LMFDB - Embedded newform 7500.2.d.c.1249.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1249,7500)]

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, 1]))

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

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

chi := DirichletCharacter("7500.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$59.8878015160$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.6724000000.12
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 16x^{6} + 86x^{4} + 181x^{2} + 121 $ x^8 + 16x^6 + 86x^4 + 181*x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.2
Root			$-2.12233i$ of defining polynomial
Character	$\chi$	$=$	7500.1249
Dual form			7500.2.d.c.1249.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.50430i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.50430i q^{7} -1.00000 q^{9} -6.17438 q^{11} -3.55634i q^{13} -0.495700i q^{17} -0.311674 q^{19} -1.50430 q^{21} -3.06064i q^{23} +1.00000i q^{27} +0.122334 q^{29} -2.94362 q^{31} +6.17438i q^{33} +4.36700i q^{37} -3.55634 q^{39} +4.25327 q^{41} -3.62663i q^{43} -5.28215i q^{47} +4.73708 q^{49} -0.495700 q^{51} -8.59978i q^{53} +0.311674i q^{57} -12.8826 q^{59} -11.3209 q^{61} +1.50430i q^{63} -13.2414i q^{67} -3.06064 q^{69} +3.26823 q^{71} +15.9668i q^{73} +9.28811i q^{77} +8.52746 q^{79} +1.00000 q^{81} +11.3381i q^{83} -0.122334i q^{87} -14.1326 q^{89} -5.34980 q^{91} +2.94362i q^{93} +11.4828i q^{97} +6.17438 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 2 q^{11} + 10 q^{19} - 8 q^{21} - 12 q^{29} + 22 q^{31} + 10 q^{39} + 8 q^{49} - 8 q^{51} - 2 q^{59} - 44 q^{61} + 18 q^{69} + 40 q^{71} + 6 q^{79} + 8 q^{81} + 30 q^{89} - 20 q^{91} + 2 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 - 2 * q^11 + 10 * q^19 - 8 * q^21 - 12 * q^29 + 22 * q^31 + 10 * q^39 + 8 * q^49 - 8 * q^51 - 2 * q^59 - 44 * q^61 + 18 * q^69 + 40 * q^71 + 6 * q^79 + 8 * q^81 + 30 * q^89 - 20 * q^91 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7500\mathbb{Z}\right)^\times$.

$n$	$2501$	$3751$	$6877$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.50430i	− 0.568572i	−0.958740	−	0.284286i	$-0.908244\pi$
$7$	− 1.50430i	− 0.568572i	0.958740	−	0.284286i	$-0.0917565\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−6.17438	−1.86164	−0.930822	−	0.365473i	$-0.880907\pi$
$11$	−6.17438	−1.86164	−0.930822	+	0.365473i	$0.880907\pi$
$12$	0	0
$13$	− 3.55634i	− 0.986352i	−0.869930	−	0.493176i	$-0.835836\pi$
$13$	− 3.55634i	− 0.986352i	0.869930	−	0.493176i	$-0.164164\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.495700i	− 0.120225i	−0.998192	−	0.0601125i	$-0.980854\pi$
$17$	− 0.495700i	− 0.120225i	0.998192	−	0.0601125i	$-0.0191459\pi$
$18$	0	0
$19$	−0.311674	−0.0715030	−0.0357515	−	0.999361i	$-0.511382\pi$
$19$	−0.311674	−0.0715030	−0.0357515	+	0.999361i	$0.511382\pi$
$20$	0	0
$21$	−1.50430	−0.328265
$22$	0	0
$23$	− 3.06064i	− 0.638188i	−0.947723	−	0.319094i	$-0.896621\pi$
$23$	− 3.06064i	− 0.638188i	0.947723	−	0.319094i	$-0.103379\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	0.122334	0.0227168	0.0113584	−	0.999935i	$-0.496384\pi$
$29$	0.122334	0.0227168	0.0113584	+	0.999935i	$0.496384\pi$
$30$	0	0
$31$	−2.94362	−0.528690	−0.264345	−	0.964428i	$-0.585156\pi$
$31$	−2.94362	−0.528690	−0.264345	+	0.964428i	$0.585156\pi$
$32$	0	0
$33$	6.17438i	1.07482i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.36700i	0.717930i	0.933351	+	0.358965i	$0.116870\pi$
$37$	4.36700i	0.717930i	−0.933351	+	0.358965i	$0.883130\pi$
$38$	0	0
$39$	−3.55634	−0.569470
$40$	0	0
$41$	4.25327	0.664249	0.332124	−	0.943236i	$-0.392235\pi$
$41$	4.25327	0.664249	0.332124	+	0.943236i	$0.392235\pi$
$42$	0	0
$43$	− 3.62663i	− 0.553056i	−0.961006	−	0.276528i	$-0.910816\pi$
$43$	− 3.62663i	− 0.553056i	0.961006	−	0.276528i	$-0.0891839\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 5.28215i	− 0.770480i	−0.922816	−	0.385240i	$-0.874119\pi$
$47$	− 5.28215i	− 0.770480i	0.922816	−	0.385240i	$-0.125881\pi$
$48$	0	0
$49$	4.73708	0.676726
$50$	0	0
$51$	−0.495700	−0.0694120
$52$	0	0
$53$	− 8.59978i	− 1.18127i	−0.806938	−	0.590636i	$-0.798877\pi$
$53$	− 8.59978i	− 1.18127i	0.806938	−	0.590636i	$-0.201123\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.311674i	0.0412823i
$58$	0	0
$59$	−12.8826	−1.67717	−0.838584	−	0.544772i	$-0.816616\pi$
$59$	−12.8826	−1.67717	−0.838584	+	0.544772i	$0.816616\pi$
$60$	0	0
$61$	−11.3209	−1.44950	−0.724748	−	0.689014i	$-0.758044\pi$
$61$	−11.3209	−1.44950	−0.724748	+	0.689014i	$0.758044\pi$
$62$	0	0
$63$	1.50430i	0.189524i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 13.2414i	− 1.61769i	−0.588020	−	0.808846i	$-0.700092\pi$
$67$	− 13.2414i	− 1.61769i	0.588020	−	0.808846i	$-0.299908\pi$
$68$	0	0
$69$	−3.06064	−0.368458
$70$	0	0
