LMFDB - Embedded newform 7616.2.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7616,2,Mod(1,7616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7616, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7616.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7616 = 2^{6} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7616.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:	$60.8140661794$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.453749.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 10x^{2} + x - 3 $ x^5 - 2x^4 - 5x^3 + 10*x^2 + x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 119)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.32183$ of defining polynomial
Character	$\chi$	$=$	7616.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.907578 q^{3} -3.03818 q^{5} -1.00000 q^{7} -2.17630 q^{9} +O(q^{10})$	$q-0.907578 q^{3} -3.03818 q^{5} -1.00000 q^{7} -2.17630 q^{9} -4.78178 q^{11} +4.39933 q^{13} +2.75739 q^{15} +1.00000 q^{17} +2.64366 q^{19} +0.907578 q^{21} -8.45881 q^{23} +4.23054 q^{25} +4.69790 q^{27} -7.04298 q^{29} -3.42063 q^{31} +4.33984 q^{33} +3.03818 q^{35} -9.66977 q^{37} -3.99273 q^{39} +1.33675 q^{41} -2.52513 q^{43} +6.61200 q^{45} -5.57082 q^{47} +1.00000 q^{49} -0.907578 q^{51} -5.17630 q^{53} +14.5279 q^{55} -2.39933 q^{57} -9.28732 q^{59} -6.66325 q^{61} +2.17630 q^{63} -13.3659 q^{65} -5.28251 q^{67} +7.67704 q^{69} -11.9310 q^{71} +12.7155 q^{73} -3.83955 q^{75} +4.78178 q^{77} +1.94051 q^{79} +2.26519 q^{81} +4.58417 q^{83} -3.03818 q^{85} +6.39206 q^{87} -13.3956 q^{89} -4.39933 q^{91} +3.10449 q^{93} -8.03191 q^{95} -15.1668 q^{97} +10.4066 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{3} - 5 q^{7} + 11 q^{9}+O(q^{10}) $ 5 * q + 2 * q^3 - 5 * q^7 + 11 * q^9	$ 5 q + 2 q^{3} - 5 q^{7} + 11 q^{9} + 2 q^{11} - 2 q^{13} + 8 q^{15} + 5 q^{17} - 6 q^{19} - 2 q^{21} - 10 q^{23} + 21 q^{25} + 26 q^{27} + 8 q^{29} + 6 q^{33} - 8 q^{37} + 14 q^{39} + 18 q^{41} - 8 q^{43} - 10 q^{47} + 5 q^{49} + 2 q^{51} - 4 q^{53} - 24 q^{55} + 12 q^{57} - 8 q^{59} - 22 q^{61} - 11 q^{63} - 30 q^{65} - 16 q^{67} + 32 q^{69} - 2 q^{71} + 10 q^{73} + 14 q^{75} - 2 q^{77} + 18 q^{79} + 25 q^{81} + 12 q^{83} - 26 q^{87} + 20 q^{89} + 2 q^{91} + 28 q^{93} - 22 q^{95} + 12 q^{97} + 62 q^{99}+O(q^{100}) $ 5 * q + 2 * q^3 - 5 * q^7 + 11 * q^9 + 2 * q^11 - 2 * q^13 + 8 * q^15 + 5 * q^17 - 6 * q^19 - 2 * q^21 - 10 * q^23 + 21 * q^25 + 26 * q^27 + 8 * q^29 + 6 * q^33 - 8 * q^37 + 14 * q^39 + 18 * q^41 - 8 * q^43 - 10 * q^47 + 5 * q^49 + 2 * q^51 - 4 * q^53 - 24 * q^55 + 12 * q^57 - 8 * q^59 - 22 * q^61 - 11 * q^63 - 30 * q^65 - 16 * q^67 + 32 * q^69 - 2 * q^71 + 10 * q^73 + 14 * q^75 - 2 * q^77 + 18 * q^79 + 25 * q^81 + 12 * q^83 - 26 * q^87 + 20 * q^89 + 2 * q^91 + 28 * q^93 - 22 * q^95 + 12 * q^97 + 62 * 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
$3$	−0.907578	−0.523991	−0.261995	−	0.965069i	$-0.584380\pi$
$3$	−0.907578	−0.523991	−0.261995	+	0.965069i	$0.584380\pi$
$4$	0	0
$5$	−3.03818	−1.35872	−0.679358	−	0.733807i	$-0.737742\pi$
$5$	−3.03818	−1.35872	−0.679358	+	0.733807i	$0.737742\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.17630	−0.725434
$10$	0	0
$11$	−4.78178	−1.44176	−0.720880	−	0.693060i	$-0.756262\pi$
$11$	−4.78178	−1.44176	−0.720880	+	0.693060i	$0.756262\pi$
$12$	0	0
$13$	4.39933	1.22015	0.610077	−	0.792342i	$-0.291139\pi$
$13$	4.39933	1.22015	0.610077	+	0.792342i	$0.291139\pi$
$14$	0	0
$15$	2.75739	0.711954
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	2.64366	0.606497	0.303248	−	0.952912i	$-0.401929\pi$
$19$	2.64366	0.606497	0.303248	+	0.952912i	$0.401929\pi$
$20$	0	0
$21$	0.907578	0.198050
$22$	0	0
$23$	−8.45881	−1.76378	−0.881892	−	0.471451i	$-0.843730\pi$
$23$	−8.45881	−1.76378	−0.881892	+	0.471451i	$0.843730\pi$
$24$	0	0
$25$	4.23054	0.846109
$26$	0	0
$27$	4.69790	0.904111
$28$	0	0
$29$	−7.04298	−1.30785	−0.653925	−	0.756560i	$-0.726879\pi$
$29$	−7.04298	−1.30785	−0.653925	+	0.756560i	$0.726879\pi$
$30$	0	0
$31$	−3.42063	−0.614364	−0.307182	−	0.951651i	$-0.599386\pi$
$31$	−3.42063	−0.614364	−0.307182	+	0.951651i	$0.599386\pi$
$32$	0	0
$33$	4.33984	0.755469
$34$	0	0
$35$	3.03818	0.513546
$36$	0	0
$37$	−9.66977	−1.58970	−0.794850	−	0.606806i	$-0.792451\pi$
$37$	−9.66977	−1.58970	−0.794850	+	0.606806i	$0.792451\pi$
$38$	0	0
$39$	−3.99273	−0.639349
$40$	0	0
$41$	1.33675	0.208766	0.104383	−	0.994537i	$-0.466713\pi$
$41$	1.33675	0.208766	0.104383	+	0.994537i	$0.466713\pi$
$42$	0	0
$43$	−2.52513	−0.385078	−0.192539	−	0.981289i	$-0.561672\pi$
$43$	−2.52513	−0.385078	−0.192539	+	0.981289i	$0.561672\pi$
$44$	0	0
$45$	6.61200	0.985659
$46$	0	0
$47$	−5.57082	−0.812588	−0.406294	−	0.913742i	$-0.633179\pi$
$47$	−5.57082	−0.812588	−0.406294	+	0.913742i	$0.633179\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.907578	−0.127086
$52$	0	0
