LMFDB - Embedded newform 5445.2.a.cc.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, 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("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 3x^{6} + 22x^{5} - 3x^{4} - 32x^{3} + 9x^{2} + 8x + 1 $ x^8 - 4x^7 - 3x^6 + 22x^5 - 3x^4 - 32x^3 + 9x^2 + 8*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 495)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.909121$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0908791 q^{2} -1.99174 q^{4} -1.00000 q^{5} +2.45732 q^{7} -0.362766 q^{8} +O(q^{10})$	$q+0.0908791 q^{2} -1.99174 q^{4} -1.00000 q^{5} +2.45732 q^{7} -0.362766 q^{8} -0.0908791 q^{10} -0.323527 q^{13} +0.223319 q^{14} +3.95051 q^{16} +3.66517 q^{17} -0.541216 q^{19} +1.99174 q^{20} +4.62543 q^{23} +1.00000 q^{25} -0.0294019 q^{26} -4.89435 q^{28} +1.47591 q^{29} +0.890928 q^{31} +1.08455 q^{32} +0.333088 q^{34} -2.45732 q^{35} -2.28547 q^{37} -0.0491852 q^{38} +0.362766 q^{40} -8.98054 q^{41} -6.31964 q^{43} +0.420355 q^{46} +1.46285 q^{47} -0.961571 q^{49} +0.0908791 q^{50} +0.644383 q^{52} +10.3294 q^{53} -0.891432 q^{56} +0.134129 q^{58} -6.29328 q^{59} +6.47540 q^{61} +0.0809667 q^{62} -7.80247 q^{64} +0.323527 q^{65} -7.05634 q^{67} -7.30008 q^{68} -0.223319 q^{70} +9.31793 q^{71} -6.19325 q^{73} -0.207702 q^{74} +1.07796 q^{76} -11.6049 q^{79} -3.95051 q^{80} -0.816144 q^{82} +17.3740 q^{83} -3.66517 q^{85} -0.574323 q^{86} -4.70270 q^{89} -0.795011 q^{91} -9.21267 q^{92} +0.132943 q^{94} +0.541216 q^{95} +11.0077 q^{97} -0.0873867 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} + 6 q^{4} - 8 q^{5} - 8 q^{7} + 12 q^{8}+O(q^{10}) $ 8 * q + 4 * q^2 + 6 * q^4 - 8 * q^5 - 8 * q^7 + 12 * q^8	$ 8 q + 4 q^{2} + 6 q^{4} - 8 q^{5} - 8 q^{7} + 12 q^{8} - 4 q^{10} - 6 q^{13} - 14 q^{14} + 14 q^{16} + 8 q^{17} + 2 q^{19} - 6 q^{20} - 4 q^{23} + 8 q^{25} + 2 q^{26} - 24 q^{28} + 22 q^{29} + 10 q^{31} + 28 q^{32} - 2 q^{34} + 8 q^{35} - 14 q^{37} + 20 q^{38} - 12 q^{40} + 22 q^{41} - 14 q^{43} + 2 q^{46} - 10 q^{47} + 4 q^{50} + 10 q^{52} + 18 q^{53} - 34 q^{56} + 12 q^{58} - 2 q^{59} + 14 q^{61} + 30 q^{62} + 30 q^{64} + 6 q^{65} + 10 q^{67} + 6 q^{68} + 14 q^{70} + 2 q^{71} - 16 q^{73} + 24 q^{74} + 22 q^{76} + 16 q^{79} - 14 q^{80} + 10 q^{82} + 46 q^{83} - 8 q^{85} + 28 q^{86} - 38 q^{89} + 8 q^{91} + 24 q^{92} + 10 q^{94} - 2 q^{95} - 4 q^{97} + 4 q^{98}+O(q^{100}) $ 8 * q + 4 * q^2 + 6 * q^4 - 8 * q^5 - 8 * q^7 + 12 * q^8 - 4 * q^10 - 6 * q^13 - 14 * q^14 + 14 * q^16 + 8 * q^17 + 2 * q^19 - 6 * q^20 - 4 * q^23 + 8 * q^25 + 2 * q^26 - 24 * q^28 + 22 * q^29 + 10 * q^31 + 28 * q^32 - 2 * q^34 + 8 * q^35 - 14 * q^37 + 20 * q^38 - 12 * q^40 + 22 * q^41 - 14 * q^43 + 2 * q^46 - 10 * q^47 + 4 * q^50 + 10 * q^52 + 18 * q^53 - 34 * q^56 + 12 * q^58 - 2 * q^59 + 14 * q^61 + 30 * q^62 + 30 * q^64 + 6 * q^65 + 10 * q^67 + 6 * q^68 + 14 * q^70 + 2 * q^71 - 16 * q^73 + 24 * q^74 + 22 * q^76 + 16 * q^79 - 14 * q^80 + 10 * q^82 + 46 * q^83 - 8 * q^85 + 28 * q^86 - 38 * q^89 + 8 * q^91 + 24 * q^92 + 10 * q^94 - 2 * q^95 - 4 * q^97 + 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.0908791	0.0642612	0.0321306	−	0.999484i	$-0.489771\pi$
$2$	0.0908791	0.0642612	0.0321306	+	0.999484i	$0.489771\pi$
$3$	0	0
$3$	0	0
$4$	−1.99174	−0.995870
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.45732	0.928780	0.464390	−	0.885631i	$-0.346274\pi$
$7$	2.45732	0.928780	0.464390	+	0.885631i	$0.346274\pi$
$8$	−0.362766	−0.128257
$9$	0	0
$10$	−0.0908791	−0.0287385
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.323527	−0.0897304	−0.0448652	−	0.998993i	$-0.514286\pi$
$13$	−0.323527	−0.0897304	−0.0448652	+	0.998993i	$0.514286\pi$
$14$	0.223319	0.0596846
$15$	0	0
$16$	3.95051	0.987629
$17$	3.66517	0.888935	0.444468	−	0.895795i	$-0.353393\pi$
$17$	3.66517	0.888935	0.444468	+	0.895795i	$0.353393\pi$
$18$	0	0
$19$	−0.541216	−0.124164	−0.0620818	−	0.998071i	$-0.519774\pi$
$19$	−0.541216	−0.124164	−0.0620818	+	0.998071i	$0.519774\pi$
$20$	1.99174	0.445367
$21$	0	0
$22$	0	0
$23$	4.62543	0.964470	0.482235	−	0.876042i	$-0.339825\pi$
$23$	4.62543	0.964470	0.482235	+	0.876042i	$0.339825\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.0294019	−0.00576618
$27$	0	0
$28$	−4.89435	−0.924945
$29$	1.47591	0.274069	0.137034	−	0.990566i	$-0.456243\pi$
$29$	1.47591	0.274069	0.137034	+	0.990566i	$0.456243\pi$
$30$	0	0
$31$	0.890928	0.160015	0.0800077	−	0.996794i	$-0.474506\pi$
$31$	0.890928	0.160015	0.0800077	+	0.996794i	$0.474506\pi$
$32$	1.08455	0.191723
$33$	0	0
$34$	0.333088	0.0571241
$35$	−2.45732	−0.415363
$36$	0	0
$37$	−2.28547	−0.375729	−0.187865	−	0.982195i	$-0.560157\pi$
$37$	−2.28547	−0.375729	−0.187865	+	0.982195i	$0.560157\pi$
$38$	−0.0491852	−0.00797890
$39$	0	0
$40$	0.362766	0.0573583
$41$	−8.98054	−1.40253	−0.701263	−	0.712903i	$-0.747380\pi$
$41$	−8.98054	−1.40253	−0.701263	+	0.712903i	$0.747380\pi$
$42$	0	0
$43$	−6.31964	−0.963736	−0.481868	−	0.876244i	$-0.660041\pi$
$43$	−6.31964	−0.963736	−0.481868	+	0.876244i	$0.660041\pi$
$44$	0	0
$45$	0	0
$46$	0.420355	0.0619780
$47$	1.46285	0.213379	0.106690	−	0.994292i	$-0.465975\pi$
$47$	1.46285	0.213379	0.106690	+	0.994292i	$0.465975\pi$
