LMFDB - Embedded newform 4225.2.a.bk.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, 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("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 1 $ x^4 - 5*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.18890$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.456850 q^{2} +1.00000 q^{3} -1.79129 q^{4} +0.456850 q^{6} +1.73205 q^{7} -1.73205 q^{8} -2.00000 q^{9} -2.64575 q^{11} -1.79129 q^{12} +0.791288 q^{14} +2.79129 q^{16} +4.58258 q^{17} -0.913701 q^{18} +1.73205 q^{19} +1.73205 q^{21} -1.20871 q^{22} -4.58258 q^{23} -1.73205 q^{24} -5.00000 q^{27} -3.10260 q^{28} +4.58258 q^{29} +6.20520 q^{31} +4.73930 q^{32} -2.64575 q^{33} +2.09355 q^{34} +3.58258 q^{36} -7.93725 q^{37} +0.791288 q^{38} -2.64575 q^{41} +0.791288 q^{42} -10.5826 q^{43} +4.73930 q^{44} -2.09355 q^{46} +1.82740 q^{47} +2.79129 q^{48} -4.00000 q^{49} +4.58258 q^{51} +7.58258 q^{53} -2.28425 q^{54} -3.00000 q^{56} +1.73205 q^{57} +2.09355 q^{58} -13.9518 q^{59} -1.41742 q^{61} +2.83485 q^{62} -3.46410 q^{63} -3.41742 q^{64} -1.20871 q^{66} -1.00905 q^{67} -8.20871 q^{68} -4.58258 q^{69} +7.02355 q^{71} +3.46410 q^{72} -3.62614 q^{74} -3.10260 q^{76} -4.58258 q^{77} -6.00000 q^{79} +1.00000 q^{81} -1.20871 q^{82} -6.01450 q^{83} -3.10260 q^{84} -4.83465 q^{86} +4.58258 q^{87} +4.58258 q^{88} -9.57395 q^{89} +8.20871 q^{92} +6.20520 q^{93} +0.834849 q^{94} +4.73930 q^{96} -11.4014 q^{97} -1.82740 q^{98} +5.29150 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 2 q^{4} - 8 q^{9} + 2 q^{12} - 6 q^{14} + 2 q^{16} - 14 q^{22} - 20 q^{27} - 4 q^{36} - 6 q^{38} - 6 q^{42} - 24 q^{43} + 2 q^{48} - 16 q^{49} + 12 q^{53} - 12 q^{56} - 24 q^{61} + 48 q^{62}+ \cdots + 40 q^{94}+O(q^{100}) $ 4 * q + 4 * q^3 + 2 * q^4 - 8 * q^9 + 2 * q^12 - 6 * q^14 + 2 * q^16 - 14 * q^22 - 20 * q^27 - 4 * q^36 - 6 * q^38 - 6 * q^42 - 24 * q^43 + 2 * q^48 - 16 * q^49 + 12 * q^53 - 12 * q^56 - 24 * q^61 + 48 * q^62 - 32 * q^64 - 14 * q^66 - 42 * q^68 - 42 * q^74 - 24 * q^79 + 4 * q^81 - 14 * q^82 + 42 * q^92 + 40 * q^94

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 $

$2$	0.456850	0.323042	0.161521	−	0.986869i	$-0.448360\pi$
$2$	0.456850	0.323042	0.161521	+	0.986869i	$0.448360\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	−1.79129	−0.895644
$5$	0	0
$5$	0	0
$6$	0.456850	0.186508
$7$	1.73205	0.654654	0.327327	−	0.944911i	$-0.393852\pi$
$7$	1.73205	0.654654	0.327327	+	0.944911i	$0.393852\pi$
$8$	−1.73205	−0.612372
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−2.64575	−0.797724	−0.398862	−	0.917011i	$-0.630595\pi$
$11$	−2.64575	−0.797724	−0.398862	+	0.917011i	$0.630595\pi$
$12$	−1.79129	−0.517100
$13$	0	0
$13$	0	0
$14$	0.791288	0.211481
$15$	0	0
$16$	2.79129	0.697822
$17$	4.58258	1.11144	0.555719	−	0.831370i	$-0.312443\pi$
$17$	4.58258	1.11144	0.555719	+	0.831370i	$0.312443\pi$
$18$	−0.913701	−0.215361
$19$	1.73205	0.397360	0.198680	−	0.980064i	$-0.436335\pi$
$19$	1.73205	0.397360	0.198680	+	0.980064i	$0.436335\pi$
$20$	0	0
$21$	1.73205	0.377964
$22$	−1.20871	−0.257698
$23$	−4.58258	−0.955533	−0.477767	−	0.878487i	$-0.658554\pi$
$23$	−4.58258	−0.955533	−0.477767	+	0.878487i	$0.658554\pi$
$24$	−1.73205	−0.353553
$25$	0	0
$26$	0	0
$27$	−5.00000	−0.962250
$28$	−3.10260	−0.586337
$29$	4.58258	0.850963	0.425481	−	0.904967i	$-0.360105\pi$
$29$	4.58258	0.850963	0.425481	+	0.904967i	$0.360105\pi$
$30$	0	0
$31$	6.20520	1.11449	0.557244	−	0.830349i	$-0.311859\pi$
$31$	6.20520	1.11449	0.557244	+	0.830349i	$0.311859\pi$
$32$	4.73930	0.837798
$33$	−2.64575	−0.460566
$34$	2.09355	0.359041
$35$	0	0
$36$	3.58258	0.597096
$37$	−7.93725	−1.30488	−0.652438	−	0.757842i	$-0.726254\pi$
$37$	−7.93725	−1.30488	−0.652438	+	0.757842i	$0.726254\pi$
$38$	0.791288	0.128364
$39$	0	0
$40$	0	0
$41$	−2.64575	−0.413197	−0.206598	−	0.978426i	$-0.566239\pi$
$41$	−2.64575	−0.413197	−0.206598	+	0.978426i	$0.566239\pi$
$42$	0.791288	0.122098
$43$	−10.5826	−1.61383	−0.806914	−	0.590669i	$-0.798864\pi$
$43$	−10.5826	−1.61383	−0.806914	+	0.590669i	$0.798864\pi$
$44$	4.73930	0.714477
$45$	0	0
$46$	−2.09355	−0.308677
$47$	1.82740	0.266554	0.133277	−	0.991079i	$-0.457450\pi$
$47$	1.82740	0.266554	0.133277	+	0.991079i	$0.457450\pi$
$48$	2.79129	0.402888
$49$	−4.00000	−0.571429
$50$	0	0
$51$	4.58258	0.641689
$52$	0	0
$53$	7.58258	1.04155	0.520773	−	0.853695i	$-0.325644\pi$
$53$	7.58258	1.04155	0.520773	+	0.853695i	$0.325644\pi$
$54$	−2.28425	−0.310847
$55$	0	0
$56$	−3.00000	−0.400892
$57$	1.73205	0.229416
$58$	2.09355	0.274897
$59$	−13.9518	−1.81636	−0.908182	−	0.418576i	$-0.862530\pi$
$59$	−13.9518	−1.81636	−0.908182	+	0.418576i	$0.862530\pi$
$60$	0	0
$61$	−1.41742	−0.181483	−0.0907413	−	0.995874i	$-0.528924\pi$
$61$	−1.41742	−0.181483	−0.0907413	+	0.995874i	$0.528924\pi$
$62$	2.83485	0.360026
$63$	−3.46410	−0.436436
$64$	−3.41742	−0.427178
$65$	0	0
$66$	−1.20871	−0.148782
$67$	−1.00905	−0.123275	−0.0616376	−	0.998099i	$-0.519632\pi$
$67$	−1.00905	−0.123275	−0.0616376	+	0.998099i	$0.519632\pi$
$68$	−8.20871	−0.995453
$69$	−4.58258	−0.551677
$70$	0	0
$71$	7.02355	0.833542	0.416771	−	0.909011i	$-0.363162\pi$
$71$	7.02355	0.833542	0.416771	+	0.909011i	$0.363162\pi$
$72$	3.46410	0.408248
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−3.62614	−0.421530
$75$	0	0
$76$	−3.10260	−0.355893
$77$	−4.58258	−0.522233
