LMFDB - Embedded newform 6004.2.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, 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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.85328 q^{3} +0.175152 q^{5} -4.13028 q^{7} +5.14119 q^{9} +O(q^{10})$	$q-2.85328 q^{3} +0.175152 q^{5} -4.13028 q^{7} +5.14119 q^{9} -0.252997 q^{11} +3.47792 q^{13} -0.499757 q^{15} -7.23003 q^{17} -1.00000 q^{19} +11.7848 q^{21} -3.54844 q^{23} -4.96932 q^{25} -6.10940 q^{27} +3.59977 q^{29} -10.0945 q^{31} +0.721870 q^{33} -0.723426 q^{35} +3.49432 q^{37} -9.92346 q^{39} +9.63660 q^{41} +5.79757 q^{43} +0.900489 q^{45} -11.6422 q^{47} +10.0592 q^{49} +20.6293 q^{51} -0.932870 q^{53} -0.0443129 q^{55} +2.85328 q^{57} -8.96095 q^{59} -4.15761 q^{61} -21.2345 q^{63} +0.609164 q^{65} -0.871767 q^{67} +10.1247 q^{69} +7.03713 q^{71} -13.7681 q^{73} +14.1788 q^{75} +1.04495 q^{77} -1.00000 q^{79} +2.00824 q^{81} -11.4901 q^{83} -1.26635 q^{85} -10.2711 q^{87} -1.61468 q^{89} -14.3648 q^{91} +28.8024 q^{93} -0.175152 q^{95} -3.16976 q^{97} -1.30070 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * 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
$3$	−2.85328	−1.64734	−0.823670	−	0.567069i	$-0.808077\pi$
$3$	−2.85328	−1.64734	−0.823670	+	0.567069i	$0.808077\pi$
$4$	0	0
$5$	0.175152	0.0783304	0.0391652	−	0.999233i	$-0.487530\pi$
$5$	0.175152	0.0783304	0.0391652	+	0.999233i	$0.487530\pi$
$6$	0	0
$7$	−4.13028	−1.56110	−0.780549	−	0.625095i	$-0.785060\pi$
$7$	−4.13028	−1.56110	−0.780549	+	0.625095i	$0.785060\pi$
$8$	0	0
$9$	5.14119	1.71373
$10$	0	0
$11$	−0.252997	−0.0762814	−0.0381407	−	0.999272i	$-0.512144\pi$
$11$	−0.252997	−0.0762814	−0.0381407	+	0.999272i	$0.512144\pi$
$12$	0	0
$13$	3.47792	0.964601	0.482301	−	0.876006i	$-0.339801\pi$
$13$	3.47792	0.964601	0.482301	+	0.876006i	$0.339801\pi$
$14$	0	0
$15$	−0.499757	−0.129037
$16$	0	0
$17$	−7.23003	−1.75354	−0.876770	−	0.480910i	$-0.840306\pi$
$17$	−7.23003	−1.75354	−0.876770	+	0.480910i	$0.840306\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	11.7848	2.57166
$22$	0	0
$23$	−3.54844	−0.739901	−0.369951	−	0.929051i	$-0.620625\pi$
$23$	−3.54844	−0.739901	−0.369951	+	0.929051i	$0.620625\pi$
$24$	0	0
$25$	−4.96932	−0.993864
$26$	0	0
$27$	−6.10940	−1.17575
$28$	0	0
$29$	3.59977	0.668461	0.334231	−	0.942491i	$-0.391524\pi$
$29$	3.59977	0.668461	0.334231	+	0.942491i	$0.391524\pi$
$30$	0	0
$31$	−10.0945	−1.81302	−0.906512	−	0.422180i	$-0.861265\pi$
$31$	−10.0945	−1.81302	−0.906512	+	0.422180i	$0.861265\pi$
$32$	0	0
$33$	0.721870	0.125661
$34$	0	0
$35$	−0.723426	−0.122281
$36$	0	0
$37$	3.49432	0.574463	0.287231	−	0.957861i	$-0.407265\pi$
$37$	3.49432	0.574463	0.287231	+	0.957861i	$0.407265\pi$
$38$	0	0
$39$	−9.92346	−1.58903
$40$	0	0
$41$	9.63660	1.50498	0.752492	−	0.658602i	$-0.228852\pi$
$41$	9.63660	1.50498	0.752492	+	0.658602i	$0.228852\pi$
$42$	0	0
$43$	5.79757	0.884121	0.442061	−	0.896985i	$-0.354248\pi$
$43$	5.79757	0.884121	0.442061	+	0.896985i	$0.354248\pi$
$44$	0	0
$45$	0.900489	0.134237
$46$	0	0
$47$	−11.6422	−1.69819	−0.849095	−	0.528240i	$-0.822852\pi$
$47$	−11.6422	−1.69819	−0.849095	+	0.528240i	$0.822852\pi$
$48$	0	0
$49$	10.0592	1.43703
$50$	0	0
$51$	20.6293	2.88868
$52$	0	0
$53$	−0.932870	−0.128140	−0.0640698	−	0.997945i	$-0.520408\pi$
$53$	−0.932870	−0.128140	−0.0640698	+	0.997945i	$0.520408\pi$
$54$	0	0
$55$	−0.0443129	−0.00597515
$56$	0	0
$57$	2.85328	0.377926
$58$	0	0
$59$	−8.96095	−1.16662	−0.583308	−	0.812251i	$-0.698242\pi$
$59$	−8.96095	−1.16662	−0.583308	+	0.812251i	$0.698242\pi$
$60$	0	0
$61$	−4.15761	−0.532328	−0.266164	−	0.963928i	$-0.585756\pi$
$61$	−4.15761	−0.532328	−0.266164	+	0.963928i	$0.585756\pi$
$62$	0	0
$63$	−21.2345	−2.67530
$64$	0	0
$65$	0.609164	0.0755575
$66$	0	0
$67$	−0.871767	−0.106503	−0.0532516	−	0.998581i	$-0.516959\pi$
$67$	−0.871767	−0.106503	−0.0532516	+	0.998581i	$0.516959\pi$
$68$	0	0
$69$	10.1247	1.21887
$70$	0	0
$71$	7.03713	0.835153	0.417577	−	0.908642i	$-0.362880\pi$
$71$	7.03713	0.835153	0.417577	+	0.908642i	$0.362880\pi$
$72$	0	0
$73$	−13.7681	−1.61144	−0.805718	−	0.592299i	$-0.798220\pi$
$73$	−13.7681	−1.61144	−0.805718	+	0.592299i	$0.798220\pi$
$74$	0	0
$75$	14.1788	1.63723
$76$	0	0
$77$	1.04495	0.119083
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	2.00824	0.223138
$82$	0	0
$83$	−11.4901	−1.26121	−0.630603	−	0.776106i	$-0.717192\pi$
$83$	−11.4901	−1.26121	−0.630603	+	0.776106i	$0.717192\pi$
