LMFDB - Embedded newform 5000.2.a.o.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5000,2,Mod(1,5000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5000, 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("5000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5000 = 2^{3} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5000.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:	$39.9252010106$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 20 x^{10} + 11 x^{9} + 144 x^{8} - 29 x^{7} - 440 x^{6} + 4 x^{5} + 556 x^{4} + \cdots + 45 $ x^12 - x^11 - 20x^10 + 11x^9 + 144x^8 - 29x^7 - 440x^6 + 4x^5 + 556x^4 + 15x^3 - 285*x^2 + 45
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.46481$ of defining polynomial
Character	$\chi$	$=$	5000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.208054 q^{3} +0.439180 q^{7} -2.95671 q^{9} +O(q^{10})$	$q-0.208054 q^{3} +0.439180 q^{7} -2.95671 q^{9} +4.80388 q^{11} -3.86324 q^{13} -3.88181 q^{17} +0.419740 q^{19} -0.0913734 q^{21} +6.40941 q^{23} +1.23932 q^{27} -2.19223 q^{29} +3.99890 q^{31} -0.999469 q^{33} -1.63568 q^{37} +0.803763 q^{39} +5.36395 q^{41} -5.14646 q^{43} -5.02241 q^{47} -6.80712 q^{49} +0.807629 q^{51} +10.1817 q^{53} -0.0873287 q^{57} +8.64806 q^{59} -2.95084 q^{61} -1.29853 q^{63} +4.73711 q^{67} -1.33351 q^{69} +6.24148 q^{71} -8.71306 q^{73} +2.10977 q^{77} -7.96960 q^{79} +8.61229 q^{81} +11.6924 q^{83} +0.456104 q^{87} +17.7984 q^{89} -1.69666 q^{91} -0.831988 q^{93} +9.95544 q^{97} -14.2037 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} + 26 q^{9}+O(q^{10}) $ 12 * q - 4 * q^3 + 26 * q^9	$ 12 q - 4 q^{3} + 26 q^{9} - q^{11} + 4 q^{13} + 8 q^{17} + 9 q^{19} + 12 q^{21} - 37 q^{27} + 8 q^{29} + 33 q^{31} + 26 q^{33} + 6 q^{37} + 14 q^{39} + 27 q^{41} - 50 q^{43} + 18 q^{47} + 12 q^{49} - 5 q^{51} + 22 q^{53} + 36 q^{57} + 33 q^{59} - 8 q^{61} - 26 q^{63} - 41 q^{67} + 3 q^{69} + 19 q^{71} - 5 q^{73} - 13 q^{77} + 58 q^{79} + 68 q^{81} - 18 q^{83} - 48 q^{87} + 44 q^{89} + 46 q^{91} + 10 q^{93} + 22 q^{97} + 5 q^{99}+O(q^{100}) $ 12 * q - 4 * q^3 + 26 * q^9 - q^11 + 4 * q^13 + 8 * q^17 + 9 * q^19 + 12 * q^21 - 37 * q^27 + 8 * q^29 + 33 * q^31 + 26 * q^33 + 6 * q^37 + 14 * q^39 + 27 * q^41 - 50 * q^43 + 18 * q^47 + 12 * q^49 - 5 * q^51 + 22 * q^53 + 36 * q^57 + 33 * q^59 - 8 * q^61 - 26 * q^63 - 41 * q^67 + 3 * q^69 + 19 * q^71 - 5 * q^73 - 13 * q^77 + 58 * q^79 + 68 * q^81 - 18 * q^83 - 48 * q^87 + 44 * q^89 + 46 * q^91 + 10 * q^93 + 22 * q^97 + 5 * 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
$3$	−0.208054	−0.120120	−0.0600601	−	0.998195i	$-0.519129\pi$
$3$	−0.208054	−0.120120	−0.0600601	+	0.998195i	$0.519129\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.439180	0.165995	0.0829973	−	0.996550i	$-0.473551\pi$
$7$	0.439180	0.165995	0.0829973	+	0.996550i	$0.473551\pi$
$8$	0	0
$9$	−2.95671	−0.985571
$10$	0	0
$11$	4.80388	1.44842	0.724212	−	0.689577i	$-0.242203\pi$
$11$	4.80388	1.44842	0.724212	+	0.689577i	$0.242203\pi$
$12$	0	0
$13$	−3.86324	−1.07147	−0.535735	−	0.844386i	$-0.679965\pi$
$13$	−3.86324	−1.07147	−0.535735	+	0.844386i	$0.679965\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.88181	−0.941478	−0.470739	−	0.882272i	$-0.656013\pi$
$17$	−3.88181	−0.941478	−0.470739	+	0.882272i	$0.656013\pi$
$18$	0	0
$19$	0.419740	0.0962949	0.0481474	−	0.998840i	$-0.484668\pi$
$19$	0.419740	0.0962949	0.0481474	+	0.998840i	$0.484668\pi$
$20$	0	0
$21$	−0.0913734	−0.0199393
$22$	0	0
$23$	6.40941	1.33645	0.668227	−	0.743957i	$-0.267054\pi$
$23$	6.40941	1.33645	0.668227	+	0.743957i	$0.267054\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.23932	0.238507
$28$	0	0
$29$	−2.19223	−0.407088	−0.203544	−	0.979066i	$-0.565246\pi$
$29$	−2.19223	−0.407088	−0.203544	+	0.979066i	$0.565246\pi$
$30$	0	0
$31$	3.99890	0.718223	0.359112	−	0.933295i	$-0.383080\pi$
$31$	3.99890	0.718223	0.359112	+	0.933295i	$0.383080\pi$
$32$	0	0
$33$	−0.999469	−0.173985
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.63568	−0.268904	−0.134452	−	0.990920i	$-0.542927\pi$
$37$	−1.63568	−0.268904	−0.134452	+	0.990920i	$0.542927\pi$
$38$	0	0
$39$	0.803763	0.128705
$40$	0	0
$41$	5.36395	0.837708	0.418854	−	0.908054i	$-0.362432\pi$
$41$	5.36395	0.837708	0.418854	+	0.908054i	$0.362432\pi$
$42$	0	0
$43$	−5.14646	−0.784827	−0.392414	−	0.919789i	$-0.628360\pi$
$43$	−5.14646	−0.784827	−0.392414	+	0.919789i	$0.628360\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.02241	−0.732593	−0.366297	−	0.930498i	$-0.619374\pi$
$47$	−5.02241	−0.732593	−0.366297	+	0.930498i	$0.619374\pi$
$48$	0	0
$49$	−6.80712	−0.972446
$50$	0	0
$51$	0.807629	0.113091
$52$	0	0
$53$	10.1817	1.39857	0.699285	−	0.714843i	$-0.253502\pi$