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−1.20871	−0.133480
$83$	−6.01450	−0.660177	−0.330089	−	0.943950i	$-0.607079\pi$
$83$	−6.01450	−0.660177	−0.330089	+	0.943950i	$0.607079\pi$
$84$	−3.10260	−0.338522
$85$	0	0
$86$	−4.83465	−0.521334
$87$	4.58258	0.491304
$88$	4.58258	0.488504
$89$	−9.57395	−1.01484	−0.507419	−	0.861700i	$-0.669400\pi$
$89$	−9.57395	−1.01484	−0.507419	+	0.861700i	$0.669400\pi$
$90$	0	0
$91$	0	0
$92$	8.20871	0.855817
$93$	6.20520	0.643450
$94$	0.834849	0.0861081
$95$	0	0
$96$	4.73930	0.483703
$97$	−11.4014	−1.15763	−0.578816	−	0.815458i	$-0.696485\pi$
$97$	−11.4014	−1.15763	−0.578816	+	0.815458i	$0.696485\pi$
$98$	−1.82740	−0.184595
$99$	5.29150	0.531816
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	2.09355	0.207292
$103$	−3.16515	−0.311872	−0.155936	−	0.987767i	$-0.549839\pi$
$103$	−3.16515	−0.311872	−0.155936	+	0.987767i	$0.549839\pi$
$104$	0	0
$105$	0	0
$106$	3.46410	0.336463
$107$	−10.5826	−1.02306	−0.511528	−	0.859267i	$-0.670920\pi$
$107$	−10.5826	−1.02306	−0.511528	+	0.859267i	$0.670920\pi$
$108$	8.95644	0.861834
$109$	13.1334	1.25795	0.628976	−	0.777425i	$-0.283474\pi$
$109$	13.1334	1.25795	0.628976	+	0.777425i	$0.283474\pi$
$110$	0	0
$111$	−7.93725	−0.753371
$112$	4.83465	0.456832
$113$	−7.41742	−0.697773	−0.348886	−	0.937165i	$-0.613440\pi$
$113$	−7.41742	−0.697773	−0.348886	+	0.937165i	$0.613440\pi$
$114$	0.791288	0.0741109
$115$	0	0
$116$	−8.20871	−0.762160
$117$	0	0
$118$	−6.37386	−0.586762
$119$	7.93725	0.727607
$120$	0	0
$121$	−4.00000	−0.363636
$122$	−0.647551	−0.0586265
$123$	−2.64575	−0.238559
$124$	−11.1153	−0.998184
$125$	0	0
$126$	−1.58258	−0.140987
$127$	−17.7477	−1.57486	−0.787428	−	0.616407i	$-0.788588\pi$
$127$	−17.7477	−1.57486	−0.787428	+	0.616407i	$0.788588\pi$
$128$	−11.0399	−0.975795
$129$	−10.5826	−0.931744
$130$	0	0
$131$	−7.58258	−0.662493	−0.331246	−	0.943544i	$-0.607469\pi$
$131$	−7.58258	−0.662493	−0.331246	+	0.943544i	$0.607469\pi$
$132$	4.73930	0.412503
$133$	3.00000	0.260133
$134$	−0.460985	−0.0398230
$135$	0	0
$136$	−7.93725	−0.680614
$137$	−10.4877	−0.896021	−0.448010	−	0.894028i	$-0.647867\pi$
$137$	−10.4877	−0.896021	−0.448010	+	0.894028i	$0.647867\pi$
$138$	−2.09355	−0.178215
$139$	−21.7477	−1.84462	−0.922309	−	0.386453i	$-0.873700\pi$
$139$	−21.7477	−1.84462	−0.922309	+	0.386453i	$0.873700\pi$
$140$	0	0
$141$	1.82740	0.153895
$142$	3.20871	0.269269
$143$	0	0
$144$	−5.58258	−0.465215
$145$	0	0
$146$	0	0
$147$	−4.00000	−0.329914
$148$	14.2179	1.16870
$149$	16.6929	1.36753	0.683766	−	0.729701i	$-0.260341\pi$
$149$	16.6929	1.36753	0.683766	+	0.729701i	$0.260341\pi$
$150$	0	0
$151$	9.66930	0.786877	0.393438	−	0.919351i	$-0.371285\pi$
$151$	9.66930	0.786877	0.393438	+	0.919351i	$0.371285\pi$
$152$	−3.00000	−0.243332
$153$	−9.16515	−0.740959
$154$	−2.09355	−0.168703
$155$	0	0
$156$	0	0
$157$	9.16515	0.731459	0.365729	−	0.930721i	$-0.380820\pi$
$157$	9.16515	0.731459	0.365729	+	0.930721i	$0.380820\pi$
$158$	−2.74110	−0.218070
$159$	7.58258	0.601337
$160$	0	0
$161$	−7.93725	−0.625543
$162$	0.456850	0.0358935
$163$	21.0707	1.65038	0.825191	−	0.564854i	$-0.191068\pi$
$163$	21.0707	1.65038	0.825191	+	0.564854i	$0.191068\pi$
$164$	4.73930	0.370077
$165$	0	0
$166$	−2.74773	−0.213265
$167$	9.57395	0.740855	0.370427	−	0.928861i	$-0.379211\pi$
$167$	9.57395	0.740855	0.370427	+	0.928861i	$0.379211\pi$
$168$	−3.00000	−0.231455
$169$	0	0
$170$	0	0
$171$	−3.46410	−0.264906
$172$	18.9564	1.44541
$173$	16.5826	1.26075	0.630375	−	0.776291i	$-0.282901\pi$
$173$	16.5826	1.26075	0.630375	+	0.776291i	$0.282901\pi$
$174$	2.09355	0.158712
$175$	0	0
$176$	−7.38505	−0.556669
$177$	−13.9518	−1.04868
$178$	−4.37386	−0.327835
$179$	−18.1652	−1.35773	−0.678864	−	0.734264i	$-0.737527\pi$
$179$	−18.1652	−1.35773	−0.678864	+	0.734264i	$0.737527\pi$
$180$	0	0
$181$	−8.74773	−0.650213	−0.325107	−	0.945677i	$-0.605400\pi$
$181$	−8.74773	−0.650213	−0.325107	+	0.945677i	$0.605400\pi$
$182$	0	0
$183$	−1.41742	−0.104779
$184$	7.93725	0.585142
$185$	0	0
$186$	2.83485	0.207861
$187$	−12.1244	−0.886621
$188$	−3.27340	−0.238737
$189$	−8.66025	−0.629941
$190$	0	0
$191$	−16.5826	−1.19987	−0.599937	−	0.800048i	$-0.704808\pi$
$191$	−16.5826	−1.19987	−0.599937	+	0.800048i	$0.704808\pi$
$192$	−3.41742	−0.246631
$193$	14.8655	1.07004	0.535020	−	0.844840i	$-0.320304\pi$
$193$	14.8655	1.07004	0.535020	+	0.844840i	$0.320304\pi$
$194$	−5.20871	−0.373964
$195$	0	0
$196$	7.16515	0.511797
$197$	−14.6748	−1.04553	−0.522767	−	0.852476i	$-0.675100\pi$
$197$	−14.6748	−1.04553	−0.522767	+	0.852476i	$0.675100\pi$
$198$	2.41742	0.171799
$199$	−10.5826	−0.750179	−0.375089	−	0.926989i	$-0.622388\pi$
$199$	−10.5826	−0.750179	−0.375089	+	0.926989i	$0.622388\pi$
$200$	0	0
$201$	−1.00905	−0.0711729
$202$	−4.11165	−0.289295
$203$	7.93725	0.557086
$204$	−8.20871	−0.574725
$205$	0	0
$206$	−1.44600	−0.100748
$207$	9.16515	0.637022
$208$	0	0
$209$	−4.58258	−0.316983
$210$	0	0
$211$	−0.165151	−0.0113695	−0.00568475	−	0.999984i	$-0.501810\pi$
$211$	−0.165151	−0.0113695	−0.00568475	+	0.999984i	$0.501810\pi$
$212$	−13.5826	−0.932855
$213$	7.02355	0.481246
$214$	−4.83465	−0.330490
$215$	0	0
$216$	8.66025	0.589256
$217$	10.7477	0.729603
$218$	6.00000	0.406371
$219$	0	0
$220$	0	0
$221$	0	0
$222$	−3.62614	−0.243370
$223$	8.66025	0.579934	0.289967	−	0.957037i	$-0.406356\pi$
$223$	8.66025	0.579934	0.289967	+	0.957037i	$0.406356\pi$