$71$	3.26823	0.387868	0.193934	−	0.981015i	$-0.437875\pi$
$71$	3.26823	0.387868	0.193934	+	0.981015i	$0.437875\pi$
$72$	0	0
$73$	15.9668i	1.86877i	0.356264	+	0.934385i	$0.384050\pi$
$73$	15.9668i	1.86877i	−0.356264	+	0.934385i	$0.615950\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9.28811i	1.05848i
$78$	0	0
$79$	8.52746	0.959414	0.479707	−	0.877429i	$-0.340743\pi$
$79$	8.52746	0.959414	0.479707	+	0.877429i	$0.340743\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.3381i	1.24452i	0.782810	+	0.622260i	$0.213785\pi$
$83$	11.3381i	1.24452i	−0.782810	+	0.622260i	$0.786215\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 0.122334i	− 0.0131155i
$88$	0	0
$89$	−14.1326	−1.49805	−0.749024	−	0.662543i	$-0.769477\pi$
$89$	−14.1326	−1.49805	−0.749024	+	0.662543i	$0.769477\pi$
$90$	0	0
$91$	−5.34980	−0.560812
$92$	0	0
$93$	2.94362i	0.305239i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.4828i	1.16590i	0.812509	+	0.582949i	$0.198101\pi$
$97$	11.4828i	1.16590i	−0.812509	+	0.582949i	$0.801899\pi$
$98$	0	0
$99$	6.17438	0.620548
$100$	0	0
$101$	−14.8359	−1.47622	−0.738111	−	0.674679i	$-0.764282\pi$
$101$	−14.8359	−1.47622	−0.738111	+	0.674679i	$0.764282\pi$
$102$	0	0
$103$	19.2645i	1.89819i	0.314987	+	0.949096i	$0.398000\pi$
$103$	19.2645i	1.89819i	−0.314987	+	0.949096i	$0.602000\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8.63523i	− 0.834799i	−0.908723	−	0.417400i	$-0.862942\pi$
$107$	− 8.63523i	− 0.834799i	0.908723	−	0.417400i	$-0.137058\pi$
$108$	0	0
$109$	18.6893	1.79011	0.895055	−	0.445955i	$-0.147136\pi$
$109$	18.6893	1.79011	0.895055	+	0.445955i	$0.147136\pi$
$110$	0	0
$111$	4.36700	0.414497
$112$	0	0
$113$	6.05638i	0.569736i	0.958567	+	0.284868i	$0.0919498\pi$
$113$	6.05638i	0.569736i	−0.958567	+	0.284868i	$0.908050\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.55634i	0.328784i
$118$	0	0
$119$	−0.745682	−0.0683566
$120$	0	0
$121$	27.1229	2.46572
$122$	0	0
$123$	− 4.25327i	− 0.383504i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.24565i	0.820418i	0.911991	+	0.410209i	$0.134544\pi$
$127$	9.24565i	0.820418i	−0.911991	+	0.410209i	$0.865456\pi$
$128$	0	0
$129$	−3.62663	−0.319307
$130$	0	0
$131$	19.0155	1.66140	0.830698	−	0.556724i	$-0.187942\pi$
$131$	19.0155	1.66140	0.830698	+	0.556724i	$0.187942\pi$
$132$	0	0
$133$	0.468851i	0.0406546i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 7.86205i	− 0.671700i	−0.941915	−	0.335850i	$-0.890976\pi$
$137$	− 7.86205i	− 0.671700i	0.941915	−	0.335850i	$-0.109024\pi$
$138$	0	0
$139$	13.8004	1.17053	0.585266	−	0.810842i	$-0.300990\pi$
$139$	13.8004	1.17053	0.585266	+	0.810842i	$0.300990\pi$
$140$	0	0
$141$	−5.28215	−0.444837
$142$	0	0
$143$	21.9582i	1.83624i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 4.73708i	− 0.390708i
$148$	0	0
$149$	−5.12168	−0.419585	−0.209792	−	0.977746i	$-0.567279\pi$
$149$	−5.12168	−0.419585	−0.209792	+	0.977746i	$0.567279\pi$
$150$	0	0
$151$	−12.9476	−1.05366	−0.526829	−	0.849972i	$-0.676619\pi$
$151$	−12.9476	−1.05366	−0.526829	+	0.849972i	$0.676619\pi$
$152$	0	0
$153$	0.495700i	0.0400750i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.02750i	0.321429i	0.987001	+	0.160715i	$0.0513799\pi$
$157$	4.02750i	0.321429i	−0.987001	+	0.160715i	$0.948620\pi$
$158$	0	0
$159$	−8.59978	−0.682007
$160$	0	0
$161$	−4.60412	−0.362856
$162$	0	0
$163$	15.6190i	1.22338i	0.791099	+	0.611688i	$0.209509\pi$
$163$	15.6190i	1.22338i	−0.791099	+	0.611688i	$0.790491\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 11.8445i	− 0.916551i	−0.888810	−	0.458276i	$-0.848467\pi$
$167$	− 11.8445i	− 0.916551i	0.888810	−	0.458276i	$-0.151533\pi$
$168$	0	0
$169$	0.352437	0.0271105
$170$	0	0
$171$	0.311674	0.0238343
$172$	0	0
$173$	14.8305i	1.12754i	0.825930	+	0.563772i	$0.190651\pi$
$173$	14.8305i	1.12754i	−0.825930	+	0.563772i	$0.809349\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.8826i	0.968314i
$178$	0	0
$179$	0.957535	0.0715695	0.0357848	−	0.999360i	$-0.488607\pi$
$179$	0.957535	0.0715695	0.0357848	+	0.999360i	$0.488607\pi$
$180$	0	0
$181$	4.14123	0.307815	0.153908	−	0.988085i	$-0.450814\pi$
$181$	4.14123	0.307815	0.153908	+	0.988085i	$0.450814\pi$
$182$	0	0
$183$	11.3209i	0.836867i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.06064i	0.223816i
$188$	0	0
$189$	1.50430	0.109422
$190$	0	0
$191$	−17.6305	−1.27570	−0.637849	−	0.770162i	$-0.720176\pi$
$191$	−17.6305	−1.27570	−0.637849	+	0.770162i	$0.720176\pi$
$192$	0	0
$193$	5.25392i	0.378185i	0.981959	+	0.189093i	$0.0605546\pi$
$193$	5.25392i	0.378185i	−0.981959	+	0.189093i	$0.939445\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.11597i	0.150756i	0.997155	+	0.0753782i	$0.0240164\pi$
$197$	2.11597i	0.150756i	−0.997155	+	0.0753782i	$0.975984\pi$
$198$	0	0
$199$	−9.07029	−0.642976	−0.321488	−	0.946914i	$-0.604183\pi$