$53$	10.1817	1.39857	0.699285	+	0.714843i	$0.253502\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.0873287	−0.0115670
$58$	0	0
$59$	8.64806	1.12588	0.562941	−	0.826497i	$-0.309670\pi$
$59$	8.64806	1.12588	0.562941	+	0.826497i	$0.309670\pi$
$60$	0	0
$61$	−2.95084	−0.377817	−0.188908	−	0.981995i	$-0.560495\pi$
$61$	−2.95084	−0.377817	−0.188908	+	0.981995i	$0.560495\pi$
$62$	0	0
$63$	−1.29853	−0.163599
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.73711	0.578730	0.289365	−	0.957219i	$-0.406556\pi$
$67$	4.73711	0.578730	0.289365	+	0.957219i	$0.406556\pi$
$68$	0	0
$69$	−1.33351	−0.160535
$70$	0	0
$71$	6.24148	0.740727	0.370364	−	0.928887i	$-0.379233\pi$
$71$	6.24148	0.740727	0.370364	+	0.928887i	$0.379233\pi$
$72$	0	0
$73$	−8.71306	−1.01979	−0.509893	−	0.860238i	$-0.670315\pi$
$73$	−8.71306	−1.01979	−0.509893	+	0.860238i	$0.670315\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.10977	0.240431
$78$	0	0
$79$	−7.96960	−0.896650	−0.448325	−	0.893871i	$-0.647979\pi$
$79$	−7.96960	−0.896650	−0.448325	+	0.893871i	$0.647979\pi$
$80$	0	0
$81$	8.61229	0.956922
$82$	0	0
$83$	11.6924	1.28340	0.641702	−	0.766954i	$-0.278229\pi$
$83$	11.6924	1.28340	0.641702	+	0.766954i	$0.278229\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.456104	0.0488995
$88$	0	0
$89$	17.7984	1.88662	0.943311	−	0.331909i	$-0.107693\pi$
$89$	17.7984	1.88662	0.943311	+	0.331909i	$0.107693\pi$
$90$	0	0
$91$	−1.69666	−0.177858
$92$	0	0
$93$	−0.831988	−0.0862731
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.95544	1.01082	0.505411	−	0.862879i	$-0.331341\pi$
$97$	9.95544	1.01082	0.505411	+	0.862879i	$0.331341\pi$
$98$	0	0
$99$	−14.2037	−1.42753
$100$	0	0
$101$	−2.12812	−0.211756	−0.105878	−	0.994379i	$-0.533765\pi$
$101$	−2.12812	−0.211756	−0.105878	+	0.994379i	$0.533765\pi$
$102$	0	0
$103$	7.77069	0.765669	0.382835	−	0.923817i	$-0.374948\pi$
$103$	7.77069	0.765669	0.382835	+	0.923817i	$0.374948\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.44195	−0.332746	−0.166373	−	0.986063i	$-0.553206\pi$
$107$	−3.44195	−0.332746	−0.166373	+	0.986063i	$0.553206\pi$
$108$	0	0
$109$	18.1308	1.73662	0.868309	−	0.496023i	$-0.165207\pi$
$109$	18.1308	1.73662	0.868309	+	0.496023i	$0.165207\pi$
$110$	0	0
$111$	0.340310	0.0323008
$112$	0	0
$113$	−8.95983	−0.842870	−0.421435	−	0.906859i	$-0.638473\pi$
$113$	−8.95983	−0.842870	−0.421435	+	0.906859i	$0.638473\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	11.4225	1.05601
$118$	0	0
$119$	−1.70482	−0.156280
$120$	0	0
$121$	12.0773	1.09793
$122$	0	0
$123$	−1.11599	−0.100626
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.2470	−0.909272	−0.454636	−	0.890677i	$-0.650231\pi$
$127$	−10.2470	−0.909272	−0.454636	+	0.890677i	$0.650231\pi$
$128$	0	0
$129$	1.07074	0.0942737
$130$	0	0
$131$	−14.8748	−1.29962	−0.649810	−	0.760097i	$-0.725151\pi$
$131$	−14.8748	−1.29962	−0.649810	+	0.760097i	$0.725151\pi$
$132$	0	0
$133$	0.184341	0.0159844
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.6364	0.994169	0.497084	−	0.867702i	$-0.334404\pi$
$137$	11.6364	0.994169	0.497084	+	0.867702i	$0.334404\pi$
$138$	0	0
$139$	16.9917	1.44122	0.720609	−	0.693342i	$-0.243862\pi$
$139$	16.9917	1.44122	0.720609	+	0.693342i	$0.243862\pi$
$140$	0	0
$141$	1.04493	0.0879993
$142$	0	0
$143$	−18.5585	−1.55194
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.41625	0.116810
$148$	0	0
$149$	−9.62295	−0.788343	−0.394171	−	0.919037i	$-0.628968\pi$
$149$	−9.62295	−0.788343	−0.394171	+	0.919037i	$0.628968\pi$
$150$	0	0
$151$	22.0763	1.79654	0.898270	−	0.439444i	$-0.144824\pi$
$151$	22.0763	1.79654	0.898270	+	0.439444i	$0.144824\pi$
$152$	0	0
$153$	11.4774	0.927894
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.12075	0.408680	0.204340	−	0.978900i	$-0.434495\pi$
$157$	5.12075	0.408680	0.204340	+	0.978900i	$0.434495\pi$
$158$	0	0
$159$	−2.11836	−0.167997
$160$	0	0
$161$	2.81489	0.221844
$162$	0	0
$163$	−21.2545	−1.66478	−0.832390	−	0.554191i	$-0.813028\pi$
$163$	−21.2545	−1.66478	−0.832390	+	0.554191i	$0.813028\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.3053	−1.41651	−0.708254	−	0.705958i	$-0.750517\pi$
$167$	−18.3053	−1.41651	−0.708254	+	0.705958i	$0.750517\pi$
$168$	0	0
$169$	1.92460	0.148046
$170$	0	0
$171$	−1.24105	−0.0949054
$172$	0	0
$173$	11.8308	0.899477	0.449738	−	0.893160i	$-0.351517\pi$
$173$	11.8308	0.899477	0.449738	+	0.893160i	$0.351517\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.79927	−0.135241
$178$	0	0
$179$	−2.37814	−0.177751	−0.0888753	−	0.996043i	$-0.528327\pi$
$179$	−2.37814	−0.177751	−0.0888753	+	0.996043i	$0.528327\pi$
$180$	0	0
$181$	24.9634	1.85552	0.927759	−	0.373181i	$-0.121733\pi$
$181$	24.9634	1.85552	0.927759	+	0.373181i	$0.121733\pi$
$182$	0	0
$183$	0.613936	0.0453834
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−18.6478	−1.36366
