LMFDB - Embedded newform 3025.2.a.bo.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3025,2,Mod(1,3025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3025, 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("3025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3025 = 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3025.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:	$24.1547466114$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 34 x^{10} + 28 x^{9} + 340 x^{8} + 44 x^{7} - 884 x^{6} - 132 x^{5} + 761 x^{4} + \cdots - 8 $ x^12 - 2x^11 - 34x^10 + 28x^9 + 340x^8 + 44x^7 - 884x^6 - 132x^5 + 761x^4 - 142x^3 - 166x^2 + 72*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 605)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$5.28545$ of defining polynomial
Character	$\chi$	$=$	3025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.328351 q^{2} -0.962000 q^{3} -1.89219 q^{4} -0.315873 q^{6} -3.27259 q^{7} -1.27800 q^{8} -2.07456 q^{9} +O(q^{10})$	\(q+0.328351 q^{2} -0.962000 q^{3} -1.89219 q^{4} -0.315873 q^{6} -3.27259 q^{7} -1.27800 q^{8} -2.07456 q^{9} +1.82028 q^{12} -5.20729 q^{13} -1.07456 q^{14} +3.36474 q^{16} -3.60094 q^{17} -0.681182 q^{18} -5.00941 q^{19} +3.14823 q^{21} -5.84372 q^{23} +1.22944 q^{24} -1.70982 q^{26} +4.88172 q^{27} +6.19234 q^{28} +2.04792 q^{29} -3.25693 q^{31} +3.66082 q^{32} -1.18237 q^{34} +3.92544 q^{36} -6.23700 q^{37} -1.64484 q^{38} +5.00941 q^{39} -10.2056 q^{41} +1.03372 q^{42} -0.596820 q^{43} -1.91879 q^{46} +0.962000 q^{47} -3.23688 q^{48} +3.70982 q^{49} +3.46410 q^{51} +9.85316 q^{52} -0.393280 q^{53} +1.60292 q^{54} +4.18237 q^{56} +4.81906 q^{57} +0.672437 q^{58} -0.527447 q^{59} +5.32529 q^{61} -1.06941 q^{62} +6.78916 q^{63} -5.52745 q^{64} -8.45057 q^{67} +6.81364 q^{68} +5.62166 q^{69} +12.5274 q^{71} +2.65129 q^{72} +8.51527 q^{73} -2.04792 q^{74} +9.47874 q^{76} +1.64484 q^{78} -2.83235 q^{79} +1.52745 q^{81} -3.35100 q^{82} -12.1407 q^{83} -5.95703 q^{84} -0.195966 q^{86} -1.97010 q^{87} +9.00000 q^{89} +17.0413 q^{91} +11.0574 q^{92} +3.13316 q^{93} +0.315873 q^{94} -3.52171 q^{96} -11.2942 q^{97} +1.21812 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 20 q^{4} + 20 q^{9}+O(q^{10}) $ 12 * q + 20 * q^4 + 20 * q^9	$ 12 q + 20 q^{4} + 20 q^{9} + 32 q^{14} + 36 q^{16} + 20 q^{26} + 8 q^{31} - 12 q^{34} + 92 q^{36} + 4 q^{49} + 48 q^{56} + 32 q^{59} - 28 q^{64} + 16 q^{69} + 112 q^{71} - 20 q^{81} - 56 q^{86} + 108 q^{89} + 72 q^{91}+O(q^{100}) $ 12 * q + 20 * q^4 + 20 * q^9 + 32 * q^14 + 36 * q^16 + 20 * q^26 + 8 * q^31 - 12 * q^34 + 92 * q^36 + 4 * q^49 + 48 * q^56 + 32 * q^59 - 28 * q^64 + 16 * q^69 + 112 * q^71 - 20 * q^81 - 56 * q^86 + 108 * q^89 + 72 * q^91

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 $

$2$	0.328351	0.232179	0.116089	−	0.993239i	$-0.462964\pi$
$2$	0.328351	0.232179	0.116089	+	0.993239i	$0.462964\pi$
$3$	−0.962000	−0.555411	−0.277706	−	0.960666i	$-0.589574\pi$
$3$	−0.962000	−0.555411	−0.277706	+	0.960666i	$0.589574\pi$
$4$	−1.89219	−0.946093
$5$	0	0
$5$	0	0
$6$	−0.315873	−0.128955
$7$	−3.27259	−1.23692	−0.618461	−	0.785816i	$-0.712243\pi$
$7$	−3.27259	−1.23692	−0.618461	+	0.785816i	$0.712243\pi$
$8$	−1.27800	−0.451842
$9$	−2.07456	−0.691518
$10$	0	0
$11$	0	0
$11$	0	0
$12$	1.82028	0.525471
$13$	−5.20729	−1.44424	−0.722121	−	0.691767i	$-0.756833\pi$
$13$	−5.20729	−1.44424	−0.722121	+	0.691767i	$0.756833\pi$
$14$	−1.07456	−0.287187
$15$	0	0
$16$	3.36474	0.841185
$17$	−3.60094	−0.873355	−0.436678	−	0.899618i	$-0.643845\pi$
$17$	−3.60094	−0.873355	−0.436678	+	0.899618i	$0.643845\pi$
$18$	−0.681182	−0.160556
$19$	−5.00941	−1.14924	−0.574619	−	0.818421i	$-0.694850\pi$
$19$	−5.00941	−1.14924	−0.574619	+	0.818421i	$0.694850\pi$
$20$	0	0
$21$	3.14823	0.687000
$22$	0	0
$23$	−5.84372	−1.21850	−0.609250	−	0.792978i	$-0.708530\pi$
$23$	−5.84372	−1.21850	−0.609250	+	0.792978i	$0.708530\pi$
$24$	1.22944	0.250958
$25$	0	0
$26$	−1.70982	−0.335323
$27$	4.88172	0.939488
$28$	6.19234	1.17024
$29$	2.04792	0.380290	0.190145	−	0.981756i	$-0.439104\pi$
$29$	2.04792	0.380290	0.190145	+	0.981756i	$0.439104\pi$
$30$	0	0
$31$	−3.25693	−0.584961	−0.292481	−	0.956271i	$-0.594481\pi$
$31$	−3.25693	−0.584961	−0.292481	+	0.956271i	$0.594481\pi$
$32$	3.66082	0.647147
$33$	0	0
$34$	−1.18237	−0.202775
$35$	0	0
$36$	3.92544	0.654241
$37$	−6.23700	−1.02536	−0.512679	−	0.858581i	$-0.671347\pi$
$37$	−6.23700	−1.02536	−0.512679	+	0.858581i	$0.671347\pi$
$38$	−1.64484	−0.266829
$39$	5.00941	0.802148
$40$	0	0
$41$	−10.2056	−1.59384	−0.796921	−	0.604084i	$-0.793539\pi$
$41$	−10.2056	−1.59384	−0.796921	+	0.604084i	$0.793539\pi$
$42$	1.03372	0.159507
$43$	−0.596820	−0.0910142	−0.0455071	−	0.998964i	$-0.514490\pi$
$43$	−0.596820	−0.0910142	−0.0455071	+	0.998964i	$0.514490\pi$
$44$	0	0
$45$	0	0
$46$	−1.91879	−0.282910
$47$	0.962000	0.140322	0.0701611	−	0.997536i	$-0.477649\pi$
$47$	0.962000	0.140322	0.0701611	+	0.997536i	$0.477649\pi$
$48$	−3.23688	−0.467203
$49$	3.70982	0.529974
$50$	0	0
$51$	3.46410	0.485071
$52$	9.85316	1.36639
$53$	−0.393280	−0.0540212	−0.0270106	−	0.999635i	$-0.508599\pi$
$53$	−0.393280	−0.0540212	−0.0270106	+	0.999635i	$0.508599\pi$
$54$	1.60292	0.218129
$55$	0	0
$56$	4.18237	0.558893
$57$	4.81906	0.638300
$58$	0.672437	0.0882953
$59$	−0.527447	−0.0686677	−0.0343339	−	0.999410i	$-0.510931\pi$