$199$	−9.07029	−0.642976	−0.321488	+	0.946914i	$0.604183\pi$
$200$	0	0
$201$	−13.2414	−0.933975
$202$	0	0
$203$	− 0.184026i	− 0.0129161i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.06064i	0.212729i
$208$	0	0
$209$	1.92439	0.133113
$210$	0	0
$211$	−19.8149	−1.36412	−0.682058	−	0.731298i	$-0.738915\pi$
$211$	−19.8149	−1.36412	−0.682058	+	0.731298i	$0.738915\pi$
$212$	0	0
$213$	− 3.26823i	− 0.223936i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.42809i	0.300598i
$218$	0	0
$219$	15.9668	1.07894
$220$	0	0
$221$	−1.76288	−0.118584
$222$	0	0
$223$	6.58214i	0.440773i	0.975413	+	0.220386i	$0.0707318\pi$
$223$	6.58214i	0.440773i	−0.975413	+	0.220386i	$0.929268\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 11.7821i	− 0.782006i	−0.920389	−	0.391003i	$-0.872128\pi$
$227$	− 11.7821i	− 0.782006i	0.920389	−	0.391003i	$-0.127872\pi$
$228$	0	0
$229$	−11.8899	−0.785705	−0.392853	−	0.919601i	$-0.628512\pi$
$229$	−11.8899	−0.785705	−0.392853	+	0.919601i	$0.628512\pi$
$230$	0	0
$231$	9.28811	0.611113
$232$	0	0
$233$	− 5.05001i	− 0.330837i	−0.986223	−	0.165419i	$-0.947102\pi$
$233$	− 5.05001i	− 0.330837i	0.986223	−	0.165419i	$-0.0528975\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 8.52746i	− 0.553918i
$238$	0	0
$239$	−2.67932	−0.173311	−0.0866556	−	0.996238i	$-0.527618\pi$
$239$	−2.67932	−0.173311	−0.0866556	+	0.996238i	$0.527618\pi$
$240$	0	0
$241$	12.6561	0.815250	0.407625	−	0.913149i	$-0.366357\pi$
$241$	12.6561	0.815250	0.407625	+	0.913149i	$0.366357\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.10842i	0.0705271i
$248$	0	0
$249$	11.3381	0.718524
$250$	0	0
$251$	−22.9068	−1.44586	−0.722932	−	0.690919i	$-0.757206\pi$
$251$	−22.9068	−1.44586	−0.722932	+	0.690919i	$0.757206\pi$
$252$	0	0
$253$	18.8975i	1.18808i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	27.7764i	1.73264i	0.499486	+	0.866322i	$0.333522\pi$
$257$	27.7764i	1.73264i	−0.499486	+	0.866322i	$0.666478\pi$
$258$	0	0
$259$	6.56928	0.408195
$260$	0	0
$261$	−0.122334	−0.00757225
$262$	0	0
$263$	11.7115i	0.722161i	0.932535	+	0.361081i	$0.117592\pi$
$263$	11.7115i	0.722161i	−0.932535	+	0.361081i	$0.882408\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	14.1326i	0.864899i
$268$	0	0
$269$	−4.20621	−0.256457	−0.128229	−	0.991745i	$-0.540929\pi$
$269$	−4.20621	−0.256457	−0.128229	+	0.991745i	$0.540929\pi$
$270$	0	0
$271$	−12.5659	−0.763325	−0.381663	−	0.924302i	$-0.624648\pi$
$271$	−12.5659	−0.763325	−0.381663	+	0.924302i	$0.624648\pi$
$272$	0	0
$273$	5.34980i	0.323785i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 31.8653i	− 1.91460i	−0.289100	−	0.957299i	$-0.593356\pi$
$277$	− 31.8653i	− 1.91460i	0.289100	−	0.957299i	$-0.406644\pi$
$278$	0	0
$279$	2.94362	0.176230
$280$	0	0
$281$	16.2998	0.972364	0.486182	−	0.873858i	$-0.338389\pi$
$281$	16.2998	0.972364	0.486182	+	0.873858i	$0.338389\pi$
$282$	0	0
$283$	− 13.6305i	− 0.810249i	−0.914262	−	0.405124i	$-0.867228\pi$
$283$	− 13.6305i	− 0.810249i	0.914262	−	0.405124i	$-0.132772\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 6.39819i	− 0.377673i
$288$	0	0
$289$	16.7543	0.985546
$290$	0	0
$291$	11.4828	0.673132
$292$	0	0
$293$	25.2652i	1.47601i	0.674796	+	0.738004i	$0.264232\pi$
$293$	25.2652i	1.47601i	−0.674796	+	0.738004i	$0.735768\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 6.17438i	− 0.358274i
$298$	0	0
$299$	−10.8847	−0.629477
$300$	0	0
$301$	−5.45554	−0.314452
$302$	0	0
$303$	14.8359i	0.852297i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	7.01023i	0.400095i	0.979786	+	0.200047i	$0.0641096\pi$
$307$	7.01023i	0.400095i	−0.979786	+	0.200047i	$0.935890\pi$
$308$	0	0
$309$	19.2645	1.09592
$310$	0	0
$311$	14.5118	0.822891	0.411446	−	0.911434i	$-0.365024\pi$
$311$	14.5118	0.822891	0.411446	+	0.911434i	$0.365024\pi$
$312$	0	0
$313$	21.9197i	1.23897i	0.785008	+	0.619486i	$0.212659\pi$
$313$	21.9197i	1.23897i	−0.785008	+	0.619486i	$0.787341\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	33.7178i	1.89378i	0.321559	+	0.946890i	$0.395793\pi$
$317$	33.7178i	1.89378i	−0.321559	+	0.946890i	$0.604207\pi$
$318$	0	0
$319$	−0.755333	−0.0422905
$320$	0	0
$321$	−8.63523	−0.481972
$322$	0	0
$323$	0.154497i	0.00859645i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 18.6893i	− 1.03352i
$328$	0	0
$329$	−7.94593	−0.438073
$330$	0	0
$331$	33.5950	1.84655	0.923275	−	0.384139i	$-0.125502\pi$
$331$	33.5950	1.84655	0.923275	+	0.384139i	$0.125502\pi$
$332$	0	0
$333$	− 4.36700i	− 0.239310i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 3.08526i	− 0.168065i	−0.996463	−	0.0840323i	$-0.973220\pi$
$337$	− 3.08526i	− 0.168065i	0.996463	−	0.0840323i	$-0.0267799\pi$
$338$	0	0
$339$	6.05638	0.328937
$340$	0	0
$341$	18.1750	0.984233
$342$	0	0
$343$	− 17.6561i	− 0.953339i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.5739i	− 0.675005i	−0.941325	−	0.337502i	$-0.890418\pi$