$224$	8.20871	0.548468
$225$	0	0
$226$	−3.38865	−0.225410
$227$	0.818350	0.0543158	0.0271579	−	0.999631i	$-0.491354\pi$
$227$	0.818350	0.0543158	0.0271579	+	0.999631i	$0.491354\pi$
$228$	−3.10260	−0.205475
$229$	26.2668	1.73576	0.867880	−	0.496774i	$-0.165482\pi$
$229$	26.2668	1.73576	0.867880	+	0.496774i	$0.165482\pi$
$230$	0	0
$231$	−4.58258	−0.301511
$232$	−7.93725	−0.521106
$233$	−2.83485	−0.185717	−0.0928586	−	0.995679i	$-0.529600\pi$
$233$	−2.83485	−0.185717	−0.0928586	+	0.995679i	$0.529600\pi$
$234$	0	0
$235$	0	0
$236$	24.9916	1.62682
$237$	−6.00000	−0.389742
$238$	3.62614	0.235048
$239$	−0.190700	−0.0123354	−0.00616769	−	0.999981i	$-0.501963\pi$
$239$	−0.190700	−0.0123354	−0.00616769	+	0.999981i	$0.501963\pi$
$240$	0	0
$241$	1.73205	0.111571	0.0557856	−	0.998443i	$-0.482234\pi$
$241$	1.73205	0.111571	0.0557856	+	0.998443i	$0.482234\pi$
$242$	−1.82740	−0.117470
$243$	16.0000	1.02640
$244$	2.53901	0.162544
$245$	0	0
$246$	−1.20871	−0.0770647
$247$	0	0
$248$	−10.7477	−0.682481
$249$	−6.01450	−0.381154
$250$	0	0
$251$	0.165151	0.0104243	0.00521213	−	0.999986i	$-0.498341\pi$
$251$	0.165151	0.0104243	0.00521213	+	0.999986i	$0.498341\pi$
$252$	6.20520	0.390891
$253$	12.1244	0.762252
$254$	−8.10805	−0.508745
$255$	0	0
$256$	1.79129	0.111955
$257$	18.1652	1.13311	0.566556	−	0.824024i	$-0.308276\pi$
$257$	18.1652	1.13311	0.566556	+	0.824024i	$0.308276\pi$
$258$	−4.83465	−0.300992
$259$	−13.7477	−0.854242
$260$	0	0
$261$	−9.16515	−0.567309
$262$	−3.46410	−0.214013
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	4.58258	0.282038
$265$	0	0
$266$	1.37055	0.0840339
$267$	−9.57395	−0.585917
$268$	1.80750	0.110411
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	0	0
$271$	8.66025	0.526073	0.263036	−	0.964786i	$-0.415276\pi$
$271$	8.66025	0.526073	0.263036	+	0.964786i	$0.415276\pi$
$272$	12.7913	0.775586
$273$	0	0
$274$	−4.79129	−0.289452
$275$	0	0
$276$	8.20871	0.494106
$277$	7.41742	0.445670	0.222835	−	0.974856i	$-0.428469\pi$
$277$	7.41742	0.445670	0.222835	+	0.974856i	$0.428469\pi$
$278$	−9.93545	−0.595889
$279$	−12.4104	−0.742992
$280$	0	0
$281$	−3.65480	−0.218027	−0.109014	−	0.994040i	$-0.534769\pi$
$281$	−3.65480	−0.218027	−0.109014	+	0.994040i	$0.534769\pi$
$282$	0.834849	0.0497145
$283$	27.7477	1.64943	0.824716	−	0.565548i	$-0.191335\pi$
$283$	27.7477	1.64943	0.824716	+	0.565548i	$0.191335\pi$
$284$	−12.5812	−0.746557
$285$	0	0
$286$	0	0
$287$	−4.58258	−0.270501
$288$	−9.47860	−0.558532
$289$	4.00000	0.235294
$290$	0	0
$291$	−11.4014	−0.668359
$292$	0	0
$293$	−18.1389	−1.05968	−0.529842	−	0.848097i	$-0.677749\pi$
$293$	−18.1389	−1.05968	−0.529842	+	0.848097i	$0.677749\pi$
$294$	−1.82740	−0.106576
$295$	0	0
$296$	13.7477	0.799070
$297$	13.2288	0.767610
$298$	7.62614	0.441770
$299$	0	0
$300$	0	0
$301$	−18.3296	−1.05650
$302$	4.41742	0.254194
$303$	−9.00000	−0.517036
$304$	4.83465	0.277286
$305$	0	0
$306$	−4.18710	−0.239361
$307$	24.2487	1.38395	0.691974	−	0.721923i	$-0.256741\pi$
$307$	24.2487	1.38395	0.691974	+	0.721923i	$0.256741\pi$
$308$	8.20871	0.467735
$309$	−3.16515	−0.180059
$310$	0	0
$311$	−7.58258	−0.429968	−0.214984	−	0.976618i	$-0.568970\pi$
$311$	−7.58258	−0.429968	−0.214984	+	0.976618i	$0.568970\pi$
$312$	0	0
$313$	−3.25227	−0.183829	−0.0919147	−	0.995767i	$-0.529299\pi$
$313$	−3.25227	−0.183829	−0.0919147	+	0.995767i	$0.529299\pi$
$314$	4.18710	0.236292
$315$	0	0
$316$	10.7477	0.604607
$317$	0.190700	0.0107108	0.00535540	−	0.999986i	$-0.498295\pi$
$317$	0.190700	0.0107108	0.00535540	+	0.999986i	$0.498295\pi$
$318$	3.46410	0.194257
$319$	−12.1244	−0.678834
$320$	0	0
$321$	−10.5826	−0.590662
$322$	−3.62614	−0.202077
$323$	7.93725	0.441641
$324$	−1.79129	−0.0995160
$325$	0	0
$326$	9.62614	0.533142
$327$	13.1334	0.726279
$328$	4.58258	0.253030
$329$	3.16515	0.174500
$330$	0	0
$331$	−4.47315	−0.245867	−0.122933	−	0.992415i	$-0.539230\pi$
$331$	−4.47315	−0.245867	−0.122933	+	0.992415i	$0.539230\pi$
$332$	10.7737	0.591284
$333$	15.8745	0.869918
$334$	4.37386	0.239327
$335$	0	0
$336$	4.83465	0.263752
$337$	−30.7477	−1.67494	−0.837468	−	0.546487i	$-0.815965\pi$
$337$	−30.7477	−1.67494	−0.837468	+	0.546487i	$0.815965\pi$
$338$	0	0
$339$	−7.41742	−0.402859
$340$	0	0
$341$	−16.4174	−0.889053
$342$	−1.58258	−0.0855759
$343$	−19.0526	−1.02874
$344$	18.3296	0.988264
$345$	0	0
$346$	7.57575	0.407275
$347$	−21.3303	−1.14507	−0.572535	−	0.819880i	$-0.694040\pi$
$347$	−21.3303	−1.14507	−0.572535	+	0.819880i	$0.694040\pi$
$348$	−8.20871	−0.440033
$349$	−2.45505	−0.131416	−0.0657079	−	0.997839i	$-0.520931\pi$
$349$	−2.45505	−0.131416	−0.0657079	+	0.997839i	$0.520931\pi$
$350$	0	0
$351$	0	0
$352$	−12.5390	−0.668332
$353$	−6.83285	−0.363676	−0.181838	−	0.983328i	$-0.558205\pi$
$353$	−6.83285	−0.363676	−0.181838	+	0.983328i	$0.558205\pi$
$354$	−6.37386	−0.338767
$355$	0	0
$356$	17.1497	0.908933
$357$	7.93725	0.420084
$358$	−8.29875	−0.438603
$359$	−19.5293	−1.03072	−0.515359	−	0.856975i	$-0.672341\pi$
$359$	−19.5293	−1.03072	−0.515359	+	0.856975i	$0.672341\pi$
$360$	0	0
$361$	−16.0000	−0.842105
$362$	−3.99640	−0.210046
$363$	−4.00000	−0.209946
$364$	0	0
$365$	0	0
$366$	−0.647551	−0.0338480
$367$	−1.74773	−0.0912306	−0.0456153	−	0.998959i	$-0.514525\pi$
$367$	−1.74773	−0.0912306	−0.0456153	+	0.998959i	$0.514525\pi$
$368$	−12.7913	−0.666792
$369$	5.29150	0.275465
$370$	0	0