$84$	0	0
$85$	−1.26635	−0.137355
$86$	0	0
$87$	−10.2711	−1.10118
$88$	0	0
$89$	−1.61468	−0.171156	−0.0855780	−	0.996331i	$-0.527274\pi$
$89$	−1.61468	−0.171156	−0.0855780	+	0.996331i	$0.527274\pi$
$90$	0	0
$91$	−14.3648	−1.50584
$92$	0	0
$93$	28.8024	2.98667
$94$	0	0
$95$	−0.175152	−0.0179702
$96$	0	0
$97$	−3.16976	−0.321840	−0.160920	−	0.986967i	$-0.551446\pi$
$97$	−3.16976	−0.321840	−0.160920	+	0.986967i	$0.551446\pi$
$98$	0	0
$99$	−1.30070	−0.130726
$100$	0	0
$101$	−17.1506	−1.70655	−0.853276	−	0.521459i	$-0.825388\pi$
$101$	−17.1506	−1.70655	−0.853276	+	0.521459i	$0.825388\pi$
$102$	0	0
$103$	−2.34784	−0.231340	−0.115670	−	0.993288i	$-0.536901\pi$
$103$	−2.34784	−0.231340	−0.115670	+	0.993288i	$0.536901\pi$
$104$	0	0
$105$	2.06413	0.201439
$106$	0	0
$107$	9.61629	0.929641	0.464821	−	0.885405i	$-0.346119\pi$
$107$	9.61629	0.929641	0.464821	+	0.885405i	$0.346119\pi$
$108$	0	0
$109$	−8.31380	−0.796318	−0.398159	−	0.917316i	$-0.630351\pi$
$109$	−8.31380	−0.796318	−0.398159	+	0.917316i	$0.630351\pi$
$110$	0	0
$111$	−9.97026	−0.946335
$112$	0	0
$113$	−14.3437	−1.34934	−0.674672	−	0.738117i	$-0.735715\pi$
$113$	−14.3437	−1.34934	−0.674672	+	0.738117i	$0.735715\pi$
$114$	0	0
$115$	−0.621517	−0.0579567
$116$	0	0
$117$	17.8806	1.65306
$118$	0	0
$119$	29.8620	2.73745
$120$	0	0
$121$	−10.9360	−0.994181
$122$	0	0
$123$	−27.4959	−2.47922
$124$	0	0
$125$	−1.74615	−0.156180
$126$	0	0
$127$	−13.6626	−1.21236	−0.606178	−	0.795329i	$-0.707298\pi$
$127$	−13.6626	−1.21236	−0.606178	+	0.795329i	$0.707298\pi$
$128$	0	0
$129$	−16.5421	−1.45645
$130$	0	0
$131$	−3.17572	−0.277464	−0.138732	−	0.990330i	$-0.544303\pi$
$131$	−3.17572	−0.277464	−0.138732	+	0.990330i	$0.544303\pi$
$132$	0	0
$133$	4.13028	0.358140
$134$	0	0
$135$	−1.07007	−0.0920972
$136$	0	0
$137$	10.3735	0.886270	0.443135	−	0.896455i	$-0.353866\pi$
$137$	10.3735	0.886270	0.443135	+	0.896455i	$0.353866\pi$
$138$	0	0
$139$	−2.63559	−0.223548	−0.111774	−	0.993734i	$-0.535653\pi$
$139$	−2.63559	−0.223548	−0.111774	+	0.993734i	$0.535653\pi$
$140$	0	0
$141$	33.2184	2.79750
$142$	0	0
$143$	−0.879902	−0.0735811
$144$	0	0
$145$	0.630507	0.0523608
$146$	0	0
$147$	−28.7016	−2.36727
$148$	0	0
$149$	12.5976	1.03203	0.516016	−	0.856579i	$-0.327414\pi$
$149$	12.5976	1.03203	0.516016	+	0.856579i	$0.327414\pi$
$150$	0	0
$151$	15.7965	1.28550	0.642749	−	0.766077i	$-0.277794\pi$
$151$	15.7965	1.28550	0.642749	+	0.766077i	$0.277794\pi$
$152$	0	0
$153$	−37.1709	−3.00509
$154$	0	0
$155$	−1.76807	−0.142015
$156$	0	0
$157$	−4.14189	−0.330559	−0.165279	−	0.986247i	$-0.552853\pi$
$157$	−4.14189	−0.330559	−0.165279	+	0.986247i	$0.552853\pi$
$158$	0	0
$159$	2.66174	0.211089
$160$	0	0
$161$	14.6560	1.15506
$162$	0	0
$163$	9.26472	0.725669	0.362834	−	0.931854i	$-0.381809\pi$
$163$	9.26472	0.725669	0.362834	+	0.931854i	$0.381809\pi$
$164$	0	0
$165$	0.126437	0.00984310
$166$	0	0
$167$	−4.90138	−0.379280	−0.189640	−	0.981854i	$-0.560732\pi$
$167$	−4.90138	−0.379280	−0.189640	+	0.981854i	$0.560732\pi$
$168$	0	0
$169$	−0.904082	−0.0695447
$170$	0	0
$171$	−5.14119	−0.393156
$172$	0	0
$173$	8.18675	0.622427	0.311213	−	0.950340i	$-0.399265\pi$
$173$	8.18675	0.622427	0.311213	+	0.950340i	$0.399265\pi$
$174$	0	0
$175$	20.5247	1.55152
$176$	0	0
$177$	25.5681	1.92181
$178$	0	0
$179$	14.2056	1.06178	0.530889	−	0.847441i	$-0.321858\pi$
$179$	14.2056	1.06178	0.530889	+	0.847441i	$0.321858\pi$
$180$	0	0
$181$	10.6850	0.794206	0.397103	−	0.917774i	$-0.370016\pi$
$181$	10.6850	0.794206	0.397103	+	0.917774i	$0.370016\pi$
$182$	0	0
$183$	11.8628	0.876925
$184$	0	0
$185$	0.612037	0.0449979
$186$	0	0
$187$	1.82917	0.133763
$188$	0	0
$189$	25.2335	1.83547
$190$	0	0
$191$	−4.61377	−0.333841	−0.166920	−	0.985970i	$-0.553382\pi$
$191$	−4.61377	−0.333841	−0.166920	+	0.985970i	$0.553382\pi$
$192$	0	0
$193$	−23.6534	−1.70261	−0.851305	−	0.524671i	$-0.824188\pi$
$193$	−23.6534	−1.70261	−0.851305	+	0.524671i	$0.824188\pi$
$194$	0	0
$195$	−1.73811	−0.124469
$196$	0	0
$197$	7.79539	0.555399	0.277699	−	0.960668i	$-0.410428\pi$
$197$	7.79539	0.555399	0.277699	+	0.960668i	$0.410428\pi$
$198$	0	0
$199$	16.6246	1.17848	0.589242	−	0.807956i	$-0.299426\pi$
$199$	16.6246	1.17848	0.589242	+	0.807956i	$0.299426\pi$
$200$	0	0
$201$	2.48739	0.175447
$202$	0	0
$203$	−14.8681	−1.04353
$204$	0	0
$205$	1.68787	0.117886
$206$	0	0
$207$	−18.2432	−1.26799
$208$	0	0
$209$	0.252997	0.0175002
$210$	0	0