$188$	0	0
$189$	0.544285	0.0395909
$190$	0	0
$191$	23.6458	1.71095	0.855474	−	0.517846i	$-0.173266\pi$
$191$	23.6458	1.71095	0.855474	+	0.517846i	$0.173266\pi$
$192$	0	0
$193$	−15.2980	−1.10118	−0.550589	−	0.834777i	$-0.685597\pi$
$193$	−15.2980	−1.10118	−0.550589	+	0.834777i	$0.685597\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.4660	−0.745675	−0.372837	−	0.927897i	$-0.621615\pi$
$197$	−10.4660	−0.745675	−0.372837	+	0.927897i	$0.621615\pi$
$198$	0	0
$199$	−2.40458	−0.170456	−0.0852280	−	0.996361i	$-0.527162\pi$
$199$	−2.40458	−0.170456	−0.0852280	+	0.996361i	$0.527162\pi$
$200$	0	0
$201$	−0.985576	−0.0695171
$202$	0	0
$203$	−0.962786	−0.0675743
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−18.9508	−1.31717
$208$	0	0
$209$	2.01638	0.139476
$210$	0	0
$211$	24.6200	1.69491	0.847456	−	0.530866i	$-0.178133\pi$
$211$	24.6200	1.69491	0.847456	+	0.530866i	$0.178133\pi$
$212$	0	0
$213$	−1.29857	−0.0889763
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.75624	0.119221
$218$	0	0
$219$	1.81279	0.122497
$220$	0	0
$221$	14.9964	1.00877
$222$	0	0
$223$	29.2952	1.96175	0.980876	−	0.194632i	$-0.0623512\pi$
$223$	29.2952	1.96175	0.980876	+	0.194632i	$0.0623512\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.85649	−0.388709	−0.194354	−	0.980931i	$-0.562261\pi$
$227$	−5.85649	−0.388709	−0.194354	+	0.980931i	$0.562261\pi$
$228$	0	0
$229$	−23.2114	−1.53385	−0.766926	−	0.641736i	$-0.778215\pi$
$229$	−23.2114	−1.53385	−0.766926	+	0.641736i	$0.778215\pi$
$230$	0	0
$231$	−0.438947	−0.0288806
$232$	0	0
$233$	−6.34558	−0.415713	−0.207856	−	0.978159i	$-0.566649\pi$
$233$	−6.34558	−0.415713	−0.207856	+	0.978159i	$0.566649\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.65811	0.107706
$238$	0	0
$239$	8.55100	0.553118	0.276559	−	0.960997i	$-0.410806\pi$
$239$	8.55100	0.553118	0.276559	+	0.960997i	$0.410806\pi$
$240$	0	0
$241$	1.95899	0.126190	0.0630949	−	0.998008i	$-0.479903\pi$
$241$	1.95899	0.126190	0.0630949	+	0.998008i	$0.479903\pi$
$242$	0	0
$243$	−5.50979	−0.353453
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.62155	−0.103177
$248$	0	0
$249$	−2.43265	−0.154163
$250$	0	0
$251$	17.8626	1.12748	0.563738	−	0.825954i	$-0.309363\pi$
$251$	17.8626	1.12748	0.563738	+	0.825954i	$0.309363\pi$
$252$	0	0
$253$	30.7900	1.93575
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.6986	1.22876	0.614381	−	0.789009i	$-0.289406\pi$
$257$	19.6986	1.22876	0.614381	+	0.789009i	$0.289406\pi$
$258$	0	0
$259$	−0.718358	−0.0446366
$260$	0	0
$261$	6.48181	0.401214
$262$	0	0
$263$	−11.4590	−0.706590	−0.353295	−	0.935512i	$-0.614939\pi$
$263$	−11.4590	−0.706590	−0.353295	+	0.935512i	$0.614939\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3.70303	−0.226622
$268$	0	0
$269$	9.31338	0.567847	0.283924	−	0.958847i	$-0.408364\pi$
$269$	9.31338	0.567847	0.283924	+	0.958847i	$0.408364\pi$
$270$	0	0
$271$	−7.94073	−0.482365	−0.241182	−	0.970480i	$-0.577535\pi$
$271$	−7.94073	−0.482365	−0.241182	+	0.970480i	$0.577535\pi$
$272$	0	0
$273$	0.352997	0.0213644
$274$	0	0
$275$	0	0
$276$	0	0
$277$	30.4930	1.83215	0.916073	−	0.401012i	$-0.131342\pi$
$277$	30.4930	1.83215	0.916073	+	0.401012i	$0.131342\pi$
$278$	0	0
$279$	−11.8236	−0.707860
$280$	0	0
$281$	16.5837	0.989298	0.494649	−	0.869093i	$-0.335297\pi$
$281$	16.5837	0.989298	0.494649	+	0.869093i	$0.335297\pi$
$282$	0	0
$283$	3.50335	0.208252	0.104126	−	0.994564i	$-0.466795\pi$
$283$	3.50335	0.208252	0.104126	+	0.994564i	$0.466795\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.35574	0.139055
$288$	0	0
$289$	−1.93151	−0.113619
$290$	0	0
$291$	−2.07127	−0.121420
$292$	0	0
$293$	7.67200	0.448203	0.224102	−	0.974566i	$-0.428055\pi$
$293$	7.67200	0.448203	0.224102	+	0.974566i	$0.428055\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.95355	0.345460
$298$	0	0
$299$	−24.7611	−1.43197
$300$	0	0
$301$	−2.26022	−0.130277
$302$	0	0
$303$	0.442765	0.0254362
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.8706	0.791638	0.395819	−	0.918329i	$-0.370461\pi$
$307$	13.8706	0.791638	0.395819	+	0.918329i	$0.370461\pi$
$308$	0	0
$309$	−1.61673	−0.0919724
$310$	0	0
$311$	23.1851	1.31471	0.657354	−	0.753582i	$-0.271676\pi$
$311$	23.1851	1.31471	0.657354	+	0.753582i	$0.271676\pi$
$312$	0	0
$313$	−15.6893	−0.886810	−0.443405	−	0.896321i	$-0.646230\pi$
$313$	−15.6893	−0.886810	−0.443405	+	0.896321i	$0.646230\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.8323	1.00156	0.500781	−	0.865574i	$-0.333046\pi$
$317$	17.8323	1.00156	0.500781	+	0.865574i	$0.333046\pi$
$318$	0	0
$319$	−10.5312	−0.589636
$320$	0	0
$321$	0.716112	0.0399695
$322$	0	0
$323$	−1.62935	−0.0906595
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.77220	−0.208603
$328$	0	0
$329$	−2.20574	−0.121607
$330$	0	0