$53$	−5.17630	−0.711020	−0.355510	−	0.934673i	$-0.615693\pi$
$53$	−5.17630	−0.711020	−0.355510	+	0.934673i	$0.615693\pi$
$54$	0	0
$55$	14.5279	1.95894
$56$	0	0
$57$	−2.39933	−0.317799
$58$	0	0
$59$	−9.28732	−1.20911	−0.604553	−	0.796565i	$-0.706648\pi$
$59$	−9.28732	−1.20911	−0.604553	+	0.796565i	$0.706648\pi$
$60$	0	0
$61$	−6.66325	−0.853141	−0.426571	−	0.904454i	$-0.640278\pi$
$61$	−6.66325	−0.853141	−0.426571	+	0.904454i	$0.640278\pi$
$62$	0	0
$63$	2.17630	0.274188
$64$	0	0
$65$	−13.3659	−1.65784
$66$	0	0
$67$	−5.28251	−0.645362	−0.322681	−	0.946508i	$-0.604584\pi$
$67$	−5.28251	−0.645362	−0.322681	+	0.946508i	$0.604584\pi$
$68$	0	0
$69$	7.67704	0.924206
$70$	0	0
$71$	−11.9310	−1.41595	−0.707973	−	0.706239i	$-0.750390\pi$
$71$	−11.9310	−1.41595	−0.707973	+	0.706239i	$0.750390\pi$
$72$	0	0
$73$	12.7155	1.48823	0.744116	−	0.668050i	$-0.232871\pi$
$73$	12.7155	1.48823	0.744116	+	0.668050i	$0.232871\pi$
$74$	0	0
$75$	−3.83955	−0.443353
$76$	0	0
$77$	4.78178	0.544934
$78$	0	0
$79$	1.94051	0.218325	0.109162	−	0.994024i	$-0.465183\pi$
$79$	1.94051	0.218325	0.109162	+	0.994024i	$0.465183\pi$
$80$	0	0
$81$	2.26519	0.251688
$82$	0	0
$83$	4.58417	0.503178	0.251589	−	0.967834i	$-0.419047\pi$
$83$	4.58417	0.503178	0.251589	+	0.967834i	$0.419047\pi$
$84$	0	0
$85$	−3.03818	−0.329537
$86$	0	0
$87$	6.39206	0.685301
$88$	0	0
$89$	−13.3956	−1.41993	−0.709965	−	0.704237i	$-0.751289\pi$
$89$	−13.3956	−1.41993	−0.709965	+	0.704237i	$0.751289\pi$
$90$	0	0
$91$	−4.39933	−0.461175
$92$	0	0
$93$	3.10449	0.321921
$94$	0	0
$95$	−8.03191	−0.824057
$96$	0	0
$97$	−15.1668	−1.53995	−0.769976	−	0.638073i	$-0.779732\pi$
$97$	−15.1668	−1.53995	−0.769976	+	0.638073i	$0.779732\pi$
$98$	0	0
$99$	10.4066	1.04590
$100$	0	0
$101$	−3.24786	−0.323174	−0.161587	−	0.986858i	$-0.551661\pi$
$101$	−3.24786	−0.323174	−0.161587	+	0.986858i	$0.551661\pi$
$102$	0	0
$103$	−12.7031	−1.25168	−0.625839	−	0.779952i	$-0.715243\pi$
$103$	−12.7031	−1.25168	−0.625839	+	0.779952i	$0.715243\pi$
$104$	0	0
$105$	−2.75739	−0.269093
$106$	0	0
$107$	−3.91410	−0.378390	−0.189195	−	0.981940i	$-0.560588\pi$
$107$	−3.91410	−0.378390	−0.189195	+	0.981940i	$0.560588\pi$
$108$	0	0
$109$	3.56356	0.341327	0.170663	−	0.985329i	$-0.445409\pi$
$109$	3.56356	0.341327	0.170663	+	0.985329i	$0.445409\pi$
$110$	0	0
$111$	8.77607	0.832988
$112$	0	0
$113$	0.966622	0.0909322	0.0454661	−	0.998966i	$-0.485523\pi$
$113$	0.966622	0.0909322	0.0454661	+	0.998966i	$0.485523\pi$
$114$	0	0
$115$	25.6994	2.39648
$116$	0	0
$117$	−9.57426	−0.885141
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	11.8654	1.07867
$122$	0	0
$123$	−1.21321	−0.109391
$124$	0	0
$125$	2.33775	0.209095
$126$	0	0
$127$	−5.55875	−0.493260	−0.246630	−	0.969110i	$-0.579323\pi$
$127$	−5.55875	−0.493260	−0.246630	+	0.969110i	$0.579323\pi$
$128$	0	0
$129$	2.29175	0.201777
$130$	0	0
$131$	3.51134	0.306787	0.153393	−	0.988165i	$-0.450980\pi$
$131$	3.51134	0.306787	0.153393	+	0.988165i	$0.450980\pi$
$132$	0	0
$133$	−2.64366	−0.229234
$134$	0	0
$135$	−14.2731	−1.22843
$136$	0	0
$137$	−1.93780	−0.165557	−0.0827786	−	0.996568i	$-0.526379\pi$
$137$	−1.93780	−0.165557	−0.0827786	+	0.996568i	$0.526379\pi$
$138$	0	0
$139$	−4.69834	−0.398508	−0.199254	−	0.979948i	$-0.563852\pi$
$139$	−4.69834	−0.398508	−0.199254	+	0.979948i	$0.563852\pi$
$140$	0	0
$141$	5.05596	0.425789
$142$	0	0
$143$	−21.0366	−1.75917
$144$	0	0
$145$	21.3979	1.77700
$146$	0	0
$147$	−0.907578	−0.0748558
$148$	0	0
$149$	−4.24162	−0.347487	−0.173743	−	0.984791i	$-0.555586\pi$
$149$	−4.24162	−0.347487	−0.173743	+	0.984791i	$0.555586\pi$
$150$	0	0
$151$	12.4584	1.01385	0.506924	−	0.861991i	$-0.330783\pi$
$151$	12.4584	1.01385	0.506924	+	0.861991i	$0.330783\pi$
$152$	0	0
$153$	−2.17630	−0.175944
$154$	0	0
$155$	10.3925	0.834746
$156$	0	0
$157$	17.6472	1.40840	0.704199	−	0.710002i	$-0.251306\pi$
$157$	17.6472	1.40840	0.704199	+	0.710002i	$0.251306\pi$
$158$	0	0
$159$	4.69790	0.372568
$160$	0	0
$161$	8.45881	0.666648
$162$	0	0
$163$	2.67557	0.209567	0.104783	−	0.994495i	$-0.466585\pi$
$163$	2.67557	0.209567	0.104783	+	0.994495i	$0.466585\pi$
$164$	0	0
$165$	−13.1852	−1.02647
$166$	0	0
$167$	−1.95783	−0.151501	−0.0757507	−	0.997127i	$-0.524135\pi$
$167$	−1.95783	−0.151501	−0.0757507	+	0.997127i	$0.524135\pi$
$168$	0	0
$169$	6.35407	0.488775
$170$	0	0
$171$	−5.75340	−0.439973
$172$	0	0
$173$	−6.01732	−0.457488	−0.228744	−	0.973487i	$-0.573462\pi$
$173$	−6.01732	−0.457488	−0.228744	+	0.973487i	$0.573462\pi$
$174$	0	0
$175$	−4.23054	−0.319799
$176$	0	0
$177$	8.42897	0.633560
$178$	0	0
$179$	−1.75112	−0.130885	−0.0654423	−	0.997856i	$-0.520846\pi$
$179$	−1.75112	−0.130885	−0.0654423	+	0.997856i	$0.520846\pi$
$180$	0	0
$181$	−2.76491	−0.205514	−0.102757	−	0.994707i	$-0.532766\pi$
$181$	−2.76491	−0.205514	−0.102757	+	0.994707i	$0.532766\pi$
$182$	0	0
$183$	6.04742	0.447038
$184$	0	0
$185$	29.3785	2.15995
