LMFDB - Embedded newform 7500.2.a.n.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1,7500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7500, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7500.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.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:	$59.8878015160$
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} - 6 x^{11} - 11 x^{10} + 94 x^{9} + 27 x^{8} - 460 x^{7} + 55 x^{6} + 812 x^{5} - 127 x^{4} + \cdots - 4 $ x^12 - 6x^11 - 11x^10 + 94x^9 + 27x^8 - 460x^7 + 55x^6 + 812x^5 - 127x^4 - 512x^3 + 40x^2 + 72*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5^{3} $
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.09160$ of defining polynomial
Character	$\chi$	$=$	7500.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.44380 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.44380 q^{7} +1.00000 q^{9} -0.576983 q^{11} +6.44645 q^{13} +1.89772 q^{17} +8.27470 q^{19} +2.44380 q^{21} +4.20898 q^{23} +1.00000 q^{27} +6.53865 q^{29} +4.86687 q^{31} -0.576983 q^{33} -0.218005 q^{37} +6.44645 q^{39} +6.45452 q^{41} -3.42419 q^{43} -9.61833 q^{47} -1.02783 q^{49} +1.89772 q^{51} -13.9922 q^{53} +8.27470 q^{57} -12.0530 q^{59} -4.81065 q^{61} +2.44380 q^{63} +3.87094 q^{67} +4.20898 q^{69} -6.48247 q^{71} -10.0509 q^{73} -1.41003 q^{77} -4.74470 q^{79} +1.00000 q^{81} +3.43418 q^{83} +6.53865 q^{87} -5.68638 q^{89} +15.7538 q^{91} +4.86687 q^{93} +6.58475 q^{97} -0.576983 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^3 + 8 * q^7 + 12 * q^9	$ 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9} + 2 q^{11} + 8 q^{17} + 10 q^{19} + 8 q^{21} + 18 q^{23} + 12 q^{27} + 8 q^{29} - 2 q^{31} + 2 q^{33} + 4 q^{37} + 10 q^{41} + 28 q^{43} + 22 q^{47} + 28 q^{49} + 8 q^{51} + 16 q^{53} + 10 q^{57} - 2 q^{59} + 34 q^{61} + 8 q^{63} + 32 q^{67} + 18 q^{69} + 24 q^{73} + 18 q^{77} + 6 q^{79} + 12 q^{81} + 28 q^{83} + 8 q^{87} + 10 q^{89} + 20 q^{91} - 2 q^{93} + 16 q^{97} + 2 q^{99}+O(q^{100}) $ 12 * q + 12 * q^3 + 8 * q^7 + 12 * q^9 + 2 * q^11 + 8 * q^17 + 10 * q^19 + 8 * q^21 + 18 * q^23 + 12 * q^27 + 8 * q^29 - 2 * q^31 + 2 * q^33 + 4 * q^37 + 10 * q^41 + 28 * q^43 + 22 * q^47 + 28 * q^49 + 8 * q^51 + 16 * q^53 + 10 * q^57 - 2 * q^59 + 34 * q^61 + 8 * q^63 + 32 * q^67 + 18 * q^69 + 24 * q^73 + 18 * q^77 + 6 * q^79 + 12 * q^81 + 28 * q^83 + 8 * q^87 + 10 * q^89 + 20 * q^91 - 2 * q^93 + 16 * q^97 + 2 * q^99

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	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.44380	0.923670	0.461835	−	0.886966i	$-0.347191\pi$
$7$	2.44380	0.923670	0.461835	+	0.886966i	$0.347191\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.576983	−0.173967	−0.0869835	−	0.996210i	$-0.527723\pi$
$11$	−0.576983	−0.173967	−0.0869835	+	0.996210i	$0.527723\pi$
$12$	0	0
$13$	6.44645	1.78792	0.893961	−	0.448145i	$-0.147915\pi$
$13$	6.44645	1.78792	0.893961	+	0.448145i	$0.147915\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.89772	0.460265	0.230133	−	0.973159i	$-0.426084\pi$
$17$	1.89772	0.460265	0.230133	+	0.973159i	$0.426084\pi$
$18$	0	0
$19$	8.27470	1.89835	0.949174	−	0.314753i	$-0.101922\pi$
$19$	8.27470	1.89835	0.949174	+	0.314753i	$0.101922\pi$
$20$	0	0
$21$	2.44380	0.533281
$22$	0	0
$23$	4.20898	0.877633	0.438816	−	0.898577i	$-0.355398\pi$
$23$	4.20898	0.877633	0.438816	+	0.898577i	$0.355398\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.53865	1.21420	0.607098	−	0.794627i	$-0.292333\pi$
$29$	6.53865	1.21420	0.607098	+	0.794627i	$0.292333\pi$
$30$	0	0
$31$	4.86687	0.874117	0.437058	−	0.899433i	$-0.356020\pi$
$31$	4.86687	0.874117	0.437058	+	0.899433i	$0.356020\pi$
$32$	0	0
$33$	−0.576983	−0.100440
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.218005	−0.0358398	−0.0179199	−	0.999839i	$-0.505704\pi$
$37$	−0.218005	−0.0358398	−0.0179199	+	0.999839i	$0.505704\pi$
$38$	0	0
$39$	6.44645	1.03226
$40$	0	0
$41$	6.45452	1.00803	0.504014	−	0.863696i	$-0.331856\pi$
$41$	6.45452	1.00803	0.504014	+	0.863696i	$0.331856\pi$
$42$	0	0
$43$	−3.42419	−0.522184	−0.261092	−	0.965314i	$-0.584083\pi$
$43$	−3.42419	−0.522184	−0.261092	+	0.965314i	$0.584083\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.61833	−1.40298	−0.701489	−	0.712680i	$-0.747481\pi$
$47$	−9.61833	−1.40298	−0.701489	+	0.712680i	$0.747481\pi$
$48$	0	0
$49$	−1.02783	−0.146833
$50$	0	0
$51$	1.89772	0.265734
$52$	0	0
$53$	−13.9922	−1.92198	−0.960988	−	0.276589i	$-0.910796\pi$
$53$	−13.9922	−1.92198	−0.960988	+	0.276589i	$0.910796\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	8.27470	1.09601
$58$	0	0
$59$	−12.0530	−1.56917	−0.784586	−	0.620020i	$-0.787124\pi$
$59$	−12.0530	−1.56917	−0.784586	+	0.620020i	$0.787124\pi$
$60$	0	0
