LMFDB - Embedded newform 5625.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1875)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.95630$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.209057 q^{2} -1.95630 q^{4} -0.591023 q^{7} +0.827091 q^{8} +O(q^{10})$	$q-0.209057 q^{2} -1.95630 q^{4} -0.591023 q^{7} +0.827091 q^{8} +0.870796 q^{11} -1.15057 q^{13} +0.123557 q^{14} +3.73968 q^{16} -4.93395 q^{17} -2.84943 q^{19} -0.182046 q^{22} +6.91259 q^{23} +0.240534 q^{26} +1.15622 q^{28} +7.48883 q^{29} +3.45991 q^{31} -2.43599 q^{32} +1.03148 q^{34} -10.1681 q^{37} +0.595693 q^{38} -9.11409 q^{41} +2.81486 q^{43} -1.70353 q^{44} -1.44512 q^{46} +6.68842 q^{47} -6.65069 q^{49} +2.25085 q^{52} +3.87238 q^{53} -0.488830 q^{56} -1.56559 q^{58} +11.5277 q^{59} -12.7564 q^{61} -0.723318 q^{62} -6.97010 q^{64} +7.60292 q^{67} +9.65227 q^{68} +15.0566 q^{71} +2.98798 q^{73} +2.12571 q^{74} +5.57433 q^{76} -0.514660 q^{77} +3.33728 q^{79} +1.90536 q^{82} -9.73310 q^{83} -0.588467 q^{86} +0.720227 q^{88} -0.645045 q^{89} +0.680012 q^{91} -13.5231 q^{92} -1.39826 q^{94} -11.6970 q^{97} +1.39037 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + q^{4} - 5 q^{7} - 3 q^{8}+O(q^{10}) $ 4 * q + q^2 + q^4 - 5 * q^7 - 3 * q^8	$ 4 q + q^{2} + q^{4} - 5 q^{7} - 3 q^{8} + 6 q^{11} - 7 q^{13} + 10 q^{14} - 9 q^{16} - 7 q^{17} - 9 q^{19} - 6 q^{22} + 10 q^{23} + 2 q^{26} - 5 q^{28} + 28 q^{29} - 10 q^{31} + 7 q^{34} + 10 q^{37} - 6 q^{38} - q^{43} + 9 q^{44} + 5 q^{46} - 23 q^{47} - 3 q^{49} - 13 q^{52} + 2 q^{58} - 4 q^{59} - 43 q^{61} - 10 q^{62} - 7 q^{64} - 8 q^{67} - 3 q^{68} + 27 q^{71} - 15 q^{73} - 5 q^{74} + 9 q^{76} - 15 q^{77} + 10 q^{79} + 20 q^{82} + 3 q^{83} - 24 q^{86} + 3 q^{88} + 9 q^{89} + 5 q^{91} - 15 q^{92} - 22 q^{94} - 13 q^{97} - 42 q^{98}+O(q^{100}) $ 4 * q + q^2 + q^4 - 5 * q^7 - 3 * q^8 + 6 * q^11 - 7 * q^13 + 10 * q^14 - 9 * q^16 - 7 * q^17 - 9 * q^19 - 6 * q^22 + 10 * q^23 + 2 * q^26 - 5 * q^28 + 28 * q^29 - 10 * q^31 + 7 * q^34 + 10 * q^37 - 6 * q^38 - q^43 + 9 * q^44 + 5 * q^46 - 23 * q^47 - 3 * q^49 - 13 * q^52 + 2 * q^58 - 4 * q^59 - 43 * q^61 - 10 * q^62 - 7 * q^64 - 8 * q^67 - 3 * q^68 + 27 * q^71 - 15 * q^73 - 5 * q^74 + 9 * q^76 - 15 * q^77 + 10 * q^79 + 20 * q^82 + 3 * q^83 - 24 * q^86 + 3 * q^88 + 9 * q^89 + 5 * q^91 - 15 * q^92 - 22 * q^94 - 13 * q^97 - 42 * 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 $

$2$	−0.209057	−0.147826	−0.0739128	−	0.997265i	$-0.523549\pi$
$2$	−0.209057	−0.147826	−0.0739128	+	0.997265i	$0.523549\pi$
$3$	0	0
$3$	0	0
$4$	−1.95630	−0.978148
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.591023	−0.223386	−0.111693	−	0.993743i	$-0.535627\pi$
$7$	−0.591023	−0.223386	−0.111693	+	0.993743i	$0.535627\pi$
$8$	0.827091	0.292421
$9$	0	0
$10$	0	0
$11$	0.870796	0.262555	0.131277	−	0.991346i	$-0.458092\pi$
$11$	0.870796	0.262555	0.131277	+	0.991346i	$0.458092\pi$
$12$	0	0
$13$	−1.15057	−0.319110	−0.159555	−	0.987189i	$-0.551006\pi$
$13$	−1.15057	−0.319110	−0.159555	+	0.987189i	$0.551006\pi$
$14$	0.123557	0.0330221
$15$	0	0
$16$	3.73968	0.934920
$17$	−4.93395	−1.19666	−0.598330	−	0.801250i	$-0.704169\pi$
$17$	−4.93395	−1.19666	−0.598330	+	0.801250i	$0.704169\pi$
$18$	0	0
$19$	−2.84943	−0.653704	−0.326852	−	0.945075i	$-0.605988\pi$
$19$	−2.84943	−0.653704	−0.326852	+	0.945075i	$0.605988\pi$
$20$	0	0
$21$	0	0
$22$	−0.182046	−0.0388123
$23$	6.91259	1.44137	0.720687	−	0.693260i	$-0.243826\pi$
$23$	6.91259	1.44137	0.720687	+	0.693260i	$0.243826\pi$
$24$	0	0
$25$	0	0
$26$	0.240534	0.0471727
$27$	0	0
$28$	1.15622	0.218504
$29$	7.48883	1.39064	0.695320	−	0.718700i	$-0.255262\pi$
$29$	7.48883	1.39064	0.695320	+	0.718700i	$0.255262\pi$
$30$	0	0
$31$	3.45991	0.621418	0.310709	−	0.950505i	$-0.399434\pi$
$31$	3.45991	0.621418	0.310709	+	0.950505i	$0.399434\pi$
$32$	−2.43599	−0.430626
$33$	0	0
$34$	1.03148	0.176897
$35$	0	0
$36$	0	0
$37$	−10.1681	−1.67163	−0.835814	−	0.549013i	$-0.815004\pi$
$37$	−10.1681	−1.67163	−0.835814	+	0.549013i	$0.815004\pi$
$38$	0.595693	0.0966342
$39$	0	0
$40$	0	0
$41$	−9.11409	−1.42338	−0.711691	−	0.702493i	$-0.752070\pi$
$41$	−9.11409	−1.42338	−0.711691	+	0.702493i	$0.752070\pi$
$42$	0	0
$43$	2.81486	0.429263	0.214631	−	0.976695i	$-0.431145\pi$
$43$	2.81486	0.429263	0.214631	+	0.976695i	$0.431145\pi$
$44$	−1.70353	−0.256817
$45$	0	0
$46$	−1.44512	−0.213072
$47$	6.68842	0.975606	0.487803	−	0.872954i	$-0.337798\pi$
$47$	6.68842	0.975606	0.487803	+	0.872954i	$0.337798\pi$
$48$	0	0
$49$	−6.65069	−0.950099
$50$	0	0
$51$	0	0
$52$	2.25085	0.312137
$53$	3.87238	0.531912	0.265956	−	0.963985i	$-0.414312\pi$
$53$	3.87238	0.531912	0.265956	+	0.963985i	$0.414312\pi$
$54$	0	0
$55$	0	0
$56$	−0.488830	−0.0653226
$57$	0	0
$58$	−1.56559	−0.205572
$59$	11.5277	1.50078	0.750392	−	0.660993i	$-0.229865\pi$
$59$	11.5277	1.50078	0.750392	+	0.660993i	$0.229865\pi$
$60$	0	0
