LMFDB - Embedded newform 4225.2.a.bk.1.1

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			1.1
Root			$0.456850$ 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)

$f(q)$	$=$	$q-2.18890 q^{2} +1.00000 q^{3} +2.79129 q^{4} -2.18890 q^{6} +1.73205 q^{7} -1.73205 q^{8} -2.00000 q^{9} +O(q^{10})$	\(q-2.18890 q^{2} +1.00000 q^{3} +2.79129 q^{4} -2.18890 q^{6} +1.73205 q^{7} -1.73205 q^{8} -2.00000 q^{9} +2.64575 q^{11} +2.79129 q^{12} -3.79129 q^{14} -1.79129 q^{16} -4.58258 q^{17} +4.37780 q^{18} +1.73205 q^{19} +1.73205 q^{21} -5.79129 q^{22} +4.58258 q^{23} -1.73205 q^{24} -5.00000 q^{27} +4.83465 q^{28} -4.58258 q^{29} -9.66930 q^{31} +7.38505 q^{32} +2.64575 q^{33} +10.0308 q^{34} -5.58258 q^{36} +7.93725 q^{37} -3.79129 q^{38} +2.64575 q^{41} -3.79129 q^{42} -1.41742 q^{43} +7.38505 q^{44} -10.0308 q^{46} -8.75560 q^{47} -1.79129 q^{48} -4.00000 q^{49} -4.58258 q^{51} -1.58258 q^{53} +10.9445 q^{54} -3.00000 q^{56} +1.73205 q^{57} +10.0308 q^{58} -3.36875 q^{59} -10.5826 q^{61} +21.1652 q^{62} -3.46410 q^{63} -12.5826 q^{64} -5.79129 q^{66} +14.8655 q^{67} -12.7913 q^{68} +4.58258 q^{69} -3.55945 q^{71} +3.46410 q^{72} -17.3739 q^{74} +4.83465 q^{76} +4.58258 q^{77} -6.00000 q^{79} +1.00000 q^{81} -5.79129 q^{82} -11.3060 q^{83} +4.83465 q^{84} +3.10260 q^{86} -4.58258 q^{87} -4.58258 q^{88} -4.28245 q^{89} +12.7913 q^{92} -9.66930 q^{93} +19.1652 q^{94} +7.38505 q^{96} +4.47315 q^{97} +8.75560 q^{98} -5.29150 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 2 q^{4} - 8 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 2 * q^4 - 8 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.18890	−1.54779	−0.773893	−	0.633316i	$-0.781693\pi$
$2$	−2.18890	−1.54779	−0.773893	+	0.633316i	$0.781693\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	2.79129	1.39564
$5$	0	0
$5$	0	0
$6$	−2.18890	−0.893615
$7$	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	$-0.369405\pi$
$11$	2.64575	0.797724	0.398862	+	0.917011i	$0.369405\pi$
$12$	2.79129	0.805775
$13$	0	0
$13$	0	0
$14$	−3.79129	−1.01326
$15$	0	0
$16$	−1.79129	−0.447822
$17$	−4.58258	−1.11144	−0.555719	−	0.831370i	$-0.687557\pi$
$17$	−4.58258	−1.11144	−0.555719	+	0.831370i	$0.687557\pi$
$18$	4.37780	1.03186
$19$	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
$22$	−5.79129	−1.23471
$23$	4.58258	0.955533	0.477767	−	0.878487i	$-0.341446\pi$
$23$	4.58258	0.955533	0.477767	+	0.878487i	$0.341446\pi$
$24$	−1.73205	−0.353553
$25$	0	0
$26$	0	0
$27$	−5.00000	−0.962250
$28$	4.83465	0.913663
$29$	−4.58258	−0.850963	−0.425481	−	0.904967i	$-0.639895\pi$
$29$	−4.58258	−0.850963	−0.425481	+	0.904967i	$0.639895\pi$
$30$	0	0
$31$	−9.66930	−1.73666	−0.868329	−	0.495988i	$-0.834806\pi$
$31$	−9.66930	−1.73666	−0.868329	+	0.495988i	$0.834806\pi$
$32$	7.38505	1.30551
$33$	2.64575	0.460566
$34$	10.0308	1.72027
$35$	0	0
$36$	−5.58258	−0.930429
$37$	7.93725	1.30488	0.652438	−	0.757842i	$-0.273746\pi$
$37$	7.93725	1.30488	0.652438	+	0.757842i	$0.273746\pi$
$38$	−3.79129	−0.615028
$39$	0	0
$40$	0	0
$41$	2.64575	0.413197	0.206598	−	0.978426i	$-0.433761\pi$
$41$	2.64575	0.413197	0.206598	+	0.978426i	$0.433761\pi$
$42$	−3.79129	−0.585008
$43$	−1.41742	−0.216155	−0.108078	−	0.994142i	$-0.534469\pi$
$43$	−1.41742	−0.216155	−0.108078	+	0.994142i	$0.534469\pi$
$44$	7.38505	1.11334
$45$	0	0
$46$	−10.0308	−1.47896
$47$	−8.75560	−1.27714	−0.638568	−	0.769565i	$-0.720473\pi$
$47$	−8.75560	−1.27714	−0.638568	+	0.769565i	$0.720473\pi$
$48$	−1.79129	−0.258550
$49$	−4.00000	−0.571429
$50$	0	0
$51$	−4.58258	−0.641689
$52$	0	0
$53$	−1.58258	−0.217383	−0.108692	−	0.994076i	$-0.534666\pi$
$53$	−1.58258	−0.217383	−0.108692	+	0.994076i	$0.534666\pi$
$54$	10.9445	1.48936
$55$	0	0
$56$	−3.00000	−0.400892
$57$	1.73205	0.229416
$58$	10.0308	1.31711
$59$	−3.36875	−0.438574	−0.219287	−	0.975660i	$-0.570373\pi$
$59$	−3.36875	−0.438574	−0.219287	+	0.975660i	$0.570373\pi$
$60$	0	0
$61$	−10.5826	−1.35496	−0.677480	−	0.735541i	$-0.736928\pi$
$61$	−10.5826	−1.35496	−0.677480	+	0.735541i	$0.736928\pi$
$62$	21.1652	2.68798
$63$	−3.46410	−0.436436
$64$	−12.5826	−1.57282
$65$	0	0
$66$	−5.79129	−0.712858
$67$	14.8655	1.81610	0.908052	−	0.418857i	$-0.137569\pi$
$67$	14.8655	1.81610	0.908052	+	0.418857i	$0.137569\pi$
$68$	−12.7913	−1.55117
$69$	4.58258	0.551677
$70$	0	0
$71$	−3.55945	−0.422429	−0.211215	−	0.977440i	$-0.567742\pi$
$71$	−3.55945	−0.422429	−0.211215	+	0.977440i	$0.567742\pi$
$72$	3.46410	0.408248
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−17.3739	−2.01967
$75$	0	0
$76$	4.83465	0.554573
$77$	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
$82$	−5.79129	−0.639541
$83$	−11.3060	−1.24100	−0.620498	−	0.784208i	$-0.713069\pi$
$83$	−11.3060	−1.24100	−0.620498	+	0.784208i	$0.713069\pi$
$84$	4.83465	0.527504
$85$	0	0
$86$	3.10260	0.334562
$87$	−4.58258	−0.491304
$88$	−4.58258	−0.488504
$89$	−4.28245	−0.453939	−0.226969	−	0.973902i	$-0.572882\pi$
$89$	−4.28245	−0.453939	−0.226969	+	0.973902i	$0.572882\pi$
$90$	0	0
$91$	0	0
$92$	12.7913	1.33358
$93$	−9.66930	−1.00266
$94$	19.1652	1.97673
$95$	0	0
$96$	7.38505	0.753734
$97$	4.47315	0.454180	0.227090	−	0.973874i	$-0.427079\pi$