$59$	−0.527447	−0.0686677	−0.0343339	+	0.999410i	$0.510931\pi$
$60$	0	0
$61$	5.32529	0.681833	0.340917	−	0.940094i	$-0.389263\pi$
$61$	5.32529	0.681833	0.340917	+	0.940094i	$0.389263\pi$
$62$	−1.06941	−0.135816
$63$	6.78916	0.855354
$64$	−5.52745	−0.690931
$65$	0	0
$66$	0	0
$67$	−8.45057	−1.03240	−0.516201	−	0.856468i	$-0.672654\pi$
$67$	−8.45057	−1.03240	−0.516201	+	0.856468i	$0.672654\pi$
$68$	6.81364	0.826275
$69$	5.62166	0.676769
$70$	0	0
$71$	12.5274	1.48673	0.743367	−	0.668884i	$-0.233228\pi$
$71$	12.5274	1.48673	0.743367	+	0.668884i	$0.233228\pi$
$72$	2.65129	0.312457
$73$	8.51527	0.996638	0.498319	−	0.866994i	$-0.333951\pi$
$73$	8.51527	0.996638	0.498319	+	0.866994i	$0.333951\pi$
$74$	−2.04792	−0.238066
$75$	0	0
$76$	9.47874	1.08729
$77$	0	0
$78$	1.64484	0.186242
$79$	−2.83235	−0.318665	−0.159332	−	0.987225i	$-0.550934\pi$
$79$	−2.83235	−0.318665	−0.159332	+	0.987225i	$0.550934\pi$
$80$	0	0
$81$	1.52745	0.169716
$82$	−3.35100	−0.370056
$83$	−12.1407	−1.33261	−0.666307	−	0.745678i	$-0.732126\pi$
$83$	−12.1407	−1.33261	−0.666307	+	0.745678i	$0.732126\pi$
$84$	−5.95703	−0.649966
$85$	0	0
$86$	−0.195966	−0.0211316
$87$	−1.97010	−0.211217
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	17.0413	1.78641
$92$	11.0574	1.15281
$93$	3.13316	0.324894
$94$	0.315873	0.0325799
$95$	0	0
$96$	−3.52171	−0.359433
$97$	−11.2942	−1.14675	−0.573374	−	0.819293i	$-0.694366\pi$
$97$	−11.2942	−1.14675	−0.573374	+	0.819293i	$0.694366\pi$
$98$	1.21812	0.123049
$99$	0	0
$100$	0	0
$101$	−13.1671	−1.31017	−0.655085	−	0.755555i	$-0.727367\pi$
$101$	−13.1671	−1.31017	−0.655085	+	0.755555i	$0.727367\pi$
$102$	1.13744	0.112623
$103$	2.78228	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$103$	2.78228	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$104$	6.65492	0.652569
$105$	0	0
$106$	−0.129134	−0.0125426
$107$	−15.9502	−1.54197	−0.770983	−	0.636856i	$-0.780235\pi$
$107$	−15.9502	−1.54197	−0.770983	+	0.636856i	$0.780235\pi$
$108$	−9.23713	−0.888843
$109$	3.90911	0.374425	0.187212	−	0.982319i	$-0.440055\pi$
$109$	3.90911	0.374425	0.187212	+	0.982319i	$0.440055\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	−11.0114	−1.04048
$113$	9.87757	0.929204	0.464602	−	0.885520i	$-0.346197\pi$
$113$	9.87757	0.929204	0.464602	+	0.885520i	$0.346197\pi$
$114$	1.58234	0.148200
$115$	0	0
$116$	−3.87505	−0.359790
$117$	10.8028	0.998720
$118$	−0.173187	−0.0159432
$119$	11.7844	1.08027
$120$	0	0
$121$	0	0
$122$	1.74856	0.158307
$123$	9.81776	0.885237
$124$	6.16271	0.553427
$125$	0	0
$126$	2.22922	0.198595
$127$	5.41588	0.480581	0.240291	−	0.970701i	$-0.422757\pi$
$127$	5.41588	0.480581	0.240291	+	0.970701i	$0.422757\pi$
$128$	−9.13658	−0.807567
$129$	0.574141	0.0505503
$130$	0	0
$131$	19.1241	1.67088	0.835440	−	0.549582i	$-0.185213\pi$
$131$	19.1241	1.67088	0.835440	+	0.549582i	$0.185213\pi$
$132$	0	0
$133$	16.3937	1.42152
$134$	−2.77475	−0.239702
$135$	0	0
$136$	4.60200	0.394618
$137$	−16.0429	−1.37063	−0.685317	−	0.728245i	$-0.740336\pi$
$137$	−16.0429	−1.37063	−0.685317	+	0.728245i	$0.740336\pi$
$138$	1.84588	0.157131
$139$	12.5694	1.06612	0.533060	−	0.846078i	$-0.321042\pi$
$139$	12.5694	1.06612	0.533060	+	0.846078i	$0.321042\pi$
$140$	0	0
$141$	−0.925445	−0.0779365
$142$	4.11339	0.345188
$143$	0	0
$144$	−6.98034	−0.581695
$145$	0	0
$146$	2.79600	0.231398
$147$	−3.56884	−0.294353
$148$	11.8016	0.970083
$149$	−13.9279	−1.14102	−0.570510	−	0.821290i	$-0.693255\pi$
$149$	−13.9279	−1.14102	−0.570510	+	0.821290i	$0.693255\pi$
$150$	0	0
$151$	−16.0335	−1.30478	−0.652392	−	0.757881i	$-0.726235\pi$
$151$	−16.0335	−1.30478	−0.652392	+	0.757881i	$0.726235\pi$
$152$	6.40204	0.519274
$153$	7.47034	0.603941
$154$	0	0
$155$	0	0
$156$	−9.47874	−0.758907
$157$	−2.27488	−0.181555	−0.0907776	−	0.995871i	$-0.528935\pi$
$157$	−2.27488	−0.181555	−0.0907776	+	0.995871i	$0.528935\pi$
$158$	−0.930005	−0.0739873
$159$	0.378336	0.0300040
$160$	0	0
$161$	19.1241	1.50719
$162$	0.501538	0.0394046
$163$	22.5694	1.76777	0.883887	−	0.467701i	$-0.154918\pi$
$163$	22.5694	1.76777	0.883887	+	0.467701i	$0.154918\pi$
$164$	19.3108	1.50792
$165$	0	0
$166$	−3.98640	−0.309405
$167$	−3.80952	−0.294790	−0.147395	−	0.989078i	$-0.547089\pi$
$167$	−3.80952	−0.294790	−0.147395	+	0.989078i	$0.547089\pi$
$168$	−4.02344	−0.310415
$169$	14.1159	1.08583
$170$	0	0
$171$	10.3923	0.794719
$172$	1.12929	0.0861079
$173$	0.949651	0.0722006	0.0361003	−	0.999348i	$-0.488506\pi$
$173$	0.949651	0.0722006	0.0361003	+	0.999348i	$0.488506\pi$
$174$	−0.646885	−0.0490402
$175$	0	0
$176$	0	0
$177$	0.507404	0.0381388
$178$	2.95516	0.221498
$179$	3.63526	0.271712	0.135856	−	0.990729i	$-0.456622\pi$
$179$	3.63526	0.271712	0.135856	+	0.990729i	$0.456622\pi$
$180$	0	0
$181$	12.5687	0.934227	0.467114	−	0.884197i	$-0.345294\pi$
$181$	12.5687	0.934227	0.467114	+	0.884197i	$0.345294\pi$
$182$	5.59552	0.414768
$183$	−5.12293	−0.378698
$184$	7.46829	0.550570
$185$	0	0
$186$	1.02878	0.0754335
$187$	0	0
$188$	−1.82028	−0.132758
$189$	−15.9759	−1.16207
$190$	0	0
$191$	−2.67656	−0.193669	−0.0968345	−	0.995301i	$-0.530872\pi$
$191$	−2.67656	−0.193669	−0.0968345	+	0.995301i	$0.530872\pi$
$192$	5.31741	0.383751
$193$	−0.388232	−0.0279455	−0.0139728	−	0.999902i	$-0.504448\pi$
$193$	−0.388232	−0.0279455	−0.0139728	+	0.999902i	$0.504448\pi$
$194$	−3.70845	−0.266251
$195$	0	0
$196$	−7.01966	−0.501404
$197$	−18.5416	−1.32104	−0.660518	−	0.750810i	$-0.729663\pi$