$61$	−12.7564	−1.63329	−0.816643	−	0.577143i	$-0.804168\pi$
$61$	−12.7564	−1.63329	−0.816643	+	0.577143i	$0.804168\pi$
$62$	−0.723318	−0.0918615
$63$	0	0
$64$	−6.97010	−0.871263
$65$	0	0
$66$	0	0
$67$	7.60292	0.928845	0.464422	−	0.885614i	$-0.346262\pi$
$67$	7.60292	0.928845	0.464422	+	0.885614i	$0.346262\pi$
$68$	9.65227	1.17051
$69$	0	0
$70$	0	0
$71$	15.0566	1.78689	0.893444	−	0.449175i	$-0.148282\pi$
$71$	15.0566	1.78689	0.893444	+	0.449175i	$0.148282\pi$
$72$	0	0
$73$	2.98798	0.349716	0.174858	−	0.984594i	$-0.444053\pi$
$73$	2.98798	0.349716	0.174858	+	0.984594i	$0.444053\pi$
$74$	2.12571	0.247109
$75$	0	0
$76$	5.57433	0.639419
$77$	−0.514660	−0.0586510
$78$	0	0
$79$	3.33728	0.375474	0.187737	−	0.982219i	$-0.439885\pi$
$79$	3.33728	0.375474	0.187737	+	0.982219i	$0.439885\pi$
$80$	0	0
$81$	0	0
$82$	1.90536	0.210412
$83$	−9.73310	−1.06835	−0.534173	−	0.845375i	$-0.679377\pi$
$83$	−9.73310	−1.06835	−0.534173	+	0.845375i	$0.679377\pi$
$84$	0	0
$85$	0	0
$86$	−0.588467	−0.0634560
$87$	0	0
$88$	0.720227	0.0767765
$89$	−0.645045	−0.0683746	−0.0341873	−	0.999415i	$-0.510884\pi$
$89$	−0.645045	−0.0683746	−0.0341873	+	0.999415i	$0.510884\pi$
$90$	0	0
$91$	0.680012	0.0712847
$92$	−13.5231	−1.40988
$93$	0	0
$94$	−1.39826	−0.144220
$95$	0	0
$96$	0	0
$97$	−11.6970	−1.18765	−0.593823	−	0.804596i	$-0.702382\pi$
$97$	−11.6970	−1.18765	−0.593823	+	0.804596i	$0.702382\pi$
$98$	1.39037	0.140449
$99$	0	0
$100$	0	0
$101$	14.7636	1.46903	0.734517	−	0.678591i	$-0.237409\pi$
$101$	14.7636	1.46903	0.734517	+	0.678591i	$0.237409\pi$
$102$	0	0
$103$	1.41969	0.139887	0.0699433	−	0.997551i	$-0.477718\pi$
$103$	1.41969	0.139887	0.0699433	+	0.997551i	$0.477718\pi$
$104$	−0.951625	−0.0933145
$105$	0	0
$106$	−0.809547	−0.0786301
$107$	−14.9579	−1.44603	−0.723016	−	0.690831i	$-0.757245\pi$
$107$	−14.9579	−1.44603	−0.723016	+	0.690831i	$0.757245\pi$
$108$	0	0
$109$	−11.0977	−1.06297	−0.531485	−	0.847068i	$-0.678366\pi$
$109$	−11.0977	−1.06297	−0.531485	+	0.847068i	$0.678366\pi$
$110$	0	0
$111$	0	0
$112$	−2.21024	−0.208848
$113$	−12.1069	−1.13892	−0.569459	−	0.822020i	$-0.692847\pi$
$113$	−12.1069	−1.13892	−0.569459	+	0.822020i	$0.692847\pi$
$114$	0	0
$115$	0	0
$116$	−14.6504	−1.36025
$117$	0	0
$118$	−2.40995	−0.221854
$119$	2.91608	0.267317
$120$	0	0
$121$	−10.2417	−0.931065
$122$	2.66681	0.241442
$123$	0	0
$124$	−6.76860	−0.607838
$125$	0	0
$126$	0	0
$127$	−0.541671	−0.0480656	−0.0240328	−	0.999711i	$-0.507651\pi$
$127$	−0.541671	−0.0480656	−0.0240328	+	0.999711i	$0.507651\pi$
$128$	6.32912	0.559421
$129$	0	0
$130$	0	0
$131$	2.46747	0.215583	0.107792	−	0.994173i	$-0.465622\pi$
$131$	2.46747	0.215583	0.107792	+	0.994173i	$0.465622\pi$
$132$	0	0
$133$	1.68408	0.146028
$134$	−1.58944	−0.137307
$135$	0	0
$136$	−4.08083	−0.349928
$137$	−5.64471	−0.482261	−0.241130	−	0.970493i	$-0.577518\pi$
$137$	−5.64471	−0.482261	−0.241130	+	0.970493i	$0.577518\pi$
$138$	0	0
$139$	−0.326034	−0.0276538	−0.0138269	−	0.999904i	$-0.504401\pi$
$139$	−0.326034	−0.0276538	−0.0138269	+	0.999904i	$0.504401\pi$
$140$	0	0
$141$	0	0
$142$	−3.14768	−0.264148
$143$	−1.00191	−0.0837839
$144$	0	0
$145$	0	0
$146$	−0.624657	−0.0516970
$147$	0	0
$148$	19.8918	1.63510
$149$	0.637614	0.0522354	0.0261177	−	0.999659i	$-0.491686\pi$
$149$	0.637614	0.0522354	0.0261177	+	0.999659i	$0.491686\pi$
$150$	0	0
$151$	11.3948	0.927299	0.463649	−	0.886019i	$-0.346540\pi$
$151$	11.3948	0.927299	0.463649	+	0.886019i	$0.346540\pi$
$152$	−2.35674	−0.191157
$153$	0	0
$154$	0.107593	0.00867011
$155$	0	0
$156$	0	0
$157$	−2.86068	−0.228307	−0.114154	−	0.993463i	$-0.536416\pi$
$157$	−2.86068	−0.228307	−0.114154	+	0.993463i	$0.536416\pi$
$158$	−0.697683	−0.0555046
$159$	0	0
$160$	0	0
$161$	−4.08550	−0.321982
$162$	0	0
$163$	−14.1106	−1.10523	−0.552614	−	0.833437i	$-0.686370\pi$
$163$	−14.1106	−1.10523	−0.552614	+	0.833437i	$0.686370\pi$
$164$	17.8299	1.39228
$165$	0	0
$166$	2.03477	0.157929
$167$	−7.49041	−0.579625	−0.289813	−	0.957083i	$-0.593593\pi$
$167$	−7.49041	−0.579625	−0.289813	+	0.957083i	$0.593593\pi$
$168$	0	0
$169$	−11.6762	−0.898169
$170$	0	0
$171$	0	0
$172$	−5.50670	−0.419882
$173$	20.3231	1.54514	0.772570	−	0.634929i	$-0.218971\pi$
$173$	20.3231	1.54514	0.772570	+	0.634929i	$0.218971\pi$
$174$	0	0
$175$	0	0
$176$	3.25650	0.245468
$177$	0	0
$178$	0.134851	0.0101075
$179$	−1.13209	−0.0846164	−0.0423082	−	0.999105i	$-0.513471\pi$
$179$	−1.13209	−0.0846164	−0.0423082	+	0.999105i	$0.513471\pi$
$180$	0	0
$181$	4.96450	0.369008	0.184504	−	0.982832i	$-0.440932\pi$
$181$	4.96450	0.369008	0.184504	+	0.982832i	$0.440932\pi$
$182$	−0.142161	−0.0105377
$183$	0	0
$184$	5.71734	0.421488
$185$	0	0
$186$	0	0
$187$	−4.29647	−0.314189
$188$	−13.0845	−0.954287
$189$	0	0
$190$	0	0
$191$	−6.44610	−0.466424	−0.233212	−	0.972426i	$-0.574924\pi$
$191$	−6.44610	−0.466424	−0.233212	+	0.972426i	$0.574924\pi$
$192$	0	0