$97$	4.47315	0.454180	0.227090	+	0.973874i	$0.427079\pi$
$98$	8.75560	0.884450
$99$	−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	$0.647781\pi$
$102$	10.0308	0.993198
$103$	15.1652	1.49427	0.747133	−	0.664674i	$-0.231430\pi$
$103$	15.1652	1.49427	0.747133	+	0.664674i	$0.231430\pi$
$104$	0	0
$105$	0	0
$106$	3.46410	0.336463
$107$	−1.41742	−0.137028	−0.0685138	−	0.997650i	$-0.521826\pi$
$107$	−1.41742	−0.137028	−0.0685138	+	0.997650i	$0.521826\pi$
$108$	−13.9564	−1.34296
$109$	−2.74110	−0.262550	−0.131275	−	0.991346i	$-0.541907\pi$
$109$	−2.74110	−0.262550	−0.131275	+	0.991346i	$0.541907\pi$
$110$	0	0
$111$	7.93725	0.753371
$112$	−3.10260	−0.293168
$113$	−16.5826	−1.55996	−0.779979	−	0.625806i	$-0.784770\pi$
$113$	−16.5826	−1.55996	−0.779979	+	0.625806i	$0.784770\pi$
$114$	−3.79129	−0.355087
$115$	0	0
$116$	−12.7913	−1.18764
$117$	0	0
$118$	7.37386	0.678819
$119$	−7.93725	−0.727607
$120$	0	0
$121$	−4.00000	−0.363636
$122$	23.1642	2.09719
$123$	2.64575	0.238559
$124$	−26.9898	−2.42376
$125$	0	0
$126$	7.58258	0.675510
$127$	9.74773	0.864971	0.432485	−	0.901641i	$-0.357637\pi$
$127$	9.74773	0.864971	0.432485	+	0.901641i	$0.357637\pi$
$128$	12.7719	1.12889
$129$	−1.41742	−0.124797
$130$	0	0
$131$	1.58258	0.138270	0.0691351	−	0.997607i	$-0.477976\pi$
$131$	1.58258	0.138270	0.0691351	+	0.997607i	$0.477976\pi$
$132$	7.38505	0.642786
$133$	3.00000	0.260133
$134$	−32.5390	−2.81094
$135$	0	0
$136$	7.93725	0.680614
$137$	0.0953502	0.00814632	0.00407316	−	0.999992i	$-0.498703\pi$
$137$	0.0953502	0.00814632	0.00407316	+	0.999992i	$0.498703\pi$
$138$	−10.0308	−0.853879
$139$	5.74773	0.487516	0.243758	−	0.969836i	$-0.421620\pi$
$139$	5.74773	0.487516	0.243758	+	0.969836i	$0.421620\pi$
$140$	0	0
$141$	−8.75560	−0.737355
$142$	7.79129	0.653830
$143$	0	0
$144$	3.58258	0.298548
$145$	0	0
$146$	0	0
$147$	−4.00000	−0.329914
$148$	22.1552	1.82114
$149$	−9.76465	−0.799952	−0.399976	−	0.916526i	$-0.630981\pi$
$149$	−9.76465	−0.799952	−0.399976	+	0.916526i	$0.630981\pi$
$150$	0	0
$151$	−6.20520	−0.504972	−0.252486	−	0.967601i	$-0.581248\pi$
$151$	−6.20520	−0.504972	−0.252486	+	0.967601i	$0.581248\pi$
$152$	−3.00000	−0.243332
$153$	9.16515	0.740959
$154$	−10.0308	−0.808305
$155$	0	0
$156$	0	0
$157$	−9.16515	−0.731459	−0.365729	−	0.930721i	$-0.619180\pi$
$157$	−9.16515	−0.731459	−0.365729	+	0.930721i	$0.619180\pi$
$158$	13.1334	1.04484
$159$	−1.58258	−0.125506
$160$	0	0
$161$	7.93725	0.625543
$162$	−2.18890	−0.171976
$163$	−10.6784	−0.836393	−0.418197	−	0.908357i	$-0.637338\pi$
$163$	−10.6784	−0.836393	−0.418197	+	0.908357i	$0.637338\pi$
$164$	7.38505	0.576676
$165$	0	0
$166$	24.7477	1.92080
$167$	4.28245	0.331386	0.165693	−	0.986177i	$-0.447014\pi$
$167$	4.28245	0.331386	0.165693	+	0.986177i	$0.447014\pi$
$168$	−3.00000	−0.231455
$169$	0	0
$170$	0	0
$171$	−3.46410	−0.264906
$172$	−3.95644	−0.301676
$173$	7.41742	0.563936	0.281968	−	0.959424i	$-0.409013\pi$
$173$	7.41742	0.563936	0.281968	+	0.959424i	$0.409013\pi$
$174$	10.0308	0.760433
$175$	0	0
$176$	−4.73930	−0.357238
$177$	−3.36875	−0.253211
$178$	9.37386	0.702601
$179$	0.165151	0.0123440	0.00617200	−	0.999981i	$-0.498035\pi$
$179$	0.165151	0.0123440	0.00617200	+	0.999981i	$0.498035\pi$
$180$	0	0
$181$	18.7477	1.39351	0.696754	−	0.717310i	$-0.254627\pi$
$181$	18.7477	1.39351	0.696754	+	0.717310i	$0.254627\pi$
$182$	0	0
$183$	−10.5826	−0.782287
$184$	−7.93725	−0.585142
$185$	0	0
$186$	21.1652	1.55190
$187$	−12.1244	−0.886621
$188$	−24.4394	−1.78243
$189$	−8.66025	−0.629941
$190$	0	0
$191$	−7.41742	−0.536706	−0.268353	−	0.963321i	$-0.586479\pi$
$191$	−7.41742	−0.536706	−0.268353	+	0.963321i	$0.586479\pi$
$192$	−12.5826	−0.908069
$193$	−1.00905	−0.0726331	−0.0363165	−	0.999340i	$-0.511562\pi$
$193$	−1.00905	−0.0726331	−0.0363165	+	0.999340i	$0.511562\pi$
$194$	−9.79129	−0.702973
$195$	0	0
$196$	−11.1652	−0.797511
$197$	−19.9663	−1.42254	−0.711269	−	0.702920i	$-0.751879\pi$
$197$	−19.9663	−1.42254	−0.711269	+	0.702920i	$0.751879\pi$
$198$	11.5826	0.823138
$199$	−1.41742	−0.100479	−0.0502393	−	0.998737i	$-0.515998\pi$
$199$	−1.41742	−0.100479	−0.0502393	+	0.998737i	$0.515998\pi$
$200$	0	0
$201$	14.8655	1.04853
$202$	19.7001	1.38609
$203$	−7.93725	−0.557086
$204$	−12.7913	−0.895569
$205$	0	0
$206$	−33.1950	−2.31281
$207$	−9.16515	−0.637022
$208$	0	0
$209$	4.58258	0.316983
$210$	0	0
$211$	18.1652	1.25054	0.625270	−	0.780408i	$-0.284989\pi$
$211$	18.1652	1.25054	0.625270	+	0.780408i	$0.284989\pi$
$212$	−4.41742	−0.303390
$213$	−3.55945	−0.243890
$214$	3.10260	0.212089
$215$	0	0
$216$	8.66025	0.589256
$217$	−16.7477	−1.13691
$218$	6.00000	0.406371
$219$	0	0
$220$	0	0
$221$	0	0
$222$	−17.3739	−1.16606
$223$	8.66025	0.579934	0.289967	−	0.957037i	$-0.406356\pi$
$223$	8.66025	0.579934	0.289967	+	0.957037i	$0.406356\pi$
$224$	12.7913	0.854654
$225$	0	0
$226$	36.2976	2.41448
$227$	6.10985	0.405525	0.202763	−	0.979228i	$-0.435008\pi$
$227$	6.10985	0.405525	0.202763	+	0.979228i	$0.435008\pi$
$228$	4.83465	0.320183
$229$	−5.48220	−0.362274	−0.181137	−	0.983458i	$-0.557978\pi$
$229$	−5.48220	−0.362274	−0.181137	+	0.983458i	$0.557978\pi$
$230$	0	0
