LMFDB - Embedded newform 936.6.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [936,6,Mod(1,936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(936, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("936.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 936 = 2^{3} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	936.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:	$150.119255345$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 186x^{2} - 529x - 95 $ x^4 - x^3 - 186x^2 - 529x - 95
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 5 $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.82810$ of defining polynomial
Character	$\chi$	$=$	936.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-94.1278 q^{5} +33.4615 q^{7} -109.791 q^{11} -169.000 q^{13} -1155.28 q^{17} +1312.40 q^{19} -52.2234 q^{23} +5735.03 q^{25} +7094.51 q^{29} +8210.74 q^{31} -3149.65 q^{35} -12812.2 q^{37} -2811.26 q^{41} +3494.85 q^{43} +4232.26 q^{47} -15687.3 q^{49} -13517.8 q^{53} +10334.4 q^{55} +42651.7 q^{59} +35971.5 q^{61} +15907.6 q^{65} +35153.7 q^{67} -8006.18 q^{71} -11656.0 q^{73} -3673.78 q^{77} +28206.0 q^{79} -3127.94 q^{83} +108744. q^{85} -92556.1 q^{89} -5654.99 q^{91} -123533. q^{95} +6867.86 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5} - 176 q^{7} - 96 q^{11} - 676 q^{13} - 1224 q^{17} + 3856 q^{19} + 368 q^{23} + 5884 q^{25} - 264 q^{29} + 10928 q^{31} - 7984 q^{35} + 12008 q^{37} - 32792 q^{41} + 24288 q^{43} - 14176 q^{47}+ \cdots + 34664 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 - 176 * q^7 - 96 * q^11 - 676 * q^13 - 1224 * q^17 + 3856 * q^19 + 368 * q^23 + 5884 * q^25 - 264 * q^29 + 10928 * q^31 - 7984 * q^35 + 12008 * q^37 - 32792 * q^41 + 24288 * q^43 - 14176 * q^47 + 40948 * q^49 - 37384 * q^53 - 34384 * q^55 - 11456 * q^59 - 15752 * q^61 - 1352 * q^65 - 14032 * q^67 - 7552 * q^71 + 76104 * q^73 - 3776 * q^77 + 61056 * q^79 - 233856 * q^83 + 116816 * q^85 - 86136 * q^89 + 29744 * q^91 - 268368 * q^95 + 34664 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	−94.1278	−1.68381	−0.841904	−	0.539627i	$-0.818565\pi$
$5$	−94.1278	−1.68381	−0.841904	+	0.539627i	$0.818565\pi$
$6$	0	0
$7$	33.4615	0.258107	0.129054	−	0.991638i	$-0.458806\pi$
$7$	33.4615	0.258107	0.129054	+	0.991638i	$0.458806\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−109.791	−0.273581	−0.136791	−	0.990600i	$-0.543679\pi$
$11$	−109.791	−0.273581	−0.136791	+	0.990600i	$0.543679\pi$
$12$	0	0
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1155.28	−0.969538	−0.484769	−	0.874642i	$-0.661096\pi$
$17$	−1155.28	−0.969538	−0.484769	+	0.874642i	$0.661096\pi$
$18$	0	0
$19$	1312.40	0.834028	0.417014	−	0.908900i	$-0.363077\pi$
$19$	1312.40	0.834028	0.417014	+	0.908900i	$0.363077\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−52.2234	−0.0205847	−0.0102924	−	0.999947i	$-0.503276\pi$
$23$	−52.2234	−0.0205847	−0.0102924	+	0.999947i	$0.503276\pi$
$24$	0	0
$25$	5735.03	1.83521
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7094.51	1.56649	0.783245	−	0.621713i	$-0.213563\pi$
$29$	7094.51	1.56649	0.783245	+	0.621713i	$0.213563\pi$
$30$	0	0
$31$	8210.74	1.53454	0.767269	−	0.641325i	$-0.221615\pi$
$31$	8210.74	1.53454	0.767269	+	0.641325i	$0.221615\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3149.65	−0.434603
$36$	0	0
$37$	−12812.2	−1.53858	−0.769289	−	0.638901i	$-0.779389\pi$
$37$	−12812.2	−1.53858	−0.769289	+	0.638901i	$0.779389\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2811.26	−0.261180	−0.130590	−	0.991436i	$-0.541687\pi$
$41$	−2811.26	−0.261180	−0.130590	+	0.991436i	$0.541687\pi$
$42$	0	0
$43$	3494.85	0.288242	0.144121	−	0.989560i	$-0.453964\pi$
$43$	3494.85	0.288242	0.144121	+	0.989560i	$0.453964\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4232.26	0.279465	0.139733	−	0.990189i	$-0.455376\pi$
$47$	4232.26	0.279465	0.139733	+	0.990189i	$0.455376\pi$
$48$	0	0
$49$	−15687.3	−0.933381
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−13517.8	−0.661025	−0.330512	−	0.943802i	$-0.607222\pi$
$53$	−13517.8	−0.661025	−0.330512	+	0.943802i	$0.607222\pi$
$54$	0	0
$55$	10334.4	0.460659
$56$	0	0
$57$	0	0
$58$	0	0
$59$	42651.7	1.59517	0.797584	−	0.603208i	$-0.206111\pi$
$59$	42651.7	1.59517	0.797584	+	0.603208i	$0.206111\pi$
$60$	0	0
$61$	35971.5	1.23775	0.618876	−	0.785488i	$-0.287588\pi$
$61$	35971.5	1.23775	0.618876	+	0.785488i	$0.287588\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	15907.6	0.467004
$66$	0	0
$67$	35153.7	0.956719	0.478359	−	0.878164i	$-0.341232\pi$
$67$	35153.7	0.956719	0.478359	+	0.878164i	$0.341232\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8006.18	−0.188486	−0.0942431	−	0.995549i	$-0.530043\pi$
