$	3.43535	0.323171	0.161585	−	0.986859i	$-0.448339\pi$
$113$	3.43535	0.323171	0.161585	+	0.986859i	$0.448339\pi$
$114$	0	0
$115$	0	0
$116$	10.3840	0.964134
$117$	0	0
$118$	15.9758	1.47069
$119$	−6.58211	−0.603381
$120$	0	0
$121$	−2.36227	−0.214752
$122$	−27.9095	−2.52680
$123$	0	0
$124$	−54.6752	−4.90997
$125$	0	0
$126$	0	0
$127$	−1.46144	−0.129681	−0.0648407	−	0.997896i	$-0.520654\pi$
$127$	−1.46144	−0.129681	−0.0648407	+	0.997896i	$0.520654\pi$
$128$	73.6848	6.51288
$129$	0	0
$130$	0	0
$131$	13.2295	1.15587	0.577934	−	0.816083i	$-0.303859\pi$
$131$	13.2295	1.15587	0.577934	+	0.816083i	$0.303859\pi$
$132$	0	0
$133$	−3.04892	−0.264375
$134$	−13.9608	−1.20603
$135$	0	0
$136$	−23.2935	−1.99740
$137$	16.1739	1.38183	0.690915	−	0.722936i	$-0.257208\pi$
$137$	16.1739	1.38183	0.690915	+	0.722936i	$0.257208\pi$
$138$	0	0
$139$	13.0978	1.11094	0.555472	−	0.831535i	$-0.312538\pi$
$139$	13.0978	1.11094	0.555472	+	0.831535i	$0.312538\pi$
$140$	0	0
$141$	0	0
$142$	−7.57002	−0.635262
$143$	−9.54288	−0.798015
$144$	0	0
$145$	0	0
$146$	−38.6045	−3.19493
$147$	0	0
$148$	6.78017	0.557326
$149$	−11.0640	−0.906397	−0.453198	−	0.891410i	$-0.649717\pi$
$149$	−11.0640	−0.906397	−0.453198	+	0.891410i	$0.649717\pi$
$150$	0	0
$151$	10.4426	0.849811	0.424905	−	0.905238i	$-0.360307\pi$
$151$	10.4426	0.849811	0.424905	+	0.905238i	$0.360307\pi$
$152$	−10.7899	−0.875173
$153$	0	0
$154$	25.1075	2.02322
$155$	0	0
$156$	0	0
$157$	−2.48427	−0.198266	−0.0991332	−	0.995074i	$-0.531607\pi$
$157$	−2.48427	−0.198266	−0.0991332	+	0.995074i	$0.531607\pi$
$158$	15.8726	1.26276
$159$	0	0
$160$	0	0
$161$	3.65279	0.287880
$162$	0	0
$163$	−3.16852	−0.248178	−0.124089	−	0.992271i	$-0.539601\pi$
$163$	−3.16852	−0.248178	−0.124089	+	0.992271i	$0.539601\pi$
$164$	−50.1420	−3.91543
$165$	0	0
$166$	−8.43296	−0.654525
$167$	4.74632	0.367281	0.183640	−	0.982993i	$-0.441212\pi$
$167$	4.74632	0.367281	0.183640	+	0.982993i	$0.441212\pi$
$168$	0	0
$169$	−2.45712	−0.189009
$170$	0	0
$171$	0	0
$172$	−30.8582	−2.35291
$173$	3.96316	0.301314	0.150657	−	0.988586i	$-0.451861\pi$
$173$	3.96316	0.301314	0.150657	+	0.988586i	$0.451861\pi$
$174$	0	0
$175$	0	0
$176$	54.4620	4.10523
$177$	0	0
$178$	28.8146	2.15975
$179$	14.0858	1.05282	0.526409	−	0.850231i	$-0.323538\pi$
$179$	14.0858	1.05282	0.526409	+	0.850231i	$0.323538\pi$
$180$	0	0
$181$	−12.2513	−0.910631	−0.455316	−	0.890330i	$-0.650474\pi$
$181$	−12.2513	−0.910631	−0.455316	+	0.890330i	$0.650474\pi$
$182$	−27.7385	−2.05612
$183$	0	0
$184$	12.9269	0.952985
$185$	0	0
$186$	0	0
$187$	−6.34481	−0.463979
$188$	13.7899	1.00573
$189$	0	0
$190$	0	0
$191$	−20.1468	−1.45777	−0.728884	−	0.684637i	$-0.759961\pi$
$191$	−20.1468	−1.45777	−0.728884	+	0.684637i	$0.759961\pi$
$192$	0	0
$193$	9.52781	0.685827	0.342913	−	0.939367i	$-0.388586\pi$
$193$	9.52781	0.685827	0.342913	+	0.939367i	$0.388586\pi$
$194$	−9.09783	−0.653186
$195$	0	0
$196$	13.4330	0.959497
$197$	4.96316	0.353611	0.176805	−	0.984246i	$-0.443424\pi$
$197$	4.96316	0.353611	0.176805	+	0.984246i	$0.443424\pi$
$198$	0	0
$199$	5.42221	0.384370	0.192185	−	0.981359i	$-0.438443\pi$
$199$	5.42221	0.384370	0.192185	+	0.981359i	$0.438443\pi$
$200$	0	0
$201$	0	0
$202$	−14.2687	−1.00395
$203$	5.41119	0.379791
$204$	0	0
$205$	0	0
$206$	40.0984	2.79379
$207$	0	0
$208$	−60.1691	−4.17198
$209$	−2.93900	−0.203295
$210$	0	0
$211$	15.9638	1.09899	0.549495	−	0.835497i	$-0.314820\pi$
$211$	15.9638	1.09899	0.549495	+	0.835497i	$0.314820\pi$
$212$	51.6262	3.54570
$213$	0	0
$214$	−22.9366	−1.56791
$215$	0	0
$216$	0	0
$217$	−28.4916	−1.93413
$218$	−31.7385	−2.14961
$219$	0	0
$220$	0	0
$221$	7.00969	0.471523
$222$	0	0
$223$	23.2150	1.55459	0.777297	−	0.629134i	$-0.216590\pi$
$223$	23.2150	1.55459	0.777297	+	0.629134i	$0.216590\pi$
$224$	92.5115	6.18119
$225$	0	0
$226$	9.62565	0.640288
$227$	−17.4795	−1.16015	−0.580077	−	0.814562i	$-0.696978\pi$
$227$	−17.4795	−1.16015	−0.580077	+	0.814562i	$0.696978\pi$
$228$	0	0
$229$	−2.32544	−0.153669	−0.0768346	−	0.997044i	$-0.524481\pi$