$231$	4.58258	0.301511
$232$	7.93725	0.521106
$233$	−21.1652	−1.38658	−0.693288	−	0.720661i	$-0.743838\pi$
$233$	−21.1652	−1.38658	−0.693288	+	0.720661i	$0.743838\pi$
$234$	0	0
$235$	0	0
$236$	−9.40315	−0.612093
$237$	−6.00000	−0.389742
$238$	17.3739	1.12618
$239$	20.9753	1.35678	0.678390	−	0.734702i	$-0.262678\pi$
$239$	20.9753	1.35678	0.678390	+	0.734702i	$0.262678\pi$
$240$	0	0
$241$	1.73205	0.111571	0.0557856	−	0.998443i	$-0.482234\pi$
$241$	1.73205	0.111571	0.0557856	+	0.998443i	$0.482234\pi$
$242$	8.75560	0.562832
$243$	16.0000	1.02640
$244$	−29.5390	−1.89104
$245$	0	0
$246$	−5.79129	−0.369239
$247$	0	0
$248$	16.7477	1.06348
$249$	−11.3060	−0.716489
$250$	0	0
$251$	−18.1652	−1.14657	−0.573287	−	0.819355i	$-0.694332\pi$
$251$	−18.1652	−1.14657	−0.573287	+	0.819355i	$0.694332\pi$
$252$	−9.66930	−0.609109
$253$	12.1244	0.762252
$254$	−21.3368	−1.33879
$255$	0	0
$256$	−2.79129	−0.174455
$257$	−0.165151	−0.0103019	−0.00515093	−	0.999987i	$-0.501640\pi$
$257$	−0.165151	−0.0103019	−0.00515093	+	0.999987i	$0.501640\pi$
$258$	3.10260	0.193160
$259$	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
$266$	−6.56670	−0.402630
$267$	−4.28245	−0.262082
$268$	41.4938	2.53464
$269$	−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	$0.415276\pi$
$272$	8.20871	0.497726
$273$	0	0
$274$	−0.208712	−0.0126088
$275$	0	0
$276$	12.7913	0.769945
$277$	16.5826	0.996350	0.498175	−	0.867076i	$-0.334004\pi$
$277$	16.5826	0.996350	0.498175	+	0.867076i	$0.334004\pi$
$278$	−12.5812	−0.754571
$279$	19.3386	1.15777
$280$	0	0
$281$	17.5112	1.04463	0.522316	−	0.852752i	$-0.325068\pi$
$281$	17.5112	1.04463	0.522316	+	0.852752i	$0.325068\pi$
$282$	19.1652	1.14127
$283$	0.252273	0.0149961	0.00749803	−	0.999972i	$-0.497613\pi$
$283$	0.252273	0.0149961	0.00749803	+	0.999972i	$0.497613\pi$
$284$	−9.93545	−0.589561
$285$	0	0
$286$	0	0
$287$	4.58258	0.270501
$288$	−14.7701	−0.870337
$289$	4.00000	0.235294
$290$	0	0
$291$	4.47315	0.262221
$292$	0	0
$293$	−23.4304	−1.36882	−0.684408	−	0.729099i	$-0.739939\pi$
$293$	−23.4304	−1.36882	−0.684408	+	0.729099i	$0.739939\pi$
$294$	8.75560	0.510637
$295$	0	0
$296$	−13.7477	−0.799070
$297$	−13.2288	−0.767610
$298$	21.3739	1.23815
$299$	0	0
$300$	0	0
$301$	−2.45505	−0.141507
$302$	13.5826	0.781589
$303$	−9.00000	−0.517036
$304$	−3.10260	−0.177946
$305$	0	0
$306$	−20.0616	−1.14685
$307$	24.2487	1.38395	0.691974	−	0.721923i	$-0.256741\pi$
$307$	24.2487	1.38395	0.691974	+	0.721923i	$0.256741\pi$
$308$	12.7913	0.728851
$309$	15.1652	0.862715
$310$	0	0
$311$	1.58258	0.0897396	0.0448698	−	0.998993i	$-0.485713\pi$
$311$	1.58258	0.0897396	0.0448698	+	0.998993i	$0.485713\pi$
$312$	0	0
$313$	−30.7477	−1.73796	−0.868982	−	0.494843i	$-0.835225\pi$
$313$	−30.7477	−1.73796	−0.868982	+	0.494843i	$0.835225\pi$
$314$	20.0616	1.13214
$315$	0	0
$316$	−16.7477	−0.942133
$317$	−20.9753	−1.17809	−0.589045	−	0.808100i	$-0.700496\pi$
$317$	−20.9753	−1.17809	−0.589045	+	0.808100i	$0.700496\pi$
$318$	3.46410	0.194257
$319$	−12.1244	−0.678834
$320$	0	0
$321$	−1.41742	−0.0791129
$322$	−17.3739	−0.968208
$323$	−7.93725	−0.441641
$324$	2.79129	0.155072
$325$	0	0
$326$	23.3739	1.29456
$327$	−2.74110	−0.151583
$328$	−4.58258	−0.253030
$329$	−15.1652	−0.836082
$330$	0	0
$331$	11.4014	0.626675	0.313338	−	0.949642i	$-0.398553\pi$
$331$	11.4014	0.626675	0.313338	+	0.949642i	$0.398553\pi$
$332$	−31.5583	−1.73199
$333$	−15.8745	−0.869918
$334$	−9.37386	−0.512915
$335$	0	0
$336$	−3.10260	−0.169261
$337$	−3.25227	−0.177163	−0.0885813	−	0.996069i	$-0.528233\pi$
$337$	−3.25227	−0.177163	−0.0885813	+	0.996069i	$0.528233\pi$
$338$	0	0
$339$	−16.5826	−0.900642
$340$	0	0
$341$	−25.5826	−1.38537
$342$	7.58258	0.410019
$343$	−19.0526	−1.02874
$344$	2.45505	0.132367
$345$	0	0
$346$	−16.2360	−0.872853
$347$	15.3303	0.822974	0.411487	−	0.911416i	$-0.365010\pi$
$347$	15.3303	0.822974	0.411487	+	0.911416i	$0.365010\pi$
$348$	−12.7913	−0.685685
$349$	−18.3296	−0.981159	−0.490579	−	0.871396i	$-0.663215\pi$
$349$	−18.3296	−0.981159	−0.490579	+	0.871396i	$0.663215\pi$
$350$	0	0
$351$	0	0
$352$	19.5390	1.04143
$353$	−17.4159	−0.926953	−0.463476	−	0.886109i	$-0.653398\pi$
$353$	−17.4159	−0.926953	−0.463476	+	0.886109i	$0.653398\pi$
$354$	7.37386	0.391916
$355$	0	0
$356$	−11.9536	−0.633537
$357$	−7.93725	−0.420084
$358$	−0.361500	−0.0191059
$359$	33.3857	1.76203	0.881015	−	0.473088i	$-0.156861\pi$
$359$	33.3857	1.76203	0.881015	+	0.473088i	$0.156861\pi$
$360$	0	0
$361$	−16.0000	−0.842105
$362$	−41.0369	−2.15685
$363$	−4.00000	−0.209946
$364$	0	0
$365$	0	0
$366$	23.1642	1.21081
$367$	25.7477	1.34402	0.672010	−	0.740542i	$-0.265431\pi$
$367$	25.7477	1.34402	0.672010	+	0.740542i	$0.265431\pi$
$368$	−8.20871	−0.427909
$369$	−5.29150	−0.275465
$370$	0	0
$371$	−2.74110	−0.142311
$372$	−26.9898	−1.39936
$373$	−13.0000	−0.673114	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000	−0.673114	−0.336557	+	0.941663i	$0.609263\pi$
$374$	26.5390	1.37230
$375$	0	0
$376$	15.1652	0.782083
$377$	0	0
$378$	18.9564	0.975014
$379$	21.0707	1.08233	0.541164	−	0.840917i	$-0.317984\pi$
$379$	21.0707	1.08233	0.541164	+	0.840917i	$0.317984\pi$
$380$	0	0