$211$	11.9314	0.821393	0.410697	−	0.911772i	$-0.365285\pi$
$211$	11.9314	0.821393	0.410697	+	0.911772i	$0.365285\pi$
$212$	0	0
$213$	−20.0789	−1.37578
$214$	0	0
$215$	1.01546	0.0692535
$216$	0	0
$217$	41.6930	2.83031
$218$	0	0
$219$	39.2843	2.65458
$220$	0	0
$221$	−25.1455	−1.69147
$222$	0	0
$223$	2.07825	0.139170	0.0695848	−	0.997576i	$-0.477833\pi$
$223$	2.07825	0.139170	0.0695848	+	0.997576i	$0.477833\pi$
$224$	0	0
$225$	−25.5482	−1.70321
$226$	0	0
$227$	27.7413	1.84126	0.920628	−	0.390441i	$-0.127678\pi$
$227$	27.7413	1.84126	0.920628	+	0.390441i	$0.127678\pi$
$228$	0	0
$229$	30.1099	1.98972	0.994858	−	0.101275i	$-0.0322922\pi$
$229$	30.1099	1.98972	0.994858	+	0.101275i	$0.0322922\pi$
$230$	0	0
$231$	−2.98152	−0.196170
$232$	0	0
$233$	−1.40706	−0.0921798	−0.0460899	−	0.998937i	$-0.514676\pi$
$233$	−1.40706	−0.0921798	−0.0460899	+	0.998937i	$0.514676\pi$
$234$	0	0
$235$	−2.03916	−0.133020
$236$	0	0
$237$	2.85328	0.185340
$238$	0	0
$239$	−30.4608	−1.97034	−0.985171	−	0.171574i	$-0.945115\pi$
$239$	−30.4608	−1.97034	−0.985171	+	0.171574i	$0.945115\pi$
$240$	0	0
$241$	3.96932	0.255686	0.127843	−	0.991794i	$-0.459195\pi$
$241$	3.96932	0.255686	0.127843	+	0.991794i	$0.459195\pi$
$242$	0	0
$243$	12.5981	0.808170
$244$	0	0
$245$	1.76189	0.112563
$246$	0	0
$247$	−3.47792	−0.221295
$248$	0	0
$249$	32.7845	2.07764
$250$	0	0
$251$	−4.53064	−0.285972	−0.142986	−	0.989725i	$-0.545670\pi$
$251$	−4.53064	−0.285972	−0.142986	+	0.989725i	$0.545670\pi$
$252$	0	0
$253$	0.897745	0.0564407
$254$	0	0
$255$	3.61326	0.226271
$256$	0	0
$257$	15.7636	0.983305	0.491652	−	0.870792i	$-0.336393\pi$
$257$	15.7636	0.983305	0.491652	+	0.870792i	$0.336393\pi$
$258$	0	0
$259$	−14.4325	−0.896792
$260$	0	0
$261$	18.5071	1.14556
$262$	0	0
$263$	−21.9724	−1.35487	−0.677437	−	0.735581i	$-0.736909\pi$
$263$	−21.9724	−1.35487	−0.677437	+	0.735581i	$0.736909\pi$
$264$	0	0
$265$	−0.163394	−0.0100372
$266$	0	0
$267$	4.60713	0.281952
$268$	0	0
$269$	2.67133	0.162874	0.0814370	−	0.996678i	$-0.474049\pi$
$269$	2.67133	0.162874	0.0814370	+	0.996678i	$0.474049\pi$
$270$	0	0
$271$	5.19213	0.315400	0.157700	−	0.987487i	$-0.449592\pi$
$271$	5.19213	0.315400	0.157700	+	0.987487i	$0.449592\pi$
$272$	0	0
$273$	40.9866	2.48062
$274$	0	0
$275$	1.25722	0.0758134
$276$	0	0
$277$	15.4044	0.925559	0.462780	−	0.886473i	$-0.346852\pi$
$277$	15.4044	0.925559	0.462780	+	0.886473i	$0.346852\pi$
$278$	0	0
$279$	−51.8977	−3.10703
$280$	0	0
$281$	−1.29847	−0.0774600	−0.0387300	−	0.999250i	$-0.512331\pi$
$281$	−1.29847	−0.0774600	−0.0387300	+	0.999250i	$0.512331\pi$
$282$	0	0
$283$	−23.3406	−1.38745	−0.693726	−	0.720239i	$-0.744032\pi$
$283$	−23.3406	−1.38745	−0.693726	+	0.720239i	$0.744032\pi$
$284$	0	0
$285$	0.499757	0.0296031
$286$	0	0
$287$	−39.8018	−2.34943
$288$	0	0
$289$	35.2734	2.07490
$290$	0	0
$291$	9.04419	0.530180
$292$	0	0
$293$	−4.32261	−0.252530	−0.126265	−	0.991997i	$-0.540299\pi$
$293$	−4.32261	−0.252530	−0.126265	+	0.991997i	$0.540299\pi$
$294$	0	0
$295$	−1.56953	−0.0913815
$296$	0	0
$297$	1.54566	0.0896882
$298$	0	0
$299$	−12.3412	−0.713710
$300$	0	0
$301$	−23.9456	−1.38020
$302$	0	0
$303$	48.9355	2.81127
$304$	0	0
$305$	−0.728214	−0.0416974
$306$	0	0
$307$	−17.8397	−1.01817	−0.509084	−	0.860717i	$-0.670016\pi$
$307$	−17.8397	−1.01817	−0.509084	+	0.860717i	$0.670016\pi$
$308$	0	0
$309$	6.69904	0.381095
$310$	0	0
$311$	28.5604	1.61951	0.809756	−	0.586766i	$-0.199599\pi$
$311$	28.5604	1.61951	0.809756	+	0.586766i	$0.199599\pi$
$312$	0	0
$313$	−29.4884	−1.66678	−0.833390	−	0.552685i	$-0.813603\pi$
$313$	−29.4884	−1.66678	−0.833390	+	0.552685i	$0.813603\pi$
$314$	0	0
$315$	−3.71927	−0.209557
$316$	0	0
$317$	4.95768	0.278451	0.139225	−	0.990261i	$-0.455539\pi$
$317$	4.95768	0.278451	0.139225	+	0.990261i	$0.455539\pi$
$318$	0	0
$319$	−0.910731	−0.0509911
$320$	0	0
$321$	−27.4379	−1.53144
$322$	0	0
$323$	7.23003	0.402290
$324$	0	0
$325$	−17.2829	−0.958683
$326$	0	0
$327$	23.7216	1.31181
$328$	0	0
$329$	48.0855	2.65104
$330$	0	0
$331$	18.4727	1.01535	0.507677	−	0.861548i	$-0.330504\pi$
$331$	18.4727	1.01535	0.507677	+	0.861548i	$0.330504\pi$
$332$	0	0
$333$	17.9650	0.984473
$334$	0	0
$335$	−0.152692	−0.00834244
$336$	0	0
$337$	33.1045	1.80331	0.901657	−	0.432451i	$-0.142351\pi$
$337$	33.1045	1.80331	0.901657	+	0.432451i	$0.142351\pi$
$338$	0	0
$339$	40.9266	2.22283
$340$	0	0
$341$	2.55387	0.138300
$342$	0	0
$343$	−12.6353	−0.682240
$344$	0	0
$345$	1.77336	0.0954744