$229$	−2.32544	−0.153669	−0.0768346	+	0.997044i	$0.524481\pi$
$230$	0	0
$231$	0	0
$232$	19.1497	1.25724
$233$	−18.9342	−1.24042	−0.620211	−	0.784435i	$-0.712953\pi$
$233$	−18.9342	−1.24042	−0.620211	+	0.784435i	$0.712953\pi$
$234$	0	0
$235$	0	0
$236$	33.3599	2.17154
$237$	0	0
$238$	−18.4426	−1.19546
$239$	−11.1685	−0.722432	−0.361216	−	0.932482i	$-0.617638\pi$
$239$	−11.1685	−0.722432	−0.361216	+	0.932482i	$0.617638\pi$
$240$	0	0
$241$	7.42998	0.478607	0.239303	−	0.970945i	$-0.423081\pi$
$241$	7.42998	0.478607	0.239303	+	0.970945i	$0.423081\pi$
$242$	−6.61894	−0.425482
$243$	0	0
$244$	−58.2790	−3.73093
$245$	0	0
$246$	0	0
$247$	3.24698	0.206600
$248$	−100.829	−6.40266
$249$	0	0
$250$	0	0
$251$	14.7409	0.930440	0.465220	−	0.885195i	$-0.345975\pi$
$251$	14.7409	0.930440	0.465220	+	0.885195i	$0.345975\pi$
$252$	0	0
$253$	3.52111	0.221370
$254$	−4.09485	−0.256934
$255$	0	0
$256$	110.548	6.90927
$257$	3.34721	0.208793	0.104397	−	0.994536i	$-0.466709\pi$
$257$	3.34721	0.208793	0.104397	+	0.994536i	$0.466709\pi$
$258$	0	0
$259$	3.53319	0.219542
$260$	0	0
$261$	0	0
$262$	37.0683	2.29009
$263$	−16.6853	−1.02886	−0.514430	−	0.857532i	$-0.671997\pi$
$263$	−16.6853	−1.02886	−0.514430	+	0.857532i	$0.671997\pi$
$264$	0	0
$265$	0	0
$266$	−8.54288	−0.523797
$267$	0	0
$268$	−29.1521	−1.78075
$269$	−15.5961	−0.950911	−0.475456	−	0.879740i	$-0.657717\pi$
$269$	−15.5961	−0.950911	−0.475456	+	0.879740i	$0.657717\pi$
$270$	0	0
$271$	−13.2131	−0.802640	−0.401320	−	0.915938i	$-0.631449\pi$
$271$	−13.2131	−0.802640	−0.401320	+	0.915938i	$0.631449\pi$
$272$	−40.0049	−2.42565
$273$	0	0
$274$	45.3183	2.73778
$275$	0	0
$276$	0	0
$277$	−12.9758	−0.779642	−0.389821	−	0.920891i	$-0.627463\pi$
$277$	−12.9758	−0.779642	−0.389821	+	0.920891i	$0.627463\pi$
$278$	36.6993	2.20108
$279$	0	0
$280$	0	0
$281$	−24.8901	−1.48482	−0.742409	−	0.669947i	$-0.766317\pi$
$281$	−24.8901	−1.48482	−0.742409	+	0.669947i	$0.766317\pi$
$282$	0	0
$283$	13.5821	0.807372	0.403686	−	0.914898i	$-0.367729\pi$
$283$	13.5821	0.807372	0.403686	+	0.914898i	$0.367729\pi$
$284$	−15.8073	−0.937992
$285$	0	0
$286$	−26.7385	−1.58108
$287$	−26.1293	−1.54236
$288$	0	0
$289$	−12.3394	−0.725849
$290$	0	0
$291$	0	0
$292$	−80.6118	−4.71745
$293$	1.27652	0.0745751	0.0372875	−	0.999305i	$-0.488128\pi$
$293$	1.27652	0.0745751	0.0372875	+	0.999305i	$0.488128\pi$
$294$	0	0
$295$	0	0
$296$	12.5036	0.726760
$297$	0	0
$298$	−31.0006	−1.79582
$299$	−3.89008	−0.224969
$300$	0	0
$301$	−16.0804	−0.926857
$302$	29.2597	1.68370
$303$	0	0
$304$	−18.5308	−1.06281
$305$	0	0
$306$	0	0
$307$	32.2295	1.83944	0.919718	−	0.392580i	$-0.128417\pi$
$307$	32.2295	1.83944	0.919718	+	0.392580i	$0.128417\pi$
$308$	52.4282	2.98737
$309$	0	0
$310$	0	0
$311$	−12.4983	−0.708712	−0.354356	−	0.935111i	$-0.615300\pi$
$311$	−12.4983	−0.708712	−0.354356	+	0.935111i	$0.615300\pi$
$312$	0	0
$313$	20.9390	1.18354	0.591771	−	0.806106i	$-0.298429\pi$
$313$	20.9390	1.18354	0.591771	+	0.806106i	$0.298429\pi$
$314$	−6.96077	−0.392819
$315$	0	0
$316$	33.1444	1.86452
$317$	15.8562	0.890575	0.445287	−	0.895388i	$-0.353102\pi$
$317$	15.8562	0.890575	0.445287	+	0.895388i	$0.353102\pi$
$318$	0	0
$319$	5.21611	0.292046
$320$	0	0
$321$	0	0
$322$	10.2349	0.570369
$323$	2.15883	0.120121
$324$	0	0
$325$	0	0
$326$	−8.87800	−0.491707
$327$	0	0
$328$	−92.4693	−5.10576
$329$	7.18598	0.396176
$330$	0	0
$331$	−13.2349	−0.727456	−0.363728	−	0.931505i	$-0.618496\pi$
$331$	−13.2349	−0.727456	−0.363728	+	0.931505i	$0.618496\pi$
$332$	−17.6093	−0.966433
$333$	0	0
$334$	13.2989	0.727682
$335$	0	0
$336$	0	0
$337$	26.0780	1.42056	0.710279	−	0.703920i	$-0.248569\pi$
$337$	26.0780	1.42056	0.710279	+	0.703920i	$0.248569\pi$
$338$	−6.88471	−0.374479
$339$	0	0
$340$	0	0
$341$	−27.4644	−1.48728
$342$	0	0
$343$	−14.3424	−0.774418
$344$	−56.9071	−3.06822
$345$	0	0
$346$	11.1045	0.596984
$347$	8.98254	0.482208	0.241104	−	0.970499i	$-0.422490\pi$
$347$	8.98254	0.482208	0.241104	+	0.970499i	$0.422490\pi$