$61$	−4.81065	−0.615941	−0.307970	−	0.951396i	$-0.599650\pi$
$61$	−4.81065	−0.615941	−0.307970	+	0.951396i	$0.599650\pi$
$62$	0	0
$63$	2.44380	0.307890
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.87094	0.472910	0.236455	−	0.971642i	$-0.424014\pi$
$67$	3.87094	0.472910	0.236455	+	0.971642i	$0.424014\pi$
$68$	0	0
$69$	4.20898	0.506701
$70$	0	0
$71$	−6.48247	−0.769328	−0.384664	−	0.923057i	$-0.625683\pi$
$71$	−6.48247	−0.769328	−0.384664	+	0.923057i	$0.625683\pi$
$72$	0	0
$73$	−10.0509	−1.17637	−0.588184	−	0.808727i	$-0.700157\pi$
$73$	−10.0509	−1.17637	−0.588184	+	0.808727i	$0.700157\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.41003	−0.160688
$78$	0	0
$79$	−4.74470	−0.533821	−0.266910	−	0.963721i	$-0.586003\pi$
$79$	−4.74470	−0.533821	−0.266910	+	0.963721i	$0.586003\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.43418	0.376950	0.188475	−	0.982078i	$-0.439646\pi$
$83$	3.43418	0.376950	0.188475	+	0.982078i	$0.439646\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.53865	0.701017
$88$	0	0
$89$	−5.68638	−0.602755	−0.301378	−	0.953505i	$-0.597446\pi$
$89$	−5.68638	−0.602755	−0.301378	+	0.953505i	$0.597446\pi$
$90$	0	0
$91$	15.7538	1.65145
$92$	0	0
$93$	4.86687	0.504671
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.58475	0.668580	0.334290	−	0.942470i	$-0.391503\pi$
$97$	6.58475	0.668580	0.334290	+	0.942470i	$0.391503\pi$
$98$	0	0
$99$	−0.576983	−0.0579890
$100$	0	0
$101$	−12.0363	−1.19766	−0.598828	−	0.800877i	$-0.704367\pi$
$101$	−12.0363	−1.19766	−0.598828	+	0.800877i	$0.704367\pi$
$102$	0	0
$103$	−5.78624	−0.570135	−0.285068	−	0.958507i	$-0.592016\pi$
$103$	−5.78624	−0.570135	−0.285068	+	0.958507i	$0.592016\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.07081	0.393540	0.196770	−	0.980450i	$-0.436955\pi$
$107$	4.07081	0.393540	0.196770	+	0.980450i	$0.436955\pi$
$108$	0	0
$109$	−1.45776	−0.139628	−0.0698139	−	0.997560i	$-0.522241\pi$
$109$	−1.45776	−0.139628	−0.0698139	+	0.997560i	$0.522241\pi$
$110$	0	0
$111$	−0.218005	−0.0206921
$112$	0	0
$113$	−5.85785	−0.551060	−0.275530	−	0.961292i	$-0.588853\pi$
$113$	−5.85785	−0.551060	−0.275530	+	0.961292i	$0.588853\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.44645	0.595974
$118$	0	0
$119$	4.63766	0.425133
$120$	0	0
$121$	−10.6671	−0.969735
$122$	0	0
$123$	6.45452	0.581985
$124$	0	0
$125$	0	0
$126$	0	0
$127$	15.2854	1.35636	0.678182	−	0.734894i	$-0.262768\pi$
$127$	15.2854	1.35636	0.678182	+	0.734894i	$0.262768\pi$
$128$	0	0
$129$	−3.42419	−0.301483
$130$	0	0
$131$	−3.75392	−0.327981	−0.163991	−	0.986462i	$-0.552437\pi$
$131$	−3.75392	−0.327981	−0.163991	+	0.986462i	$0.552437\pi$
$132$	0	0
$133$	20.2217	1.75345
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.2511	1.73017	0.865084	−	0.501628i	$-0.167265\pi$
$137$	20.2511	1.73017	0.865084	+	0.501628i	$0.167265\pi$
$138$	0	0
$139$	−0.928868	−0.0787855	−0.0393928	−	0.999224i	$-0.512542\pi$
$139$	−0.928868	−0.0787855	−0.0393928	+	0.999224i	$0.512542\pi$
$140$	0	0
$141$	−9.61833	−0.810010
$142$	0	0
$143$	−3.71949	−0.311040
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.02783	−0.0847740
$148$	0	0
$149$	13.1432	1.07673	0.538364	−	0.842712i	$-0.319042\pi$
$149$	13.1432	1.07673	0.538364	+	0.842712i	$0.319042\pi$
$150$	0	0
$151$	17.5864	1.43116	0.715580	−	0.698531i	$-0.246163\pi$
$151$	17.5864	1.43116	0.715580	+	0.698531i	$0.246163\pi$
$152$	0	0
$153$	1.89772	0.153422
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.68198	0.692897	0.346449	−	0.938069i	$-0.387387\pi$
$157$	8.68198	0.692897	0.346449	+	0.938069i	$0.387387\pi$
$158$	0	0
$159$	−13.9922	−1.10965
$160$	0	0
$161$	10.2859	0.810643
$162$	0	0
$163$	−1.40791	−0.110276	−0.0551380	−	0.998479i	$-0.517560\pi$
$163$	−1.40791	−0.110276	−0.0551380	+	0.998479i	$0.517560\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.81475	−0.682106	−0.341053	−	0.940044i	$-0.610784\pi$
$167$	−8.81475	−0.682106	−0.341053	+	0.940044i	$0.610784\pi$
$168$	0	0
$169$	28.5567	2.19667
$170$	0	0
$171$	8.27470	0.632782
$172$	0	0
$173$	−7.90310	−0.600862	−0.300431	−	0.953804i	$-0.597130\pi$
$173$	−7.90310	−0.600862	−0.300431	+	0.953804i	$0.597130\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.0530	−0.905962
$178$	0	0
$179$	−11.4886	−0.858701	−0.429351	−	0.903138i	$-0.641258\pi$
$179$	−11.4886	−0.858701	−0.429351	+	0.903138i	$0.641258\pi$
$180$	0	0
$181$	3.92307	0.291600	0.145800	−	0.989314i	$-0.453424\pi$
$181$	3.92307	0.291600	0.145800	+	0.989314i	$0.453424\pi$
$182$	0	0
$183$	−4.81065	−0.355614
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.09495	−0.0800709
$188$	0	0
$189$	2.44380	0.177760
$190$	0	0
$191$	−25.5456	−1.84841	−0.924207	−	0.381893i	$-0.875272\pi$
$191$	−25.5456	−1.84841	−0.924207	+	0.381893i	$0.875272\pi$
$192$	0	0
$193$	0.421651	0.0303511	0.0151756	−	0.999885i	$-0.495169\pi$
