LMFDB - Embedded newform 16.30.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,30,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 30, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 30);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 30 $
Character orbit:	$[\chi]$	$=$	16.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:	$85.2448678129$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{51349}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 12837 $ x^2 - x - 12837
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9}\cdot 3 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-112.802$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.44920e7 q^{3} -6.21544e9 q^{5} +3.40335e10 q^{7} +1.41387e14 q^{9} +O(q^{10})$	$q+1.44920e7 q^{3} -6.21544e9 q^{5} +3.40335e10 q^{7} +1.41387e14 q^{9} +7.97271e14 q^{11} +1.10490e16 q^{13} -9.00740e16 q^{15} -7.47691e17 q^{17} +2.49040e18 q^{19} +4.93212e17 q^{21} +1.63712e18 q^{23} -1.47633e20 q^{25} +1.05439e21 q^{27} +1.87750e21 q^{29} +4.79819e21 q^{31} +1.15540e22 q^{33} -2.11533e20 q^{35} +7.91036e22 q^{37} +1.60122e23 q^{39} -6.06439e21 q^{41} -6.72526e23 q^{43} -8.78782e23 q^{45} +1.90366e24 q^{47} -3.21875e24 q^{49} -1.08355e25 q^{51} -1.65597e24 q^{53} -4.95539e24 q^{55} +3.60909e25 q^{57} +8.93685e25 q^{59} +4.59924e25 q^{61} +4.81189e24 q^{63} -6.86745e25 q^{65} +1.04774e26 q^{67} +2.37252e25 q^{69} -1.75243e26 q^{71} -4.56674e26 q^{73} -2.13949e27 q^{75} +2.71339e25 q^{77} +3.28137e27 q^{79} +5.57671e27 q^{81} +4.40086e27 q^{83} +4.64723e27 q^{85} +2.72087e28 q^{87} -6.06897e26 q^{89} +3.76036e26 q^{91} +6.95352e28 q^{93} -1.54790e28 q^{95} +1.44471e28 q^{97} +1.12724e29 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4967640 q^{3} - 17477788500 q^{5} + 3020312682800 q^{7} + 163469569523706 q^{9}+O(q^{10}) $ 2 * q + 4967640 * q^3 - 17477788500 * q^5 + 3020312682800 * q^7 + 163469569523706 * q^9	$ 2 q + 4967640 q^{3} - 17477788500 q^{5} + 3020312682800 q^{7} + 163469569523706 q^{9} + 20\!\cdots\!16 q^{11}+ \cdots + 14\!\cdots\!48 q^{99}+O(q^{100}) $ 2 * q + 4967640 * q^3 - 17477788500 * q^5 + 3020312682800 * q^7 + 163469569523706 * q^9 + 2055380519588616 * q^11 + 17139371179332220 * q^13 + 17192351392506000 * q^15 - 664795927907749980 * q^17 - 1232169445452155080 * q^19 - 27949113476248821696 * q^21 + 18588430504374379920 * q^23 - 207056854502617281250 * q^25 + 1497723629507205824880 * q^27 + 996096417765761595420 * q^29 + 1087258664181924030656 * q^31 - 428627571739162757280 * q^33 - 33844046536902649068000 * q^35 + 98878127759336138311660 * q^37 + 102115380027539490485328 * q^39 - 106699837259610025757676 * q^41 - 510640433114974031366200 * q^43 - 1127484267514124881450500 * q^45 + 4521107671109825795039520 * q^47 + 2479210387392854586312114 * q^49 - 11625042659032475755122384 * q^51 - 16141544629666790250663540 * q^53 - 19124657798421120971058000 * q^55 + 71545878452212360967003040 * q^57 + 83497291515660150276062760 * q^59 - 15876762943230179624113316 * q^61 + 70756669640265175617870960 * q^63 - 137266226551726798673511000 * q^65 - 24468754434128844890782120 * q^67 - 137724750477330905859409728 * q^69 + 188428428253360067209577136 * q^71 + 995872721297568533052080980 * q^73 - 1573516410323456985690375000 * q^75 + 3784200478484579386131729600 * q^77 + 7218460852587735797823642080 * q^79 - 161317068789001769884865358 * q^81 - 2210846225248155523694669640 * q^83 + 3713635892367734904614943000 * q^85 + 35603430680717182696869516240 * q^87 + 5872964746600969363962031860 * q^89 + 18563550073842319069958420896 * q^91 + 104879397265778648689061310720 * q^93 + 26445947767121183571013410000 * q^95 + 109649982219471997093227005380 * q^97 + 140506044809020243496724809448 * 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
$3$	1.44920e7	1.74932	0.874660	−	0.484736i	$-0.161084\pi$
$3$	1.44920e7	1.74932	0.874660	+	0.484736i	$0.161084\pi$
$4$	0	0
$5$	−6.21544e9	−0.455415	−0.227707	−	0.973730i	$-0.573123\pi$
$5$	−6.21544e9	−0.455415	−0.227707	+	0.973730i	$0.573123\pi$
$6$	0	0
$7$	3.40335e10	0.0189664	0.00948319	−	0.999955i	$-0.496981\pi$
$7$	3.40335e10	0.0189664	0.00948319	+	0.999955i	$0.496981\pi$
$8$	0	0
$9$	1.41387e14	2.06012
$10$	0	0
$11$	7.97271e14	0.633012	0.316506	−	0.948590i	$-0.397490\pi$
$11$	7.97271e14	0.633012	0.316506	+	0.948590i	$0.397490\pi$
$12$	0	0
$13$	1.10490e16	0.778296	0.389148	−	0.921175i	$-0.372769\pi$
$13$	1.10490e16	0.778296	0.389148	+	0.921175i	$0.372769\pi$
$14$	0	0
$15$	−9.00740e16	−0.796666
$16$	0	0
$17$	−7.47691e17	−1.07699	−0.538496	−	0.842628i	$-0.681007\pi$
$17$	−7.47691e17	−1.07699	−0.538496	+	0.842628i	$0.681007\pi$
$18$	0	0
$19$	2.49040e18	0.715060	0.357530	−	0.933902i	$-0.383619\pi$
$19$	2.49040e18	0.715060	0.357530	+	0.933902i	$0.383619\pi$
$20$	0	0
$21$	4.93212e17	0.0331783
$22$	0	0
$23$	1.63712e18	0.0294461	0.0147231	−	0.999892i	$-0.495313\pi$
$23$	1.63712e18	0.0294461	0.0147231	+	0.999892i	$0.495313\pi$
$24$	0	0
$25$	−1.47633e20	−0.792598
$26$	0	0
$27$	1.05439e21	1.85449
$28$	0	0
$29$	1.87750e21	1.17168	0.585839	−	0.810427i	$-0.300765\pi$
$29$	1.87750e21	1.17168	0.585839	+	0.810427i	$0.300765\pi$
$30$	0	0
$31$	4.79819e21	1.13850	0.569250	−	0.822165i	$-0.307234\pi$
$31$	4.79819e21	1.13850	0.569250	+	0.822165i	$0.307234\pi$
$32$	0	0
$33$	1.15540e22	1.10734
$34$	0	0
$35$	−2.11533e20	−0.00863757
$36$	0	0
$37$	7.91036e22	1.44302	0.721508	−	0.692406i	$-0.243449\pi$
$37$	7.91036e22	1.44302	0.721508	+	0.692406i	$0.243449\pi$
$38$	0	0
$39$	1.60122e23	1.36149
$40$	0	0
$41$	−6.06439e21	−0.0249702	−0.0124851	−	0.999922i	$-0.503974\pi$
$41$	−6.06439e21	−0.0249702	−0.0124851	+	0.999922i	$0.503974\pi$
$42$	0	0
$43$	−6.72526e23	−1.38809	−0.694044	−	0.719933i	$-0.744173\pi$
