LMFDB - Embedded newform 3332.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, 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("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3332 = 2^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3332.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:	$26.6061539535$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	3332.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.381966 q^{3} +1.38197 q^{5} -2.85410 q^{9} +O(q^{10})$	$q+0.381966 q^{3} +1.38197 q^{5} -2.85410 q^{9} -3.23607 q^{13} +0.527864 q^{15} +1.00000 q^{17} -2.47214 q^{19} +5.23607 q^{23} -3.09017 q^{25} -2.23607 q^{27} +2.76393 q^{29} +9.56231 q^{31} +3.70820 q^{37} -1.23607 q^{39} -1.14590 q^{41} +4.85410 q^{43} -3.94427 q^{45} +10.9443 q^{47} +0.381966 q^{51} +12.3262 q^{53} -0.944272 q^{57} +5.23607 q^{59} +12.5623 q^{61} -4.47214 q^{65} -5.09017 q^{67} +2.00000 q^{69} -4.76393 q^{71} -2.85410 q^{73} -1.18034 q^{75} -1.70820 q^{79} +7.70820 q^{81} -1.52786 q^{83} +1.38197 q^{85} +1.05573 q^{87} +13.2361 q^{89} +3.65248 q^{93} -3.41641 q^{95} +3.85410 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} + 5 q^{5} + q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 + 5 * q^5 + q^9	$ 2 q + 3 q^{3} + 5 q^{5} + q^{9} - 2 q^{13} + 10 q^{15} + 2 q^{17} + 4 q^{19} + 6 q^{23} + 5 q^{25} + 10 q^{29} - q^{31} - 6 q^{37} + 2 q^{39} - 9 q^{41} + 3 q^{43} + 10 q^{45} + 4 q^{47} + 3 q^{51} + 9 q^{53} + 16 q^{57} + 6 q^{59} + 5 q^{61} + q^{67} + 4 q^{69} - 14 q^{71} + q^{73} + 20 q^{75} + 10 q^{79} + 2 q^{81} - 12 q^{83} + 5 q^{85} + 20 q^{87} + 22 q^{89} - 24 q^{93} + 20 q^{95} + q^{97}+O(q^{100}) $ 2 * q + 3 * q^3 + 5 * q^5 + q^9 - 2 * q^13 + 10 * q^15 + 2 * q^17 + 4 * q^19 + 6 * q^23 + 5 * q^25 + 10 * q^29 - q^31 - 6 * q^37 + 2 * q^39 - 9 * q^41 + 3 * q^43 + 10 * q^45 + 4 * q^47 + 3 * q^51 + 9 * q^53 + 16 * q^57 + 6 * q^59 + 5 * q^61 + q^67 + 4 * q^69 - 14 * q^71 + q^73 + 20 * q^75 + 10 * q^79 + 2 * q^81 - 12 * q^83 + 5 * q^85 + 20 * q^87 + 22 * q^89 - 24 * q^93 + 20 * q^95 + q^97

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
$3$	0.381966	0.220528	0.110264	−	0.993902i	$-0.464830\pi$
$3$	0.381966	0.220528	0.110264	+	0.993902i	$0.464830\pi$
$4$	0	0
$5$	1.38197	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$5$	1.38197	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.85410	−0.951367
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−3.23607	−0.897524	−0.448762	−	0.893651i	$-0.648135\pi$
$13$	−3.23607	−0.897524	−0.448762	+	0.893651i	$0.648135\pi$
$14$	0	0
$15$	0.527864	0.136294
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.47214	−0.567147	−0.283573	−	0.958951i	$-0.591520\pi$
$19$	−2.47214	−0.567147	−0.283573	+	0.958951i	$0.591520\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.23607	1.09180	0.545898	−	0.837852i	$-0.316189\pi$
$23$	5.23607	1.09180	0.545898	+	0.837852i	$0.316189\pi$
$24$	0	0
$25$	−3.09017	−0.618034
$26$	0	0
$27$	−2.23607	−0.430331
$28$	0	0
$29$	2.76393	0.513249	0.256625	−	0.966511i	$-0.417390\pi$
$29$	2.76393	0.513249	0.256625	+	0.966511i	$0.417390\pi$
$30$	0	0
$31$	9.56231	1.71744	0.858720	−	0.512444i	$-0.171260\pi$
$31$	9.56231	1.71744	0.858720	+	0.512444i	$0.171260\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.70820	0.609625	0.304812	−	0.952412i	$-0.401406\pi$
$37$	3.70820	0.609625	0.304812	+	0.952412i	$0.401406\pi$
$38$	0	0
$39$	−1.23607	−0.197929
$40$	0	0
$41$	−1.14590	−0.178959	−0.0894796	−	0.995989i	$-0.528520\pi$
$41$	−1.14590	−0.178959	−0.0894796	+	0.995989i	$0.528520\pi$
$42$	0	0
$43$	4.85410	0.740244	0.370122	−	0.928983i	$-0.379316\pi$
$43$	4.85410	0.740244	0.370122	+	0.928983i	$0.379316\pi$
$44$	0	0
$45$	−3.94427	−0.587977
$46$	0	0
$47$	10.9443	1.59639	0.798193	−	0.602402i	$-0.205789\pi$
$47$	10.9443	1.59639	0.798193	+	0.602402i	$0.205789\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.381966	0.0534859
$52$	0	0
$53$	12.3262	1.69314	0.846569	−	0.532278i	$-0.178664\pi$
$53$	12.3262	1.69314	0.846569	+	0.532278i	$0.178664\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.944272	−0.125072
$58$	0	0
$59$	5.23607	0.681678	0.340839	−	0.940122i	$-0.389289\pi$
$59$	5.23607	0.681678	0.340839	+	0.940122i	$0.389289\pi$
$60$	0	0
$61$	12.5623	1.60844	0.804219	−	0.594333i	$-0.202584\pi$
$61$	12.5623	1.60844	0.804219	+	0.594333i	$0.202584\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.47214	−0.554700
$66$	0	0