$348$	0	0
$349$	0.599564	0.0320939	0.0160470	−	0.999871i	$-0.494892\pi$
$349$	0.599564	0.0320939	0.0160470	+	0.999871i	$0.494892\pi$
$350$	0	0
$351$	0	0
$352$	89.1764	4.75312
$353$	31.8896	1.69731	0.848656	−	0.528945i	$-0.177412\pi$
$353$	31.8896	1.69731	0.848656	+	0.528945i	$0.177412\pi$
$354$	0	0
$355$	0	0
$356$	60.1691	3.18896
$357$	0	0
$358$	39.4674	2.08592
$359$	18.3763	0.969863	0.484931	−	0.874552i	$-0.338845\pi$
$359$	18.3763	0.969863	0.484931	+	0.874552i	$0.338845\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−34.3274	−1.80421
$363$	0	0
$364$	−57.9221	−3.03594
$365$	0	0
$366$	0	0
$367$	14.6377	0.764083	0.382042	−	0.924145i	$-0.375221\pi$
$367$	14.6377	0.764083	0.382042	+	0.924145i	$0.375221\pi$
$368$	22.2010	1.15731
$369$	0	0
$370$	0	0
$371$	26.9028	1.39672
$372$	0	0
$373$	1.02715	0.0531837	0.0265918	−	0.999646i	$-0.491535\pi$
$373$	1.02715	0.0531837	0.0265918	+	0.999646i	$0.491535\pi$
$374$	−17.7778	−0.919267
$375$	0	0
$376$	25.4306	1.31148
$377$	−5.76271	−0.296795
$378$	0	0
$379$	−22.5284	−1.15721	−0.578603	−	0.815609i	$-0.696402\pi$
$379$	−22.5284	−1.15721	−0.578603	+	0.815609i	$0.696402\pi$
$380$	0	0
$381$	0	0
$382$	−56.4499	−2.88823
$383$	32.6698	1.66935	0.834674	−	0.550745i	$-0.185656\pi$
$383$	32.6698	1.66935	0.834674	+	0.550745i	$0.185656\pi$
$384$	0	0
$385$	0	0
$386$	26.6963	1.35881
$387$	0	0
$388$	−18.9976	−0.964457
$389$	8.16421	0.413942	0.206971	−	0.978347i	$-0.433639\pi$
$389$	8.16421	0.413942	0.206971	+	0.978347i	$0.433639\pi$
$390$	0	0
$391$	−2.58642	−0.130801
$392$	24.7724	1.25120
$393$	0	0
$394$	13.9065	0.700598
$395$	0	0
$396$	0	0
$397$	−37.5478	−1.88447	−0.942235	−	0.334954i	$-0.891279\pi$
$397$	−37.5478	−1.88447	−0.942235	+	0.334954i	$0.891279\pi$
$398$	15.1927	0.761541
$399$	0	0
$400$	0	0
$401$	−32.2519	−1.61058	−0.805291	−	0.592880i	$-0.797991\pi$
$401$	−32.2519	−1.61058	−0.805291	+	0.592880i	$0.797991\pi$
$402$	0	0
$403$	30.3424	1.51146
$404$	−29.7952	−1.48237
$405$	0	0
$406$	15.1618	0.752468
$407$	3.40581	0.168820
$408$	0	0
$409$	−31.5459	−1.55984	−0.779921	−	0.625878i	$-0.784741\pi$
$409$	−31.5459	−1.55984	−0.779921	+	0.625878i	$0.784741\pi$
$410$	0	0
$411$	0	0
$412$	83.7314	4.12515
$413$	17.3840	0.855413
$414$	0	0
$415$	0	0
$416$	−98.5212	−4.83040
$417$	0	0
$418$	−8.23490	−0.402782
$419$	14.6866	0.717490	0.358745	−	0.933436i	$-0.383205\pi$
$419$	14.6866	0.717490	0.358745	+	0.933436i	$0.383205\pi$
$420$	0	0
$421$	15.1142	0.736622	0.368311	−	0.929703i	$-0.379936\pi$
$421$	15.1142	0.736622	0.368311	+	0.929703i	$0.379936\pi$
$422$	44.7294	2.17740
$423$	0	0
$424$	95.2065	4.62364
$425$	0	0
$426$	0	0
$427$	−30.3696	−1.46969
$428$	−47.8950	−2.31509
$429$	0	0
$430$	0	0
$431$	−1.67696	−0.0807761	−0.0403881	−	0.999184i	$-0.512859\pi$
$431$	−1.67696	−0.0807761	−0.0403881	+	0.999184i	$0.512859\pi$
$432$	0	0
$433$	13.1545	0.632166	0.316083	−	0.948732i	$-0.397632\pi$
$433$	13.1545	0.632166	0.316083	+	0.948732i	$0.397632\pi$
$434$	−79.8316	−3.83204
$435$	0	0
$436$	−66.2747	−3.17398
$437$	−1.19806	−0.0573111
$438$	0	0
$439$	6.45580	0.308118	0.154059	−	0.988062i	$-0.450765\pi$
$439$	6.45580	0.308118	0.154059	+	0.988062i	$0.450765\pi$
$440$	0	0
$441$	0	0
$442$	19.6407	0.934213
$443$	−25.3709	−1.20541	−0.602704	−	0.797965i	$-0.705910\pi$
$443$	−25.3709	−1.20541	−0.602704	+	0.797965i	$0.705910\pi$
$444$	0	0
$445$	0	0
$446$	65.0471	3.08007
$447$	0	0
$448$	146.214	6.90795
$449$	3.59956	0.169874	0.0849370	−	0.996386i	$-0.472931\pi$
$449$	3.59956	0.169874	0.0849370	+	0.996386i	$0.472931\pi$
$450$	0	0
$451$	−25.1873	−1.18602
$452$	20.0998	0.945413
$453$	0	0
$454$	−48.9764	−2.29858
$455$	0	0
$456$	0	0
$457$	12.2784	0.574361	0.287181	−	0.957876i	$-0.407282\pi$
$457$	12.2784	0.574361	0.287181	+	0.957876i	$0.407282\pi$
$458$	−6.51573	−0.304460
$459$	0	0
$460$	0	0
$461$	−39.4935	−1.83939	−0.919697	−	0.392628i	$-0.871566\pi$
$461$	−39.4935	−1.83939	−0.919697	+	0.392628i	$0.871566\pi$
$462$	0	0
$463$	−18.3991	−0.855079	−0.427540	−	0.903997i	$-0.640620\pi$