$43$	−6.72526e23	−1.38809	−0.694044	+	0.719933i	$0.744173\pi$
$44$	0	0
$45$	−8.78782e23	−0.938210
$46$	0	0
$47$	1.90366e24	1.08186	0.540928	−	0.841069i	$-0.318073\pi$
$47$	1.90366e24	1.08186	0.540928	+	0.841069i	$0.318073\pi$
$48$	0	0
$49$	−3.21875e24	−0.999640
$50$	0	0
$51$	−1.08355e25	−1.88401
$52$	0	0
$53$	−1.65597e24	−0.164836	−0.0824181	−	0.996598i	$-0.526264\pi$
$53$	−1.65597e24	−0.164836	−0.0824181	+	0.996598i	$0.526264\pi$
$54$	0	0
$55$	−4.95539e24	−0.288283
$56$	0	0
$57$	3.60909e25	1.25087
$58$	0	0
$59$	8.93685e25	1.87858	0.939292	−	0.343120i	$-0.111484\pi$
$59$	8.93685e25	1.87858	0.939292	+	0.343120i	$0.111484\pi$
$60$	0	0
$61$	4.59924e25	0.596216	0.298108	−	0.954532i	$-0.403644\pi$
$61$	4.59924e25	0.596216	0.298108	+	0.954532i	$0.403644\pi$
$62$	0	0
$63$	4.81189e24	0.0390731
$64$	0	0
$65$	−6.86745e25	−0.354447
$66$	0	0
$67$	1.04774e26	0.348472	0.174236	−	0.984704i	$-0.444254\pi$
$67$	1.04774e26	0.348472	0.174236	+	0.984704i	$0.444254\pi$
$68$	0	0
$69$	2.37252e25	0.0515107
$70$	0	0
$71$	−1.75243e26	−0.251417	−0.125709	−	0.992067i	$-0.540120\pi$
$71$	−1.75243e26	−0.251417	−0.125709	+	0.992067i	$0.540120\pi$
$72$	0	0
$73$	−4.56674e26	−0.437950	−0.218975	−	0.975730i	$-0.570271\pi$
$73$	−4.56674e26	−0.437950	−0.218975	+	0.975730i	$0.570271\pi$
$74$	0	0
$75$	−2.13949e27	−1.38651
$76$	0	0
$77$	2.71339e25	0.0120059
$78$	0	0
$79$	3.28137e27	1.00107	0.500533	−	0.865717i	$-0.333137\pi$
$79$	3.28137e27	1.00107	0.500533	+	0.865717i	$0.333137\pi$
$80$	0	0
$81$	5.57671e27	1.18398
$82$	0	0
$83$	4.40086e27	0.656002	0.328001	−	0.944677i	$-0.393625\pi$
$83$	4.40086e27	0.656002	0.328001	+	0.944677i	$0.393625\pi$
$84$	0	0
$85$	4.64723e27	0.490478
$86$	0	0
$87$	2.72087e28	2.04964
$88$	0	0
$89$	−6.06897e26	−0.0328821	−0.0164411	−	0.999865i	$-0.505234\pi$
$89$	−6.06897e26	−0.0328821	−0.0164411	+	0.999865i	$0.505234\pi$
$90$	0	0
$91$	3.76036e26	0.0147615
$92$	0	0
$93$	6.95352e28	1.99160
$94$	0	0
$95$	−1.54790e28	−0.325649
$96$	0	0
$97$	1.44471e28	0.224694	0.112347	−	0.993669i	$-0.464163\pi$
$97$	1.44471e28	0.224694	0.112347	+	0.993669i	$0.464163\pi$
$98$	0	0
$99$	1.12724e29	1.30408
$100$	0	0
$101$	−4.90236e28	−0.424371	−0.212185	−	0.977229i	$-0.568058\pi$
$101$	−4.90236e28	−0.424371	−0.212185	+	0.977229i	$0.568058\pi$
$102$	0	0
$103$	−1.31966e29	−0.859654	−0.429827	−	0.902911i	$-0.641426\pi$
$103$	−1.31966e29	−0.859654	−0.429827	+	0.902911i	$0.641426\pi$
$104$	0	0
$105$	−3.06553e27	−0.0151099
$106$	0	0
$107$	2.31447e29	0.867733	0.433867	−	0.900977i	$-0.357149\pi$
$107$	2.31447e29	0.867733	0.433867	+	0.900977i	$0.357149\pi$
$108$	0	0
$109$	6.73768e29	1.93119	0.965597	−	0.260043i	$-0.0837367\pi$
$109$	6.73768e29	1.93119	0.965597	+	0.260043i	$0.0837367\pi$
$110$	0	0
$111$	1.14637e30	2.52430
$112$	0	0
$113$	5.10156e28	0.0867092	0.0433546	−	0.999060i	$-0.486195\pi$
$113$	5.10156e28	0.0867092	0.0433546	+	0.999060i	$0.486195\pi$