$186$	0	0
$187$	−4.78178	−0.349678
$188$	0	0
$189$	−4.69790	−0.341722
$190$	0	0
$191$	18.8461	1.36365	0.681827	−	0.731514i	$-0.261186\pi$
$191$	18.8461	1.36365	0.681827	+	0.731514i	$0.261186\pi$
$192$	0	0
$193$	11.9107	0.857348	0.428674	−	0.903459i	$-0.358981\pi$
$193$	11.9107	0.857348	0.428674	+	0.903459i	$0.358981\pi$
$194$	0	0
$195$	12.1306	0.868693
$196$	0	0
$197$	17.2682	1.23031	0.615153	−	0.788408i	$-0.289094\pi$
$197$	17.2682	1.23031	0.615153	+	0.788408i	$0.289094\pi$
$198$	0	0
$199$	−15.2451	−1.08070	−0.540350	−	0.841440i	$-0.681708\pi$
$199$	−15.2451	−1.08070	−0.540350	+	0.841440i	$0.681708\pi$
$200$	0	0
$201$	4.79429	0.338163
$202$	0	0
$203$	7.04298	0.494321
$204$	0	0
$205$	−4.06130	−0.283653
$206$	0	0
$207$	18.4089	1.27951
$208$	0	0
$209$	−12.6414	−0.874423
$210$	0	0
$211$	−8.51331	−0.586080	−0.293040	−	0.956100i	$-0.594667\pi$
$211$	−8.51331	−0.586080	−0.293040	+	0.956100i	$0.594667\pi$
$212$	0	0
$213$	10.8283	0.741942
$214$	0	0
$215$	7.67179	0.523212
$216$	0	0
$217$	3.42063	0.232208
$218$	0	0
$219$	−11.5403	−0.779820
$220$	0	0
$221$	4.39933	0.295931
$222$	0	0
$223$	−2.49093	−0.166805	−0.0834027	−	0.996516i	$-0.526579\pi$
$223$	−2.49093	−0.166805	−0.0834027	+	0.996516i	$0.526579\pi$
$224$	0	0
$225$	−9.20694	−0.613796
$226$	0	0
$227$	16.9842	1.12728	0.563640	−	0.826020i	$-0.309400\pi$
$227$	16.9842	1.12728	0.563640	+	0.826020i	$0.309400\pi$
$228$	0	0
$229$	16.3303	1.07914	0.539568	−	0.841942i	$-0.318587\pi$
$229$	16.3303	1.07914	0.539568	+	0.841942i	$0.318587\pi$
$230$	0	0
$231$	−4.33984	−0.285540
$232$	0	0
$233$	−5.77825	−0.378546	−0.189273	−	0.981925i	$-0.560613\pi$
$233$	−5.77825	−0.378546	−0.189273	+	0.981925i	$0.560613\pi$
$234$	0	0
$235$	16.9252	1.10408
$236$	0	0
$237$	−1.76117	−0.114400
$238$	0	0
$239$	7.44275	0.481432	0.240716	−	0.970596i	$-0.422618\pi$
$239$	7.44275	0.481432	0.240716	+	0.970596i	$0.422618\pi$
$240$	0	0
$241$	21.9036	1.41093	0.705467	−	0.708743i	$-0.250737\pi$
$241$	21.9036	1.41093	0.705467	+	0.708743i	$0.250737\pi$
$242$	0	0
$243$	−16.1495	−1.03599
$244$	0	0
$245$	−3.03818	−0.194102
$246$	0	0
$247$	11.6303	0.740019
$248$	0	0
$249$	−4.16049	−0.263661
$250$	0	0
$251$	−16.5769	−1.04632	−0.523162	−	0.852233i	$-0.675248\pi$
$251$	−16.5769	−1.04632	−0.523162	+	0.852233i	$0.675248\pi$
$252$	0	0
$253$	40.4482	2.54296
$254$	0	0
$255$	2.75739	0.172674
$256$	0	0
$257$	12.1158	0.755764	0.377882	−	0.925854i	$-0.376653\pi$
$257$	12.1158	0.755764	0.377882	+	0.925854i	$0.376653\pi$
$258$	0	0
$259$	9.66977	0.600850
$260$	0	0
$261$	15.3277	0.948758
$262$	0	0
$263$	3.20135	0.197404	0.0987018	−	0.995117i	$-0.468531\pi$
$263$	3.20135	0.197404	0.0987018	+	0.995117i	$0.468531\pi$
$264$	0	0
$265$	15.7265	0.966074
$266$	0	0
$267$	12.1575	0.744030
$268$	0	0
$269$	−15.4400	−0.941396	−0.470698	−	0.882294i	$-0.655998\pi$
$269$	−15.4400	−0.941396	−0.470698	+	0.882294i	$0.655998\pi$
$270$	0	0
$271$	−24.8878	−1.51182	−0.755912	−	0.654673i	$-0.772806\pi$
$271$	−24.8878	−1.51182	−0.755912	+	0.654673i	$0.772806\pi$
$272$	0	0
$273$	3.99273	0.241651
$274$	0	0
$275$	−20.2295	−1.21989
$276$	0	0
$277$	11.2870	0.678172	0.339086	−	0.940755i	$-0.389882\pi$
$277$	11.2870	0.678172	0.339086	+	0.940755i	$0.389882\pi$
$278$	0	0
$279$	7.44433	0.445680
$280$	0	0
$281$	13.0332	0.777495	0.388747	−	0.921344i	$-0.372908\pi$
$281$	13.0332	0.777495	0.388747	+	0.921344i	$0.372908\pi$
$282$	0	0
$283$	−21.7190	−1.29106	−0.645531	−	0.763734i	$-0.723364\pi$
$283$	−21.7190	−1.29106	−0.645531	+	0.763734i	$0.723364\pi$
$284$	0	0
$285$	7.28959	0.431798
$286$	0	0
$287$	−1.33675	−0.0789061
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	13.7650	0.806920
$292$	0	0
$293$	28.9400	1.69069	0.845346	−	0.534219i	$-0.179394\pi$
$293$	28.9400	1.69069	0.845346	+	0.534219i	$0.179394\pi$
$294$	0	0
$295$	28.2165	1.64283
$296$	0	0
$297$	−22.4643	−1.30351
$298$	0	0
$299$	−37.2131	−2.15209
$300$	0	0
$301$	2.52513	0.145546
$302$	0	0
$303$	2.94769	0.169340
$304$	0	0
$305$	20.2441	1.15918
$306$	0	0
$307$	11.5836	0.661110	0.330555	−	0.943787i	$-0.392764\pi$
$307$	11.5836	0.661110	0.330555	+	0.943787i	$0.392764\pi$
$308$	0	0
$309$	11.5291	0.655867
$310$	0	0
$311$	−15.7999	−0.895932	−0.447966	−	0.894051i	$-0.647851\pi$
$311$	−15.7999	−0.895932	−0.447966	+	0.894051i	$0.647851\pi$
$312$	0	0
$313$	20.2658	1.14549	0.572745	−	0.819733i	$-0.305878\pi$
$313$	20.2658	1.14549	0.572745	+	0.819733i	$0.305878\pi$
$314$	0	0
$315$	−6.61200	−0.372544
$316$	0	0
$317$	−6.36537	−0.357515	−0.178757	−	0.983893i	$-0.557208\pi$
$317$	−6.36537	−0.357515	−0.178757	+	0.983893i	$0.557208\pi$
$318$	0	0
$319$	33.6780	1.88561
$320$	0	0
$321$	3.55235	0.198273
$322$	0	0
$323$	2.64366	0.147097
$324$	0	0
$325$	18.6115	1.03238
$326$	0	0
$327$	−3.23421	−0.178852
$328$	0	0
$329$	5.57082	0.307130
$330$	0	0
$331$	−24.5910	−1.35165	−0.675823	−	0.737064i	$-0.736212\pi$
$331$	−24.5910	−1.35165	−0.675823	+	0.737064i	$0.736212\pi$