$193$	0.421651	0.0303511	0.0151756	+	0.999885i	$0.495169\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.22140	0.514504	0.257252	−	0.966344i	$-0.417183\pi$
$197$	7.22140	0.514504	0.257252	+	0.966344i	$0.417183\pi$
$198$	0	0
$199$	3.93505	0.278949	0.139474	−	0.990226i	$-0.455459\pi$
$199$	3.93505	0.278949	0.139474	+	0.990226i	$0.455459\pi$
$200$	0	0
$201$	3.87094	0.273035
$202$	0	0
$203$	15.9792	1.12152
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.20898	0.292544
$208$	0	0
$209$	−4.77437	−0.330250
$210$	0	0
$211$	−24.7610	−1.70462	−0.852308	−	0.523041i	$-0.824798\pi$
$211$	−24.7610	−1.70462	−0.852308	+	0.523041i	$0.824798\pi$
$212$	0	0
$213$	−6.48247	−0.444172
$214$	0	0
$215$	0	0
$216$	0	0
$217$	11.8937	0.807396
$218$	0	0
$219$	−10.0509	−0.679176
$220$	0	0
$221$	12.2336	0.822918
$222$	0	0
$223$	19.6727	1.31738	0.658690	−	0.752415i	$-0.271111\pi$
$223$	19.6727	1.31738	0.658690	+	0.752415i	$0.271111\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.3748	1.75055	0.875277	−	0.483622i	$-0.160679\pi$
$227$	26.3748	1.75055	0.875277	+	0.483622i	$0.160679\pi$
$228$	0	0
$229$	−16.0212	−1.05871	−0.529355	−	0.848400i	$-0.677566\pi$
$229$	−16.0212	−1.05871	−0.529355	+	0.848400i	$0.677566\pi$
$230$	0	0
$231$	−1.41003	−0.0927734
$232$	0	0
$233$	−29.3108	−1.92021	−0.960107	−	0.279632i	$-0.909787\pi$
$233$	−29.3108	−1.92021	−0.960107	+	0.279632i	$0.909787\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.74470	−0.308201
$238$	0	0
$239$	25.8558	1.67247	0.836237	−	0.548368i	$-0.184750\pi$
$239$	25.8558	1.67247	0.836237	+	0.548368i	$0.184750\pi$
$240$	0	0
$241$	−9.34158	−0.601744	−0.300872	−	0.953665i	$-0.597278\pi$
$241$	−9.34158	−0.601744	−0.300872	+	0.953665i	$0.597278\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	53.3424	3.39410
$248$	0	0
$249$	3.43418	0.217632
$250$	0	0
$251$	−13.5088	−0.852666	−0.426333	−	0.904566i	$-0.640195\pi$
$251$	−13.5088	−0.852666	−0.426333	+	0.904566i	$0.640195\pi$
$252$	0	0
$253$	−2.42851	−0.152679
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.0662	−0.815044	−0.407522	−	0.913195i	$-0.633607\pi$
$257$	−13.0662	−0.815044	−0.407522	+	0.913195i	$0.633607\pi$
$258$	0	0
$259$	−0.532761	−0.0331042
$260$	0	0
$261$	6.53865	0.404732
$262$	0	0
$263$	4.43065	0.273206	0.136603	−	0.990626i	$-0.456382\pi$
$263$	4.43065	0.273206	0.136603	+	0.990626i	$0.456382\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−5.68638	−0.348001
$268$	0	0
$269$	28.0509	1.71029	0.855146	−	0.518387i	$-0.173467\pi$
$269$	28.0509	1.71029	0.855146	+	0.518387i	$0.173467\pi$
$270$	0	0
$271$	5.98512	0.363570	0.181785	−	0.983338i	$-0.441812\pi$
$271$	5.98512	0.363570	0.181785	+	0.983338i	$0.441812\pi$
$272$	0	0
$273$	15.7538	0.953466
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5.07072	−0.304670	−0.152335	−	0.988329i	$-0.548679\pi$
$277$	−5.07072	−0.304670	−0.152335	+	0.988329i	$0.548679\pi$
$278$	0	0
$279$	4.86687	0.291372
$280$	0	0
$281$	−18.1559	−1.08309	−0.541546	−	0.840671i	$-0.682161\pi$
$281$	−18.1559	−1.08309	−0.541546	+	0.840671i	$0.682161\pi$
$282$	0	0
$283$	4.69717	0.279218	0.139609	−	0.990207i	$-0.455416\pi$
$283$	4.69717	0.279218	0.139609	+	0.990207i	$0.455416\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.7736	0.931085
$288$	0	0
$289$	−13.3987	−0.788156
$290$	0	0
$291$	6.58475	0.386005
$292$	0	0
$293$	−9.60771	−0.561288	−0.280644	−	0.959812i	$-0.590548\pi$
$293$	−9.60771	−0.561288	−0.280644	+	0.959812i	$0.590548\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−0.576983	−0.0334800
$298$	0	0
$299$	27.1329	1.56914
$300$	0	0
$301$	−8.36804	−0.482326
$302$	0	0
$303$	−12.0363	−0.691467
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−13.5400	−0.772771	−0.386386	−	0.922337i	$-0.626277\pi$
$307$	−13.5400	−0.772771	−0.386386	+	0.922337i	$0.626277\pi$
$308$	0	0
$309$	−5.78624	−0.329168
$310$	0	0
$311$	−2.96725	−0.168257	−0.0841286	−	0.996455i	$-0.526811\pi$
$311$	−2.96725	−0.168257	−0.0841286	+	0.996455i	$0.526811\pi$
$312$	0	0
$313$	14.0058	0.791652	0.395826	−	0.918326i	$-0.370458\pi$
$313$	14.0058	0.791652	0.395826	+	0.918326i	$0.370458\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.0176	1.51746	0.758730	−	0.651405i	$-0.225820\pi$
$317$	27.0176	1.51746	0.758730	+	0.651405i	$0.225820\pi$
$318$	0	0
$319$	−3.77269	−0.211230
$320$	0	0
$321$	4.07081	0.227211
$322$	0	0
$323$	15.7031	0.873743
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.45776	−0.0806141
$328$	0	0
$329$	−23.5053	−1.29589
$330$	0	0
$331$	−6.65492	−0.365787	−0.182894	−	0.983133i	$-0.558546\pi$
$331$	−6.65492	−0.365787	−0.182894	+	0.983133i	$0.558546\pi$
$332$	0	0
$333$	−0.218005	−0.0119466
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−19.2028	−1.04604	−0.523021	−	0.852320i	$-0.675195\pi$