$71$	−8006.18	−0.188486	−0.0942431	+	0.995549i	$0.530043\pi$
$72$	0	0
$73$	−11656.0	−0.256001	−0.128000	−	0.991774i	$-0.540856\pi$
$73$	−11656.0	−0.256001	−0.128000	+	0.991774i	$0.540856\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3673.78	−0.0706133
$78$	0	0
$79$	28206.0	0.508480	0.254240	−	0.967141i	$-0.418175\pi$
$79$	28206.0	0.508480	0.254240	+	0.967141i	$0.418175\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3127.94	−0.0498383	−0.0249192	−	0.999689i	$-0.507933\pi$
$83$	−3127.94	−0.0498383	−0.0249192	+	0.999689i	$0.507933\pi$
$84$	0	0
$85$	108744.	1.63252
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−92556.1	−1.23860	−0.619299	−	0.785156i	$-0.712583\pi$
$89$	−92556.1	−1.23860	−0.619299	+	0.785156i	$0.712583\pi$
$90$	0	0
$91$	−5654.99	−0.0715861
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−123533.	−1.40434
$96$	0	0
$97$	6867.86	0.0741126	0.0370563	−	0.999313i	$-0.488202\pi$
$97$	6867.86	0.0741126	0.0370563	+	0.999313i	$0.488202\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−39694.2	−0.387189	−0.193595	−	0.981082i	$-0.562015\pi$
$101$	−39694.2	−0.387189	−0.193595	+	0.981082i	$0.562015\pi$
$102$	0	0
$103$	60349.8	0.560509	0.280255	−	0.959926i	$-0.409581\pi$
$103$	60349.8	0.560509	0.280255	+	0.959926i	$0.409581\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3692.75	−0.0311810	−0.0155905	−	0.999878i	$-0.504963\pi$
$107$	−3692.75	−0.0311810	−0.0155905	+	0.999878i	$0.504963\pi$
$108$	0	0
$109$	−185664.	−1.49679	−0.748395	−	0.663254i	$-0.769175\pi$
$109$	−185664.	−1.49679	−0.748395	+	0.663254i	$0.769175\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	62430.5	0.459939	0.229970	−	0.973198i	$-0.426137\pi$
$113$	62430.5	0.459939	0.229970	+	0.973198i	$0.426137\pi$
$114$	0	0
$115$	4915.67	0.0346608
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−38657.4	−0.250245
$120$	0	0
$121$	−148997.	−0.925153
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−245677.	−1.40634
$126$	0	0
$127$	283633.	1.56044	0.780220	−	0.625505i	$-0.215107\pi$
$127$	283633.	1.56044	0.780220	+	0.625505i	$0.215107\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−65395.1	−0.332941	−0.166470	−	0.986046i	$-0.553237\pi$
$131$	−65395.1	−0.332941	−0.166470	+	0.986046i	$0.553237\pi$
$132$	0	0
$133$	43914.7	0.215269
$134$	0	0
$135$	0	0
$136$	0	0
$137$	69085.3	0.314474	0.157237	−	0.987561i	$-0.449741\pi$
$137$	69085.3	0.314474	0.157237	+	0.987561i	$0.449741\pi$
$138$	0	0
$139$	207136.	0.909322	0.454661	−	0.890665i	$-0.349760\pi$
$139$	207136.	0.909322	0.454661	+	0.890665i	$0.349760\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	18554.7	0.0758778
$144$	0	0
$145$	−667791.	−2.63767
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−410160.	−1.51352	−0.756759	−	0.653694i	$-0.773219\pi$
$149$	−410160.	−1.51352	−0.756759	+	0.653694i	$0.773219\pi$
$150$	0	0
$151$	−312600.	−1.11570	−0.557849	−	0.829943i	$-0.688373\pi$
$151$	−312600.	−1.11570	−0.557849	+	0.829943i	$0.688373\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−772858.	−2.58387
$156$	0	0
$157$	−462608.	−1.49784	−0.748918	−	0.662662i	$-0.769426\pi$
$157$	−462608.	−1.49784	−0.748918	+	0.662662i	$0.769426\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1747.47	−0.00531307
$162$	0	0
$163$	505955.	1.49157	0.745783	−	0.666188i	$-0.232075\pi$
$163$	505955.	1.49157	0.745783	+	0.666188i	$0.232075\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−72857.3	−0.202154	−0.101077	−	0.994879i	$-0.532229\pi$
$167$	−72857.3	−0.202154	−0.101077	+	0.994879i	$0.532229\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−15026.2	−0.0381711	−0.0190856	−	0.999818i	$-0.506075\pi$
$173$	−15026.2	−0.0381711	−0.0190856	+	0.999818i	$0.506075\pi$
$174$	0	0
$175$	191903.	0.473681
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−196586.	−0.458586	−0.229293	−	0.973357i	$-0.573641\pi$
$179$	−196586.	−0.458586	−0.229293	+	0.973357i	$0.573641\pi$
$180$	0	0
$181$	−268073.	−0.608215	−0.304107	−	0.952638i	$-0.598358\pi$
$181$	−268073.	−0.608215	−0.304107	+	0.952638i	$0.598358\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.20598e6	2.59067
$186$	0	0
$187$	126840.	0.265248
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−702006.	−1.39238	−0.696189	−	0.717858i	$-0.745123\pi$
$191$	−702006.	−1.39238	−0.696189	+	0.717858i	$0.745123\pi$
$192$	0	0
$193$	−614642.	−1.18776	−0.593881	−	0.804553i	$-0.702405\pi$
$193$	−614642.	−1.18776	−0.593881	+	0.804553i	$0.702405\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	725714.	1.33229	0.666147	−	0.745821i	$-0.267942\pi$