$381$	9.74773	0.499391
$382$	16.2360	0.830706
$383$	2.83645	0.144936	0.0724680	−	0.997371i	$-0.476912\pi$
$383$	2.83645	0.144936	0.0724680	+	0.997371i	$0.476912\pi$
$384$	12.7719	0.651764
$385$	0	0
$386$	2.20871	0.112420
$387$	2.83485	0.144103
$388$	12.4859	0.633873
$389$	−15.1652	−0.768904	−0.384452	−	0.923145i	$-0.625610\pi$
$389$	−15.1652	−0.768904	−0.384452	+	0.923145i	$0.625610\pi$
$390$	0	0
$391$	−21.0000	−1.06202
$392$	6.92820	0.349927
$393$	1.58258	0.0798304
$394$	43.7042	2.20178
$395$	0	0
$396$	−14.7701	−0.742226
$397$	−27.2759	−1.36894	−0.684468	−	0.729043i	$-0.739966\pi$
$397$	−27.2759	−1.36894	−0.684468	+	0.729043i	$0.739966\pi$
$398$	3.10260	0.155519
$399$	3.00000	0.150188
$400$	0	0
$401$	12.5058	0.624508	0.312254	−	0.949999i	$-0.398916\pi$
$401$	12.5058	0.624508	0.312254	+	0.949999i	$0.398916\pi$
$402$	−32.5390	−1.62290
$403$	0	0
$404$	−25.1216	−1.24985
$405$	0	0
$406$	17.3739	0.862250
$407$	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
$411$	0.0953502	0.00470328
$412$	42.3303	2.08546
$413$	−5.83485	−0.287114
$414$	20.0616	0.985974
$415$	0	0
$416$	0	0
$417$	5.74773	0.281467
$418$	−10.0308	−0.490623
$419$	−24.1652	−1.18054	−0.590272	−	0.807204i	$-0.700980\pi$
$419$	−24.1652	−1.18054	−0.590272	+	0.807204i	$0.700980\pi$
$420$	0	0
$421$	−26.2668	−1.28017	−0.640083	−	0.768306i	$-0.721100\pi$
$421$	−26.2668	−1.28017	−0.640083	+	0.768306i	$0.721100\pi$
$422$	−39.7617	−1.93557
$423$	17.5112	0.851424
$424$	2.74110	0.133120
$425$	0	0
$426$	7.79129	0.377489
$427$	−18.3296	−0.887030
$428$	−3.95644	−0.191242
$429$	0	0
$430$	0	0
$431$	−29.6356	−1.42749	−0.713747	−	0.700403i	$-0.753004\pi$
$431$	−29.6356	−1.42749	−0.713747	+	0.700403i	$0.753004\pi$
$432$	8.95644	0.430917
$433$	17.7477	0.852901	0.426451	−	0.904511i	$-0.359764\pi$
$433$	17.7477	0.852901	0.426451	+	0.904511i	$0.359764\pi$
$434$	36.6591	1.75969
$435$	0	0
$436$	−7.65120	−0.366426
$437$	7.93725	0.379690
$438$	0	0
$439$	−40.4955	−1.93274	−0.966371	−	0.257151i	$-0.917216\pi$
$439$	−40.4955	−1.93274	−0.966371	+	0.257151i	$0.917216\pi$
$440$	0	0
$441$	8.00000	0.380952
$442$	0	0
$443$	−25.9129	−1.23116	−0.615579	−	0.788075i	$-0.711078\pi$
$443$	−25.9129	−1.23116	−0.615579	+	0.788075i	$0.711078\pi$
$444$	22.1552	1.05144
$445$	0	0
$446$	−18.9564	−0.897613
$447$	−9.76465	−0.461852
$448$	−21.7937	−1.02965
$449$	37.4775	1.76867	0.884336	−	0.466852i	$-0.154612\pi$
$449$	37.4775	1.76867	0.884336	+	0.466852i	$0.154612\pi$
$450$	0	0
$451$	7.00000	0.329617
$452$	−46.2867	−2.17715
$453$	−6.20520	−0.291546
$454$	−13.3739	−0.627667
$455$	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
$461$	−1.19975	−0.0558780	−0.0279390	−	0.999610i	$-0.508894\pi$
$461$	−1.19975	−0.0558780	−0.0279390	+	0.999610i	$0.508894\pi$
$462$	−10.0308	−0.466675
$463$	8.22330	0.382169	0.191085	−	0.981574i	$-0.438799\pi$
$463$	8.22330	0.382169	0.191085	+	0.981574i	$0.438799\pi$
$464$	8.20871	0.381080
$465$	0	0
$466$	46.3284	2.14612
$467$	−12.3303	−0.570578	−0.285289	−	0.958441i	$-0.592090\pi$
$467$	−12.3303	−0.570578	−0.285289	+	0.958441i	$0.592090\pi$
$468$	0	0
$469$	25.7477	1.18892
$470$	0	0
$471$	−9.16515	−0.422308
$472$	5.83485	0.268571
$473$	−3.75015	−0.172432
$474$	13.1334	0.603237
$475$	0	0
$476$	−22.1552	−1.01548
$477$	3.16515	0.144922
$478$	−45.9129	−2.10001
$479$	32.3767	1.47933	0.739664	−	0.672977i	$-0.234985\pi$
$479$	32.3767	1.47933	0.739664	+	0.672977i	$0.234985\pi$
$480$	0	0
$481$	0	0
$482$	−3.79129	−0.172688
$483$	7.93725	0.361158
$484$	−11.1652	−0.507507
$485$	0	0
$486$	−35.0224	−1.58865
$487$	−21.0707	−0.954803	−0.477401	−	0.878685i	$-0.658421\pi$
$487$	−21.0707	−0.954803	−0.477401	+	0.878685i	$0.658421\pi$
$488$	18.3296	0.829740
$489$	−10.6784	−0.482892
$490$	0	0
$491$	−28.5826	−1.28991	−0.644957	−	0.764219i	$-0.723125\pi$
$491$	−28.5826	−1.28991	−0.644957	+	0.764219i	$0.723125\pi$
$492$	7.38505	0.332944
$493$	21.0000	0.945792
$494$	0	0
$495$	0	0
$496$	17.3205	0.777714
$497$	−6.16515	−0.276545
$498$	24.7477	1.10897
$499$	−16.5975	−0.743006	−0.371503	−	0.928432i	$-0.621158\pi$
$499$	−16.5975	−0.743006	−0.371503	+	0.928432i	$0.621158\pi$
$500$	0	0
$501$	4.28245	0.191326
$502$	39.7617	1.77465
$503$	−18.1652	−0.809944	−0.404972	−	0.914329i	$-0.632719\pi$
$503$	−18.1652	−0.809944	−0.404972	+	0.914329i	$0.632719\pi$
$504$	6.00000	0.267261
$505$	0	0
$506$	−26.5390	−1.17980
$507$	0	0
$508$	27.2087	1.20719
$509$	−29.6356	−1.31357	−0.656787	−	0.754076i	$-0.728085\pi$
$509$	−29.6356	−1.31357	−0.656787	+	0.754076i	$0.728085\pi$
$510$	0	0
$511$	0	0
$512$	−19.4340	−0.858868
$513$	−8.66025	−0.382360
$514$	0.361500	0.0159451
$515$	0	0
$516$	−3.95644	−0.174173
$517$	−23.1652	−1.01880
$518$	−30.0924	−1.32218
$519$	7.41742	0.325589
$520$	0	0
$521$	−27.4955	−1.20460	−0.602299	−	0.798271i	$-0.705748\pi$
$521$	−27.4955	−1.20460	−0.602299	+	0.798271i	$0.705748\pi$
$522$	−20.0616	−0.878073
$523$	−18.1652	−0.794307	−0.397153	−	0.917752i	$-0.630002\pi$
$523$	−18.1652	−0.794307	−0.397153	+	0.917752i	$0.630002\pi$
$524$	4.41742	0.192976
$525$	0	0
$526$	−19.7001	−0.858966
$527$	44.3103	1.93019
$528$	−4.73930	−0.206252
$529$	−2.00000	−0.0869565
$530$	0	0