$371$	13.1334	0.681852
$372$	−11.1153	−0.576302
$373$	−13.0000	−0.673114	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000	−0.673114	−0.336557	+	0.941663i	$0.609263\pi$
$374$	−5.53901	−0.286416
$375$	0	0
$376$	−3.16515	−0.163230
$377$	0	0
$378$	−3.95644	−0.203497
$379$	−10.6784	−0.548510	−0.274255	−	0.961657i	$-0.588431\pi$
$379$	−10.6784	−0.548510	−0.274255	+	0.961657i	$0.588431\pi$
$380$	0	0
$381$	−17.7477	−0.909244
$382$	−7.57575	−0.387609
$383$	−23.6211	−1.20698	−0.603490	−	0.797371i	$-0.706224\pi$
$383$	−23.6211	−1.20698	−0.603490	+	0.797371i	$0.706224\pi$
$384$	−11.0399	−0.563375
$385$	0	0
$386$	6.79129	0.345667
$387$	21.1652	1.07589
$388$	20.4231	1.03683
$389$	3.16515	0.160480	0.0802398	−	0.996776i	$-0.474431\pi$
$389$	3.16515	0.160480	0.0802398	+	0.996776i	$0.474431\pi$
$390$	0	0
$391$	−21.0000	−1.06202
$392$	6.92820	0.349927
$393$	−7.58258	−0.382490
$394$	−6.70417	−0.337751
$395$	0	0
$396$	−9.47860	−0.476318
$397$	20.3477	1.02122	0.510610	−	0.859812i	$-0.329420\pi$
$397$	20.3477	1.02122	0.510610	+	0.859812i	$0.329420\pi$
$398$	−4.83465	−0.242339
$399$	3.00000	0.150188
$400$	0	0
$401$	−29.8263	−1.48945	−0.744726	−	0.667370i	$-0.767420\pi$
$401$	−29.8263	−1.48945	−0.744726	+	0.667370i	$0.767420\pi$
$402$	−0.460985	−0.0229918
$403$	0	0
$404$	16.1216	0.802079
$405$	0	0
$406$	3.62614	0.179962
$407$	21.0000	1.04093
$408$	−7.93725	−0.392953
$409$	−8.66025	−0.428222	−0.214111	−	0.976809i	$-0.568685\pi$
$409$	−8.66025	−0.428222	−0.214111	+	0.976809i	$0.568685\pi$
$410$	0	0
$411$	−10.4877	−0.517318
$412$	5.66970	0.279326
$413$	−24.1652	−1.18909
$414$	4.18710	0.205785
$415$	0	0
$416$	0	0
$417$	−21.7477	−1.06499
$418$	−2.09355	−0.102399
$419$	−5.83485	−0.285051	−0.142526	−	0.989791i	$-0.545522\pi$
$419$	−5.83485	−0.285051	−0.142526	+	0.989791i	$0.545522\pi$
$420$	0	0
$421$	5.48220	0.267186	0.133593	−	0.991036i	$-0.457348\pi$
$421$	5.48220	0.267186	0.133593	+	0.991036i	$0.457348\pi$
$422$	−0.0754495	−0.00367282
$423$	−3.65480	−0.177703
$424$	−13.1334	−0.637815
$425$	0	0
$426$	3.20871	0.155463
$427$	−2.45505	−0.118808
$428$	18.9564	0.916294
$429$	0	0
$430$	0	0
$431$	−8.46955	−0.407964	−0.203982	−	0.978975i	$-0.565388\pi$
$431$	−8.46955	−0.407964	−0.203982	+	0.978975i	$0.565388\pi$
$432$	−13.9564	−0.671479
$433$	−9.74773	−0.468446	−0.234223	−	0.972183i	$-0.575255\pi$
$433$	−9.74773	−0.468446	−0.234223	+	0.972183i	$0.575255\pi$
$434$	4.91010	0.235692
$435$	0	0
$436$	−23.5257	−1.12668
$437$	−7.93725	−0.379690
$438$	0	0
$439$	14.4955	0.691830	0.345915	−	0.938266i	$-0.387569\pi$
$439$	14.4955	0.691830	0.345915	+	0.938266i	$0.387569\pi$
$440$	0	0
$441$	8.00000	0.380952
$442$	0	0
$443$	19.9129	0.946089	0.473045	−	0.881038i	$-0.343155\pi$
$443$	19.9129	0.946089	0.473045	+	0.881038i	$0.343155\pi$
$444$	14.2179	0.674752
$445$	0	0
$446$	3.95644	0.187343
$447$	16.6929	0.789545
$448$	−5.91915	−0.279654
$449$	11.0200	0.520064	0.260032	−	0.965600i	$-0.416267\pi$
$449$	11.0200	0.520064	0.260032	+	0.965600i	$0.416267\pi$
$450$	0	0
$451$	7.00000	0.329617
$452$	13.2867	0.624956
$453$	9.66930	0.454304
$454$	0.373864	0.0175463
$455$	0	0
$456$	−3.00000	−0.140488
$457$	1.73205	0.0810219	0.0405110	−	0.999179i	$-0.487101\pi$
$457$	1.73205	0.0810219	0.0405110	+	0.999179i	$0.487101\pi$
$458$	12.0000	0.560723
$459$	−22.9129	−1.06948
$460$	0	0
$461$	35.8408	1.66927	0.834635	−	0.550803i	$-0.185678\pi$
$461$	35.8408	1.66927	0.834635	+	0.550803i	$0.185678\pi$
$462$	−2.09355	−0.0974008
$463$	−39.4002	−1.83108	−0.915542	−	0.402223i	$-0.868238\pi$
$463$	−39.4002	−1.83108	−0.915542	+	0.402223i	$0.868238\pi$
$464$	12.7913	0.593821
$465$	0	0
$466$	−1.29510	−0.0599944
$467$	24.3303	1.12587	0.562936	−	0.826500i	$-0.309672\pi$
$467$	24.3303	1.12587	0.562936	+	0.826500i	$0.309672\pi$
$468$	0	0
$469$	−1.74773	−0.0807025
$470$	0	0
$471$	9.16515	0.422308
$472$	24.1652	1.11229
$473$	27.9989	1.28739
$474$	−2.74110	−0.125903
$475$	0	0
$476$	−14.2179	−0.651677
$477$	−15.1652	−0.694365
$478$	−0.0871215	−0.00398485
$479$	−4.66385	−0.213097	−0.106548	−	0.994308i	$-0.533980\pi$
$479$	−4.66385	−0.213097	−0.106548	+	0.994308i	$0.533980\pi$
$480$	0	0
$481$	0	0
$482$	0.791288	0.0360422
$483$	−7.93725	−0.361158
$484$	7.16515	0.325689
$485$	0	0
$486$	7.30960	0.331570
$487$	10.6784	0.483882	0.241941	−	0.970291i	$-0.422216\pi$
$487$	10.6784	0.483882	0.241941	+	0.970291i	$0.422216\pi$
$488$	2.45505	0.111135
$489$	21.0707	0.952848
$490$	0	0
$491$	−19.4174	−0.876296	−0.438148	−	0.898903i	$-0.644365\pi$
$491$	−19.4174	−0.876296	−0.438148	+	0.898903i	$0.644365\pi$
$492$	4.73930	0.213664
$493$	21.0000	0.945792
$494$	0	0
$495$	0	0
$496$	17.3205	0.777714
$497$	12.1652	0.545682
$498$	−2.74773	−0.123129
$499$	−0.723000	−0.0323659	−0.0161830	−	0.999869i	$-0.505151\pi$
$499$	−0.723000	−0.0323659	−0.0161830	+	0.999869i	$0.505151\pi$
$500$	0	0
$501$	9.57395	0.427733
$502$	0.0754495	0.00336747
$503$	0.165151	0.00736374	0.00368187	−	0.999993i	$-0.498828\pi$
$503$	0.165151	0.00736374	0.00368187	+	0.999993i	$0.498828\pi$
$504$	6.00000	0.267261
$505$	0	0
$506$	5.53901	0.246239
$507$	0	0
$508$	31.7913	1.41051
$509$	−8.46955	−0.375406	−0.187703	−	0.982226i	$-0.560104\pi$
$509$	−8.46955	−0.375406	−0.187703	+	0.982226i	$0.560104\pi$
$510$	0	0
$511$	0	0
$512$	22.8981	1.01196
$513$	−8.66025	−0.382360
$514$	8.29875	0.366042
$515$	0	0
$516$	18.9564	0.834511
$517$	−4.83485	−0.212636