$197$	725714.	1.33229	0.666147	+	0.745821i	$0.267942\pi$
$198$	0	0
$199$	−623315.	−1.11577	−0.557885	−	0.829918i	$-0.688387\pi$
$199$	−623315.	−1.11577	−0.557885	+	0.829918i	$0.688387\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	237393.	0.404322
$204$	0	0
$205$	264617.	0.439778
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−144090.	−0.228174
$210$	0	0
$211$	−1.17911e6	−1.82326	−0.911631	−	0.411010i	$-0.865176\pi$
$211$	−1.17911e6	−1.82326	−0.911631	+	0.411010i	$0.865176\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−328963.	−0.485345
$216$	0	0
$217$	274743.	0.396076
$218$	0	0
$219$	0	0
$220$	0	0
$221$	195242.	0.268902
$222$	0	0
$223$	−1.33293e6	−1.79492	−0.897461	−	0.441094i	$-0.854591\pi$
$223$	−1.33293e6	−1.79492	−0.897461	+	0.441094i	$0.854591\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−28437.9	−0.0366296	−0.0183148	−	0.999832i	$-0.505830\pi$
$227$	−28437.9	−0.0366296	−0.0183148	+	0.999832i	$0.505830\pi$
$228$	0	0
$229$	−140585.	−0.177154	−0.0885772	−	0.996069i	$-0.528232\pi$
$229$	−140585.	−0.177154	−0.0885772	+	0.996069i	$0.528232\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.18087e6	1.42499	0.712497	−	0.701675i	$-0.247564\pi$
$233$	1.18087e6	1.42499	0.712497	+	0.701675i	$0.247564\pi$
$234$	0	0
$235$	−398373.	−0.470566
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.40333e6	1.58915	0.794577	−	0.607164i	$-0.207693\pi$
$239$	1.40333e6	1.58915	0.794577	+	0.607164i	$0.207693\pi$
$240$	0	0
$241$	−1.55388e6	−1.72335	−0.861676	−	0.507459i	$-0.830585\pi$
$241$	−1.55388e6	−1.72335	−0.861676	+	0.507459i	$0.830585\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.47661e6	1.57163
$246$	0	0
$247$	−221795.	−0.231318
$248$	0	0
$249$	0	0
$250$	0	0
$251$	679718.	0.680996	0.340498	−	0.940245i	$-0.389404\pi$
$251$	679718.	0.680996	0.340498	+	0.940245i	$0.389404\pi$
$252$	0	0
$253$	5733.68	0.00563160
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.16742e6	−1.10254	−0.551272	−	0.834326i	$-0.685857\pi$
$257$	−1.16742e6	−1.10254	−0.551272	+	0.834326i	$0.685857\pi$
$258$	0	0
$259$	−428715.	−0.397118
$260$	0	0
$261$	0	0
$262$	0	0
$263$	274301.	0.244533	0.122267	−	0.992497i	$-0.460984\pi$
$263$	274301.	0.244533	0.122267	+	0.992497i	$0.460984\pi$
$264$	0	0
$265$	1.27240e6	1.11304
$266$	0	0
$267$	0	0
$268$	0	0
$269$	481307.	0.405547	0.202773	−	0.979226i	$-0.435005\pi$
$269$	481307.	0.405547	0.202773	+	0.979226i	$0.435005\pi$
$270$	0	0
$271$	−458496.	−0.379238	−0.189619	−	0.981858i	$-0.560725\pi$
$271$	−458496.	−0.379238	−0.189619	+	0.981858i	$0.560725\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−629657.	−0.502080
$276$	0	0
$277$	463861.	0.363236	0.181618	−	0.983369i	$-0.441867\pi$
$277$	463861.	0.363236	0.181618	+	0.983369i	$0.441867\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−287065.	−0.216878	−0.108439	−	0.994103i	$-0.534585\pi$
$281$	−287065.	−0.216878	−0.108439	+	0.994103i	$0.534585\pi$
$282$	0	0
$283$	1.52222e6	1.12983	0.564914	−	0.825150i	$-0.308909\pi$
$283$	1.52222e6	1.12983	0.564914	+	0.825150i	$0.308909\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−94068.8	−0.0674126
$288$	0	0
$289$	−85185.2	−0.0599956
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−451386.	−0.307170	−0.153585	−	0.988135i	$-0.549082\pi$
$293$	−451386.	−0.307170	−0.153585	+	0.988135i	$0.549082\pi$
$294$	0	0
$295$	−4.01471e6	−2.68596
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8825.75	0.00570918
$300$	0	0
$301$	116943.	0.0743974
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.38592e6	−2.08414
$306$	0	0
$307$	−101365.	−0.0613821	−0.0306911	−	0.999529i	$-0.509771\pi$
$307$	−101365.	−0.0613821	−0.0306911	+	0.999529i	$0.509771\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3.20785e6	−1.88067	−0.940337	−	0.340243i	$-0.889491\pi$
$311$	−3.20785e6	−1.88067	−0.940337	+	0.340243i	$0.889491\pi$
$312$	0	0
$313$	379116.	0.218732	0.109366	−	0.994002i	$-0.465118\pi$
$313$	379116.	0.218732	0.109366	+	0.994002i	$0.465118\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.46265e6	1.93536	0.967678	−	0.252189i	$-0.0811504\pi$
$317$	3.46265e6	1.93536	0.967678	+	0.252189i	$0.0811504\pi$
$318$	0	0
$319$	−778916.	−0.428562
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.51618e6	−0.808622
$324$	0	0
$325$	−969221.	−0.508996
$326$	0	0
$327$	0	0
$328$	0	0
$329$	141618.	0.0721320
$330$	0	0
$331$	2.04899e6	1.02794	0.513972	−	0.857807i	$-0.328173\pi$
$331$	2.04899e6	1.02794	0.513972	+	0.857807i	$0.328173\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.30894e6	−1.61093
$336$	0	0
$337$	520864.	0.249833	0.124917	−	0.992167i	$-0.460134\pi$