$347$	− 12.5739i	− 0.675005i	0.941325	−	0.337502i	$-0.109582\pi$
$348$	0	0
$349$	−13.3100	−0.712466	−0.356233	−	0.934397i	$-0.615939\pi$
$349$	−13.3100	−0.712466	−0.356233	+	0.934397i	$0.615939\pi$
$350$	0	0
$351$	3.55634	0.189823
$352$	0	0
$353$	− 11.2914i	− 0.600980i	−0.953785	−	0.300490i	$-0.902850\pi$
$353$	− 11.2914i	− 0.600980i	0.953785	−	0.300490i	$-0.0971502\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.745682i	0.0394657i
$358$	0	0
$359$	−19.4072	−1.02427	−0.512135	−	0.858905i	$-0.671145\pi$
$359$	−19.4072	−1.02427	−0.512135	+	0.858905i	$0.671145\pi$
$360$	0	0
$361$	−18.9029	−0.994887
$362$	0	0
$363$	− 27.1229i	− 1.42358i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 17.5703i	− 0.917160i	−0.888653	−	0.458580i	$-0.848358\pi$
$367$	− 17.5703i	− 0.917160i	0.888653	−	0.458580i	$-0.151642\pi$
$368$	0	0
$369$	−4.25327	−0.221416
$370$	0	0
$371$	−12.9367	−0.671637
$372$	0	0
$373$	− 28.2065i	− 1.46048i	−0.683193	−	0.730238i	$-0.739409\pi$
$373$	− 28.2065i	− 1.46048i	0.683193	−	0.730238i	$-0.260591\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 0.435060i	− 0.0224067i
$378$	0	0
$379$	−3.62822	−0.186369	−0.0931845	−	0.995649i	$-0.529705\pi$
$379$	−3.62822	−0.186369	−0.0931845	+	0.995649i	$0.529705\pi$
$380$	0	0
$381$	9.24565	0.473669
$382$	0	0
$383$	− 13.8348i	− 0.706925i	−0.935449	−	0.353463i	$-0.885004\pi$
$383$	− 13.8348i	− 0.706925i	0.935449	−	0.353463i	$-0.114996\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.62663i	0.184352i
$388$	0	0
$389$	20.9014	1.05974	0.529871	−	0.848079i	$-0.322241\pi$
$389$	20.9014	1.05974	0.529871	+	0.848079i	$0.322241\pi$
$390$	0	0
$391$	−1.51716	−0.0767261
$392$	0	0
$393$	− 19.0155i	− 0.959207i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	15.4522i	0.775523i	0.921760	+	0.387761i	$0.126752\pi$
$397$	15.4522i	0.775523i	−0.921760	+	0.387761i	$0.873248\pi$
$398$	0	0
$399$	0.468851	0.0234719
$400$	0	0
$401$	2.25590	0.112654	0.0563271	−	0.998412i	$-0.482061\pi$
$401$	2.25590	0.112654	0.0563271	+	0.998412i	$0.482061\pi$
$402$	0	0
$403$	10.4685i	0.521474i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 26.9635i	− 1.33653i
$408$	0	0
$409$	2.33260	0.115340	0.0576699	−	0.998336i	$-0.481633\pi$
$409$	2.33260	0.115340	0.0576699	+	0.998336i	$0.481633\pi$
$410$	0	0
$411$	−7.86205	−0.387806
$412$	0	0
$413$	19.3793i	0.953591i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 13.8004i	− 0.675806i
$418$	0	0
$419$	5.04076	0.246257	0.123129	−	0.992391i	$-0.460707\pi$
$419$	5.04076	0.246257	0.123129	+	0.992391i	$0.460707\pi$
$420$	0	0
$421$	12.9386	0.630587	0.315293	−	0.948994i	$-0.397897\pi$
$421$	12.9386	0.630587	0.315293	+	0.948994i	$0.397897\pi$
$422$	0	0
$423$	5.28215i	0.256827i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	17.0301i	0.824142i
$428$	0	0
$429$	21.9582	1.06015
$430$	0	0
$431$	1.56501	0.0753841	0.0376920	−	0.999289i	$-0.487999\pi$
$431$	1.56501	0.0753841	0.0376920	+	0.999289i	$0.487999\pi$
$432$	0	0
$433$	25.8729i	1.24337i	0.783266	+	0.621687i	$0.213552\pi$
$433$	25.8729i	1.24337i	−0.783266	+	0.621687i	$0.786448\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.953923i	0.0456323i
$438$	0	0
$439$	−16.3713	−0.781358	−0.390679	−	0.920527i	$-0.627760\pi$
$439$	−16.3713	−0.781358	−0.390679	+	0.920527i	$0.627760\pi$
$440$	0	0
$441$	−4.73708	−0.225575
$442$	0	0
$443$	28.1180i	1.33593i	0.744194	+	0.667963i	$0.232834\pi$
$443$	28.1180i	1.33593i	−0.744194	+	0.667963i	$0.767166\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.12168i	0.242247i
$448$	0	0
$449$	2.65681	0.125383	0.0626914	−	0.998033i	$-0.480032\pi$
$449$	2.65681	0.125383	0.0626914	+	0.998033i	$0.480032\pi$
$450$	0	0
$451$	−26.2613	−1.23659
$452$	0	0
$453$	12.9476i	0.608329i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.321574i	0.0150426i	0.999972	+	0.00752129i	$0.00239412\pi$
$457$	0.321574i	0.0150426i	−0.999972	+	0.00752129i	$0.997606\pi$
$458$	0	0
$459$	0.495700	0.0231373
$460$	0	0
$461$	−7.68562	−0.357955	−0.178977	−	0.983853i	$-0.557279\pi$
$461$	−7.68562	−0.357955	−0.178977	+	0.983853i	$0.557279\pi$
$462$	0	0
$463$	19.1041i	0.887842i	0.896066	+	0.443921i	$0.146413\pi$
$463$	19.1041i	0.887842i	−0.896066	+	0.443921i	$0.853587\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.41944i	0.297056i	0.988908	+	0.148528i	$0.0474536\pi$
$467$	6.41944i	0.297056i	−0.988908	+	0.148528i	$0.952546\pi$
$468$	0	0
$469$	−19.9190	−0.919774
$470$	0	0
$471$	4.02750	0.185577
$472$	0	0
$473$	22.3922i	1.02959i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.59978i	0.393757i
$478$	0	0
$479$	−33.5609	−1.53344	−0.766719	−	0.641983i	$-0.778112\pi$
$479$	−33.5609	−1.53344	−0.766719	+	0.641983i	$0.778112\pi$
$480$	0	0
$481$	15.5305	0.708132
$482$	0	0