$67$	−5.09017	−0.621863	−0.310932	−	0.950432i	$-0.600641\pi$
$67$	−5.09017	−0.621863	−0.310932	+	0.950432i	$0.600641\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	−4.76393	−0.565375	−0.282687	−	0.959212i	$-0.591226\pi$
$71$	−4.76393	−0.565375	−0.282687	+	0.959212i	$0.591226\pi$
$72$	0	0
$73$	−2.85410	−0.334047	−0.167024	−	0.985953i	$-0.553416\pi$
$73$	−2.85410	−0.334047	−0.167024	+	0.985953i	$0.553416\pi$
$74$	0	0
$75$	−1.18034	−0.136294
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1.70820	−0.192188	−0.0960940	−	0.995372i	$-0.530635\pi$
$79$	−1.70820	−0.192188	−0.0960940	+	0.995372i	$0.530635\pi$
$80$	0	0
$81$	7.70820	0.856467
$82$	0	0
$83$	−1.52786	−0.167705	−0.0838524	−	0.996478i	$-0.526722\pi$
$83$	−1.52786	−0.167705	−0.0838524	+	0.996478i	$0.526722\pi$
$84$	0	0
$85$	1.38197	0.149895
$86$	0	0
$87$	1.05573	0.113186
$88$	0	0
$89$	13.2361	1.40302	0.701510	−	0.712659i	$-0.252509\pi$
$89$	13.2361	1.40302	0.701510	+	0.712659i	$0.252509\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.65248	0.378744
$94$	0	0
$95$	−3.41641	−0.350516
$96$	0	0
$97$	3.85410	0.391325	0.195662	−	0.980671i	$-0.437314\pi$
$97$	3.85410	0.391325	0.195662	+	0.980671i	$0.437314\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.4721	1.63904	0.819519	−	0.573051i	$-0.194240\pi$
$101$	16.4721	1.63904	0.819519	+	0.573051i	$0.194240\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−10.1803	−0.975100	−0.487550	−	0.873095i	$-0.662109\pi$
$109$	−10.1803	−0.975100	−0.487550	+	0.873095i	$0.662109\pi$
$110$	0	0
$111$	1.41641	0.134439
$112$	0	0
$113$	−4.94427	−0.465118	−0.232559	−	0.972582i	$-0.574710\pi$
$113$	−4.94427	−0.465118	−0.232559	+	0.972582i	$0.574710\pi$
$114$	0	0
$115$	7.23607	0.674767
$116$	0	0
$117$	9.23607	0.853875
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−0.437694	−0.0394655
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	−13.3820	−1.18746	−0.593729	−	0.804665i	$-0.702345\pi$
$127$	−13.3820	−1.18746	−0.593729	+	0.804665i	$0.702345\pi$
$128$	0	0
$129$	1.85410	0.163245
$130$	0	0
$131$	−2.47214	−0.215992	−0.107996	−	0.994151i	$-0.534443\pi$
$131$	−2.47214	−0.215992	−0.107996	+	0.994151i	$0.534443\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3.09017	−0.265959
$136$	0	0
$137$	−0.0901699	−0.00770374	−0.00385187	−	0.999993i	$-0.501226\pi$
$137$	−0.0901699	−0.00770374	−0.00385187	+	0.999993i	$0.501226\pi$
$138$	0	0
$139$	−11.3820	−0.965406	−0.482703	−	0.875784i	$-0.660345\pi$
$139$	−11.3820	−0.965406	−0.482703	+	0.875784i	$0.660345\pi$
$140$	0	0
$141$	4.18034	0.352048
$142$	0	0
$143$	0	0
$144$	0	0
$145$	3.81966	0.317206
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.61803	−0.460247	−0.230124	−	0.973161i	$-0.573913\pi$
$149$	−5.61803	−0.460247	−0.230124	+	0.973161i	$0.573913\pi$
$150$	0	0
$151$	12.2705	0.998560	0.499280	−	0.866441i	$-0.333598\pi$
$151$	12.2705	0.998560	0.499280	+	0.866441i	$0.333598\pi$
$152$	0	0
$153$	−2.85410	−0.230740
$154$	0	0
$155$	13.2148	1.06144
$156$	0	0
$157$	10.2918	0.821375	0.410687	−	0.911776i	$-0.365289\pi$
$157$	10.2918	0.821375	0.410687	+	0.911776i	$0.365289\pi$
$158$	0	0
$159$	4.70820	0.373385
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−20.5623	−1.59116	−0.795580	−	0.605849i	$-0.792833\pi$
$167$	−20.5623	−1.59116	−0.795580	+	0.605849i	$0.792833\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	0	0
$171$	7.05573	0.539565
$172$	0	0
$173$	6.67376	0.507397	0.253698	−	0.967283i	$-0.418353\pi$
$173$	6.67376	0.507397	0.253698	+	0.967283i	$0.418353\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.00000	0.150329
$178$	0	0
$179$	8.09017	0.604688	0.302344	−	0.953199i	$-0.402231\pi$
$179$	8.09017	0.604688	0.302344	+	0.953199i	$0.402231\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	4.79837	0.354706
$184$	0	0
$185$	5.12461	0.376769
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.9098	0.789408	0.394704	−	0.918808i	$-0.370847\pi$
$191$	10.9098	0.789408	0.394704	+	0.918808i	$0.370847\pi$
$192$	0	0
$193$	−14.4721	−1.04173	−0.520864	−	0.853640i	$-0.674390\pi$
$193$	−14.4721	−1.04173	−0.520864	+	0.853640i	$0.674390\pi$
$194$	0	0
$195$	−1.70820	−0.122327
$196$	0	0
$197$	−2.29180	−0.163284	−0.0816419	−	0.996662i	$-0.526016\pi$
$197$	−2.29180	−0.163284	−0.0816419	+	0.996662i	$0.526016\pi$
$198$	0	0
$199$	18.8541	1.33653	0.668266	−	0.743922i	$-0.267037\pi$
$199$	18.8541	1.33653	0.668266	+	0.743922i	$0.267037\pi$
$200$	0	0