$331$	2.39331	0.131548	0.0657742	−	0.997835i	$-0.479048\pi$
$331$	2.39331	0.131548	0.0657742	+	0.997835i	$0.479048\pi$
$332$	0	0
$333$	4.83623	0.265024
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0.0223423	0.00121706	0.000608532	−	1.00000i	$-0.499806\pi$
$337$	0.0223423	0.00121706	0.000608532		1.00000i	$0.499806\pi$
$338$	0	0
$339$	1.86413	0.101246
$340$	0	0
$341$	19.2102	1.04029
$342$	0	0
$343$	−6.06382	−0.327415
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.1110	−0.864884	−0.432442	−	0.901662i	$-0.642348\pi$
$347$	−16.1110	−0.864884	−0.432442	+	0.901662i	$0.642348\pi$
$348$	0	0
$349$	−11.9729	−0.640893	−0.320446	−	0.947267i	$-0.603833\pi$
$349$	−11.9729	−0.640893	−0.320446	+	0.947267i	$0.603833\pi$
$350$	0	0
$351$	−4.78779	−0.255553
$352$	0	0
$353$	36.3370	1.93402	0.967012	−	0.254731i	$-0.0819871\pi$
$353$	36.3370	1.93402	0.967012	+	0.254731i	$0.0819871\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.354695	0.0187724
$358$	0	0
$359$	19.2490	1.01593	0.507963	−	0.861379i	$-0.330399\pi$
$359$	19.2490	1.01593	0.507963	+	0.861379i	$0.330399\pi$
$360$	0	0
$361$	−18.8238	−0.990727
$362$	0	0
$363$	−2.51273	−0.131884
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−18.6974	−0.975996	−0.487998	−	0.872845i	$-0.662273\pi$
$367$	−18.6974	−0.975996	−0.487998	+	0.872845i	$0.662273\pi$
$368$	0	0
$369$	−15.8597	−0.825621
$370$	0	0
$371$	4.47162	0.232155
$372$	0	0
$373$	−37.5211	−1.94277	−0.971384	−	0.237513i	$-0.923668\pi$
$373$	−37.5211	−1.94277	−0.971384	+	0.237513i	$0.923668\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.46912	0.436182
$378$	0	0
$379$	13.8931	0.713638	0.356819	−	0.934173i	$-0.383861\pi$
$379$	13.8931	0.713638	0.356819	+	0.934173i	$0.383861\pi$
$380$	0	0
$381$	2.13193	0.109222
$382$	0	0
$383$	12.7229	0.650107	0.325054	−	0.945696i	$-0.394618\pi$
$383$	12.7229	0.650107	0.325054	+	0.945696i	$0.394618\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	15.2166	0.773503
$388$	0	0
$389$	19.5283	0.990125	0.495062	−	0.868857i	$-0.335145\pi$
$389$	19.5283	0.990125	0.495062	+	0.868857i	$0.335145\pi$
$390$	0	0
$391$	−24.8801	−1.25824
$392$	0	0
$393$	3.09477	0.156111
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.8951	1.34982	0.674912	−	0.737898i	$-0.264181\pi$
$397$	26.8951	1.34982	0.674912	+	0.737898i	$0.264181\pi$
$398$	0	0
$399$	−0.0383530	−0.00192005
$400$	0	0
$401$	−24.0774	−1.20237	−0.601184	−	0.799111i	$-0.705304\pi$
$401$	−24.0774	−1.20237	−0.601184	+	0.799111i	$0.705304\pi$
$402$	0	0
$403$	−15.4487	−0.769554
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.85760	−0.389487
$408$	0	0
$409$	17.0162	0.841399	0.420699	−	0.907200i	$-0.361785\pi$
$409$	17.0162	0.841399	0.420699	+	0.907200i	$0.361785\pi$
$410$	0	0
$411$	−2.42101	−0.119420
$412$	0	0
$413$	3.79806	0.186890
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−3.53520	−0.173120
$418$	0	0
$419$	8.12712	0.397036	0.198518	−	0.980097i	$-0.436387\pi$
$419$	8.12712	0.397036	0.198518	+	0.980097i	$0.436387\pi$
$420$	0	0
$421$	−1.25114	−0.0609769	−0.0304885	−	0.999535i	$-0.509706\pi$
$421$	−1.25114	−0.0609769	−0.0304885	+	0.999535i	$0.509706\pi$
$422$	0	0
$423$	14.8498	0.722023
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.29595	−0.0627155
$428$	0	0
$429$	3.86118	0.186420
$430$	0	0
$431$	−9.09453	−0.438068	−0.219034	−	0.975717i	$-0.570291\pi$
$431$	−9.09453	−0.438068	−0.219034	+	0.975717i	$0.570291\pi$
$432$	0	0
$433$	−6.29255	−0.302401	−0.151200	−	0.988503i	$-0.548314\pi$
$433$	−6.29255	−0.302401	−0.151200	+	0.988503i	$0.548314\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.69028	0.128694
$438$	0	0
$439$	32.6087	1.55633	0.778163	−	0.628062i	$-0.216152\pi$
$439$	32.6087	1.55633	0.778163	+	0.628062i	$0.216152\pi$
$440$	0	0
$441$	20.1267	0.958415
$442$	0	0
$443$	−40.2755	−1.91354	−0.956772	−	0.290838i	$-0.906066\pi$
$443$	−40.2755	−1.91354	−0.956772	+	0.290838i	$0.906066\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.00210	0.0946959
$448$	0	0
$449$	−19.7756	−0.933268	−0.466634	−	0.884451i	$-0.654533\pi$
$449$	−19.7756	−0.933268	−0.466634	+	0.884451i	$0.654533\pi$
$450$	0	0
$451$	25.7678	1.21336
$452$	0	0
$453$	−4.59306	−0.215801
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−14.9856	−0.700998	−0.350499	−	0.936563i	$-0.613988\pi$
$457$	−14.9856	−0.700998	−0.350499	+	0.936563i	$0.613988\pi$
$458$	0	0
$459$	−4.81081	−0.224549
$460$	0	0
$461$	−25.2118	−1.17423	−0.587115	−	0.809503i	$-0.699737\pi$
$461$	−25.2118	−1.17423	−0.587115	+	0.809503i	$0.699737\pi$
$462$	0	0
$463$	37.0541	1.72205	0.861026	−	0.508561i	$-0.169822\pi$
$463$	37.0541	1.72205	0.861026	+	0.508561i	$0.169822\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.0743	1.39167	0.695836	−	0.718201i	$-0.255034\pi$
$467$	30.0743	1.39167	0.695836	+	0.718201i	$0.255034\pi$