$114$	0	0
$115$	−1.01755e28	−0.0134102
$116$	0	0
$117$	1.56219e30	1.60339
$118$	0	0
$119$	−2.54465e28	−0.0204267
$120$	0	0
$121$	−9.50668e29	−0.599296
$122$	0	0
$123$	−8.78850e28	−0.0436808
$124$	0	0
$125$	2.07532e30	0.816375
$126$	0	0
$127$	−3.46836e30	−1.08385	−0.541926	−	0.840426i	$-0.682305\pi$
$127$	−3.46836e30	−1.08385	−0.541926	+	0.840426i	$0.682305\pi$
$128$	0	0
$129$	−9.74624e30	−2.42821
$130$	0	0
$131$	1.16207e30	0.231631	0.115816	−	0.993271i	$-0.463052\pi$
$131$	1.16207e30	0.231631	0.115816	+	0.993271i	$0.463052\pi$
$132$	0	0
$133$	8.47570e28	0.0135621
$134$	0	0
$135$	−6.55348e30	−0.844564
$136$	0	0
$137$	−1.12214e31	−1.16842	−0.584210	−	0.811603i	$-0.698596\pi$
$137$	−1.12214e31	−1.16842	−0.584210	+	0.811603i	$0.698596\pi$
$138$	0	0
$139$	2.10245e30	0.177423	0.0887117	−	0.996057i	$-0.471725\pi$
$139$	2.10245e30	0.177423	0.0887117	+	0.996057i	$0.471725\pi$
$140$	0	0
$141$	2.75877e31	1.89251
$142$	0	0
$143$	8.80906e30	0.492671
$144$	0	0
$145$	−1.16695e31	−0.533600
$146$	0	0
$147$	−4.66460e31	−1.74869
$148$	0	0
$149$	−4.28030e30	−0.131909	−0.0659543	−	0.997823i	$-0.521009\pi$
$149$	−4.28030e30	−0.131909	−0.0659543	+	0.997823i	$0.521009\pi$
$150$	0	0
$151$	9.32731e30	0.236914	0.118457	−	0.992959i	$-0.462205\pi$
$151$	9.32731e30	0.236914	0.118457	+	0.992959i	$0.462205\pi$
$152$	0	0
$153$	−1.05714e32	−2.21874
$154$	0	0
$155$	−2.98229e31	−0.518489
$156$	0	0
$157$	2.58823e31	0.373644	0.186822	−	0.982394i	$-0.440181\pi$
$157$	2.58823e31	0.373644	0.186822	+	0.982394i	$0.440181\pi$
$158$	0	0
$159$	−2.39983e31	−0.288351
$160$	0	0
$161$	5.57170e28	0.000558486 0
$162$	0	0
$163$	−1.19680e31	−0.100300	−0.0501501	−	0.998742i	$-0.515970\pi$
$163$	−1.19680e31	−0.100300	−0.0501501	+	0.998742i	$0.515970\pi$
$164$	0	0
$165$	−7.18134e31	−0.504299
$166$	0	0
$167$	−1.07804e32	−0.635692	−0.317846	−	0.948142i	$-0.602959\pi$
$167$	−1.07804e32	−0.635692	−0.317846	+	0.948142i	$0.602959\pi$
$168$	0	0
$169$	−7.94575e31	−0.394255
$170$	0	0
$171$	3.52110e32	1.47311
$172$	0	0
$173$	3.82198e32	1.35089	0.675446	−	0.737410i	$-0.263951\pi$
$173$	3.82198e32	1.35089	0.675446	+	0.737410i	$0.263951\pi$
$174$	0	0
$175$	−5.02445e30	−0.0150327
$176$	0	0
$177$	1.29513e33	3.28624
$178$	0	0
$179$	5.24041e32	1.12979	0.564893	−	0.825164i	$-0.308917\pi$
$179$	5.24041e32	1.12979	0.564893	+	0.825164i	$0.308917\pi$
$180$	0	0
$181$	−7.54402e32	−1.38441	−0.692203	−	0.721703i	$-0.743360\pi$
$181$	−7.54402e32	−1.38441	−0.692203	+	0.721703i	$0.743360\pi$
$182$	0	0
$183$	6.66521e32	1.04297
$184$	0	0
$185$	−4.91664e32	−0.657170
$186$	0	0
$187$	−5.96112e32	−0.681750
$188$	0	0
$189$	3.58844e31	0.0351730
$190$	0	0
$191$	1.08024e33	0.908947	0.454473	−	0.890760i	$-0.349827\pi$
$191$	1.08024e33	0.908947	0.454473	+	0.890760i	$0.349827\pi$
$192$	0	0
$193$	1.91169e33	1.38305	0.691524	−	0.722354i	$-0.256940\pi$