$201$	−1.94427	−0.137138
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.58359	−0.110603
$206$	0	0
$207$	−14.9443	−1.03870
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−25.5967	−1.76215	−0.881076	−	0.472974i	$-0.843180\pi$
$211$	−25.5967	−1.76215	−0.881076	+	0.472974i	$0.843180\pi$
$212$	0	0
$213$	−1.81966	−0.124681
$214$	0	0
$215$	6.70820	0.457496
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.09017	−0.0736669
$220$	0	0
$221$	−3.23607	−0.217681
$222$	0	0
$223$	−19.8885	−1.33184	−0.665918	−	0.746025i	$-0.731960\pi$
$223$	−19.8885	−1.33184	−0.665918	+	0.746025i	$0.731960\pi$
$224$	0	0
$225$	8.81966	0.587977
$226$	0	0
$227$	−13.8541	−0.919529	−0.459765	−	0.888041i	$-0.652066\pi$
$227$	−13.8541	−0.919529	−0.459765	+	0.888041i	$0.652066\pi$
$228$	0	0
$229$	9.41641	0.622254	0.311127	−	0.950368i	$-0.399294\pi$
$229$	9.41641	0.622254	0.311127	+	0.950368i	$0.399294\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.1803	1.32206	0.661029	−	0.750360i	$-0.270120\pi$
$233$	20.1803	1.32206	0.661029	+	0.750360i	$0.270120\pi$
$234$	0	0
$235$	15.1246	0.986621
$236$	0	0
$237$	−0.652476	−0.0423829
$238$	0	0
$239$	−21.6180	−1.39835	−0.699177	−	0.714948i	$-0.746450\pi$
$239$	−21.6180	−1.39835	−0.699177	+	0.714948i	$0.746450\pi$
$240$	0	0
$241$	1.38197	0.0890203	0.0445101	−	0.999009i	$-0.485827\pi$
$241$	1.38197	0.0890203	0.0445101	+	0.999009i	$0.485827\pi$
$242$	0	0
$243$	9.65248	0.619207
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	−0.583592	−0.0369836
$250$	0	0
$251$	2.18034	0.137622	0.0688109	−	0.997630i	$-0.478079\pi$
$251$	2.18034	0.137622	0.0688109	+	0.997630i	$0.478079\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0.527864	0.0330561
$256$	0	0
$257$	10.4721	0.653234	0.326617	−	0.945157i	$-0.394091\pi$
$257$	10.4721	0.653234	0.326617	+	0.945157i	$0.394091\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−7.88854	−0.488289
$262$	0	0
$263$	−7.41641	−0.457315	−0.228658	−	0.973507i	$-0.573434\pi$
$263$	−7.41641	−0.457315	−0.228658	+	0.973507i	$0.573434\pi$
$264$	0	0
$265$	17.0344	1.04642
$266$	0	0
$267$	5.05573	0.309406
$268$	0	0
$269$	−13.4164	−0.818013	−0.409006	−	0.912532i	$-0.634125\pi$
$269$	−13.4164	−0.818013	−0.409006	+	0.912532i	$0.634125\pi$
$270$	0	0
$271$	−11.5279	−0.700268	−0.350134	−	0.936700i	$-0.613864\pi$
$271$	−11.5279	−0.700268	−0.350134	+	0.936700i	$0.613864\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.12461	0.548245	0.274122	−	0.961695i	$-0.411613\pi$
$277$	9.12461	0.548245	0.274122	+	0.961695i	$0.411613\pi$
$278$	0	0
$279$	−27.2918	−1.63392
$280$	0	0
$281$	14.5066	0.865390	0.432695	−	0.901540i	$-0.357563\pi$
$281$	14.5066	0.865390	0.432695	+	0.901540i	$0.357563\pi$
$282$	0	0
$283$	25.8541	1.53687	0.768433	−	0.639930i	$-0.221037\pi$
$283$	25.8541	1.53687	0.768433	+	0.639930i	$0.221037\pi$
$284$	0	0
$285$	−1.30495	−0.0772987
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	1.47214	0.0862981
$292$	0	0
$293$	21.5967	1.26170	0.630848	−	0.775907i	$-0.282707\pi$
$293$	21.5967	1.26170	0.630848	+	0.775907i	$0.282707\pi$
$294$	0	0
$295$	7.23607	0.421300
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−16.9443	−0.979913
$300$	0	0
$301$	0	0
$302$	0	0
$303$	6.29180	0.361454
$304$	0	0
$305$	17.3607	0.994070
$306$	0	0
$307$	17.4164	0.994007	0.497003	−	0.867749i	$-0.334434\pi$
$307$	17.4164	0.994007	0.497003	+	0.867749i	$0.334434\pi$
$308$	0	0
$309$	2.29180	0.130376
$310$	0	0
$311$	11.7426	0.665864	0.332932	−	0.942951i	$-0.391962\pi$
$311$	11.7426	0.665864	0.332932	+	0.942951i	$0.391962\pi$
$312$	0	0
$313$	23.8541	1.34831	0.674157	−	0.738588i	$-0.264507\pi$
$313$	23.8541	1.34831	0.674157	+	0.738588i	$0.264507\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.8885	−0.667727	−0.333864	−	0.942621i	$-0.608352\pi$
$317$	−11.8885	−0.667727	−0.333864	+	0.942621i	$0.608352\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.47214	−0.137553
$324$	0	0
$325$	10.0000	0.554700
$326$	0	0
$327$	−3.88854	−0.215037
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.32624	−0.127862	−0.0639308	−	0.997954i	$-0.520364\pi$
$331$	−2.32624	−0.127862	−0.0639308	+	0.997954i	$0.520364\pi$
$332$	0	0
$333$	−10.5836	−0.579977
$334$	0	0
$335$	−7.03444	−0.384333
$336$	0	0
$337$	13.4164	0.730838	0.365419	−	0.930843i	$-0.380926\pi$
$337$	13.4164	0.730838	0.365419	+	0.930843i	$0.380926\pi$
$338$	0	0
$339$	−1.88854	−0.102572
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2.76393	0.148805