$197$	−18.5416	−1.32104	−0.660518	+	0.750810i	$0.729663\pi$
$198$	0	0
$199$	10.1491	0.719451	0.359726	−	0.933058i	$-0.382870\pi$
$199$	10.1491	0.719451	0.359726	+	0.933058i	$0.382870\pi$
$200$	0	0
$201$	8.12945	0.573407
$202$	−4.32341	−0.304194
$203$	−6.70201	−0.470389
$204$	−6.55472	−0.458922
$205$	0	0
$206$	0.913565	0.0636511
$207$	12.1231	0.842616
$208$	−17.5212	−1.21487
$209$	0	0
$210$	0	0
$211$	−8.47351	−0.583341	−0.291670	−	0.956519i	$-0.594211\pi$
$211$	−8.47351	−0.583341	−0.291670	+	0.956519i	$0.594211\pi$
$212$	0.744160	0.0511091
$213$	−12.0514	−0.825749
$214$	−5.23726	−0.358012
$215$	0	0
$216$	−6.23885	−0.424500
$217$	10.6586	0.723551
$218$	1.28356	0.0869335
$219$	−8.19170	−0.553544
$220$	0	0
$221$	18.7511	1.26134
$222$	1.97010	0.132225
$223$	−23.7069	−1.58753	−0.793764	−	0.608225i	$-0.791882\pi$
$223$	−23.7069	−1.58753	−0.793764	+	0.608225i	$0.791882\pi$
$224$	−11.9803	−0.800470
$225$	0	0
$226$	3.24331	0.215742
$227$	0.716582	0.0475612	0.0237806	−	0.999717i	$-0.492430\pi$
$227$	0.716582	0.0475612	0.0237806	+	0.999717i	$0.492430\pi$
$228$	−9.11855	−0.603891
$229$	13.1491	0.868918	0.434459	−	0.900692i	$-0.356940\pi$
$229$	13.1491	0.868918	0.434459	+	0.900692i	$0.356940\pi$
$230$	0	0
$231$	0	0
$232$	−2.61725	−0.171831
$233$	−26.9327	−1.76442	−0.882209	−	0.470858i	$-0.843944\pi$
$233$	−26.9327	−1.76442	−0.882209	+	0.470858i	$0.843944\pi$
$234$	3.54711	0.231882
$235$	0	0
$236$	0.998027	0.0649660
$237$	2.72473	0.176990
$238$	3.86941	0.250816
$239$	−5.00941	−0.324032	−0.162016	−	0.986788i	$-0.551800\pi$
$239$	−5.00941	−0.324032	−0.162016	+	0.986788i	$0.551800\pi$
$240$	0	0
$241$	4.69354	0.302337	0.151169	−	0.988508i	$-0.451696\pi$
$241$	4.69354	0.302337	0.151169	+	0.988508i	$0.451696\pi$
$242$	0	0
$243$	−16.1146	−1.03375
$244$	−10.0764	−0.645077
$245$	0	0
$246$	3.22367	0.205533
$247$	26.0855	1.65978
$248$	4.16236	0.264310
$249$	11.6793	0.740149
$250$	0	0
$251$	4.37834	0.276358	0.138179	−	0.990407i	$-0.455875\pi$
$251$	4.37834	0.276358	0.138179	+	0.990407i	$0.455875\pi$
$252$	−12.8464	−0.809244
$253$	0	0
$254$	1.77831	0.111581
$255$	0	0
$256$	8.05489	0.503431
$257$	6.51616	0.406467	0.203233	−	0.979130i	$-0.434855\pi$
$257$	6.51616	0.406467	0.203233	+	0.979130i	$0.434855\pi$
$258$	0.188520	0.0117367
$259$	20.4111	1.26829
$260$	0	0
$261$	−4.24853	−0.262978
$262$	6.27941	0.387943
$263$	4.28212	0.264047	0.132023	−	0.991247i	$-0.457853\pi$
$263$	4.28212	0.264047	0.132023	+	0.991247i	$0.457853\pi$
$264$	0	0
$265$	0	0
$266$	5.38289	0.330046
$267$	−8.65800	−0.529861
$268$	15.9900	0.976747
$269$	10.6217	0.647614	0.323807	−	0.946123i	$-0.395037\pi$
$269$	10.6217	0.647614	0.323807	+	0.946123i	$0.395037\pi$
$270$	0	0
$271$	−10.6506	−0.646976	−0.323488	−	0.946232i	$-0.604856\pi$
$271$	−10.6506	−0.646976	−0.323488	+	0.946232i	$0.604856\pi$
$272$	−12.1162	−0.734653
$273$	−16.3937	−0.992194
$274$	−5.26768	−0.318232
$275$	0	0
$276$	−10.6372	−0.640286
$277$	7.35058	0.441654	0.220827	−	0.975313i	$-0.429124\pi$
$277$	7.35058	0.441654	0.220827	+	0.975313i	$0.429124\pi$
$278$	4.12716	0.247530
$279$	6.75667	0.404511
$280$	0	0
$281$	−28.2293	−1.68402	−0.842011	−	0.539461i	$-0.818628\pi$
$281$	−28.2293	−1.68402	−0.842011	+	0.539461i	$0.818628\pi$
$282$	−0.303870	−0.0180952
$283$	−10.0617	−0.598109	−0.299054	−	0.954236i	$-0.596671\pi$
$283$	−10.0617	−0.598109	−0.299054	+	0.954236i	$0.596671\pi$
$284$	−23.7043	−1.40659
$285$	0	0
$286$	0	0
$287$	33.3986	1.97146
$288$	−7.59457	−0.447514
$289$	−4.03326	−0.237251
$290$	0	0
$291$	10.8650	0.636917
$292$	−16.1125	−0.942912
$293$	0.0244803	0.00143016	0.000715078	−	1.00000i	$-0.499772\pi$
$293$	0.0244803	0.00143016	0.000715078		1.00000i	$0.499772\pi$
$294$	−1.17183	−0.0683426
$295$	0	0
$296$	7.97090	0.463299
$297$	0	0
$298$	−4.57325	−0.264921
$299$	30.4300	1.75981
$300$	0	0
$301$	1.95314	0.112577
$302$	−5.26460	−0.302944
$303$	12.6667	0.727683
$304$	−16.8554	−0.966722
$305$	0	0
$306$	2.45289	0.140222
$307$	32.9698	1.88169	0.940844	−	0.338839i	$-0.110034\pi$
$307$	32.9698	1.88169	0.940844	+	0.338839i	$0.110034\pi$
$308$	0	0
$309$	−2.67656	−0.152264
$310$	0	0
$311$	−4.82567	−0.273639	−0.136819	−	0.990596i	$-0.543688\pi$
$311$	−4.82567	−0.273639	−0.136819	+	0.990596i	$0.543688\pi$
$312$	−6.40204	−0.362444
$313$	−17.0662	−0.964637	−0.482318	−	0.875996i	$-0.660205\pi$
$313$	−17.0662	−0.964637	−0.482318	+	0.875996i	$0.660205\pi$
$314$	−0.746958	−0.0421533
$315$	0	0
$316$	5.35934	0.301487
$317$	1.71657	0.0964120	0.0482060	−	0.998837i	$-0.484650\pi$
$317$	1.71657	0.0964120	0.0482060	+	0.998837i	$0.484650\pi$
$318$	0.124227	0.00696629
$319$	0	0
$320$	0	0
$321$	15.3441	0.856425
$322$	6.27941	0.349938
$323$	18.0386	1.00369
$324$	−2.89021	−0.160567
$325$	0	0
$326$	7.41068	0.410440
$327$	−3.76056	−0.207960
$328$	13.0427	0.720164
$329$	−3.14823	−0.173567
$330$	0	0
$331$	0.892186	0.0490390	0.0245195	−	0.999699i	$-0.492194\pi$
$331$	0.892186	0.0490390	0.0245195	+	0.999699i	$0.492194\pi$
$332$	22.9724	1.26078
$333$	12.9390	0.709053
$334$	−1.25086	−0.0684440
$335$	0	0
$336$	10.5930	0.577894
$337$	−18.7856	−1.02332	−0.511659	−	0.859189i	$-0.670969\pi$
$337$	−18.7856	−1.02332	−0.511659	+	0.859189i	$0.670969\pi$
$338$	4.63495	0.252108
$339$	−9.50223	−0.516090
$340$	0	0
$341$	0	0
$342$	3.41232	0.184517
$343$	10.7674	0.581385
$344$	0.762737	0.0411240
$345$	0	0
$346$	0.311818	0.0167635
$347$	−20.9598	−1.12518	−0.562591	−	0.826735i	$-0.690196\pi$