$193$	1.91169e33	1.38305	0.691524	+	0.722354i	$0.256940\pi$
$194$	0	0
$195$	−9.95229e32	−0.620042
$196$	0	0
$197$	−2.82411e33	−1.51747	−0.758737	−	0.651397i	$-0.774183\pi$
$197$	−2.82411e33	−1.51747	−0.758737	+	0.651397i	$0.774183\pi$
$198$	0	0
$199$	−2.05181e33	−0.952288	−0.476144	−	0.879367i	$-0.657966\pi$
$199$	−2.05181e33	−0.952288	−0.476144	+	0.879367i	$0.657966\pi$
$200$	0	0
$201$	1.51838e33	0.609590
$202$	0	0
$203$	6.38978e31	0.0222225
$204$	0	0
$205$	3.76929e31	0.0113718
$206$	0	0
$207$	2.31468e32	0.0606626
$208$	0	0
$209$	1.98553e33	0.452641
$210$	0	0
$211$	4.24073e33	0.842063	0.421032	−	0.907046i	$-0.361668\pi$
$211$	4.24073e33	0.842063	0.421032	+	0.907046i	$0.361668\pi$
$212$	0	0
$213$	−2.53962e33	−0.439809
$214$	0	0
$215$	4.18005e33	0.632155
$216$	0	0
$217$	1.63299e32	0.0215932
$218$	0	0
$219$	−6.61810e33	−0.766115
$220$	0	0
$221$	−8.26125e33	−0.838219
$222$	0	0
$223$	−1.95826e34	−1.74362	−0.871808	−	0.489848i	$-0.837052\pi$
$223$	−1.95826e34	−1.74362	−0.871808	+	0.489848i	$0.837052\pi$
$224$	0	0
$225$	−2.08734e34	−1.63285
$226$	0	0
$227$	−1.22424e34	−0.842345	−0.421173	−	0.906980i	$-0.638381\pi$
$227$	−1.22424e34	−0.842345	−0.421173	+	0.906980i	$0.638381\pi$
$228$	0	0
$229$	5.07275e33	0.307347	0.153673	−	0.988122i	$-0.450890\pi$
$229$	5.07275e33	0.307347	0.153673	+	0.988122i	$0.450890\pi$
$230$	0	0
$231$	3.93224e32	0.0210023
$232$	0	0
$233$	−2.33456e34	−1.10038	−0.550192	−	0.835038i	$-0.685446\pi$
$233$	−2.33456e34	−1.10038	−0.550192	+	0.835038i	$0.685446\pi$
$234$	0	0
$235$	−1.18321e34	−0.492693
$236$	0	0
$237$	4.75536e34	1.75119
$238$	0	0
$239$	1.69274e34	0.551849	0.275925	−	0.961179i	$-0.411016\pi$
$239$	1.69274e34	0.551849	0.275925	+	0.961179i	$0.411016\pi$
$240$	0	0
$241$	−2.26884e34	−0.655476	−0.327738	−	0.944769i	$-0.606286\pi$
$241$	−2.26884e34	−0.655476	−0.327738	+	0.944769i	$0.606286\pi$
$242$	0	0
$243$	8.45454e33	0.216670
$244$	0	0
$245$	2.00059e34	0.455251
$246$	0	0
$247$	2.75165e34	0.556528
$248$	0	0
$249$	6.37771e34	1.14756
$250$	0	0
$251$	9.82834e34	1.57475	0.787374	−	0.616475i	$-0.211440\pi$
$251$	9.82834e34	1.57475	0.787374	+	0.616475i	$0.211440\pi$
$252$	0	0
$253$	1.30523e33	0.0186398
$254$	0	0
$255$	6.73475e34	0.858004
$256$	0	0
$257$	−8.53561e34	−0.970966	−0.485483	−	0.874246i	$-0.661356\pi$
$257$	−8.53561e34	−0.970966	−0.485483	+	0.874246i	$0.661356\pi$
$258$	0	0
$259$	2.69217e33	0.0273688
$260$	0	0
$261$	2.65454e35	2.41380
$262$	0	0
$263$	−2.07906e35	−1.69243	−0.846213	−	0.532845i	$-0.821123\pi$
$263$	−2.07906e35	−1.69243	−0.846213	+	0.532845i	$0.821123\pi$
$264$	0	0
$265$	1.02926e34	0.0750688
$266$	0	0
$267$	−8.79513e33	−0.0575214
$268$	0	0
$269$	−2.34883e35	−1.37862	−0.689309	−	0.724468i	$-0.742086\pi$
$269$	−2.34883e35	−1.37862	−0.689309	+	0.724468i	$0.742086\pi$
$270$	0	0
$271$	8.82959e34	0.465463	0.232732	−	0.972541i	$-0.425234\pi$
$271$	8.82959e34	0.465463	0.232732	+	0.972541i	$0.425234\pi$