$346$	0	0
$347$	−9.88854	−0.530845	−0.265422	−	0.964132i	$-0.585511\pi$
$347$	−9.88854	−0.530845	−0.265422	+	0.964132i	$0.585511\pi$
$348$	0	0
$349$	−0.472136	−0.0252729	−0.0126364	−	0.999920i	$-0.504022\pi$
$349$	−0.472136	−0.0252729	−0.0126364	+	0.999920i	$0.504022\pi$
$350$	0	0
$351$	7.23607	0.386233
$352$	0	0
$353$	−24.0689	−1.28106	−0.640529	−	0.767934i	$-0.721285\pi$
$353$	−24.0689	−1.28106	−0.640529	+	0.767934i	$0.721285\pi$
$354$	0	0
$355$	−6.58359	−0.349421
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.14590	0.218812	0.109406	−	0.993997i	$-0.465105\pi$
$359$	4.14590	0.218812	0.109406	+	0.993997i	$0.465105\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	0	0
$363$	−4.20163	−0.220528
$364$	0	0
$365$	−3.94427	−0.206453
$366$	0	0
$367$	−15.6180	−0.815255	−0.407627	−	0.913148i	$-0.633644\pi$
$367$	−15.6180	−0.815255	−0.407627	+	0.913148i	$0.633644\pi$
$368$	0	0
$369$	3.27051	0.170256
$370$	0	0
$371$	0	0
$372$	0	0
$373$	7.50658	0.388676	0.194338	−	0.980935i	$-0.437744\pi$
$373$	7.50658	0.388676	0.194338	+	0.980935i	$0.437744\pi$
$374$	0	0
$375$	−4.27051	−0.220528
$376$	0	0
$377$	−8.94427	−0.460653
$378$	0	0
$379$	31.1246	1.59876	0.799382	−	0.600823i	$-0.205160\pi$
$379$	31.1246	1.59876	0.799382	+	0.600823i	$0.205160\pi$
$380$	0	0
$381$	−5.11146	−0.261868
$382$	0	0
$383$	25.8885	1.32284	0.661421	−	0.750014i	$-0.269954\pi$
$383$	25.8885	1.32284	0.661421	+	0.750014i	$0.269954\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−13.8541	−0.704244
$388$	0	0
$389$	−10.5066	−0.532705	−0.266352	−	0.963876i	$-0.585818\pi$
$389$	−10.5066	−0.532705	−0.266352	+	0.963876i	$0.585818\pi$
$390$	0	0
$391$	5.23607	0.264799
$392$	0	0
$393$	−0.944272	−0.0476322
$394$	0	0
$395$	−2.36068	−0.118779
$396$	0	0
$397$	−1.67376	−0.0840037	−0.0420019	−	0.999118i	$-0.513374\pi$
$397$	−1.67376	−0.0840037	−0.0420019	+	0.999118i	$0.513374\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	28.9443	1.44541	0.722704	−	0.691158i	$-0.242899\pi$
$401$	28.9443	1.44541	0.722704	+	0.691158i	$0.242899\pi$
$402$	0	0
$403$	−30.9443	−1.54144
$404$	0	0
$405$	10.6525	0.529326
$406$	0	0
$407$	0	0
$408$	0	0
$409$	16.0000	0.791149	0.395575	−	0.918434i	$-0.370545\pi$
$409$	16.0000	0.791149	0.395575	+	0.918434i	$0.370545\pi$
$410$	0	0
$411$	−0.0344419	−0.00169889
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−2.11146	−0.103647
$416$	0	0
$417$	−4.34752	−0.212899
$418$	0	0
$419$	−2.50658	−0.122454	−0.0612272	−	0.998124i	$-0.519501\pi$
$419$	−2.50658	−0.122454	−0.0612272	+	0.998124i	$0.519501\pi$
$420$	0	0
$421$	−16.8541	−0.821419	−0.410709	−	0.911766i	$-0.634719\pi$
$421$	−16.8541	−0.821419	−0.410709	+	0.911766i	$0.634719\pi$
$422$	0	0
$423$	−31.2361	−1.51875
$424$	0	0
$425$	−3.09017	−0.149895
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.9443	0.719840	0.359920	−	0.932983i	$-0.382804\pi$
$431$	14.9443	0.719840	0.359920	+	0.932983i	$0.382804\pi$
$432$	0	0
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	0	0
$435$	1.45898	0.0699528
$436$	0	0
$437$	−12.9443	−0.619208
$438$	0	0
$439$	−22.8541	−1.09077	−0.545383	−	0.838187i	$-0.683616\pi$
$439$	−22.8541	−1.09077	−0.545383	+	0.838187i	$0.683616\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−24.9443	−1.18514	−0.592569	−	0.805520i	$-0.701886\pi$
$443$	−24.9443	−1.18514	−0.592569	+	0.805520i	$0.701886\pi$
$444$	0	0
$445$	18.2918	0.867114
$446$	0	0
$447$	−2.14590	−0.101497
$448$	0	0
$449$	−16.9443	−0.799650	−0.399825	−	0.916592i	$-0.630929\pi$
$449$	−16.9443	−0.799650	−0.399825	+	0.916592i	$0.630929\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	4.68692	0.220211
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−21.2705	−0.994992	−0.497496	−	0.867466i	$-0.665747\pi$
$457$	−21.2705	−0.994992	−0.497496	+	0.867466i	$0.665747\pi$
$458$	0	0
$459$	−2.23607	−0.104371
$460$	0	0
$461$	26.8328	1.24973	0.624864	−	0.780733i	$-0.285154\pi$
$461$	26.8328	1.24973	0.624864	+	0.780733i	$0.285154\pi$
$462$	0	0
$463$	21.7984	1.01306	0.506528	−	0.862223i	$-0.330929\pi$
$463$	21.7984	1.01306	0.506528	+	0.862223i	$0.330929\pi$
$464$	0	0
$465$	5.04760	0.234077
$466$	0	0
$467$	−33.2361	−1.53798	−0.768991	−	0.639260i	$-0.779241\pi$
$467$	−33.2361	−1.53798	−0.768991	+	0.639260i	$0.779241\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	3.93112	0.181136
$472$	0	0
$473$	0	0
$474$	0	0
$475$	7.63932	0.350516
$476$	0	0
$477$	−35.1803	−1.61080
$478$	0	0