$332$	0	0
$333$	21.0443	1.15322
$334$	0	0
$335$	16.0492	0.876863
$336$	0	0
$337$	7.11956	0.387827	0.193913	−	0.981019i	$-0.437882\pi$
$337$	7.11956	0.387827	0.193913	+	0.981019i	$0.437882\pi$
$338$	0	0
$339$	−0.877285	−0.0476476
$340$	0	0
$341$	16.3567	0.885766
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−23.3242	−1.25573
$346$	0	0
$347$	−1.51477	−0.0813173	−0.0406586	−	0.999173i	$-0.512946\pi$
$347$	−1.51477	−0.0813173	−0.0406586	+	0.999173i	$0.512946\pi$
$348$	0	0
$349$	−24.7937	−1.32717	−0.663587	−	0.748099i	$-0.730967\pi$
$349$	−24.7937	−1.32717	−0.663587	+	0.748099i	$0.730967\pi$
$350$	0	0
$351$	20.6676	1.10315
$352$	0	0
$353$	−11.8416	−0.630267	−0.315133	−	0.949047i	$-0.602049\pi$
$353$	−11.8416	−0.630267	−0.315133	+	0.949047i	$0.602049\pi$
$354$	0	0
$355$	36.2485	1.92387
$356$	0	0
$357$	0.907578	0.0480341
$358$	0	0
$359$	−26.6743	−1.40781	−0.703907	−	0.710292i	$-0.748563\pi$
$359$	−26.6743	−1.40781	−0.703907	+	0.710292i	$0.748563\pi$
$360$	0	0
$361$	−12.0111	−0.632162
$362$	0	0
$363$	−10.7688	−0.565215
$364$	0	0
$365$	−38.6319	−2.02209
$366$	0	0
$367$	−28.6897	−1.49759	−0.748796	−	0.662801i	$-0.769368\pi$
$367$	−28.6897	−1.49759	−0.748796	+	0.662801i	$0.769368\pi$
$368$	0	0
$369$	−2.90918	−0.151446
$370$	0	0
$371$	5.17630	0.268740
$372$	0	0
$373$	26.8946	1.39255	0.696275	−	0.717775i	$-0.254839\pi$
$373$	26.8946	1.39255	0.696275	+	0.717775i	$0.254839\pi$
$374$	0	0
$375$	−2.12169	−0.109564
$376$	0	0
$377$	−30.9844	−1.59578
$378$	0	0
$379$	−0.960453	−0.0493351	−0.0246676	−	0.999696i	$-0.507853\pi$
$379$	−0.960453	−0.0493351	−0.0246676	+	0.999696i	$0.507853\pi$
$380$	0	0
$381$	5.04500	0.258463
$382$	0	0
$383$	−10.9913	−0.561627	−0.280814	−	0.959762i	$-0.590604\pi$
$383$	−10.9913	−0.561627	−0.280814	+	0.959762i	$0.590604\pi$
$384$	0	0
$385$	−14.5279	−0.740411
$386$	0	0
$387$	5.49544	0.279349
$388$	0	0
$389$	25.4239	1.28904	0.644520	−	0.764587i	$-0.277057\pi$
$389$	25.4239	1.28904	0.644520	+	0.764587i	$0.277057\pi$
$390$	0	0
$391$	−8.45881	−0.427781
$392$	0	0
$393$	−3.18681	−0.160753
$394$	0	0
$395$	−5.89563	−0.296641
$396$	0	0
$397$	15.0140	0.753533	0.376767	−	0.926308i	$-0.377036\pi$
$397$	15.0140	0.753533	0.376767	+	0.926308i	$0.377036\pi$
$398$	0	0
$399$	2.39933	0.120117
$400$	0	0
$401$	−11.0656	−0.552588	−0.276294	−	0.961073i	$-0.589106\pi$
$401$	−11.0656	−0.552588	−0.276294	+	0.961073i	$0.589106\pi$
$402$	0	0
$403$	−15.0485	−0.749618
$404$	0	0
$405$	−6.88207	−0.341973
$406$	0	0
$407$	46.2387	2.29197
$408$	0	0
$409$	28.5650	1.41245	0.706225	−	0.707988i	$-0.250397\pi$
$409$	28.5650	1.41245	0.706225	+	0.707988i	$0.250397\pi$
$410$	0	0
$411$	1.75870	0.0867504
$412$	0	0
$413$	9.28732	0.456999
$414$	0	0
$415$	−13.9275	−0.683676
$416$	0	0
$417$	4.26411	0.208815
$418$	0	0
$419$	1.98566	0.0970057	0.0485029	−	0.998823i	$-0.484555\pi$
$419$	1.98566	0.0970057	0.0485029	+	0.998823i	$0.484555\pi$
$420$	0	0
$421$	−25.1984	−1.22810	−0.614048	−	0.789269i	$-0.710460\pi$
$421$	−25.1984	−1.22810	−0.614048	+	0.789269i	$0.710460\pi$
$422$	0	0
$423$	12.1238	0.589479
$424$	0	0
$425$	4.23054	0.205211
$426$	0	0
$427$	6.66325	0.322457
$428$	0	0
$429$	19.0924	0.921788
$430$	0	0
$431$	−27.4984	−1.32455	−0.662275	−	0.749261i	$-0.730409\pi$
$431$	−27.4984	−1.32455	−0.662275	+	0.749261i	$0.730409\pi$
$432$	0	0
$433$	28.1977	1.35509	0.677547	−	0.735480i	$-0.263043\pi$
$433$	28.1977	1.35509	0.677547	+	0.735480i	$0.263043\pi$
$434$	0	0
$435$	−19.4202	−0.931129
$436$	0	0
$437$	−22.3622	−1.06973
$438$	0	0
$439$	6.31298	0.301302	0.150651	−	0.988587i	$-0.451863\pi$
$439$	6.31298	0.301302	0.150651	+	0.988587i	$0.451863\pi$
$440$	0	0
$441$	−2.17630	−0.103633
$442$	0	0
$443$	−26.4557	−1.25695	−0.628473	−	0.777831i	$-0.716320\pi$
$443$	−26.4557	−1.25695	−0.628473	+	0.777831i	$0.716320\pi$
$444$	0	0
$445$	40.6982	1.92928
$446$	0	0
$447$	3.84960	0.182080
$448$	0	0
$449$	−34.5555	−1.63077	−0.815387	−	0.578916i	$-0.803476\pi$
$449$	−34.5555	−1.63077	−0.815387	+	0.578916i	$0.803476\pi$
$450$	0	0
$451$	−6.39206	−0.300990
$452$	0	0
$453$	−11.3069	−0.531247
$454$	0	0
$455$	13.3659	0.626605
$456$	0	0
$457$	26.5922	1.24393	0.621965	−	0.783045i	$-0.286334\pi$
$457$	26.5922	1.24393	0.621965	+	0.783045i	$0.286334\pi$
$458$	0	0
$459$	4.69790	0.219279
$460$	0	0
$461$	0.310199	0.0144474	0.00722371	−	0.999974i	$-0.497701\pi$
$461$	0.310199	0.0144474	0.00722371	+	0.999974i	$0.497701\pi$
$462$	0	0
$463$	22.4654	1.04406	0.522028	−	0.852928i	$-0.325175\pi$
$463$	22.4654	1.04406	0.522028	+	0.852928i	$0.325175\pi$
$464$	0	0
$465$	−9.43201	−0.437399
$466$	0	0
$467$	11.6126	0.537365	0.268682	−	0.963229i	$-0.413412\pi$
$467$	11.6126	0.537365	0.268682	+	0.963229i	$0.413412\pi$
$468$	0	0
$469$	5.28251	0.243924
$470$	0	0
$471$	−16.0162	−0.737988
$472$	0	0
$473$	12.0746	0.555190
$474$	0	0
$475$	11.1841	0.513162
$476$	0	0
$477$	11.2652	0.515798
$478$	0	0