$346$	0	0
$347$	27.1224	1.45601	0.728005	−	0.685572i	$-0.240448\pi$
$347$	27.1224	1.45601	0.728005	+	0.685572i	$0.240448\pi$
$348$	0	0
$349$	−13.1464	−0.703710	−0.351855	−	0.936055i	$-0.614449\pi$
$349$	−13.1464	−0.703710	−0.351855	+	0.936055i	$0.614449\pi$
$350$	0	0
$351$	−21.2480	−1.13413
$352$	0	0
$353$	26.2312	1.39615	0.698073	−	0.716026i	$-0.254041\pi$
$353$	26.2312	1.39615	0.698073	+	0.716026i	$0.254041\pi$
$354$	0	0
$355$	1.23257	0.0654179
$356$	0	0
$357$	−85.2046	−4.50951
$358$	0	0
$359$	7.57728	0.399914	0.199957	−	0.979805i	$-0.435920\pi$
$359$	7.57728	0.399914	0.199957	+	0.979805i	$0.435920\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	31.2034	1.63775
$364$	0	0
$365$	−2.41151	−0.126224
$366$	0	0
$367$	−1.18457	−0.0618342	−0.0309171	−	0.999522i	$-0.509843\pi$
$367$	−1.18457	−0.0618342	−0.0309171	+	0.999522i	$0.509843\pi$
$368$	0	0
$369$	49.5435	2.57913
$370$	0	0
$371$	3.85301	0.200038
$372$	0	0
$373$	−1.18887	−0.0615572	−0.0307786	−	0.999526i	$-0.509799\pi$
$373$	−1.18887	−0.0615572	−0.0307786	+	0.999526i	$0.509799\pi$
$374$	0	0
$375$	4.98224	0.257282
$376$	0	0
$377$	12.5197	0.644798
$378$	0	0
$379$	−8.55226	−0.439300	−0.219650	−	0.975579i	$-0.570492\pi$
$379$	−8.55226	−0.439300	−0.219650	+	0.975579i	$0.570492\pi$
$380$	0	0
$381$	38.9831	1.99716
$382$	0	0
$383$	−10.2083	−0.521621	−0.260810	−	0.965390i	$-0.583990\pi$
$383$	−10.2083	−0.521621	−0.260810	+	0.965390i	$0.583990\pi$
$384$	0	0
$385$	0.183025	0.00932779
$386$	0	0
$387$	29.8064	1.51514
$388$	0	0
$389$	17.4715	0.885840	0.442920	−	0.896561i	$-0.353943\pi$
$389$	17.4715	0.885840	0.442920	+	0.896561i	$0.353943\pi$
$390$	0	0
$391$	25.6553	1.29745
$392$	0	0
$393$	9.06121	0.457078
$394$	0	0
$395$	−0.175152	−0.00881285
$396$	0	0
$397$	−8.58532	−0.430885	−0.215442	−	0.976517i	$-0.569119\pi$
$397$	−8.58532	−0.430885	−0.215442	+	0.976517i	$0.569119\pi$
$398$	0	0
$399$	−11.7848	−0.589979
$400$	0	0
$401$	−16.3141	−0.814687	−0.407343	−	0.913275i	$-0.633545\pi$
$401$	−16.3141	−0.814687	−0.407343	+	0.913275i	$0.633545\pi$
$402$	0	0
$403$	−35.1078	−1.74884
$404$	0	0
$405$	0.351748	0.0174785
$406$	0	0
$407$	−0.884052	−0.0438208
$408$	0	0
$409$	31.5315	1.55913	0.779567	−	0.626319i	$-0.215439\pi$
$409$	31.5315	1.55913	0.779567	+	0.626319i	$0.215439\pi$
$410$	0	0
$411$	−29.5985	−1.45999
$412$	0	0
$413$	37.0112	1.82120
$414$	0	0
$415$	−2.01252	−0.0987907
$416$	0	0
$417$	7.52008	0.368260
$418$	0	0
$419$	14.9247	0.729119	0.364559	−	0.931180i	$-0.381220\pi$
$419$	14.9247	0.729119	0.364559	+	0.931180i	$0.381220\pi$
$420$	0	0
$421$	3.92569	0.191326	0.0956631	−	0.995414i	$-0.469503\pi$
$421$	3.92569	0.191326	0.0956631	+	0.995414i	$0.469503\pi$
$422$	0	0
$423$	−59.8548	−2.91024
$424$	0	0
$425$	35.9284	1.74278
$426$	0	0
$427$	17.1721	0.831016
$428$	0	0
$429$	2.51060	0.121213
$430$	0	0
$431$	4.01693	0.193489	0.0967443	−	0.995309i	$-0.469157\pi$
$431$	4.01693	0.193489	0.0967443	+	0.995309i	$0.469157\pi$
$432$	0	0
$433$	28.6587	1.37725	0.688626	−	0.725117i	$-0.258214\pi$
$433$	28.6587	1.37725	0.688626	+	0.725117i	$0.258214\pi$
$434$	0	0
$435$	−1.79901	−0.0862560
$436$	0	0
$437$	3.54844	0.169745
$438$	0	0
$439$	−1.72552	−0.0823545	−0.0411773	−	0.999152i	$-0.513111\pi$
$439$	−1.72552	−0.0823545	−0.0411773	+	0.999152i	$0.513111\pi$
$440$	0	0
$441$	51.7161	2.46267
$442$	0	0
$443$	−27.8819	−1.32471	−0.662354	−	0.749191i	$-0.730442\pi$
$443$	−27.8819	−1.32471	−0.662354	+	0.749191i	$0.730442\pi$
$444$	0	0
$445$	−0.282815	−0.0134067
$446$	0	0
$447$	−35.9443	−1.70011
$448$	0	0
$449$	−28.6883	−1.35388	−0.676942	−	0.736037i	$-0.736695\pi$
$449$	−28.6883	−1.35388	−0.676942	+	0.736037i	$0.736695\pi$
$450$	0	0
$451$	−2.43803	−0.114802
$452$	0	0
$453$	−45.0717	−2.11765
$454$	0	0
$455$	−2.51602	−0.117953
$456$	0	0
$457$	13.9533	0.652709	0.326354	−	0.945248i	$-0.394180\pi$
$457$	13.9533	0.652709	0.326354	+	0.945248i	$0.394180\pi$
$458$	0	0
$459$	44.1711	2.06173
$460$	0	0
$461$	25.0141	1.16502	0.582512	−	0.812822i	$-0.302070\pi$
$461$	25.0141	1.16502	0.582512	+	0.812822i	$0.302070\pi$
$462$	0	0
$463$	21.9652	1.02081	0.510405	−	0.859934i	$-0.329495\pi$
$463$	21.9652	1.02081	0.510405	+	0.859934i	$0.329495\pi$
$464$	0	0
$465$	5.04479	0.233947
$466$	0	0
$467$	−38.9384	−1.80186	−0.900928	−	0.433969i	$-0.857113\pi$
$467$	−38.9384	−1.80186	−0.900928	+	0.433969i	$0.857113\pi$
$468$	0	0
$469$	3.60064	0.166262
$470$	0	0
$471$	11.8180	0.544543
$472$	0	0