$347$	−20.9598	−1.12518	−0.562591	+	0.826735i	$0.690196\pi$
$348$	3.72780	0.199831
$349$	−27.1866	−1.45527	−0.727634	−	0.685966i	$-0.759380\pi$
$349$	−27.1866	−1.45527	−0.727634	+	0.685966i	$0.759380\pi$
$350$	0	0
$351$	−25.4205	−1.35685
$352$	0	0
$353$	8.66841	0.461373	0.230686	−	0.973028i	$-0.425903\pi$
$353$	8.66841	0.461373	0.230686	+	0.973028i	$0.425903\pi$
$354$	0.166606	0.00885503
$355$	0	0
$356$	−17.0297	−0.902571
$357$	−11.3366	−0.599995
$358$	1.19364	0.0630858
$359$	−6.90465	−0.364414	−0.182207	−	0.983260i	$-0.558324\pi$
$359$	−6.90465	−0.364414	−0.182207	+	0.983260i	$0.558324\pi$
$360$	0	0
$361$	6.09422	0.320748
$362$	4.12695	0.216908
$363$	0	0
$364$	−32.2453	−1.69011
$365$	0	0
$366$	−1.68212	−0.0879256
$367$	16.2900	0.850332	0.425166	−	0.905115i	$-0.360216\pi$
$367$	16.2900	0.850332	0.425166	+	0.905115i	$0.360216\pi$
$368$	−19.6626	−1.02498
$369$	21.1720	1.10217
$370$	0	0
$371$	1.28704	0.0668200
$372$	−5.92853	−0.307380
$373$	−3.69622	−0.191383	−0.0956915	−	0.995411i	$-0.530506\pi$
$373$	−3.69622	−0.191383	−0.0956915	+	0.995411i	$0.530506\pi$
$374$	0	0
$375$	0	0
$376$	−1.22944	−0.0634034
$377$	−10.6641	−0.549231
$378$	−5.24568	−0.269809
$379$	−3.70178	−0.190148	−0.0950738	−	0.995470i	$-0.530309\pi$
$379$	−3.70178	−0.190148	−0.0950738	+	0.995470i	$0.530309\pi$
$380$	0	0
$381$	−5.21007	−0.266920
$382$	−0.878849	−0.0449658
$383$	−31.3709	−1.60298	−0.801488	−	0.598011i	$-0.795958\pi$
$383$	−31.3709	−1.60298	−0.801488	+	0.598011i	$0.795958\pi$
$384$	8.78939	0.448532
$385$	0	0
$386$	−0.127476	−0.00648836
$387$	1.23814	0.0629380
$388$	21.3707	1.08493
$389$	−13.8257	−0.700989	−0.350495	−	0.936565i	$-0.613987\pi$
$389$	−13.8257	−0.700989	−0.350495	+	0.936565i	$0.613987\pi$
$390$	0	0
$391$	21.0429	1.06418
$392$	−4.74115	−0.239464
$393$	−18.3974	−0.928025
$394$	−6.08815	−0.306717
$395$	0	0
$396$	0	0
$397$	−9.73413	−0.488542	−0.244271	−	0.969707i	$-0.578549\pi$
$397$	−9.73413	−0.488542	−0.244271	+	0.969707i	$0.578549\pi$
$398$	3.33247	0.167041
$399$	−15.7708	−0.789526
$400$	0	0
$401$	−26.1768	−1.30721	−0.653604	−	0.756837i	$-0.726744\pi$
$401$	−26.1768	−1.30721	−0.653604	+	0.756837i	$0.726744\pi$
$402$	2.66931	0.133133
$403$	16.9597	0.844825
$404$	24.9145	1.23954
$405$	0	0
$406$	−2.20061	−0.109214
$407$	0	0
$408$	−4.42713	−0.219175
$409$	2.64562	0.130817	0.0654086	−	0.997859i	$-0.479165\pi$
$409$	2.64562	0.130817	0.0654086	+	0.997859i	$0.479165\pi$
$410$	0	0
$411$	15.4332	0.761265
$412$	−5.26460	−0.259368
$413$	1.72611	0.0849365
$414$	3.98064	0.195638
$415$	0	0
$416$	−19.0629	−0.934637
$417$	−12.0917	−0.592135
$418$	0	0
$419$	24.3118	1.18771	0.593855	−	0.804572i	$-0.297605\pi$
$419$	24.3118	1.18771	0.593855	+	0.804572i	$0.297605\pi$
$420$	0	0
$421$	−27.8922	−1.35938	−0.679691	−	0.733499i	$-0.737886\pi$
$421$	−27.8922	−1.35938	−0.679691	+	0.733499i	$0.737886\pi$
$422$	−2.78228	−0.135439
$423$	−1.99572	−0.0970354
$424$	0.502613	0.0244090
$425$	0	0
$426$	−3.95709	−0.191721
$427$	−17.4275	−0.843374
$428$	30.1808	1.45884
$429$	0	0
$430$	0	0
$431$	20.0377	0.965180	0.482590	−	0.875846i	$-0.339696\pi$
$431$	20.0377	0.965180	0.482590	+	0.875846i	$0.339696\pi$
$432$	16.4257	0.790283
$433$	−4.38473	−0.210716	−0.105358	−	0.994434i	$-0.533599\pi$
$433$	−4.38473	−0.210716	−0.105358	+	0.994434i	$0.533599\pi$
$434$	3.49975	0.167993
$435$	0	0
$436$	−7.39676	−0.354241
$437$	29.2736	1.40035
$438$	−2.68975	−0.128521
$439$	−34.2675	−1.63550	−0.817750	−	0.575573i	$-0.804779\pi$
$439$	−34.2675	−1.63550	−0.817750	+	0.575573i	$0.804779\pi$
$440$	0	0
$441$	−7.69622	−0.366487
$442$	6.15694	0.292856
$443$	−18.4932	−0.878637	−0.439319	−	0.898331i	$-0.644780\pi$
$443$	−18.4932	−0.878637	−0.439319	+	0.898331i	$0.644780\pi$
$444$	−11.3531	−0.538795
$445$	0	0
$446$	−7.78416	−0.368591
$447$	13.3987	0.633736
$448$	18.0890	0.854627
$449$	18.7708	0.885848	0.442924	−	0.896559i	$-0.353941\pi$
$449$	18.7708	0.885848	0.442924	+	0.896559i	$0.353941\pi$
$450$	0	0
$451$	0	0
$452$	−18.6902	−0.879113
$453$	15.4242	0.724692
$454$	0.235290	0.0110427
$455$	0	0
$456$	−6.15876	−0.288410
$457$	14.4990	0.678236	0.339118	−	0.940744i	$-0.389871\pi$
$457$	14.4990	0.678236	0.339118	+	0.940744i	$0.389871\pi$
$458$	4.31752	0.201744
$459$	−17.5788	−0.820507
$460$	0	0
$461$	−15.2150	−0.708632	−0.354316	−	0.935126i	$-0.615286\pi$
$461$	−15.2150	−0.708632	−0.354316	+	0.935126i	$0.615286\pi$
$462$	0	0
$463$	9.08060	0.422011	0.211006	−	0.977485i	$-0.432326\pi$
$463$	9.08060	0.422011	0.211006	+	0.977485i	$0.432326\pi$
$464$	6.89073	0.319894
$465$	0	0
$466$	−8.84336	−0.409661
$467$	17.2840	0.799809	0.399904	−	0.916557i	$-0.369043\pi$
$467$	17.2840	0.799809	0.399904	+	0.916557i	$0.369043\pi$
$468$	−20.4409	−0.944882
$469$	27.6552	1.27700
$470$	0	0
$471$	2.18843	0.100838
$472$	0.674078	0.0310269
$473$	0	0
$474$	0.894665	0.0410933
$475$	0	0
$476$	−22.2982	−1.02204
$477$	0.815882	0.0373567
$478$	−1.64484	−0.0752334
$479$	−32.6306	−1.49093	−0.745464	−	0.666546i	$-0.767772\pi$
$479$	−32.6306	−1.49093	−0.745464	+	0.666546i	$0.767772\pi$
$480$	0	0
$481$	32.4779	1.48086
$482$	1.54113	0.0701964
$483$	−18.3974	−0.837110
$484$	0	0
$485$	0	0
$486$	−5.29123	−0.240015
$487$	15.2563	0.691329	0.345664	−	0.938358i	$-0.387654\pi$
$487$	15.2563	0.691329	0.345664	+	0.938358i	$0.387654\pi$
$488$	−6.80572	−0.308081
$489$	−21.7118	−0.981841
$490$	0	0
$491$	−6.66994	−0.301010	−0.150505	−	0.988609i	$-0.548090\pi$