$337$	−19.2028	−1.04604	−0.523021	+	0.852320i	$0.675195\pi$
$338$	0	0
$339$	−5.85785	−0.318155
$340$	0	0
$341$	−2.80811	−0.152067
$342$	0	0
$343$	−19.6184	−1.05930
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.7591	0.899676	0.449838	−	0.893110i	$-0.351482\pi$
$347$	16.7591	0.899676	0.449838	+	0.893110i	$0.351482\pi$
$348$	0	0
$349$	22.9371	1.22780	0.613898	−	0.789385i	$-0.289601\pi$
$349$	22.9371	1.22780	0.613898	+	0.789385i	$0.289601\pi$
$350$	0	0
$351$	6.44645	0.344086
$352$	0	0
$353$	7.01413	0.373324	0.186662	−	0.982424i	$-0.440233\pi$
$353$	7.01413	0.373324	0.186662	+	0.982424i	$0.440233\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.63766	0.245451
$358$	0	0
$359$	−14.9199	−0.787443	−0.393722	−	0.919230i	$-0.628813\pi$
$359$	−14.9199	−0.787443	−0.393722	+	0.919230i	$0.628813\pi$
$360$	0	0
$361$	49.4707	2.60372
$362$	0	0
$363$	−10.6671	−0.559877
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0.680967	0.0355462	0.0177731	−	0.999842i	$-0.494342\pi$
$367$	0.680967	0.0355462	0.0177731	+	0.999842i	$0.494342\pi$
$368$	0	0
$369$	6.45452	0.336009
$370$	0	0
$371$	−34.1942	−1.77527
$372$	0	0
$373$	−28.6371	−1.48277	−0.741386	−	0.671079i	$-0.765831\pi$
$373$	−28.6371	−1.48277	−0.741386	+	0.671079i	$0.765831\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	42.1510	2.17089
$378$	0	0
$379$	−21.9020	−1.12503	−0.562514	−	0.826787i	$-0.690166\pi$
$379$	−21.9020	−1.12503	−0.562514	+	0.826787i	$0.690166\pi$
$380$	0	0
$381$	15.2854	0.783097
$382$	0	0
$383$	−4.56593	−0.233308	−0.116654	−	0.993173i	$-0.537217\pi$
$383$	−4.56593	−0.233308	−0.116654	+	0.993173i	$0.537217\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.42419	−0.174061
$388$	0	0
$389$	21.7420	1.10236	0.551182	−	0.834385i	$-0.314177\pi$
$389$	21.7420	1.10236	0.551182	+	0.834385i	$0.314177\pi$
$390$	0	0
$391$	7.98747	0.403944
$392$	0	0
$393$	−3.75392	−0.189360
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−15.6234	−0.784114	−0.392057	−	0.919941i	$-0.628236\pi$
$397$	−15.6234	−0.784114	−0.392057	+	0.919941i	$0.628236\pi$
$398$	0	0
$399$	20.2217	1.01235
$400$	0	0
$401$	−19.9417	−0.995841	−0.497920	−	0.867223i	$-0.665903\pi$
$401$	−19.9417	−0.995841	−0.497920	+	0.867223i	$0.665903\pi$
$402$	0	0
$403$	31.3740	1.56285
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.125785	0.00623495
$408$	0	0
$409$	−17.2159	−0.851273	−0.425636	−	0.904894i	$-0.639950\pi$
$409$	−17.2159	−0.851273	−0.425636	+	0.904894i	$0.639950\pi$
$410$	0	0
$411$	20.2511	0.998913
$412$	0	0
$413$	−29.4552	−1.44940
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−0.928868	−0.0454869
$418$	0	0
$419$	13.1307	0.641477	0.320738	−	0.947168i	$-0.396069\pi$
$419$	13.1307	0.641477	0.320738	+	0.947168i	$0.396069\pi$
$420$	0	0
$421$	17.0222	0.829610	0.414805	−	0.909910i	$-0.363850\pi$
$421$	17.0222	0.829610	0.414805	+	0.909910i	$0.363850\pi$
$422$	0	0
$423$	−9.61833	−0.467659
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−11.7563	−0.568927
$428$	0	0
$429$	−3.71949	−0.179579
$430$	0	0
$431$	−12.5147	−0.602811	−0.301405	−	0.953496i	$-0.597456\pi$
$431$	−12.5147	−0.602811	−0.301405	+	0.953496i	$0.597456\pi$
$432$	0	0
$433$	−20.7471	−0.997044	−0.498522	−	0.866877i	$-0.666124\pi$
$433$	−20.7471	−0.997044	−0.498522	+	0.866877i	$0.666124\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	34.8280	1.66605
$438$	0	0
$439$	−33.0806	−1.57885	−0.789425	−	0.613847i	$-0.789621\pi$
$439$	−33.0806	−1.57885	−0.789425	+	0.613847i	$0.789621\pi$
$440$	0	0
$441$	−1.02783	−0.0489443
$442$	0	0
$443$	23.4802	1.11558	0.557788	−	0.829984i	$-0.311650\pi$
$443$	23.4802	1.11558	0.557788	+	0.829984i	$0.311650\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	13.1432	0.621650
$448$	0	0
$449$	−31.6965	−1.49585	−0.747925	−	0.663783i	$-0.768950\pi$
$449$	−31.6965	−1.49585	−0.747925	+	0.663783i	$0.768950\pi$
$450$	0	0
$451$	−3.72415	−0.175363
$452$	0	0
$453$	17.5864	0.826281
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.2267	1.32039	0.660196	−	0.751094i	$-0.270473\pi$
$457$	28.2267	1.32039	0.660196	+	0.751094i	$0.270473\pi$
$458$	0	0
$459$	1.89772	0.0885780
$460$	0	0
$461$	−4.94266	−0.230203	−0.115101	−	0.993354i	$-0.536719\pi$
$461$	−4.94266	−0.230203	−0.115101	+	0.993354i	$0.536719\pi$
$462$	0	0
$463$	−18.4463	−0.857272	−0.428636	−	0.903477i	$-0.641006\pi$
$463$	−18.4463	−0.857272	−0.428636	+	0.903477i	$0.641006\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.3011	0.800600	0.400300	−	0.916384i	$-0.368906\pi$
$467$	17.3011	0.800600	0.400300	+	0.916384i	$0.368906\pi$
$468$	0	0
$469$	9.45980	0.436813
$470$	0	0
$471$	8.68198	0.400044
$472$	0	0
$473$	1.97570	0.0908427
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−13.9922	−0.640659
$478$	0	0
$479$	−34.3756	−1.57066	−0.785331	−	0.619076i	$-0.787507\pi$