$468$	0	0
$469$	2.08044	0.0960660
$470$	0	0
$471$	−1.06539	−0.0490908
$472$	0	0
$473$	−24.7230	−1.13676
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−30.1045	−1.37839
$478$	0	0
$479$	−15.8518	−0.724286	−0.362143	−	0.932123i	$-0.617955\pi$
$479$	−15.8518	−0.724286	−0.362143	+	0.932123i	$0.617955\pi$
$480$	0	0
$481$	6.31901	0.288122
$482$	0	0
$483$	−0.585649	−0.0266480
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−2.52753	−0.114533	−0.0572666	−	0.998359i	$-0.518239\pi$
$487$	−2.52753	−0.114533	−0.0572666	+	0.998359i	$0.518239\pi$
$488$	0	0
$489$	4.42209	0.199974
$490$	0	0
$491$	−7.72546	−0.348645	−0.174323	−	0.984689i	$-0.555774\pi$
$491$	−7.72546	−0.348645	−0.174323	+	0.984689i	$0.555774\pi$
$492$	0	0
$493$	8.50984	0.383264
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.74113	0.122957
$498$	0	0
$499$	−2.60431	−0.116585	−0.0582925	−	0.998300i	$-0.518566\pi$
$499$	−2.60431	−0.116585	−0.0582925	+	0.998300i	$0.518566\pi$
$500$	0	0
$501$	3.80850	0.170151
$502$	0	0
$503$	−11.4734	−0.511575	−0.255788	−	0.966733i	$-0.582335\pi$
$503$	−11.4734	−0.511575	−0.255788	+	0.966733i	$0.582335\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−0.400422	−0.0177834
$508$	0	0
$509$	−14.4898	−0.642251	−0.321125	−	0.947037i	$-0.604061\pi$
$509$	−14.4898	−0.642251	−0.321125	+	0.947037i	$0.604061\pi$
$510$	0	0
$511$	−3.82660	−0.169279
$512$	0	0
$513$	0.520192	0.0229670
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−24.1271	−1.06111
$518$	0	0
$519$	−2.46144	−0.108045
$520$	0	0
$521$	3.26715	0.143136	0.0715682	−	0.997436i	$-0.477200\pi$
$521$	3.26715	0.143136	0.0715682	+	0.997436i	$0.477200\pi$
$522$	0	0
$523$	36.7617	1.60748	0.803738	−	0.594983i	$-0.202841\pi$
$523$	36.7617	1.60748	0.803738	+	0.594983i	$0.202841\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.5230	−0.676192
$528$	0	0
$529$	18.0805	0.786109
$530$	0	0
$531$	−25.5698	−1.10964
$532$	0	0
$533$	−20.7222	−0.897578
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.494783	0.0213514
$538$	0	0
$539$	−32.7006	−1.40851
$540$	0	0
$541$	−7.30964	−0.314266	−0.157133	−	0.987577i	$-0.550225\pi$
$541$	−7.30964	−0.314266	−0.157133	+	0.987577i	$0.550225\pi$
$542$	0	0
$543$	−5.19375	−0.222885
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−25.9477	−1.10944	−0.554721	−	0.832037i	$-0.687175\pi$
$547$	−25.9477	−1.10944	−0.554721	+	0.832037i	$0.687175\pi$
$548$	0	0
$549$	8.72480	0.372365
$550$	0	0
$551$	−0.920167	−0.0392004
$552$	0	0
$553$	−3.50009	−0.148839
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.94894	−0.209693	−0.104847	−	0.994488i	$-0.533435\pi$
$557$	−4.94894	−0.209693	−0.104847	+	0.994488i	$0.533435\pi$
$558$	0	0
$559$	19.8820	0.840918
$560$	0	0
$561$	3.87975	0.163803
$562$	0	0
$563$	−16.7972	−0.707918	−0.353959	−	0.935261i	$-0.615165\pi$
$563$	−16.7972	−0.707918	−0.353959	+	0.935261i	$0.615165\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.78235	0.158844
$568$	0	0
$569$	−4.18204	−0.175320	−0.0876601	−	0.996150i	$-0.527939\pi$
$569$	−4.18204	−0.175320	−0.0876601	+	0.996150i	$0.527939\pi$
$570$	0	0
$571$	6.27637	0.262658	0.131329	−	0.991339i	$-0.458076\pi$
$571$	6.27637	0.262658	0.131329	+	0.991339i	$0.458076\pi$
$572$	0	0
$573$	−4.91961	−0.205519
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15.5995	0.649417	0.324709	−	0.945814i	$-0.394734\pi$
$577$	15.5995	0.649417	0.324709	+	0.945814i	$0.394734\pi$
$578$	0	0
$579$	3.18283	0.132274
$580$	0	0
$581$	5.13506	0.213038
$582$	0	0
$583$	48.9119	2.02572
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−40.4389	−1.66909	−0.834545	−	0.550940i	$-0.814269\pi$
$587$	−40.4389	−1.66909	−0.834545	+	0.550940i	$0.814269\pi$
$588$	0	0
$589$	1.67850	0.0691612
$590$	0	0
$591$	2.17751	0.0895707
$592$	0	0
$593$	33.8595	1.39044	0.695221	−	0.718796i	$-0.255306\pi$
$593$	33.8595	1.39044	0.695221	+	0.718796i	$0.255306\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.500283	0.0204752
$598$	0	0
$599$	41.6471	1.70166	0.850828	−	0.525445i	$-0.176101\pi$
$599$	41.6471	1.70166	0.850828	+	0.525445i	$0.176101\pi$
$600$	0	0
$601$	26.2025	1.06882	0.534412	−	0.845224i	$-0.320533\pi$
$601$	26.2025	1.06882	0.534412	+	0.845224i	$0.320533\pi$
$602$	0	0
$603$	−14.0063	−0.570379
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−23.1509	−0.939667	−0.469833	−	0.882755i	$-0.655686\pi$
$607$	−23.1509	−0.939667	−0.469833	+	0.882755i	$0.655686\pi$
$608$	0	0
$609$	0.200312	0.00811705
$610$	0	0
$611$	19.4028	0.784951
$612$	0	0
$613$	−18.4426	−0.744888	−0.372444	−	0.928055i	$-0.621480\pi$
$613$	−18.4426	−0.744888	−0.372444	+	0.928055i	$0.621480\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.5871	1.55346	0.776729	−	0.629835i	$-0.216878\pi$
$617$	38.5871	1.55346	0.776729	+	0.629835i	$0.216878\pi$