$473$	−1.46677	−0.0674420
$474$	0	0
$475$	4.96932	0.228008
$476$	0	0
$477$	−4.79606	−0.219596
$478$	0	0
$479$	13.7634	0.628864	0.314432	−	0.949280i	$-0.398186\pi$
$479$	13.7634	0.628864	0.314432	+	0.949280i	$0.398186\pi$
$480$	0	0
$481$	12.1530	0.554127
$482$	0	0
$483$	−41.8178	−1.90277
$484$	0	0
$485$	−0.555189	−0.0252098
$486$	0	0
$487$	−26.1113	−1.18322	−0.591608	−	0.806225i	$-0.701507\pi$
$487$	−26.1113	−1.18322	−0.591608	+	0.806225i	$0.701507\pi$
$488$	0	0
$489$	−26.4348	−1.19542
$490$	0	0
$491$	12.7845	0.576956	0.288478	−	0.957486i	$-0.406851\pi$
$491$	12.7845	0.576956	0.288478	+	0.957486i	$0.406851\pi$
$492$	0	0
$493$	−26.0265	−1.17217
$494$	0	0
$495$	−0.227821	−0.0102398
$496$	0	0
$497$	−29.0653	−1.30376
$498$	0	0
$499$	17.5338	0.784919	0.392460	−	0.919769i	$-0.371624\pi$
$499$	17.5338	0.784919	0.392460	+	0.919769i	$0.371624\pi$
$500$	0	0
$501$	13.9850	0.624803
$502$	0	0
$503$	−18.9269	−0.843910	−0.421955	−	0.906617i	$-0.638656\pi$
$503$	−18.9269	−0.843910	−0.421955	+	0.906617i	$0.638656\pi$
$504$	0	0
$505$	−3.00397	−0.133675
$506$	0	0
$507$	2.57960	0.114564
$508$	0	0
$509$	−28.4413	−1.26064	−0.630319	−	0.776336i	$-0.717076\pi$
$509$	−28.4413	−1.26064	−0.630319	+	0.776336i	$0.717076\pi$
$510$	0	0
$511$	56.8661	2.51561
$512$	0	0
$513$	6.10940	0.269737
$514$	0	0
$515$	−0.411229	−0.0181209
$516$	0	0
$517$	2.94544	0.129540
$518$	0	0
$519$	−23.3591	−1.02535
$520$	0	0
$521$	8.00890	0.350876	0.175438	−	0.984490i	$-0.443866\pi$
$521$	8.00890	0.350876	0.175438	+	0.984490i	$0.443866\pi$
$522$	0	0
$523$	−6.08654	−0.266146	−0.133073	−	0.991106i	$-0.542484\pi$
$523$	−6.08654	−0.266146	−0.133073	+	0.991106i	$0.542484\pi$
$524$	0	0
$525$	−58.5626	−2.55588
$526$	0	0
$527$	72.9835	3.17921
$528$	0	0
$529$	−10.4086	−0.452546
$530$	0	0
$531$	−46.0699	−1.99926
$532$	0	0
$533$	33.5153	1.45171
$534$	0	0
$535$	1.68431	0.0728191
$536$	0	0
$537$	−40.5326	−1.74911
$538$	0	0
$539$	−2.54494	−0.109618
$540$	0	0
$541$	−22.9908	−0.988452	−0.494226	−	0.869334i	$-0.664548\pi$
$541$	−22.9908	−0.988452	−0.494226	+	0.869334i	$0.664548\pi$
$542$	0	0
$543$	−30.4871	−1.30833
$544$	0	0
$545$	−1.45618	−0.0623758
$546$	0	0
$547$	−18.2258	−0.779281	−0.389640	−	0.920967i	$-0.627401\pi$
$547$	−18.2258	−0.779281	−0.389640	+	0.920967i	$0.627401\pi$
$548$	0	0
$549$	−21.3751	−0.912265
$550$	0	0
$551$	−3.59977	−0.153355
$552$	0	0
$553$	4.13028	0.175637
$554$	0	0
$555$	−1.74631	−0.0741268
$556$	0	0
$557$	4.54732	0.192676	0.0963381	−	0.995349i	$-0.469287\pi$
$557$	4.54732	0.192676	0.0963381	+	0.995349i	$0.469287\pi$
$558$	0	0
$559$	20.1635	0.852824
$560$	0	0
$561$	−5.21914	−0.220352
$562$	0	0
$563$	−21.0256	−0.886124	−0.443062	−	0.896491i	$-0.646108\pi$
$563$	−21.0256	−0.886124	−0.443062	+	0.896491i	$0.646108\pi$
$564$	0	0
$565$	−2.51233	−0.105695
$566$	0	0
$567$	−8.29459	−0.348340
$568$	0	0
$569$	−9.15160	−0.383655	−0.191828	−	0.981429i	$-0.561441\pi$
$569$	−9.15160	−0.383655	−0.191828	+	0.981429i	$0.561441\pi$
$570$	0	0
$571$	9.87586	0.413292	0.206646	−	0.978416i	$-0.433745\pi$
$571$	9.87586	0.413292	0.206646	+	0.978416i	$0.433745\pi$
$572$	0	0
$573$	13.1644	0.549949
$574$	0	0
$575$	17.6334	0.735362
$576$	0	0
$577$	−35.4101	−1.47414	−0.737070	−	0.675816i	$-0.763791\pi$
$577$	−35.4101	−1.47414	−0.737070	+	0.675816i	$0.763791\pi$
$578$	0	0
$579$	67.4898	2.80478
$580$	0	0
$581$	47.4574	1.96887
$582$	0	0
$583$	0.236013	0.00977466
$584$	0	0
$585$	3.13183	0.129485
$586$	0	0
$587$	30.6452	1.26486	0.632431	−	0.774617i	$-0.282057\pi$
$587$	30.6452	1.26486	0.632431	+	0.774617i	$0.282057\pi$
$588$	0	0
$589$	10.0945	0.415936
$590$	0	0
$591$	−22.2424	−0.914931
$592$	0	0
$593$	6.69072	0.274755	0.137378	−	0.990519i	$-0.456133\pi$
$593$	6.69072	0.274755	0.137378	+	0.990519i	$0.456133\pi$
$594$	0	0
$595$	5.23039	0.214425
$596$	0	0
$597$	−47.4345	−1.94136
$598$	0	0
$599$	12.0222	0.491213	0.245606	−	0.969370i	$-0.421013\pi$
$599$	12.0222	0.491213	0.245606	+	0.969370i	$0.421013\pi$
$600$	0	0
$601$	39.4499	1.60919	0.804597	−	0.593821i	$-0.202381\pi$
$601$	39.4499	1.60919	0.804597	+	0.593821i	$0.202381\pi$
$602$	0	0
$603$	−4.48192	−0.182518
$604$	0	0
$605$	−1.91546	−0.0778746
$606$	0	0
$607$	27.4444	1.11393	0.556966	−	0.830535i	$-0.311965\pi$
$607$	27.4444	1.11393	0.556966	+	0.830535i	$0.311965\pi$
$608$	0	0
$609$	42.4227	1.71905
$610$	0	0
$611$	−40.4907	−1.63808
$612$	0	0
$613$	−12.6244	−0.509894	−0.254947	−	0.966955i	$-0.582058\pi$