$463$	−18.3991	−0.855079	−0.427540	+	0.903997i	$0.640620\pi$
$464$	32.8883	1.52680
$465$	0	0
$466$	−53.0525	−2.45761
$467$	−36.2282	−1.67644	−0.838220	−	0.545332i	$-0.816404\pi$
$467$	−36.2282	−1.67644	−0.838220	+	0.545332i	$0.816404\pi$
$468$	0	0
$469$	−15.1914	−0.701472
$470$	0	0
$471$	0	0
$472$	61.5206	2.83172
$473$	−15.5007	−0.712721
$474$	0	0
$475$	0	0
$476$	−38.5109	−1.76515
$477$	0	0
$478$	−31.2935	−1.43133
$479$	34.3129	1.56780	0.783898	−	0.620890i	$-0.213229\pi$
$479$	34.3129	1.56780	0.783898	+	0.620890i	$0.213229\pi$
$480$	0	0
$481$	−3.76271	−0.171565
$482$	20.8183	0.948249
$483$	0	0
$484$	−13.8213	−0.628242
$485$	0	0
$486$	0	0
$487$	24.4722	1.10894	0.554470	−	0.832203i	$-0.312921\pi$
$487$	24.4722	1.10894	0.554470	+	0.832203i	$0.312921\pi$
$488$	−107.475	−4.86518
$489$	0	0
$490$	0	0
$491$	−18.3067	−0.826168	−0.413084	−	0.910693i	$-0.635548\pi$
$491$	−18.3067	−0.826168	−0.413084	+	0.910693i	$0.635548\pi$
$492$	0	0
$493$	−3.83148	−0.172561
$494$	9.09783	0.409331
$495$	0	0
$496$	−173.167	−7.77542
$497$	−8.23729	−0.369493
$498$	0	0
$499$	−25.3424	−1.13448	−0.567241	−	0.823552i	$-0.691989\pi$
$499$	−25.3424	−1.13448	−0.567241	+	0.823552i	$0.691989\pi$
$500$	0	0
$501$	0	0
$502$	41.3032	1.84345
$503$	14.1105	0.629156	0.314578	−	0.949232i	$-0.398137\pi$
$503$	14.1105	0.629156	0.314578	+	0.949232i	$0.398137\pi$
$504$	0	0
$505$	0	0
$506$	9.86592	0.438594
$507$	0	0
$508$	−8.55065	−0.379374
$509$	17.8019	0.789057	0.394529	−	0.918884i	$-0.370908\pi$
$509$	17.8019	0.789057	0.394529	+	0.918884i	$0.370908\pi$
$510$	0	0
$511$	−42.0073	−1.85829
$512$	162.380	7.17625
$513$	0	0
$514$	9.37867	0.413675
$515$	0	0
$516$	0	0
$517$	6.92692	0.304646
$518$	9.89977	0.434971
$519$	0	0
$520$	0	0
$521$	−13.3948	−0.586837	−0.293418	−	0.955984i	$-0.594793\pi$
$521$	−13.3948	−0.586837	−0.293418	+	0.955984i	$0.594793\pi$
$522$	0	0
$523$	1.46921	0.0642439	0.0321219	−	0.999484i	$-0.489774\pi$
$523$	1.46921	0.0642439	0.0321219	+	0.999484i	$0.489774\pi$
$524$	77.4040	3.38141
$525$	0	0
$526$	−46.7512	−2.03845
$527$	20.1739	0.878789
$528$	0	0
$529$	−21.5646	−0.937593
$530$	0	0
$531$	0	0
$532$	−17.8388	−0.773409
$533$	27.8267	1.20531
$534$	0	0
$535$	0	0
$536$	−53.7609	−2.32212
$537$	0	0
$538$	−43.6993	−1.88401
$539$	6.74764	0.290642
$540$	0	0
$541$	−42.7144	−1.83643	−0.918217	−	0.396077i	$-0.870371\pi$
$541$	−42.7144	−1.83643	−0.918217	+	0.396077i	$0.870371\pi$
$542$	−37.0224	−1.59025
$543$	0	0
$544$	−65.5042	−2.80847
$545$	0	0
$546$	0	0
$547$	−10.6866	−0.456928	−0.228464	−	0.973552i	$-0.573370\pi$
$547$	−10.6866	−0.456928	−0.228464	+	0.973552i	$0.573370\pi$
$548$	94.6311	4.04244
$549$	0	0
$550$	0	0
$551$	−1.77479	−0.0756086
$552$	0	0
$553$	17.2717	0.734469
$554$	−36.3575	−1.54468
$555$	0	0
$556$	76.6335	3.24999
$557$	−34.2083	−1.44945	−0.724727	−	0.689036i	$-0.758034\pi$
$557$	−34.2083	−1.44945	−0.724727	+	0.689036i	$0.758034\pi$
$558$	0	0
$559$	17.1250	0.724310
$560$	0	0
$561$	0	0
$562$	−69.7405	−2.94182
$563$	−5.46788	−0.230444	−0.115222	−	0.993340i	$-0.536758\pi$
$563$	−5.46788	−0.230444	−0.115222	+	0.993340i	$0.536758\pi$
$564$	0	0
$565$	0	0
$566$	38.0562	1.59962
$567$	0	0
$568$	−29.1511	−1.22315
$569$	17.4819	0.732878	0.366439	−	0.930442i	$-0.380577\pi$
$569$	17.4819	0.732878	0.366439	+	0.930442i	$0.380577\pi$
$570$	0	0
$571$	−7.21877	−0.302096	−0.151048	−	0.988526i	$-0.548265\pi$
$571$	−7.21877	−0.302096	−0.151048	+	0.988526i	$0.548265\pi$
$572$	−55.8340	−2.33454
$573$	0	0
$574$	−73.2127	−3.05584
$575$	0	0
$576$	0	0
$577$	−1.80625	−0.0751952	−0.0375976	−	0.999293i	$-0.511971\pi$
$577$	−1.80625	−0.0751952	−0.0375976	+	0.999293i	$0.511971\pi$
$578$	−34.5743	−1.43810
$579$	0	0
$580$	0	0
$581$	−9.17629	−0.380697
$582$	0	0
$583$	25.9329	1.07403
$584$	−148.660	−6.15160
$585$	0	0
$586$	3.57673	0.147753
$587$	2.39075	0.0986767	0.0493384	−	0.998782i	$-0.484289\pi$
$587$	2.39075	0.0986767	0.0493384	+	0.998782i	$0.484289\pi$
$588$	0	0
$589$	9.34481	0.385046
$590$	0	0
$591$	0	0
$592$	21.4741	0.882580