$193$	18.4312	1.32671	0.663353	−	0.748307i	$-0.269133\pi$
$193$	18.4312	1.32671	0.663353	+	0.748307i	$0.269133\pi$
$194$	2.44533	0.175564
$195$	0	0
$196$	13.0107	0.929337
$197$	−16.8343	−1.19940	−0.599698	−	0.800227i	$-0.704713\pi$
$197$	−16.8343	−1.19940	−0.599698	+	0.800227i	$0.704713\pi$
$198$	0	0
$199$	16.9968	1.20487	0.602435	−	0.798168i	$-0.294197\pi$
$199$	16.9968	1.20487	0.602435	+	0.798168i	$0.294197\pi$
$200$	0	0
$201$	0	0
$202$	−3.08643	−0.217161
$203$	−4.42607	−0.310649
$204$	0	0
$205$	0	0
$206$	−0.296797	−0.0206788
$207$	0	0
$208$	−4.30276	−0.298343
$209$	−2.48127	−0.171633
$210$	0	0
$211$	5.00874	0.344816	0.172408	−	0.985026i	$-0.444845\pi$
$211$	5.00874	0.344816	0.172408	+	0.985026i	$0.444845\pi$
$212$	−7.57551	−0.520288
$213$	0	0
$214$	3.12705	0.213761
$215$	0	0
$216$	0	0
$217$	−2.04489	−0.138816
$218$	2.32006	0.157134
$219$	0	0
$220$	0	0
$221$	5.67685	0.381866
$222$	0	0
$223$	−15.9452	−1.06777	−0.533886	−	0.845556i	$-0.679269\pi$
$223$	−15.9452	−1.06777	−0.533886	+	0.845556i	$0.679269\pi$
$224$	1.43972	0.0961957
$225$	0	0
$226$	2.53102	0.168361
$227$	−26.8967	−1.78520	−0.892600	−	0.450850i	$-0.851121\pi$
$227$	−26.8967	−1.78520	−0.892600	+	0.450850i	$0.851121\pi$
$228$	0	0
$229$	−2.94598	−0.194676	−0.0973378	−	0.995251i	$-0.531033\pi$
$229$	−2.94598	−0.194676	−0.0973378	+	0.995251i	$0.531033\pi$
$230$	0	0
$231$	0	0
$232$	6.19394	0.406652
$233$	−23.4054	−1.53334	−0.766670	−	0.642041i	$-0.778088\pi$
$233$	−23.4054	−1.53334	−0.766670	+	0.642041i	$0.778088\pi$
$234$	0	0
$235$	0	0
$236$	−22.5517	−1.46799
$237$	0	0
$238$	−0.609627	−0.0395162
$239$	−14.7790	−0.955973	−0.477987	−	0.878367i	$-0.658633\pi$
$239$	−14.7790	−0.955973	−0.477987	+	0.878367i	$0.658633\pi$
$240$	0	0
$241$	−14.3894	−0.926903	−0.463451	−	0.886122i	$-0.653389\pi$
$241$	−14.3894	−0.926903	−0.463451	+	0.886122i	$0.653389\pi$
$242$	2.14110	0.137635
$243$	0	0
$244$	24.9552	1.59760
$245$	0	0
$246$	0	0
$247$	3.27847	0.208604
$248$	2.86166	0.181716
$249$	0	0
$250$	0	0
$251$	18.7342	1.18249	0.591247	−	0.806490i	$-0.298636\pi$
$251$	18.7342	1.18249	0.591247	+	0.806490i	$0.298636\pi$
$252$	0	0
$253$	6.01945	0.378440
$254$	0.113240	0.00710532
$255$	0	0
$256$	12.6171	0.788566
$257$	−6.42218	−0.400605	−0.200302	−	0.979734i	$-0.564192\pi$
$257$	−6.42218	−0.400605	−0.200302	+	0.979734i	$0.564192\pi$
$258$	0	0
$259$	6.00959	0.373418
$260$	0	0
$261$	0	0
$262$	−0.515841	−0.0318687
$263$	−14.2071	−0.876050	−0.438025	−	0.898963i	$-0.644322\pi$
$263$	−14.2071	−0.876050	−0.438025	+	0.898963i	$0.644322\pi$
$264$	0	0
$265$	0	0
$266$	−0.352068	−0.0215867
$267$	0	0
$268$	−14.8736	−0.908547
$269$	8.67115	0.528689	0.264345	−	0.964428i	$-0.414844\pi$
$269$	8.67115	0.528689	0.264345	+	0.964428i	$0.414844\pi$
$270$	0	0
$271$	11.5697	0.702807	0.351403	−	0.936224i	$-0.385705\pi$
$271$	11.5697	0.702807	0.351403	+	0.936224i	$0.385705\pi$
$272$	−18.4514	−1.11878
$273$	0	0
$274$	1.18007	0.0712904
$275$	0	0
$276$	0	0
$277$	−19.2540	−1.15686	−0.578431	−	0.815732i	$-0.696335\pi$
$277$	−19.2540	−1.15686	−0.578431	+	0.815732i	$0.696335\pi$
$278$	0.0681596	0.00408794
$279$	0	0
$280$	0	0
$281$	3.21917	0.192040	0.0960198	−	0.995379i	$-0.469389\pi$
$281$	3.21917	0.192040	0.0960198	+	0.995379i	$0.469389\pi$
$282$	0	0
$283$	−30.6086	−1.81949	−0.909745	−	0.415167i	$-0.863723\pi$
$283$	−30.6086	−1.81949	−0.909745	+	0.415167i	$0.863723\pi$
$284$	−29.4551	−1.74784
$285$	0	0
$286$	0.209456	0.0123854
$287$	5.38664	0.317963
$288$	0	0
$289$	7.34391	0.431995
$290$	0	0
$291$	0	0
$292$	−5.84536	−0.342074
$293$	−12.6725	−0.740333	−0.370166	−	0.928965i	$-0.620699\pi$
$293$	−12.6725	−0.740333	−0.370166	+	0.928965i	$0.620699\pi$
$294$	0	0
$295$	0	0
$296$	−8.40995	−0.488819
$297$	0	0
$298$	−0.133298	−0.00772173
$299$	−7.95341	−0.459958
$300$	0	0
$301$	−1.66365	−0.0958911
$302$	−2.38217	−0.137078
$303$	0	0
$304$	−10.6560	−0.611162
$305$	0	0
$306$	0	0
$307$	−28.0055	−1.59836	−0.799180	−	0.601092i	$-0.794732\pi$
$307$	−28.0055	−1.59836	−0.799180	+	0.601092i	$0.794732\pi$
$308$	1.00683	0.0573693
$309$	0	0
$310$	0	0
$311$	−29.1437	−1.65259	−0.826293	−	0.563241i	$-0.809554\pi$
$311$	−29.1437	−1.65259	−0.826293	+	0.563241i	$0.809554\pi$
$312$	0	0
$313$	−19.1782	−1.08402	−0.542009	−	0.840373i	$-0.682336\pi$
$313$	−19.1782	−1.08402	−0.542009	+	0.840373i	$0.682336\pi$
$314$	0.598045	0.0337497
$315$	0	0
$316$	−6.52871	−0.367269
$317$	19.8985	1.11761	0.558804	−	0.829300i	$-0.311260\pi$
$317$	19.8985	1.11761	0.558804	+	0.829300i	$0.311260\pi$
$318$	0	0
$319$	6.52124	0.365119
$320$	0	0
$321$	0	0
$322$	0.854102	0.0475972
$323$	14.0590	0.782262
$324$	0	0
$325$	0	0
$326$	2.94992	0.163381
$327$	0	0
$328$	−7.53818	−0.416226
$329$	−3.95301	−0.217936
$330$	0	0
$331$	−27.8128	−1.52873	−0.764366	−	0.644783i	$-0.776948\pi$
$331$	−27.8128	−1.52873	−0.764366	+	0.644783i	$0.776948\pi$
$332$	19.0408	1.04500
$333$	0	0
$334$	1.56592	0.0856834