$48$	0	0
$49$	−0.961571	−0.137367
$50$	0.0908791	0.0128522
$51$	0	0
$52$	0.644383	0.0893598
$53$	10.3294	1.41885	0.709427	−	0.704779i	$-0.248954\pi$
$53$	10.3294	1.41885	0.709427	+	0.704779i	$0.248954\pi$
$54$	0	0
$55$	0	0
$56$	−0.891432	−0.119123
$57$	0	0
$58$	0.134129	0.0176120
$59$	−6.29328	−0.819315	−0.409657	−	0.912239i	$-0.634352\pi$
$59$	−6.29328	−0.819315	−0.409657	+	0.912239i	$0.634352\pi$
$60$	0	0
$61$	6.47540	0.829090	0.414545	−	0.910029i	$-0.363941\pi$
$61$	6.47540	0.829090	0.414545	+	0.910029i	$0.363941\pi$
$62$	0.0809667	0.0102828
$63$	0	0
$64$	−7.80247	−0.975308
$65$	0.323527	0.0401286
$66$	0	0
$67$	−7.05634	−0.862069	−0.431035	−	0.902335i	$-0.641851\pi$
$67$	−7.05634	−0.862069	−0.431035	+	0.902335i	$0.641851\pi$
$68$	−7.30008	−0.885264
$69$	0	0
$70$	−0.223319	−0.0266917
$71$	9.31793	1.10583	0.552917	−	0.833236i	$-0.313515\pi$
$71$	9.31793	1.10583	0.552917	+	0.833236i	$0.313515\pi$
$72$	0	0
$73$	−6.19325	−0.724865	−0.362432	−	0.932010i	$-0.618054\pi$
$73$	−6.19325	−0.724865	−0.362432	+	0.932010i	$0.618054\pi$
$74$	−0.207702	−0.0241448
$75$	0	0
$76$	1.07796	0.123651
$77$	0	0
$78$	0	0
$79$	−11.6049	−1.30565	−0.652827	−	0.757507i	$-0.726417\pi$
$79$	−11.6049	−1.30565	−0.652827	+	0.757507i	$0.726417\pi$
$80$	−3.95051	−0.441681
$81$	0	0
$82$	−0.816144	−0.0901280
$83$	17.3740	1.90705	0.953524	−	0.301316i	$-0.0974258\pi$
$83$	17.3740	1.90705	0.953524	+	0.301316i	$0.0974258\pi$
$84$	0	0
$85$	−3.66517	−0.397544
$86$	−0.574323	−0.0619308
$87$	0	0
$88$	0	0
$89$	−4.70270	−0.498485	−0.249242	−	0.968441i	$-0.580182\pi$
$89$	−4.70270	−0.498485	−0.249242	+	0.968441i	$0.580182\pi$
$90$	0	0
$91$	−0.795011	−0.0833398
$92$	−9.21267	−0.960487
$93$	0	0
$94$	0.132943	0.0137120
$95$	0.541216	0.0555276
$96$	0	0
$97$	11.0077	1.11766	0.558829	−	0.829283i	$-0.311251\pi$
$97$	11.0077	1.11766	0.558829	+	0.829283i	$0.311251\pi$
$98$	−0.0873867	−0.00882739
$99$	0	0
$100$	−1.99174	−0.199174
$101$	6.71144	0.667814	0.333907	−	0.942606i	$-0.391633\pi$
$101$	6.71144	0.667814	0.333907	+	0.942606i	$0.391633\pi$
$102$	0	0
$103$	15.6033	1.53744	0.768720	−	0.639586i	$-0.220894\pi$
$103$	15.6033	1.53744	0.768720	+	0.639586i	$0.220894\pi$
$104$	0.117365	0.0115086
$105$	0	0
$106$	0.938727	0.0911773
$107$	−8.98415	−0.868531	−0.434265	−	0.900785i	$-0.642992\pi$
$107$	−8.98415	−0.868531	−0.434265	+	0.900785i	$0.642992\pi$
$108$	0	0
$109$	10.7685	1.03144	0.515719	−	0.856758i	$-0.327525\pi$
$109$	10.7685	1.03144	0.515719	+	0.856758i	$0.327525\pi$
$110$	0	0
$111$	0	0
$112$	9.70768	0.917290
$113$	−19.8949	−1.87156	−0.935778	−	0.352591i	$-0.885301\pi$
$113$	−19.8949	−1.87156	−0.935778	+	0.352591i	$0.885301\pi$
$114$	0	0
$115$	−4.62543	−0.431324
$116$	−2.93962	−0.272937
$117$	0	0
$118$	−0.571927	−0.0526502
$119$	9.00651	0.825625
$120$	0	0
$121$	0	0
$122$	0.588478	0.0532783
$123$	0	0
$124$	−1.77450	−0.159355
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.1101	1.07460	0.537300	−	0.843391i	$-0.319444\pi$
$127$	12.1101	1.07460	0.537300	+	0.843391i	$0.319444\pi$
$128$	−2.87818	−0.254398
$129$	0	0
$130$	0.0294019	0.00257872
$131$	18.7043	1.63420	0.817099	−	0.576497i	$-0.195581\pi$
$131$	18.7043	1.63420	0.817099	+	0.576497i	$0.195581\pi$
$132$	0	0
$133$	−1.32994	−0.115321
$134$	−0.641274	−0.0553976
$135$	0	0
$136$	−1.32960	−0.114012
$137$	15.7854	1.34864	0.674319	−	0.738440i	$-0.264437\pi$
$137$	15.7854	1.34864	0.674319	+	0.738440i	$0.264437\pi$
$138$	0	0
$139$	−13.8356	−1.17352	−0.586758	−	0.809762i	$-0.699596\pi$
$139$	−13.8356	−1.17352	−0.586758	+	0.809762i	$0.699596\pi$
$140$	4.89435	0.413648
$141$	0	0
$142$	0.846805	0.0710623
$143$	0	0
$144$	0	0
$145$	−1.47591	−0.122567
$146$	−0.562837	−0.0465807
$147$	0	0
$148$	4.55207	0.374178
$149$	9.03793	0.740416	0.370208	−	0.928949i	$-0.379286\pi$
$149$	9.03793	0.740416	0.370208	+	0.928949i	$0.379286\pi$
$150$	0	0
$151$	8.85151	0.720325	0.360163	−	0.932890i	$-0.382721\pi$
$151$	8.85151	0.720325	0.360163	+	0.932890i	$0.382721\pi$
$152$	0.196335	0.0159249
$153$	0	0
$154$	0	0
$155$	−0.890928	−0.0715611
$156$	0	0
$157$	11.7391	0.936884	0.468442	−	0.883494i	$-0.344816\pi$
$157$	11.7391	0.936884	0.468442	+	0.883494i	$0.344816\pi$
$158$	−1.05464	−0.0839030
$159$	0	0
$160$	−1.08455	−0.0857413
$161$	11.3662	0.895780
$162$	0	0
$163$	7.62018	0.596858	0.298429	−	0.954432i	$-0.403537\pi$
$163$	7.62018	0.596858	0.298429	+	0.954432i	$0.403537\pi$
$164$	17.8869	1.39673
$165$	0	0
$166$	1.57894	0.122549
$167$	10.1173	0.782897	0.391448	−	0.920200i	$-0.371974\pi$
$167$	10.1173	0.782897	0.391448	+	0.920200i	$0.371974\pi$
$168$	0	0
$169$	−12.8953	−0.991948
$170$	−0.333088	−0.0255467
$171$	0	0
$172$	12.5871	0.959756
$173$	9.69518	0.737111	0.368556	−	0.929606i	$-0.379852\pi$
$173$	9.69518	0.737111	0.368556	+	0.929606i	$0.379852\pi$
$174$	0	0
$175$	2.45732	0.185756
$176$	0	0
$177$	0	0
$178$	−0.427377	−0.0320332
$179$	−18.1583	−1.35722	−0.678608	−	0.734501i	$-0.737416\pi$
$179$	−18.1583	−1.35722	−0.678608	+	0.734501i	$0.737416\pi$
$180$	0	0
$181$	−4.40926	−0.327738	−0.163869	−	0.986482i	$-0.552397\pi$
$181$	−4.40926	−0.327738	−0.163869	+	0.986482i	$0.552397\pi$
$182$	−0.0722499	−0.00535552
$183$	0	0
$184$	−1.67795	−0.123700