$483$	4.60412i	0.209495i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	11.9105i	0.539715i	0.962900	+	0.269858i	$0.0869767\pi$
$487$	11.9105i	0.539715i	−0.962900	+	0.269858i	$0.913023\pi$
$488$	0	0
$489$	15.6190	0.706316
$490$	0	0
$491$	15.6130	0.704607	0.352303	−	0.935886i	$-0.385398\pi$
$491$	15.6130	0.704607	0.352303	+	0.935886i	$0.385398\pi$
$492$	0	0
$493$	− 0.0606408i	− 0.00273112i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4.91640i	− 0.220531i
$498$	0	0
$499$	−30.9130	−1.38386	−0.691929	−	0.721966i	$-0.743239\pi$
$499$	−30.9130	−1.38386	−0.691929	+	0.721966i	$0.743239\pi$
$500$	0	0
$501$	−11.8445	−0.529171
$502$	0	0
$503$	31.0396i	1.38399i	0.721903	+	0.691994i	$0.243268\pi$
$503$	31.0396i	1.38399i	−0.721903	+	0.691994i	$0.756732\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 0.352437i	− 0.0156523i
$508$	0	0
$509$	13.3428	0.591409	0.295704	−	0.955280i	$-0.404446\pi$
$509$	13.3428	0.591409	0.295704	+	0.955280i	$0.404446\pi$
$510$	0	0
$511$	24.0188	1.06253
$512$	0	0
$513$	− 0.311674i	− 0.0137608i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	32.6139i	1.43436i
$518$	0	0
$519$	14.8305	0.650988
$520$	0	0
$521$	26.5176	1.16176	0.580879	−	0.813990i	$-0.302709\pi$
$521$	26.5176	1.16176	0.580879	+	0.813990i	$0.302709\pi$
$522$	0	0
$523$	− 17.4147i	− 0.761492i	−0.924680	−	0.380746i	$-0.875667\pi$
$523$	− 17.4147i	− 0.761492i	0.924680	−	0.380746i	$-0.124333\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.45915i	0.0635618i
$528$	0	0
$529$	13.6325	0.592716
$530$	0	0
$531$	12.8826	0.559056
$532$	0	0
$533$	− 15.1261i	− 0.655183i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 0.957535i	− 0.0413207i
$538$	0	0
$539$	−29.2485	−1.25982
$540$	0	0
$541$	5.46682	0.235037	0.117519	−	0.993071i	$-0.462506\pi$
$541$	5.46682	0.235037	0.117519	+	0.993071i	$0.462506\pi$
$542$	0	0
$543$	− 4.14123i	− 0.177717i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 21.6459i	− 0.925510i	−0.886486	−	0.462755i	$-0.846861\pi$
$547$	− 21.6459i	− 0.925510i	0.886486	−	0.462755i	$-0.153139\pi$
$548$	0	0
$549$	11.3209	0.483165
$550$	0	0
$551$	−0.0381282	−0.00162432
$552$	0	0
$553$	− 12.8279i	− 0.545496i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.58830i	0.109670i	0.998495	+	0.0548349i	$0.0174633\pi$
$557$	2.58830i	0.109670i	−0.998495	+	0.0548349i	$0.982537\pi$
$558$	0	0
$559$	−12.8975	−0.545508
$560$	0	0
$561$	3.06064	0.129220
$562$	0	0
$563$	19.8939i	0.838426i	0.907888	+	0.419213i	$0.137694\pi$
$563$	19.8939i	0.838426i	−0.907888	+	0.419213i	$0.862306\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.50430i	− 0.0631746i
$568$	0	0
$569$	−32.4303	−1.35955	−0.679775	−	0.733421i	$-0.737923\pi$
$569$	−32.4303	−1.35955	−0.679775	+	0.733421i	$0.737923\pi$
$570$	0	0
$571$	13.7182	0.574088	0.287044	−	0.957917i	$-0.407327\pi$
$571$	13.7182	0.574088	0.287044	+	0.957917i	$0.407327\pi$
$572$	0	0
$573$	17.6305i	0.736524i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 24.6432i	− 1.02591i	−0.858415	−	0.512955i	$-0.828551\pi$
$577$	− 24.6432i	− 1.02591i	0.858415	−	0.512955i	$-0.171449\pi$
$578$	0	0
$579$	5.25392	0.218345
$580$	0	0
$581$	17.0559	0.707599
$582$	0	0
$583$	53.0983i	2.19911i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 8.32165i	− 0.343471i	−0.985143	−	0.171736i	$-0.945062\pi$
$587$	− 8.32165i	− 0.343471i	0.985143	−	0.171736i	$-0.0549375\pi$
$588$	0	0
$589$	0.917451	0.0378029
$590$	0	0
$591$	2.11597	0.0870393
$592$	0	0
$593$	21.7904i	0.894826i	0.894328	+	0.447413i	$0.147654\pi$
$593$	21.7904i	0.894826i	−0.894328	+	0.447413i	$0.852346\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.07029i	0.371222i
$598$	0	0
$599$	−10.6482	−0.435074	−0.217537	−	0.976052i	$-0.569802\pi$
$599$	−10.6482	−0.435074	−0.217537	+	0.976052i	$0.569802\pi$
$600$	0	0
$601$	2.60740	0.106358	0.0531791	−	0.998585i	$-0.483065\pi$
$601$	2.60740	0.106358	0.0531791	+	0.998585i	$0.483065\pi$
$602$	0	0
$603$	13.2414i	0.539231i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 32.9820i	− 1.33870i	−0.742948	−	0.669349i	$-0.766573\pi$
$607$	− 32.9820i	− 1.33870i	0.742948	−	0.669349i	$-0.233427\pi$
$608$	0	0
$609$	−0.184026	−0.00745712
$610$	0	0
$611$	−18.7851	−0.759964
$612$	0	0
$613$	− 18.7891i	− 0.758886i	−0.925215	−	0.379443i	$-0.876116\pi$
$613$	− 18.7891i	− 0.758886i	0.925215	−	0.379443i	$-0.123884\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 39.3278i	− 1.58328i	−0.610990	−	0.791639i	$-0.709228\pi$
$617$	− 39.3278i	− 1.58328i	0.610990	−	0.791639i	$-0.290772\pi$
$618$	0	0
$619$	16.0365	0.644560	0.322280	−	0.946644i	$-0.395551\pi$
$619$	16.0365	0.644560	0.322280	+	0.946644i	$0.395551\pi$
$620$	0	0
$621$	3.06064	0.122819
$622$	0	0
$623$	21.2596i	0.851748i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.92439i	− 0.0768529i
$628$	0	0
$629$	2.16472	0.0863132
$630$	0	0