$272$	0	0
$273$	5.44951e33	0.0258225
$274$	0	0
$275$	−1.17703e35	−0.501724
$276$	0	0
$277$	−7.48695e34	−0.287308	−0.143654	−	0.989628i	$-0.545885\pi$
$277$	−7.48695e34	−0.287308	−0.143654	+	0.989628i	$0.545885\pi$
$278$	0	0
$279$	6.78402e35	2.34545
$280$	0	0
$281$	3.79424e35	1.18272	0.591361	−	0.806407i	$-0.298591\pi$
$281$	3.79424e35	1.18272	0.591361	+	0.806407i	$0.298591\pi$
$282$	0	0
$283$	−5.23811e35	−1.47323	−0.736616	−	0.676311i	$-0.763578\pi$
$283$	−5.23811e35	−1.47323	−0.736616	+	0.676311i	$0.763578\pi$
$284$	0	0
$285$	−2.24321e35	−0.569664
$286$	0	0
$287$	−2.06392e32	−0.000473593 0
$288$	0	0
$289$	7.70734e34	0.159914
$290$	0	0
$291$	2.09367e35	0.393061
$292$	0	0
$293$	−8.91980e35	−1.51626	−0.758131	−	0.652102i	$-0.773887\pi$
$293$	−8.91980e35	−1.51626	−0.758131	+	0.652102i	$0.773887\pi$
$294$	0	0
$295$	−5.55465e35	−0.855534
$296$	0	0
$297$	8.40632e35	1.17392
$298$	0	0
$299$	1.80886e34	0.0229178
$300$	0	0
$301$	−2.28884e34	−0.0263270
$302$	0	0
$303$	−7.10449e35	−0.742360
$304$	0	0
$305$	−2.85863e35	−0.271525
$306$	0	0
$307$	−6.34791e35	−0.548434	−0.274217	−	0.961668i	$-0.588419\pi$
$307$	−6.34791e35	−0.548434	−0.274217	+	0.961668i	$0.588419\pi$
$308$	0	0
$309$	−1.91246e36	−1.50381
$310$	0	0
$311$	−1.79660e36	−1.28655	−0.643273	−	0.765637i	$-0.722424\pi$
$311$	−1.79660e36	−1.28655	−0.643273	+	0.765637i	$0.722424\pi$
$312$	0	0
$313$	−2.68729e36	−1.75357	−0.876783	−	0.480886i	$-0.840315\pi$
$313$	−2.68729e36	−1.75357	−0.876783	+	0.480886i	$0.840315\pi$
$314$	0	0
$315$	−2.99080e34	−0.0177944
$316$	0	0
$317$	1.03006e36	0.559119	0.279559	−	0.960128i	$-0.409812\pi$
$317$	1.03006e36	0.559119	0.279559	+	0.960128i	$0.409812\pi$
$318$	0	0
$319$	1.49687e36	0.741687
$320$	0	0
$321$	3.35412e36	1.51794
$322$	0	0
$323$	−1.86205e36	−0.770114
$324$	0	0
$325$	−1.63120e36	−0.616876
$326$	0	0
$327$	9.76422e36	3.37828
$328$	0	0
$329$	6.47880e34	0.0205189
$330$	0	0
$331$	−1.57492e36	−0.456827	−0.228414	−	0.973564i	$-0.573354\pi$
$331$	−1.57492e36	−0.456827	−0.228414	+	0.973564i	$0.573354\pi$
$332$	0	0
$333$	1.11842e37	2.97279
$334$	0	0
$335$	−6.51216e35	−0.158699
$336$	0	0
$337$	−3.02595e36	−0.676438	−0.338219	−	0.941067i	$-0.609824\pi$
$337$	−3.02595e36	−0.676438	−0.338219	+	0.941067i	$0.609824\pi$
$338$	0	0
$339$	7.39316e35	0.151682
$340$	0	0
$341$	3.82546e36	0.720684
$342$	0	0
$343$	−2.19130e35	−0.0379259
$344$	0	0
$345$	−1.47462e35	−0.0234587
$346$	0	0
$347$	−4.44778e36	−0.650678	−0.325339	−	0.945597i	$-0.605478\pi$
$347$	−4.44778e36	−0.650678	−0.325339	+	0.945597i	$0.605478\pi$
$348$	0	0
$349$	−9.68519e35	−0.130359	−0.0651793	−	0.997874i	$-0.520762\pi$
$349$	−9.68519e35	−0.130359	−0.0651793	+	0.997874i	$0.520762\pi$
$350$	0	0
$351$	1.16499e37	1.44335
$352$	0	0
$353$	−9.40660e36	−1.07325	−0.536623	−	0.843822i	$-0.680300\pi$
$353$	−9.40660e36	−1.07325	−0.536623	+	0.843822i	$0.680300\pi$