$479$	31.7976	1.45287	0.726435	−	0.687235i	$-0.241176\pi$
$479$	31.7976	1.45287	0.726435	+	0.687235i	$0.241176\pi$
$480$	0	0
$481$	−42.5405	−1.93968
$482$	0	0
$483$	−7.67704	−0.349317
$484$	0	0
$485$	46.0794	2.09236
$486$	0	0
$487$	−23.0166	−1.04298	−0.521490	−	0.853257i	$-0.674624\pi$
$487$	−23.0166	−1.04298	−0.521490	+	0.853257i	$0.674624\pi$
$488$	0	0
$489$	−2.42829	−0.109811
$490$	0	0
$491$	33.3176	1.50360	0.751802	−	0.659389i	$-0.229185\pi$
$491$	33.3176	1.50360	0.751802	+	0.659389i	$0.229185\pi$
$492$	0	0
$493$	−7.04298	−0.317200
$494$	0	0
$495$	−31.6171	−1.42108
$496$	0	0
$497$	11.9310	0.535177
$498$	0	0
$499$	−37.0236	−1.65741	−0.828703	−	0.559689i	$-0.810921\pi$
$499$	−37.0236	−1.65741	−0.828703	+	0.559689i	$0.810921\pi$
$500$	0	0
$501$	1.77688	0.0793853
$502$	0	0
$503$	−12.4172	−0.553654	−0.276827	−	0.960920i	$-0.589283\pi$
$503$	−12.4172	−0.553654	−0.276827	+	0.960920i	$0.589283\pi$
$504$	0	0
$505$	9.86759	0.439102
$506$	0	0
$507$	−5.76682	−0.256113
$508$	0	0
$509$	−11.9257	−0.528597	−0.264299	−	0.964441i	$-0.585140\pi$
$509$	−11.9257	−0.528597	−0.264299	+	0.964441i	$0.585140\pi$
$510$	0	0
$511$	−12.7155	−0.562499
$512$	0	0
$513$	12.4196	0.548340
$514$	0	0
$515$	38.5945	1.70067
$516$	0	0
$517$	26.6385	1.17156
$518$	0	0
$519$	5.46119	0.239719
$520$	0	0
$521$	26.3980	1.15652	0.578259	−	0.815853i	$-0.303732\pi$
$521$	26.3980	1.15652	0.578259	+	0.815853i	$0.303732\pi$
$522$	0	0
$523$	−13.6272	−0.595874	−0.297937	−	0.954586i	$-0.596299\pi$
$523$	−13.6272	−0.595874	−0.297937	+	0.954586i	$0.596299\pi$
$524$	0	0
$525$	3.83955	0.167572
$526$	0	0
$527$	−3.42063	−0.149005
$528$	0	0
$529$	48.5515	2.11094
$530$	0	0
$531$	20.2120	0.877126
$532$	0	0
$533$	5.88081	0.254726
$534$	0	0
$535$	11.8917	0.514125
$536$	0	0
$537$	1.58927	0.0685823
$538$	0	0
$539$	−4.78178	−0.205966
$540$	0	0
$541$	−26.3358	−1.13226	−0.566132	−	0.824314i	$-0.691561\pi$
$541$	−26.3358	−1.13226	−0.566132	+	0.824314i	$0.691561\pi$
$542$	0	0
$543$	2.50937	0.107687
$544$	0	0
$545$	−10.8267	−0.463766
$546$	0	0
$547$	2.25139	0.0962624	0.0481312	−	0.998841i	$-0.484673\pi$
$547$	2.25139	0.0962624	0.0481312	+	0.998841i	$0.484673\pi$
$548$	0	0
$549$	14.5012	0.618898
$550$	0	0
$551$	−18.6192	−0.793206
$552$	0	0
$553$	−1.94051	−0.0825190
$554$	0	0
$555$	−26.6633	−1.13179
$556$	0	0
$557$	−30.3355	−1.28536	−0.642679	−	0.766136i	$-0.722177\pi$
$557$	−30.3355	−1.28536	−0.642679	+	0.766136i	$0.722177\pi$
$558$	0	0
$559$	−11.1089	−0.469854
$560$	0	0
$561$	4.33984	0.183228
$562$	0	0
$563$	−10.9867	−0.463035	−0.231518	−	0.972831i	$-0.574369\pi$
$563$	−10.9867	−0.463035	−0.231518	+	0.972831i	$0.574369\pi$
$564$	0	0
$565$	−2.93677	−0.123551
$566$	0	0
$567$	−2.26519	−0.0951292
$568$	0	0
$569$	−28.8479	−1.20937	−0.604684	−	0.796465i	$-0.706701\pi$
$569$	−28.8479	−1.20937	−0.604684	+	0.796465i	$0.706701\pi$
$570$	0	0
$571$	10.5165	0.440100	0.220050	−	0.975489i	$-0.429378\pi$
$571$	10.5165	0.440100	0.220050	+	0.975489i	$0.429378\pi$
$572$	0	0
$573$	−17.1043	−0.714542
$574$	0	0
$575$	−35.7854	−1.49235
$576$	0	0
$577$	−36.8288	−1.53320	−0.766601	−	0.642123i	$-0.778054\pi$
$577$	−36.8288	−1.53320	−0.766601	+	0.642123i	$0.778054\pi$
$578$	0	0
$579$	−10.8099	−0.449242
$580$	0	0
$581$	−4.58417	−0.190183
$582$	0	0
$583$	24.7519	1.02512
$584$	0	0
$585$	29.0883	1.20265
$586$	0	0
$587$	−30.1420	−1.24409	−0.622047	−	0.782980i	$-0.713699\pi$
$587$	−30.1420	−1.24409	−0.622047	+	0.782980i	$0.713699\pi$
$588$	0	0
$589$	−9.04298	−0.372610
$590$	0	0
$591$	−15.6722	−0.644669
$592$	0	0
$593$	−7.88925	−0.323973	−0.161986	−	0.986793i	$-0.551790\pi$
$593$	−7.88925	−0.323973	−0.161986	+	0.986793i	$0.551790\pi$
$594$	0	0
$595$	3.03818	0.124553
$596$	0	0
$597$	13.8362	0.566276
$598$	0	0
$599$	29.8059	1.21783	0.608917	−	0.793234i	$-0.291604\pi$
$599$	29.8059	1.21783	0.608917	+	0.793234i	$0.291604\pi$
$600$	0	0
$601$	11.9432	0.487175	0.243587	−	0.969879i	$-0.421676\pi$
$601$	11.9432	0.487175	0.243587	+	0.969879i	$0.421676\pi$
$602$	0	0
$603$	11.4963	0.468167
$604$	0	0
$605$	−36.0493	−1.46561
$606$	0	0
$607$	−4.67281	−0.189664	−0.0948318	−	0.995493i	$-0.530231\pi$
$607$	−4.67281	−0.189664	−0.0948318	+	0.995493i	$0.530231\pi$
$608$	0	0
$609$	−6.39206	−0.259019
$610$	0	0
$611$	−24.5079	−0.991483
$612$	0	0
$613$	−42.3894	−1.71209	−0.856046	−	0.516900i	$-0.827086\pi$
$613$	−42.3894	−1.71209	−0.856046	+	0.516900i	$0.827086\pi$
$614$	0	0
$615$	3.68595	0.148632
$616$	0	0
$617$	−23.2809	−0.937255	−0.468628	−	0.883396i	$-0.655251\pi$
$617$	−23.2809	−0.937255	−0.468628	+	0.883396i	$0.655251\pi$
$618$	0	0
$619$	43.2201	1.73716	0.868582	−	0.495545i	$-0.165032\pi$
$619$	43.2201	1.73716	0.868582	+	0.495545i	$0.165032\pi$
$620$	0	0
$621$	−39.7387	−1.59466
$622$	0	0
$623$	13.3956	0.536683
$624$	0	0
$625$	−28.2552	−1.13021
$626$	0	0
$627$	11.4730	0.458189
$628$	0	0
$629$	−9.66977	−0.385559
$630$	0	0
$631$	30.6825	1.22145	0.610725	−	0.791843i	$-0.290878\pi$