$185$	2.28547	0.168031
$186$	0	0
$187$	0	0
$188$	−2.91363	−0.212498
$189$	0	0
$190$	0.0491852	0.00356827
$191$	−12.2858	−0.888967	−0.444483	−	0.895787i	$-0.646613\pi$
$191$	−12.2858	−0.888967	−0.444483	+	0.895787i	$0.646613\pi$
$192$	0	0
$193$	−23.2963	−1.67691	−0.838453	−	0.544973i	$-0.816540\pi$
$193$	−23.2963	−1.67691	−0.838453	+	0.544973i	$0.816540\pi$
$194$	1.00037	0.0718221
$195$	0	0
$196$	1.91520	0.136800
$197$	19.6929	1.40306	0.701532	−	0.712638i	$-0.252500\pi$
$197$	19.6929	1.40306	0.701532	+	0.712638i	$0.252500\pi$
$198$	0	0
$199$	22.2083	1.57431	0.787154	−	0.616757i	$-0.211554\pi$
$199$	22.2083	1.57431	0.787154	+	0.616757i	$0.211554\pi$
$200$	−0.362766	−0.0256514
$201$	0	0
$202$	0.609930	0.0429145
$203$	3.62677	0.254550
$204$	0	0
$205$	8.98054	0.627228
$206$	1.41801	0.0987978
$207$	0	0
$208$	−1.27810	−0.0886203
$209$	0	0
$210$	0	0
$211$	12.9918	0.894391	0.447195	−	0.894436i	$-0.352423\pi$
$211$	12.9918	0.894391	0.447195	+	0.894436i	$0.352423\pi$
$212$	−20.5735	−1.41299
$213$	0	0
$214$	−0.816471	−0.0558128
$215$	6.31964	0.430996
$216$	0	0
$217$	2.18930	0.148619
$218$	0.978634	0.0662815
$219$	0	0
$220$	0	0
$221$	−1.18578	−0.0797645
$222$	0	0
$223$	−22.6211	−1.51482	−0.757409	−	0.652941i	$-0.773535\pi$
$223$	−22.6211	−1.51482	−0.757409	+	0.652941i	$0.773535\pi$
$224$	2.66509	0.178069
$225$	0	0
$226$	−1.80803	−0.120268
$227$	15.7324	1.04420	0.522098	−	0.852886i	$-0.325150\pi$
$227$	15.7324	1.04420	0.522098	+	0.852886i	$0.325150\pi$
$228$	0	0
$229$	−24.3011	−1.60586	−0.802930	−	0.596074i	$-0.796727\pi$
$229$	−24.3011	−1.60586	−0.802930	+	0.596074i	$0.796727\pi$
$230$	−0.420355	−0.0277174
$231$	0	0
$232$	−0.535408	−0.0351513
$233$	−5.35751	−0.350982	−0.175491	−	0.984481i	$-0.556151\pi$
$233$	−5.35751	−0.350982	−0.175491	+	0.984481i	$0.556151\pi$
$234$	0	0
$235$	−1.46285	−0.0954261
$236$	12.5346	0.815932
$237$	0	0
$238$	0.818503	0.0530557
$239$	0.921658	0.0596171	0.0298085	−	0.999556i	$-0.490510\pi$
$239$	0.921658	0.0596171	0.0298085	+	0.999556i	$0.490510\pi$
$240$	0	0
$241$	−11.4029	−0.734523	−0.367262	−	0.930118i	$-0.619705\pi$
$241$	−11.4029	−0.734523	−0.367262	+	0.930118i	$0.619705\pi$
$242$	0	0
$243$	0	0
$244$	−12.8973	−0.825666
$245$	0.961571	0.0614325
$246$	0	0
$247$	0.175098	0.0111412
$248$	−0.323198	−0.0205231
$249$	0	0
$250$	−0.0908791	−0.00574770
$251$	0.889679	0.0561561	0.0280780	−	0.999606i	$-0.491061\pi$
$251$	0.889679	0.0561561	0.0280780	+	0.999606i	$0.491061\pi$
$252$	0	0
$253$	0	0
$254$	1.10056	0.0690551
$255$	0	0
$256$	15.3434	0.958960
$257$	16.0126	0.998838	0.499419	−	0.866361i	$-0.333547\pi$
$257$	16.0126	0.998838	0.499419	+	0.866361i	$0.333547\pi$
$258$	0	0
$259$	−5.61614	−0.348970
$260$	−0.644383	−0.0399629
$261$	0	0
$262$	1.69983	0.105016
$263$	21.2954	1.31313	0.656565	−	0.754269i	$-0.272009\pi$
$263$	21.2954	1.31313	0.656565	+	0.754269i	$0.272009\pi$
$264$	0	0
$265$	−10.3294	−0.634531
$266$	−0.120864	−0.00741064
$267$	0	0
$268$	14.0544	0.858510
$269$	30.2834	1.84641	0.923206	−	0.384306i	$-0.125559\pi$
$269$	30.2834	1.84641	0.923206	+	0.384306i	$0.125559\pi$
$270$	0	0
$271$	−6.01860	−0.365604	−0.182802	−	0.983150i	$-0.558517\pi$
$271$	−6.01860	−0.365604	−0.182802	+	0.983150i	$0.558517\pi$
$272$	14.4793	0.877938
$273$	0	0
$274$	1.43456	0.0866651
$275$	0	0
$276$	0	0
$277$	28.0368	1.68457	0.842285	−	0.539032i	$-0.181210\pi$
$277$	28.0368	1.68457	0.842285	+	0.539032i	$0.181210\pi$
$278$	−1.25736	−0.0754116
$279$	0	0
$280$	0.891432	0.0532733
$281$	26.2291	1.56470	0.782348	−	0.622842i	$-0.214022\pi$
$281$	26.2291	1.56470	0.782348	+	0.622842i	$0.214022\pi$
$282$	0	0
$283$	−12.4007	−0.737147	−0.368574	−	0.929599i	$-0.620154\pi$
$283$	−12.4007	−0.737147	−0.368574	+	0.929599i	$0.620154\pi$
$284$	−18.5589	−1.10127
$285$	0	0
$286$	0	0
$287$	−22.0681	−1.30264
$288$	0	0
$289$	−3.56650	−0.209794
$290$	−0.134129	−0.00787632
$291$	0	0
$292$	12.3353	0.721871
$293$	29.7792	1.73972	0.869859	−	0.493301i	$-0.164210\pi$
$293$	29.7792	1.73972	0.869859	+	0.493301i	$0.164210\pi$
$294$	0	0
$295$	6.29328	0.366409
$296$	0.829091	0.0481899
$297$	0	0
$298$	0.821359	0.0475801
$299$	−1.49645	−0.0865422
$300$	0	0
$301$	−15.5294	−0.895099
$302$	0.804417	0.0462890
$303$	0	0
$304$	−2.13808	−0.122627
$305$	−6.47540	−0.370780
$306$	0	0
$307$	−13.5474	−0.773194	−0.386597	−	0.922249i	$-0.626350\pi$
$307$	−13.5474	−0.773194	−0.386597	+	0.922249i	$0.626350\pi$
$308$	0	0
$309$	0	0
$310$	−0.0809667	−0.00459860
$311$	−8.75200	−0.496281	−0.248140	−	0.968724i	$-0.579819\pi$
$311$	−8.75200	−0.496281	−0.248140	+	0.968724i	$0.579819\pi$
$312$	0	0
$313$	16.2116	0.916333	0.458166	−	0.888866i	$-0.348506\pi$
$313$	16.2116	0.916333	0.458166	+	0.888866i	$0.348506\pi$
$314$	1.06684	0.0602053
$315$	0	0
$316$	23.1140	1.30026
$317$	5.88258	0.330399	0.165199	−	0.986260i	$-0.447173\pi$
$317$	5.88258	0.330399	0.165199	+	0.986260i	$0.447173\pi$
$318$	0	0
$319$	0	0
$320$	7.80247	0.436171
$321$	0	0
$322$	1.03295	0.0575639
$323$	−1.98365	−0.110373
$324$	0	0
$325$	−0.323527	−0.0179461
$326$	0.692515	0.0383549
$327$	0	0
$328$	3.25783	0.179884
$329$	3.59470	0.198182
$330$	0	0
$331$	13.4772	0.740775	0.370387	−	0.928877i	$-0.379225\pi$
$331$	13.4772	0.740775	0.370387	+	0.928877i	$0.379225\pi$