$631$	4.82685	0.192154	0.0960770	−	0.995374i	$-0.469371\pi$
$631$	4.82685	0.192154	0.0960770	+	0.995374i	$0.469371\pi$
$632$	0	0
$633$	19.8149i	0.787572i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 16.8467i	− 0.667490i
$638$	0	0
$639$	−3.26823	−0.129289
$640$	0	0
$641$	−36.5645	−1.44421	−0.722106	−	0.691782i	$-0.756826\pi$
$641$	−36.5645	−1.44421	−0.722106	+	0.691782i	$0.756826\pi$
$642$	0	0
$643$	− 18.3769i	− 0.724714i	−0.932039	−	0.362357i	$-0.881972\pi$
$643$	− 18.3769i	− 0.724714i	0.932039	−	0.362357i	$-0.118028\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	41.1338i	1.61714i	0.588403	+	0.808568i	$0.299757\pi$
$647$	41.1338i	1.61714i	−0.588403	+	0.808568i	$0.700243\pi$
$648$	0	0
$649$	79.5419	3.12229
$650$	0	0
$651$	4.42809	0.173550
$652$	0	0
$653$	− 21.9447i	− 0.858761i	−0.903124	−	0.429381i	$-0.858732\pi$
$653$	− 21.9447i	− 0.858761i	0.903124	−	0.429381i	$-0.141268\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 15.9668i	− 0.622924i
$658$	0	0
$659$	−21.2048	−0.826020	−0.413010	−	0.910726i	$-0.635523\pi$
$659$	−21.2048	−0.826020	−0.413010	+	0.910726i	$0.635523\pi$
$660$	0	0
$661$	17.9442	0.697948	0.348974	−	0.937132i	$-0.386530\pi$
$661$	17.9442	0.697948	0.348974	+	0.937132i	$0.386530\pi$
$662$	0	0
$663$	1.76288i	0.0684646i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 0.374419i	− 0.0144976i
$668$	0	0
$669$	6.58214	0.254480
$670$	0	0
$671$	69.8996	2.69845
$672$	0	0
$673$	34.3014i	1.32222i	0.750288	+	0.661111i	$0.229915\pi$
$673$	34.3014i	1.32222i	−0.750288	+	0.661111i	$0.770085\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.14695i	0.159380i	0.996820	+	0.0796901i	$0.0253931\pi$
$677$	4.14695i	0.159380i	−0.996820	+	0.0796901i	$0.974607\pi$
$678$	0	0
$679$	17.2735	0.662897
$680$	0	0
$681$	−11.7821	−0.451491
$682$	0	0
$683$	2.11840i	0.0810583i	0.999178	+	0.0405291i	$0.0129044\pi$
$683$	2.11840i	0.0810583i	−0.999178	+	0.0405291i	$0.987096\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	11.8899i	0.453627i
$688$	0	0
$689$	−30.5838	−1.16515
$690$	0	0
$691$	−3.97453	−0.151198	−0.0755991	−	0.997138i	$-0.524087\pi$
$691$	−3.97453	−0.151198	−0.0755991	+	0.997138i	$0.524087\pi$
$692$	0	0
$693$	− 9.28811i	− 0.352826i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 2.10835i	− 0.0798593i
$698$	0	0
$699$	−5.05001	−0.191009
$700$	0	0
$701$	−10.1954	−0.385076	−0.192538	−	0.981289i	$-0.561672\pi$
$701$	−10.1954	−0.385076	−0.192538	+	0.981289i	$0.561672\pi$
$702$	0	0
$703$	− 1.36108i	− 0.0513342i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	22.3176i	0.839338i
$708$	0	0
$709$	−29.8413	−1.12071	−0.560357	−	0.828251i	$-0.689336\pi$
$709$	−29.8413	−1.12071	−0.560357	+	0.828251i	$0.689336\pi$
$710$	0	0
$711$	−8.52746	−0.319805
$712$	0	0
$713$	9.00937i	0.337404i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.67932i	0.100061i
$718$	0	0
$719$	−7.43434	−0.277254	−0.138627	−	0.990345i	$-0.544269\pi$
$719$	−7.43434	−0.277254	−0.138627	+	0.990345i	$0.544269\pi$
$720$	0	0
$721$	28.9796	1.07926
$722$	0	0
$723$	− 12.6561i	− 0.470685i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	15.9195i	0.590420i	0.955432	+	0.295210i	$0.0953896\pi$
$727$	15.9195i	0.590420i	−0.955432	+	0.295210i	$0.904610\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−1.79772	−0.0664912
$732$	0	0
$733$	30.6103i	1.13062i	0.824880	+	0.565308i	$0.191243\pi$
$733$	30.6103i	1.13062i	−0.824880	+	0.565308i	$0.808757\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	81.7573i	3.01157i
$738$	0	0
$739$	−25.9379	−0.954141	−0.477071	−	0.878865i	$-0.658301\pi$
$739$	−25.9379	−0.954141	−0.477071	+	0.878865i	$0.658301\pi$
$740$	0	0
$741$	1.10842	0.0407188
$742$	0	0
$743$	17.2136i	0.631504i	0.948842	+	0.315752i	$0.102257\pi$
$743$	17.2136i	0.631504i	−0.948842	+	0.315752i	$0.897743\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 11.3381i	− 0.414840i
$748$	0	0
$749$	−12.9900	−0.474643
$750$	0	0
$751$	10.4020	0.379576	0.189788	−	0.981825i	$-0.439220\pi$
$751$	10.4020	0.379576	0.189788	+	0.981825i	$0.439220\pi$
$752$	0	0
$753$	22.9068i	0.834770i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	16.9118i	0.614669i	0.951601	+	0.307335i	$0.0994371\pi$
$757$	16.9118i	0.614669i	−0.951601	+	0.307335i	$0.900563\pi$
$758$	0	0
$759$	18.8975	0.685937
$760$	0	0
$761$	0.522347	0.0189351	0.00946753	−	0.999955i	$-0.496986\pi$
$761$	0.522347	0.0189351	0.00946753	+	0.999955i	$0.496986\pi$
$762$	0	0
$763$	− 28.1143i	− 1.01781i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	45.8148i	1.65428i
$768$	0	0
$769$	38.0791	1.37317	0.686584	−	0.727050i	$-0.259109\pi$
$769$	38.0791	1.37317	0.686584	+	0.727050i	$0.259109\pi$
$770$	0	0
$771$	27.7764	1.00034
$772$	0	0
$773$	− 22.6913i	− 0.816151i	−0.912948	−	0.408075i	$-0.866200\pi$