$491$	−6.66994	−0.301010	−0.150505	+	0.988609i	$0.548090\pi$
$492$	−18.5770	−0.837517
$493$	−7.37444	−0.332128
$494$	8.56518	0.385365
$495$	0	0
$496$	−10.9587	−0.492060
$497$	−40.9971	−1.83897
$498$	3.83492	0.171847
$499$	−6.14911	−0.275272	−0.137636	−	0.990483i	$-0.543950\pi$
$499$	−6.14911	−0.275272	−0.137636	+	0.990483i	$0.543950\pi$
$500$	0	0
$501$	3.66476	0.163730
$502$	1.43763	0.0641645
$503$	25.7081	1.14627	0.573134	−	0.819462i	$-0.305728\pi$
$503$	25.7081	1.14627	0.573134	+	0.819462i	$0.305728\pi$
$504$	−8.67656	−0.386485
$505$	0	0
$506$	0	0
$507$	−13.5795	−0.603085
$508$	−10.2478	−0.454675
$509$	−16.8453	−0.746656	−0.373328	−	0.927699i	$-0.621783\pi$
$509$	−16.8453	−0.746656	−0.373328	+	0.927699i	$0.621783\pi$
$510$	0	0
$511$	−27.8670	−1.23276
$512$	20.9180	0.924453
$513$	−24.4546	−1.07970
$514$	2.13959	0.0943731
$515$	0	0
$516$	−1.08638	−0.0478253
$517$	0	0
$518$	6.70201	0.294469
$519$	−0.913565	−0.0401010
$520$	0	0
$521$	−25.1707	−1.10275	−0.551375	−	0.834257i	$-0.685897\pi$
$521$	−25.1707	−1.10275	−0.551375	+	0.834257i	$0.685897\pi$
$522$	−1.39501	−0.0610578
$523$	−17.9094	−0.783123	−0.391562	−	0.920152i	$-0.628065\pi$
$523$	−17.9094	−0.783123	−0.391562	+	0.920152i	$0.628065\pi$
$524$	−36.1863	−1.58081
$525$	0	0
$526$	1.40604	0.0613061
$527$	11.7280	0.510879
$528$	0	0
$529$	11.1491	0.484744
$530$	0	0
$531$	1.09422	0.0474850
$532$	−31.0200	−1.34489
$533$	53.1433	2.30189
$534$	−2.84286	−0.123023
$535$	0	0
$536$	10.7998	0.466482
$537$	−3.49712	−0.150912
$538$	3.48763	0.150362
$539$	0	0
$540$	0	0
$541$	9.44468	0.406059	0.203029	−	0.979173i	$-0.434921\pi$
$541$	9.44468	0.406059	0.203029	+	0.979173i	$0.434921\pi$
$542$	−3.49712	−0.150214
$543$	−12.0911	−0.518880
$544$	−13.1824	−0.565189
$545$	0	0
$546$	−5.38289	−0.230367
$547$	15.3044	0.654370	0.327185	−	0.944960i	$-0.393900\pi$
$547$	15.3044	0.654370	0.327185	+	0.944960i	$0.393900\pi$
$548$	30.3561	1.29675
$549$	−11.0476	−0.471500
$550$	0	0
$551$	−10.2589	−0.437044
$552$	−7.18450	−0.305792
$553$	9.26912	0.394163
$554$	2.41357	0.102543
$555$	0	0
$556$	−23.7836	−1.00865
$557$	9.94844	0.421529	0.210764	−	0.977537i	$-0.432405\pi$
$557$	9.94844	0.421529	0.210764	+	0.977537i	$0.432405\pi$
$558$	2.21856	0.0939190
$559$	3.10781	0.131447
$560$	0	0
$561$	0	0
$562$	−9.26912	−0.390994
$563$	28.0419	1.18183	0.590914	−	0.806735i	$-0.298767\pi$
$563$	28.0419	1.18183	0.590914	+	0.806735i	$0.298767\pi$
$564$	1.75111	0.0737352
$565$	0	0
$566$	−3.30378	−0.138868
$567$	−4.99870	−0.209926
$568$	−16.0101	−0.671769
$569$	20.3771	0.854251	0.427126	−	0.904192i	$-0.359526\pi$
$569$	20.3771	0.854251	0.427126	+	0.904192i	$0.359526\pi$
$570$	0	0
$571$	−43.6546	−1.82689	−0.913444	−	0.406964i	$-0.866588\pi$
$571$	−43.6546	−1.82689	−0.913444	+	0.406964i	$0.866588\pi$
$572$	0	0
$573$	2.57485	0.107566
$574$	10.9664	0.457731
$575$	0	0
$576$	11.4670	0.477791
$577$	−37.4644	−1.55966	−0.779832	−	0.625989i	$-0.784696\pi$
$577$	−37.4644	−1.55966	−0.779832	+	0.625989i	$0.784696\pi$
$578$	−1.32432	−0.0550846
$579$	0.373479	0.0155213
$580$	0	0
$581$	39.7315	1.64834
$582$	3.56753	0.147879
$583$	0	0
$584$	−10.8825	−0.450322
$585$	0	0
$586$	0.00803813	0.000332052 0
$587$	−6.01916	−0.248437	−0.124219	−	0.992255i	$-0.539642\pi$
$587$	−6.01916	−0.248437	−0.124219	+	0.992255i	$0.539642\pi$
$588$	6.75292	0.278486
$589$	16.3153	0.672259
$590$	0	0
$591$	17.8370	0.733718
$592$	−20.9859	−0.862515
$593$	−14.3574	−0.589589	−0.294794	−	0.955561i	$-0.595251\pi$
$593$	−14.3574	−0.589589	−0.294794	+	0.955561i	$0.595251\pi$
$594$	0	0
$595$	0	0
$596$	26.3542	1.07951
$597$	−9.76345	−0.399591
$598$	9.99169	0.408591
$599$	27.9471	1.14189	0.570943	−	0.820989i	$-0.306578\pi$
$599$	27.9471	1.14189	0.570943	+	0.820989i	$0.306578\pi$
$600$	0	0
$601$	20.5638	0.838815	0.419408	−	0.907798i	$-0.362238\pi$
$601$	20.5638	0.838815	0.419408	+	0.907798i	$0.362238\pi$
$602$	0.641316	0.0261381
$603$	17.5312	0.713925
$604$	30.3383	1.23445
$605$	0	0
$606$	4.15912	0.168953
$607$	−19.1738	−0.778242	−0.389121	−	0.921187i	$-0.627221\pi$
$607$	−19.1738	−0.778242	−0.389121	+	0.921187i	$0.627221\pi$
$608$	−18.3385	−0.743726
$609$	6.44733	0.261259
$610$	0	0
$611$	−5.00941	−0.202659
$612$	−14.1353	−0.571385
$613$	−18.0536	−0.729180	−0.364590	−	0.931168i	$-0.618791\pi$
$613$	−18.0536	−0.729180	−0.364590	+	0.931168i	$0.618791\pi$
$614$	10.8257	0.436888
$615$	0	0
$616$	0	0
$617$	−32.0433	−1.29001	−0.645007	−	0.764176i	$-0.723146\pi$
$617$	−32.0433	−1.29001	−0.645007	+	0.764176i	$0.723146\pi$
$618$	−0.878849	−0.0353525
$619$	−5.30985	−0.213421	−0.106710	−	0.994290i	$-0.534032\pi$
$619$	−5.30985	−0.213421	−0.106710	+	0.994290i	$0.534032\pi$
$620$	0	0
$621$	−28.5274	−1.14477
$622$	−1.58451	−0.0635331
$623$	−29.4533	−1.18002
$624$	16.8554	0.674755
$625$	0	0
$626$	−5.60369	−0.223968
$627$	0	0
$628$	4.30450	0.171768
$629$	22.4591	0.895501
$630$	0	0
$631$	39.3924	1.56819	0.784094	−	0.620642i	$-0.213128\pi$
$631$	39.3924	1.56819	0.784094	+	0.620642i	$0.213128\pi$
$632$	3.61975	0.143986
$633$	8.15152	0.323994
$634$	0.563636	0.0223848
$635$	0	0
$636$	−0.715882	−0.0283866
$637$	−19.3181	−0.765410
$638$	0	0
$639$	−25.9889	−1.02810
$640$	0	0
$641$	−47.2650	−1.86685	−0.933427	−	0.358768i	$-0.883197\pi$
$641$	−47.2650	−1.86685	−0.933427	+	0.358768i	$0.883197\pi$
$642$	5.03825	0.198844
$643$	2.37860	0.0938027	0.0469014	−	0.998900i	$-0.485065\pi$