$332$	−34.6046	−1.89917
$333$	0	0
$334$	0.919447	0.0503099
$335$	7.05634	0.385529
$336$	0	0
$337$	15.1603	0.825837	0.412918	−	0.910768i	$-0.364509\pi$
$337$	15.1603	0.825837	0.412918	+	0.910768i	$0.364509\pi$
$338$	−1.17192	−0.0637438
$339$	0	0
$340$	7.30008	0.395902
$341$	0	0
$342$	0	0
$343$	−19.5641	−1.05636
$344$	2.29255	0.123606
$345$	0	0
$346$	0.881090	0.0473677
$347$	17.3583	0.931841	0.465920	−	0.884827i	$-0.345723\pi$
$347$	17.3583	0.931841	0.465920	+	0.884827i	$0.345723\pi$
$348$	0	0
$349$	4.89237	0.261882	0.130941	−	0.991390i	$-0.458200\pi$
$349$	4.89237	0.261882	0.130941	+	0.991390i	$0.458200\pi$
$350$	0.223319	0.0119369
$351$	0	0
$352$	0	0
$353$	−9.32665	−0.496408	−0.248204	−	0.968708i	$-0.579840\pi$
$353$	−9.32665	−0.496408	−0.248204	+	0.968708i	$0.579840\pi$
$354$	0	0
$355$	−9.31793	−0.494544
$356$	9.36655	0.496426
$357$	0	0
$358$	−1.65021	−0.0872163
$359$	28.9762	1.52931	0.764654	−	0.644441i	$-0.222910\pi$
$359$	28.9762	1.52931	0.764654	+	0.644441i	$0.222910\pi$
$360$	0	0
$361$	−18.7071	−0.984583
$362$	−0.400710	−0.0210608
$363$	0	0
$364$	1.58346	0.0829956
$365$	6.19325	0.324169
$366$	0	0
$367$	−13.9476	−0.728060	−0.364030	−	0.931387i	$-0.618599\pi$
$367$	−13.9476	−0.728060	−0.364030	+	0.931387i	$0.618599\pi$
$368$	18.2728	0.952538
$369$	0	0
$370$	0.207702	0.0107979
$371$	25.3827	1.31780
$372$	0	0
$373$	22.0121	1.13974	0.569871	−	0.821734i	$-0.306993\pi$
$373$	22.0121	1.13974	0.569871	+	0.821734i	$0.306993\pi$
$374$	0	0
$375$	0	0
$376$	−0.530674	−0.0273674
$377$	−0.477496	−0.0245923
$378$	0	0
$379$	−1.89456	−0.0973169	−0.0486584	−	0.998815i	$-0.515495\pi$
$379$	−1.89456	−0.0973169	−0.0486584	+	0.998815i	$0.515495\pi$
$380$	−1.07796	−0.0552983
$381$	0	0
$382$	−1.11652	−0.0571261
$383$	−0.958764	−0.0489905	−0.0244953	−	0.999700i	$-0.507798\pi$
$383$	−0.958764	−0.0489905	−0.0244953	+	0.999700i	$0.507798\pi$
$384$	0	0
$385$	0	0
$386$	−2.11715	−0.107760
$387$	0	0
$388$	−21.9244	−1.11304
$389$	6.52874	0.331020	0.165510	−	0.986208i	$-0.447073\pi$
$389$	6.52874	0.331020	0.165510	+	0.986208i	$0.447073\pi$
$390$	0	0
$391$	16.9530	0.857351
$392$	0.348825	0.0176183
$393$	0	0
$394$	1.78968	0.0901626
$395$	11.6049	0.583907
$396$	0	0
$397$	−21.3621	−1.07213	−0.536066	−	0.844176i	$-0.680090\pi$
$397$	−21.3621	−1.07213	−0.536066	+	0.844176i	$0.680090\pi$
$398$	2.01827	0.101167
$399$	0	0
$400$	3.95051	0.197526
$401$	−12.6526	−0.631842	−0.315921	−	0.948786i	$-0.602313\pi$
$401$	−12.6526	−0.631842	−0.315921	+	0.948786i	$0.602313\pi$
$402$	0	0
$403$	−0.288240	−0.0143582
$404$	−13.3675	−0.665056
$405$	0	0
$406$	0.329598	0.0163577
$407$	0	0
$408$	0	0
$409$	−5.00765	−0.247612	−0.123806	−	0.992306i	$-0.539510\pi$
$409$	−5.00765	−0.247612	−0.123806	+	0.992306i	$0.539510\pi$
$410$	0.816144	0.0403065
$411$	0	0
$412$	−31.0777	−1.53109
$413$	−15.4646	−0.760963
$414$	0	0
$415$	−17.3740	−0.852858
$416$	−0.350882	−0.0172034
$417$	0	0
$418$	0	0
$419$	9.31728	0.455179	0.227589	−	0.973757i	$-0.426916\pi$
$419$	9.31728	0.455179	0.227589	+	0.973757i	$0.426916\pi$
$420$	0	0
$421$	23.6784	1.15401	0.577007	−	0.816739i	$-0.304220\pi$
$421$	23.6784	1.15401	0.577007	+	0.816739i	$0.304220\pi$
$422$	1.18068	0.0574746
$423$	0	0
$424$	−3.74716	−0.181978
$425$	3.66517	0.177787
$426$	0	0
$427$	15.9121	0.770042
$428$	17.8941	0.864944
$429$	0	0
$430$	0.574323	0.0276963
$431$	−25.7069	−1.23826	−0.619130	−	0.785289i	$-0.712515\pi$
$431$	−25.7069	−1.23826	−0.619130	+	0.785289i	$0.712515\pi$
$432$	0	0
$433$	19.8945	0.956066	0.478033	−	0.878342i	$-0.341350\pi$
$433$	19.8945	0.956066	0.478033	+	0.878342i	$0.341350\pi$
$434$	0.198961	0.00955045
$435$	0	0
$436$	−21.4481	−1.02718
$437$	−2.50336	−0.119752
$438$	0	0
$439$	2.83831	0.135465	0.0677327	−	0.997704i	$-0.478423\pi$
$439$	2.83831	0.135465	0.0677327	+	0.997704i	$0.478423\pi$
$440$	0	0
$441$	0	0
$442$	−0.107763	−0.00512576
$443$	27.0934	1.28725	0.643624	−	0.765342i	$-0.277430\pi$
$443$	27.0934	1.28725	0.643624	+	0.765342i	$0.277430\pi$
$444$	0	0
$445$	4.70270	0.222929
$446$	−2.05578	−0.0973440
$447$	0	0
$448$	−19.1732	−0.905847
$449$	26.1493	1.23406	0.617031	−	0.786939i	$-0.288335\pi$
$449$	26.1493	1.23406	0.617031	+	0.786939i	$0.288335\pi$
$450$	0	0
$451$	0	0
$452$	39.6255	1.86383
$453$	0	0
$454$	1.42975	0.0671013
$455$	0.795011	0.0372707
$456$	0	0
$457$	14.0946	0.659319	0.329660	−	0.944100i	$-0.393066\pi$
$457$	14.0946	0.659319	0.329660	+	0.944100i	$0.393066\pi$
$458$	−2.20846	−0.103194
$459$	0	0
$460$	9.21267	0.429543
$461$	−0.212479	−0.00989612	−0.00494806	−	0.999988i	$-0.501575\pi$
$461$	−0.212479	−0.00989612	−0.00494806	+	0.999988i	$0.501575\pi$
$462$	0	0
$463$	−9.01059	−0.418758	−0.209379	−	0.977835i	$-0.567144\pi$
$463$	−9.01059	−0.418758	−0.209379	+	0.977835i	$0.567144\pi$
$464$	5.83058	0.270678
$465$	0	0
$466$	−0.486886	−0.0225545
$467$	−5.93499	−0.274638	−0.137319	−	0.990527i	$-0.543849\pi$
$467$	−5.93499	−0.274638	−0.137319	+	0.990527i	$0.543849\pi$
$468$	0	0
$469$	−17.3397	−0.800673
$470$	−0.132943	−0.00613220
$471$	0	0
$472$	2.28299	0.105083
$473$	0	0
$474$	0	0
$475$	−0.541216	−0.0248327
$476$	−17.9386	−0.822216
$477$	0	0
$478$	0.0837594	0.00383107
$479$	30.6871	1.40213	0.701065	−	0.713097i	$-0.252708\pi$