$479$	−8.38197	−0.382982	−0.191491	−	0.981494i	$-0.561332\pi$
$479$	−8.38197	−0.382982	−0.191491	+	0.981494i	$0.561332\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.32624	0.241852
$486$	0	0
$487$	36.9443	1.67410	0.837052	−	0.547123i	$-0.184277\pi$
$487$	36.9443	1.67410	0.837052	+	0.547123i	$0.184277\pi$
$488$	0	0
$489$	−2.29180	−0.103639
$490$	0	0
$491$	−15.2016	−0.686040	−0.343020	−	0.939328i	$-0.611450\pi$
$491$	−15.2016	−0.686040	−0.343020	+	0.939328i	$0.611450\pi$
$492$	0	0
$493$	2.76393	0.124481
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	27.3050	1.22234	0.611169	−	0.791500i	$-0.290700\pi$
$499$	27.3050	1.22234	0.611169	+	0.791500i	$0.290700\pi$
$500$	0	0
$501$	−7.85410	−0.350895
$502$	0	0
$503$	−21.2705	−0.948405	−0.474203	−	0.880416i	$-0.657264\pi$
$503$	−21.2705	−0.948405	−0.474203	+	0.880416i	$0.657264\pi$
$504$	0	0
$505$	22.7639	1.01298
$506$	0	0
$507$	−0.965558	−0.0428819
$508$	0	0
$509$	−18.7639	−0.831697	−0.415848	−	0.909434i	$-0.636515\pi$
$509$	−18.7639	−0.831697	−0.415848	+	0.909434i	$0.636515\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	5.52786	0.244061
$514$	0	0
$515$	8.29180	0.365380
$516$	0	0
$517$	0	0
$518$	0	0
$519$	2.54915	0.111895
$520$	0	0
$521$	−1.90983	−0.0836712	−0.0418356	−	0.999125i	$-0.513321\pi$
$521$	−1.90983	−0.0836712	−0.0418356	+	0.999125i	$0.513321\pi$
$522$	0	0
$523$	−38.6525	−1.69015	−0.845077	−	0.534644i	$-0.820446\pi$
$523$	−38.6525	−1.69015	−0.845077	+	0.534644i	$0.820446\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.56231	0.416541
$528$	0	0
$529$	4.41641	0.192018
$530$	0	0
$531$	−14.9443	−0.648526
$532$	0	0
$533$	3.70820	0.160620
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3.09017	0.133351
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	5.34752	0.229484
$544$	0	0
$545$	−14.0689	−0.602645
$546$	0	0
$547$	27.1246	1.15976	0.579882	−	0.814700i	$-0.303099\pi$
$547$	27.1246	1.15976	0.579882	+	0.814700i	$0.303099\pi$
$548$	0	0
$549$	−35.8541	−1.53022
$550$	0	0
$551$	−6.83282	−0.291088
$552$	0	0
$553$	0	0
$554$	0	0
$555$	1.95743	0.0830882
$556$	0	0
$557$	15.5279	0.657937	0.328968	−	0.944341i	$-0.393299\pi$
$557$	15.5279	0.657937	0.328968	+	0.944341i	$0.393299\pi$
$558$	0	0
$559$	−15.7082	−0.664386
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.1246	0.974586	0.487293	−	0.873238i	$-0.337984\pi$
$563$	23.1246	0.974586	0.487293	+	0.873238i	$0.337984\pi$
$564$	0	0
$565$	−6.83282	−0.287459
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.09017	−0.171469	−0.0857344	−	0.996318i	$-0.527324\pi$
$569$	−4.09017	−0.171469	−0.0857344	+	0.996318i	$0.527324\pi$
$570$	0	0
$571$	−2.65248	−0.111003	−0.0555013	−	0.998459i	$-0.517676\pi$
$571$	−2.65248	−0.111003	−0.0555013	+	0.998459i	$0.517676\pi$
$572$	0	0
$573$	4.16718	0.174087
$574$	0	0
$575$	−16.1803	−0.674767
$576$	0	0
$577$	18.5836	0.773645	0.386823	−	0.922154i	$-0.373573\pi$
$577$	18.5836	0.773645	0.386823	+	0.922154i	$0.373573\pi$
$578$	0	0
$579$	−5.52786	−0.229730
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	12.7639	0.527724
$586$	0	0
$587$	−30.0000	−1.23823	−0.619116	−	0.785299i	$-0.712509\pi$
$587$	−30.0000	−1.23823	−0.619116	+	0.785299i	$0.712509\pi$
$588$	0	0
$589$	−23.6393	−0.974041
$590$	0	0
$591$	−0.875388	−0.0360087
$592$	0	0
$593$	−19.5967	−0.804742	−0.402371	−	0.915477i	$-0.631814\pi$
$593$	−19.5967	−0.804742	−0.402371	+	0.915477i	$0.631814\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.20163	0.294743
$598$	0	0
$599$	17.0344	0.696008	0.348004	−	0.937493i	$-0.386859\pi$
$599$	17.0344	0.696008	0.348004	+	0.937493i	$0.386859\pi$
$600$	0	0
$601$	32.4721	1.32457	0.662283	−	0.749254i	$-0.269588\pi$
$601$	32.4721	1.32457	0.662283	+	0.749254i	$0.269588\pi$
$602$	0	0
$603$	14.5279	0.591620
$604$	0	0
$605$	−15.2016	−0.618034
$606$	0	0
$607$	−18.1459	−0.736519	−0.368260	−	0.929723i	$-0.620046\pi$
$607$	−18.1459	−0.736519	−0.368260	+	0.929723i	$0.620046\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−35.4164	−1.43279
$612$	0	0
$613$	5.85410	0.236445	0.118222	−	0.992987i	$-0.462280\pi$
$613$	5.85410	0.236445	0.118222	+	0.992987i	$0.462280\pi$
$614$	0	0
$615$	−0.604878	−0.0243911
$616$	0	0
$617$	−3.59675	−0.144800	−0.0723998	−	0.997376i	$-0.523066\pi$
$617$	−3.59675	−0.144800	−0.0723998	+	0.997376i	$0.523066\pi$
$618$	0	0
$619$	−6.83282	−0.274634	−0.137317	−	0.990527i	$-0.543848\pi$
$619$	−6.83282	−0.274634	−0.137317	+	0.990527i	$0.543848\pi$
$620$	0	0
$621$	−11.7082	−0.469834