$335$	0	0
$336$	0	0
$337$	31.7776	1.73104	0.865518	−	0.500877i	$-0.166989\pi$
$337$	31.7776	1.73104	0.865518	+	0.500877i	$0.166989\pi$
$338$	2.44099	0.132772
$339$	0	0
$340$	0	0
$341$	3.01287	0.163156
$342$	0	0
$343$	8.06787	0.435624
$344$	2.32815	0.125525
$345$	0	0
$346$	−4.24869	−0.228411
$347$	−27.6962	−1.48681	−0.743404	−	0.668842i	$-0.766790\pi$
$347$	−27.6962	−1.48681	−0.743404	+	0.668842i	$0.766790\pi$
$348$	0	0
$349$	−11.3435	−0.607206	−0.303603	−	0.952799i	$-0.598190\pi$
$349$	−11.3435	−0.607206	−0.303603	+	0.952799i	$0.598190\pi$
$350$	0	0
$351$	0	0
$352$	−2.12125	−0.113063
$353$	−13.2761	−0.706614	−0.353307	−	0.935507i	$-0.614943\pi$
$353$	−13.2761	−0.706614	−0.353307	+	0.935507i	$0.614943\pi$
$354$	0	0
$355$	0	0
$356$	1.26190	0.0668805
$357$	0	0
$358$	0.236671	0.0125085
$359$	−0.547922	−0.0289182	−0.0144591	−	0.999895i	$-0.504603\pi$
$359$	−0.547922	−0.0289182	−0.0144591	+	0.999895i	$0.504603\pi$
$360$	0	0
$361$	−10.8807	−0.572671
$362$	−1.03786	−0.0545489
$363$	0	0
$364$	−1.33030	−0.0697269
$365$	0	0
$366$	0	0
$367$	−32.3147	−1.68681	−0.843406	−	0.537277i	$-0.819453\pi$
$367$	−32.3147	−1.68681	−0.843406	+	0.537277i	$0.819453\pi$
$368$	25.8509	1.34757
$369$	0	0
$370$	0	0
$371$	−2.28866	−0.118821
$372$	0	0
$373$	15.2740	0.790860	0.395430	−	0.918496i	$-0.370596\pi$
$373$	15.2740	0.790860	0.395430	+	0.918496i	$0.370596\pi$
$374$	0.898206	0.0464451
$375$	0	0
$376$	5.53193	0.285288
$377$	−8.61641	−0.443768
$378$	0	0
$379$	11.5928	0.595482	0.297741	−	0.954647i	$-0.403767\pi$
$379$	11.5928	0.595482	0.297741	+	0.954647i	$0.403767\pi$
$380$	0	0
$381$	0	0
$382$	1.34760	0.0689493
$383$	−27.1864	−1.38916	−0.694580	−	0.719416i	$-0.744410\pi$
$383$	−27.1864	−1.38916	−0.694580	+	0.719416i	$0.744410\pi$
$384$	0	0
$385$	0	0
$386$	−3.85317	−0.196121
$387$	0	0
$388$	22.8827	1.16169
$389$	−36.2825	−1.83959	−0.919797	−	0.392394i	$-0.871647\pi$
$389$	−36.2825	−1.83959	−0.919797	+	0.392394i	$0.871647\pi$
$390$	0	0
$391$	−34.1064	−1.72484
$392$	−5.50073	−0.277829
$393$	0	0
$394$	3.51933	0.177301
$395$	0	0
$396$	0	0
$397$	24.6879	1.23905	0.619525	−	0.784977i	$-0.287325\pi$
$397$	24.6879	1.23905	0.619525	+	0.784977i	$0.287325\pi$
$398$	−3.55330	−0.178111
$399$	0	0
$400$	0	0
$401$	34.4435	1.72003	0.860014	−	0.510271i	$-0.170455\pi$
$401$	34.4435	1.72003	0.860014	+	0.510271i	$0.170455\pi$
$402$	0	0
$403$	−3.98086	−0.198301
$404$	−28.8820	−1.43693
$405$	0	0
$406$	0.925301	0.0459219
$407$	−8.85435	−0.438894
$408$	0	0
$409$	−12.9896	−0.642294	−0.321147	−	0.947029i	$-0.604068\pi$
$409$	−12.9896	−0.642294	−0.321147	+	0.947029i	$0.604068\pi$
$410$	0	0
$411$	0	0
$412$	−2.77734	−0.136830
$413$	−6.81316	−0.335254
$414$	0	0
$415$	0	0
$416$	2.80277	0.137417
$417$	0	0
$418$	0.518727	0.0253718
$419$	−18.5116	−0.904353	−0.452177	−	0.891928i	$-0.649352\pi$
$419$	−18.5116	−0.904353	−0.452177	+	0.891928i	$0.649352\pi$
$420$	0	0
$421$	23.8357	1.16168	0.580841	−	0.814017i	$-0.302724\pi$
$421$	23.8357	1.16168	0.580841	+	0.814017i	$0.302724\pi$
$422$	−1.04711	−0.0509726
$423$	0	0
$424$	3.20281	0.155542
$425$	0	0
$426$	0	0
$427$	7.53931	0.364853
$428$	29.2620	1.41443
$429$	0	0
$430$	0	0
$431$	29.6800	1.42963	0.714817	−	0.699312i	$-0.246510\pi$
$431$	29.6800	1.42963	0.714817	+	0.699312i	$0.246510\pi$
$432$	0	0
$433$	18.0164	0.865811	0.432906	−	0.901439i	$-0.357488\pi$
$433$	18.0164	0.865811	0.432906	+	0.901439i	$0.357488\pi$
$434$	0.427497	0.0205205
$435$	0	0
$436$	21.7104	1.03974
$437$	−19.6970	−0.942233
$438$	0	0
$439$	21.0442	1.00438	0.502191	−	0.864757i	$-0.332527\pi$
$439$	21.0442	1.00438	0.502191	+	0.864757i	$0.332527\pi$
$440$	0	0
$441$	0	0
$442$	−1.18679	−0.0564496
$443$	−4.15877	−0.197589	−0.0987946	−	0.995108i	$-0.531499\pi$
$443$	−4.15877	−0.197589	−0.0987946	+	0.995108i	$0.531499\pi$
$444$	0	0
$445$	0	0
$446$	3.33346	0.157844
$447$	0	0
$448$	4.11949	0.194628
$449$	7.39256	0.348876	0.174438	−	0.984668i	$-0.444189\pi$
$449$	7.39256	0.348876	0.174438	+	0.984668i	$0.444189\pi$
$450$	0	0
$451$	−7.93651	−0.373716
$452$	23.6846	1.11403
$453$	0	0
$454$	5.62295	0.263898
$455$	0	0
$456$	0	0
$457$	0.573353	0.0268203	0.0134102	−	0.999910i	$-0.495731\pi$
$457$	0.573353	0.0268203	0.0134102	+	0.999910i	$0.495731\pi$
$458$	0.615877	0.0287780
$459$	0	0
$460$	0	0
$461$	−3.74666	−0.174499	−0.0872497	−	0.996186i	$-0.527808\pi$
$461$	−3.74666	−0.174499	−0.0872497	+	0.996186i	$0.527808\pi$
$462$	0	0
$463$	−17.3803	−0.807732	−0.403866	−	0.914818i	$-0.632334\pi$
$463$	−17.3803	−0.807732	−0.403866	+	0.914818i	$0.632334\pi$
$464$	28.0058	1.30014
$465$	0	0
$466$	4.89307	0.226667
$467$	10.0211	0.463720	0.231860	−	0.972749i	$-0.425519\pi$
$467$	10.0211	0.463720	0.231860	+	0.972749i	$0.425519\pi$
$468$	0	0
$469$	−4.49350	−0.207491
$470$	0	0
$471$	0	0
$472$	9.53449	0.438860
$473$	2.45117	0.112705
$474$	0	0
$475$	0	0
$476$	−5.70471	−0.261475
$477$	0	0
$478$	3.08965	0.141317