$613$	−12.6244	−0.509894	−0.254947	+	0.966955i	$0.582058\pi$
$614$	0	0
$615$	−4.81596	−0.194198
$616$	0	0
$617$	19.2391	0.774538	0.387269	−	0.921967i	$-0.373418\pi$
$617$	19.2391	0.774538	0.387269	+	0.921967i	$0.373418\pi$
$618$	0	0
$619$	37.1222	1.49207	0.746033	−	0.665909i	$-0.231956\pi$
$619$	37.1222	1.49207	0.746033	+	0.665909i	$0.231956\pi$
$620$	0	0
$621$	21.6788	0.869942
$622$	0	0
$623$	6.66908	0.267191
$624$	0	0
$625$	24.5408	0.981631
$626$	0	0
$627$	−0.721870	−0.0288287
$628$	0	0
$629$	−25.2640	−1.00734
$630$	0	0
$631$	−35.0253	−1.39434	−0.697168	−	0.716908i	$-0.745557\pi$
$631$	−35.0253	−1.39434	−0.697168	+	0.716908i	$0.745557\pi$
$632$	0	0
$633$	−34.0437	−1.35311
$634$	0	0
$635$	−2.39302	−0.0949643
$636$	0	0
$637$	34.9850	1.38616
$638$	0	0
$639$	36.1792	1.43123
$640$	0	0
$641$	−32.8722	−1.29837	−0.649187	−	0.760628i	$-0.724891\pi$
$641$	−32.8722	−1.29837	−0.649187	+	0.760628i	$0.724891\pi$
$642$	0	0
$643$	−17.9591	−0.708238	−0.354119	−	0.935200i	$-0.615219\pi$
$643$	−17.9591	−0.708238	−0.354119	+	0.935200i	$0.615219\pi$
$644$	0	0
$645$	−2.89738	−0.114084
$646$	0	0
$647$	12.0444	0.473513	0.236757	−	0.971569i	$-0.423916\pi$
$647$	12.0444	0.473513	0.236757	+	0.971569i	$0.423916\pi$
$648$	0	0
$649$	2.26709	0.0889912
$650$	0	0
$651$	−118.962	−4.66248
$652$	0	0
$653$	46.2093	1.80831	0.904154	−	0.427206i	$-0.140502\pi$
$653$	46.2093	1.80831	0.904154	+	0.427206i	$0.140502\pi$
$654$	0	0
$655$	−0.556234	−0.0217339
$656$	0	0
$657$	−70.7845	−2.76157
$658$	0	0
$659$	−22.8313	−0.889380	−0.444690	−	0.895685i	$-0.646686\pi$
$659$	−22.8313	−0.889380	−0.444690	+	0.895685i	$0.646686\pi$
$660$	0	0
$661$	−48.8252	−1.89908	−0.949540	−	0.313647i	$-0.898449\pi$
$661$	−48.8252	−1.89908	−0.949540	+	0.313647i	$0.898449\pi$
$662$	0	0
$663$	71.7470	2.78642
$664$	0	0
$665$	0.723426	0.0280533
$666$	0	0
$667$	−12.7736	−0.494595
$668$	0	0
$669$	−5.92981	−0.229260
$670$	0	0
$671$	1.05186	0.0406067
$672$	0	0
$673$	32.5121	1.25325	0.626626	−	0.779321i	$-0.284436\pi$
$673$	32.5121	1.25325	0.626626	+	0.779321i	$0.284436\pi$
$674$	0	0
$675$	30.3596	1.16854
$676$	0	0
$677$	−17.1697	−0.659886	−0.329943	−	0.944001i	$-0.607030\pi$
$677$	−17.1697	−0.659886	−0.329943	+	0.944001i	$0.607030\pi$
$678$	0	0
$679$	13.0920	0.502423
$680$	0	0
$681$	−79.1536	−3.03317
$682$	0	0
$683$	−1.42118	−0.0543799	−0.0271900	−	0.999630i	$-0.508656\pi$
$683$	−1.42118	−0.0543799	−0.0271900	+	0.999630i	$0.508656\pi$
$684$	0	0
$685$	1.81694	0.0694218
$686$	0	0
$687$	−85.9118	−3.27774
$688$	0	0
$689$	−3.24445	−0.123604
$690$	0	0
$691$	−7.75527	−0.295024	−0.147512	−	0.989060i	$-0.547127\pi$
$691$	−7.75527	−0.295024	−0.147512	+	0.989060i	$0.547127\pi$
$692$	0	0
$693$	5.37227	0.204076
$694$	0	0
$695$	−0.461629	−0.0175106
$696$	0	0
$697$	−69.6729	−2.63905
$698$	0	0
$699$	4.01474	0.151851
$700$	0	0
$701$	15.3118	0.578318	0.289159	−	0.957281i	$-0.406624\pi$
$701$	15.3118	0.578318	0.289159	+	0.957281i	$0.406624\pi$
$702$	0	0
$703$	−3.49432	−0.131791
$704$	0	0
$705$	5.81828	0.219129
$706$	0	0
$707$	70.8369	2.66410
$708$	0	0
$709$	0.371266	0.0139432	0.00697159	−	0.999976i	$-0.497781\pi$
$709$	0.371266	0.0139432	0.00697159	+	0.999976i	$0.497781\pi$
$710$	0	0
$711$	−5.14119	−0.192810
$712$	0	0
$713$	35.8197	1.34146
$714$	0	0
$715$	−0.154117	−0.00576364
$716$	0	0
$717$	86.9130	3.24582
$718$	0	0
$719$	4.35221	0.162310	0.0811550	−	0.996701i	$-0.474139\pi$
$719$	4.35221	0.162310	0.0811550	+	0.996701i	$0.474139\pi$
$720$	0	0
$721$	9.69723	0.361144
$722$	0	0
$723$	−11.3256	−0.421202
$724$	0	0
$725$	−17.8884	−0.664360
$726$	0	0
$727$	46.5526	1.72654	0.863270	−	0.504742i	$-0.168412\pi$
$727$	46.5526	1.72654	0.863270	+	0.504742i	$0.168412\pi$
$728$	0	0
$729$	−41.9707	−1.55447
$730$	0	0
$731$	−41.9166	−1.55034
$732$	0	0
$733$	−1.34771	−0.0497788	−0.0248894	−	0.999690i	$-0.507923\pi$
$733$	−1.34771	−0.0497788	−0.0248894	+	0.999690i	$0.507923\pi$
$734$	0	0
$735$	−5.02715	−0.185429
$736$	0	0
$737$	0.220554	0.00812422
$738$	0	0
$739$	5.49353	0.202083	0.101041	−	0.994882i	$-0.467783\pi$
$739$	5.49353	0.202083	0.101041	+	0.994882i	$0.467783\pi$
$740$	0	0
$741$	9.92346	0.364548
$742$	0	0
$743$	33.4358	1.22664	0.613321	−	0.789834i	$-0.289833\pi$
$743$	33.4358	1.22664	0.613321	+	0.789834i	$0.289833\pi$
$744$	0	0
$745$	2.20649	0.0808395
$746$	0	0
$747$	−59.0729	−2.16137
$748$	0	0
$749$	−39.7179	−1.45126
$750$	0	0
$751$	27.1375	0.990261	0.495130	−	0.868819i	$-0.335120\pi$
$751$	27.1375	0.990261	0.495130	+	0.868819i	$0.335120\pi$