$337$	520864.	0.249833	0.124917	+	0.992167i	$0.460134\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−901468.	−0.419821
$342$	0	0
$343$	−1.08731e6	−0.499019
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.56918e6	−1.14544	−0.572718	−	0.819752i	$-0.694111\pi$
$347$	−2.56918e6	−1.14544	−0.572718	+	0.819752i	$0.694111\pi$
$348$	0	0
$349$	−3.42386e6	−1.50471	−0.752354	−	0.658759i	$-0.771082\pi$
$349$	−3.42386e6	−1.50471	−0.752354	+	0.658759i	$0.771082\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	739548.	0.315886	0.157943	−	0.987448i	$-0.449514\pi$
$353$	739548.	0.315886	0.157943	+	0.987448i	$0.449514\pi$
$354$	0	0
$355$	753604.	0.317375
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−586748.	−0.240279	−0.120139	−	0.992757i	$-0.538334\pi$
$359$	−586748.	−0.240279	−0.120139	+	0.992757i	$0.538334\pi$
$360$	0	0
$361$	−753718.	−0.304397
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.09715e6	0.431056
$366$	0	0
$367$	−4.85860e6	−1.88298	−0.941492	−	0.337037i	$-0.890575\pi$
$367$	−4.85860e6	−1.88298	−0.941492	+	0.337037i	$0.890575\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−452327.	−0.170615
$372$	0	0
$373$	−229665.	−0.0854717	−0.0427359	−	0.999086i	$-0.513607\pi$
$373$	−229665.	−0.0854717	−0.0427359	+	0.999086i	$0.513607\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.19897e6	−0.434466
$378$	0	0
$379$	−1.69343e6	−0.605576	−0.302788	−	0.953058i	$-0.597917\pi$
$379$	−1.69343e6	−0.605576	−0.302788	+	0.953058i	$0.597917\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−447870.	−0.156011	−0.0780055	−	0.996953i	$-0.524855\pi$
$383$	−447870.	−0.156011	−0.0780055	+	0.996953i	$0.524855\pi$
$384$	0	0
$385$	345805.	0.118899
$386$	0	0
$387$	0	0
$388$	0	0
$389$	965484.	0.323498	0.161749	−	0.986832i	$-0.448287\pi$
$389$	965484.	0.323498	0.161749	+	0.986832i	$0.448287\pi$
$390$	0	0
$391$	60332.6	0.0199577
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.65497e6	−0.856183
$396$	0	0
$397$	−1.59726e6	−0.508627	−0.254313	−	0.967122i	$-0.581849\pi$
$397$	−1.59726e6	−0.508627	−0.254313	+	0.967122i	$0.581849\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.67502e6	−0.830741	−0.415371	−	0.909652i	$-0.636348\pi$
$401$	−2.67502e6	−0.830741	−0.415371	+	0.909652i	$0.636348\pi$
$402$	0	0
$403$	−1.38761e6	−0.425605
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.40667e6	0.420926
$408$	0	0
$409$	3.87780e6	1.14625	0.573123	−	0.819470i	$-0.305732\pi$
$409$	3.87780e6	1.14625	0.573123	+	0.819470i	$0.305732\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.42719e6	0.411724
$414$	0	0
$415$	294426.	0.0839182
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.05001e6	0.570453	0.285227	−	0.958460i	$-0.407931\pi$
$419$	2.05001e6	0.570453	0.285227	+	0.958460i	$0.407931\pi$
$420$	0	0
$421$	−5.60798e6	−1.54206	−0.771030	−	0.636799i	$-0.780258\pi$
$421$	−5.60798e6	−1.54206	−0.771030	+	0.636799i	$0.780258\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.62557e6	−1.77931
$426$	0	0
$427$	1.20366e6	0.319473
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.44139e6	−1.41097	−0.705483	−	0.708727i	$-0.749270\pi$
$431$	−5.44139e6	−1.41097	−0.705483	+	0.708727i	$0.749270\pi$
$432$	0	0
$433$	4.82997e6	1.23801	0.619005	−	0.785387i	$-0.287536\pi$
$433$	4.82997e6	1.23801	0.619005	+	0.785387i	$0.287536\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−68537.7	−0.0171683
$438$	0	0
$439$	−1.69685e6	−0.420224	−0.210112	−	0.977677i	$-0.567383\pi$
$439$	−1.69685e6	−0.420224	−0.210112	+	0.977677i	$0.567383\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.65667e6	0.885272	0.442636	−	0.896701i	$-0.354043\pi$
$443$	3.65667e6	0.885272	0.442636	+	0.896701i	$0.354043\pi$
$444$	0	0
$445$	8.71210e6	2.08556
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.20796e6	−1.21913	−0.609567	−	0.792734i	$-0.708657\pi$
$449$	−5.20796e6	−1.21913	−0.609567	+	0.792734i	$0.708657\pi$
$450$	0	0
$451$	308652.	0.0714541
$452$	0	0
$453$	0	0
$454$	0	0
$455$	532292.	0.120537
$456$	0	0
$457$	6.72690e6	1.50669	0.753346	−	0.657624i	$-0.228438\pi$
$457$	6.72690e6	1.50669	0.753346	+	0.657624i	$0.228438\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.54845e6	−0.777653	−0.388827	−	0.921311i	$-0.627120\pi$
$461$	−3.54845e6	−0.777653	−0.388827	+	0.921311i	$0.627120\pi$
$462$	0	0
$463$	4.97769e6	1.07913	0.539567	−	0.841943i	$-0.318588\pi$
$463$	4.97769e6	1.07913	0.539567	+	0.841943i	$0.318588\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.32958e6	0.706475	0.353238	−	0.935534i	$-0.385081\pi$
$467$	3.32958e6	0.706475	0.353238	+	0.935534i	$0.385081\pi$
$468$	0	0
$469$	1.17630e6	0.246936
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−383705.	−0.0788577
$474$	0	0