$531$	6.73750	0.292383
$532$	8.37386	0.363053
$533$	0	0
$534$	9.37386	0.405647
$535$	0	0
$536$	−25.7477	−1.11213
$537$	0.165151	0.00712681
$538$	32.8335	1.41555
$539$	−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	$0.428284\pi$
$542$	−18.9564	−0.814249
$543$	18.7477	0.804542
$544$	−33.8426	−1.45099
$545$	0	0
$546$	0	0
$547$	1.25227	0.0535433	0.0267717	−	0.999642i	$-0.491477\pi$
$547$	1.25227	0.0535433	0.0267717	+	0.999642i	$0.491477\pi$
$548$	0.266150	0.0113694
$549$	21.1652	0.903307
$550$	0	0
$551$	−7.93725	−0.338138
$552$	−7.93725	−0.337832
$553$	−10.3923	−0.441926
$554$	−36.2976	−1.54214
$555$	0	0
$556$	16.0436	0.680399
$557$	−13.0381	−0.552440	−0.276220	−	0.961094i	$-0.589082\pi$
$557$	−13.0381	−0.552440	−0.276220	+	0.961094i	$0.589082\pi$
$558$	−42.3303	−1.79198
$559$	0	0
$560$	0	0
$561$	−12.1244	−0.511891
$562$	−38.3303	−1.61687
$563$	9.00000	0.379305	0.189652	−	0.981851i	$-0.439264\pi$
$563$	9.00000	0.379305	0.189652	+	0.981851i	$0.439264\pi$
$564$	−24.4394	−1.02908
$565$	0	0
$566$	−0.552200	−0.0232107
$567$	1.73205	0.0727393
$568$	6.16515	0.258684
$569$	19.7477	0.827868	0.413934	−	0.910307i	$-0.364154\pi$
$569$	19.7477	0.827868	0.413934	+	0.910307i	$0.364154\pi$
$570$	0	0
$571$	−29.0780	−1.21688	−0.608439	−	0.793601i	$-0.708204\pi$
$571$	−29.0780	−1.21688	−0.608439	+	0.793601i	$0.708204\pi$
$572$	0	0
$573$	−7.41742	−0.309867
$574$	−10.0308	−0.418678
$575$	0	0
$576$	25.1652	1.04855
$577$	6.92820	0.288425	0.144212	−	0.989547i	$-0.453935\pi$
$577$	6.92820	0.288425	0.144212	+	0.989547i	$0.453935\pi$
$578$	−8.75560	−0.364185
$579$	−1.00905	−0.0419347
$580$	0	0
$581$	−19.5826	−0.812422
$582$	−9.79129	−0.405862
$583$	−4.18710	−0.173412
$584$	0	0
$585$	0	0
$586$	51.2867	2.11864
$587$	−18.7110	−0.772284	−0.386142	−	0.922439i	$-0.626193\pi$
$587$	−18.7110	−0.772284	−0.386142	+	0.922439i	$0.626193\pi$
$588$	−11.1652	−0.460443
$589$	−16.7477	−0.690078
$590$	0	0
$591$	−19.9663	−0.821302
$592$	−14.2179	−0.584352
$593$	−21.1660	−0.869184	−0.434592	−	0.900627i	$-0.643107\pi$
$593$	−21.1660	−0.869184	−0.434592	+	0.900627i	$0.643107\pi$
$594$	28.9564	1.18810
$595$	0	0
$596$	−27.2560	−1.11645
$597$	−1.41742	−0.0580113
$598$	0	0
$599$	−39.4955	−1.61374	−0.806870	−	0.590729i	$-0.798840\pi$
$599$	−39.4955	−1.61374	−0.806870	+	0.590729i	$0.798840\pi$
$600$	0	0
$601$	−28.9129	−1.17938	−0.589690	−	0.807629i	$-0.700750\pi$
$601$	−28.9129	−1.17938	−0.589690	+	0.807629i	$0.700750\pi$
$602$	5.37386	0.219022
$603$	−29.7309	−1.21074
$604$	−17.3205	−0.704761
$605$	0	0
$606$	19.7001	0.800262
$607$	−19.7477	−0.801536	−0.400768	−	0.916180i	$-0.631257\pi$
$607$	−19.7477	−0.801536	−0.400768	+	0.916180i	$0.631257\pi$
$608$	12.7913	0.518755
$609$	−7.93725	−0.321634
$610$	0	0
$611$	0	0
$612$	25.5826	1.03411
$613$	21.7937	0.880238	0.440119	−	0.897940i	$-0.354936\pi$
$613$	21.7937	0.880238	0.440119	+	0.897940i	$0.354936\pi$
$614$	−53.0780	−2.14205
$615$	0	0
$616$	−7.93725	−0.319801
$617$	−3.36875	−0.135621	−0.0678104	−	0.997698i	$-0.521601\pi$
$617$	−3.36875	−0.135621	−0.0678104	+	0.997698i	$0.521601\pi$
$618$	−33.1950	−1.33530
$619$	2.01810	0.0811143	0.0405572	−	0.999177i	$-0.487087\pi$
$619$	2.01810	0.0811143	0.0405572	+	0.999177i	$0.487087\pi$
$620$	0	0
$621$	−22.9129	−0.919462
$622$	−3.46410	−0.138898
$623$	−7.41742	−0.297173
$624$	0	0
$625$	0	0
$626$	67.3037	2.69000
$627$	4.58258	0.183010
$628$	−25.5826	−1.02086
$629$	−36.3731	−1.45029
$630$	0	0
$631$	21.7937	0.867592	0.433796	−	0.901011i	$-0.357174\pi$
$631$	21.7937	0.867592	0.433796	+	0.901011i	$0.357174\pi$
$632$	10.3923	0.413384
$633$	18.1652	0.722000
$634$	45.9129	1.82343
$635$	0	0
$636$	−4.41742	−0.175162
$637$	0	0
$638$	26.5390	1.05069
$639$	7.11890	0.281619
$640$	0	0
$641$	0.165151	0.00652309	0.00326154	−	0.999995i	$-0.498962\pi$
$641$	0.165151	0.00652309	0.00326154	+	0.999995i	$0.498962\pi$
$642$	3.10260	0.122450
$643$	5.91915	0.233429	0.116714	−	0.993166i	$-0.462764\pi$
$643$	5.91915	0.233429	0.116714	+	0.993166i	$0.462764\pi$
$644$	22.1552	0.873036
$645$	0	0
$646$	17.3739	0.683566
$647$	−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
$649$	−8.91288	−0.349861
$650$	0	0
$651$	−16.7477	−0.656395
$652$	−29.8064	−1.16731
$653$	48.8258	1.91070	0.955350	−	0.295477i	$-0.0954787\pi$
$653$	48.8258	1.91070	0.955350	+	0.295477i	$0.0954787\pi$
$654$	6.00000	0.234619
$655$	0	0
$656$	−4.73930	−0.185039
$657$	0	0
$658$	33.1950	1.29408
$659$	−24.4955	−0.954207	−0.477104	−	0.878847i	$-0.658313\pi$
$659$	−24.4955	−0.954207	−0.477104	+	0.878847i	$0.658313\pi$
$660$	0	0
$661$	2.45505	0.0954904	0.0477452	−	0.998860i	$-0.484796\pi$
$661$	2.45505	0.0954904	0.0477452	+	0.998860i	$0.484796\pi$
$662$	−24.9564	−0.969960
$663$	0	0
$664$	19.5826	0.759951
$665$	0	0
$666$	34.7477	1.34645
$667$	−21.0000	−0.813123
$668$	11.9536	0.462497
$669$	8.66025	0.334825
$670$	0	0
$671$	−27.9989	−1.08088
$672$	12.7913	0.493435
$673$	5.83485	0.224917	0.112458	−	0.993656i	$-0.464127\pi$
$673$	5.83485	0.224917	0.112458	+	0.993656i	$0.464127\pi$
$674$	7.11890	0.274210
$675$	0	0
$676$	0	0
$677$	−21.1652	−0.813443	−0.406721	−	0.913552i	$-0.633328\pi$
$677$	−21.1652	−0.813443	−0.406721	+	0.913552i	$0.633328\pi$