$479$	30.6871	1.40213	0.701065	+	0.713097i	$0.252708\pi$
$480$	0	0
$481$	0.739413	0.0337143
$482$	−1.03628	−0.0472014
$483$	0	0
$484$	0	0
$485$	−11.0077	−0.499832
$486$	0	0
$487$	40.0484	1.81476	0.907382	−	0.420306i	$-0.138077\pi$
$487$	40.0484	1.81476	0.907382	+	0.420306i	$0.138077\pi$
$488$	−2.34905	−0.106337
$489$	0	0
$490$	0.0873867	0.00394773
$491$	19.5208	0.880961	0.440481	−	0.897762i	$-0.354808\pi$
$491$	19.5208	0.880961	0.440481	+	0.897762i	$0.354808\pi$
$492$	0	0
$493$	5.40945	0.243629
$494$	0.0159128	0.000715950 0
$495$	0	0
$496$	3.51962	0.158036
$497$	22.8971	1.02708
$498$	0	0
$499$	9.27904	0.415387	0.207693	−	0.978194i	$-0.433404\pi$
$499$	9.27904	0.415387	0.207693	+	0.978194i	$0.433404\pi$
$500$	1.99174	0.0890734
$501$	0	0
$502$	0.0808532	0.00360866
$503$	−19.5067	−0.869760	−0.434880	−	0.900488i	$-0.643209\pi$
$503$	−19.5067	−0.869760	−0.434880	+	0.900488i	$0.643209\pi$
$504$	0	0
$505$	−6.71144	−0.298655
$506$	0	0
$507$	0	0
$508$	−24.1202	−1.07016
$509$	−10.5738	−0.468676	−0.234338	−	0.972155i	$-0.575292\pi$
$509$	−10.5738	−0.468676	−0.234338	+	0.972155i	$0.575292\pi$
$510$	0	0
$511$	−15.2188	−0.673240
$512$	7.15076	0.316022
$513$	0	0
$514$	1.45521	0.0641865
$515$	−15.6033	−0.687564
$516$	0	0
$517$	0	0
$518$	−0.510390	−0.0224252
$519$	0	0
$520$	−0.117365	−0.00514678
$521$	8.37614	0.366965	0.183483	−	0.983023i	$-0.441263\pi$
$521$	8.37614	0.366965	0.183483	+	0.983023i	$0.441263\pi$
$522$	0	0
$523$	−34.4717	−1.50734	−0.753672	−	0.657251i	$-0.771719\pi$
$523$	−34.4717	−1.50734	−0.753672	+	0.657251i	$0.771719\pi$
$524$	−37.2540	−1.62745
$525$	0	0
$526$	1.93531	0.0843834
$527$	3.26541	0.142243
$528$	0	0
$529$	−1.60536	−0.0697983
$530$	−0.938727	−0.0407757
$531$	0	0
$532$	2.64890	0.114844
$533$	2.90545	0.125849
$534$	0	0
$535$	8.98415	0.388419
$536$	2.55980	0.110567
$537$	0	0
$538$	2.75213	0.118653
$539$	0	0
$540$	0	0
$541$	24.0619	1.03450	0.517250	−	0.855834i	$-0.326956\pi$
$541$	24.0619	1.03450	0.517250	+	0.855834i	$0.326956\pi$
$542$	−0.546965	−0.0234942
$543$	0	0
$544$	3.97507	0.170430
$545$	−10.7685	−0.461273
$546$	0	0
$547$	22.1544	0.947252	0.473626	−	0.880726i	$-0.342945\pi$
$547$	22.1544	0.947252	0.473626	+	0.880726i	$0.342945\pi$
$548$	−31.4404	−1.34307
$549$	0	0
$550$	0	0
$551$	−0.798784	−0.0340293
$552$	0	0
$553$	−28.5170	−1.21267
$554$	2.54796	0.108253
$555$	0	0
$556$	27.5568	1.16867
$557$	−7.69514	−0.326053	−0.163027	−	0.986622i	$-0.552126\pi$
$557$	−7.69514	−0.326053	−0.163027	+	0.986622i	$0.552126\pi$
$558$	0	0
$559$	2.04458	0.0864763
$560$	−9.70768	−0.410224
$561$	0	0
$562$	2.38367	0.100549
$563$	32.6446	1.37581	0.687903	−	0.725802i	$-0.258531\pi$
$563$	32.6446	1.37581	0.687903	+	0.725802i	$0.258531\pi$
$564$	0	0
$565$	19.8949	0.836985
$566$	−1.12697	−0.0473700
$567$	0	0
$568$	−3.38023	−0.141831
$569$	16.3704	0.686284	0.343142	−	0.939284i	$-0.388509\pi$
$569$	16.3704	0.686284	0.343142	+	0.939284i	$0.388509\pi$
$570$	0	0
$571$	−14.2204	−0.595107	−0.297554	−	0.954705i	$-0.596171\pi$
$571$	−14.2204	−0.595107	−0.297554	+	0.954705i	$0.596171\pi$
$572$	0	0
$573$	0	0
$574$	−2.00553	−0.0837091
$575$	4.62543	0.192894
$576$	0	0
$577$	21.8547	0.909825	0.454913	−	0.890536i	$-0.349671\pi$
$577$	21.8547	0.909825	0.454913	+	0.890536i	$0.349671\pi$
$578$	−0.324120	−0.0134816
$579$	0	0
$580$	2.93962	0.122061
$581$	42.6936	1.77123
$582$	0	0
$583$	0	0
$584$	2.24670	0.0929690
$585$	0	0
$586$	2.70630	0.111796
$587$	−38.5391	−1.59068	−0.795339	−	0.606165i	$-0.792707\pi$
$587$	−38.5391	−1.59068	−0.795339	+	0.606165i	$0.792707\pi$
$588$	0	0
$589$	−0.482185	−0.0198681
$590$	0.571927	0.0235459
$591$	0	0
$592$	−9.02879	−0.371081
$593$	−30.7989	−1.26476	−0.632379	−	0.774659i	$-0.717921\pi$
$593$	−30.7989	−1.26476	−0.632379	+	0.774659i	$0.717921\pi$
$594$	0	0
$595$	−9.00651	−0.369231
$596$	−18.0012	−0.737359
$597$	0	0
$598$	−0.135996	−0.00556131
$599$	−37.7384	−1.54195	−0.770975	−	0.636865i	$-0.780231\pi$
$599$	−37.7384	−1.54195	−0.770975	+	0.636865i	$0.780231\pi$
$600$	0	0
$601$	22.9568	0.936426	0.468213	−	0.883616i	$-0.344898\pi$
$601$	22.9568	0.936426	0.468213	+	0.883616i	$0.344898\pi$
$602$	−1.41130	−0.0575201
$603$	0	0
$604$	−17.6299	−0.717351
$605$	0	0
$606$	0	0
$607$	−11.2519	−0.456701	−0.228350	−	0.973579i	$-0.573333\pi$
$607$	−11.2519	−0.456701	−0.228350	+	0.973579i	$0.573333\pi$
$608$	−0.586976	−0.0238050
$609$	0	0
$610$	−0.588478	−0.0238268
$611$	−0.473274	−0.0191466
$612$	0	0
$613$	−44.7721	−1.80833	−0.904164	−	0.427186i	$-0.859505\pi$
$613$	−44.7721	−1.80833	−0.904164	+	0.427186i	$0.859505\pi$
$614$	−1.23118	−0.0496864
$615$	0	0
$616$	0	0
$617$	−16.4108	−0.660673	−0.330337	−	0.943863i	$-0.607162\pi$
$617$	−16.4108	−0.660673	−0.330337	+	0.943863i	$0.607162\pi$
$618$	0	0
$619$	1.50701	0.0605720	0.0302860	−	0.999541i	$-0.490358\pi$
$619$	1.50701	0.0605720	0.0302860	+	0.999541i	$0.490358\pi$
$620$	1.77450	0.0712656
$621$	0	0
$622$	−0.795374	−0.0318916
$623$	−11.5560	−0.462983
$624$	0	0
$625$	1.00000	0.0400000
$626$	1.47329	0.0588847
$627$	0	0
$628$	−23.3813	−0.933015
$629$	−8.37665	−0.333999
$630$	0	0
$631$	−33.3576	−1.32795	−0.663973	−	0.747756i	$-0.731131\pi$
$631$	−33.3576	−1.32795	−0.663973	+	0.747756i	$0.731131\pi$