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.70820	0.147856
$630$	0	0
$631$	−30.9787	−1.23324	−0.616622	−	0.787260i	$-0.711499\pi$
$631$	−30.9787	−1.23324	−0.616622	+	0.787260i	$0.711499\pi$
$632$	0	0
$633$	−9.77709	−0.388604
$634$	0	0
$635$	−18.4934	−0.733889
$636$	0	0
$637$	0	0
$638$	0	0
$639$	13.5967	0.537879
$640$	0	0
$641$	−29.7771	−1.17612	−0.588062	−	0.808816i	$-0.700109\pi$
$641$	−29.7771	−1.17612	−0.588062	+	0.808816i	$0.700109\pi$
$642$	0	0
$643$	31.6180	1.24689	0.623447	−	0.781866i	$-0.285732\pi$
$643$	31.6180	1.24689	0.623447	+	0.781866i	$0.285732\pi$
$644$	0	0
$645$	2.56231	0.100891
$646$	0	0
$647$	−21.5967	−0.849056	−0.424528	−	0.905415i	$-0.639560\pi$
$647$	−21.5967	−0.849056	−0.424528	+	0.905415i	$0.639560\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20.7639	−0.812555	−0.406278	−	0.913750i	$-0.633173\pi$
$653$	−20.7639	−0.812555	−0.406278	+	0.913750i	$0.633173\pi$
$654$	0	0
$655$	−3.41641	−0.133490
$656$	0	0
$657$	8.14590	0.317802
$658$	0	0
$659$	−21.0902	−0.821556	−0.410778	−	0.911735i	$-0.634743\pi$
$659$	−21.0902	−0.821556	−0.410778	+	0.911735i	$0.634743\pi$
$660$	0	0
$661$	−31.4164	−1.22196	−0.610978	−	0.791647i	$-0.709224\pi$
$661$	−31.4164	−1.22196	−0.610978	+	0.791647i	$0.709224\pi$
$662$	0	0
$663$	−1.23607	−0.0480049
$664$	0	0
$665$	0	0
$666$	0	0
$667$	14.4721	0.560363
$668$	0	0
$669$	−7.59675	−0.293707
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−16.9443	−0.653154	−0.326577	−	0.945171i	$-0.605895\pi$
$673$	−16.9443	−0.653154	−0.326577	+	0.945171i	$0.605895\pi$
$674$	0	0
$675$	6.90983	0.265959
$676$	0	0
$677$	32.8328	1.26187	0.630934	−	0.775837i	$-0.282672\pi$
$677$	32.8328	1.26187	0.630934	+	0.775837i	$0.282672\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−5.29180	−0.202782
$682$	0	0
$683$	−10.1115	−0.386904	−0.193452	−	0.981110i	$-0.561968\pi$
$683$	−10.1115	−0.386904	−0.193452	+	0.981110i	$0.561968\pi$
$684$	0	0
$685$	−0.124612	−0.00476117
$686$	0	0
$687$	3.59675	0.137224
$688$	0	0
$689$	−39.8885	−1.51963
$690$	0	0
$691$	−35.9787	−1.36869	−0.684347	−	0.729156i	$-0.739913\pi$
$691$	−35.9787	−1.36869	−0.684347	+	0.729156i	$0.739913\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.7295	−0.596654
$696$	0	0
$697$	−1.14590	−0.0434040
$698$	0	0
$699$	7.70820	0.291551
$700$	0	0
$701$	39.3050	1.48453	0.742264	−	0.670108i	$-0.233752\pi$
$701$	39.3050	1.48453	0.742264	+	0.670108i	$0.233752\pi$
$702$	0	0
$703$	−9.16718	−0.345747
$704$	0	0
$705$	5.77709	0.217578
$706$	0	0
$707$	0	0
$708$	0	0
$709$	28.6525	1.07607	0.538033	−	0.842924i	$-0.319168\pi$
$709$	28.6525	1.07607	0.538033	+	0.842924i	$0.319168\pi$
$710$	0	0
$711$	4.87539	0.182841
$712$	0	0
$713$	50.0689	1.87509
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.25735	−0.308377
$718$	0	0
$719$	−22.0902	−0.823824	−0.411912	−	0.911224i	$-0.635139\pi$
$719$	−22.0902	−0.823824	−0.411912	+	0.911224i	$0.635139\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0.527864	0.0196315
$724$	0	0
$725$	−8.54102	−0.317206
$726$	0	0
$727$	−0.291796	−0.0108221	−0.00541106	−	0.999985i	$-0.501722\pi$
$727$	−0.291796	−0.0108221	−0.00541106	+	0.999985i	$0.501722\pi$
$728$	0	0
$729$	−19.4377	−0.719915
$730$	0	0
$731$	4.85410	0.179535
$732$	0	0
$733$	−34.2492	−1.26502	−0.632512	−	0.774551i	$-0.717976\pi$
$733$	−34.2492	−1.26502	−0.632512	+	0.774551i	$0.717976\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	9.61803	0.353805	0.176903	−	0.984228i	$-0.443392\pi$
$739$	9.61803	0.353805	0.176903	+	0.984228i	$0.443392\pi$
$740$	0	0
$741$	3.05573	0.112255
$742$	0	0
$743$	35.8885	1.31662	0.658311	−	0.752746i	$-0.271271\pi$
$743$	35.8885	1.31662	0.658311	+	0.752746i	$0.271271\pi$
$744$	0	0
$745$	−7.76393	−0.284448
$746$	0	0
$747$	4.36068	0.159549
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−2.83282	−0.103371	−0.0516855	−	0.998663i	$-0.516459\pi$
$751$	−2.83282	−0.103371	−0.0516855	+	0.998663i	$0.516459\pi$
$752$	0	0
$753$	0.832816	0.0303495
$754$	0	0
$755$	16.9574	0.617144
$756$	0	0
$757$	14.5623	0.529276	0.264638	−	0.964348i	$-0.414748\pi$
$757$	14.5623	0.529276	0.264638	+	0.964348i	$0.414748\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	53.8885	1.95346	0.976729	−	0.214477i	$-0.0688046\pi$
$761$	53.8885	1.95346	0.976729	+	0.214477i	$0.0688046\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3.94427	−0.142605
$766$	0	0
$767$	−16.9443	−0.611822
$768$	0	0
$769$	−13.5967	−0.490311	−0.245156	−	0.969484i	$-0.578839\pi$