$479$	−18.6136	−0.850476	−0.425238	−	0.905082i	$-0.639810\pi$
$479$	−18.6136	−0.850476	−0.425238	+	0.905082i	$0.639810\pi$
$480$	0	0
$481$	11.6991	0.533433
$482$	3.00820	0.137020
$483$	0	0
$484$	20.0358	0.910719
$485$	0	0
$486$	0	0
$487$	−2.07209	−0.0938954	−0.0469477	−	0.998897i	$-0.514949\pi$
$487$	−2.07209	−0.0938954	−0.0469477	+	0.998897i	$0.514949\pi$
$488$	−10.5507	−0.477607
$489$	0	0
$490$	0	0
$491$	22.2355	1.00348	0.501738	−	0.865020i	$-0.332694\pi$
$491$	22.2355	1.00348	0.501738	+	0.865020i	$0.332694\pi$
$492$	0	0
$493$	−36.9495	−1.66412
$494$	−0.685386	−0.0308370
$495$	0	0
$496$	12.9390	0.580976
$497$	−8.89878	−0.399165
$498$	0	0
$499$	28.2713	1.26560	0.632798	−	0.774317i	$-0.281906\pi$
$499$	28.2713	1.26560	0.632798	+	0.774317i	$0.281906\pi$
$500$	0	0
$501$	0	0
$502$	−3.91652	−0.174803
$503$	−3.39793	−0.151506	−0.0757531	−	0.997127i	$-0.524136\pi$
$503$	−3.39793	−0.151506	−0.0757531	+	0.997127i	$0.524136\pi$
$504$	0	0
$505$	0	0
$506$	−1.25841	−0.0559431
$507$	0	0
$508$	1.05967	0.0470152
$509$	10.8405	0.480497	0.240248	−	0.970711i	$-0.422771\pi$
$509$	10.8405	0.480497	0.240248	+	0.970711i	$0.422771\pi$
$510$	0	0
$511$	−1.76596	−0.0781216
$512$	−15.2959	−0.675991
$513$	0	0
$514$	1.34260	0.0592196
$515$	0	0
$516$	0	0
$517$	5.82425	0.256150
$518$	−1.25635	−0.0552007
$519$	0	0
$520$	0	0
$521$	−15.6473	−0.685519	−0.342760	−	0.939423i	$-0.611362\pi$
$521$	−15.6473	−0.685519	−0.342760	+	0.939423i	$0.611362\pi$
$522$	0	0
$523$	18.3578	0.802729	0.401364	−	0.915918i	$-0.368536\pi$
$523$	18.3578	0.802729	0.401364	+	0.915918i	$0.368536\pi$
$524$	−4.82709	−0.210872
$525$	0	0
$526$	2.97010	0.129503
$527$	−17.0710	−0.743626
$528$	0	0
$529$	24.7839	1.07756
$530$	0	0
$531$	0	0
$532$	−3.29456	−0.142837
$533$	10.4864	0.454216
$534$	0	0
$535$	0	0
$536$	6.28831	0.271613
$537$	0	0
$538$	−1.81276	−0.0781538
$539$	−5.79139	−0.249453
$540$	0	0
$541$	−13.8856	−0.596988	−0.298494	−	0.954412i	$-0.596484\pi$
$541$	−13.8856	−0.596988	−0.298494	+	0.954412i	$0.596484\pi$
$542$	−2.41872	−0.103893
$543$	0	0
$544$	12.0191	0.515313
$545$	0	0
$546$	0	0
$547$	−41.9134	−1.79209	−0.896043	−	0.443966i	$-0.853571\pi$
$547$	−41.9134	−1.79209	−0.896043	+	0.443966i	$0.853571\pi$
$548$	11.0427	0.471722
$549$	0	0
$550$	0	0
$551$	−21.3389	−0.909068
$552$	0	0
$553$	−1.97241	−0.0838755
$554$	4.02518	0.171014
$555$	0	0
$556$	0.637818	0.0270495
$557$	−8.04257	−0.340774	−0.170387	−	0.985377i	$-0.554502\pi$
$557$	−8.04257	−0.340774	−0.170387	+	0.985377i	$0.554502\pi$
$558$	0	0
$559$	−3.23869	−0.136982
$560$	0	0
$561$	0	0
$562$	−0.672990	−0.0283884
$563$	5.67464	0.239158	0.119579	−	0.992825i	$-0.461846\pi$
$563$	5.67464	0.239158	0.119579	+	0.992825i	$0.461846\pi$
$564$	0	0
$565$	0	0
$566$	6.39893	0.268967
$567$	0	0
$568$	12.4532	0.522523
$569$	−8.25303	−0.345985	−0.172993	−	0.984923i	$-0.555344\pi$
$569$	−8.25303	−0.345985	−0.172993	+	0.984923i	$0.555344\pi$
$570$	0	0
$571$	−17.5703	−0.735293	−0.367646	−	0.929966i	$-0.619836\pi$
$571$	−17.5703	−0.735293	−0.367646	+	0.929966i	$0.619836\pi$
$572$	1.96003	0.0819531
$573$	0	0
$574$	−1.12611	−0.0470031
$575$	0	0
$576$	0	0
$577$	−29.6207	−1.23313	−0.616564	−	0.787305i	$-0.711476\pi$
$577$	−29.6207	−1.23313	−0.616564	+	0.787305i	$0.711476\pi$
$578$	−1.53529	−0.0638598
$579$	0	0
$580$	0	0
$581$	5.75249	0.238653
$582$	0	0
$583$	3.37205	0.139656
$584$	2.47133	0.102264
$585$	0	0
$586$	2.64926	0.109440
$587$	−18.1475	−0.749027	−0.374513	−	0.927222i	$-0.622190\pi$
$587$	−18.1475	−0.749027	−0.374513	+	0.927222i	$0.622190\pi$
$588$	0	0
$589$	−9.85877	−0.406224
$590$	0	0
$591$	0	0
$592$	−38.0255	−1.56284
$593$	−26.0022	−1.06778	−0.533891	−	0.845553i	$-0.679271\pi$
$593$	−26.0022	−1.06778	−0.533891	+	0.845553i	$0.679271\pi$
$594$	0	0
$595$	0	0
$596$	−1.24736	−0.0510939
$597$	0	0
$598$	1.66272	0.0679935
$599$	−20.6259	−0.842753	−0.421376	−	0.906886i	$-0.638453\pi$
$599$	−20.6259	−0.842753	−0.421376	+	0.906886i	$0.638453\pi$
$600$	0	0
$601$	−17.7602	−0.724452	−0.362226	−	0.932090i	$-0.617983\pi$
$601$	−17.7602	−0.724452	−0.362226	+	0.932090i	$0.617983\pi$
$602$	0.347797	0.0141752
$603$	0	0
$604$	−22.2917	−0.907035
$605$	0	0
$606$	0	0
$607$	13.0143	0.528232	0.264116	−	0.964491i	$-0.414920\pi$
$607$	13.0143	0.528232	0.264116	+	0.964491i	$0.414920\pi$
$608$	6.94118	0.281502
$609$	0	0
$610$	0	0
$611$	−7.69548	−0.311326
$612$	0	0
$613$	21.9959	0.888407	0.444204	−	0.895926i	$-0.353487\pi$
$613$	21.9959	0.888407	0.444204	+	0.895926i	$0.353487\pi$
$614$	5.85475	0.236278
$615$	0	0
$616$	−0.425671	−0.0171508
$617$	33.3091	1.34097	0.670487	−	0.741922i	$-0.266085\pi$
$617$	33.3091	1.34097	0.670487	+	0.741922i	$0.266085\pi$
$618$	0	0
$619$	−27.9714	−1.12426	−0.562132	−	0.827048i	$-0.690019\pi$
$619$	−27.9714	−1.12426	−0.562132	+	0.827048i	$0.690019\pi$
$620$	0	0
$621$	0	0
$622$	6.09268	0.244294
$623$	0.381236	0.0152739
$624$	0	0
$625$	0	0
$626$	4.00934	0.160245