$631$	30.6825	1.22145	0.610725	+	0.791843i	$0.290878\pi$
$632$	0	0
$633$	7.72649	0.307100
$634$	0	0
$635$	16.8885	0.670200
$636$	0	0
$637$	4.39933	0.174308
$638$	0	0
$639$	25.9654	1.02718
$640$	0	0
$641$	−21.3561	−0.843517	−0.421758	−	0.906708i	$-0.638587\pi$
$641$	−21.3561	−0.843517	−0.421758	+	0.906708i	$0.638587\pi$
$642$	0	0
$643$	−22.7013	−0.895253	−0.447626	−	0.894221i	$-0.647731\pi$
$643$	−22.7013	−0.895253	−0.447626	+	0.894221i	$0.647731\pi$
$644$	0	0
$645$	−6.96275	−0.274158
$646$	0	0
$647$	35.2630	1.38633	0.693165	−	0.720779i	$-0.256216\pi$
$647$	35.2630	1.38633	0.693165	+	0.720779i	$0.256216\pi$
$648$	0	0
$649$	44.4099	1.74324
$650$	0	0
$651$	−3.10449	−0.121675
$652$	0	0
$653$	30.0593	1.17631	0.588155	−	0.808748i	$-0.299854\pi$
$653$	30.0593	1.17631	0.588155	+	0.808748i	$0.299854\pi$
$654$	0	0
$655$	−10.6681	−0.416836
$656$	0	0
$657$	−27.6727	−1.07961
$658$	0	0
$659$	18.9747	0.739148	0.369574	−	0.929201i	$-0.379504\pi$
$659$	18.9747	0.739148	0.369574	+	0.929201i	$0.379504\pi$
$660$	0	0
$661$	−25.8038	−1.00365	−0.501825	−	0.864969i	$-0.667338\pi$
$661$	−25.8038	−1.00365	−0.501825	+	0.864969i	$0.667338\pi$
$662$	0	0
$663$	−3.99273	−0.155065
$664$	0	0
$665$	8.03191	0.311464
$666$	0	0
$667$	59.5753	2.30676
$668$	0	0
$669$	2.26072	0.0874044
$670$	0	0
$671$	31.8622	1.23003
$672$	0	0
$673$	4.38401	0.168991	0.0844956	−	0.996424i	$-0.473072\pi$
$673$	4.38401	0.168991	0.0844956	+	0.996424i	$0.473072\pi$
$674$	0	0
$675$	19.8747	0.764976
$676$	0	0
$677$	32.1115	1.23415	0.617073	−	0.786906i	$-0.288318\pi$
$677$	32.1115	1.23415	0.617073	+	0.786906i	$0.288318\pi$
$678$	0	0
$679$	15.1668	0.582047
$680$	0	0
$681$	−15.4145	−0.590684
$682$	0	0
$683$	18.7093	0.715892	0.357946	−	0.933742i	$-0.383477\pi$
$683$	18.7093	0.715892	0.357946	+	0.933742i	$0.383477\pi$
$684$	0	0
$685$	5.88738	0.224945
$686$	0	0
$687$	−14.8210	−0.565457
$688$	0	0
$689$	−22.7722	−0.867553
$690$	0	0
$691$	40.9131	1.55641	0.778205	−	0.628011i	$-0.216131\pi$
$691$	40.9131	1.55641	0.778205	+	0.628011i	$0.216131\pi$
$692$	0	0
$693$	−10.4066	−0.395314
$694$	0	0
$695$	14.2744	0.541459
$696$	0	0
$697$	1.33675	0.0506331
$698$	0	0
$699$	5.24421	0.198354
$700$	0	0
$701$	17.6240	0.665649	0.332825	−	0.942989i	$-0.391998\pi$
$701$	17.6240	0.665649	0.332825	+	0.942989i	$0.391998\pi$
$702$	0	0
$703$	−25.5636	−0.964148
$704$	0	0
$705$	−15.3609	−0.578526
$706$	0	0
$707$	3.24786	0.122148
$708$	0	0
$709$	8.18690	0.307466	0.153733	−	0.988112i	$-0.450870\pi$
$709$	8.18690	0.307466	0.153733	+	0.988112i	$0.450870\pi$
$710$	0	0
$711$	−4.22314	−0.158380
$712$	0	0
$713$	28.9345	1.08361
$714$	0	0
$715$	63.9130	2.39021
$716$	0	0
$717$	−6.75488	−0.252266
$718$	0	0
$719$	42.6093	1.58906	0.794530	−	0.607225i	$-0.207717\pi$
$719$	42.6093	1.58906	0.794530	+	0.607225i	$0.207717\pi$
$720$	0	0
$721$	12.7031	0.473090
$722$	0	0
$723$	−19.8792	−0.739316
$724$	0	0
$725$	−29.7956	−1.10658
$726$	0	0
$727$	−10.5470	−0.391166	−0.195583	−	0.980687i	$-0.562660\pi$
$727$	−10.5470	−0.391166	−0.195583	+	0.980687i	$0.562660\pi$
$728$	0	0
$729$	7.86138	0.291162
$730$	0	0
$731$	−2.52513	−0.0933951
$732$	0	0
$733$	−6.61491	−0.244327	−0.122164	−	0.992510i	$-0.538983\pi$
$733$	−6.61491	−0.244327	−0.122164	+	0.992510i	$0.538983\pi$
$734$	0	0
$735$	2.75739	0.101708
$736$	0	0
$737$	25.2598	0.930457
$738$	0	0
$739$	0.834376	0.0306930	0.0153465	−	0.999882i	$-0.495115\pi$
$739$	0.834376	0.0306930	0.0153465	+	0.999882i	$0.495115\pi$
$740$	0	0
$741$	−10.5554	−0.387763
$742$	0	0
$743$	52.9082	1.94101	0.970506	−	0.241077i	$-0.0775006\pi$
$743$	52.9082	1.94101	0.970506	+	0.241077i	$0.0775006\pi$
$744$	0	0
$745$	12.8868	0.472136
$746$	0	0
$747$	−9.97654	−0.365022
$748$	0	0
$749$	3.91410	0.143018
$750$	0	0
$751$	35.3681	1.29060	0.645300	−	0.763929i	$-0.276732\pi$
$751$	35.3681	1.29060	0.645300	+	0.763929i	$0.276732\pi$
$752$	0	0
$753$	15.0448	0.548264
$754$	0	0
$755$	−37.8508	−1.37753
$756$	0	0
$757$	19.6550	0.714372	0.357186	−	0.934033i	$-0.383736\pi$
$757$	19.6550	0.714372	0.357186	+	0.934033i	$0.383736\pi$
$758$	0	0
$759$	−36.7099	−1.33248
$760$	0	0
$761$	−6.80776	−0.246781	−0.123390	−	0.992358i	$-0.539377\pi$
$761$	−6.80776	−0.246781	−0.123390	+	0.992358i	$0.539377\pi$
$762$	0	0
$763$	−3.56356	−0.129009
$764$	0	0
$765$	6.61200	0.239057
$766$	0	0
$767$	−40.8579	−1.47529
$768$	0	0
$769$	44.3645	1.59983	0.799913	−	0.600116i	$-0.204879\pi$
$769$	44.3645	1.59983	0.799913	+	0.600116i	$0.204879\pi$
$770$	0	0
$771$	−10.9961	−0.396013
$772$	0	0
$773$	25.6974	0.924270	0.462135	−	0.886810i	$-0.347084\pi$
$773$	25.6974	0.924270	0.462135	+	0.886810i	$0.347084\pi$
$774$	0	0
$775$	−14.4711	−0.519819
$776$	0	0
$777$	−8.77607	−0.314840
$778$	0	0
$779$	3.53392	0.126616
$780$	0	0
$781$	57.0513	2.04146
$782$	0	0
$783$	−33.0872	−1.18244
$784$	0	0
$785$	−53.6153	−1.91361
$786$	0	0
$787$	−25.6566	−0.914558	−0.457279	−	0.889323i	$-0.651176\pi$