$773$	− 22.6913i	− 0.816151i	0.912948	−	0.408075i	$-0.133800\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 6.56928i	− 0.235671i
$778$	0	0
$779$	−1.32563	−0.0474957
$780$	0	0
$781$	−20.1793	−0.722072
$782$	0	0
$783$	0.122334i	0.00437184i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 2.27911i	− 0.0812415i	−0.999175	−	0.0406207i	$-0.987066\pi$
$787$	− 2.27911i	− 0.0812415i	0.999175	−	0.0406207i	$-0.0129335\pi$
$788$	0	0
$789$	11.7115	0.416940
$790$	0	0
$791$	9.11061	0.323936
$792$	0	0
$793$	40.2611i	1.42971i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 19.6409i	− 0.695715i	−0.937547	−	0.347858i	$-0.886909\pi$
$797$	− 19.6409i	− 0.695715i	0.937547	−	0.347858i	$-0.113091\pi$
$798$	0	0
$799$	−2.61836	−0.0926310
$800$	0	0
$801$	14.1326	0.499349
$802$	0	0
$803$	− 98.5849i	− 3.47899i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.20621i	0.148066i
$808$	0	0
$809$	−11.1951	−0.393599	−0.196799	−	0.980444i	$-0.563055\pi$
$809$	−11.1951	−0.393599	−0.196799	+	0.980444i	$0.563055\pi$
$810$	0	0
$811$	−19.2989	−0.677677	−0.338839	−	0.940844i	$-0.610034\pi$
$811$	−19.2989	−0.677677	−0.338839	+	0.940844i	$0.610034\pi$
$812$	0	0
$813$	12.5659i	0.440706i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.13033i	0.0395452i
$818$	0	0
$819$	5.34980	0.186937
$820$	0	0
$821$	−48.1255	−1.67959	−0.839797	−	0.542901i	$-0.817326\pi$
$821$	−48.1255	−1.67959	−0.839797	+	0.542901i	$0.817326\pi$
$822$	0	0
$823$	34.9012i	1.21658i	0.793716	+	0.608289i	$0.208144\pi$
$823$	34.9012i	1.21658i	−0.793716	+	0.608289i	$0.791856\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 23.1640i	− 0.805491i	−0.915312	−	0.402746i	$-0.868056\pi$
$827$	− 23.1640i	− 0.805491i	0.915312	−	0.402746i	$-0.131944\pi$
$828$	0	0
$829$	−21.7129	−0.754121	−0.377060	−	0.926189i	$-0.623065\pi$
$829$	−21.7129	−0.754121	−0.377060	+	0.926189i	$0.623065\pi$
$830$	0	0
$831$	−31.8653	−1.10539
$832$	0	0
$833$	− 2.34817i	− 0.0813594i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2.94362i	− 0.101746i
$838$	0	0
$839$	−10.3890	−0.358668	−0.179334	−	0.983788i	$-0.557394\pi$
$839$	−10.3890	−0.358668	−0.179334	+	0.983788i	$0.557394\pi$
$840$	0	0
$841$	−28.9850	−0.999484
$842$	0	0
$843$	− 16.2998i	− 0.561395i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 40.8010i	− 1.40194i
$848$	0	0
$849$	−13.6305	−0.467797
$850$	0	0
$851$	13.3658	0.458174
$852$	0	0
$853$	33.6101i	1.15079i	0.817876	+	0.575395i	$0.195152\pi$
$853$	33.6101i	1.15079i	−0.817876	+	0.575395i	$0.804848\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 46.0078i	− 1.57160i	−0.618482	−	0.785799i	$-0.712252\pi$
$857$	− 46.0078i	− 1.57160i	0.618482	−	0.785799i	$-0.287748\pi$
$858$	0	0
$859$	30.3516	1.03558	0.517791	−	0.855507i	$-0.326755\pi$
$859$	30.3516	1.03558	0.517791	+	0.855507i	$0.326755\pi$
$860$	0	0
$861$	−6.39819	−0.218050
$862$	0	0
$863$	− 7.32052i	− 0.249193i	−0.992207	−	0.124597i	$-0.960236\pi$
$863$	− 7.32052i	− 0.249193i	0.992207	−	0.124597i	$-0.0397637\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 16.7543i	− 0.569005i
$868$	0	0
$869$	−52.6518	−1.78609
$870$	0	0
$871$	−47.0909	−1.59561
$872$	0	0
$873$	− 11.4828i	− 0.388633i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.5437i	1.57167i	0.618438	+	0.785834i	$0.287766\pi$
$877$	46.5437i	1.57167i	−0.618438	+	0.785834i	$0.712234\pi$
$878$	0	0
$879$	25.2652	0.852174
$880$	0	0
$881$	−14.9721	−0.504423	−0.252211	−	0.967672i	$-0.581158\pi$
$881$	−14.9721	−0.504423	−0.252211	+	0.967672i	$0.581158\pi$
$882$	0	0
$883$	− 8.74240i	− 0.294205i	−0.989121	−	0.147103i	$-0.953005\pi$
$883$	− 8.74240i	− 0.294205i	0.989121	−	0.147103i	$-0.0469947\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 16.4865i	− 0.553563i	−0.960933	−	0.276782i	$-0.910732\pi$
$887$	− 16.4865i	− 0.553563i	0.960933	−	0.276782i	$-0.0892679\pi$
$888$	0	0
$889$	13.9082	0.466467
$890$	0	0
$891$	−6.17438	−0.206849
$892$	0	0
$893$	1.64631i	0.0550916i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	10.8847i	0.363429i
$898$	0	0
$899$	−0.360104	−0.0120101
$900$	0	0
$901$	−4.26292	−0.142018
$902$	0	0
$903$	5.45554i	0.181549i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 26.0243i	− 0.864124i	−0.901844	−	0.432062i	$-0.857786\pi$
$907$	− 26.0243i	− 0.864124i	0.901844	−	0.432062i	$-0.142214\pi$
$908$	0	0
$909$	14.8359	0.492074
$910$	0	0
$911$	−51.6447	−1.71107	−0.855533	−	0.517748i	$-0.826771\pi$
$911$	−51.6447	−1.71107	−0.855533	+	0.517748i	$0.826771\pi$
$912$	0	0
$913$	− 70.0058i	− 2.31685i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 28.6051i	− 0.944623i
$918$	0	0
$919$	27.6307	0.911452	0.455726	−	0.890120i	$-0.349380\pi$
$919$	27.6307	0.911452	0.455726	+	0.890120i	$0.349380\pi$
$920$	0	0
$921$	7.01023	0.230995
$922$	0	0
$923$	− 11.6229i	− 0.382574i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 19.2645i	− 0.632731i
$928$	0	0