$627$	0	0
$628$	5.59634	0.223318
$629$	50.1690	2.00037
$630$	0	0
$631$	−23.0172	−0.916301	−0.458150	−	0.888875i	$-0.651488\pi$
$631$	−23.0172	−0.916301	−0.458150	+	0.888875i	$0.651488\pi$
$632$	2.76024	0.109796
$633$	0	0
$634$	−4.15991	−0.165211
$635$	0	0
$636$	0	0
$637$	7.65208	0.303186
$638$	−1.36331	−0.0539740
$639$	0	0
$640$	0	0
$641$	−29.1915	−1.15300	−0.576498	−	0.817099i	$-0.695581\pi$
$641$	−29.1915	−1.15300	−0.576498	+	0.817099i	$0.695581\pi$
$642$	0	0
$643$	29.6236	1.16824	0.584119	−	0.811668i	$-0.301440\pi$
$643$	29.6236	1.16824	0.584119	+	0.811668i	$0.301440\pi$
$644$	7.99244	0.314946
$645$	0	0
$646$	−2.93912	−0.115638
$647$	49.0862	1.92978	0.964889	−	0.262657i	$-0.0845987\pi$
$647$	49.0862	1.92978	0.964889	+	0.262657i	$0.0845987\pi$
$648$	0	0
$649$	10.0383	0.394038
$650$	0	0
$651$	0	0
$652$	27.6045	1.08108
$653$	−11.6946	−0.457645	−0.228823	−	0.973468i	$-0.573488\pi$
$653$	−11.6946	−0.457645	−0.228823	+	0.973468i	$0.573488\pi$
$654$	0	0
$655$	0	0
$656$	−34.0838	−1.33075
$657$	0	0
$658$	0.826404	0.0322166
$659$	−1.18709	−0.0462424	−0.0231212	−	0.999733i	$-0.507360\pi$
$659$	−1.18709	−0.0462424	−0.0231212	+	0.999733i	$0.507360\pi$
$660$	0	0
$661$	3.72234	0.144782	0.0723912	−	0.997376i	$-0.476937\pi$
$661$	3.72234	0.144782	0.0723912	+	0.997376i	$0.476937\pi$
$662$	5.81446	0.225986
$663$	0	0
$664$	−8.05016	−0.312407
$665$	0	0
$666$	0	0
$667$	51.7672	2.00443
$668$	14.6535	0.566959
$669$	0	0
$670$	0	0
$671$	−11.1082	−0.428827
$672$	0	0
$673$	−46.9935	−1.81147	−0.905734	−	0.423847i	$-0.860680\pi$
$673$	−46.9935	−1.81147	−0.905734	+	0.423847i	$0.860680\pi$
$674$	−6.64333	−0.255892
$675$	0	0
$676$	22.8421	0.878541
$677$	36.4081	1.39928	0.699639	−	0.714497i	$-0.253344\pi$
$677$	36.4081	1.39928	0.699639	+	0.714497i	$0.253344\pi$
$678$	0	0
$679$	6.91317	0.265303
$680$	0	0
$681$	0	0
$682$	−0.629862	−0.0241187
$683$	4.66507	0.178504	0.0892519	−	0.996009i	$-0.471552\pi$
$683$	4.66507	0.178504	0.0892519	+	0.996009i	$0.471552\pi$
$684$	0	0
$685$	0	0
$686$	−1.68664	−0.0643964
$687$	0	0
$688$	10.5267	0.401326
$689$	−4.45543	−0.169738
$690$	0	0
$691$	31.9254	1.21450	0.607249	−	0.794511i	$-0.292273\pi$
$691$	31.9254	1.21450	0.607249	+	0.794511i	$0.292273\pi$
$692$	−39.7581	−1.51138
$693$	0	0
$694$	5.79008	0.219788
$695$	0	0
$696$	0	0
$697$	44.9685	1.70330
$698$	2.37144	0.0897605
$699$	0	0
$700$	0	0
$701$	3.09598	0.116933	0.0584667	−	0.998289i	$-0.481379\pi$
$701$	3.09598	0.116933	0.0584667	+	0.998289i	$0.481379\pi$
$702$	0	0
$703$	28.9733	1.09275
$704$	−6.06954	−0.228754
$705$	0	0
$706$	2.77546	0.104456
$707$	−8.72563	−0.328161
$708$	0	0
$709$	47.0490	1.76696	0.883481	−	0.468468i	$-0.155194\pi$
$709$	47.0490	1.76696	0.883481	+	0.468468i	$0.155194\pi$
$710$	0	0
$711$	0	0
$712$	−0.533511	−0.0199942
$713$	23.9169	0.895696
$714$	0	0
$715$	0	0
$716$	2.21470	0.0827674
$717$	0	0
$718$	0.114547	0.00427485
$719$	−25.7083	−0.958757	−0.479378	−	0.877608i	$-0.659138\pi$
$719$	−25.7083	−0.958757	−0.479378	+	0.877608i	$0.659138\pi$
$720$	0	0
$721$	−0.839072	−0.0312487
$722$	2.27469	0.0846553
$723$	0	0
$724$	−9.71202	−0.360945
$725$	0	0
$726$	0	0
$727$	35.4025	1.31301	0.656504	−	0.754323i	$-0.272035\pi$
$727$	35.4025	1.31301	0.656504	+	0.754323i	$0.272035\pi$
$728$	0.562432	0.0208451
$729$	0	0
$730$	0	0
$731$	−13.8884	−0.513681
$732$	0	0
$733$	−25.7565	−0.951336	−0.475668	−	0.879625i	$-0.657794\pi$
$733$	−25.7565	−0.951336	−0.475668	+	0.879625i	$0.657794\pi$
$734$	6.75560	0.249354
$735$	0	0
$736$	−16.8390	−0.620693
$737$	6.62059	0.243873
$738$	0	0
$739$	−22.1071	−0.813222	−0.406611	−	0.913601i	$-0.633289\pi$
$739$	−22.1071	−0.813222	−0.406611	+	0.913601i	$0.633289\pi$
$740$	0	0
$741$	0	0
$742$	0.478461	0.0175648
$743$	20.2268	0.742049	0.371025	−	0.928623i	$-0.379007\pi$
$743$	20.2268	0.742049	0.371025	+	0.928623i	$0.379007\pi$
$744$	0	0
$745$	0	0
$746$	−3.19314	−0.116909
$747$	0	0
$748$	8.40516	0.307323
$749$	8.84045	0.323023
$750$	0	0
$751$	−1.23206	−0.0449584	−0.0224792	−	0.999747i	$-0.507156\pi$
$751$	−1.23206	−0.0449584	−0.0224792	+	0.999747i	$0.507156\pi$
$752$	25.0126	0.912114
$753$	0	0
$754$	1.80132	0.0656002
$755$	0	0
$756$	0	0
$757$	−15.2062	−0.552678	−0.276339	−	0.961060i	$-0.589121\pi$
$757$	−15.2062	−0.552678	−0.276339	+	0.961060i	$0.589121\pi$
$758$	−2.42356	−0.0880275
$759$	0	0
$760$	0	0
$761$	32.3209	1.17163	0.585817	−	0.810444i	$-0.300774\pi$
$761$	32.3209	1.17163	0.585817	+	0.810444i	$0.300774\pi$
$762$	0	0
$763$	6.55901	0.237452
$764$	12.6105	0.456231
$765$	0	0
$766$	5.68350	0.205353
$767$	−13.2635	−0.478915
$768$	0	0
$769$	−28.8068	−1.03880	−0.519400	−	0.854531i	$-0.673845\pi$
$769$	−28.8068	−1.03880	−0.519400	+	0.854531i	$0.673845\pi$
$770$	0	0
$771$	0	0
$772$	−36.0569	−1.29771
$773$	−38.2458	−1.37561	−0.687803	−	0.725897i	$-0.741425\pi$
$773$	−38.2458	−1.37561	−0.687803	+	0.725897i	$0.741425\pi$
$774$	0	0
$775$	0	0