$479$	−34.3756	−1.57066	−0.785331	+	0.619076i	$0.787507\pi$
$480$	0	0
$481$	−1.40536	−0.0640788
$482$	0	0
$483$	10.2859	0.468025
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−5.04208	−0.228478	−0.114239	−	0.993453i	$-0.536443\pi$
$487$	−5.04208	−0.228478	−0.114239	+	0.993453i	$0.536443\pi$
$488$	0	0
$489$	−1.40791	−0.0636678
$490$	0	0
$491$	−26.0163	−1.17410	−0.587051	−	0.809550i	$-0.699711\pi$
$491$	−26.0163	−1.17410	−0.587051	+	0.809550i	$0.699711\pi$
$492$	0	0
$493$	12.4085	0.558852
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−15.8419	−0.710605
$498$	0	0
$499$	2.49658	0.111762	0.0558812	−	0.998437i	$-0.482203\pi$
$499$	2.49658	0.111762	0.0558812	+	0.998437i	$0.482203\pi$
$500$	0	0
$501$	−8.81475	−0.393814
$502$	0	0
$503$	14.2600	0.635824	0.317912	−	0.948120i	$-0.397018\pi$
$503$	14.2600	0.635824	0.317912	+	0.948120i	$0.397018\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	28.5567	1.26825
$508$	0	0
$509$	−15.6163	−0.692179	−0.346089	−	0.938202i	$-0.612491\pi$
$509$	−15.6163	−0.692179	−0.346089	+	0.938202i	$0.612491\pi$
$510$	0	0
$511$	−24.5624	−1.08658
$512$	0	0
$513$	8.27470	0.365337
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.54962	0.244072
$518$	0	0
$519$	−7.90310	−0.346908
$520$	0	0
$521$	14.3727	0.629678	0.314839	−	0.949145i	$-0.398049\pi$
$521$	14.3727	0.629678	0.314839	+	0.949145i	$0.398049\pi$
$522$	0	0
$523$	−4.41486	−0.193048	−0.0965241	−	0.995331i	$-0.530772\pi$
$523$	−4.41486	−0.193048	−0.0965241	+	0.995331i	$0.530772\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.23597	0.402325
$528$	0	0
$529$	−5.28451	−0.229761
$530$	0	0
$531$	−12.0530	−0.523057
$532$	0	0
$533$	41.6087	1.80227
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−11.4886	−0.495771
$538$	0	0
$539$	0.593041	0.0255441
$540$	0	0
$541$	36.0371	1.54936	0.774679	−	0.632355i	$-0.217912\pi$
$541$	36.0371	1.54936	0.774679	+	0.632355i	$0.217912\pi$
$542$	0	0
$543$	3.92307	0.168355
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−12.1032	−0.517497	−0.258748	−	0.965945i	$-0.583310\pi$
$547$	−12.1032	−0.517497	−0.258748	+	0.965945i	$0.583310\pi$
$548$	0	0
$549$	−4.81065	−0.205314
$550$	0	0
$551$	54.1054	2.30497
$552$	0	0
$553$	−11.5951	−0.493074
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24.0437	−1.01876	−0.509382	−	0.860541i	$-0.670126\pi$
$557$	−24.0437	−1.01876	−0.509382	+	0.860541i	$0.670126\pi$
$558$	0	0
$559$	−22.0738	−0.933624
$560$	0	0
$561$	−1.09495	−0.0462290
$562$	0	0
$563$	18.9430	0.798352	0.399176	−	0.916874i	$-0.369296\pi$
$563$	18.9430	0.798352	0.399176	+	0.916874i	$0.369296\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.44380	0.102630
$568$	0	0
$569$	−31.2220	−1.30889	−0.654447	−	0.756108i	$-0.727098\pi$
$569$	−31.2220	−1.30889	−0.654447	+	0.756108i	$0.727098\pi$
$570$	0	0
$571$	−0.0391729	−0.00163934	−0.000819668	−	1.00000i	$-0.500261\pi$
$571$	−0.0391729	−0.00163934	−0.000819668		1.00000i	$0.500261\pi$
$572$	0	0
$573$	−25.5456	−1.06718
$574$	0	0
$575$	0	0
$576$	0	0
$577$	20.1295	0.838001	0.419000	−	0.907986i	$-0.362381\pi$
$577$	20.1295	0.838001	0.419000	+	0.907986i	$0.362381\pi$
$578$	0	0
$579$	0.421651	0.0175232
$580$	0	0
$581$	8.39245	0.348178
$582$	0	0
$583$	8.07327	0.334361
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.96937	0.411480	0.205740	−	0.978607i	$-0.434040\pi$
$587$	9.96937	0.411480	0.205740	+	0.978607i	$0.434040\pi$
$588$	0	0
$589$	40.2719	1.65938
$590$	0	0
$591$	7.22140	0.297049
$592$	0	0
$593$	20.3619	0.836163	0.418082	−	0.908410i	$-0.362703\pi$
$593$	20.3619	0.836163	0.418082	+	0.908410i	$0.362703\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.93505	0.161051
$598$	0	0
$599$	−24.1075	−0.985007	−0.492503	−	0.870311i	$-0.663918\pi$
$599$	−24.1075	−0.985007	−0.492503	+	0.870311i	$0.663918\pi$
$600$	0	0
$601$	37.2054	1.51764	0.758820	−	0.651300i	$-0.225776\pi$
$601$	37.2054	1.51764	0.758820	+	0.651300i	$0.225776\pi$
$602$	0	0
$603$	3.87094	0.157637
$604$	0	0
$605$	0	0
$606$	0	0
$607$	40.5752	1.64690	0.823448	−	0.567392i	$-0.192048\pi$
$607$	40.5752	1.64690	0.823448	+	0.567392i	$0.192048\pi$
$608$	0	0
$609$	15.9792	0.647508
$610$	0	0
$611$	−62.0041	−2.50842
$612$	0	0
$613$	−5.13559	−0.207424	−0.103712	−	0.994607i	$-0.533072\pi$
$613$	−5.13559	−0.207424	−0.103712	+	0.994607i	$0.533072\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16.3817	0.659504	0.329752	−	0.944068i	$-0.393035\pi$
$617$	16.3817	0.659504	0.329752	+	0.944068i	$0.393035\pi$
$618$	0	0
$619$	18.4485	0.741507	0.370754	−	0.928731i	$-0.379099\pi$
$619$	18.4485	0.741507	0.370754	+	0.928731i	$0.379099\pi$
$620$	0	0
$621$	4.20898	0.168900
$622$	0	0
$623$	−13.8964	−0.556747
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.77437	−0.190670
$628$	0	0
$629$	−0.413713	−0.0164958