$354$	0	0
$355$	1.08921e36	0.114499
$356$	0	0
$357$	−3.68770e35	−0.0357328
$358$	0	0
$359$	−2.90920e36	−0.259958	−0.129979	−	0.991517i	$-0.541491\pi$
$359$	−2.90920e36	−0.259958	−0.129979	+	0.991517i	$0.541491\pi$
$360$	0	0
$361$	−5.92772e36	−0.488690
$362$	0	0
$363$	−1.37771e37	−1.04836
$364$	0	0
$365$	2.83843e36	0.199449
$366$	0	0
$367$	−3.00435e36	−0.195026	−0.0975131	−	0.995234i	$-0.531089\pi$
$367$	−3.00435e36	−0.195026	−0.0975131	+	0.995234i	$0.531089\pi$
$368$	0	0
$369$	−8.57426e35	−0.0514416
$370$	0	0
$371$	−5.63584e34	−0.00312634
$372$	0	0
$373$	2.82375e37	1.44893	0.724466	−	0.689310i	$-0.242086\pi$
$373$	2.82375e37	1.44893	0.724466	+	0.689310i	$0.242086\pi$
$374$	0	0
$375$	3.00755e37	1.42810
$376$	0	0
$377$	2.07445e37	0.911913
$378$	0	0
$379$	−1.51814e37	−0.618079	−0.309039	−	0.951049i	$-0.600007\pi$
$379$	−1.51814e37	−0.618079	−0.309039	+	0.951049i	$0.600007\pi$
$380$	0	0
$381$	−5.02634e37	−1.89601
$382$	0	0
$383$	2.59896e37	0.908692	0.454346	−	0.890825i	$-0.349873\pi$
$383$	2.59896e37	0.908692	0.454346	+	0.890825i	$0.349873\pi$
$384$	0	0
$385$	−1.68649e35	−0.00546768
$386$	0	0
$387$	−9.50865e37	−2.85963
$388$	0	0
$389$	−4.80732e37	−1.34163	−0.670816	−	0.741623i	$-0.734056\pi$
$389$	−4.80732e37	−1.34163	−0.670816	+	0.741623i	$0.734056\pi$
$390$	0	0
$391$	−1.22406e36	−0.0317133
$392$	0	0
$393$	1.68406e37	0.405197
$394$	0	0
$395$	−2.03952e37	−0.455900
$396$	0	0
$397$	−4.33277e37	−0.900126	−0.450063	−	0.892997i	$-0.648599\pi$
$397$	−4.33277e37	−0.900126	−0.450063	+	0.892997i	$0.648599\pi$
$398$	0	0
$399$	1.22830e36	0.0237245
$400$	0	0
$401$	−2.87417e37	−0.516320	−0.258160	−	0.966102i	$-0.583116\pi$
$401$	−2.87417e37	−0.516320	−0.258160	+	0.966102i	$0.583116\pi$
$402$	0	0
$403$	5.30153e37	0.886090
$404$	0	0
$405$	−3.46617e37	−0.539203
$406$	0	0
$407$	6.30670e37	0.913446
$408$	0	0
$409$	8.54677e37	1.15296	0.576480	−	0.817112i	$-0.304426\pi$
$409$	8.54677e37	1.15296	0.576480	+	0.817112i	$0.304426\pi$
$410$	0	0
$411$	−1.62620e38	−2.04394
$412$	0	0
$413$	3.04152e36	0.0356299
$414$	0	0
$415$	−2.73533e37	−0.298753
$416$	0	0
$417$	3.04686e37	0.310370
$418$	0	0
$419$	5.26548e37	0.500420	0.250210	−	0.968192i	$-0.419500\pi$
$419$	5.26548e37	0.500420	0.250210	+	0.968192i	$0.419500\pi$
$420$	0	0
$421$	9.94249e37	0.881870	0.440935	−	0.897539i	$-0.354647\pi$
$421$	9.94249e37	0.881870	0.440935	+	0.897539i	$0.354647\pi$
$422$	0	0
$423$	2.69152e38	2.22876
$424$	0	0
$425$	1.10384e38	0.853622
$426$	0	0
$427$	1.56528e36	0.0113081
$428$	0	0
$429$	1.27661e38	0.861839
$430$	0	0
$431$	1.96737e38	1.24155	0.620777	−	0.783987i	$-0.286817\pi$
$431$	1.96737e38	1.24155	0.620777	+	0.783987i	$0.286817\pi$
$432$	0	0
$433$	8.93739e37	0.527396	0.263698	−	0.964605i	$-0.415058\pi$
$433$	8.93739e37	0.527396	0.263698	+	0.964605i	$0.415058\pi$
$434$	0	0
$435$	−1.69114e38	−0.933437
$436$	0	0
$437$	4.07710e36	0.0210557
$438$	0	0