$618$	0	0
$619$	37.5969	1.51115	0.755573	−	0.655064i	$-0.227358\pi$
$619$	37.5969	1.51115	0.755573	+	0.655064i	$0.227358\pi$
$620$	0	0
$621$	7.94331	0.318754
$622$	0	0
$623$	7.81669	0.313169
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.419517	−0.0167539
$628$	0	0
$629$	6.34940	0.253167
$630$	0	0
$631$	−34.0511	−1.35555	−0.677777	−	0.735268i	$-0.737056\pi$
$631$	−34.0511	−1.35555	−0.677777	+	0.735268i	$0.737056\pi$
$632$	0	0
$633$	−5.12230	−0.203593
$634$	0	0
$635$	0	0
$636$	0	0
$637$	26.2975	1.04195
$638$	0	0
$639$	−18.4543	−0.730039
$640$	0	0
$641$	−8.27812	−0.326966	−0.163483	−	0.986546i	$-0.552273\pi$
$641$	−8.27812	−0.326966	−0.163483	+	0.986546i	$0.552273\pi$
$642$	0	0
$643$	13.0742	0.515597	0.257799	−	0.966199i	$-0.417003\pi$
$643$	13.0742	0.515597	0.257799	+	0.966199i	$0.417003\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.0273	1.25912	0.629562	−	0.776950i	$-0.283234\pi$
$647$	32.0273	1.25912	0.629562	+	0.776950i	$0.283234\pi$
$648$	0	0
$649$	41.5443	1.63076
$650$	0	0
$651$	−0.365393	−0.0143209
$652$	0	0
$653$	−4.79802	−0.187761	−0.0938804	−	0.995583i	$-0.529927\pi$
$653$	−4.79802	−0.187761	−0.0938804	+	0.995583i	$0.529927\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	25.7620	1.00507
$658$	0	0
$659$	−24.7222	−0.963040	−0.481520	−	0.876435i	$-0.659915\pi$
$659$	−24.7222	−0.963040	−0.481520	+	0.876435i	$0.659915\pi$
$660$	0	0
$661$	−5.95450	−0.231603	−0.115802	−	0.993272i	$-0.536944\pi$
$661$	−5.95450	−0.231603	−0.115802	+	0.993272i	$0.536944\pi$
$662$	0	0
$663$	−3.12006	−0.121173
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−14.0509	−0.544054
$668$	0	0
$669$	−6.09500	−0.235646
$670$	0	0
$671$	−14.1755	−0.547239
$672$	0	0
$673$	19.2332	0.741384	0.370692	−	0.928756i	$-0.379121\pi$
$673$	19.2332	0.741384	0.370692	+	0.928756i	$0.379121\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12.3542	−0.474809	−0.237405	−	0.971411i	$-0.576297\pi$
$677$	−12.3542	−0.474809	−0.237405	+	0.971411i	$0.576297\pi$
$678$	0	0
$679$	4.37224	0.167791
$680$	0	0
$681$	1.21847	0.0466918
$682$	0	0
$683$	−35.8877	−1.37321	−0.686603	−	0.727033i	$-0.740899\pi$
$683$	−35.8877	−1.37321	−0.686603	+	0.727033i	$0.740899\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.82923	0.184247
$688$	0	0
$689$	−39.3345	−1.49852
$690$	0	0
$691$	−3.90849	−0.148686	−0.0743430	−	0.997233i	$-0.523686\pi$
$691$	−3.90849	−0.148686	−0.0743430	+	0.997233i	$0.523686\pi$
$692$	0	0
$693$	−6.23799	−0.236962
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−20.8219	−0.788684
$698$	0	0
$699$	1.32023	0.0499355
$700$	0	0
$701$	−32.1168	−1.21303	−0.606517	−	0.795070i	$-0.707434\pi$
$701$	−32.1168	−1.21303	−0.606517	+	0.795070i	$0.707434\pi$
$702$	0	0
$703$	−0.686559	−0.0258941
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.934628	−0.0351503
$708$	0	0
$709$	−23.1985	−0.871240	−0.435620	−	0.900131i	$-0.643471\pi$
$709$	−23.1985	−0.871240	−0.435620	+	0.900131i	$0.643471\pi$
$710$	0	0
$711$	23.5638	0.883712
$712$	0	0
$713$	25.6306	0.959872
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.77907	−0.0664407
$718$	0	0
$719$	−31.7394	−1.18368	−0.591841	−	0.806055i	$-0.701599\pi$
$719$	−31.7394	−1.18368	−0.591841	+	0.806055i	$0.701599\pi$
$720$	0	0
$721$	3.41274	0.127097
$722$	0	0
$723$	−0.407577	−0.0151579
$724$	0	0
$725$	0	0
$726$	0	0
$727$	46.3732	1.71989	0.859944	−	0.510389i	$-0.170499\pi$
$727$	46.3732	1.71989	0.859944	+	0.510389i	$0.170499\pi$
$728$	0	0
$729$	−24.6905	−0.914465
$730$	0	0
$731$	19.9776	0.738898
$732$	0	0
$733$	−36.5335	−1.34940	−0.674698	−	0.738094i	$-0.735726\pi$
$733$	−36.5335	−1.34940	−0.674698	+	0.738094i	$0.735726\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22.7565	0.838246
$738$	0	0
$739$	−19.3502	−0.711809	−0.355904	−	0.934522i	$-0.615827\pi$
$739$	−19.3502	−0.711809	−0.355904	+	0.934522i	$0.615827\pi$
$740$	0	0
$741$	0.337371	0.0123936
$742$	0	0
$743$	41.9184	1.53784	0.768918	−	0.639347i	$-0.220795\pi$
$743$	41.9184	1.53784	0.768918	+	0.639347i	$0.220795\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−34.5710	−1.26489
$748$	0	0
$749$	−1.51164	−0.0552340
$750$	0	0
$751$	9.60342	0.350434	0.175217	−	0.984530i	$-0.443937\pi$
$751$	9.60342	0.350434	0.175217	+	0.984530i	$0.443937\pi$
$752$	0	0
$753$	−3.71639	−0.135433
$754$	0	0
$755$	0	0
$756$	0	0
$757$	38.0538	1.38309	0.691544	−	0.722334i	$-0.256931\pi$
$757$	38.0538	1.38309	0.691544	+	0.722334i	$0.256931\pi$
$758$	0	0
$759$	−6.40600	−0.232523
$760$	0	0
$761$	−9.15679	−0.331933	−0.165967	−	0.986131i	$-0.553074\pi$
$761$	−9.15679	−0.331933	−0.165967	+	0.986131i	$0.553074\pi$
$762$	0	0
$763$	7.96271	0.288269
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−33.4095	−1.20635
$768$	0	0
$769$	42.3506	1.52720	0.763602	−	0.645688i	$-0.223429\pi$