$630$	0	0
$631$	38.5616	1.53511	0.767556	−	0.640982i	$-0.221473\pi$
$631$	38.5616	1.53511	0.767556	+	0.640982i	$0.221473\pi$
$632$	0	0
$633$	−24.7610	−0.984160
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.62585	−0.262526
$638$	0	0
$639$	−6.48247	−0.256443
$640$	0	0
$641$	18.9989	0.750412	0.375206	−	0.926941i	$-0.377572\pi$
$641$	18.9989	0.750412	0.375206	+	0.926941i	$0.377572\pi$
$642$	0	0
$643$	37.6504	1.48479	0.742393	−	0.669964i	$-0.233691\pi$
$643$	37.6504	1.48479	0.742393	+	0.669964i	$0.233691\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.73458	−0.225449	−0.112725	−	0.993626i	$-0.535958\pi$
$647$	−5.73458	−0.225449	−0.112725	+	0.993626i	$0.535958\pi$
$648$	0	0
$649$	6.95440	0.272984
$650$	0	0
$651$	11.8937	0.466150
$652$	0	0
$653$	−40.0810	−1.56849	−0.784246	−	0.620450i	$-0.786950\pi$
$653$	−40.0810	−1.56849	−0.784246	+	0.620450i	$0.786950\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−10.0509	−0.392123
$658$	0	0
$659$	−31.5748	−1.22998	−0.614989	−	0.788536i	$-0.710840\pi$
$659$	−31.5748	−1.22998	−0.614989	+	0.788536i	$0.710840\pi$
$660$	0	0
$661$	31.3504	1.21939	0.609695	−	0.792636i	$-0.291292\pi$
$661$	31.3504	1.21939	0.609695	+	0.792636i	$0.291292\pi$
$662$	0	0
$663$	12.2336	0.475112
$664$	0	0
$665$	0	0
$666$	0	0
$667$	27.5210	1.06562
$668$	0	0
$669$	19.6727	0.760589
$670$	0	0
$671$	2.77567	0.107153
$672$	0	0
$673$	−4.47476	−0.172489	−0.0862446	−	0.996274i	$-0.527487\pi$
$673$	−4.47476	−0.172489	−0.0862446	+	0.996274i	$0.527487\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14.0828	0.541244	0.270622	−	0.962686i	$-0.412771\pi$
$677$	14.0828	0.541244	0.270622	+	0.962686i	$0.412771\pi$
$678$	0	0
$679$	16.0918	0.617548
$680$	0	0
$681$	26.3748	1.01068
$682$	0	0
$683$	11.8357	0.452880	0.226440	−	0.974025i	$-0.427291\pi$
$683$	11.8357	0.452880	0.226440	+	0.974025i	$0.427291\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−16.0212	−0.611247
$688$	0	0
$689$	−90.2000	−3.43634
$690$	0	0
$691$	−37.7571	−1.43635	−0.718173	−	0.695865i	$-0.755021\pi$
$691$	−37.7571	−1.43635	−0.718173	+	0.695865i	$0.755021\pi$
$692$	0	0
$693$	−1.41003	−0.0535627
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.2489	0.463960
$698$	0	0
$699$	−29.3108	−1.10864
$700$	0	0
$701$	−39.1678	−1.47935	−0.739674	−	0.672965i	$-0.765020\pi$
$701$	−39.1678	−1.47935	−0.739674	+	0.672965i	$0.765020\pi$
$702$	0	0
$703$	−1.80393	−0.0680364
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−29.4143	−1.10624
$708$	0	0
$709$	−42.4884	−1.59569	−0.797843	−	0.602865i	$-0.794026\pi$
$709$	−42.4884	−1.59569	−0.797843	+	0.602865i	$0.794026\pi$
$710$	0	0
$711$	−4.74470	−0.177940
$712$	0	0
$713$	20.4846	0.767153
$714$	0	0
$715$	0	0
$716$	0	0
$717$	25.8558	0.965604
$718$	0	0
$719$	29.6979	1.10754	0.553772	−	0.832668i	$-0.313188\pi$
$719$	29.6979	1.10754	0.553772	+	0.832668i	$0.313188\pi$
$720$	0	0
$721$	−14.1404	−0.526617
$722$	0	0
$723$	−9.34158	−0.347417
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−41.9319	−1.55517	−0.777583	−	0.628780i	$-0.783555\pi$
$727$	−41.9319	−1.55517	−0.777583	+	0.628780i	$0.783555\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.49815	−0.240343
$732$	0	0
$733$	−29.2104	−1.07891	−0.539456	−	0.842014i	$-0.681370\pi$
$733$	−29.2104	−1.07891	−0.539456	+	0.842014i	$0.681370\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.23347	−0.0822707
$738$	0	0
$739$	25.9202	0.953491	0.476746	−	0.879041i	$-0.341816\pi$
$739$	25.9202	0.953491	0.476746	+	0.879041i	$0.341816\pi$
$740$	0	0
$741$	53.3424	1.95958
$742$	0	0
$743$	9.22935	0.338592	0.169296	−	0.985565i	$-0.445851\pi$
$743$	9.22935	0.338592	0.169296	+	0.985565i	$0.445851\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.43418	0.125650
$748$	0	0
$749$	9.94826	0.363502
$750$	0	0
$751$	−37.2805	−1.36038	−0.680192	−	0.733034i	$-0.738103\pi$
$751$	−37.2805	−1.36038	−0.680192	+	0.733034i	$0.738103\pi$
$752$	0	0
$753$	−13.5088	−0.492287
$754$	0	0
$755$	0	0
$756$	0	0
$757$	27.6758	1.00589	0.502947	−	0.864317i	$-0.332249\pi$
$757$	27.6758	1.00589	0.502947	+	0.864317i	$0.332249\pi$
$758$	0	0
$759$	−2.42851	−0.0881493
$760$	0	0
$761$	−13.6324	−0.494174	−0.247087	−	0.968993i	$-0.579473\pi$
$761$	−13.6324	−0.494174	−0.247087	+	0.968993i	$0.579473\pi$
$762$	0	0
$763$	−3.56247	−0.128970
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−77.6992	−2.80556
$768$	0	0
$769$	−30.5109	−1.10025	−0.550126	−	0.835082i	$-0.685420\pi$
$769$	−30.5109	−1.10025	−0.550126	+	0.835082i	$0.685420\pi$
$770$	0	0
$771$	−13.0662	−0.470566
$772$	0	0
$773$	8.44505	0.303748	0.151874	−	0.988400i	$-0.451469\pi$
$773$	8.44505	0.303748	0.151874	+	0.988400i	$0.451469\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.532761	−0.0191127
$778$	0	0
$779$	53.4093	1.91359
$780$	0	0
$781$	3.74028	0.133838