$475$	7.52663e6	1.53062
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8.05780e6	−1.60464	−0.802320	−	0.596894i	$-0.796401\pi$
$479$	−8.05780e6	−1.60464	−0.802320	+	0.596894i	$0.796401\pi$
$480$	0	0
$481$	2.16526e6	0.426725
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−646456.	−0.124791
$486$	0	0
$487$	−9.96655e6	−1.90424	−0.952122	−	0.305719i	$-0.901103\pi$
$487$	−9.96655e6	−1.90424	−0.952122	+	0.305719i	$0.901103\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8.50234e6	1.59160	0.795801	−	0.605558i	$-0.207050\pi$
$491$	8.50234e6	1.59160	0.795801	+	0.605558i	$0.207050\pi$
$492$	0	0
$493$	−8.19615e6	−1.51877
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−267899.	−0.0486497
$498$	0	0
$499$	2.30152e6	0.413774	0.206887	−	0.978365i	$-0.433667\pi$
$499$	2.30152e6	0.413774	0.206887	+	0.978365i	$0.433667\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.15778e6	1.43765	0.718823	−	0.695193i	$-0.244681\pi$
$503$	8.15778e6	1.43765	0.718823	+	0.695193i	$0.244681\pi$
$504$	0	0
$505$	3.73632e6	0.651952
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.01951e6	1.02983	0.514917	−	0.857240i	$-0.327823\pi$
$509$	6.01951e6	1.02983	0.514917	+	0.857240i	$0.327823\pi$
$510$	0	0
$511$	−390026.	−0.0660756
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5.68059e6	−0.943790
$516$	0	0
$517$	−464666.	−0.0764565
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.05732e7	1.70653	0.853263	−	0.521480i	$-0.174620\pi$
$521$	1.05732e7	1.70653	0.853263	+	0.521480i	$0.174620\pi$
$522$	0	0
$523$	−98658.5	−0.0157718	−0.00788588	−	0.999969i	$-0.502510\pi$
$523$	−98658.5	−0.0157718	−0.00788588	+	0.999969i	$0.502510\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.48570e6	−1.48779
$528$	0	0
$529$	−6.43362e6	−0.999576
$530$	0	0
$531$	0	0
$532$	0	0
$533$	475102.	0.0724384
$534$	0	0
$535$	347590.	0.0525029
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.72233e6	0.255356
$540$	0	0
$541$	6.90663e6	1.01455	0.507275	−	0.861785i	$-0.330653\pi$
$541$	6.90663e6	1.01455	0.507275	+	0.861785i	$0.330653\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.74761e7	2.52031
$546$	0	0
$547$	−1.01800e7	−1.45471	−0.727357	−	0.686259i	$-0.759252\pi$
$547$	−1.01800e7	−1.45471	−0.727357	+	0.686259i	$0.759252\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9.31081e6	1.30650
$552$	0	0
$553$	943815.	0.131242
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.00546e6	−0.410463	−0.205231	−	0.978714i	$-0.565795\pi$
$557$	−3.00546e6	−0.410463	−0.205231	+	0.978714i	$0.565795\pi$
$558$	0	0
$559$	−590630.	−0.0799441
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.12827e7	1.50018	0.750089	−	0.661337i	$-0.230011\pi$
$563$	1.12827e7	1.50018	0.750089	+	0.661337i	$0.230011\pi$
$564$	0	0
$565$	−5.87644e6	−0.774450
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.52382e6	1.10371	0.551853	−	0.833941i	$-0.313921\pi$
$569$	8.52382e6	1.10371	0.551853	+	0.833941i	$0.313921\pi$
$570$	0	0
$571$	9.54675e6	1.22536	0.612682	−	0.790329i	$-0.290091\pi$
$571$	9.54675e6	1.22536	0.612682	+	0.790329i	$0.290091\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−299503.	−0.0377774
$576$	0	0
$577$	7.22142e6	0.902991	0.451495	−	0.892273i	$-0.350891\pi$
$577$	7.22142e6	0.902991	0.451495	+	0.892273i	$0.350891\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−104666.	−0.0128636
$582$	0	0
$583$	1.48414e6	0.180844
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.37884e6	0.764094	0.382047	−	0.924143i	$-0.375219\pi$
$587$	6.37884e6	0.764094	0.382047	+	0.924143i	$0.375219\pi$
$588$	0	0
$589$	1.07757e7	1.27985
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4.67591e6	−0.546046	−0.273023	−	0.962008i	$-0.588023\pi$
$593$	−4.67591e6	−0.546046	−0.273023	+	0.962008i	$0.588023\pi$
$594$	0	0
$595$	3.63873e6	0.421364
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5.78485e6	−0.658757	−0.329378	−	0.944198i	$-0.606839\pi$
$599$	−5.78485e6	−0.658757	−0.329378	+	0.944198i	$0.606839\pi$
$600$	0	0
$601$	603630.	0.0681687	0.0340843	−	0.999419i	$-0.489149\pi$
$601$	603630.	0.0681687	0.0340843	+	0.999419i	$0.489149\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.40247e7	1.55778
$606$	0	0
$607$	1.17363e6	0.129288	0.0646439	−	0.997908i	$-0.479409\pi$
$607$	1.17363e6	0.129288	0.0646439	+	0.997908i	$0.479409\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−715252.	−0.0775097
$612$	0	0
$613$	−6.02374e6	−0.647463	−0.323732	−	0.946149i	$-0.604938\pi$
$613$	−6.02374e6	−0.647463	−0.323732	+	0.946149i	$0.604938\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	371323.	0.0392680	0.0196340	−	0.999807i	$-0.493750\pi$
$617$	371323.	0.0392680	0.0196340	+	0.999807i	$0.493750\pi$