$643$	2.37860	0.0938027	0.0469014	+	0.998900i	$0.485065\pi$
$644$	−36.1863	−1.42594
$645$	0	0
$646$	5.92298	0.233036
$647$	−0.890278	−0.0350004	−0.0175002	−	0.999847i	$-0.505571\pi$
$647$	−0.890278	−0.0350004	−0.0175002	+	0.999847i	$0.505571\pi$
$648$	−1.95208	−0.0766849
$649$	0	0
$650$	0	0
$651$	−10.2535	−0.401868
$652$	−42.7055	−1.67248
$653$	−13.0531	−0.510809	−0.255404	−	0.966834i	$-0.582209\pi$
$653$	−13.0531	−0.510809	−0.255404	+	0.966834i	$0.582209\pi$
$654$	−1.23478	−0.0482839
$655$	0	0
$656$	−34.3391	−1.34072
$657$	−17.6654	−0.689193
$658$	−1.03372	−0.0402987
$659$	−24.8805	−0.969205	−0.484603	−	0.874734i	$-0.661036\pi$
$659$	−24.8805	−0.969205	−0.484603	+	0.874734i	$0.661036\pi$
$660$	0	0
$661$	−33.1788	−1.29051	−0.645253	−	0.763969i	$-0.723248\pi$
$661$	−33.1788	−1.29051	−0.645253	+	0.763969i	$0.723248\pi$
$662$	0.292950	0.0113858
$663$	−18.0386	−0.700560
$664$	15.5158	0.602131
$665$	0	0
$666$	4.24853	0.164627
$667$	−11.9675	−0.463384
$668$	7.20833	0.278899
$669$	22.8060	0.881731
$670$	0	0
$671$	0	0
$672$	11.5251	0.444590
$673$	−23.8197	−0.918182	−0.459091	−	0.888389i	$-0.651825\pi$
$673$	−23.8197	−0.918182	−0.459091	+	0.888389i	$0.651825\pi$
$674$	−6.16827	−0.237593
$675$	0	0
$676$	−26.7098	−1.02730
$677$	1.87036	0.0718837	0.0359418	−	0.999354i	$-0.488557\pi$
$677$	1.87036	0.0718837	0.0359418	+	0.999354i	$0.488557\pi$
$678$	−3.12006	−0.119825
$679$	36.9611	1.41844
$680$	0	0
$681$	−0.689352	−0.0264160
$682$	0	0
$683$	−9.51629	−0.364131	−0.182065	−	0.983286i	$-0.558278\pi$
$683$	−9.51629	−0.364131	−0.182065	+	0.983286i	$0.558278\pi$
$684$	−19.6642	−0.751878
$685$	0	0
$686$	3.53548	0.134985
$687$	−12.6494	−0.482607
$688$	−2.00814	−0.0765598
$689$	2.04792	0.0780197
$690$	0	0
$691$	−4.56677	−0.173728	−0.0868641	−	0.996220i	$-0.527685\pi$
$691$	−4.56677	−0.173728	−0.0868641	+	0.996220i	$0.527685\pi$
$692$	−1.79692	−0.0683085
$693$	0	0
$694$	−6.88217	−0.261244
$695$	0	0
$696$	2.51780	0.0954368
$697$	36.7496	1.39199
$698$	−8.92675	−0.337883
$699$	25.9092	0.979977
$700$	0	0
$701$	28.7555	1.08608	0.543040	−	0.839707i	$-0.317273\pi$
$701$	28.7555	1.08608	0.543040	+	0.839707i	$0.317273\pi$
$702$	−8.34685	−0.315032
$703$	31.2437	1.17838
$704$	0	0
$705$	0	0
$706$	2.84628	0.107121
$707$	43.0903	1.62058
$708$	−0.960102	−0.0360829
$709$	−36.4141	−1.36756	−0.683780	−	0.729689i	$-0.739665\pi$
$709$	−36.4141	−1.36756	−0.683780	+	0.729689i	$0.739665\pi$
$710$	0	0
$711$	5.87588	0.220363
$712$	−11.5020	−0.431056
$713$	19.0326	0.712775
$714$	−3.72237	−0.139306
$715$	0	0
$716$	−6.87859	−0.257065
$717$	4.81906	0.179971
$718$	−2.26715	−0.0846092
$719$	39.1239	1.45907	0.729537	−	0.683941i	$-0.239735\pi$
$719$	39.1239	1.45907	0.729537	+	0.683941i	$0.239735\pi$
$720$	0	0
$721$	−9.10526	−0.339098
$722$	2.00104	0.0744710
$723$	−4.51519	−0.167922
$724$	−23.7824	−0.883866
$725$	0	0
$726$	0	0
$727$	32.8912	1.21987	0.609933	−	0.792453i	$-0.291196\pi$
$727$	32.8912	1.21987	0.609933	+	0.792453i	$0.291196\pi$
$728$	−21.7788	−0.807176
$729$	10.9199	0.404440
$730$	0	0
$731$	2.14911	0.0794877
$732$	9.69353	0.358283
$733$	11.9256	0.440484	0.220242	−	0.975445i	$-0.429315\pi$
$733$	11.9256	0.440484	0.220242	+	0.975445i	$0.429315\pi$
$734$	5.34884	0.197429
$735$	0	0
$736$	−21.3928	−0.788549
$737$	0	0
$738$	6.95184	0.255901
$739$	−37.2429	−1.37000	−0.685002	−	0.728541i	$-0.740199\pi$
$739$	−37.2429	−1.37000	−0.685002	+	0.728541i	$0.740199\pi$
$740$	0	0
$741$	−25.0942	−0.921859
$742$	0.422602	0.0155142
$743$	13.3453	0.489590	0.244795	−	0.969575i	$-0.421279\pi$
$743$	13.3453	0.489590	0.244795	+	0.969575i	$0.421279\pi$
$744$	−4.00419	−0.146801
$745$	0	0
$746$	−1.21366	−0.0444351
$747$	25.1865	0.921527
$748$	0	0
$749$	52.1984	1.90729
$750$	0	0
$751$	0.795996	0.0290463	0.0145231	−	0.999895i	$-0.495377\pi$
$751$	0.795996	0.0290463	0.0145231	+	0.999895i	$0.495377\pi$
$752$	3.23688	0.118037
$753$	−4.21196	−0.153492
$754$	−3.50157	−0.127520
$755$	0	0
$756$	30.2293	1.09943
$757$	40.6693	1.47815	0.739076	−	0.673622i	$-0.235263\pi$
$757$	40.6693	1.47815	0.739076	+	0.673622i	$0.235263\pi$
$758$	−1.21548	−0.0441483
$759$	0	0
$760$	0	0
$761$	−5.77029	−0.209173	−0.104586	−	0.994516i	$-0.533352\pi$
$761$	−5.77029	−0.209173	−0.104586	+	0.994516i	$0.533352\pi$
$762$	−1.71073	−0.0619733
$763$	−12.7929	−0.463134
$764$	5.06454	0.183229
$765$	0	0
$766$	−10.3006	−0.372177
$767$	2.74657	0.0991728
$768$	−7.74881	−0.279611
$769$	−16.4209	−0.592152	−0.296076	−	0.955164i	$-0.595678\pi$
$769$	−16.4209	−0.592152	−0.296076	+	0.955164i	$0.595678\pi$
$770$	0	0
$771$	−6.26855	−0.225756
$772$	0.734607	0.0264391
$773$	24.5841	0.884227	0.442114	−	0.896959i	$-0.354229\pi$
$773$	24.5841	0.884227	0.442114	+	0.896959i	$0.354229\pi$
$774$	0.406543	0.0146129
$775$	0	0
$776$	14.4340	0.518149
$777$	−19.6355	−0.704420
$778$	−4.53967	−0.162755
$779$	51.1239	1.83170
$780$	0	0
$781$	0	0
$782$	6.90944	0.247081
$783$	9.99740	0.357278
$784$	12.4826	0.445806
$785$	0	0
$786$	−6.04079	−0.215468
$787$	−1.83496	−0.0654091	−0.0327046	−	0.999465i	$-0.510412\pi$
$787$	−1.83496	−0.0654091	−0.0327046	+	0.999465i	$0.510412\pi$
$788$	35.0842	1.24982
$789$	−4.11940	−0.146654
$790$	0	0
$791$	−32.3252	−1.14935
$792$	0	0
$793$	−27.7303	−0.984732
$794$	−3.19621	−0.113429
$795$	0	0
$796$	−19.2040	−0.680668
$797$	42.0057	1.48792	0.743959	−	0.668226i	$-0.232946\pi$
$797$	42.0057	1.48792	0.743959	+	0.668226i	$0.232946\pi$