$752$	0	0
$753$	12.9272	0.471093
$754$	0	0
$755$	2.76678	0.100693
$756$	0	0
$757$	−34.3342	−1.24790	−0.623949	−	0.781465i	$-0.714473\pi$
$757$	−34.3342	−1.24790	−0.623949	+	0.781465i	$0.714473\pi$
$758$	0	0
$759$	−2.56151	−0.0929770
$760$	0	0
$761$	31.0275	1.12475	0.562373	−	0.826884i	$-0.309889\pi$
$761$	31.0275	1.12475	0.562373	+	0.826884i	$0.309889\pi$
$762$	0	0
$763$	34.3383	1.24313
$764$	0	0
$765$	−6.51057	−0.235390
$766$	0	0
$767$	−31.1655	−1.12532
$768$	0	0
$769$	17.6126	0.635126	0.317563	−	0.948237i	$-0.397136\pi$
$769$	17.6126	0.635126	0.317563	+	0.948237i	$0.397136\pi$
$770$	0	0
$771$	−44.9778	−1.61984
$772$	0	0
$773$	−21.7139	−0.780994	−0.390497	−	0.920604i	$-0.627697\pi$
$773$	−21.7139	−0.780994	−0.390497	+	0.920604i	$0.627697\pi$
$774$	0	0
$775$	50.1628	1.80190
$776$	0	0
$777$	41.1799	1.47732
$778$	0	0
$779$	−9.63660	−0.345267
$780$	0	0
$781$	−1.78037	−0.0637067
$782$	0	0
$783$	−21.9924	−0.785946
$784$	0	0
$785$	−0.725460	−0.0258928
$786$	0	0
$787$	5.90170	0.210373	0.105187	−	0.994453i	$-0.466456\pi$
$787$	5.90170	0.210373	0.105187	+	0.994453i	$0.466456\pi$
$788$	0	0
$789$	62.6932	2.23194
$790$	0	0
$791$	59.2436	2.10646
$792$	0	0
$793$	−14.4598	−0.513484
$794$	0	0
$795$	0.466208	0.0165347
$796$	0	0
$797$	−23.5522	−0.834263	−0.417132	−	0.908846i	$-0.636965\pi$
$797$	−23.5522	−0.834263	−0.417132	+	0.908846i	$0.636965\pi$
$798$	0	0
$799$	84.1735	2.97785
$800$	0	0
$801$	−8.30138	−0.293315
$802$	0	0
$803$	3.48329	0.122923
$804$	0	0
$805$	2.56704	0.0904761
$806$	0	0
$807$	−7.62205	−0.268309
$808$	0	0
$809$	−39.3119	−1.38213	−0.691067	−	0.722791i	$-0.742859\pi$
$809$	−39.3119	−1.38213	−0.691067	+	0.722791i	$0.742859\pi$
$810$	0	0
$811$	40.1673	1.41046	0.705232	−	0.708977i	$-0.250843\pi$
$811$	40.1673	1.41046	0.705232	+	0.708977i	$0.250843\pi$
$812$	0	0
$813$	−14.8146	−0.519571
$814$	0	0
$815$	1.62273	0.0568419
$816$	0	0
$817$	−5.79757	−0.202831
$818$	0	0
$819$	−73.8519	−2.58060
$820$	0	0
$821$	40.0959	1.39936	0.699679	−	0.714458i	$-0.253326\pi$
$821$	40.0959	1.39936	0.699679	+	0.714458i	$0.253326\pi$
$822$	0	0
$823$	19.4891	0.679346	0.339673	−	0.940544i	$-0.389684\pi$
$823$	19.4891	0.679346	0.339673	+	0.940544i	$0.389684\pi$
$824$	0	0
$825$	−3.58720	−0.124890
$826$	0	0
$827$	13.8750	0.482481	0.241241	−	0.970465i	$-0.422446\pi$
$827$	13.8750	0.482481	0.241241	+	0.970465i	$0.422446\pi$
$828$	0	0
$829$	−14.2615	−0.495323	−0.247661	−	0.968847i	$-0.579662\pi$
$829$	−14.2615	−0.495323	−0.247661	+	0.968847i	$0.579662\pi$
$830$	0	0
$831$	−43.9529	−1.52471
$832$	0	0
$833$	−72.7282	−2.51988
$834$	0	0
$835$	−0.858486	−0.0297091
$836$	0	0
$837$	61.6712	2.13167
$838$	0	0
$839$	−0.563037	−0.0194382	−0.00971911	−	0.999953i	$-0.503094\pi$
$839$	−0.563037	−0.0194382	−0.00971911	+	0.999953i	$0.503094\pi$
$840$	0	0
$841$	−16.0416	−0.553160
$842$	0	0
$843$	3.70488	0.127603
$844$	0	0
$845$	−0.158352	−0.00544746
$846$	0	0
$847$	45.1687	1.55201
$848$	0	0
$849$	66.5970	2.28560
$850$	0	0
$851$	−12.3994	−0.425046
$852$	0	0
$853$	−43.2561	−1.48106	−0.740530	−	0.672023i	$-0.765426\pi$
$853$	−43.2561	−1.48106	−0.740530	+	0.672023i	$0.765426\pi$
$854$	0	0
$855$	−0.900489	−0.0307961
$856$	0	0
$857$	26.6871	0.911613	0.455807	−	0.890079i	$-0.349351\pi$
$857$	26.6871	0.911613	0.455807	+	0.890079i	$0.349351\pi$
$858$	0	0
$859$	−43.2209	−1.47468	−0.737339	−	0.675522i	$-0.763918\pi$
$859$	−43.2209	−1.47468	−0.737339	+	0.675522i	$0.763918\pi$
$860$	0	0
$861$	113.566	3.87030
$862$	0	0
$863$	54.6960	1.86187	0.930937	−	0.365181i	$-0.118993\pi$
$863$	54.6960	1.86187	0.930937	+	0.365181i	$0.118993\pi$
$864$	0	0
$865$	1.43393	0.0487549
$866$	0	0
$867$	−100.645	−3.41807
$868$	0	0
$869$	0.252997	0.00858233
$870$	0	0
$871$	−3.03193	−0.102733
$872$	0	0
$873$	−16.2963	−0.551546
$874$	0	0
$875$	7.21207	0.243812
$876$	0	0
$877$	−12.2912	−0.415045	−0.207522	−	0.978230i	$-0.566540\pi$
$877$	−12.2912	−0.415045	−0.207522	+	0.978230i	$0.566540\pi$
$878$	0	0
$879$	12.3336	0.416002
$880$	0	0
$881$	12.5054	0.421318	0.210659	−	0.977560i	$-0.432439\pi$
$881$	12.5054	0.421318	0.210659	+	0.977560i	$0.432439\pi$
$882$	0	0
$883$	19.5592	0.658219	0.329110	−	0.944292i	$-0.393251\pi$
$883$	19.5592	0.658219	0.329110	+	0.944292i	$0.393251\pi$
$884$	0	0
$885$	4.47830	0.150536
$886$	0	0
$887$	−5.31621	−0.178501	−0.0892505	−	0.996009i	$-0.528447\pi$
$887$	−5.31621	−0.178501	−0.0892505	+	0.996009i	$0.528447\pi$
$888$	0	0