$787$	−25.6566	−0.914558	−0.457279	+	0.889323i	$0.651176\pi$
$788$	0	0
$789$	−2.90547	−0.103438
$790$	0	0
$791$	−0.966622	−0.0343691
$792$	0	0
$793$	−29.3138	−1.04096
$794$	0	0
$795$	−14.2731	−0.506213
$796$	0	0
$797$	−20.3683	−0.721483	−0.360741	−	0.932666i	$-0.617476\pi$
$797$	−20.3683	−0.721483	−0.360741	+	0.932666i	$0.617476\pi$
$798$	0	0
$799$	−5.57082	−0.197082
$800$	0	0
$801$	29.1528	1.03006
$802$	0	0
$803$	−60.8026	−2.14568
$804$	0	0
$805$	−25.6994	−0.905785
$806$	0	0
$807$	14.0130	0.493282
$808$	0	0
$809$	−36.5480	−1.28496	−0.642480	−	0.766303i	$-0.722094\pi$
$809$	−36.5480	−1.28496	−0.642480	+	0.766303i	$0.722094\pi$
$810$	0	0
$811$	−7.33037	−0.257404	−0.128702	−	0.991683i	$-0.541081\pi$
$811$	−7.33037	−0.257404	−0.128702	+	0.991683i	$0.541081\pi$
$812$	0	0
$813$	22.5876	0.792182
$814$	0	0
$815$	−8.12886	−0.284742
$816$	0	0
$817$	−6.67557	−0.233549
$818$	0	0
$819$	9.57426	0.334552
$820$	0	0
$821$	−1.76321	−0.0615366	−0.0307683	−	0.999527i	$-0.509795\pi$
$821$	−1.76321	−0.0615366	−0.0307683	+	0.999527i	$0.509795\pi$
$822$	0	0
$823$	0.636323	0.0221808	0.0110904	−	0.999938i	$-0.496470\pi$
$823$	0.636323	0.0221808	0.0110904	+	0.999938i	$0.496470\pi$
$824$	0	0
$825$	18.3599	0.639209
$826$	0	0
$827$	6.81869	0.237109	0.118554	−	0.992948i	$-0.462174\pi$
$827$	6.81869	0.237109	0.118554	+	0.992948i	$0.462174\pi$
$828$	0	0
$829$	−50.9480	−1.76950	−0.884749	−	0.466068i	$-0.845670\pi$
$829$	−50.9480	−1.76950	−0.884749	+	0.466068i	$0.845670\pi$
$830$	0	0
$831$	−10.2439	−0.355356
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	5.94824	0.205847
$836$	0	0
$837$	−16.0698	−0.555453
$838$	0	0
$839$	2.95064	0.101867	0.0509336	−	0.998702i	$-0.483780\pi$
$839$	2.95064	0.101867	0.0509336	+	0.998702i	$0.483780\pi$
$840$	0	0
$841$	20.6036	0.710470
$842$	0	0
$843$	−11.8286	−0.407400
$844$	0	0
$845$	−19.3048	−0.664106
$846$	0	0
$847$	−11.8654	−0.407700
$848$	0	0
$849$	19.7117	0.676504
$850$	0	0
$851$	81.7948	2.80389
$852$	0	0
$853$	28.8764	0.988709	0.494355	−	0.869260i	$-0.335404\pi$
$853$	28.8764	0.988709	0.494355	+	0.869260i	$0.335404\pi$
$854$	0	0
$855$	17.4799	0.597799
$856$	0	0
$857$	−9.83977	−0.336120	−0.168060	−	0.985777i	$-0.553750\pi$
$857$	−9.83977	−0.336120	−0.168060	+	0.985777i	$0.553750\pi$
$858$	0	0
$859$	11.1344	0.379900	0.189950	−	0.981794i	$-0.439167\pi$
$859$	11.1344	0.379900	0.189950	+	0.981794i	$0.439167\pi$
$860$	0	0
$861$	1.21321	0.0413460
$862$	0	0
$863$	22.7030	0.772818	0.386409	−	0.922328i	$-0.373715\pi$
$863$	22.7030	0.772818	0.386409	+	0.922328i	$0.373715\pi$
$864$	0	0
$865$	18.2817	0.621596
$866$	0	0
$867$	−0.907578	−0.0308230
$868$	0	0
$869$	−9.27910	−0.314772
$870$	0	0
$871$	−23.2395	−0.787440
$872$	0	0
$873$	33.0075	1.11713
$874$	0	0
$875$	−2.33775	−0.0790304
$876$	0	0
$877$	−1.55717	−0.0525819	−0.0262910	−	0.999654i	$-0.508370\pi$
$877$	−1.55717	−0.0525819	−0.0262910	+	0.999654i	$0.508370\pi$
$878$	0	0
$879$	−26.2653	−0.885907
$880$	0	0
$881$	−10.3326	−0.348113	−0.174056	−	0.984736i	$-0.555687\pi$
$881$	−10.3326	−0.348113	−0.174056	+	0.984736i	$0.555687\pi$
$882$	0	0
$883$	−23.6160	−0.794743	−0.397371	−	0.917658i	$-0.630078\pi$
$883$	−23.6160	−0.794743	−0.397371	+	0.917658i	$0.630078\pi$
$884$	0	0
$885$	−25.6087	−0.860828
$886$	0	0
$887$	13.9260	0.467588	0.233794	−	0.972286i	$-0.424886\pi$
$887$	13.9260	0.467588	0.233794	+	0.972286i	$0.424886\pi$
$888$	0	0
$889$	5.55875	0.186435
$890$	0	0
$891$	−10.8317	−0.362874
$892$	0	0
$893$	−14.7274	−0.492832
$894$	0	0
$895$	5.32021	0.177835
$896$	0	0
$897$	33.7738	1.12767
$898$	0	0
$899$	24.0915	0.803495
$900$	0	0
$901$	−5.17630	−0.172448
$902$	0	0
$903$	−2.29175	−0.0762646
$904$	0	0
$905$	8.40028	0.279235
$906$	0	0
$907$	−11.3628	−0.377295	−0.188648	−	0.982045i	$-0.560410\pi$
$907$	−11.3628	−0.377295	−0.188648	+	0.982045i	$0.560410\pi$
$908$	0	0
$909$	7.06832	0.234442
$910$	0	0
$911$	4.75507	0.157543	0.0787713	−	0.996893i	$-0.474900\pi$
$911$	4.75507	0.157543	0.0787713	+	0.996893i	$0.474900\pi$
$912$	0	0
$913$	−21.9205	−0.725462
$914$	0	0
$915$	−18.3731	−0.607398
$916$	0	0
$917$	−3.51134	−0.115955
$918$	0	0
$919$	−32.6303	−1.07637	−0.538187	−	0.842825i	$-0.680891\pi$
$919$	−32.6303	−1.07637	−0.538187	+	0.842825i	$0.680891\pi$
$920$	0	0
$921$	−10.5130	−0.346416
$922$	0	0
$923$	−52.4882	−1.72767
$924$	0	0
$925$	−40.9084	−1.34506
$926$	0	0
$927$	27.6459	0.908010
$928$	0	0
$929$	−47.3546	−1.55365	−0.776827	−	0.629714i	$-0.783172\pi$
$929$	−47.3546	−1.55365	−0.776827	+	0.629714i	$0.783172\pi$
$930$	0	0
$931$	2.64366	0.0866424
$932$	0	0
$933$	14.3397	0.469460
$934$	0	0
$935$	14.5279	0.475113
$936$	0	0
$937$	−51.6525	−1.68741	−0.843707	−	0.536803i	$-0.819632\pi$
$937$	−51.6525	−1.68741	−0.843707	+	0.536803i	$0.819632\pi$
$938$	0	0
$939$	−18.3928	−0.600226
$940$	0	0
$941$	54.7240	1.78395	0.891975	−	0.452084i	$-0.149319\pi$