$769$	−13.5967	−0.490311	−0.245156	+	0.969484i	$0.578839\pi$
$770$	0	0
$771$	4.00000	0.144056
$772$	0	0
$773$	−20.4721	−0.736332	−0.368166	−	0.929760i	$-0.620014\pi$
$773$	−20.4721	−0.736332	−0.368166	+	0.929760i	$0.620014\pi$
$774$	0	0
$775$	−29.5492	−1.06144
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.83282	0.101496
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−6.18034	−0.220867
$784$	0	0
$785$	14.2229	0.507638
$786$	0	0
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	0	0
$789$	−2.83282	−0.100851
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−40.6525	−1.44361
$794$	0	0
$795$	6.50658	0.230765
$796$	0	0
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	0	0
$799$	10.9443	0.387181
$800$	0	0
$801$	−37.7771	−1.33479
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.12461	−0.180395
$808$	0	0
$809$	−0.875388	−0.0307770	−0.0153885	−	0.999882i	$-0.504899\pi$
$809$	−0.875388	−0.0307770	−0.0153885	+	0.999882i	$0.504899\pi$
$810$	0	0
$811$	31.9787	1.12292	0.561462	−	0.827502i	$-0.310239\pi$
$811$	31.9787	1.12292	0.561462	+	0.827502i	$0.310239\pi$
$812$	0	0
$813$	−4.40325	−0.154429
$814$	0	0
$815$	−8.29180	−0.290449
$816$	0	0
$817$	−12.0000	−0.419827
$818$	0	0
$819$	0	0
$820$	0	0
$821$	46.4721	1.62189	0.810944	−	0.585123i	$-0.198954\pi$
$821$	46.4721	1.62189	0.810944	+	0.585123i	$0.198954\pi$
$822$	0	0
$823$	24.5410	0.855446	0.427723	−	0.903910i	$-0.359316\pi$
$823$	24.5410	0.855446	0.427723	+	0.903910i	$0.359316\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−41.2361	−1.43392	−0.716959	−	0.697115i	$-0.754467\pi$
$827$	−41.2361	−1.43392	−0.716959	+	0.697115i	$0.754467\pi$
$828$	0	0
$829$	37.5967	1.30579	0.652895	−	0.757449i	$-0.273554\pi$
$829$	37.5967	1.30579	0.652895	+	0.757449i	$0.273554\pi$
$830$	0	0
$831$	3.48529	0.120903
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−28.4164	−0.983390
$836$	0	0
$837$	−21.3820	−0.739069
$838$	0	0
$839$	−15.0557	−0.519781	−0.259891	−	0.965638i	$-0.583687\pi$
$839$	−15.0557	−0.519781	−0.259891	+	0.965638i	$0.583687\pi$
$840$	0	0
$841$	−21.3607	−0.736575
$842$	0	0
$843$	5.54102	0.190843
$844$	0	0
$845$	−3.49342	−0.120177
$846$	0	0
$847$	0	0
$848$	0	0
$849$	9.87539	0.338922
$850$	0	0
$851$	19.4164	0.665586
$852$	0	0
$853$	45.7771	1.56738	0.783689	−	0.621154i	$-0.213336\pi$
$853$	45.7771	1.56738	0.783689	+	0.621154i	$0.213336\pi$
$854$	0	0
$855$	9.75078	0.333470
$856$	0	0
$857$	−44.7984	−1.53028	−0.765142	−	0.643862i	$-0.777331\pi$
$857$	−44.7984	−1.53028	−0.765142	+	0.643862i	$0.777331\pi$
$858$	0	0
$859$	−36.8328	−1.25672	−0.628360	−	0.777923i	$-0.716273\pi$
$859$	−36.8328	−1.25672	−0.628360	+	0.777923i	$0.716273\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−26.3262	−0.896156	−0.448078	−	0.893995i	$-0.647891\pi$
$863$	−26.3262	−0.896156	−0.448078	+	0.893995i	$0.647891\pi$
$864$	0	0
$865$	9.22291	0.313588
$866$	0	0
$867$	0.381966	0.0129722
$868$	0	0
$869$	0	0
$870$	0	0
$871$	16.4721	0.558137
$872$	0	0
$873$	−11.0000	−0.372294
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−17.3050	−0.584347	−0.292173	−	0.956365i	$-0.594378\pi$
$877$	−17.3050	−0.584347	−0.292173	+	0.956365i	$0.594378\pi$
$878$	0	0
$879$	8.24922	0.278239
$880$	0	0
$881$	−21.1591	−0.712867	−0.356433	−	0.934321i	$-0.616007\pi$
$881$	−21.1591	−0.712867	−0.356433	+	0.934321i	$0.616007\pi$
$882$	0	0
$883$	−14.6869	−0.494254	−0.247127	−	0.968983i	$-0.579487\pi$
$883$	−14.6869	−0.494254	−0.247127	+	0.968983i	$0.579487\pi$
$884$	0	0
$885$	2.76393	0.0929086
$886$	0	0
$887$	−11.7426	−0.394279	−0.197140	−	0.980375i	$-0.563165\pi$
$887$	−11.7426	−0.394279	−0.197140	+	0.980375i	$0.563165\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−27.0557	−0.905385
$894$	0	0
$895$	11.1803	0.373718
$896$	0	0
$897$	−6.47214	−0.216098
$898$	0	0
$899$	26.4296	0.881475
$900$	0	0
$901$	12.3262	0.410647
$902$	0	0
$903$	0	0
$904$	0	0
$905$	19.3475	0.643133
$906$	0	0
$907$	−36.1803	−1.20135	−0.600674	−	0.799494i	$-0.705101\pi$
$907$	−36.1803	−1.20135	−0.600674	+	0.799494i	$0.705101\pi$
$908$	0	0
$909$	−47.0132	−1.55933
$910$	0	0
$911$	50.7214	1.68047	0.840237	−	0.542220i	$-0.182416\pi$
$911$	50.7214	1.68047	0.840237	+	0.542220i	$0.182416\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	6.63119	0.219220
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−23.6869	−0.781359	−0.390680	−	0.920527i	$-0.627760\pi$
$919$	−23.6869	−0.781359	−0.390680	+	0.920527i	$0.627760\pi$