$593$	−42.2650	−1.73562	−0.867808	−	0.496899i	$-0.834472\pi$
$593$	−42.2650	−1.73562	−0.867808	+	0.496899i	$0.834472\pi$
$594$	0	0
$595$	0	0
$596$	−64.7338	−2.65160
$597$	0	0
$598$	−10.8998	−0.445725
$599$	−8.95838	−0.366029	−0.183015	−	0.983110i	$-0.558586\pi$
$599$	−8.95838	−0.366029	−0.183015	+	0.983110i	$0.558586\pi$
$600$	0	0
$601$	−19.8998	−0.811729	−0.405864	−	0.913933i	$-0.633029\pi$
$601$	−19.8998	−0.811729	−0.405864	+	0.913933i	$0.633029\pi$
$602$	−45.0562	−1.83635
$603$	0	0
$604$	61.0984	2.48606
$605$	0	0
$606$	0	0
$607$	−26.0084	−1.05565	−0.527823	−	0.849354i	$-0.676992\pi$
$607$	−26.0084	−1.05565	−0.527823	+	0.849354i	$0.676992\pi$
$608$	−30.3424	−1.23055
$609$	0	0
$610$	0	0
$611$	−7.65279	−0.309599
$612$	0	0
$613$	−41.3763	−1.67117	−0.835586	−	0.549360i	$-0.814872\pi$
$613$	−41.3763	−1.67117	−0.835586	+	0.549360i	$0.814872\pi$
$614$	90.3051	3.64442
$615$	0	0
$616$	96.6854	3.89557
$617$	43.9845	1.77075	0.885374	−	0.464880i	$-0.153902\pi$
$617$	43.9845	1.77075	0.885374	+	0.464880i	$0.153902\pi$
$618$	0	0
$619$	23.4553	0.942749	0.471374	−	0.881933i	$-0.343758\pi$
$619$	23.4553	0.942749	0.471374	+	0.881933i	$0.343758\pi$
$620$	0	0
$621$	0	0
$622$	−35.0194	−1.40415
$623$	31.3545	1.25619
$624$	0	0
$625$	0	0
$626$	58.6698	2.34492
$627$	0	0
$628$	−14.5351	−0.580014
$629$	−2.50173	−0.0997505
$630$	0	0
$631$	−34.8877	−1.38886	−0.694429	−	0.719562i	$-0.744343\pi$
$631$	−34.8877	−1.38886	−0.694429	+	0.719562i	$0.744343\pi$
$632$	61.1232	2.43135
$633$	0	0
$634$	44.4282	1.76447
$635$	0	0
$636$	0	0
$637$	−7.45473	−0.295367
$638$	14.6152	0.578622
$639$	0	0
$640$	0	0
$641$	38.5314	1.52190	0.760949	−	0.648812i	$-0.224734\pi$
$641$	38.5314	1.52190	0.760949	+	0.648812i	$0.224734\pi$
$642$	0	0
$643$	18.6353	0.734906	0.367453	−	0.930042i	$-0.380230\pi$
$643$	18.6353	0.734906	0.367453	+	0.930042i	$0.380230\pi$
$644$	21.3720	0.842173
$645$	0	0
$646$	6.04892	0.237991
$647$	19.2282	0.755938	0.377969	−	0.925818i	$-0.376623\pi$
$647$	19.2282	0.755938	0.377969	+	0.925818i	$0.376623\pi$
$648$	0	0
$649$	16.7573	0.657783
$650$	0	0
$651$	0	0
$652$	−18.5386	−0.726026
$653$	−3.98553	−0.155966	−0.0779828	−	0.996955i	$-0.524848\pi$
$653$	−3.98553	−0.155966	−0.0779828	+	0.996955i	$0.524848\pi$
$654$	0	0
$655$	0	0
$656$	−158.809	−6.20046
$657$	0	0
$658$	20.1347	0.784931
$659$	−33.5719	−1.30778	−0.653889	−	0.756591i	$-0.726864\pi$
$659$	−33.5719	−1.30778	−0.653889	+	0.756591i	$0.726864\pi$
$660$	0	0
$661$	36.9909	1.43878	0.719390	−	0.694607i	$-0.244422\pi$
$661$	36.9909	1.43878	0.719390	+	0.694607i	$0.244422\pi$
$662$	−37.0834	−1.44129
$663$	0	0
$664$	−32.4741	−1.26024
$665$	0	0
$666$	0	0
$667$	2.12631	0.0823310
$668$	27.7700	1.07445
$669$	0	0
$670$	0	0
$671$	−29.2747	−1.13014
$672$	0	0
$673$	−12.0747	−0.465447	−0.232723	−	0.972543i	$-0.574764\pi$
$673$	−12.0747	−0.465447	−0.232723	+	0.972543i	$0.574764\pi$
$674$	73.0689	2.81451
$675$	0	0
$676$	−14.3763	−0.552934
$677$	41.9135	1.61087	0.805434	−	0.592686i	$-0.201933\pi$
$677$	41.9135	1.61087	0.805434	+	0.592686i	$0.201933\pi$
$678$	0	0
$679$	−9.89977	−0.379918
$680$	0	0
$681$	0	0
$682$	−76.9536	−2.94671
$683$	9.17092	0.350915	0.175458	−	0.984487i	$-0.443859\pi$
$683$	9.17092	0.350915	0.175458	+	0.984487i	$0.443859\pi$
$684$	0	0
$685$	0	0
$686$	−40.1866	−1.53433
$687$	0	0
$688$	−97.7338	−3.72606
$689$	−28.6504	−1.09149
$690$	0	0
$691$	−0.111244	−0.00423193	−0.00211596	−	0.999998i	$-0.500674\pi$
$691$	−0.111244	−0.00423193	−0.00211596	+	0.999998i	$0.500674\pi$
$692$	23.1879	0.881472
$693$	0	0
$694$	25.1685	0.955384
$695$	0	0
$696$	0	0
$697$	18.5013	0.700785
$698$	1.67994	0.0635867
$699$	0	0
$700$	0	0
$701$	27.7832	1.04936	0.524678	−	0.851301i	$-0.324186\pi$
$701$	27.7832	1.04936	0.524678	+	0.851301i	$0.324186\pi$
$702$	0	0
$703$	−1.15883	−0.0437062
$704$	140.943	5.31198
$705$	0	0
$706$	89.3527	3.36283
$707$	−15.5265	−0.583933
$708$	0	0
$709$	37.4668	1.40710	0.703548	−	0.710648i	$-0.251598\pi$
$709$	37.4668	1.40710	0.703548	+	0.710648i	$0.251598\pi$
$710$	0	0