$618$	0	0
$619$	−1.54077e6	−0.161626	−0.0808131	−	0.996729i	$-0.525752\pi$
$619$	−1.54077e6	−0.161626	−0.0808131	+	0.996729i	$0.525752\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.09706e6	−0.319691
$624$	0	0
$625$	5.20301e6	0.532789
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.48017e7	1.49171
$630$	0	0
$631$	−1.06031e7	−1.06013	−0.530066	−	0.847957i	$-0.677833\pi$
$631$	−1.06031e7	−1.06013	−0.530066	+	0.847957i	$0.677833\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.66977e7	−2.62748
$636$	0	0
$637$	2.65116e6	0.258873
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7.63599e6	−0.734041	−0.367021	−	0.930213i	$-0.619622\pi$
$641$	−7.63599e6	−0.734041	−0.367021	+	0.930213i	$0.619622\pi$
$642$	0	0
$643$	−1.34581e7	−1.28368	−0.641838	−	0.766840i	$-0.721828\pi$
$643$	−1.34581e7	−1.28368	−0.641838	+	0.766840i	$0.721828\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.65614e7	1.55538	0.777690	−	0.628648i	$-0.216391\pi$
$647$	1.65614e7	1.55538	0.777690	+	0.628648i	$0.216391\pi$
$648$	0	0
$649$	−4.68279e6	−0.436408
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.51489e6	−0.230800	−0.115400	−	0.993319i	$-0.536815\pi$
$653$	−2.51489e6	−0.230800	−0.115400	+	0.993319i	$0.536815\pi$
$654$	0	0
$655$	6.15549e6	0.560609
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.38636e6	−0.124355	−0.0621774	−	0.998065i	$-0.519804\pi$
$659$	−1.38636e6	−0.124355	−0.0621774	+	0.998065i	$0.519804\pi$
$660$	0	0
$661$	1.46224e6	0.130171	0.0650857	−	0.997880i	$-0.479268\pi$
$661$	1.46224e6	0.130171	0.0650857	+	0.997880i	$0.479268\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.13359e6	−0.362471
$666$	0	0
$667$	−370500.	−0.0322458
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.94936e6	−0.338626
$672$	0	0
$673$	−5.05023e6	−0.429807	−0.214903	−	0.976635i	$-0.568944\pi$
$673$	−5.05023e6	−0.429807	−0.214903	+	0.976635i	$0.568944\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−92645.8	−0.00776880	−0.00388440	−	0.999992i	$-0.501236\pi$
$677$	−92645.8	−0.00776880	−0.00388440	+	0.999992i	$0.501236\pi$
$678$	0	0
$679$	229809.	0.0191290
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−3.72081e6	−0.305201	−0.152600	−	0.988288i	$-0.548765\pi$
$683$	−3.72081e6	−0.305201	−0.152600	+	0.988288i	$0.548765\pi$
$684$	0	0
$685$	−6.50284e6	−0.529513
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.28452e6	0.183335
$690$	0	0
$691$	7.18845e6	0.572717	0.286358	−	0.958123i	$-0.407555\pi$
$691$	7.18845e6	0.572717	0.286358	+	0.958123i	$0.407555\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.94972e7	−1.53112
$696$	0	0
$697$	3.24779e6	0.253224
$698$	0	0
$699$	0	0
$700$	0	0
$701$	15297.9	0.00117581	0.000587905	−	1.00000i	$-0.499813\pi$
$701$	15297.9	0.00117581	0.000587905		1.00000i	$0.499813\pi$
$702$	0	0
$703$	−1.68147e7	−1.28322
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.32823e6	−0.0999363
$708$	0	0
$709$	1.39863e6	0.104493	0.0522466	−	0.998634i	$-0.483362\pi$
$709$	1.39863e6	0.104493	0.0522466	+	0.998634i	$0.483362\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−428793.	−0.0315881
$714$	0	0
$715$	−1.74652e6	−0.127764
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.35207e7	0.975384	0.487692	−	0.873016i	$-0.337839\pi$
$719$	1.35207e7	0.975384	0.487692	+	0.873016i	$0.337839\pi$
$720$	0	0
$721$	2.01939e6	0.144671
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.06873e7	2.87484
$726$	0	0
$727$	−1.03561e7	−0.726708	−0.363354	−	0.931651i	$-0.618368\pi$
$727$	−1.03561e7	−0.726708	−0.363354	+	0.931651i	$0.618368\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.03753e6	−0.279462
$732$	0	0
$733$	2.72409e7	1.87267	0.936335	−	0.351107i	$-0.114195\pi$
$733$	2.72409e7	1.87267	0.936335	+	0.351107i	$0.114195\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.85958e6	−0.261740
$738$	0	0
$739$	6.18785e6	0.416801	0.208401	−	0.978044i	$-0.433174\pi$
$739$	6.18785e6	0.416801	0.208401	+	0.978044i	$0.433174\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.57724e7	−1.71270	−0.856352	−	0.516392i	$-0.827275\pi$
$743$	−2.57724e7	−1.71270	−0.856352	+	0.516392i	$0.827275\pi$
$744$	0	0
$745$	3.86075e7	2.54848
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−123565.	−0.00804804
$750$	0	0
$751$	4.06370e6	0.262919	0.131459	−	0.991322i	$-0.458034\pi$
$751$	4.06370e6	0.262919	0.131459	+	0.991322i	$0.458034\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.94243e7	1.87862
$756$	0	0
$757$	−3.90503e6	−0.247677	−0.123838	−	0.992302i	$-0.539520\pi$
$757$	−3.90503e6	−0.247677	−0.123838	+	0.992302i	$0.539520\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.93874e7	−1.21355	−0.606774	−	0.794874i	$-0.707537\pi$