$776$	−9.67444	−0.347292
$777$	0	0
$778$	7.58510	0.271939
$779$	25.9700	0.930471
$780$	0	0
$781$	13.1112	0.469156
$782$	7.13018	0.254975
$783$	0	0
$784$	−24.8715	−0.888267
$785$	0	0
$786$	0	0
$787$	−36.7934	−1.31154	−0.655771	−	0.754960i	$-0.727656\pi$
$787$	−36.7934	−1.31154	−0.655771	+	0.754960i	$0.727656\pi$
$788$	32.9329	1.17319
$789$	0	0
$790$	0	0
$791$	7.15543	0.254418
$792$	0	0
$793$	14.6771	0.521199
$794$	−5.16117	−0.183163
$795$	0	0
$796$	−33.2507	−1.17854
$797$	36.6698	1.29891	0.649456	−	0.760399i	$-0.274997\pi$
$797$	36.6698	1.29891	0.649456	+	0.760399i	$0.274997\pi$
$798$	0	0
$799$	−33.0004	−1.16747
$800$	0	0
$801$	0	0
$802$	−7.20066	−0.254264
$803$	2.60192	0.0918197
$804$	0	0
$805$	0	0
$806$	0.832227	0.0293139
$807$	0	0
$808$	12.2108	0.429576
$809$	39.2361	1.37947	0.689735	−	0.724062i	$-0.257727\pi$
$809$	39.2361	1.37947	0.689735	+	0.724062i	$0.257727\pi$
$810$	0	0
$811$	6.70444	0.235425	0.117712	−	0.993048i	$-0.462444\pi$
$811$	6.70444	0.235425	0.117712	+	0.993048i	$0.462444\pi$
$812$	8.65870	0.303861
$813$	0	0
$814$	1.85106	0.0648797
$815$	0	0
$816$	0	0
$817$	−8.02076	−0.280611
$818$	2.71557	0.0949475
$819$	0	0
$820$	0	0
$821$	30.1557	1.05244	0.526220	−	0.850348i	$-0.323609\pi$
$821$	30.1557	1.05244	0.526220	+	0.850348i	$0.323609\pi$
$822$	0	0
$823$	1.90341	0.0663487	0.0331744	−	0.999450i	$-0.489438\pi$
$823$	1.90341	0.0663487	0.0331744	+	0.999450i	$0.489438\pi$
$824$	1.17422	0.0409057
$825$	0	0
$826$	1.42434	0.0495590
$827$	−17.5652	−0.610803	−0.305402	−	0.952224i	$-0.598791\pi$
$827$	−17.5652	−0.610803	−0.305402	+	0.952224i	$0.598791\pi$
$828$	0	0
$829$	−24.0291	−0.834566	−0.417283	−	0.908777i	$-0.637018\pi$
$829$	−24.0291	−0.834566	−0.417283	+	0.908777i	$0.637018\pi$
$830$	0	0
$831$	0	0
$832$	8.01958	0.278029
$833$	32.8142	1.13695
$834$	0	0
$835$	0	0
$836$	4.85410	0.167883
$837$	0	0
$838$	3.86999	0.133687
$839$	−1.07320	−0.0370511	−0.0185255	−	0.999828i	$-0.505897\pi$
$839$	−1.07320	−0.0370511	−0.0185255	+	0.999828i	$0.505897\pi$
$840$	0	0
$841$	27.0826	0.933882
$842$	−4.98302	−0.171726
$843$	0	0
$844$	−9.79857	−0.337281
$845$	0	0
$846$	0	0
$847$	6.05309	0.207987
$848$	14.4815	0.497295
$849$	0	0
$850$	0	0
$851$	−70.2880	−2.40944
$852$	0	0
$853$	−17.2818	−0.591716	−0.295858	−	0.955232i	$-0.595606\pi$
$853$	−17.2818	−0.591716	−0.295858	+	0.955232i	$0.595606\pi$
$854$	−1.57615	−0.0539346
$855$	0	0
$856$	−12.3715	−0.422850
$857$	−16.2742	−0.555917	−0.277959	−	0.960593i	$-0.589658\pi$
$857$	−16.2742	−0.555917	−0.277959	+	0.960593i	$0.589658\pi$
$858$	0	0
$859$	−11.2429	−0.383604	−0.191802	−	0.981434i	$-0.561433\pi$
$859$	−11.2429	−0.383604	−0.191802	+	0.981434i	$0.561433\pi$
$860$	0	0
$861$	0	0
$862$	−6.20480	−0.211336
$863$	33.3846	1.13642	0.568212	−	0.822882i	$-0.307635\pi$
$863$	33.3846	1.13642	0.568212	+	0.822882i	$0.307635\pi$
$864$	0	0
$865$	0	0
$866$	−3.76645	−0.127989
$867$	0	0
$868$	4.00040	0.135782
$869$	2.90609	0.0985825
$870$	0	0
$871$	−8.74768	−0.296404
$872$	−9.17883	−0.310834
$873$	0	0
$874$	4.11778	0.139286
$875$	0	0
$876$	0	0
$877$	16.1140	0.544130	0.272065	−	0.962279i	$-0.412293\pi$
$877$	16.1140	0.544130	0.272065	+	0.962279i	$0.412293\pi$
$878$	−4.39943	−0.148473
$879$	0	0
$880$	0	0
$881$	−25.2799	−0.851701	−0.425851	−	0.904793i	$-0.640025\pi$
$881$	−25.2799	−0.851701	−0.425851	+	0.904793i	$0.640025\pi$
$882$	0	0
$883$	31.0260	1.04411	0.522053	−	0.852913i	$-0.325166\pi$
$883$	31.0260	1.04411	0.522053	+	0.852913i	$0.325166\pi$
$884$	−11.1056	−0.373522
$885$	0	0
$886$	0.869420	0.0292087
$887$	38.1973	1.28254	0.641270	−	0.767316i	$-0.278408\pi$
$887$	38.1973	1.28254	0.641270	+	0.767316i	$0.278408\pi$
$888$	0	0
$889$	0.320140	0.0107372
$890$	0	0
$891$	0	0
$892$	31.1936	1.04444
$893$	−19.0582	−0.637758
$894$	0	0
$895$	0	0
$896$	−3.74066	−0.124967
$897$	0	0
$898$	−1.54547	−0.0515728
$899$	25.9107	0.864169
$900$	0	0
$901$	−19.1061	−0.636517
$902$	1.65918	0.0552447
$903$	0	0
$904$	−10.0135	−0.333043
$905$	0	0
$906$	0	0
$907$	40.9212	1.35877	0.679383	−	0.733784i	$-0.262247\pi$
$907$	40.9212	1.35877	0.679383	+	0.733784i	$0.262247\pi$
$908$	52.6180	1.74619
$909$	0	0
$910$	0	0
$911$	45.6189	1.51142	0.755710	−	0.654906i	$-0.227292\pi$
$911$	45.6189	1.51142	0.755710	+	0.654906i	$0.227292\pi$
$912$	0	0
$913$	−8.47554	−0.280500
$914$	−0.119863	−0.00396473
$915$	0	0
$916$	5.76320	0.190422
$917$	−1.45833	−0.0481583
$918$	0	0
$919$	20.6582	0.681451	0.340725	−	0.940163i	$-0.389327\pi$
$919$	20.6582	0.681451	0.340725	+	0.940163i	$0.389327\pi$
$920$	0	0
$921$	0	0
$922$	0.783265	0.0257955
$923$	−17.3236	−0.570214
$924$	0	0
$925$	0	0
$926$	3.63348	0.119404
$927$	0	0
$928$	−18.2427	−0.598846
$929$	23.2327	0.762239	0.381119	−	0.924526i	$-0.375539\pi$
$929$	23.2327	0.762239	0.381119	+	0.924526i	$0.375539\pi$
$930$	0	0
$931$	18.9507	0.621084
$932$	45.7879	1.49983
$933$	0	0
$934$	−2.09498	−0.0685497
$935$	0	0
$936$	0	0