$518$	−6.28065	−0.275956
$519$	16.5826	0.727894
$520$	0	0
$521$	27.4955	1.20460	0.602299	−	0.798271i	$-0.294252\pi$
$521$	27.4955	1.20460	0.602299	+	0.798271i	$0.294252\pi$
$522$	−4.18710	−0.183264
$523$	0.165151	0.00722157	0.00361078	−	0.999993i	$-0.498851\pi$
$523$	0.165151	0.00722157	0.00361078	+	0.999993i	$0.498851\pi$
$524$	13.5826	0.593358
$525$	0	0
$526$	4.11165	0.179277
$527$	28.4358	1.23868
$528$	−7.38505	−0.321393
$529$	−2.00000	−0.0869565
$530$	0	0
$531$	27.9035	1.21091
$532$	−5.37386	−0.232987
$533$	0	0
$534$	−4.37386	−0.189276
$535$	0	0
$536$	1.74773	0.0754903
$537$	−18.1652	−0.783884
$538$	−6.85275	−0.295443
$539$	10.5830	0.455842
$540$	0	0
$541$	10.3923	0.446800	0.223400	−	0.974727i	$-0.428284\pi$
$541$	10.3923	0.446800	0.223400	+	0.974727i	$0.428284\pi$
$542$	3.95644	0.169944
$543$	−8.74773	−0.375401
$544$	21.7182	0.931161
$545$	0	0
$546$	0	0
$547$	28.7477	1.22916	0.614582	−	0.788853i	$-0.289325\pi$
$547$	28.7477	1.22916	0.614582	+	0.788853i	$0.289325\pi$
$548$	18.7864	0.802516
$549$	2.83485	0.120988
$550$	0	0
$551$	7.93725	0.338138
$552$	7.93725	0.337832
$553$	−10.3923	−0.441926
$554$	3.38865	0.143970
$555$	0	0
$556$	38.9564	1.65212
$557$	−7.74655	−0.328232	−0.164116	−	0.986441i	$-0.552477\pi$
$557$	−7.74655	−0.328232	−0.164116	+	0.986441i	$0.552477\pi$
$558$	−5.66970	−0.240017
$559$	0	0
$560$	0	0
$561$	−12.1244	−0.511891
$562$	−1.66970	−0.0704319
$563$	9.00000	0.379305	0.189652	−	0.981851i	$-0.439264\pi$
$563$	9.00000	0.379305	0.189652	+	0.981851i	$0.439264\pi$
$564$	−3.27340	−0.137835
$565$	0	0
$566$	12.6766	0.532835
$567$	1.73205	0.0727393
$568$	−12.1652	−0.510438
$569$	−7.74773	−0.324802	−0.162401	−	0.986725i	$-0.551924\pi$
$569$	−7.74773	−0.324802	−0.162401	+	0.986725i	$0.551924\pi$
$570$	0	0
$571$	35.0780	1.46797	0.733985	−	0.679166i	$-0.237658\pi$
$571$	35.0780	1.46797	0.733985	+	0.679166i	$0.237658\pi$
$572$	0	0
$573$	−16.5826	−0.692747
$574$	−2.09355	−0.0873831
$575$	0	0
$576$	6.83485	0.284785
$577$	6.92820	0.288425	0.144212	−	0.989547i	$-0.453935\pi$
$577$	6.92820	0.288425	0.144212	+	0.989547i	$0.453935\pi$
$578$	1.82740	0.0760099
$579$	14.8655	0.617787
$580$	0	0
$581$	−10.4174	−0.432188
$582$	−5.20871	−0.215908
$583$	−20.0616	−0.830867
$584$	0	0
$585$	0	0
$586$	−8.28674	−0.342322
$587$	39.4956	1.63016	0.815078	−	0.579351i	$-0.196694\pi$
$587$	39.4956	1.63016	0.815078	+	0.579351i	$0.196694\pi$
$588$	7.16515	0.295486
$589$	10.7477	0.442852
$590$	0	0
$591$	−14.6748	−0.603639
$592$	−22.1552	−0.910571
$593$	21.1660	0.869184	0.434592	−	0.900627i	$-0.356893\pi$
$593$	21.1660	0.869184	0.434592	+	0.900627i	$0.356893\pi$
$594$	6.04356	0.247970
$595$	0	0
$596$	−29.9017	−1.22482
$597$	−10.5826	−0.433116
$598$	0	0
$599$	15.4955	0.633127	0.316564	−	0.948571i	$-0.397471\pi$
$599$	15.4955	0.633127	0.316564	+	0.948571i	$0.397471\pi$
$600$	0	0
$601$	16.9129	0.689891	0.344945	−	0.938623i	$-0.387897\pi$
$601$	16.9129	0.689891	0.344945	+	0.938623i	$0.387897\pi$
$602$	−8.37386	−0.341293
$603$	2.01810	0.0821834
$604$	−17.3205	−0.704761
$605$	0	0
$606$	−4.11165	−0.167024
$607$	7.74773	0.314471	0.157235	−	0.987561i	$-0.449742\pi$
$607$	7.74773	0.314471	0.157235	+	0.987561i	$0.449742\pi$
$608$	8.20871	0.332907
$609$	7.93725	0.321634
$610$	0	0
$611$	0	0
$612$	16.4174	0.663635
$613$	5.91915	0.239072	0.119536	−	0.992830i	$-0.461859\pi$
$613$	5.91915	0.239072	0.119536	+	0.992830i	$0.461859\pi$
$614$	11.0780	0.447073
$615$	0	0
$616$	7.93725	0.319801
$617$	−13.9518	−0.561677	−0.280838	−	0.959755i	$-0.590612\pi$
$617$	−13.9518	−0.561677	−0.280838	+	0.959755i	$0.590612\pi$
$618$	−1.44600	−0.0581667
$619$	−29.7309	−1.19499	−0.597493	−	0.801874i	$-0.703836\pi$
$619$	−29.7309	−1.19499	−0.597493	+	0.801874i	$0.703836\pi$
$620$	0	0
$621$	22.9129	0.919462
$622$	−3.46410	−0.138898
$623$	−16.5826	−0.664367
$624$	0	0
$625$	0	0
$626$	−1.48580	−0.0593846
$627$	−4.58258	−0.183010
$628$	−16.4174	−0.655127
$629$	−36.3731	−1.45029
$630$	0	0
$631$	5.91915	0.235638	0.117819	−	0.993035i	$-0.462410\pi$
$631$	5.91915	0.235638	0.117819	+	0.993035i	$0.462410\pi$
$632$	10.3923	0.413384
$633$	−0.165151	−0.00656418
$634$	0.0871215	0.00346004
$635$	0	0
$636$	−13.5826	−0.538584
$637$	0	0
$638$	−5.53901	−0.219292
$639$	−14.0471	−0.555695
$640$	0	0
$641$	−18.1652	−0.717480	−0.358740	−	0.933437i	$-0.616794\pi$
$641$	−18.1652	−0.717480	−0.358740	+	0.933437i	$0.616794\pi$
$642$	−4.83465	−0.190809
$643$	21.7937	0.859458	0.429729	−	0.902958i	$-0.358609\pi$
$643$	21.7937	0.859458	0.429729	+	0.902958i	$0.358609\pi$
$644$	14.2179	0.560264
$645$	0	0
$646$	3.62614	0.142668
$647$	−27.0000	−1.06148	−0.530740	−	0.847535i	$-0.678086\pi$
$647$	−27.0000	−1.06148	−0.530740	+	0.847535i	$0.678086\pi$
$648$	−1.73205	−0.0680414
$649$	36.9129	1.44896
$650$	0	0
$651$	10.7477	0.421237
$652$	−37.7436	−1.47815
$653$	−42.8258	−1.67590	−0.837951	−	0.545746i	$-0.816246\pi$
$653$	−42.8258	−1.67590	−0.837951	+	0.545746i	$0.816246\pi$
$654$	6.00000	0.234619
$655$	0	0
$656$	−7.38505	−0.288338
$657$	0	0
$658$	1.44600	0.0563710
$659$	30.4955	1.18793	0.593967	−	0.804489i	$-0.297561\pi$
$659$	30.4955	1.18793	0.593967	+	0.804489i	$0.297561\pi$
$660$	0	0
$661$	18.3296	0.712937	0.356469	−	0.934307i	$-0.383981\pi$
$661$	18.3296	0.712937	0.356469	+	0.934307i	$0.383981\pi$
$662$	−2.04356	−0.0794252
$663$	0	0
$664$	10.4174	0.404274
$665$	0	0
$666$	7.25227	0.281020
$667$	−21.0000	−0.813123