$761$	−1.93874e7	−1.21355	−0.606774	+	0.794874i	$0.707537\pi$
$762$	0	0
$763$	−6.21258e6	−0.386332
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7.20814e6	−0.442420
$768$	0	0
$769$	−1.75737e6	−0.107163	−0.0535817	−	0.998563i	$-0.517064\pi$
$769$	−1.75737e6	−0.107163	−0.0535817	+	0.998563i	$0.517064\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.62448e6	−0.398752	−0.199376	−	0.979923i	$-0.563892\pi$
$773$	−6.62448e6	−0.398752	−0.199376	+	0.979923i	$0.563892\pi$
$774$	0	0
$775$	4.70889e7	2.81620
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.68948e6	−0.217832
$780$	0	0
$781$	879010.	0.0515663
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.35443e7	2.52207
$786$	0	0
$787$	−7.44210e6	−0.428310	−0.214155	−	0.976800i	$-0.568700\pi$
$787$	−7.44210e6	−0.428310	−0.214155	+	0.976800i	$0.568700\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.08902e6	0.118714
$792$	0	0
$793$	−6.07918e6	−0.343291
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.09663e7	1.16917	0.584583	−	0.811334i	$-0.301258\pi$
$797$	2.09663e7	1.16917	0.584583	+	0.811334i	$0.301258\pi$
$798$	0	0
$799$	−4.88945e6	−0.270952
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.27972e6	0.0700370
$804$	0	0
$805$	164486.	0.00894620
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8.39942e6	−0.451209	−0.225605	−	0.974219i	$-0.572436\pi$
$809$	−8.39942e6	−0.451209	−0.225605	+	0.974219i	$0.572436\pi$
$810$	0	0
$811$	1.60701e7	0.857961	0.428981	−	0.903314i	$-0.358873\pi$
$811$	1.60701e7	0.857961	0.428981	+	0.903314i	$0.358873\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.76244e7	−2.51151
$816$	0	0
$817$	4.58663e6	0.240402
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.01528e6	0.259679	0.129840	−	0.991535i	$-0.458554\pi$
$821$	5.01528e6	0.259679	0.129840	+	0.991535i	$0.458554\pi$
$822$	0	0
$823$	3.15405e7	1.62319	0.811594	−	0.584221i	$-0.198600\pi$
$823$	3.15405e7	1.62319	0.811594	+	0.584221i	$0.198600\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.14143e7	−1.59721	−0.798607	−	0.601853i	$-0.794429\pi$
$827$	−3.14143e7	−1.59721	−0.798607	+	0.601853i	$0.794429\pi$
$828$	0	0
$829$	−1.83920e7	−0.929488	−0.464744	−	0.885445i	$-0.653854\pi$
$829$	−1.83920e7	−0.929488	−0.464744	+	0.885445i	$0.653854\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.81233e7	0.904948
$834$	0	0
$835$	6.85789e6	0.340388
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.43095e7	−0.701810	−0.350905	−	0.936411i	$-0.614126\pi$
$839$	−1.43095e7	−0.701810	−0.350905	+	0.936411i	$0.614126\pi$
$840$	0	0
$841$	2.98210e7	1.45389
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.68838e6	−0.129524
$846$	0	0
$847$	−4.98566e6	−0.238789
$848$	0	0
$849$	0	0
$850$	0	0
$851$	669096.	0.0316712
$852$	0	0
$853$	−5.57700e6	−0.262439	−0.131219	−	0.991353i	$-0.541889\pi$
$853$	−5.57700e6	−0.262439	−0.131219	+	0.991353i	$0.541889\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.69291e7	−1.25248	−0.626240	−	0.779631i	$-0.715407\pi$
$857$	−2.69291e7	−1.25248	−0.626240	+	0.779631i	$0.715407\pi$
$858$	0	0
$859$	1.84576e7	0.853480	0.426740	−	0.904374i	$-0.359662\pi$
$859$	1.84576e7	0.853480	0.426740	+	0.904374i	$0.359662\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.14937e6	−0.143945	−0.0719725	−	0.997407i	$-0.522929\pi$
$863$	−3.14937e6	−0.143945	−0.0719725	+	0.997407i	$0.522929\pi$
$864$	0	0
$865$	1.41439e6	0.0642729
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.09678e6	−0.139111
$870$	0	0
$871$	−5.94098e6	−0.265346
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−8.22071e6	−0.362985
$876$	0	0
$877$	9.37513e6	0.411603	0.205801	−	0.978594i	$-0.434020\pi$
$877$	9.37513e6	0.411603	0.205801	+	0.978594i	$0.434020\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.09946e7	0.477245	0.238623	−	0.971112i	$-0.423304\pi$
$881$	1.09946e7	0.477245	0.238623	+	0.971112i	$0.423304\pi$
$882$	0	0
$883$	−7.04746e6	−0.304180	−0.152090	−	0.988367i	$-0.548600\pi$
$883$	−7.04746e6	−0.304180	−0.152090	+	0.988367i	$0.548600\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.64956e7	1.55751	0.778757	−	0.627326i	$-0.215851\pi$
$887$	3.64956e7	1.55751	0.778757	+	0.627326i	$0.215851\pi$
$888$	0	0
$889$	9.49078e6	0.402761
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.55440e6	0.233082
$894$	0	0
$895$	1.85042e7	0.772170
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.82512e7	2.40384
$900$	0	0
$901$	1.56169e7	0.640889
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.52331e7	1.02412
$906$	0	0
$907$	4.03594e7	1.62902	0.814510	−	0.580150i	$-0.197006\pi$