$937$	−18.3122	−0.598232	−0.299116	−	0.954217i	$-0.596692\pi$
$937$	−18.3122	−0.598232	−0.299116	+	0.954217i	$0.596692\pi$
$938$	0.939397	0.0306724
$939$	0	0
$940$	0	0
$941$	−17.3075	−0.564209	−0.282104	−	0.959384i	$-0.591032\pi$
$941$	−17.3075	−0.564209	−0.282104	+	0.959384i	$0.591032\pi$
$942$	0	0
$943$	−63.0020	−2.05163
$944$	43.1101	1.40311
$945$	0	0
$946$	−0.512434	−0.0166607
$947$	18.4829	0.600613	0.300307	−	0.953843i	$-0.402911\pi$
$947$	18.4829	0.600613	0.300307	+	0.953843i	$0.402911\pi$
$948$	0	0
$949$	−3.43787	−0.111598
$950$	0	0
$951$	0	0
$952$	2.41186	0.0781689
$953$	20.8872	0.676604	0.338302	−	0.941038i	$-0.390148\pi$
$953$	20.8872	0.676604	0.338302	+	0.941038i	$0.390148\pi$
$954$	0	0
$955$	0	0
$956$	28.9121	0.935083
$957$	0	0
$958$	3.89130	0.125722
$959$	3.33616	0.107730
$960$	0	0
$961$	−19.0290	−0.613840
$962$	−2.44578	−0.0788551
$963$	0	0
$964$	28.1499	0.906647
$965$	0	0
$966$	0	0
$967$	51.2510	1.64812	0.824061	−	0.566501i	$-0.191703\pi$
$967$	51.2510	1.64812	0.824061	+	0.566501i	$0.191703\pi$
$968$	−8.47083	−0.272263
$969$	0	0
$970$	0	0
$971$	38.6012	1.23877	0.619385	−	0.785087i	$-0.287382\pi$
$971$	38.6012	1.23877	0.619385	+	0.785087i	$0.287382\pi$
$972$	0	0
$973$	0.192693	0.00617747
$974$	0.433185	0.0138801
$975$	0	0
$976$	−47.7048	−1.52699
$977$	2.91913	0.0933912	0.0466956	−	0.998909i	$-0.485131\pi$
$977$	2.91913	0.0933912	0.0466956	+	0.998909i	$0.485131\pi$
$978$	0	0
$979$	−0.561702	−0.0179521
$980$	0	0
$981$	0	0
$982$	−4.64849	−0.148339
$983$	0.980138	0.0312615	0.0156308	−	0.999878i	$-0.495024\pi$
$983$	0.980138	0.0312615	0.0156308	+	0.999878i	$0.495024\pi$
$984$	0	0
$985$	0	0
$986$	7.72456	0.246000
$987$	0	0
$988$	−6.41365	−0.204045
$989$	19.4580	0.618728
$990$	0	0
$991$	37.0913	1.17824	0.589122	−	0.808044i	$-0.299474\pi$
$991$	37.0913	1.17824	0.589122	+	0.808044i	$0.299474\pi$
$992$	−8.42830	−0.267599
$993$	0	0
$994$	1.86035	0.0590068
$995$	0	0
$996$	0	0
$997$	−16.7675	−0.531031	−0.265515	−	0.964107i	$-0.585542\pi$
$997$	−16.7675	−0.531031	−0.265515	+	0.964107i	$0.585542\pi$
$998$	−5.91031	−0.187088
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.m.1.2		4
3.2	odd	2		1875.2.a.f.1.3	✓	4
5.4	even	2		5625.2.a.j.1.3		4
15.2	even	4		1875.2.b.d.1249.5		8
15.8	even	4		1875.2.b.d.1249.4		8
15.14	odd	2		1875.2.a.g.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.f.1.3	✓	4	3.2	odd	2
1875.2.a.g.1.2	yes	4	15.14	odd	2
1875.2.b.d.1249.4		8	15.8	even	4
1875.2.b.d.1249.5		8	15.2	even	4
5625.2.a.j.1.3		4	5.4	even	2
5625.2.a.m.1.2		4	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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-0.209057 q^{2} -1.95630 q^{4} -0.591023 q^{7} +0.827091 q^{8} +O(q^{10})\)	\(q-0.209057 q^{2} -1.95630 q^{4} -0.591023 q^{7} +0.827091 q^{8} +0.870796 q^{11} -1.15057 q^{13} +0.123557 q^{14} +3.73968 q^{16} -4.93395 q^{17} -2.84943 q^{19} -0.182046 q^{22} +6.91259 q^{23} +0.240534 q^{26} +1.15622 q^{28} +7.48883 q^{29} +3.45991 q^{31} -2.43599 q^{32} +1.03148 q^{34} -10.1681 q^{37} +0.595693 q^{38} -9.11409 q^{41} +2.81486 q^{43} -1.70353 q^{44} -1.44512 q^{46} +6.68842 q^{47} -6.65069 q^{49} +2.25085 q^{52} +3.87238 q^{53} -0.488830 q^{56} -1.56559 q^{58} +11.5277 q^{59} -12.7564 q^{61} -0.723318 q^{62} -6.97010 q^{64} +7.60292 q^{67} +9.65227 q^{68} +15.0566 q^{71} +2.98798 q^{73} +2.12571 q^{74} +5.57433 q^{76} -0.514660 q^{77} +3.33728 q^{79} +1.90536 q^{82} -9.73310 q^{83} -0.588467 q^{86} +0.720227 q^{88} -0.645045 q^{89} +0.680012 q^{91} -13.5231 q^{92} -1.39826 q^{94} -11.6970 q^{97} +1.39037 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + q^{4} - 5 q^{7} - 3 q^{8}+O(q^{10}) \) 4 * q + q^2 + q^4 - 5 * q^7 - 3 * q^8	\( 4 q + q^{2} + q^{4} - 5 q^{7} - 3 q^{8} + 6 q^{11} - 7 q^{13} + 10 q^{14} - 9 q^{16} - 7 q^{17} - 9 q^{19} - 6 q^{22} + 10 q^{23} + 2 q^{26} - 5 q^{28} + 28 q^{29} - 10 q^{31} + 7 q^{34} + 10 q^{37} - 6 q^{38} - q^{43} + 9 q^{44} + 5 q^{46} - 23 q^{47} - 3 q^{49} - 13 q^{52} + 2 q^{58} - 4 q^{59} - 43 q^{61} - 10 q^{62} - 7 q^{64} - 8 q^{67} - 3 q^{68} + 27 q^{71} - 15 q^{73} - 5 q^{74} + 9 q^{76} - 15 q^{77} + 10 q^{79} + 20 q^{82} + 3 q^{83} - 24 q^{86} + 3 q^{88} + 9 q^{89} + 5 q^{91} - 15 q^{92} - 22 q^{94} - 13 q^{97} - 42 q^{98}+O(q^{100}) \) 4 * q + q^2 + q^4 - 5 * q^7 - 3 * q^8 + 6 * q^11 - 7 * q^13 + 10 * q^14 - 9 * q^16 - 7 * q^17 - 9 * q^19 - 6 * q^22 + 10 * q^23 + 2 * q^26 - 5 * q^28 + 28 * q^29 - 10 * q^31 + 7 * q^34 + 10 * q^37 - 6 * q^38 - q^43 + 9 * q^44 + 5 * q^46 - 23 * q^47 - 3 * q^49 - 13 * q^52 + 2 * q^58 - 4 * q^59 - 43 * q^61 - 10 * q^62 - 7 * q^64 - 8 * q^67 - 3 * q^68 + 27 * q^71 - 15 * q^73 - 5 * q^74 + 9 * q^76 - 15 * q^77 + 10 * q^79 + 20 * q^82 + 3 * q^83 - 24 * q^86 + 3 * q^88 + 9 * q^89 + 5 * q^91 - 15 * q^92 - 22 * q^94 - 13 * q^97 - 42 * q^98

Introduction

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