$769$	42.3506	1.52720	0.763602	+	0.645688i	$0.223429\pi$
$770$	0	0
$771$	−4.09837	−0.147599
$772$	0	0
$773$	−10.2208	−0.367616	−0.183808	−	0.982962i	$-0.558843\pi$
$773$	−10.2208	−0.367616	−0.183808	+	0.982962i	$0.558843\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.149457	0.00536176
$778$	0	0
$779$	2.25146	0.0806670
$780$	0	0
$781$	29.9833	1.07289
$782$	0	0
$783$	−2.71688	−0.0970934
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−53.3742	−1.90258	−0.951292	−	0.308292i	$-0.900243\pi$
$787$	−53.3742	−1.90258	−0.951292	+	0.308292i	$0.900243\pi$
$788$	0	0
$789$	2.38409	0.0848757
$790$	0	0
$791$	−3.93498	−0.139912
$792$	0	0
$793$	11.3998	0.404819
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32.2834	1.14354	0.571768	−	0.820415i	$-0.306258\pi$
$797$	32.2834	1.14354	0.571768	+	0.820415i	$0.306258\pi$
$798$	0	0
$799$	19.4961	0.689721
$800$	0	0
$801$	−52.6247	−1.85940
$802$	0	0
$803$	−41.8565	−1.47708
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.93769	−0.0682099
$808$	0	0
$809$	−26.3111	−0.925051	−0.462525	−	0.886606i	$-0.653057\pi$
$809$	−26.3111	−0.925051	−0.462525	+	0.886606i	$0.653057\pi$
$810$	0	0
$811$	−46.1357	−1.62005	−0.810023	−	0.586399i	$-0.800545\pi$
$811$	−46.1357	−1.62005	−0.810023	+	0.586399i	$0.800545\pi$
$812$	0	0
$813$	1.65210	0.0579418
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.16017	−0.0755748
$818$	0	0
$819$	5.01653	0.175292
$820$	0	0
$821$	−35.3577	−1.23399	−0.616996	−	0.786967i	$-0.711650\pi$
$821$	−35.3577	−1.23399	−0.616996	+	0.786967i	$0.711650\pi$
$822$	0	0
$823$	−37.8865	−1.32064	−0.660321	−	0.750984i	$-0.729580\pi$
$823$	−37.8865	−1.32064	−0.660321	+	0.750984i	$0.729580\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−39.8763	−1.38663	−0.693317	−	0.720633i	$-0.743852\pi$
$827$	−39.8763	−1.38663	−0.693317	+	0.720633i	$0.743852\pi$
$828$	0	0
$829$	47.5902	1.65288	0.826438	−	0.563028i	$-0.190364\pi$
$829$	47.5902	1.65288	0.826438	+	0.563028i	$0.190364\pi$
$830$	0	0
$831$	−6.34420	−0.220078
$832$	0	0
$833$	26.4240	0.915537
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.95591	0.171301
$838$	0	0
$839$	−24.5026	−0.845924	−0.422962	−	0.906147i	$-0.639010\pi$
$839$	−24.5026	−0.845924	−0.422962	+	0.906147i	$0.639010\pi$
$840$	0	0
$841$	−24.1941	−0.834280
$842$	0	0
$843$	−3.45030	−0.118835
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.30410	0.182251
$848$	0	0
$849$	−0.728887	−0.0250153
$850$	0	0
$851$	−10.4837	−0.359378
$852$	0	0
$853$	−17.0184	−0.582698	−0.291349	−	0.956617i	$-0.594104\pi$
$853$	−17.0184	−0.582698	−0.291349	+	0.956617i	$0.594104\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25.5937	−0.874265	−0.437133	−	0.899397i	$-0.644006\pi$
$857$	−25.5937	−0.874265	−0.437133	+	0.899397i	$0.644006\pi$
$858$	0	0
$859$	−33.2153	−1.13329	−0.566646	−	0.823962i	$-0.691759\pi$
$859$	−33.2153	−1.13329	−0.566646	+	0.823962i	$0.691759\pi$
$860$	0	0
$861$	−0.490122	−0.0167033
$862$	0	0
$863$	24.0186	0.817604	0.408802	−	0.912623i	$-0.365947\pi$
$863$	24.0186	0.817604	0.408802	+	0.912623i	$0.365947\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0.401860	0.0136479
$868$	0	0
$869$	−38.2850	−1.29873
$870$	0	0
$871$	−18.3006	−0.620091
$872$	0	0
$873$	−29.4354	−0.996237
$874$	0	0
$875$	0	0
$876$	0	0
$877$	24.9124	0.841232	0.420616	−	0.907239i	$-0.361814\pi$
$877$	24.9124	0.841232	0.420616	+	0.907239i	$0.361814\pi$
$878$	0	0
$879$	−1.59619	−0.0538383
$880$	0	0
$881$	16.8795	0.568686	0.284343	−	0.958723i	$-0.408224\pi$
$881$	16.8795	0.568686	0.284343	+	0.958723i	$0.408224\pi$
$882$	0	0
$883$	−15.6862	−0.527883	−0.263942	−	0.964539i	$-0.585023\pi$
$883$	−15.6862	−0.527883	−0.263942	+	0.964539i	$0.585023\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	55.6261	1.86774	0.933871	−	0.357611i	$-0.116408\pi$
$887$	55.6261	1.86774	0.933871	+	0.357611i	$0.116408\pi$
$888$	0	0
$889$	−4.50027	−0.150934
$890$	0	0
$891$	41.3724	1.38603
$892$	0	0
$893$	−2.10810	−0.0705450
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.15165	0.172009
$898$	0	0
$899$	−8.76652	−0.292380
$900$	0	0
$901$	−39.5236	−1.31672
$902$	0	0
$903$	0.470249	0.0156489
$904$	0	0
$905$	0	0
$906$	0	0
$907$	9.85553	0.327247	0.163624	−	0.986523i	$-0.447682\pi$
$907$	9.85553	0.327247	0.163624	+	0.986523i	$0.447682\pi$
$908$	0	0
$909$	6.29224	0.208700
$910$	0	0
$911$	−19.7058	−0.652882	−0.326441	−	0.945218i	$-0.605849\pi$
$911$	−19.7058	−0.652882	−0.326441	+	0.945218i	$0.605849\pi$
$912$	0	0
$913$	56.1688	1.85891
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.53273	−0.215730
$918$	0	0
$919$	−34.4946	−1.13787	−0.568936	−	0.822382i	$-0.692645\pi$
$919$	−34.4946	−1.13787	−0.568936	+	0.822382i	$0.692645\pi$
$920$	0	0
$921$	−2.88584	−0.0950918
$922$	0	0
$923$	−24.1123	−0.793666