$782$	0	0
$783$	6.53865	0.233672
$784$	0	0
$785$	0	0
$786$	0	0
$787$	40.8557	1.45635	0.728174	−	0.685393i	$-0.240369\pi$
$787$	40.8557	1.45635	0.728174	+	0.685393i	$0.240369\pi$
$788$	0	0
$789$	4.43065	0.157735
$790$	0	0
$791$	−14.3154	−0.508998
$792$	0	0
$793$	−31.0116	−1.10125
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−29.5521	−1.04679	−0.523395	−	0.852090i	$-0.675335\pi$
$797$	−29.5521	−1.04679	−0.523395	+	0.852090i	$0.675335\pi$
$798$	0	0
$799$	−18.2529	−0.645742
$800$	0	0
$801$	−5.68638	−0.200918
$802$	0	0
$803$	5.79920	0.204649
$804$	0	0
$805$	0	0
$806$	0	0
$807$	28.0509	0.987438
$808$	0	0
$809$	24.8982	0.875373	0.437687	−	0.899128i	$-0.355798\pi$
$809$	24.8982	0.875373	0.437687	+	0.899128i	$0.355798\pi$
$810$	0	0
$811$	24.3586	0.855347	0.427674	−	0.903933i	$-0.359333\pi$
$811$	24.3586	0.855347	0.427674	+	0.903933i	$0.359333\pi$
$812$	0	0
$813$	5.98512	0.209907
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−28.3341	−0.991286
$818$	0	0
$819$	15.7538	0.550484
$820$	0	0
$821$	−55.3787	−1.93273	−0.966365	−	0.257175i	$-0.917208\pi$
$821$	−55.3787	−1.93273	−0.966365	+	0.257175i	$0.917208\pi$
$822$	0	0
$823$	17.4649	0.608789	0.304394	−	0.952546i	$-0.401546\pi$
$823$	17.4649	0.608789	0.304394	+	0.952546i	$0.401546\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.3838	0.813136	0.406568	−	0.913621i	$-0.366725\pi$
$827$	23.3838	0.813136	0.406568	+	0.913621i	$0.366725\pi$
$828$	0	0
$829$	36.0766	1.25299	0.626497	−	0.779424i	$-0.284488\pi$
$829$	36.0766	1.25299	0.626497	+	0.779424i	$0.284488\pi$
$830$	0	0
$831$	−5.07072	−0.175901
$832$	0	0
$833$	−1.95053	−0.0675820
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.86687	0.168224
$838$	0	0
$839$	19.0416	0.657390	0.328695	−	0.944436i	$-0.393391\pi$
$839$	19.0416	0.657390	0.328695	+	0.944436i	$0.393391\pi$
$840$	0	0
$841$	13.7539	0.474273
$842$	0	0
$843$	−18.1559	−0.625323
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−26.0683	−0.895716
$848$	0	0
$849$	4.69717	0.161206
$850$	0	0
$851$	−0.917579	−0.0314542
$852$	0	0
$853$	−18.0499	−0.618016	−0.309008	−	0.951059i	$-0.599997\pi$
$853$	−18.0499	−0.618016	−0.309008	+	0.951059i	$0.599997\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.4435	0.425061	0.212531	−	0.977154i	$-0.431829\pi$
$857$	12.4435	0.425061	0.212531	+	0.977154i	$0.431829\pi$
$858$	0	0
$859$	46.0026	1.56959	0.784794	−	0.619757i	$-0.212769\pi$
$859$	46.0026	1.56959	0.784794	+	0.619757i	$0.212769\pi$
$860$	0	0
$861$	15.7736	0.537562
$862$	0	0
$863$	47.3472	1.61172	0.805858	−	0.592109i	$-0.201704\pi$
$863$	47.3472	1.61172	0.805858	+	0.592109i	$0.201704\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−13.3987	−0.455042
$868$	0	0
$869$	2.73761	0.0928672
$870$	0	0
$871$	24.9538	0.845526
$872$	0	0
$873$	6.58475	0.222860
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−31.9576	−1.07913	−0.539565	−	0.841944i	$-0.681411\pi$
$877$	−31.9576	−1.07913	−0.539565	+	0.841944i	$0.681411\pi$
$878$	0	0
$879$	−9.60771	−0.324060
$880$	0	0
$881$	−7.92960	−0.267155	−0.133578	−	0.991038i	$-0.542647\pi$
$881$	−7.92960	−0.267155	−0.133578	+	0.991038i	$0.542647\pi$
$882$	0	0
$883$	−37.2505	−1.25358	−0.626790	−	0.779188i	$-0.715632\pi$
$883$	−37.2505	−1.25358	−0.626790	+	0.779188i	$0.715632\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.4148	0.551155	0.275577	−	0.961279i	$-0.411131\pi$
$887$	16.4148	0.551155	0.275577	+	0.961279i	$0.411131\pi$
$888$	0	0
$889$	37.3546	1.25283
$890$	0	0
$891$	−0.576983	−0.0193297
$892$	0	0
$893$	−79.5889	−2.66334
$894$	0	0
$895$	0	0
$896$	0	0
$897$	27.1329	0.905943
$898$	0	0
$899$	31.8228	1.06135
$900$	0	0
$901$	−26.5533	−0.884619
$902$	0	0
$903$	−8.36804	−0.278471
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.85782	−0.0616880	−0.0308440	−	0.999524i	$-0.509819\pi$
$907$	−1.85782	−0.0616880	−0.0308440	+	0.999524i	$0.509819\pi$
$908$	0	0
$909$	−12.0363	−0.399219
$910$	0	0
$911$	−22.5064	−0.745672	−0.372836	−	0.927897i	$-0.621615\pi$
$911$	−22.5064	−0.745672	−0.372836	+	0.927897i	$0.621615\pi$
$912$	0	0
$913$	−1.98146	−0.0655769
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.17384	−0.302947
$918$	0	0
$919$	20.9049	0.689590	0.344795	−	0.938678i	$-0.387948\pi$
$919$	20.9049	0.689590	0.344795	+	0.938678i	$0.387948\pi$
$920$	0	0
$921$	−13.5400	−0.446160
$922$	0	0
$923$	−41.7889	−1.37550
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−5.78624	−0.190045
$928$	0	0
$929$	13.9748	0.458498	0.229249	−	0.973368i	$-0.426373\pi$
$929$	13.9748	0.458498	0.229249	+	0.973368i	$0.426373\pi$
$930$	0	0
$931$	−8.50499	−0.278740
$932$	0	0
$933$	−2.96725	−0.0971434
$934$	0	0
$935$	0	0
$936$	0	0
$937$	32.7584	1.07017	0.535086	−	0.844798i	$-0.320279\pi$
$937$	32.7584	1.07017	0.535086	+	0.844798i	$0.320279\pi$
$938$	0	0