$798$	−5.17834	−0.183311
$799$	−3.46410	−0.122551
$800$	0	0
$801$	−18.6710	−0.659707
$802$	−8.59517	−0.303506
$803$	0	0
$804$	−15.3824	−0.542496
$805$	0	0
$806$	5.56874	0.196151
$807$	−10.2180	−0.359692
$808$	16.8275	0.591990
$809$	−27.0811	−0.952120	−0.476060	−	0.879413i	$-0.657935\pi$
$809$	−27.0811	−0.952120	−0.476060	+	0.879413i	$0.657935\pi$
$810$	0	0
$811$	1.02878	0.0361252	0.0180626	−	0.999837i	$-0.494250\pi$
$811$	1.02878	0.0361252	0.0180626	+	0.999837i	$0.494250\pi$
$812$	12.6814	0.445031
$813$	10.2459	0.359338
$814$	0	0
$815$	0	0
$816$	11.6558	0.408035
$817$	2.98972	0.104597
$818$	0.868689	0.0303730
$819$	−35.3531	−1.23534
$820$	0	0
$821$	−11.5065	−0.401581	−0.200790	−	0.979634i	$-0.564351\pi$
$821$	−11.5065	−0.401581	−0.200790	+	0.979634i	$0.564351\pi$
$822$	5.06751	0.176750
$823$	−28.7640	−1.00265	−0.501325	−	0.865259i	$-0.667154\pi$
$823$	−28.7640	−1.00265	−0.501325	+	0.865259i	$0.667154\pi$
$824$	−3.55576	−0.123871
$825$	0	0
$826$	0.566771	0.0197205
$827$	54.0494	1.87948	0.939742	−	0.341885i	$-0.111065\pi$
$827$	54.0494	1.87948	0.939742	+	0.341885i	$0.111065\pi$
$828$	−22.9392	−0.797193
$829$	4.64886	0.161461	0.0807307	−	0.996736i	$-0.474275\pi$
$829$	4.64886	0.161461	0.0807307	+	0.996736i	$0.474275\pi$
$830$	0	0
$831$	−7.07126	−0.245299
$832$	28.7830	0.997871
$833$	−13.3588	−0.462855
$834$	−3.97033	−0.137481
$835$	0	0
$836$	0	0
$837$	−15.8994	−0.549564
$838$	7.98280	0.275761
$839$	4.94511	0.170724	0.0853620	−	0.996350i	$-0.472795\pi$
$839$	4.94511	0.170724	0.0853620	+	0.996350i	$0.472795\pi$
$840$	0	0
$841$	−24.8060	−0.855380
$842$	−9.15842	−0.315620
$843$	27.1566	0.935324
$844$	16.0335	0.551895
$845$	0	0
$846$	−0.655297	−0.0225296
$847$	0	0
$848$	−1.32329	−0.0454418
$849$	9.67940	0.332196
$850$	0	0
$851$	36.4473	1.24940
$852$	22.8035	0.781235
$853$	29.3155	1.00374	0.501872	−	0.864942i	$-0.332645\pi$
$853$	29.3155	1.00374	0.501872	+	0.864942i	$0.332645\pi$
$854$	−5.72232	−0.195814
$855$	0	0
$856$	20.3844	0.696724
$857$	−0.656701	−0.0224325	−0.0112162	−	0.999937i	$-0.503570\pi$
$857$	−0.656701	−0.0224325	−0.0112162	+	0.999937i	$0.503570\pi$
$858$	0	0
$859$	−40.0569	−1.36672	−0.683361	−	0.730080i	$-0.739483\pi$
$859$	−40.0569	−1.36672	−0.683361	+	0.730080i	$0.739483\pi$
$860$	0	0
$861$	−32.1294	−1.09497
$862$	6.57938	0.224094
$863$	−19.9306	−0.678445	−0.339222	−	0.940706i	$-0.610164\pi$
$863$	−19.9306	−0.678445	−0.339222	+	0.940706i	$0.610164\pi$
$864$	17.8711	0.607987
$865$	0	0
$866$	−1.43973	−0.0489239
$867$	3.88000	0.131772
$868$	−20.1680	−0.684546
$869$	0	0
$870$	0	0
$871$	44.0045	1.49104
$872$	−4.99585	−0.169181
$873$	23.4304	0.792998
$874$	9.61201	0.325131
$875$	0	0
$876$	15.5002	0.523704
$877$	−4.79011	−0.161751	−0.0808753	−	0.996724i	$-0.525772\pi$
$877$	−4.79011	−0.161751	−0.0808753	+	0.996724i	$0.525772\pi$
$878$	−11.2518	−0.379729
$879$	−0.0235501	−0.000794325 0
$880$	0	0
$881$	33.9748	1.14464	0.572320	−	0.820031i	$-0.306044\pi$
$881$	33.9748	1.14464	0.572320	+	0.820031i	$0.306044\pi$
$882$	−2.52706	−0.0850905
$883$	39.2743	1.32169	0.660843	−	0.750524i	$-0.270199\pi$
$883$	39.2743	1.32169	0.660843	+	0.750524i	$0.270199\pi$
$884$	−35.4806	−1.19334
$885$	0	0
$886$	−6.07224	−0.204001
$887$	33.2519	1.11649	0.558244	−	0.829677i	$-0.311475\pi$
$887$	33.2519	1.11649	0.558244	+	0.829677i	$0.311475\pi$
$888$	−7.66801	−0.257322
$889$	−17.7239	−0.594441
$890$	0	0
$891$	0	0
$892$	44.8578	1.50195
$893$	−4.81906	−0.161264
$894$	4.39946	0.147140
$895$	0	0
$896$	29.9002	0.998896
$897$	−29.2736	−0.977418
$898$	6.16340	0.205675
$899$	−6.66994	−0.222455
$900$	0	0
$901$	1.41618	0.0471797
$902$	0	0
$903$	−1.87893	−0.0625267
$904$	−12.6236	−0.419853
$905$	0	0
$906$	5.06454	0.168258
$907$	18.3497	0.609293	0.304646	−	0.952466i	$-0.401462\pi$
$907$	18.3497	0.609293	0.304646	+	0.952466i	$0.401462\pi$
$908$	−1.35591	−0.0449974
$909$	27.3158	0.906007
$910$	0	0
$911$	38.8121	1.28590	0.642951	−	0.765908i	$-0.277710\pi$
$911$	38.8121	1.28590	0.642951	+	0.765908i	$0.277710\pi$
$912$	16.2149	0.536928
$913$	0	0
$914$	4.76076	0.157472
$915$	0	0
$916$	−24.8806	−0.822077
$917$	−62.5852	−2.06675
$918$	−5.77200	−0.190504
$919$	−0.981676	−0.0323825	−0.0161912	−	0.999869i	$-0.505154\pi$
$919$	−0.981676	−0.0323825	−0.0161912	+	0.999869i	$0.505154\pi$
$920$	0	0
$921$	−31.7170	−1.04511
$922$	−4.99585	−0.164529
$923$	−65.2340	−2.14720
$924$	0	0
$925$	0	0
$926$	2.98162	0.0979822
$927$	−5.77200	−0.189577
$928$	7.49708	0.246104
$929$	35.0100	1.14864	0.574321	−	0.818630i	$-0.305266\pi$
$929$	35.0100	1.14864	0.574321	+	0.818630i	$0.305266\pi$
$930$	0	0
$931$	−18.5840	−0.609066
$932$	50.9616	1.66930
$933$	4.64229	0.151982
$934$	5.67522	0.185699
$935$	0	0
$936$	−13.8060	−0.451263
$937$	30.8555	1.00801	0.504003	−	0.863702i	$-0.331860\pi$
$937$	30.8555	1.00801	0.504003	+	0.863702i	$0.331860\pi$
$938$	9.08060	0.296492
$939$	16.4177	0.535770
$940$	0	0
$941$	−30.0766	−0.980469	−0.490235	−	0.871590i	$-0.663089\pi$
$941$	−30.0766	−0.980469	−0.490235	+	0.871590i	$0.663089\pi$
$942$	0.718574	0.0234124
$943$	59.6385	1.94210
$944$	−1.77472	−0.0577622
$945$	0	0
$946$	0	0
$947$	−10.1860	−0.331002	−0.165501	−	0.986210i	$-0.552924\pi$
$947$	−10.1860	−0.331002	−0.165501	+	0.986210i	$0.552924\pi$
$948$	−5.15569	−0.167449
$949$	−44.3415	−1.43939
$950$	0	0
$951$	−1.65134	−0.0535483
$952$	−15.0604	−0.488112
$953$	40.1472	1.30050	0.650248	−	0.759722i	$-0.274665\pi$