$668$	−17.1497	−0.663542
$669$	8.66025	0.334825
$670$	0	0
$671$	3.75015	0.144773
$672$	8.20871	0.316658
$673$	24.1652	0.931498	0.465749	−	0.884917i	$-0.345785\pi$
$673$	24.1652	0.931498	0.465749	+	0.884917i	$0.345785\pi$
$674$	−14.0471	−0.541074
$675$	0	0
$676$	0	0
$677$	−2.83485	−0.108952	−0.0544760	−	0.998515i	$-0.517349\pi$
$677$	−2.83485	−0.108952	−0.0544760	+	0.998515i	$0.517349\pi$
$678$	−3.38865	−0.130140
$679$	−19.7477	−0.757848
$680$	0	0
$681$	0.818350	0.0313593
$682$	−7.50030	−0.287202
$683$	33.0997	1.26652	0.633262	−	0.773938i	$-0.281716\pi$
$683$	33.0997	1.26652	0.633262	+	0.773938i	$0.281716\pi$
$684$	6.20520	0.237262
$685$	0	0
$686$	−8.70417	−0.332327
$687$	26.2668	1.00214
$688$	−29.5390	−1.12616
$689$	0	0
$690$	0	0
$691$	−19.7756	−0.752298	−0.376149	−	0.926559i	$-0.622752\pi$
$691$	−19.7756	−0.752298	−0.376149	+	0.926559i	$0.622752\pi$
$692$	−29.7042	−1.12918
$693$	9.16515	0.348155
$694$	−9.74475	−0.369906
$695$	0	0
$696$	−7.93725	−0.300861
$697$	−12.1244	−0.459243
$698$	−1.12159	−0.0424528
$699$	−2.83485	−0.107224
$700$	0	0
$701$	21.1652	0.799397	0.399698	−	0.916647i	$-0.369115\pi$
$701$	21.1652	0.799397	0.399698	+	0.916647i	$0.369115\pi$
$702$	0	0
$703$	−13.7477	−0.518505
$704$	9.04165	0.340770
$705$	0	0
$706$	−3.12159	−0.117483
$707$	−15.5885	−0.586264
$708$	24.9916	0.939242
$709$	36.3731	1.36602	0.683010	−	0.730409i	$-0.260671\pi$
$709$	36.3731	1.36602	0.683010	+	0.730409i	$0.260671\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	16.5826	0.621458
$713$	−28.4358	−1.06493
$714$	3.62614	0.135705
$715$	0	0
$716$	32.5390	1.21604
$717$	−0.190700	−0.00712184
$718$	−8.92197	−0.332965
$719$	24.4955	0.913526	0.456763	−	0.889588i	$-0.349009\pi$
$719$	24.4955	0.913526	0.456763	+	0.889588i	$0.349009\pi$
$720$	0	0
$721$	−5.48220	−0.204168
$722$	−7.30960	−0.272035
$723$	1.73205	0.0644157
$724$	15.6697	0.582360
$725$	0	0
$726$	−1.82740	−0.0678212
$727$	15.2523	0.565675	0.282838	−	0.959168i	$-0.408724\pi$
$727$	15.2523	0.565675	0.282838	+	0.959168i	$0.408724\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−48.4955	−1.79367
$732$	2.53901	0.0938447
$733$	22.8027	0.842237	0.421119	−	0.907006i	$-0.361638\pi$
$733$	22.8027	0.842237	0.421119	+	0.907006i	$0.361638\pi$
$734$	−0.798450	−0.0294713
$735$	0	0
$736$	−21.7182	−0.800544
$737$	2.66970	0.0983396
$738$	2.41742	0.0889866
$739$	17.0345	0.626623	0.313311	−	0.949650i	$-0.398562\pi$
$739$	17.0345	0.626623	0.313311	+	0.949650i	$0.398562\pi$
$740$	0	0
$741$	0	0
$742$	6.00000	0.220267
$743$	−5.72845	−0.210157	−0.105078	−	0.994464i	$-0.533509\pi$
$743$	−5.72845	−0.210157	−0.105078	+	0.994464i	$0.533509\pi$
$744$	−10.7477	−0.394031
$745$	0	0
$746$	−5.93905	−0.217444
$747$	12.0290	0.440118
$748$	21.7182	0.794096
$749$	−18.3296	−0.669748
$750$	0	0
$751$	−11.7477	−0.428681	−0.214340	−	0.976759i	$-0.568760\pi$
$751$	−11.7477	−0.428681	−0.214340	+	0.976759i	$0.568760\pi$
$752$	5.10080	0.186007
$753$	0.165151	0.00601845
$754$	0	0
$755$	0	0
$756$	15.5130	0.564203
$757$	9.74773	0.354287	0.177144	−	0.984185i	$-0.443314\pi$
$757$	9.74773	0.354287	0.177144	+	0.984185i	$0.443314\pi$
$758$	−4.87841	−0.177192
$759$	12.1244	0.440086
$760$	0	0
$761$	35.4594	1.28540	0.642701	−	0.766118i	$-0.277814\pi$
$761$	35.4594	1.28540	0.642701	+	0.766118i	$0.277814\pi$
$762$	−8.10805	−0.293724
$763$	22.7477	0.823523
$764$	29.7042	1.07466
$765$	0	0
$766$	−10.7913	−0.389905
$767$	0	0
$768$	1.79129	0.0646375
$769$	−15.5885	−0.562134	−0.281067	−	0.959688i	$-0.590688\pi$
$769$	−15.5885	−0.562134	−0.281067	+	0.959688i	$0.590688\pi$
$770$	0	0
$771$	18.1652	0.654202
$772$	−26.6283	−0.958374
$773$	24.1534	0.868736	0.434368	−	0.900735i	$-0.356972\pi$
$773$	24.1534	0.868736	0.434368	+	0.900735i	$0.356972\pi$
$774$	9.66930	0.347556
$775$	0	0
$776$	19.7477	0.708902
$777$	−13.7477	−0.493197
$778$	1.44600	0.0518416
$779$	−4.58258	−0.164188
$780$	0	0
$781$	−18.5826	−0.664937
$782$	−9.59386	−0.343076
$783$	−22.9129	−0.818839
$784$	−11.1652	−0.398755
$785$	0	0
$786$	−3.46410	−0.123560
$787$	−16.3115	−0.581441	−0.290720	−	0.956808i	$-0.593895\pi$
$787$	−16.3115	−0.581441	−0.290720	+	0.956808i	$0.593895\pi$
$788$	26.2867	0.936425
$789$	9.00000	0.320408
$790$	0	0
$791$	−12.8474	−0.456799
$792$	−9.16515	−0.325669
$793$	0	0
$794$	9.29583	0.329897
$795$	0	0
$796$	18.9564	0.671893
$797$	−44.0780	−1.56132	−0.780662	−	0.624954i	$-0.785118\pi$
$797$	−44.0780	−1.56132	−0.780662	+	0.624954i	$0.785118\pi$
$798$	1.37055	0.0485170
$799$	8.37420	0.296258
$800$	0	0
$801$	19.1479	0.676558
$802$	−13.6261	−0.481156
$803$	0	0
$804$	1.80750	0.0637456
$805$	0	0
$806$	0	0
$807$	−15.0000	−0.528025
$808$	15.5885	0.548400
$809$	−54.8258	−1.92757	−0.963785	−	0.266679i	$-0.914074\pi$
$809$	−54.8258	−1.92757	−0.963785	+	0.266679i	$0.914074\pi$
$810$	0	0
$811$	−50.5155	−1.77384	−0.886920	−	0.461923i	$-0.847160\pi$
$811$	−50.5155	−1.77384	−0.886920	+	0.461923i	$0.847160\pi$
$812$	−14.2179	−0.498951
$813$	8.66025	0.303728
$814$	9.59386	0.336264
$815$	0	0
$816$	12.7913	0.447785
$817$	−18.3296	−0.641270
$818$	−3.95644	−0.138334
$819$	0	0
$820$	0	0
$821$	−18.1389	−0.633051	−0.316525	−	0.948584i	$-0.602516\pi$
$821$	−18.1389	−0.633051	−0.316525	+	0.948584i	$0.602516\pi$
$822$	−4.79129	−0.167115
$823$	31.4174	1.09514	0.547571	−	0.836759i	$-0.315552\pi$
$823$	31.4174	1.09514	0.547571	+	0.836759i	$0.315552\pi$
$824$	5.48220	0.190982