$678$	36.2976	1.39400
$679$	7.74773	0.297330
$680$	0	0
$681$	6.10985	0.234130
$682$	55.9977	2.14426
$683$	11.9337	0.456629	0.228314	−	0.973587i	$-0.426679\pi$
$683$	11.9337	0.456629	0.228314	+	0.973587i	$0.426679\pi$
$684$	−9.66930	−0.369715
$685$	0	0
$686$	41.7042	1.59227
$687$	−5.48220	−0.209159
$688$	2.53901	0.0967990
$689$	0	0
$690$	0	0
$691$	−35.6501	−1.35619	−0.678096	−	0.734973i	$-0.737195\pi$
$691$	−35.6501	−1.35619	−0.678096	+	0.734973i	$0.737195\pi$
$692$	20.7042	0.787054
$693$	−9.16515	−0.348155
$694$	−33.5565	−1.27379
$695$	0	0
$696$	7.93725	0.300861
$697$	−12.1244	−0.459243
$698$	40.1216	1.51862
$699$	−21.1652	−0.800540
$700$	0	0
$701$	2.83485	0.107071	0.0535354	−	0.998566i	$-0.482951\pi$
$701$	2.83485	0.107071	0.0535354	+	0.998566i	$0.482951\pi$
$702$	0	0
$703$	13.7477	0.518505
$704$	−33.2904	−1.25468
$705$	0	0
$706$	38.1216	1.43472
$707$	−15.5885	−0.586264
$708$	−9.40315	−0.353392
$709$	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
$712$	7.41742	0.277980
$713$	−44.3103	−1.65943
$714$	17.3739	0.650201
$715$	0	0
$716$	0.460985	0.0172278
$717$	20.9753	0.783337
$718$	−73.0780	−2.72725
$719$	−30.4955	−1.13729	−0.568644	−	0.822584i	$-0.692532\pi$
$719$	−30.4955	−1.13729	−0.568644	+	0.822584i	$0.692532\pi$
$720$	0	0
$721$	26.2668	0.978227
$722$	35.0224	1.30340
$723$	1.73205	0.0644157
$724$	52.3303	1.94484
$725$	0	0
$726$	8.75560	0.324951
$727$	42.7477	1.58543	0.792713	−	0.609595i	$-0.208668\pi$
$727$	42.7477	1.58543	0.792713	+	0.609595i	$0.208668\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	6.49545	0.240243
$732$	−29.5390	−1.09179
$733$	−8.94630	−0.330439	−0.165220	−	0.986257i	$-0.552833\pi$
$733$	−8.94630	−0.330439	−0.165220	+	0.986257i	$0.552833\pi$
$734$	−56.3592	−2.08026
$735$	0	0
$736$	33.8426	1.24745
$737$	39.3303	1.44875
$738$	11.5826	0.426361
$739$	48.7835	1.79453	0.897265	−	0.441493i	$-0.145551\pi$
$739$	48.7835	1.79453	0.897265	+	0.441493i	$0.145551\pi$
$740$	0	0
$741$	0	0
$742$	6.00000	0.220267
$743$	−42.7690	−1.56904	−0.784521	−	0.620103i	$-0.787091\pi$
$743$	−42.7690	−1.56904	−0.784521	+	0.620103i	$0.787091\pi$
$744$	16.7477	0.614001
$745$	0	0
$746$	28.4557	1.04184
$747$	22.6120	0.827330
$748$	−33.8426	−1.23741
$749$	−2.45505	−0.0897056
$750$	0	0
$751$	15.7477	0.574643	0.287321	−	0.957834i	$-0.407235\pi$
$751$	15.7477	0.574643	0.287321	+	0.957834i	$0.407235\pi$
$752$	15.6838	0.571930
$753$	−18.1652	−0.661975
$754$	0	0
$755$	0	0
$756$	−24.1733	−0.879173
$757$	−17.7477	−0.645052	−0.322526	−	0.946561i	$-0.604532\pi$
$757$	−17.7477	−0.645052	−0.322526	+	0.946561i	$0.604532\pi$
$758$	−46.1216	−1.67521
$759$	12.1244	0.440086
$760$	0	0
$761$	40.7509	1.47722	0.738609	−	0.674134i	$-0.235483\pi$
$761$	40.7509	1.47722	0.738609	+	0.674134i	$0.235483\pi$
$762$	−21.3368	−0.772951
$763$	−4.74773	−0.171879
$764$	−20.7042	−0.749050
$765$	0	0
$766$	−6.20871	−0.224330
$767$	0	0
$768$	−2.79129	−0.100722
$769$	−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
$771$	−0.165151	−0.00594778
$772$	−2.81655	−0.101370
$773$	34.7364	1.24938	0.624690	−	0.780873i	$-0.285225\pi$
$773$	34.7364	1.24938	0.624690	+	0.780873i	$0.285225\pi$
$774$	−6.20520	−0.223041
$775$	0	0
$776$	−7.74773	−0.278127
$777$	13.7477	0.493197
$778$	33.1950	1.19010
$779$	4.58258	0.164188
$780$	0	0
$781$	−9.41742	−0.336982
$782$	45.9669	1.64377
$783$	22.9129	0.818839
$784$	7.16515	0.255898
$785$	0	0
$786$	−3.46410	−0.123560
$787$	−32.1860	−1.14731	−0.573653	−	0.819099i	$-0.694474\pi$
$787$	−32.1860	−1.14731	−0.573653	+	0.819099i	$0.694474\pi$
$788$	−55.7316	−1.98536
$789$	9.00000	0.320408
$790$	0	0
$791$	−28.7219	−1.02123
$792$	9.16515	0.325669
$793$	0	0
$794$	59.7042	2.11882
$795$	0	0
$796$	−3.95644	−0.140232
$797$	20.0780	0.711200	0.355600	−	0.934638i	$-0.384276\pi$
$797$	20.0780	0.711200	0.355600	+	0.934638i	$0.384276\pi$
$798$	−6.56670	−0.232459
$799$	40.1232	1.41946
$800$	0	0
$801$	8.56490	0.302626
$802$	−27.3739	−0.966605
$803$	0	0
$804$	41.4938	1.46337
$805$	0	0
$806$	0	0
$807$	−15.0000	−0.528025
$808$	15.5885	0.548400
$809$	36.8258	1.29472	0.647362	−	0.762182i	$-0.275872\pi$
$809$	36.8258	1.29472	0.647362	+	0.762182i	$0.275872\pi$
$810$	0	0
$811$	−18.7665	−0.658981	−0.329491	−	0.944159i	$-0.606877\pi$
$811$	−18.7665	−0.658981	−0.329491	+	0.944159i	$0.606877\pi$
$812$	−22.1552	−0.777494
$813$	8.66025	0.303728
$814$	−45.9669	−1.61114
$815$	0	0
$816$	8.20871	0.287362
$817$	−2.45505	−0.0858914
$818$	18.9564	0.662796
$819$	0	0
$820$	0	0
$821$	−23.4304	−0.817725	−0.408863	−	0.912596i	$-0.634074\pi$
$821$	−23.4304	−0.817725	−0.408863	+	0.912596i	$0.634074\pi$
$822$	−0.208712	−0.00727967
$823$	40.5826	1.41462	0.707310	−	0.706904i	$-0.249909\pi$
$823$	40.5826	1.41462	0.707310	+	0.706904i	$0.249909\pi$
$824$	−26.2668	−0.915048
$825$	0	0
$826$	12.7719	0.444391
$827$	−31.5583	−1.09739	−0.548695	−	0.836023i	$-0.684875\pi$
$827$	−31.5583	−1.09739	−0.548695	+	0.836023i	$0.684875\pi$
$828$	−25.5826	−0.889056
$829$	−3.33030	−0.115666	−0.0578331	−	0.998326i	$-0.518419\pi$
$829$	−3.33030	−0.115666	−0.0578331	+	0.998326i	$0.518419\pi$
$830$	0	0
$831$	16.5826	0.575243
$832$	0	0
$833$	18.3303	0.635107