$929$	−53.7012	−1.76188	−0.880939	−	0.473230i	$-0.843088\pi$
$929$	−53.7012	−1.76188	−0.880939	+	0.473230i	$0.843088\pi$
$930$	0	0
$931$	−1.47643	−0.0483879
$932$	0	0
$933$	− 14.5118i	− 0.475097i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	30.3213i	0.990552i	0.868736	+	0.495276i	$0.164933\pi$
$937$	30.3213i	0.990552i	−0.868736	+	0.495276i	$0.835067\pi$
$938$	0	0
$939$	21.9197	0.715321
$940$	0	0
$941$	47.3682	1.54416	0.772080	−	0.635525i	$-0.219216\pi$
$941$	47.3682	1.54416	0.772080	+	0.635525i	$0.219216\pi$
$942$	0	0
$943$	− 13.0177i	− 0.423915i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21.6419i	0.703266i	0.936138	+	0.351633i	$0.114373\pi$
$947$	21.6419i	0.703266i	−0.936138	+	0.351633i	$0.885627\pi$
$948$	0	0
$949$	56.7833	1.84327
$950$	0	0
$951$	33.7178	1.09337
$952$	0	0
$953$	− 45.8917i	− 1.48658i	−0.668970	−	0.743289i	$-0.733265\pi$
$953$	− 45.8917i	− 1.48658i	0.668970	−	0.743289i	$-0.266735\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.755333i	0.0244164i
$958$	0	0
$959$	−11.8269	−0.381910
$960$	0	0
$961$	−22.3351	−0.720487
$962$	0	0
$963$	8.63523i	0.278266i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	14.9696i	0.481391i	0.970601	+	0.240695i	$0.0773754\pi$
$967$	14.9696i	0.481391i	−0.970601	+	0.240695i	$0.922625\pi$
$968$	0	0
$969$	0.154497	0.00496316
$970$	0	0
$971$	−22.3795	−0.718191	−0.359096	−	0.933301i	$-0.616915\pi$
$971$	−22.3795	−0.718191	−0.359096	+	0.933301i	$0.616915\pi$
$972$	0	0
$973$	− 20.7599i	− 0.665531i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	40.3686i	1.29150i	0.763547	+	0.645752i	$0.223456\pi$
$977$	40.3686i	1.29150i	−0.763547	+	0.645752i	$0.776544\pi$
$978$	0	0
$979$	87.2597	2.78883
$980$	0	0
$981$	−18.6893	−0.596704
$982$	0	0
$983$	− 23.3730i	− 0.745482i	−0.927935	−	0.372741i	$-0.878418\pi$
$983$	− 23.3730i	− 0.745482i	0.927935	−	0.372741i	$-0.121582\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	7.94593i	0.252922i
$988$	0	0
$989$	−11.0998	−0.352954
$990$	0	0
$991$	33.8454	1.07514	0.537568	−	0.843221i	$-0.319343\pi$
$991$	33.8454	1.07514	0.537568	+	0.843221i	$0.319343\pi$
$992$	0	0
$993$	− 33.5950i	− 1.06611i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 5.54503i	− 0.175613i	−0.996138	−	0.0878065i	$-0.972014\pi$
$997$	− 5.54503i	− 0.175613i	0.996138	−	0.0878065i	$-0.0279857\pi$
$998$	0	0
$999$	−4.36700	−0.138166

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.d.c.1249.2		8
5.2	odd	4		7500.2.a.e.1.3		4
5.3	odd	4		7500.2.a.f.1.2		4
5.4	even	2	inner	7500.2.d.c.1249.7		8
25.3	odd	20		1500.2.m.a.1201.1		8
25.4	even	10		1500.2.o.b.49.2		16
25.6	even	5		1500.2.o.b.949.1		16
25.8	odd	20		1500.2.m.a.301.1		8
25.17	odd	20		300.2.m.b.61.2	✓	8
25.19	even	10		1500.2.o.b.949.4		16
25.21	even	5		1500.2.o.b.49.3		16
25.22	odd	20		300.2.m.b.241.2	yes	8
75.17	even	20		900.2.n.b.361.1		8
75.47	even	20		900.2.n.b.541.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.m.b.61.2	✓	8	25.17	odd	20
300.2.m.b.241.2	yes	8	25.22	odd	20
900.2.n.b.361.1		8	75.17	even	20
900.2.n.b.541.1		8	75.47	even	20
1500.2.m.a.301.1		8	25.8	odd	20
1500.2.m.a.1201.1		8	25.3	odd	20
1500.2.o.b.49.2		16	25.4	even	10
1500.2.o.b.49.3		16	25.21	even	5
1500.2.o.b.949.1		16	25.6	even	5
1500.2.o.b.949.4		16	25.19	even	10
7500.2.a.e.1.3		4	5.2	odd	4
7500.2.a.f.1.2		4	5.3	odd	4
7500.2.d.c.1249.2		8	1.1	even	1	trivial
7500.2.d.c.1249.7		8	5.4	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:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.50430i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.50430i q^{7} -1.00000 q^{9} -6.17438 q^{11} -3.55634i q^{13} -0.495700i q^{17} -0.311674 q^{19} -1.50430 q^{21} -3.06064i q^{23} +1.00000i q^{27} +0.122334 q^{29} -2.94362 q^{31} +6.17438i q^{33} +4.36700i q^{37} -3.55634 q^{39} +4.25327 q^{41} -3.62663i q^{43} -5.28215i q^{47} +4.73708 q^{49} -0.495700 q^{51} -8.59978i q^{53} +0.311674i q^{57} -12.8826 q^{59} -11.3209 q^{61} +1.50430i q^{63} -13.2414i q^{67} -3.06064 q^{69} +3.26823 q^{71} +15.9668i q^{73} +9.28811i q^{77} +8.52746 q^{79} +1.00000 q^{81} +11.3381i q^{83} -0.122334i q^{87} -14.1326 q^{89} -5.34980 q^{91} +2.94362i q^{93} +11.4828i q^{97} +6.17438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 2 q^{11} + 10 q^{19} - 8 q^{21} - 12 q^{29} + 22 q^{31} + 10 q^{39} + 8 q^{49} - 8 q^{51} - 2 q^{59} - 44 q^{61} + 18 q^{69} + 40 q^{71} + 6 q^{79} + 8 q^{81} + 30 q^{89} - 20 q^{91} + 2 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 - 2 * q^11 + 10 * q^19 - 8 * q^21 - 12 * q^29 + 22 * q^31 + 10 * q^39 + 8 * q^49 - 8 * q^51 - 2 * q^59 - 44 * q^61 + 18 * q^69 + 40 * q^71 + 6 * q^79 + 8 * q^81 + 30 * q^89 - 20 * q^91 + 2 * q^99

Introduction

L-functions

Modular forms

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Database

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