$711$	0	0
$712$	110.961	4.15844
$713$	−11.1957	−0.419281
$714$	0	0
$715$	0	0
$716$	82.4137	3.07994
$717$	0	0
$718$	51.4892	1.92156
$719$	21.0513	0.785081	0.392541	−	0.919735i	$-0.371596\pi$
$719$	21.0513	0.785081	0.392541	+	0.919735i	$0.371596\pi$
$720$	0	0
$721$	43.6329	1.62498
$722$	2.80194	0.104277
$723$	0	0
$724$	−71.6805	−2.66399
$725$	0	0
$726$	0	0
$727$	44.7023	1.65792	0.828958	−	0.559310i	$-0.188934\pi$
$727$	44.7023	1.65792	0.828958	+	0.559310i	$0.188934\pi$
$728$	−106.817	−3.95891
$729$	0	0
$730$	0	0
$731$	11.3860	0.421125
$732$	0	0
$733$	−25.0301	−0.924509	−0.462254	−	0.886747i	$-0.652959\pi$
$733$	−25.0301	−0.924509	−0.462254	+	0.886747i	$0.652959\pi$
$734$	41.0140	1.51385
$735$	0	0
$736$	36.3521	1.33996
$737$	−14.6437	−0.539407
$738$	0	0
$739$	−23.9982	−0.882788	−0.441394	−	0.897313i	$-0.645516\pi$
$739$	−23.9982	−0.882788	−0.441394	+	0.897313i	$0.645516\pi$
$740$	0	0
$741$	0	0
$742$	75.3798	2.76728
$743$	10.4601	0.383744	0.191872	−	0.981420i	$-0.438544\pi$
$743$	10.4601	0.383744	0.191872	+	0.981420i	$0.438544\pi$
$744$	0	0
$745$	0	0
$746$	2.87800	0.105371
$747$	0	0
$748$	−37.1226	−1.35734
$749$	−24.9584	−0.911959
$750$	0	0
$751$	−29.9420	−1.09260	−0.546299	−	0.837590i	$-0.683964\pi$
$751$	−29.9420	−1.09260	−0.546299	+	0.837590i	$0.683964\pi$
$752$	43.6752	1.59267
$753$	0	0
$754$	−16.1468	−0.588030
$755$	0	0
$756$	0	0
$757$	−27.2174	−0.989235	−0.494617	−	0.869111i	$-0.664692\pi$
$757$	−27.2174	−0.989235	−0.494617	+	0.869111i	$0.664692\pi$
$758$	−63.1232	−2.29274
$759$	0	0
$760$	0	0
$761$	−18.6813	−0.677195	−0.338598	−	0.940931i	$-0.609953\pi$
$761$	−18.6813	−0.677195	−0.338598	+	0.940931i	$0.609953\pi$
$762$	0	0
$763$	−34.5362	−1.25029
$764$	−117.876	−4.26459
$765$	0	0
$766$	91.5387	3.30743
$767$	−18.5133	−0.668478
$768$	0	0
$769$	6.34780	0.228907	0.114454	−	0.993429i	$-0.463488\pi$
$769$	6.34780	0.228907	0.114454	+	0.993429i	$0.463488\pi$
$770$	0	0
$771$	0	0
$772$	55.7458	2.00634
$773$	49.7851	1.79064	0.895322	−	0.445419i	$-0.146945\pi$
$773$	49.7851	1.79064	0.895322	+	0.445419i	$0.146945\pi$
$774$	0	0
$775$	0	0
$776$	−35.0344	−1.25766
$777$	0	0
$778$	22.8756	0.820130
$779$	8.57002	0.307053
$780$	0	0
$781$	−7.94033	−0.284127
$782$	−7.24698	−0.259151
$783$	0	0
$784$	42.5448	1.51946
$785$	0	0
$786$	0	0
$787$	9.19865	0.327897	0.163948	−	0.986469i	$-0.447577\pi$
$787$	9.19865	0.327897	0.163948	+	0.986469i	$0.447577\pi$
$788$	29.0388	1.03446
$789$	0	0
$790$	0	0
$791$	10.4741	0.372416
$792$	0	0
$793$	32.3424	1.14851
$794$	−105.207	−3.73364
$795$	0	0
$796$	31.7245	1.12445
$797$	13.5084	0.478493	0.239247	−	0.970959i	$-0.423100\pi$
$797$	13.5084	0.478493	0.239247	+	0.970959i	$0.423100\pi$
$798$	0	0
$799$	−5.08815	−0.180006
$800$	0	0
$801$	0	0
$802$	−90.3678	−3.19100
$803$	−40.4929	−1.42896
$804$	0	0
$805$	0	0
$806$	85.0176	2.99462
$807$	0	0
$808$	−54.9469	−1.93302
$809$	−31.9560	−1.12351	−0.561756	−	0.827303i	$-0.689874\pi$
$809$	−31.9560	−1.12351	−0.561756	+	0.827303i	$0.689874\pi$
$810$	0	0
$811$	40.7012	1.42921	0.714607	−	0.699526i	$-0.246606\pi$
$811$	40.7012	1.42921	0.714607	+	0.699526i	$0.246606\pi$
$812$	31.6601	1.11105
$813$	0	0
$814$	9.54288	0.334478
$815$	0	0
$816$	0	0
$817$	5.27413	0.184518
$818$	−88.3895	−3.09047
$819$	0	0
$820$	0	0
$821$	13.4300	0.468709	0.234355	−	0.972151i	$-0.424702\pi$
$821$	13.4300	0.468709	0.234355	+	0.972151i	$0.424702\pi$
$822$	0	0
$823$	26.7275	0.931663	0.465832	−	0.884873i	$-0.345755\pi$
$823$	26.7275	0.931663	0.465832	+	0.884873i	$0.345755\pi$
$824$	154.413	5.37924
$825$	0	0
$826$	48.7090	1.69480
$827$	33.3435	1.15947	0.579733	−	0.814806i	$-0.303157\pi$
$827$	33.3435	1.15947	0.579733	+	0.814806i	$0.303157\pi$
$828$	0	0
$829$	12.2849	0.426672	0.213336	−	0.976979i	$-0.431567\pi$
$829$	12.2849	0.426672	0.213336	+	0.976979i	$0.431567\pi$
$830$	0	0
$831$	0	0
$832$	−155.712	−5.39835
$833$	−4.95646	−0.171731
$834$	0	0
$835$	0	0
$836$	−17.1957	−0.594725
$837$	0	0
$838$	41.1511	1.42154
$839$	−36.9922	−1.27711	−0.638557	−	0.769575i	$-0.720468\pi$