$632$	4.20987	0.167459
$633$	0	0
$634$	0.534604	0.0212318
$635$	−12.1101	−0.480576
$636$	0	0
$637$	0.311095	0.0123260
$638$	0	0
$639$	0	0
$640$	2.87818	0.113770
$641$	−8.60811	−0.340000	−0.170000	−	0.985444i	$-0.554377\pi$
$641$	−8.60811	−0.340000	−0.170000	+	0.985444i	$0.554377\pi$
$642$	0	0
$643$	−11.8455	−0.467139	−0.233570	−	0.972340i	$-0.575041\pi$
$643$	−11.8455	−0.467139	−0.233570	+	0.972340i	$0.575041\pi$
$644$	−22.6385	−0.892081
$645$	0	0
$646$	−0.180272	−0.00709272
$647$	22.2169	0.873437	0.436719	−	0.899598i	$-0.356141\pi$
$647$	22.2169	0.873437	0.436719	+	0.899598i	$0.356141\pi$
$648$	0	0
$649$	0	0
$650$	−0.0294019	−0.00115324
$651$	0	0
$652$	−15.1774	−0.594394
$653$	−6.85023	−0.268070	−0.134035	−	0.990977i	$-0.542793\pi$
$653$	−6.85023	−0.268070	−0.134035	+	0.990977i	$0.542793\pi$
$654$	0	0
$655$	−18.7043	−0.730836
$656$	−35.4778	−1.38517
$657$	0	0
$658$	0.326683	0.0127354
$659$	−10.7882	−0.420248	−0.210124	−	0.977675i	$-0.567387\pi$
$659$	−10.7882	−0.420248	−0.210124	+	0.977675i	$0.567387\pi$
$660$	0	0
$661$	24.3271	0.946214	0.473107	−	0.881005i	$-0.343132\pi$
$661$	24.3271	0.946214	0.473107	+	0.881005i	$0.343132\pi$
$662$	1.22480	0.0476031
$663$	0	0
$664$	−6.30271	−0.244593
$665$	1.32994	0.0515729
$666$	0	0
$667$	6.82670	0.264331
$668$	−20.1509	−0.779664
$669$	0	0
$670$	0.641274	0.0247746
$671$	0	0
$672$	0	0
$673$	−34.2444	−1.32003	−0.660013	−	0.751254i	$-0.729449\pi$
$673$	−34.2444	−1.32003	−0.660013	+	0.751254i	$0.729449\pi$
$674$	1.37776	0.0530693
$675$	0	0
$676$	25.6842	0.987852
$677$	−17.7171	−0.680922	−0.340461	−	0.940259i	$-0.610583\pi$
$677$	−17.7171	−0.680922	−0.340461	+	0.940259i	$0.610583\pi$
$678$	0	0
$679$	27.0494	1.03806
$680$	1.32960	0.0509878
$681$	0	0
$682$	0	0
$683$	−19.2788	−0.737683	−0.368842	−	0.929492i	$-0.620246\pi$
$683$	−19.2788	−0.737683	−0.368842	+	0.929492i	$0.620246\pi$
$684$	0	0
$685$	−15.7854	−0.603129
$686$	−1.77797	−0.0678833
$687$	0	0
$688$	−24.9658	−0.951813
$689$	−3.34185	−0.127314
$690$	0	0
$691$	−27.2720	−1.03748	−0.518738	−	0.854933i	$-0.673598\pi$
$691$	−27.2720	−1.03748	−0.518738	+	0.854933i	$0.673598\pi$
$692$	−19.3103	−0.734067
$693$	0	0
$694$	1.57750	0.0598812
$695$	13.8356	0.524812
$696$	0	0
$697$	−32.9152	−1.24675
$698$	0.444614	0.0168289
$699$	0	0
$700$	−4.89435	−0.184989
$701$	−39.2145	−1.48111	−0.740555	−	0.671996i	$-0.765437\pi$
$701$	−39.2145	−1.48111	−0.740555	+	0.671996i	$0.765437\pi$
$702$	0	0
$703$	1.23693	0.0466519
$704$	0	0
$705$	0	0
$706$	−0.847598	−0.0318998
$707$	16.4922	0.620252
$708$	0	0
$709$	28.2333	1.06032	0.530162	−	0.847896i	$-0.322131\pi$
$709$	28.2333	1.06032	0.530162	+	0.847896i	$0.322131\pi$
$710$	−0.846805	−0.0317800
$711$	0	0
$712$	1.70598	0.0639342
$713$	4.12093	0.154330
$714$	0	0
$715$	0	0
$716$	36.1666	1.35161
$717$	0	0
$718$	2.63333	0.0982752
$719$	−42.2198	−1.57453	−0.787266	−	0.616614i	$-0.788504\pi$
$719$	−42.2198	−1.57453	−0.787266	+	0.616614i	$0.788504\pi$
$720$	0	0
$721$	38.3423	1.42794
$722$	−1.70008	−0.0632705
$723$	0	0
$724$	8.78211	0.326384
$725$	1.47591	0.0548137
$726$	0	0
$727$	5.10543	0.189350	0.0946750	−	0.995508i	$-0.469819\pi$
$727$	5.10543	0.189350	0.0946750	+	0.995508i	$0.469819\pi$
$728$	0.288403	0.0106889
$729$	0	0
$730$	0.562837	0.0208315
$731$	−23.1626	−0.856698
$732$	0	0
$733$	19.2822	0.712204	0.356102	−	0.934447i	$-0.384106\pi$
$733$	19.2822	0.712204	0.356102	+	0.934447i	$0.384106\pi$
$734$	−1.26755	−0.0467860
$735$	0	0
$736$	5.01652	0.184911
$737$	0	0
$738$	0	0
$739$	35.9006	1.32063	0.660313	−	0.750991i	$-0.270424\pi$
$739$	35.9006	1.32063	0.660313	+	0.750991i	$0.270424\pi$
$740$	−4.55207	−0.167337
$741$	0	0
$742$	2.30675	0.0846836
$743$	12.3368	0.452593	0.226297	−	0.974058i	$-0.427338\pi$
$743$	12.3368	0.452593	0.226297	+	0.974058i	$0.427338\pi$
$744$	0	0
$745$	−9.03793	−0.331124
$746$	2.00044	0.0732412
$747$	0	0
$748$	0	0
$749$	−22.0769	−0.806674
$750$	0	0
$751$	6.79867	0.248087	0.124044	−	0.992277i	$-0.460414\pi$
$751$	6.79867	0.248087	0.124044	+	0.992277i	$0.460414\pi$
$752$	5.77903	0.210739
$753$	0	0
$754$	−0.0433944	−0.00158033
$755$	−8.85151	−0.322139
$756$	0	0
$757$	−10.2286	−0.371766	−0.185883	−	0.982572i	$-0.559515\pi$
$757$	−10.2286	−0.371766	−0.185883	+	0.982572i	$0.559515\pi$
$758$	−0.172176	−0.00625370
$759$	0	0
$760$	−0.196335	−0.00712181
$761$	24.5133	0.888607	0.444304	−	0.895876i	$-0.353451\pi$
$761$	24.5133	0.888607	0.444304	+	0.895876i	$0.353451\pi$
$762$	0	0
$763$	26.4617	0.957979
$764$	24.4701	0.885296
$765$	0	0
$766$	−0.0871316	−0.00314819
$767$	2.03605	0.0735174
$768$	0	0
$769$	−27.2852	−0.983930	−0.491965	−	0.870615i	$-0.663721\pi$
$769$	−27.2852	−0.983930	−0.491965	+	0.870615i	$0.663721\pi$
$770$	0	0
$771$	0	0
$772$	46.4003	1.66998
$773$	8.67553	0.312037	0.156019	−	0.987754i	$-0.450134\pi$
$773$	8.67553	0.312037	0.156019	+	0.987754i	$0.450134\pi$
$774$	0	0
$775$	0.890928	0.0320031
$776$	−3.99320	−0.143348
$777$	0	0
$778$	0.593326	0.0212718
$779$	4.86042	0.174142
$780$	0	0
$781$	0	0
$782$	1.54067	0.0550944
$783$	0	0
$784$	−3.79870	−0.135668
$785$	−11.7391	−0.418987
$786$	0	0
$787$	30.0936	1.07272	0.536361	−	0.843988i	$-0.319798\pi$