$889$	56.4301	1.89261
$890$	0	0
$891$	−0.508079	−0.0170213
$892$	0	0
$893$	11.6422	0.389592
$894$	0	0
$895$	2.48814	0.0831695
$896$	0	0
$897$	35.2128	1.17572
$898$	0	0
$899$	−36.3379	−1.21194
$900$	0	0
$901$	6.74468	0.224698
$902$	0	0
$903$	68.3233	2.27366
$904$	0	0
$905$	1.87149	0.0622105
$906$	0	0
$907$	28.1256	0.933894	0.466947	−	0.884285i	$-0.345354\pi$
$907$	28.1256	0.933894	0.466947	+	0.884285i	$0.345354\pi$
$908$	0	0
$909$	−88.1747	−2.92457
$910$	0	0
$911$	1.04256	0.0345417	0.0172708	−	0.999851i	$-0.494502\pi$
$911$	1.04256	0.0345417	0.0172708	+	0.999851i	$0.494502\pi$
$912$	0	0
$913$	2.90697	0.0962066
$914$	0	0
$915$	2.07780	0.0686898
$916$	0	0
$917$	13.1166	0.433149
$918$	0	0
$919$	−49.2174	−1.62353	−0.811766	−	0.583983i	$-0.801493\pi$
$919$	−49.2174	−1.62353	−0.811766	+	0.583983i	$0.801493\pi$
$920$	0	0
$921$	50.9017	1.67727
$922$	0	0
$923$	24.4746	0.805590
$924$	0	0
$925$	−17.3644	−0.570938
$926$	0	0
$927$	−12.0707	−0.396453
$928$	0	0
$929$	30.2639	0.992925	0.496463	−	0.868058i	$-0.334632\pi$
$929$	30.2639	0.992925	0.496463	+	0.868058i	$0.334632\pi$
$930$	0	0
$931$	−10.0592	−0.329676
$932$	0	0
$933$	−81.4908	−2.66789
$934$	0	0
$935$	0.320384	0.0104777
$936$	0	0
$937$	10.1115	0.330329	0.165165	−	0.986266i	$-0.447184\pi$
$937$	10.1115	0.330329	0.165165	+	0.986266i	$0.447184\pi$
$938$	0	0
$939$	84.1384	2.74575
$940$	0	0
$941$	43.1245	1.40582	0.702910	−	0.711279i	$-0.251884\pi$
$941$	43.1245	1.40582	0.702910	+	0.711279i	$0.251884\pi$
$942$	0	0
$943$	−34.1949	−1.11354
$944$	0	0
$945$	4.41970	0.143773
$946$	0	0
$947$	−34.5328	−1.12217	−0.561083	−	0.827760i	$-0.689615\pi$
$947$	−34.5328	−1.12217	−0.561083	+	0.827760i	$0.689615\pi$
$948$	0	0
$949$	−47.8844	−1.55439
$950$	0	0
$951$	−14.1456	−0.458703
$952$	0	0
$953$	11.5446	0.373966	0.186983	−	0.982363i	$-0.440129\pi$
$953$	11.5446	0.373966	0.186983	+	0.982363i	$0.440129\pi$
$954$	0	0
$955$	−0.808111	−0.0261499
$956$	0	0
$957$	2.59857	0.0839998
$958$	0	0
$959$	−42.8455	−1.38355
$960$	0	0
$961$	70.8987	2.28705
$962$	0	0
$963$	49.4391	1.59315
$964$	0	0
$965$	−4.14294	−0.133366
$966$	0	0
$967$	−36.0989	−1.16086	−0.580431	−	0.814310i	$-0.697116\pi$
$967$	−36.0989	−1.16086	−0.580431	+	0.814310i	$0.697116\pi$
$968$	0	0
$969$	−20.6293	−0.662708
$970$	0	0
$971$	−32.7799	−1.05196	−0.525979	−	0.850498i	$-0.676301\pi$
$971$	−32.7799	−1.05196	−0.525979	+	0.850498i	$0.676301\pi$
$972$	0	0
$973$	10.8857	0.348980
$974$	0	0
$975$	49.3129	1.57928
$976$	0	0
$977$	−22.9732	−0.734978	−0.367489	−	0.930028i	$-0.619782\pi$
$977$	−22.9732	−0.734978	−0.367489	+	0.930028i	$0.619782\pi$
$978$	0	0
$979$	0.408509	0.0130560
$980$	0	0
$981$	−42.7428	−1.36467
$982$	0	0
$983$	−56.3670	−1.79783	−0.898915	−	0.438124i	$-0.855643\pi$
$983$	−56.3670	−1.79783	−0.898915	+	0.438124i	$0.855643\pi$
$984$	0	0
$985$	1.36538	0.0435046
$986$	0	0
$987$	−137.201	−4.36717
$988$	0	0
$989$	−20.5723	−0.654162
$990$	0	0
$991$	0.485390	0.0154189	0.00770947	−	0.999970i	$-0.497546\pi$
$991$	0.485390	0.0154189	0.00770947	+	0.999970i	$0.497546\pi$
$992$	0	0
$993$	−52.7078	−1.67263
$994$	0	0
$995$	2.91183	0.0923111
$996$	0	0
$997$	−45.1959	−1.43137	−0.715685	−	0.698423i	$-0.753885\pi$
$997$	−45.1959	−1.43137	−0.715685	+	0.698423i	$0.753885\pi$
$998$	0	0
$999$	−21.3482	−0.675427

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.h.1.4	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.h.1.4	✓	31	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q-2.85328 q^{3} +0.175152 q^{5} -4.13028 q^{7} +5.14119 q^{9} +O(q^{10})\)	\(q-2.85328 q^{3} +0.175152 q^{5} -4.13028 q^{7} +5.14119 q^{9} -0.252997 q^{11} +3.47792 q^{13} -0.499757 q^{15} -7.23003 q^{17} -1.00000 q^{19} +11.7848 q^{21} -3.54844 q^{23} -4.96932 q^{25} -6.10940 q^{27} +3.59977 q^{29} -10.0945 q^{31} +0.721870 q^{33} -0.723426 q^{35} +3.49432 q^{37} -9.92346 q^{39} +9.63660 q^{41} +5.79757 q^{43} +0.900489 q^{45} -11.6422 q^{47} +10.0592 q^{49} +20.6293 q^{51} -0.932870 q^{53} -0.0443129 q^{55} +2.85328 q^{57} -8.96095 q^{59} -4.15761 q^{61} -21.2345 q^{63} +0.609164 q^{65} -0.871767 q^{67} +10.1247 q^{69} +7.03713 q^{71} -13.7681 q^{73} +14.1788 q^{75} +1.04495 q^{77} -1.00000 q^{79} +2.00824 q^{81} -11.4901 q^{83} -1.26635 q^{85} -10.2711 q^{87} -1.61468 q^{89} -14.3648 q^{91} +28.8024 q^{93} -0.175152 q^{95} -3.16976 q^{97} -1.30070 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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