$839$	−36.9922	−1.27711	−0.638557	+	0.769575i	$0.720468\pi$
$840$	0	0
$841$	−25.8501	−0.891383
$842$	42.3491	1.45945
$843$	0	0
$844$	93.4016	3.21502
$845$	0	0
$846$	0	0
$847$	−7.20237	−0.247477
$848$	163.510	5.61497
$849$	0	0
$850$	0	0
$851$	1.38835	0.0475922
$852$	0	0
$853$	−7.16959	−0.245482	−0.122741	−	0.992439i	$-0.539168\pi$
$853$	−7.16959	−0.245482	−0.122741	+	0.992439i	$0.539168\pi$
$854$	−85.0936	−2.91184
$855$	0	0
$856$	−88.3256	−3.01891
$857$	2.55124	0.0871486	0.0435743	−	0.999050i	$-0.486125\pi$
$857$	2.55124	0.0871486	0.0435743	+	0.999050i	$0.486125\pi$
$858$	0	0
$859$	53.9493	1.84073	0.920363	−	0.391065i	$-0.127893\pi$
$859$	53.9493	1.84073	0.920363	+	0.391065i	$0.127893\pi$
$860$	0	0
$861$	0	0
$862$	−4.69873	−0.160039
$863$	21.4698	0.730840	0.365420	−	0.930843i	$-0.380925\pi$
$863$	21.4698	0.730840	0.365420	+	0.930843i	$0.380925\pi$
$864$	0	0
$865$	0	0
$866$	36.8582	1.25249
$867$	0	0
$868$	−166.700	−5.65817
$869$	16.6491	0.564781
$870$	0	0
$871$	16.1782	0.548178
$872$	−122.221	−4.13891
$873$	0	0
$874$	−3.35690	−0.113549
$875$	0	0
$876$	0	0
$877$	38.9638	1.31571	0.657856	−	0.753144i	$-0.271463\pi$
$877$	38.9638	1.31571	0.657856	+	0.753144i	$0.271463\pi$
$878$	18.0887	0.610465
$879$	0	0
$880$	0	0
$881$	13.4373	0.452713	0.226357	−	0.974045i	$-0.427319\pi$
$881$	13.4373	0.452713	0.226357	+	0.974045i	$0.427319\pi$
$882$	0	0
$883$	27.8799	0.938234	0.469117	−	0.883136i	$-0.344572\pi$
$883$	27.8799	0.938234	0.469117	+	0.883136i	$0.344572\pi$
$884$	41.0127	1.37941
$885$	0	0
$886$	−71.0877	−2.38824
$887$	−28.9135	−0.970821	−0.485410	−	0.874286i	$-0.661330\pi$
$887$	−28.9135	−0.970821	−0.485410	+	0.874286i	$0.661330\pi$
$888$	0	0
$889$	−4.45580	−0.149443
$890$	0	0
$891$	0	0
$892$	135.828	4.54785
$893$	−2.35690	−0.0788705
$894$	0	0
$895$	0	0
$896$	224.659	7.50533
$897$	0	0
$898$	10.0858	0.336566
$899$	−16.5851	−0.553144
$900$	0	0
$901$	−19.0489	−0.634611
$902$	−70.5733	−2.34983
$903$	0	0
$904$	37.0670	1.23283
$905$	0	0
$906$	0	0
$907$	−30.7127	−1.01980	−0.509900	−	0.860234i	$-0.670317\pi$
$907$	−30.7127	−1.01980	−0.509900	+	0.860234i	$0.670317\pi$
$908$	−102.270	−3.39395
$909$	0	0
$910$	0	0
$911$	−39.3690	−1.30435	−0.652176	−	0.758067i	$-0.726144\pi$
$911$	−39.3690	−1.30435	−0.652176	+	0.758067i	$0.726144\pi$
$912$	0	0
$913$	−8.84548	−0.292743
$914$	34.4034	1.13796
$915$	0	0
$916$	−13.6058	−0.449548
$917$	40.3357	1.33200
$918$	0	0
$919$	−7.87023	−0.259615	−0.129808	−	0.991539i	$-0.541436\pi$
$919$	−7.87023	−0.259615	−0.129808	+	0.991539i	$0.541436\pi$
$920$	0	0
$921$	0	0
$922$	−110.658	−3.64434
$923$	8.77240	0.288747
$924$	0	0
$925$	0	0
$926$	−51.5532	−1.69414
$927$	0	0
$928$	53.8514	1.76776
$929$	−39.2924	−1.28914	−0.644572	−	0.764544i	$-0.722964\pi$
$929$	−39.2924	−1.28914	−0.644572	+	0.764544i	$0.722964\pi$
$930$	0	0
$931$	−2.29590	−0.0752450
$932$	−110.781	−3.62876
$933$	0	0
$934$	−101.509	−3.32148
$935$	0	0
$936$	0	0
$937$	26.7554	0.874061	0.437031	−	0.899447i	$-0.356030\pi$
$937$	26.7554	0.874061	0.437031	+	0.899447i	$0.356030\pi$
$938$	−42.5652	−1.38980
$939$	0	0
$940$	0	0
$941$	41.9191	1.36653	0.683263	−	0.730173i	$-0.260560\pi$
$941$	41.9191	1.36653	0.683263	+	0.730173i	$0.260560\pi$
$942$	0	0
$943$	−10.2674	−0.334353
$944$	105.657	3.43885
$945$	0	0
$946$	−43.4319	−1.41209
$947$	55.4868	1.80308	0.901539	−	0.432698i	$-0.142438\pi$
$947$	55.4868	1.80308	0.901539	+	0.432698i	$0.142438\pi$
$948$	0	0
$949$	44.7362	1.45220
$950$	0	0
$951$	0	0
$952$	−71.0200	−2.30177
$953$	8.48081	0.274720	0.137360	−	0.990521i	$-0.456138\pi$
$953$	8.48081	0.274720	0.137360	+	0.990521i	$0.456138\pi$
$954$	0	0
$955$	0	0
$956$	−65.3454	−2.11342
$957$	0	0
$958$	96.1426	3.10623
$959$	49.3129	1.59240
$960$	0	0
$961$	56.3256	1.81695
$962$	−10.5429	−0.339916
$963$	0	0
$964$	43.4717	1.40013
$965$	0	0
$966$	0	0
$967$	7.63102	0.245397	0.122699	−	0.992444i	$-0.460845\pi$
$967$	7.63102	0.245397	0.122699	+	0.992444i	$0.460845\pi$
$968$	−25.4886	−0.819234
$969$	0	0
$970$	0	0