$939$	14.0058	0.457060
$940$	0	0
$941$	−25.8709	−0.843368	−0.421684	−	0.906743i	$-0.638561\pi$
$941$	−25.8709	−0.843368	−0.421684	+	0.906743i	$0.638561\pi$
$942$	0	0
$943$	27.1669	0.884677
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−55.7717	−1.81234	−0.906168	−	0.422918i	$-0.861006\pi$
$947$	−55.7717	−1.81234	−0.906168	+	0.422918i	$0.861006\pi$
$948$	0	0
$949$	−64.7925	−2.10325
$950$	0	0
$951$	27.0176	0.876106
$952$	0	0
$953$	24.9297	0.807551	0.403775	−	0.914858i	$-0.367698\pi$
$953$	24.9297	0.807551	0.403775	+	0.914858i	$0.367698\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−3.77269	−0.121954
$958$	0	0
$959$	49.4897	1.59810
$960$	0	0
$961$	−7.31353	−0.235920
$962$	0	0
$963$	4.07081	0.131180
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.68802	0.279388	0.139694	−	0.990195i	$-0.455388\pi$
$967$	8.68802	0.279388	0.139694	+	0.990195i	$0.455388\pi$
$968$	0	0
$969$	15.7031	0.504456
$970$	0	0
$971$	24.6797	0.792011	0.396005	−	0.918248i	$-0.370396\pi$
$971$	24.6797	0.792011	0.396005	+	0.918248i	$0.370396\pi$
$972$	0	0
$973$	−2.26997	−0.0727719
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35.5977	1.13887	0.569435	−	0.822036i	$-0.307162\pi$
$977$	35.5977	1.13887	0.569435	+	0.822036i	$0.307162\pi$
$978$	0	0
$979$	3.28095	0.104860
$980$	0	0
$981$	−1.45776	−0.0465426
$982$	0	0
$983$	5.94123	0.189496	0.0947479	−	0.995501i	$-0.469795\pi$
$983$	5.94123	0.189496	0.0947479	+	0.995501i	$0.469795\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−23.5053	−0.748182
$988$	0	0
$989$	−14.4123	−0.458285
$990$	0	0
$991$	4.22674	0.134267	0.0671335	−	0.997744i	$-0.478615\pi$
$991$	4.22674	0.134267	0.0671335	+	0.997744i	$0.478615\pi$
$992$	0	0
$993$	−6.65492	−0.211187
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−12.5455	−0.397319	−0.198660	−	0.980069i	$-0.563659\pi$
$997$	−12.5455	−0.397319	−0.198660	+	0.980069i	$0.563659\pi$
$998$	0	0
$999$	−0.218005	−0.00689738

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.a.n.1.8		12
5.2	odd	4		7500.2.d.g.1249.8		24
5.3	odd	4		7500.2.d.g.1249.17		24
5.4	even	2		7500.2.a.m.1.5		12
25.2	odd	20		1500.2.o.c.649.1		24
25.9	even	10		1500.2.m.d.901.3		24
25.11	even	5		1500.2.m.c.601.4		24
25.12	odd	20		300.2.o.a.169.6	✓	24
25.13	odd	20		1500.2.o.c.349.1		24
25.14	even	10		1500.2.m.d.601.3		24
25.16	even	5		1500.2.m.c.901.4		24
25.23	odd	20		300.2.o.a.229.6	yes	24
75.23	even	20		900.2.w.c.829.2		24
75.62	even	20		900.2.w.c.469.2		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.169.6	✓	24	25.12	odd	20
300.2.o.a.229.6	yes	24	25.23	odd	20
900.2.w.c.469.2		24	75.62	even	20
900.2.w.c.829.2		24	75.23	even	20
1500.2.m.c.601.4		24	25.11	even	5
1500.2.m.c.901.4		24	25.16	even	5
1500.2.m.d.601.3		24	25.14	even	10
1500.2.m.d.901.3		24	25.9	even	10
1500.2.o.c.349.1		24	25.13	odd	20
1500.2.o.c.649.1		24	25.2	odd	20
7500.2.a.m.1.5		12	5.4	even	2
7500.2.a.n.1.8		12	1.1	even	1	trivial
7500.2.d.g.1249.8		24	5.2	odd	4
7500.2.d.g.1249.17		24	5.3	odd	4

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:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.44380 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.44380 q^{7} +1.00000 q^{9} -0.576983 q^{11} +6.44645 q^{13} +1.89772 q^{17} +8.27470 q^{19} +2.44380 q^{21} +4.20898 q^{23} +1.00000 q^{27} +6.53865 q^{29} +4.86687 q^{31} -0.576983 q^{33} -0.218005 q^{37} +6.44645 q^{39} +6.45452 q^{41} -3.42419 q^{43} -9.61833 q^{47} -1.02783 q^{49} +1.89772 q^{51} -13.9922 q^{53} +8.27470 q^{57} -12.0530 q^{59} -4.81065 q^{61} +2.44380 q^{63} +3.87094 q^{67} +4.20898 q^{69} -6.48247 q^{71} -10.0509 q^{73} -1.41003 q^{77} -4.74470 q^{79} +1.00000 q^{81} +3.43418 q^{83} +6.53865 q^{87} -5.68638 q^{89} +15.7538 q^{91} +4.86687 q^{93} +6.58475 q^{97} -0.576983 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^3 + 8 * q^7 + 12 * q^9	\( 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9} + 2 q^{11} + 8 q^{17} + 10 q^{19} + 8 q^{21} + 18 q^{23} + 12 q^{27} + 8 q^{29} - 2 q^{31} + 2 q^{33} + 4 q^{37} + 10 q^{41} + 28 q^{43} + 22 q^{47} + 28 q^{49} + 8 q^{51} + 16 q^{53} + 10 q^{57} - 2 q^{59} + 34 q^{61} + 8 q^{63} + 32 q^{67} + 18 q^{69} + 24 q^{73} + 18 q^{77} + 6 q^{79} + 12 q^{81} + 28 q^{83} + 8 q^{87} + 10 q^{89} + 20 q^{91} - 2 q^{93} + 16 q^{97} + 2 q^{99}+O(q^{100}) \) 12 * q + 12 * q^3 + 8 * q^7 + 12 * q^9 + 2 * q^11 + 8 * q^17 + 10 * q^19 + 8 * q^21 + 18 * q^23 + 12 * q^27 + 8 * q^29 - 2 * q^31 + 2 * q^33 + 4 * q^37 + 10 * q^41 + 28 * q^43 + 22 * q^47 + 28 * q^49 + 8 * q^51 + 16 * q^53 + 10 * q^57 - 2 * q^59 + 34 * q^61 + 8 * q^63 + 32 * q^67 + 18 * q^69 + 24 * q^73 + 18 * q^77 + 6 * q^79 + 12 * q^81 + 28 * q^83 + 8 * q^87 + 10 * q^89 + 20 * q^91 - 2 * q^93 + 16 * q^97 + 2 * q^99

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