$834$	−12.5812	−0.435652
$835$	0	0
$836$	12.7913	0.442396
$837$	48.3465	1.67110
$838$	52.8951	1.82723
$839$	1.35065	0.0466296	0.0233148	−	0.999728i	$-0.492578\pi$
$839$	1.35065	0.0466296	0.0233148	+	0.999728i	$0.492578\pi$
$840$	0	0
$841$	−8.00000	−0.275862
$842$	57.4955	1.98142
$843$	17.5112	0.603118
$844$	50.7042	1.74531
$845$	0	0
$846$	−38.3303	−1.31782
$847$	−6.92820	−0.238056
$848$	2.83485	0.0973491
$849$	0.252273	0.00865798
$850$	0	0
$851$	36.3731	1.24685
$852$	−9.93545	−0.340383
$853$	53.2566	1.82347	0.911736	−	0.410777i	$-0.134742\pi$
$853$	53.2566	1.82347	0.911736	+	0.410777i	$0.134742\pi$
$854$	40.1216	1.37293
$855$	0	0
$856$	2.45505	0.0839119
$857$	−22.7477	−0.777048	−0.388524	−	0.921439i	$-0.627015\pi$
$857$	−22.7477	−0.777048	−0.388524	+	0.921439i	$0.627015\pi$
$858$	0	0
$859$	38.2432	1.30484	0.652420	−	0.757857i	$-0.273754\pi$
$859$	38.2432	1.30484	0.652420	+	0.757857i	$0.273754\pi$
$860$	0	0
$861$	4.58258	0.156174
$862$	64.8693	2.20946
$863$	34.8317	1.18569	0.592843	−	0.805318i	$-0.298006\pi$
$863$	34.8317	1.18569	0.592843	+	0.805318i	$0.298006\pi$
$864$	−36.9253	−1.25622
$865$	0	0
$866$	−38.8480	−1.32011
$867$	4.00000	0.135847
$868$	−46.7477	−1.58672
$869$	−15.8745	−0.538506
$870$	0	0
$871$	0	0
$872$	4.74773	0.160778
$873$	−8.94630	−0.302787
$874$	−17.3739	−0.587680
$875$	0	0
$876$	0	0
$877$	7.93725	0.268022	0.134011	−	0.990980i	$-0.457214\pi$
$877$	7.93725	0.268022	0.134011	+	0.990980i	$0.457214\pi$
$878$	88.6405	2.99147
$879$	−23.4304	−0.790286
$880$	0	0
$881$	18.4955	0.623128	0.311564	−	0.950225i	$-0.399147\pi$
$881$	18.4955	0.623128	0.311564	+	0.950225i	$0.399147\pi$
$882$	−17.5112	−0.589633
$883$	46.2432	1.55621	0.778103	−	0.628136i	$-0.216182\pi$
$883$	46.2432	1.55621	0.778103	+	0.628136i	$0.216182\pi$
$884$	0	0
$885$	0	0
$886$	56.7207	1.90557
$887$	−0.495454	−0.0166357	−0.00831786	−	0.999965i	$-0.502648\pi$
$887$	−0.495454	−0.0166357	−0.00831786	+	0.999965i	$0.502648\pi$
$888$	−13.7477	−0.461344
$889$	16.8836	0.566256
$890$	0	0
$891$	2.64575	0.0886360
$892$	24.1733	0.809381
$893$	−15.1652	−0.507482
$894$	21.3739	0.714849
$895$	0	0
$896$	22.1216	0.739030
$897$	0	0
$898$	−82.0345	−2.73753
$899$	44.3103	1.47783
$900$	0	0
$901$	7.25227	0.241608
$902$	−15.3223	−0.510177
$903$	−2.45505	−0.0816990
$904$	28.7219	0.955275
$905$	0	0
$906$	13.5826	0.451251
$907$	33.7477	1.12057	0.560287	−	0.828298i	$-0.310691\pi$
$907$	33.7477	1.12057	0.560287	+	0.828298i	$0.310691\pi$
$908$	17.0544	0.565969
$909$	18.0000	0.597022
$910$	0	0
$911$	37.9129	1.25611	0.628055	−	0.778169i	$-0.283851\pi$
$911$	37.9129	1.25611	0.628055	+	0.778169i	$0.283851\pi$
$912$	−3.10260	−0.102737
$913$	−29.9129	−0.989972
$914$	−3.79129	−0.125405
$915$	0	0
$916$	−15.3024	−0.505606
$917$	2.74110	0.0905191
$918$	−50.1540	−1.65533
$919$	35.8348	1.18208	0.591041	−	0.806641i	$-0.298717\pi$
$919$	35.8348	1.18208	0.591041	+	0.806641i	$0.298717\pi$
$920$	0	0
$921$	24.2487	0.799022
$922$	2.62614	0.0864872
$923$	0	0
$924$	12.7913	0.420802
$925$	0	0
$926$	−18.0000	−0.591517
$927$	−30.3303	−0.996178
$928$	−33.8426	−1.11094
$929$	15.9699	0.523954	0.261977	−	0.965074i	$-0.415626\pi$
$929$	15.9699	0.523954	0.261977	+	0.965074i	$0.415626\pi$
$930$	0	0
$931$	−6.92820	−0.227063
$932$	−59.0780	−1.93517
$933$	1.58258	0.0518112
$934$	26.9898	0.883134
$935$	0	0
$936$	0	0
$937$	−23.4955	−0.767563	−0.383782	−	0.923424i	$-0.625378\pi$
$937$	−23.4955	−0.767563	−0.383782	+	0.923424i	$0.625378\pi$
$938$	−56.3592	−1.84019
$939$	−30.7477	−1.00341
$940$	0	0
$941$	26.4575	0.862490	0.431245	−	0.902235i	$-0.358074\pi$
$941$	26.4575	0.862490	0.431245	+	0.902235i	$0.358074\pi$
$942$	20.0616	0.653643
$943$	12.1244	0.394823
$944$	6.03440	0.196403
$945$	0	0
$946$	8.20871	0.266888
$947$	38.5819	1.25374	0.626871	−	0.779123i	$-0.284335\pi$
$947$	38.5819	1.25374	0.626871	+	0.779123i	$0.284335\pi$
$948$	−16.7477	−0.543941
$949$	0	0
$950$	0	0
$951$	−20.9753	−0.680171
$952$	13.7477	0.445566
$953$	−56.0780	−1.81655	−0.908273	−	0.418378i	$-0.862599\pi$
$953$	−56.0780	−1.81655	−0.908273	+	0.418378i	$0.862599\pi$
$954$	−6.92820	−0.224309
$955$	0	0
$956$	58.5481	1.89358
$957$	−12.1244	−0.391925
$958$	−70.8693	−2.28968
$959$	0.165151	0.00533302
$960$	0	0
$961$	62.4955	2.01598
$962$	0	0
$963$	2.83485	0.0913517
$964$	4.83465	0.155714
$965$	0	0
$966$	−17.3739	−0.558995
$967$	21.5076	0.691638	0.345819	−	0.938301i	$-0.387601\pi$
$967$	21.5076	0.691638	0.345819	+	0.938301i	$0.387601\pi$
$968$	6.92820	0.222681
$969$	−7.93725	−0.254981
$970$	0	0
$971$	36.4955	1.17119	0.585597	−	0.810602i	$-0.300860\pi$
$971$	36.4955	1.17119	0.585597	+	0.810602i	$0.300860\pi$
$972$	44.6606	1.43249
$973$	9.95536	0.319154
$974$	46.1216	1.47783
$975$	0	0
$976$	18.9564	0.606781
$977$	38.7726	1.24044	0.620222	−	0.784426i	$-0.287042\pi$
$977$	38.7726	1.24044	0.620222	+	0.784426i	$0.287042\pi$
$978$	23.3739	0.747414
$979$	−11.3303	−0.362118
$980$	0	0
$981$	5.48220	0.175033
$982$	62.5644	1.99651
$983$	3.12250	0.0995924	0.0497962	−	0.998759i	$-0.484143\pi$
$983$	3.12250	0.0995924	0.0497962	+	0.998759i	$0.484143\pi$
$984$	−4.58258	−0.146087
$985$	0	0
$986$	−45.9669	−1.46389