$971$	19.3599	0.621288	0.310644	−	0.950526i	$-0.399455\pi$
$971$	19.3599	0.621288	0.310644	+	0.950526i	$0.399455\pi$
$972$	0	0
$973$	39.9342	1.28023
$974$	68.5695	2.19711
$975$	0	0
$976$	−184.581	−5.90829
$977$	55.8883	1.78802	0.894012	−	0.448042i	$-0.147879\pi$
$977$	55.8883	1.78802	0.894012	+	0.448042i	$0.147879\pi$
$978$	0	0
$979$	30.2241	0.965968
$980$	0	0
$981$	0	0
$982$	−51.2941	−1.63686
$983$	−4.58450	−0.146223	−0.0731114	−	0.997324i	$-0.523293\pi$
$983$	−4.58450	−0.146223	−0.0731114	+	0.997324i	$0.523293\pi$
$984$	0	0
$985$	0	0
$986$	−10.7356	−0.341890
$987$	0	0
$988$	18.9976	0.604394
$989$	−6.31873	−0.200924
$990$	0	0
$991$	27.5666	0.875681	0.437840	−	0.899053i	$-0.355743\pi$
$991$	27.5666	0.875681	0.437840	+	0.899053i	$0.355743\pi$
$992$	−283.544	−9.00254
$993$	0	0
$994$	−23.0804	−0.732065
$995$	0	0
$996$	0	0
$997$	−15.9119	−0.503933	−0.251967	−	0.967736i	$-0.581077\pi$
$997$	−15.9119	−0.503933	−0.251967	+	0.967736i	$0.581077\pi$
$998$	−71.0079	−2.24772
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bn.1.3		3
3.2	odd	2		475.2.a.d.1.1	✓	3
5.4	even	2		4275.2.a.z.1.1		3
12.11	even	2		7600.2.a.bw.1.2		3
15.2	even	4		475.2.b.c.324.1		6
15.8	even	4		475.2.b.c.324.6		6
15.14	odd	2		475.2.a.h.1.3	yes	3
57.56	even	2		9025.2.a.be.1.3		3
60.59	even	2		7600.2.a.bn.1.2		3
285.284	even	2		9025.2.a.w.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.d.1.1	✓	3	3.2	odd	2
475.2.a.h.1.3	yes	3	15.14	odd	2
475.2.b.c.324.1		6	15.2	even	4
475.2.b.c.324.6		6	15.8	even	4
4275.2.a.z.1.1		3	5.4	even	2
4275.2.a.bn.1.3		3	1.1	even	1	trivial
7600.2.a.bn.1.2		3	60.59	even	2
7600.2.a.bw.1.2		3	12.11	even	2
9025.2.a.w.1.1		3	285.284	even	2
9025.2.a.be.1.3		3	57.56	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.80194 q^{2} +5.85086 q^{4} +3.04892 q^{7} +10.7899 q^{8} +O(q^{10})\)	\(q+2.80194 q^{2} +5.85086 q^{4} +3.04892 q^{7} +10.7899 q^{8} +2.93900 q^{11} -3.24698 q^{13} +8.54288 q^{14} +18.5308 q^{16} -2.15883 q^{17} -1.00000 q^{19} +8.23490 q^{22} +1.19806 q^{23} -9.09783 q^{26} +17.8388 q^{28} +1.77479 q^{29} -9.34481 q^{31} +30.3424 q^{32} -6.04892 q^{34} +1.15883 q^{37} -2.80194 q^{38} -8.57002 q^{41} -5.27413 q^{43} +17.1957 q^{44} +3.35690 q^{46} +2.35690 q^{47} +2.29590 q^{49} -18.9976 q^{52} +8.82371 q^{53} +32.8974 q^{56} +4.97285 q^{58} +5.70171 q^{59} -9.96077 q^{61} -26.1836 q^{62} +47.9560 q^{64} -4.98254 q^{67} -12.6310 q^{68} -2.70171 q^{71} -13.7778 q^{73} +3.24698 q^{74} -5.85086 q^{76} +8.96077 q^{77} +5.66487 q^{79} -24.0127 q^{82} -3.00969 q^{83} -14.7778 q^{86} +31.7114 q^{88} +10.2838 q^{89} -9.89977 q^{91} +7.00969 q^{92} +6.60388 q^{94} -3.24698 q^{97} +6.43296 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{2} + 4 q^{4} + 9 q^{8}+O(q^{10}) \) 3 * q + 4 * q^2 + 4 * q^4 + 9 * q^8	\( 3 q + 4 q^{2} + 4 q^{4} + 9 q^{8} - q^{11} - 5 q^{13} + 7 q^{14} + 18 q^{16} + 2 q^{17} - 3 q^{19} + q^{22} + 8 q^{23} - 9 q^{26} + 21 q^{28} + 7 q^{29} - 5 q^{31} + 27 q^{32} - 9 q^{34} - 5 q^{37} - 4 q^{38} - q^{41} - 5 q^{43} + 15 q^{44} + 6 q^{46} + 3 q^{47} - 7 q^{49} - 16 q^{52} + 19 q^{53} + 35 q^{56} + 21 q^{58} - 10 q^{59} - 17 q^{61} - 23 q^{62} + 49 q^{64} + q^{67} - 23 q^{68} + 19 q^{71} + q^{73} + 5 q^{74} - 4 q^{76} + 14 q^{77} + 18 q^{79} - 6 q^{82} + 13 q^{83} - 2 q^{86} + 46 q^{88} - 2 q^{89} - 7 q^{91} - q^{92} + 11 q^{94} - 5 q^{97}+O(q^{100}) \) 3 * q + 4 * q^2 + 4 * q^4 + 9 * q^8 - q^11 - 5 * q^13 + 7 * q^14 + 18 * q^16 + 2 * q^17 - 3 * q^19 + q^22 + 8 * q^23 - 9 * q^26 + 21 * q^28 + 7 * q^29 - 5 * q^31 + 27 * q^32 - 9 * q^34 - 5 * q^37 - 4 * q^38 - q^41 - 5 * q^43 + 15 * q^44 + 6 * q^46 + 3 * q^47 - 7 * q^49 - 16 * q^52 + 19 * q^53 + 35 * q^56 + 21 * q^58 - 10 * q^59 - 17 * q^61 - 23 * q^62 + 49 * q^64 + q^67 - 23 * q^68 + 19 * q^71 + q^73 + 5 * q^74 - 4 * q^76 + 14 * q^77 + 18 * q^79 - 6 * q^82 + 13 * q^83 - 2 * q^86 + 46 * q^88 - 2 * q^89 - 7 * q^91 - q^92 + 11 * q^94 - 5 * q^97

Introduction

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