$920$	0	0
$921$	6.65248	0.219207
$922$	0	0
$923$	15.4164	0.507437
$924$	0	0
$925$	−11.4590	−0.376769
$926$	0	0
$927$	−17.1246	−0.562446
$928$	0	0
$929$	−44.7984	−1.46979	−0.734893	−	0.678183i	$-0.762768\pi$
$929$	−44.7984	−1.46979	−0.734893	+	0.678183i	$0.762768\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	4.48529	0.146842
$934$	0	0
$935$	0	0
$936$	0	0
$937$	31.4164	1.02633	0.513165	−	0.858290i	$-0.328473\pi$
$937$	31.4164	1.02633	0.513165	+	0.858290i	$0.328473\pi$
$938$	0	0
$939$	9.11146	0.297341
$940$	0	0
$941$	−43.7984	−1.42779	−0.713893	−	0.700255i	$-0.753070\pi$
$941$	−43.7984	−1.42779	−0.713893	+	0.700255i	$0.753070\pi$
$942$	0	0
$943$	−6.00000	−0.195387
$944$	0	0
$945$	0	0
$946$	0	0
$947$	57.1935	1.85854	0.929269	−	0.369403i	$-0.120438\pi$
$947$	57.1935	1.85854	0.929269	+	0.369403i	$0.120438\pi$
$948$	0	0
$949$	9.23607	0.299815
$950$	0	0
$951$	−4.54102	−0.147253
$952$	0	0
$953$	−31.0902	−1.00711	−0.503555	−	0.863963i	$-0.667975\pi$
$953$	−31.0902	−1.00711	−0.503555	+	0.863963i	$0.667975\pi$
$954$	0	0
$955$	15.0770	0.487881
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	60.4377	1.94960
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−20.0000	−0.643823
$966$	0	0
$967$	−36.0902	−1.16058	−0.580291	−	0.814409i	$-0.697061\pi$
$967$	−36.0902	−1.16058	−0.580291	+	0.814409i	$0.697061\pi$
$968$	0	0
$969$	−0.944272	−0.0303344
$970$	0	0
$971$	27.2361	0.874047	0.437024	−	0.899450i	$-0.356033\pi$
$971$	27.2361	0.874047	0.437024	+	0.899450i	$0.356033\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	3.81966	0.122327
$976$	0	0
$977$	19.4508	0.622288	0.311144	−	0.950363i	$-0.399288\pi$
$977$	19.4508	0.622288	0.311144	+	0.950363i	$0.399288\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	29.0557	0.927678
$982$	0	0
$983$	−33.2705	−1.06116	−0.530582	−	0.847633i	$-0.678027\pi$
$983$	−33.2705	−1.06116	−0.530582	+	0.847633i	$0.678027\pi$
$984$	0	0
$985$	−3.16718	−0.100915
$986$	0	0
$987$	0	0
$988$	0	0
$989$	25.4164	0.808195
$990$	0	0
$991$	−58.5410	−1.85962	−0.929808	−	0.368044i	$-0.880028\pi$
$991$	−58.5410	−1.85962	−0.929808	+	0.368044i	$0.880028\pi$
$992$	0	0
$993$	−0.888544	−0.0281971
$994$	0	0
$995$	26.0557	0.826022
$996$	0	0
$997$	−12.7426	−0.403564	−0.201782	−	0.979430i	$-0.564673\pi$
$997$	−12.7426	−0.403564	−0.201782	+	0.979430i	$0.564673\pi$
$998$	0	0
$999$	−8.29180	−0.262341

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.2.a.o.1.1	yes	2
7.6	odd	2		3332.2.a.g.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3332.2.a.g.1.2	✓	2	7.6	odd	2
3332.2.a.o.1.1	yes	2	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:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3332.a (trivial)

\(f(q)\)	\(=\)	\(q+0.381966 q^{3} +1.38197 q^{5} -2.85410 q^{9} +O(q^{10})\)	\(q+0.381966 q^{3} +1.38197 q^{5} -2.85410 q^{9} -3.23607 q^{13} +0.527864 q^{15} +1.00000 q^{17} -2.47214 q^{19} +5.23607 q^{23} -3.09017 q^{25} -2.23607 q^{27} +2.76393 q^{29} +9.56231 q^{31} +3.70820 q^{37} -1.23607 q^{39} -1.14590 q^{41} +4.85410 q^{43} -3.94427 q^{45} +10.9443 q^{47} +0.381966 q^{51} +12.3262 q^{53} -0.944272 q^{57} +5.23607 q^{59} +12.5623 q^{61} -4.47214 q^{65} -5.09017 q^{67} +2.00000 q^{69} -4.76393 q^{71} -2.85410 q^{73} -1.18034 q^{75} -1.70820 q^{79} +7.70820 q^{81} -1.52786 q^{83} +1.38197 q^{85} +1.05573 q^{87} +13.2361 q^{89} +3.65248 q^{93} -3.41641 q^{95} +3.85410 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + 5 q^{5} + q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 + 5 * q^5 + q^9	\( 2 q + 3 q^{3} + 5 q^{5} + q^{9} - 2 q^{13} + 10 q^{15} + 2 q^{17} + 4 q^{19} + 6 q^{23} + 5 q^{25} + 10 q^{29} - q^{31} - 6 q^{37} + 2 q^{39} - 9 q^{41} + 3 q^{43} + 10 q^{45} + 4 q^{47} + 3 q^{51} + 9 q^{53} + 16 q^{57} + 6 q^{59} + 5 q^{61} + q^{67} + 4 q^{69} - 14 q^{71} + q^{73} + 20 q^{75} + 10 q^{79} + 2 q^{81} - 12 q^{83} + 5 q^{85} + 20 q^{87} + 22 q^{89} - 24 q^{93} + 20 q^{95} + q^{97}+O(q^{100}) \) 2 * q + 3 * q^3 + 5 * q^5 + q^9 - 2 * q^13 + 10 * q^15 + 2 * q^17 + 4 * q^19 + 6 * q^23 + 5 * q^25 + 10 * q^29 - q^31 - 6 * q^37 + 2 * q^39 - 9 * q^41 + 3 * q^43 + 10 * q^45 + 4 * q^47 + 3 * q^51 + 9 * q^53 + 16 * q^57 + 6 * q^59 + 5 * q^61 + q^67 + 4 * q^69 - 14 * q^71 + q^73 + 20 * q^75 + 10 * q^79 + 2 * q^81 - 12 * q^83 + 5 * q^85 + 20 * q^87 + 22 * q^89 - 24 * q^93 + 20 * q^95 + q^97

Introduction

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