$787$	30.0936	1.07272	0.536361	+	0.843988i	$0.319798\pi$
$788$	−39.2232	−1.39727
$789$	0	0
$790$	1.05464	0.0375226
$791$	−48.8882	−1.73826
$792$	0	0
$793$	−2.09497	−0.0743945
$794$	−1.94137	−0.0688965
$795$	0	0
$796$	−44.2333	−1.56781
$797$	−32.1847	−1.14004	−0.570020	−	0.821631i	$-0.693065\pi$
$797$	−32.1847	−1.14004	−0.570020	+	0.821631i	$0.693065\pi$
$798$	0	0
$799$	5.36162	0.189680
$800$	1.08455	0.0383447
$801$	0	0
$802$	−1.14986	−0.0406029
$803$	0	0
$804$	0	0
$805$	−11.3662	−0.400605
$806$	−0.0261950	−0.000922678 0
$807$	0	0
$808$	−2.43468	−0.0856518
$809$	−42.2435	−1.48520	−0.742602	−	0.669733i	$-0.766409\pi$
$809$	−42.2435	−1.48520	−0.742602	+	0.669733i	$0.766409\pi$
$810$	0	0
$811$	47.3043	1.66108	0.830539	−	0.556961i	$-0.188033\pi$
$811$	47.3043	1.66108	0.830539	+	0.556961i	$0.188033\pi$
$812$	−7.22359	−0.253498
$813$	0	0
$814$	0	0
$815$	−7.62018	−0.266923
$816$	0	0
$817$	3.42029	0.119661
$818$	−0.455091	−0.0159119
$819$	0	0
$820$	−17.8869	−0.624638
$821$	−26.7356	−0.933078	−0.466539	−	0.884501i	$-0.654499\pi$
$821$	−26.7356	−0.933078	−0.466539	+	0.884501i	$0.654499\pi$
$822$	0	0
$823$	−18.6123	−0.648785	−0.324392	−	0.945923i	$-0.605160\pi$
$823$	−18.6123	−0.648785	−0.324392	+	0.945923i	$0.605160\pi$
$824$	−5.66035	−0.197188
$825$	0	0
$826$	−1.40541	−0.0489004
$827$	−4.46798	−0.155367	−0.0776835	−	0.996978i	$-0.524752\pi$
$827$	−4.46798	−0.155367	−0.0776835	+	0.996978i	$0.524752\pi$
$828$	0	0
$829$	37.5259	1.30333	0.651664	−	0.758508i	$-0.274071\pi$
$829$	37.5259	1.30333	0.651664	+	0.758508i	$0.274071\pi$
$830$	−1.57894	−0.0548057
$831$	0	0
$832$	2.52431	0.0875148
$833$	−3.52433	−0.122111
$834$	0	0
$835$	−10.1173	−0.350122
$836$	0	0
$837$	0	0
$838$	0.846746	0.0292503
$839$	31.9129	1.10175	0.550877	−	0.834586i	$-0.314293\pi$
$839$	31.9129	1.10175	0.550877	+	0.834586i	$0.314293\pi$
$840$	0	0
$841$	−26.8217	−0.924886
$842$	2.15187	0.0741584
$843$	0	0
$844$	−25.8762	−0.890697
$845$	12.8953	0.443613
$846$	0	0
$847$	0	0
$848$	40.8065	1.40130
$849$	0	0
$850$	0.333088	0.0114248
$851$	−10.5713	−0.362380
$852$	0	0
$853$	−52.9135	−1.81172	−0.905862	−	0.423573i	$-0.860776\pi$
$853$	−52.9135	−1.81172	−0.905862	+	0.423573i	$0.860776\pi$
$854$	1.44608	0.0494838
$855$	0	0
$856$	3.25914	0.111395
$857$	−17.8209	−0.608750	−0.304375	−	0.952552i	$-0.598448\pi$
$857$	−17.8209	−0.608750	−0.304375	+	0.952552i	$0.598448\pi$
$858$	0	0
$859$	7.61904	0.259958	0.129979	−	0.991517i	$-0.458509\pi$
$859$	7.61904	0.259958	0.129979	+	0.991517i	$0.458509\pi$
$860$	−12.5871	−0.429216
$861$	0	0
$862$	−2.33622	−0.0795721
$863$	−2.01709	−0.0686624	−0.0343312	−	0.999411i	$-0.510930\pi$
$863$	−2.01709	−0.0686624	−0.0343312	+	0.999411i	$0.510930\pi$
$864$	0	0
$865$	−9.69518	−0.329646
$866$	1.80799	0.0614380
$867$	0	0
$868$	−4.36051	−0.148005
$869$	0	0
$870$	0	0
$871$	2.28292	0.0773538
$872$	−3.90645	−0.132289
$873$	0	0
$874$	−0.227503	−0.00769541
$875$	−2.45732	−0.0830726
$876$	0	0
$877$	−1.14869	−0.0387886	−0.0193943	−	0.999812i	$-0.506174\pi$
$877$	−1.14869	−0.0387886	−0.0193943	+	0.999812i	$0.506174\pi$
$878$	0.257943	0.00870517
$879$	0	0
$880$	0	0
$881$	−36.3252	−1.22383	−0.611913	−	0.790925i	$-0.709600\pi$
$881$	−36.3252	−1.22383	−0.611913	+	0.790925i	$0.709600\pi$
$882$	0	0
$883$	−22.9679	−0.772933	−0.386467	−	0.922303i	$-0.626305\pi$
$883$	−22.9679	−0.772933	−0.386467	+	0.922303i	$0.626305\pi$
$884$	2.36178	0.0794351
$885$	0	0
$886$	2.46223	0.0827201
$887$	−19.7299	−0.662465	−0.331233	−	0.943549i	$-0.607464\pi$
$887$	−19.7299	−0.662465	−0.331233	+	0.943549i	$0.607464\pi$
$888$	0	0
$889$	29.7585	0.998067
$890$	0.427377	0.0143257
$891$	0	0
$892$	45.0553	1.50856
$893$	−0.791721	−0.0264939
$894$	0	0
$895$	18.1583	0.606965
$896$	−7.07262	−0.236280
$897$	0	0
$898$	2.37642	0.0793023
$899$	1.31493	0.0438552
$900$	0	0
$901$	37.8591	1.26127
$902$	0	0
$903$	0	0
$904$	7.21719	0.240040
$905$	4.40926	0.146569
$906$	0	0
$907$	7.58622	0.251896	0.125948	−	0.992037i	$-0.459803\pi$
$907$	7.58622	0.251896	0.125948	+	0.992037i	$0.459803\pi$
$908$	−31.3349	−1.03988
$909$	0	0
$910$	0.0722499	0.00239506
$911$	−32.2776	−1.06940	−0.534702	−	0.845041i	$-0.679576\pi$
$911$	−32.2776	−1.06940	−0.534702	+	0.845041i	$0.679576\pi$
$912$	0	0
$913$	0	0
$914$	1.28091	0.0423687
$915$	0	0
$916$	48.4014	1.59923
$917$	45.9624	1.51781
$918$	0	0
$919$	7.13382	0.235323	0.117661	−	0.993054i	$-0.462460\pi$
$919$	7.13382	0.235323	0.117661	+	0.993054i	$0.462460\pi$
$920$	1.67795	0.0553203
$921$	0	0
$922$	−0.0193099	−0.000635937 0
$923$	−3.01461	−0.0992269
$924$	0	0
$925$	−2.28547	−0.0751459
$926$	−0.818874	−0.0269099
$927$	0	0
$928$	1.60069	0.0525454
$929$	−51.8610	−1.70150	−0.850752	−	0.525568i	$-0.823853\pi$
$929$	−51.8610	−1.70150	−0.850752	+	0.525568i	$0.823853\pi$
$930$	0	0
$931$	0.520418	0.0170560
$932$	10.6708	0.349533
$933$	0	0
$934$	−0.539366	−0.0176486
$935$	0	0
$936$	0	0
$937$	−43.5374	−1.42231	−0.711153	−	0.703037i	$-0.751827\pi$
$937$	−43.5374	−1.42231	−0.711153	+	0.703037i	$0.751827\pi$
$938$	−1.57582	−0.0514522
$939$	0	0
$940$	2.91363	0.0950320
$941$	−15.6825	−0.511236	−0.255618	−	0.966778i	$-0.582279\pi$
$941$	−15.6825	−0.511236	−0.255618	+	0.966778i	$0.582279\pi$
$942$	0	0
$943$	−41.5389	−1.35269