$907$	4.03594e7	1.62902	0.814510	+	0.580150i	$0.197006\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.09136e7	−0.834896	−0.417448	−	0.908701i	$-0.637075\pi$
$911$	−2.09136e7	−0.834896	−0.417448	+	0.908701i	$0.637075\pi$
$912$	0	0
$913$	343421.	0.0136348
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.18822e6	−0.0859344
$918$	0	0
$919$	−3.22946e7	−1.26136	−0.630682	−	0.776041i	$-0.717225\pi$
$919$	−3.22946e7	−1.26136	−0.630682	+	0.776041i	$0.717225\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.35305e6	0.0522767
$924$	0	0
$925$	−7.34784e7	−2.82361
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.53890e7	−0.965177	−0.482588	−	0.875847i	$-0.660303\pi$
$929$	−2.53890e7	−0.965177	−0.482588	+	0.875847i	$0.660303\pi$
$930$	0	0
$931$	−2.05880e7	−0.778466
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.19391e7	−0.446626
$936$	0	0
$937$	4.48170e7	1.66761	0.833803	−	0.552062i	$-0.186159\pi$
$937$	4.48170e7	1.66761	0.833803	+	0.552062i	$0.186159\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.68202e7	1.35554	0.677769	−	0.735275i	$-0.262947\pi$
$941$	3.68202e7	1.35554	0.677769	+	0.735275i	$0.262947\pi$
$942$	0	0
$943$	146813.	0.00537633
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.43343e7	−1.60644	−0.803220	−	0.595682i	$-0.796882\pi$
$947$	−4.43343e7	−1.60644	−0.803220	+	0.595682i	$0.796882\pi$
$948$	0	0
$949$	1.96986e6	0.0710018
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.26322e7	−0.450553	−0.225277	−	0.974295i	$-0.572329\pi$
$953$	−1.26322e7	−0.450553	−0.225277	+	0.974295i	$0.572329\pi$
$954$	0	0
$955$	6.60782e7	2.34450
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.31170e6	0.0811679
$960$	0	0
$961$	3.87870e7	1.35481
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.78549e7	1.99996
$966$	0	0
$967$	1.08031e7	0.371520	0.185760	−	0.982595i	$-0.440525\pi$
$967$	1.08031e7	0.371520	0.185760	+	0.982595i	$0.440525\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−2.18020e7	−0.742076	−0.371038	−	0.928618i	$-0.620998\pi$
$971$	−2.18020e7	−0.742076	−0.371038	+	0.928618i	$0.620998\pi$
$972$	0	0
$973$	6.93107e6	0.234703
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.68196e7	−0.563741	−0.281871	−	0.959452i	$-0.590955\pi$
$977$	−1.68196e7	−0.563741	−0.281871	+	0.959452i	$0.590955\pi$
$978$	0	0
$979$	1.01619e7	0.338857
$980$	0	0
$981$	0	0
$982$	0	0
$983$	5.00455e7	1.65189	0.825945	−	0.563751i	$-0.190642\pi$
$983$	5.00455e7	1.65189	0.825945	+	0.563751i	$0.190642\pi$
$984$	0	0
$985$	−6.83098e7	−2.24333
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−182513.	−0.00593340
$990$	0	0
$991$	3.81950e6	0.123544	0.0617721	−	0.998090i	$-0.480325\pi$
$991$	3.81950e6	0.123544	0.0617721	+	0.998090i	$0.480325\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.86712e7	1.87874
$996$	0	0
$997$	−2.52979e7	−0.806021	−0.403011	−	0.915195i	$-0.632036\pi$
$997$	−2.52979e7	−0.806021	−0.403011	+	0.915195i	$0.632036\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	936.6.a.m.1.1		4
3.2	odd	2		312.6.a.i.1.4	✓	4
12.11	even	2		624.6.a.v.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.6.a.i.1.4	✓	4	3.2	odd	2
624.6.a.v.1.4		4	12.11	even	2
936.6.a.m.1.1		4	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:	\( N \)	\(=\)	\( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	936.a (trivial)

\(f(q)\)	\(=\)	\(q-94.1278 q^{5} +33.4615 q^{7} -109.791 q^{11} -169.000 q^{13} -1155.28 q^{17} +1312.40 q^{19} -52.2234 q^{23} +5735.03 q^{25} +7094.51 q^{29} +8210.74 q^{31} -3149.65 q^{35} -12812.2 q^{37} -2811.26 q^{41} +3494.85 q^{43} +4232.26 q^{47} -15687.3 q^{49} -13517.8 q^{53} +10334.4 q^{55} +42651.7 q^{59} +35971.5 q^{61} +15907.6 q^{65} +35153.7 q^{67} -8006.18 q^{71} -11656.0 q^{73} -3673.78 q^{77} +28206.0 q^{79} -3127.94 q^{83} +108744. q^{85} -92556.1 q^{89} -5654.99 q^{91} -123533. q^{95} +6867.86 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5} - 176 q^{7} - 96 q^{11} - 676 q^{13} - 1224 q^{17} + 3856 q^{19} + 368 q^{23} + 5884 q^{25} - 264 q^{29} + 10928 q^{31} - 7984 q^{35} + 12008 q^{37} - 32792 q^{41} + 24288 q^{43} - 14176 q^{47}+ \cdots + 34664 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 - 176 * q^7 - 96 * q^11 - 676 * q^13 - 1224 * q^17 + 3856 * q^19 + 368 * q^23 + 5884 * q^25 - 264 * q^29 + 10928 * q^31 - 7984 * q^35 + 12008 * q^37 - 32792 * q^41 + 24288 * q^43 - 14176 * q^47 + 40948 * q^49 - 37384 * q^53 - 34384 * q^55 - 11456 * q^59 - 15752 * q^61 - 1352 * q^65 - 14032 * q^67 - 7552 * q^71 + 76104 * q^73 - 3776 * q^77 + 61056 * q^79 - 233856 * q^83 + 116816 * q^85 - 86136 * q^89 + 29744 * q^91 - 268368 * q^95 + 34664 * q^97

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