$825$	0	0
$826$	−11.0399	−0.384126
$827$	10.7737	0.374638	0.187319	−	0.982299i	$-0.440020\pi$
$827$	10.7737	0.374638	0.187319	+	0.982299i	$0.440020\pi$
$828$	−16.4174	−0.570545
$829$	33.3303	1.15761	0.578805	−	0.815466i	$-0.303519\pi$
$829$	33.3303	1.15761	0.578805	+	0.815466i	$0.303519\pi$
$830$	0	0
$831$	7.41742	0.257308
$832$	0	0
$833$	−18.3303	−0.635107
$834$	−9.93545	−0.344037
$835$	0	0
$836$	8.20871	0.283904
$837$	−31.0260	−1.07242
$838$	−2.66565	−0.0920834
$839$	43.6827	1.50809	0.754047	−	0.656821i	$-0.228099\pi$
$839$	43.6827	1.50809	0.754047	+	0.656821i	$0.228099\pi$
$840$	0	0
$841$	−8.00000	−0.275862
$842$	2.50455	0.0863123
$843$	−3.65480	−0.125878
$844$	0.295834	0.0101830
$845$	0	0
$846$	−1.66970	−0.0574054
$847$	−6.92820	−0.238056
$848$	21.1652	0.726814
$849$	27.7477	0.952300
$850$	0	0
$851$	36.3731	1.24685
$852$	−12.5812	−0.431025
$853$	5.63310	0.192874	0.0964369	−	0.995339i	$-0.469255\pi$
$853$	5.63310	0.192874	0.0964369	+	0.995339i	$0.469255\pi$
$854$	−1.12159	−0.0383800
$855$	0	0
$856$	18.3296	0.626491
$857$	4.74773	0.162179	0.0810896	−	0.996707i	$-0.474160\pi$
$857$	4.74773	0.162179	0.0810896	+	0.996707i	$0.474160\pi$
$858$	0	0
$859$	−44.2432	−1.50956	−0.754779	−	0.655979i	$-0.772256\pi$
$859$	−44.2432	−1.50956	−0.754779	+	0.655979i	$0.772256\pi$
$860$	0	0
$861$	−4.58258	−0.156174
$862$	−3.86932	−0.131789
$863$	13.6657	0.465186	0.232593	−	0.972574i	$-0.425279\pi$
$863$	13.6657	0.465186	0.232593	+	0.972574i	$0.425279\pi$
$864$	−23.6965	−0.806172
$865$	0	0
$866$	−4.45325	−0.151328
$867$	4.00000	0.135847
$868$	−19.2523	−0.653465
$869$	15.8745	0.538506
$870$	0	0
$871$	0	0
$872$	−22.7477	−0.770335
$873$	22.8027	0.771755
$874$	−3.62614	−0.122656
$875$	0	0
$876$	0	0
$877$	−7.93725	−0.268022	−0.134011	−	0.990980i	$-0.542786\pi$
$877$	−7.93725	−0.268022	−0.134011	+	0.990980i	$0.542786\pi$
$878$	6.62225	0.223490
$879$	−18.1389	−0.611809
$880$	0	0
$881$	−36.4955	−1.22956	−0.614782	−	0.788697i	$-0.710756\pi$
$881$	−36.4955	−1.22956	−0.614782	+	0.788697i	$0.710756\pi$
$882$	3.65480	0.123064
$883$	−36.2432	−1.21968	−0.609840	−	0.792524i	$-0.708766\pi$
$883$	−36.2432	−1.21968	−0.609840	+	0.792524i	$0.708766\pi$
$884$	0	0
$885$	0	0
$886$	9.09720	0.305627
$887$	54.4955	1.82978	0.914889	−	0.403705i	$-0.132278\pi$
$887$	54.4955	1.82978	0.914889	+	0.403705i	$0.132278\pi$
$888$	13.7477	0.461344
$889$	−30.7400	−1.03099
$890$	0	0
$891$	−2.64575	−0.0886360
$892$	−15.5130	−0.519414
$893$	3.16515	0.105918
$894$	7.62614	0.255056
$895$	0	0
$896$	−19.1216	−0.638808
$897$	0	0
$898$	5.03447	0.168002
$899$	28.4358	0.948387
$900$	0	0
$901$	34.7477	1.15761
$902$	3.19795	0.106480
$903$	−18.3296	−0.609970
$904$	12.8474	0.427297
$905$	0	0
$906$	4.41742	0.146759
$907$	6.25227	0.207603	0.103802	−	0.994598i	$-0.466899\pi$
$907$	6.25227	0.207603	0.103802	+	0.994598i	$0.466899\pi$
$908$	−1.46590	−0.0486476
$909$	18.0000	0.597022
$910$	0	0
$911$	−7.91288	−0.262165	−0.131083	−	0.991371i	$-0.541845\pi$
$911$	−7.91288	−0.262165	−0.131083	+	0.991371i	$0.541845\pi$
$912$	4.83465	0.160091
$913$	15.9129	0.526639
$914$	0.791288	0.0261735
$915$	0	0
$916$	−47.0514	−1.55462
$917$	−13.1334	−0.433703
$918$	−10.4678	−0.345487
$919$	54.1652	1.78674	0.893372	−	0.449318i	$-0.148333\pi$
$919$	54.1652	1.78674	0.893372	+	0.449318i	$0.148333\pi$
$920$	0	0
$921$	24.2487	0.799022
$922$	16.3739	0.539244
$923$	0	0
$924$	8.20871	0.270047
$925$	0	0
$926$	−18.0000	−0.591517
$927$	6.33030	0.207914
$928$	21.7182	0.712935
$929$	−26.3622	−0.864915	−0.432457	−	0.901654i	$-0.642353\pi$
$929$	−26.3622	−0.864915	−0.432457	+	0.901654i	$0.642353\pi$
$930$	0	0
$931$	−6.92820	−0.227063
$932$	5.07803	0.166336
$933$	−7.58258	−0.248242
$934$	11.1153	0.363704
$935$	0	0
$936$	0	0
$937$	31.4955	1.02891	0.514456	−	0.857517i	$-0.327994\pi$
$937$	31.4955	1.02891	0.514456	+	0.857517i	$0.327994\pi$
$938$	−0.798450	−0.0260703
$939$	−3.25227	−0.106134
$940$	0	0
$941$	−26.4575	−0.862490	−0.431245	−	0.902235i	$-0.641926\pi$
$941$	−26.4575	−0.862490	−0.431245	+	0.902235i	$0.641926\pi$
$942$	4.18710	0.136423
$943$	12.1244	0.394823
$944$	−38.9434	−1.26750
$945$	0	0
$946$	12.7913	0.415881
$947$	−14.3332	−0.465765	−0.232883	−	0.972505i	$-0.574816\pi$
$947$	−14.3332	−0.465765	−0.232883	+	0.972505i	$0.574816\pi$
$948$	10.7477	0.349070
$949$	0	0
$950$	0	0
$951$	0.190700	0.00618388
$952$	−13.7477	−0.445566
$953$	8.07803	0.261673	0.130837	−	0.991404i	$-0.458234\pi$
$953$	8.07803	0.261673	0.130837	+	0.991404i	$0.458234\pi$
$954$	−6.92820	−0.224309
$955$	0	0
$956$	0.341599	0.0110481
$957$	−12.1244	−0.391925
$958$	−2.13068	−0.0688392
$959$	−18.1652	−0.586583
$960$	0	0
$961$	7.50455	0.242082
$962$	0	0
$963$	21.1652	0.682037
$964$	−3.10260	−0.0999281
$965$	0	0
$966$	−3.62614	−0.116669
$967$	37.3821	1.20213	0.601064	−	0.799201i	$-0.294744\pi$
$967$	37.3821	1.20213	0.601064	+	0.799201i	$0.294744\pi$
$968$	6.92820	0.222681
$969$	7.93725	0.254981
$970$	0	0
$971$	−18.4955	−0.593547	−0.296774	−	0.954948i	$-0.595911\pi$
$971$	−18.4955	−0.593547	−0.296774	+	0.954948i	$0.595911\pi$
$972$	−28.6606	−0.919289
$973$	−37.6682	−1.20759
$974$	4.87841	0.156314
$975$	0	0
$976$	−3.95644	−0.126643
$977$	−35.3085	−1.12962	−0.564809	−	0.825222i	$-0.691050\pi$
$977$	−35.3085	−1.12962	−0.564809	+	0.825222i	$0.691050\pi$
$978$	9.62614	0.307810
$979$	25.3303	0.809560
$980$	0	0
$981$	−26.2668	−0.838635