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−22.9757	−0.754621
$928$	0	0
$929$	−19.4971	−0.639680	−0.319840	−	0.947472i	$-0.603629\pi$
$929$	−19.4971	−0.639680	−0.319840	+	0.947472i	$0.603629\pi$
$930$	0	0
$931$	−2.85722	−0.0936415
$932$	0	0
$933$	−4.82377	−0.157923
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−45.4809	−1.48580	−0.742898	−	0.669404i	$-0.766549\pi$
$937$	−45.4809	−1.48580	−0.742898	+	0.669404i	$0.766549\pi$
$938$	0	0
$939$	3.26422	0.106524
$940$	0	0
$941$	−34.7603	−1.13315	−0.566577	−	0.824009i	$-0.691733\pi$
$941$	−34.7603	−1.13315	−0.566577	+	0.824009i	$0.691733\pi$
$942$	0	0
$943$	34.3797	1.11956
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.76422	0.187312	0.0936559	−	0.995605i	$-0.470145\pi$
$947$	5.76422	0.187312	0.0936559	+	0.995605i	$0.470145\pi$
$948$	0	0
$949$	33.6606	1.09267
$950$	0	0
$951$	−3.71009	−0.120308
$952$	0	0
$953$	−11.9020	−0.385543	−0.192771	−	0.981244i	$-0.561748\pi$
$953$	−11.9020	−0.385543	−0.192771	+	0.981244i	$0.561748\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.19107	0.0708272
$958$	0	0
$959$	5.11050	0.165027
$960$	0	0
$961$	−15.0088	−0.484156
$962$	0	0
$963$	10.1769	0.327944
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−40.3476	−1.29749	−0.648745	−	0.761005i	$-0.724706\pi$
$967$	−40.3476	−1.29749	−0.648745	+	0.761005i	$0.724706\pi$
$968$	0	0
$969$	0.338994	0.0108900
$970$	0	0
$971$	−45.0121	−1.44451	−0.722253	−	0.691629i	$-0.756894\pi$
$971$	−45.0121	−1.44451	−0.722253	+	0.691629i	$0.756894\pi$
$972$	0	0
$973$	7.46243	0.239234
$974$	0	0
$975$	0	0
$976$	0	0
$977$	25.5135	0.816251	0.408125	−	0.912926i	$-0.366183\pi$
$977$	25.5135	0.816251	0.408125	+	0.912926i	$0.366183\pi$
$978$	0	0
$979$	85.5012	2.73263
$980$	0	0
$981$	−53.6077	−1.71156
$982$	0	0
$983$	14.1598	0.451626	0.225813	−	0.974171i	$-0.427496\pi$
$983$	14.1598	0.451626	0.225813	+	0.974171i	$0.427496\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.458914	0.0146074
$988$	0	0
$989$	−32.9857	−1.04889
$990$	0	0
$991$	−56.6099	−1.79827	−0.899136	−	0.437670i	$-0.855804\pi$
$991$	−56.6099	−1.79827	−0.899136	+	0.437670i	$0.855804\pi$
$992$	0	0
$993$	−0.497939	−0.0158016
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−9.60527	−0.304202	−0.152101	−	0.988365i	$-0.548604\pi$
$997$	−9.60527	−0.304202	−0.152101	+	0.988365i	$0.548604\pi$
$998$	0	0
$999$	−2.02713	−0.0641355

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5000.2.a.o.1.7	✓	12
4.3	odd	2		10000.2.a.bp.1.6		12
5.4	even	2		5000.2.a.p.1.6	yes	12
20.19	odd	2		10000.2.a.bo.1.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5000.2.a.o.1.7	✓	12	1.1	even	1	trivial
5000.2.a.p.1.6	yes	12	5.4	even	2
10000.2.a.bo.1.7		12	20.19	odd	2
10000.2.a.bp.1.6		12	4.3	odd	2

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:	\( N \)	\(=\)	\( 5000 = 2^{3} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5000.a (trivial)

\(f(q)\)	\(=\)	\(q-0.208054 q^{3} +0.439180 q^{7} -2.95671 q^{9} +O(q^{10})\)	\(q-0.208054 q^{3} +0.439180 q^{7} -2.95671 q^{9} +4.80388 q^{11} -3.86324 q^{13} -3.88181 q^{17} +0.419740 q^{19} -0.0913734 q^{21} +6.40941 q^{23} +1.23932 q^{27} -2.19223 q^{29} +3.99890 q^{31} -0.999469 q^{33} -1.63568 q^{37} +0.803763 q^{39} +5.36395 q^{41} -5.14646 q^{43} -5.02241 q^{47} -6.80712 q^{49} +0.807629 q^{51} +10.1817 q^{53} -0.0873287 q^{57} +8.64806 q^{59} -2.95084 q^{61} -1.29853 q^{63} +4.73711 q^{67} -1.33351 q^{69} +6.24148 q^{71} -8.71306 q^{73} +2.10977 q^{77} -7.96960 q^{79} +8.61229 q^{81} +11.6924 q^{83} +0.456104 q^{87} +17.7984 q^{89} -1.69666 q^{91} -0.831988 q^{93} +9.95544 q^{97} -14.2037 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 26 q^{9}+O(q^{10}) \) 12 * q - 4 * q^3 + 26 * q^9	\( 12 q - 4 q^{3} + 26 q^{9} - q^{11} + 4 q^{13} + 8 q^{17} + 9 q^{19} + 12 q^{21} - 37 q^{27} + 8 q^{29} + 33 q^{31} + 26 q^{33} + 6 q^{37} + 14 q^{39} + 27 q^{41} - 50 q^{43} + 18 q^{47} + 12 q^{49} - 5 q^{51} + 22 q^{53} + 36 q^{57} + 33 q^{59} - 8 q^{61} - 26 q^{63} - 41 q^{67} + 3 q^{69} + 19 q^{71} - 5 q^{73} - 13 q^{77} + 58 q^{79} + 68 q^{81} - 18 q^{83} - 48 q^{87} + 44 q^{89} + 46 q^{91} + 10 q^{93} + 22 q^{97} + 5 q^{99}+O(q^{100}) \) 12 * q - 4 * q^3 + 26 * q^9 - q^11 + 4 * q^13 + 8 * q^17 + 9 * q^19 + 12 * q^21 - 37 * q^27 + 8 * q^29 + 33 * q^31 + 26 * q^33 + 6 * q^37 + 14 * q^39 + 27 * q^41 - 50 * q^43 + 18 * q^47 + 12 * q^49 - 5 * q^51 + 22 * q^53 + 36 * q^57 + 33 * q^59 - 8 * q^61 - 26 * q^63 - 41 * q^67 + 3 * q^69 + 19 * q^71 - 5 * q^73 - 13 * q^77 + 58 * q^79 + 68 * q^81 - 18 * q^83 - 48 * q^87 + 44 * q^89 + 46 * q^91 + 10 * q^93 + 22 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

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Database

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