$953$	40.1472	1.30050	0.650248	+	0.759722i	$0.274665\pi$
$954$	0.267895	0.00867343
$955$	0	0
$956$	9.47874	0.306564
$957$	0	0
$958$	−10.7143	−0.346162
$959$	52.5016	1.69537
$960$	0	0
$961$	−20.3924	−0.657821
$962$	10.6641	0.343825
$963$	33.0896	1.06630
$964$	−8.88105	−0.286039
$965$	0	0
$966$	−6.04079	−0.194359
$967$	−58.0856	−1.86791	−0.933953	−	0.357395i	$-0.883665\pi$
$967$	−58.0856	−1.86791	−0.933953	+	0.357395i	$0.883665\pi$
$968$	0	0
$969$	−17.3531	−0.557462
$970$	0	0
$971$	32.8121	1.05299	0.526495	−	0.850178i	$-0.323506\pi$
$971$	32.8121	1.05299	0.526495	+	0.850178i	$0.323506\pi$
$972$	30.4918	0.978024
$973$	−41.1343	−1.31871
$974$	5.00941	0.160512
$975$	0	0
$976$	17.9182	0.573548
$977$	17.9245	0.573454	0.286727	−	0.958012i	$-0.407433\pi$
$977$	17.9245	0.573454	0.286727	+	0.958012i	$0.407433\pi$
$978$	−7.12908	−0.227963
$979$	0	0
$980$	0	0
$981$	−8.10966	−0.258922
$982$	−2.19008	−0.0698882
$983$	17.0766	0.544658	0.272329	−	0.962204i	$-0.412206\pi$
$983$	17.0766	0.544658	0.272329	+	0.962204i	$0.412206\pi$
$984$	−12.5471	−0.399987
$985$	0	0
$986$	−2.42140	−0.0771132
$987$	3.02860	0.0964013
$988$	−49.3585	−1.57030
$989$	3.48765	0.110901
$990$	0	0
$991$	17.5551	0.557658	0.278829	−	0.960341i	$-0.410054\pi$
$991$	17.5551	0.557658	0.278829	+	0.960341i	$0.410054\pi$
$992$	−11.9230	−0.378556
$993$	−0.858283	−0.0272368
$994$	−13.4614	−0.426971
$995$	0	0
$996$	−22.0995	−0.700249
$997$	15.2047	0.481537	0.240769	−	0.970583i	$-0.422600\pi$
$997$	15.2047	0.481537	0.240769	+	0.970583i	$0.422600\pi$
$998$	−2.01906	−0.0639124
$999$	−30.4473	−0.963311

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.bo.1.7		12
5.2	odd	4		605.2.b.h.364.8	yes	12
5.3	odd	4		605.2.b.h.364.5	✓	12
5.4	even	2	inner	3025.2.a.bo.1.6		12
11.10	odd	2	inner	3025.2.a.bo.1.5		12
55.2	even	20		605.2.j.k.444.6		48
55.3	odd	20		605.2.j.k.9.6		48
55.7	even	20		605.2.j.k.269.8		48
55.8	even	20		605.2.j.k.9.8		48
55.13	even	20		605.2.j.k.444.7		48
55.17	even	20		605.2.j.k.124.7		48
55.18	even	20		605.2.j.k.269.5		48
55.27	odd	20		605.2.j.k.124.5		48
55.28	even	20		605.2.j.k.124.6		48
55.32	even	4		605.2.b.h.364.6	yes	12
55.37	odd	20		605.2.j.k.269.6		48
55.38	odd	20		605.2.j.k.124.8		48
55.42	odd	20		605.2.j.k.444.8		48
55.43	even	4		605.2.b.h.364.7	yes	12
55.47	odd	20		605.2.j.k.9.7		48
55.48	odd	20		605.2.j.k.269.7		48
55.52	even	20		605.2.j.k.9.5		48
55.53	odd	20		605.2.j.k.444.5		48
55.54	odd	2	inner	3025.2.a.bo.1.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.b.h.364.5	✓	12	5.3	odd	4
605.2.b.h.364.6	yes	12	55.32	even	4
605.2.b.h.364.7	yes	12	55.43	even	4
605.2.b.h.364.8	yes	12	5.2	odd	4
605.2.j.k.9.5		48	55.52	even	20
605.2.j.k.9.6		48	55.3	odd	20
605.2.j.k.9.7		48	55.47	odd	20
605.2.j.k.9.8		48	55.8	even	20
605.2.j.k.124.5		48	55.27	odd	20
605.2.j.k.124.6		48	55.28	even	20
605.2.j.k.124.7		48	55.17	even	20
605.2.j.k.124.8		48	55.38	odd	20
605.2.j.k.269.5		48	55.18	even	20
605.2.j.k.269.6		48	55.37	odd	20
605.2.j.k.269.7		48	55.48	odd	20
605.2.j.k.269.8		48	55.7	even	20
605.2.j.k.444.5		48	55.53	odd	20
605.2.j.k.444.6		48	55.2	even	20
605.2.j.k.444.7		48	55.13	even	20
605.2.j.k.444.8		48	55.42	odd	20
3025.2.a.bo.1.5		12	11.10	odd	2	inner
3025.2.a.bo.1.6		12	5.4	even	2	inner
3025.2.a.bo.1.7		12	1.1	even	1	trivial
3025.2.a.bo.1.8		12	55.54	odd	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:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

\(f(q)\)	\(=\)	\(q+0.328351 q^{2} -0.962000 q^{3} -1.89219 q^{4} -0.315873 q^{6} -3.27259 q^{7} -1.27800 q^{8} -2.07456 q^{9} +O(q^{10})\)	\(q+0.328351 q^{2} -0.962000 q^{3} -1.89219 q^{4} -0.315873 q^{6} -3.27259 q^{7} -1.27800 q^{8} -2.07456 q^{9} +1.82028 q^{12} -5.20729 q^{13} -1.07456 q^{14} +3.36474 q^{16} -3.60094 q^{17} -0.681182 q^{18} -5.00941 q^{19} +3.14823 q^{21} -5.84372 q^{23} +1.22944 q^{24} -1.70982 q^{26} +4.88172 q^{27} +6.19234 q^{28} +2.04792 q^{29} -3.25693 q^{31} +3.66082 q^{32} -1.18237 q^{34} +3.92544 q^{36} -6.23700 q^{37} -1.64484 q^{38} +5.00941 q^{39} -10.2056 q^{41} +1.03372 q^{42} -0.596820 q^{43} -1.91879 q^{46} +0.962000 q^{47} -3.23688 q^{48} +3.70982 q^{49} +3.46410 q^{51} +9.85316 q^{52} -0.393280 q^{53} +1.60292 q^{54} +4.18237 q^{56} +4.81906 q^{57} +0.672437 q^{58} -0.527447 q^{59} +5.32529 q^{61} -1.06941 q^{62} +6.78916 q^{63} -5.52745 q^{64} -8.45057 q^{67} +6.81364 q^{68} +5.62166 q^{69} +12.5274 q^{71} +2.65129 q^{72} +8.51527 q^{73} -2.04792 q^{74} +9.47874 q^{76} +1.64484 q^{78} -2.83235 q^{79} +1.52745 q^{81} -3.35100 q^{82} -12.1407 q^{83} -5.95703 q^{84} -0.195966 q^{86} -1.97010 q^{87} +9.00000 q^{89} +17.0413 q^{91} +11.0574 q^{92} +3.13316 q^{93} +0.315873 q^{94} -3.52171 q^{96} -11.2942 q^{97} +1.21812 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 20 q^{4} + 20 q^{9}+O(q^{10}) \) 12 * q + 20 * q^4 + 20 * q^9	\( 12 q + 20 q^{4} + 20 q^{9} + 32 q^{14} + 36 q^{16} + 20 q^{26} + 8 q^{31} - 12 q^{34} + 92 q^{36} + 4 q^{49} + 48 q^{56} + 32 q^{59} - 28 q^{64} + 16 q^{69} + 112 q^{71} - 20 q^{81} - 56 q^{86} + 108 q^{89} + 72 q^{91}+O(q^{100}) \) 12 * q + 20 * q^4 + 20 * q^9 + 32 * q^14 + 36 * q^16 + 20 * q^26 + 8 * q^31 - 12 * q^34 + 92 * q^36 + 4 * q^49 + 48 * q^56 + 32 * q^59 - 28 * q^64 + 16 * q^69 + 112 * q^71 - 20 * q^81 - 56 * q^86 + 108 * q^89 + 72 * q^91

Introduction

L-functions

Modular forms

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