$982$	−8.87086	−0.283080
$983$	−55.0840	−1.75691	−0.878454	−	0.477827i	$-0.841424\pi$
$983$	−55.0840	−1.75691	−0.878454	+	0.477827i	$0.841424\pi$
$984$	4.58258	0.146087
$985$	0	0
$986$	9.59386	0.305531
$987$	3.16515	0.100748
$988$	0	0
$989$	48.4955	1.54207
$990$	0	0
$991$	−13.0000	−0.412959	−0.206479	−	0.978451i	$-0.566201\pi$
$991$	−13.0000	−0.412959	−0.206479	+	0.978451i	$0.566201\pi$
$992$	29.4083	0.933715
$993$	−4.47315	−0.141951
$994$	5.55765	0.176278
$995$	0	0
$996$	10.7737	0.341378
$997$	0.165151	0.00523040	0.00261520	−	0.999997i	$-0.499168\pi$
$997$	0.165151	0.00523040	0.00261520	+	0.999997i	$0.499168\pi$
$998$	−0.330303	−0.0104556
$999$	39.6863	1.25562

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.bk.1.3		4
5.2	odd	4		845.2.b.f.339.5		8
5.3	odd	4		845.2.b.f.339.4		8
5.4	even	2		4225.2.a.bj.1.2		4
13.6	odd	12		325.2.n.c.101.1		4
13.11	odd	12		325.2.n.c.251.1		4
13.12	even	2	inner	4225.2.a.bk.1.2		4
65.2	even	12		845.2.l.c.654.2		8
65.3	odd	12		845.2.n.c.529.3		8
65.7	even	12		845.2.l.c.699.3		8
65.8	even	4		845.2.d.c.844.6		8
65.12	odd	4		845.2.b.f.339.3		8
65.17	odd	12		845.2.n.d.484.1		8
65.18	even	4		845.2.d.c.844.4		8
65.19	odd	12		325.2.n.b.101.2		4
65.22	odd	12		845.2.n.c.484.3		8
65.23	odd	12		845.2.n.d.529.1		8
65.24	odd	12		325.2.n.b.251.2		4
65.28	even	12		845.2.l.c.654.3		8
65.32	even	12		65.2.l.a.49.2	yes	8
65.33	even	12		845.2.l.c.699.2		8
65.37	even	12		65.2.l.a.4.3	yes	8
65.38	odd	4		845.2.b.f.339.6		8
65.42	odd	12		845.2.n.d.529.2		8
65.43	odd	12		845.2.n.c.484.4		8
65.47	even	4		845.2.d.c.844.3		8
65.48	odd	12		845.2.n.d.484.2		8
65.57	even	4		845.2.d.c.844.5		8
65.58	even	12		65.2.l.a.49.3	yes	8
65.62	odd	12		845.2.n.c.529.4		8
65.63	even	12		65.2.l.a.4.2	✓	8
65.64	even	2		4225.2.a.bj.1.3		4
195.32	odd	12		585.2.bf.a.244.3		8
195.128	odd	12		585.2.bf.a.199.3		8
195.167	odd	12		585.2.bf.a.199.2		8
195.188	odd	12		585.2.bf.a.244.2		8
260.63	odd	12		1040.2.df.b.849.4		8
260.123	odd	12		1040.2.df.b.49.1		8
260.167	odd	12		1040.2.df.b.849.1		8
260.227	odd	12		1040.2.df.b.49.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.l.a.4.2	✓	8	65.63	even	12
65.2.l.a.4.3	yes	8	65.37	even	12
65.2.l.a.49.2	yes	8	65.32	even	12
65.2.l.a.49.3	yes	8	65.58	even	12
325.2.n.b.101.2		4	65.19	odd	12
325.2.n.b.251.2		4	65.24	odd	12
325.2.n.c.101.1		4	13.6	odd	12
325.2.n.c.251.1		4	13.11	odd	12
585.2.bf.a.199.2		8	195.167	odd	12
585.2.bf.a.199.3		8	195.128	odd	12
585.2.bf.a.244.2		8	195.188	odd	12
585.2.bf.a.244.3		8	195.32	odd	12
845.2.b.f.339.3		8	65.12	odd	4
845.2.b.f.339.4		8	5.3	odd	4
845.2.b.f.339.5		8	5.2	odd	4
845.2.b.f.339.6		8	65.38	odd	4
845.2.d.c.844.3		8	65.47	even	4
845.2.d.c.844.4		8	65.18	even	4
845.2.d.c.844.5		8	65.57	even	4
845.2.d.c.844.6		8	65.8	even	4
845.2.l.c.654.2		8	65.2	even	12
845.2.l.c.654.3		8	65.28	even	12
845.2.l.c.699.2		8	65.33	even	12
845.2.l.c.699.3		8	65.7	even	12
845.2.n.c.484.3		8	65.22	odd	12
845.2.n.c.484.4		8	65.43	odd	12
845.2.n.c.529.3		8	65.3	odd	12
845.2.n.c.529.4		8	65.62	odd	12
845.2.n.d.484.1		8	65.17	odd	12
845.2.n.d.484.2		8	65.48	odd	12
845.2.n.d.529.1		8	65.23	odd	12
845.2.n.d.529.2		8	65.42	odd	12
1040.2.df.b.49.1		8	260.123	odd	12
1040.2.df.b.49.4		8	260.227	odd	12
1040.2.df.b.849.1		8	260.167	odd	12
1040.2.df.b.849.4		8	260.63	odd	12
4225.2.a.bj.1.2		4	5.4	even	2
4225.2.a.bj.1.3		4	65.64	even	2
4225.2.a.bk.1.2		4	13.12	even	2	inner
4225.2.a.bk.1.3		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:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q+0.456850 q^{2} +1.00000 q^{3} -1.79129 q^{4} +0.456850 q^{6} +1.73205 q^{7} -1.73205 q^{8} -2.00000 q^{9} -2.64575 q^{11} -1.79129 q^{12} +0.791288 q^{14} +2.79129 q^{16} +4.58258 q^{17} -0.913701 q^{18} +1.73205 q^{19} +1.73205 q^{21} -1.20871 q^{22} -4.58258 q^{23} -1.73205 q^{24} -5.00000 q^{27} -3.10260 q^{28} +4.58258 q^{29} +6.20520 q^{31} +4.73930 q^{32} -2.64575 q^{33} +2.09355 q^{34} +3.58258 q^{36} -7.93725 q^{37} +0.791288 q^{38} -2.64575 q^{41} +0.791288 q^{42} -10.5826 q^{43} +4.73930 q^{44} -2.09355 q^{46} +1.82740 q^{47} +2.79129 q^{48} -4.00000 q^{49} +4.58258 q^{51} +7.58258 q^{53} -2.28425 q^{54} -3.00000 q^{56} +1.73205 q^{57} +2.09355 q^{58} -13.9518 q^{59} -1.41742 q^{61} +2.83485 q^{62} -3.46410 q^{63} -3.41742 q^{64} -1.20871 q^{66} -1.00905 q^{67} -8.20871 q^{68} -4.58258 q^{69} +7.02355 q^{71} +3.46410 q^{72} -3.62614 q^{74} -3.10260 q^{76} -4.58258 q^{77} -6.00000 q^{79} +1.00000 q^{81} -1.20871 q^{82} -6.01450 q^{83} -3.10260 q^{84} -4.83465 q^{86} +4.58258 q^{87} +4.58258 q^{88} -9.57395 q^{89} +8.20871 q^{92} +6.20520 q^{93} +0.834849 q^{94} +4.73930 q^{96} -11.4014 q^{97} -1.82740 q^{98} +5.29150 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 2 q^{4} - 8 q^{9} + 2 q^{12} - 6 q^{14} + 2 q^{16} - 14 q^{22} - 20 q^{27} - 4 q^{36} - 6 q^{38} - 6 q^{42} - 24 q^{43} + 2 q^{48} - 16 q^{49} + 12 q^{53} - 12 q^{56} - 24 q^{61} + 48 q^{62}+ \cdots + 40 q^{94}+O(q^{100}) \) 4 * q + 4 * q^3 + 2 * q^4 - 8 * q^9 + 2 * q^12 - 6 * q^14 + 2 * q^16 - 14 * q^22 - 20 * q^27 - 4 * q^36 - 6 * q^38 - 6 * q^42 - 24 * q^43 + 2 * q^48 - 16 * q^49 + 12 * q^53 - 12 * q^56 - 24 * q^61 + 48 * q^62 - 32 * q^64 - 14 * q^66 - 42 * q^68 - 42 * q^74 - 24 * q^79 + 4 * q^81 - 14 * q^82 + 42 * q^92 + 40 * q^94

Introduction

L-functions

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