$987$	−15.1652	−0.482712
$988$	0	0
$989$	−6.49545	−0.206543
$990$	0	0
$991$	−13.0000	−0.412959	−0.206479	−	0.978451i	$-0.566201\pi$
$991$	−13.0000	−0.412959	−0.206479	+	0.978451i	$0.566201\pi$
$992$	−71.4083	−2.26722
$993$	11.4014	0.361811
$994$	13.4949	0.428032
$995$	0	0
$996$	−31.5583	−0.999963
$997$	−18.1652	−0.575296	−0.287648	−	0.957736i	$-0.592873\pi$
$997$	−18.1652	−0.575296	−0.287648	+	0.957736i	$0.592873\pi$
$998$	36.3303	1.15002
$999$	−39.6863	−1.25562

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.bk.1.1		4
5.2	odd	4		845.2.b.f.339.1		8
5.3	odd	4		845.2.b.f.339.8		8
5.4	even	2		4225.2.a.bj.1.4		4
13.6	odd	12		325.2.n.c.101.2		4
13.11	odd	12		325.2.n.c.251.2		4
13.12	even	2	inner	4225.2.a.bk.1.4		4
65.2	even	12		845.2.l.c.654.4		8
65.3	odd	12		845.2.n.c.529.1		8
65.7	even	12		845.2.l.c.699.1		8
65.8	even	4		845.2.d.c.844.2		8
65.12	odd	4		845.2.b.f.339.7		8
65.17	odd	12		845.2.n.d.484.3		8
65.18	even	4		845.2.d.c.844.8		8
65.19	odd	12		325.2.n.b.101.1		4
65.22	odd	12		845.2.n.c.484.1		8
65.23	odd	12		845.2.n.d.529.3		8
65.24	odd	12		325.2.n.b.251.1		4
65.28	even	12		845.2.l.c.654.1		8
65.32	even	12		65.2.l.a.49.4	yes	8
65.33	even	12		845.2.l.c.699.4		8
65.37	even	12		65.2.l.a.4.1	✓	8
65.38	odd	4		845.2.b.f.339.2		8
65.42	odd	12		845.2.n.d.529.4		8
65.43	odd	12		845.2.n.c.484.2		8
65.47	even	4		845.2.d.c.844.7		8
65.48	odd	12		845.2.n.d.484.4		8
65.57	even	4		845.2.d.c.844.1		8
65.58	even	12		65.2.l.a.49.1	yes	8
65.62	odd	12		845.2.n.c.529.2		8
65.63	even	12		65.2.l.a.4.4	yes	8
65.64	even	2		4225.2.a.bj.1.1		4
195.32	odd	12		585.2.bf.a.244.1		8
195.128	odd	12		585.2.bf.a.199.1		8
195.167	odd	12		585.2.bf.a.199.4		8
195.188	odd	12		585.2.bf.a.244.4		8
260.63	odd	12		1040.2.df.b.849.3		8
260.123	odd	12		1040.2.df.b.49.2		8
260.167	odd	12		1040.2.df.b.849.2		8
260.227	odd	12		1040.2.df.b.49.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.l.a.4.1	✓	8	65.37	even	12
65.2.l.a.4.4	yes	8	65.63	even	12
65.2.l.a.49.1	yes	8	65.58	even	12
65.2.l.a.49.4	yes	8	65.32	even	12
325.2.n.b.101.1		4	65.19	odd	12
325.2.n.b.251.1		4	65.24	odd	12
325.2.n.c.101.2		4	13.6	odd	12
325.2.n.c.251.2		4	13.11	odd	12
585.2.bf.a.199.1		8	195.128	odd	12
585.2.bf.a.199.4		8	195.167	odd	12
585.2.bf.a.244.1		8	195.32	odd	12
585.2.bf.a.244.4		8	195.188	odd	12
845.2.b.f.339.1		8	5.2	odd	4
845.2.b.f.339.2		8	65.38	odd	4
845.2.b.f.339.7		8	65.12	odd	4
845.2.b.f.339.8		8	5.3	odd	4
845.2.d.c.844.1		8	65.57	even	4
845.2.d.c.844.2		8	65.8	even	4
845.2.d.c.844.7		8	65.47	even	4
845.2.d.c.844.8		8	65.18	even	4
845.2.l.c.654.1		8	65.28	even	12
845.2.l.c.654.4		8	65.2	even	12
845.2.l.c.699.1		8	65.7	even	12
845.2.l.c.699.4		8	65.33	even	12
845.2.n.c.484.1		8	65.22	odd	12
845.2.n.c.484.2		8	65.43	odd	12
845.2.n.c.529.1		8	65.3	odd	12
845.2.n.c.529.2		8	65.62	odd	12
845.2.n.d.484.3		8	65.17	odd	12
845.2.n.d.484.4		8	65.48	odd	12
845.2.n.d.529.3		8	65.23	odd	12
845.2.n.d.529.4		8	65.42	odd	12
1040.2.df.b.49.2		8	260.123	odd	12
1040.2.df.b.49.3		8	260.227	odd	12
1040.2.df.b.849.2		8	260.167	odd	12
1040.2.df.b.849.3		8	260.63	odd	12
4225.2.a.bj.1.1		4	65.64	even	2
4225.2.a.bj.1.4		4	5.4	even	2
4225.2.a.bk.1.1		4	1.1	even	1	trivial
4225.2.a.bk.1.4		4	13.12	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-2.18890 q^{2} +1.00000 q^{3} +2.79129 q^{4} -2.18890 q^{6} +1.73205 q^{7} -1.73205 q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q-2.18890 q^{2} +1.00000 q^{3} +2.79129 q^{4} -2.18890 q^{6} +1.73205 q^{7} -1.73205 q^{8} -2.00000 q^{9} +2.64575 q^{11} +2.79129 q^{12} -3.79129 q^{14} -1.79129 q^{16} -4.58258 q^{17} +4.37780 q^{18} +1.73205 q^{19} +1.73205 q^{21} -5.79129 q^{22} +4.58258 q^{23} -1.73205 q^{24} -5.00000 q^{27} +4.83465 q^{28} -4.58258 q^{29} -9.66930 q^{31} +7.38505 q^{32} +2.64575 q^{33} +10.0308 q^{34} -5.58258 q^{36} +7.93725 q^{37} -3.79129 q^{38} +2.64575 q^{41} -3.79129 q^{42} -1.41742 q^{43} +7.38505 q^{44} -10.0308 q^{46} -8.75560 q^{47} -1.79129 q^{48} -4.00000 q^{49} -4.58258 q^{51} -1.58258 q^{53} +10.9445 q^{54} -3.00000 q^{56} +1.73205 q^{57} +10.0308 q^{58} -3.36875 q^{59} -10.5826 q^{61} +21.1652 q^{62} -3.46410 q^{63} -12.5826 q^{64} -5.79129 q^{66} +14.8655 q^{67} -12.7913 q^{68} +4.58258 q^{69} -3.55945 q^{71} +3.46410 q^{72} -17.3739 q^{74} +4.83465 q^{76} +4.58258 q^{77} -6.00000 q^{79} +1.00000 q^{81} -5.79129 q^{82} -11.3060 q^{83} +4.83465 q^{84} +3.10260 q^{86} -4.58258 q^{87} -4.58258 q^{88} -4.28245 q^{89} +12.7913 q^{92} -9.66930 q^{93} +19.1652 q^{94} +7.38505 q^{96} +4.47315 q^{97} +8.75560 q^{98} -5.29150 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 2 q^{4} - 8 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 2 * q^4 - 8 * q^9	\( 4 q + 4 q^{3} + 2 q^{4} - 8 q^{9} + 2 q^{12} - 6 q^{14} + 2 q^{16} - 14 q^{22} - 20 q^{27} - 4 q^{36} - 6 q^{38} - 6 q^{42} - 24 q^{43} + 2 q^{48} - 16 q^{49} + 12 q^{53} - 12 q^{56} - 24 q^{61} + 48 q^{62} - 32 q^{64} - 14 q^{66} - 42 q^{68} - 42 q^{74} - 24 q^{79} + 4 q^{81} - 14 q^{82} + 42 q^{92} + 40 q^{94}+O(q^{100}) \) 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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