$941$	54.7240	1.78395	0.891975	+	0.452084i	$0.149319\pi$
$942$	0	0
$943$	−11.3073	−0.368218
$944$	0	0
$945$	14.2731	0.464303
$946$	0	0
$947$	36.9728	1.20145	0.600727	−	0.799454i	$-0.294878\pi$
$947$	36.9728	1.20145	0.600727	+	0.799454i	$0.294878\pi$
$948$	0	0
$949$	55.9395	1.81587
$950$	0	0
$951$	5.77707	0.187334
$952$	0	0
$953$	−7.91313	−0.256331	−0.128166	−	0.991753i	$-0.540909\pi$
$953$	−7.91313	−0.256331	−0.128166	+	0.991753i	$0.540909\pi$
$954$	0	0
$955$	−57.2578	−1.85282
$956$	0	0
$957$	−30.5654	−0.988039
$958$	0	0
$959$	1.93780	0.0625747
$960$	0	0
$961$	−19.2993	−0.622557
$962$	0	0
$963$	8.51826	0.274497
$964$	0	0
$965$	−36.1867	−1.16489
$966$	0	0
$967$	8.05906	0.259162	0.129581	−	0.991569i	$-0.458637\pi$
$967$	8.05906	0.259162	0.129581	+	0.991569i	$0.458637\pi$
$968$	0	0
$969$	−2.39933	−0.0770775
$970$	0	0
$971$	1.81193	0.0581476	0.0290738	−	0.999577i	$-0.490744\pi$
$971$	1.81193	0.0581476	0.0290738	+	0.999577i	$0.490744\pi$
$972$	0	0
$973$	4.69834	0.150622
$974$	0	0
$975$	−16.8914	−0.540959
$976$	0	0
$977$	15.1934	0.486079	0.243040	−	0.970016i	$-0.421855\pi$
$977$	15.1934	0.486079	0.243040	+	0.970016i	$0.421855\pi$
$978$	0	0
$979$	64.0547	2.04720
$980$	0	0
$981$	−7.75538	−0.247610
$982$	0	0
$983$	11.6472	0.371488	0.185744	−	0.982598i	$-0.440530\pi$
$983$	11.6472	0.371488	0.185744	+	0.982598i	$0.440530\pi$
$984$	0	0
$985$	−52.4638	−1.67164
$986$	0	0
$987$	−5.05596	−0.160933
$988$	0	0
$989$	21.3596	0.679195
$990$	0	0
$991$	−47.1759	−1.49859	−0.749296	−	0.662235i	$-0.769608\pi$
$991$	−47.1759	−1.49859	−0.749296	+	0.662235i	$0.769608\pi$
$992$	0	0
$993$	22.3183	0.708249
$994$	0	0
$995$	46.3175	1.46836
$996$	0	0
$997$	−39.7574	−1.25913	−0.629564	−	0.776949i	$-0.716766\pi$
$997$	−39.7574	−1.25913	−0.629564	+	0.776949i	$0.716766\pi$
$998$	0	0
$999$	−45.4276	−1.43727

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.bt.1.2		5
4.3	odd	2		7616.2.a.bq.1.4		5
8.3	odd	2		1904.2.a.t.1.2		5
8.5	even	2		119.2.a.b.1.3	✓	5
24.5	odd	2		1071.2.a.m.1.3		5
40.29	even	2		2975.2.a.m.1.3		5
56.5	odd	6		833.2.e.h.18.3		10
56.13	odd	2		833.2.a.g.1.3		5
56.37	even	6		833.2.e.i.18.3		10
56.45	odd	6		833.2.e.h.324.3		10
56.53	even	6		833.2.e.i.324.3		10
136.101	even	2		2023.2.a.j.1.3		5
168.125	even	2		7497.2.a.br.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
119.2.a.b.1.3	✓	5	8.5	even	2
833.2.a.g.1.3		5	56.13	odd	2
833.2.e.h.18.3		10	56.5	odd	6
833.2.e.h.324.3		10	56.45	odd	6
833.2.e.i.18.3		10	56.37	even	6
833.2.e.i.324.3		10	56.53	even	6
1071.2.a.m.1.3		5	24.5	odd	2
1904.2.a.t.1.2		5	8.3	odd	2
2023.2.a.j.1.3		5	136.101	even	2
2975.2.a.m.1.3		5	40.29	even	2
7497.2.a.br.1.3		5	168.125	even	2
7616.2.a.bq.1.4		5	4.3	odd	2
7616.2.a.bt.1.2		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7616.a (trivial)

\(f(q)\)	\(=\)	\(q-0.907578 q^{3} -3.03818 q^{5} -1.00000 q^{7} -2.17630 q^{9} +O(q^{10})\)	\(q-0.907578 q^{3} -3.03818 q^{5} -1.00000 q^{7} -2.17630 q^{9} -4.78178 q^{11} +4.39933 q^{13} +2.75739 q^{15} +1.00000 q^{17} +2.64366 q^{19} +0.907578 q^{21} -8.45881 q^{23} +4.23054 q^{25} +4.69790 q^{27} -7.04298 q^{29} -3.42063 q^{31} +4.33984 q^{33} +3.03818 q^{35} -9.66977 q^{37} -3.99273 q^{39} +1.33675 q^{41} -2.52513 q^{43} +6.61200 q^{45} -5.57082 q^{47} +1.00000 q^{49} -0.907578 q^{51} -5.17630 q^{53} +14.5279 q^{55} -2.39933 q^{57} -9.28732 q^{59} -6.66325 q^{61} +2.17630 q^{63} -13.3659 q^{65} -5.28251 q^{67} +7.67704 q^{69} -11.9310 q^{71} +12.7155 q^{73} -3.83955 q^{75} +4.78178 q^{77} +1.94051 q^{79} +2.26519 q^{81} +4.58417 q^{83} -3.03818 q^{85} +6.39206 q^{87} -13.3956 q^{89} -4.39933 q^{91} +3.10449 q^{93} -8.03191 q^{95} -15.1668 q^{97} +10.4066 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} - 5 q^{7} + 11 q^{9}+O(q^{10}) \) 5 * q + 2 * q^3 - 5 * q^7 + 11 * q^9	\( 5 q + 2 q^{3} - 5 q^{7} + 11 q^{9} + 2 q^{11} - 2 q^{13} + 8 q^{15} + 5 q^{17} - 6 q^{19} - 2 q^{21} - 10 q^{23} + 21 q^{25} + 26 q^{27} + 8 q^{29} + 6 q^{33} - 8 q^{37} + 14 q^{39} + 18 q^{41} - 8 q^{43} - 10 q^{47} + 5 q^{49} + 2 q^{51} - 4 q^{53} - 24 q^{55} + 12 q^{57} - 8 q^{59} - 22 q^{61} - 11 q^{63} - 30 q^{65} - 16 q^{67} + 32 q^{69} - 2 q^{71} + 10 q^{73} + 14 q^{75} - 2 q^{77} + 18 q^{79} + 25 q^{81} + 12 q^{83} - 26 q^{87} + 20 q^{89} + 2 q^{91} + 28 q^{93} - 22 q^{95} + 12 q^{97} + 62 q^{99}+O(q^{100}) \) 5 * q + 2 * q^3 - 5 * q^7 + 11 * q^9 + 2 * q^11 - 2 * q^13 + 8 * q^15 + 5 * q^17 - 6 * q^19 - 2 * q^21 - 10 * q^23 + 21 * q^25 + 26 * q^27 + 8 * q^29 + 6 * q^33 - 8 * q^37 + 14 * q^39 + 18 * q^41 - 8 * q^43 - 10 * q^47 + 5 * q^49 + 2 * q^51 - 4 * q^53 - 24 * q^55 + 12 * q^57 - 8 * q^59 - 22 * q^61 - 11 * q^63 - 30 * q^65 - 16 * q^67 + 32 * q^69 - 2 * q^71 + 10 * q^73 + 14 * q^75 - 2 * q^77 + 18 * q^79 + 25 * q^81 + 12 * q^83 - 26 * q^87 + 20 * q^89 + 2 * q^91 + 28 * q^93 - 22 * q^95 + 12 * q^97 + 62 * q^99

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