$944$	−24.8617	−0.809179
$945$	0	0
$946$	0	0
$947$	10.8305	0.351944	0.175972	−	0.984395i	$-0.443693\pi$
$947$	10.8305	0.351944	0.175972	+	0.984395i	$0.443693\pi$
$948$	0	0
$949$	2.00369	0.0650424
$950$	−0.0491852	−0.00159578
$951$	0	0
$952$	−3.26725	−0.105892
$953$	32.8952	1.06558	0.532790	−	0.846247i	$-0.321144\pi$
$953$	32.8952	1.06558	0.532790	+	0.846247i	$0.321144\pi$
$954$	0	0
$955$	12.2858	0.397558
$956$	−1.83570	−0.0593709
$957$	0	0
$958$	2.78882	0.0901026
$959$	38.7898	1.25259
$960$	0	0
$961$	−30.2062	−0.974395
$962$	0.0671972	0.00216652
$963$	0	0
$964$	22.7116	0.731490
$965$	23.2963	0.749935
$966$	0	0
$967$	7.65301	0.246104	0.123052	−	0.992400i	$-0.460732\pi$
$967$	7.65301	0.246104	0.123052	+	0.992400i	$0.460732\pi$
$968$	0	0
$969$	0	0
$970$	−1.00037	−0.0321198
$971$	43.2132	1.38678	0.693388	−	0.720564i	$-0.256117\pi$
$971$	43.2132	1.38678	0.693388	+	0.720564i	$0.256117\pi$
$972$	0	0
$973$	−33.9984	−1.08994
$974$	3.63956	0.116619
$975$	0	0
$976$	25.5811	0.818833
$977$	−17.6554	−0.564845	−0.282423	−	0.959290i	$-0.591138\pi$
$977$	−17.6554	−0.564845	−0.282423	+	0.959290i	$0.591138\pi$
$978$	0	0
$979$	0	0
$980$	−1.91520	−0.0611789
$981$	0	0
$982$	1.77403	0.0566117
$983$	12.0992	0.385903	0.192952	−	0.981208i	$-0.438194\pi$
$983$	12.0992	0.385903	0.192952	+	0.981208i	$0.438194\pi$
$984$	0	0
$985$	−19.6929	−0.627469
$986$	0.491606	0.0156559
$987$	0	0
$988$	−0.348750	−0.0110952
$989$	−29.2311	−0.929494
$990$	0	0
$991$	−8.45499	−0.268581	−0.134291	−	0.990942i	$-0.542876\pi$
$991$	−8.45499	−0.268581	−0.134291	+	0.990942i	$0.542876\pi$
$992$	0.966257	0.0306787
$993$	0	0
$994$	2.08087	0.0660012
$995$	−22.2083	−0.704052
$996$	0	0
$997$	−54.5603	−1.72794	−0.863970	−	0.503542i	$-0.832030\pi$
$997$	−54.5603	−1.72794	−0.863970	+	0.503542i	$0.832030\pi$
$998$	0.843270	0.0266933
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.cc.1.4		8
3.2	odd	2		5445.2.a.cb.1.5		8
11.7	odd	10		495.2.n.h.181.3	yes	16
11.8	odd	10		495.2.n.h.361.3	yes	16
11.10	odd	2		5445.2.a.ca.1.5		8
33.8	even	10		495.2.n.g.361.2	yes	16
33.29	even	10		495.2.n.g.181.2	✓	16
33.32	even	2		5445.2.a.cd.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.n.g.181.2	✓	16	33.29	even	10
495.2.n.g.361.2	yes	16	33.8	even	10
495.2.n.h.181.3	yes	16	11.7	odd	10
495.2.n.h.361.3	yes	16	11.8	odd	10
5445.2.a.ca.1.5		8	11.10	odd	2
5445.2.a.cb.1.5		8	3.2	odd	2
5445.2.a.cc.1.4		8	1.1	even	1	trivial
5445.2.a.cd.1.4		8	33.32	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0908791 q^{2} -1.99174 q^{4} -1.00000 q^{5} +2.45732 q^{7} -0.362766 q^{8} +O(q^{10})\)	\(q+0.0908791 q^{2} -1.99174 q^{4} -1.00000 q^{5} +2.45732 q^{7} -0.362766 q^{8} -0.0908791 q^{10} -0.323527 q^{13} +0.223319 q^{14} +3.95051 q^{16} +3.66517 q^{17} -0.541216 q^{19} +1.99174 q^{20} +4.62543 q^{23} +1.00000 q^{25} -0.0294019 q^{26} -4.89435 q^{28} +1.47591 q^{29} +0.890928 q^{31} +1.08455 q^{32} +0.333088 q^{34} -2.45732 q^{35} -2.28547 q^{37} -0.0491852 q^{38} +0.362766 q^{40} -8.98054 q^{41} -6.31964 q^{43} +0.420355 q^{46} +1.46285 q^{47} -0.961571 q^{49} +0.0908791 q^{50} +0.644383 q^{52} +10.3294 q^{53} -0.891432 q^{56} +0.134129 q^{58} -6.29328 q^{59} +6.47540 q^{61} +0.0809667 q^{62} -7.80247 q^{64} +0.323527 q^{65} -7.05634 q^{67} -7.30008 q^{68} -0.223319 q^{70} +9.31793 q^{71} -6.19325 q^{73} -0.207702 q^{74} +1.07796 q^{76} -11.6049 q^{79} -3.95051 q^{80} -0.816144 q^{82} +17.3740 q^{83} -3.66517 q^{85} -0.574323 q^{86} -4.70270 q^{89} -0.795011 q^{91} -9.21267 q^{92} +0.132943 q^{94} +0.541216 q^{95} +11.0077 q^{97} -0.0873867 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 6 q^{4} - 8 q^{5} - 8 q^{7} + 12 q^{8}+O(q^{10}) \) 8 * q + 4 * q^2 + 6 * q^4 - 8 * q^5 - 8 * q^7 + 12 * q^8	\( 8 q + 4 q^{2} + 6 q^{4} - 8 q^{5} - 8 q^{7} + 12 q^{8} - 4 q^{10} - 6 q^{13} - 14 q^{14} + 14 q^{16} + 8 q^{17} + 2 q^{19} - 6 q^{20} - 4 q^{23} + 8 q^{25} + 2 q^{26} - 24 q^{28} + 22 q^{29} + 10 q^{31} + 28 q^{32} - 2 q^{34} + 8 q^{35} - 14 q^{37} + 20 q^{38} - 12 q^{40} + 22 q^{41} - 14 q^{43} + 2 q^{46} - 10 q^{47} + 4 q^{50} + 10 q^{52} + 18 q^{53} - 34 q^{56} + 12 q^{58} - 2 q^{59} + 14 q^{61} + 30 q^{62} + 30 q^{64} + 6 q^{65} + 10 q^{67} + 6 q^{68} + 14 q^{70} + 2 q^{71} - 16 q^{73} + 24 q^{74} + 22 q^{76} + 16 q^{79} - 14 q^{80} + 10 q^{82} + 46 q^{83} - 8 q^{85} + 28 q^{86} - 38 q^{89} + 8 q^{91} + 24 q^{92} + 10 q^{94} - 2 q^{95} - 4 q^{97} + 4 q^{98}+O(q^{100}) \) 8 * q + 4 * q^2 + 6 * q^4 - 8 * q^5 - 8 * q^7 + 12 * q^8 - 4 * q^10 - 6 * q^13 - 14 * q^14 + 14 * q^16 + 8 * q^17 + 2 * q^19 - 6 * q^20 - 4 * q^23 + 8 * q^25 + 2 * q^26 - 24 * q^28 + 22 * q^29 + 10 * q^31 + 28 * q^32 - 2 * q^34 + 8 * q^35 - 14 * q^37 + 20 * q^38 - 12 * q^40 + 22 * q^41 - 14 * q^43 + 2 * q^46 - 10 * q^47 + 4 * q^50 + 10 * q^52 + 18 * q^53 - 34 * q^56 + 12 * q^58 - 2 * q^59 + 14 * q^61 + 30 * q^62 + 30 * q^64 + 6 * q^65 + 10 * q^67 + 6 * q^68 + 14 * q^70 + 2 * q^71 - 16 * q^73 + 24 * q^74 + 22 * q^76 + 16 * q^79 - 14 * q^80 + 10 * q^82 + 46 * q^83 - 8 * q^85 + 28 * q^86 - 38 * q^89 + 8 * q^91 + 24 * q^92 + 10 * q^94 - 2 * q^95 - 4 * q^97 + 4 * q^98

Introduction

L-functions

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