$439$	3.31662e38	1.60310	0.801548	−	0.597930i	$-0.204010\pi$
$439$	3.31662e38	1.60310	0.801548	+	0.597930i	$0.204010\pi$
$440$	0	0
$441$	−4.55089e38	−2.05938
$442$	0	0
$443$	−2.62823e38	−1.11380	−0.556902	−	0.830578i	$-0.688010\pi$
$443$	−2.62823e38	−1.11380	−0.556902	+	0.830578i	$0.688010\pi$
$444$	0	0
$445$	3.77213e36	0.0149750
$446$	0	0
$447$	−6.20299e37	−0.230750
$448$	0	0
$449$	1.08508e38	0.378348	0.189174	−	0.981944i	$-0.439419\pi$
$449$	1.08508e38	0.378348	0.189174	+	0.981944i	$0.439419\pi$
$450$	0	0
$451$	−4.83496e36	−0.0158064
$452$	0	0
$453$	1.35171e38	0.414439
$454$	0	0
$455$	−2.33723e36	−0.00672258
$456$	0	0
$457$	−3.49142e38	−0.942360	−0.471180	−	0.882037i	$-0.656172\pi$
$457$	−3.49142e38	−0.942360	−0.471180	+	0.882037i	$0.656172\pi$
$458$	0	0
$459$	−7.88356e38	−1.99728
$460$	0	0
$461$	5.84500e38	1.39034	0.695171	−	0.718845i	$-0.255329\pi$
$461$	5.84500e38	1.39034	0.695171	+	0.718845i	$0.255329\pi$
$462$	0	0
$463$	−2.77160e38	−0.619164	−0.309582	−	0.950873i	$-0.600189\pi$
$463$	−2.77160e38	−0.619164	−0.309582	+	0.950873i	$0.600189\pi$
$464$	0	0
$465$	−4.32192e38	−0.907004
$466$	0	0
$467$	9.01377e38	1.77751	0.888754	−	0.458384i	$-0.151571\pi$
$467$	9.01377e38	1.77751	0.888754	+	0.458384i	$0.151571\pi$
$468$	0	0
$469$	3.56582e36	0.00660926
$470$	0	0
$471$	3.75086e38	0.653623
$472$	0	0
$473$	−5.36186e38	−0.878676
$474$	0	0
$475$	−3.67665e38	−0.566755
$476$	0	0
$477$	−2.34133e38	−0.339583
$478$	0	0
$479$	−5.62143e38	−0.767330	−0.383665	−	0.923472i	$-0.625338\pi$
$479$	−5.62143e38	−0.767330	−0.383665	+	0.923472i	$0.625338\pi$
$480$	0	0
$481$	8.74016e38	1.12309
$482$	0	0
$483$	8.07450e35	0.000976972 0
$484$	0	0
$485$	−8.97952e37	−0.102329
$486$	0	0
$487$	1.05085e39	1.12816	0.564081	−	0.825720i	$-0.309231\pi$
$487$	1.05085e39	1.12816	0.564081	+	0.825720i	$0.309231\pi$
$488$	0	0
$489$	−1.73440e38	−0.175457
$490$	0	0
$491$	−7.99399e38	−0.762223	−0.381112	−	0.924529i	$-0.624459\pi$
$491$	−7.99399e38	−0.762223	−0.381112	+	0.924529i	$0.624459\pi$
$492$	0	0
$493$	−1.40379e39	−1.26189
$494$	0	0
$495$	−7.00628e38	−0.593898
$496$	0	0
$497$	−5.96413e36	−0.00476848
$498$	0	0
$499$	−1.14067e39	−0.860401	−0.430201	−	0.902733i	$-0.641557\pi$
$499$	−1.14067e39	−0.860401	−0.430201	+	0.902733i	$0.641557\pi$
$500$	0	0
$501$	−1.56229e39	−1.11203
$502$	0	0
$503$	−3.45164e38	−0.231894	−0.115947	−	0.993255i	$-0.536990\pi$
$503$	−3.45164e38	−0.231894	−0.115947	+	0.993255i	$0.536990\pi$
$504$	0	0
$505$	3.04703e38	0.193265
$506$	0	0
$507$	−1.15150e39	−0.689679
$508$	0	0
$509$	−7.06121e37	−0.0399458	−0.0199729	−	0.999801i	$-0.506358\pi$
$509$	−7.06121e37	−0.0399458	−0.0199729	+	0.999801i	$0.506358\pi$
$510$	0	0
$511$	−1.55422e37	−0.00830633
$512$	0	0
$513$	2.62585e39	1.32607
$514$	0	0
$515$	8.20230e38	0.391499
$516$	0	0
$517$	1.51773e39	0.684828
$518$	0	0
$519$	5.53881e39	2.36314
$520$	0	0
$521$	3.46255e39	1.39717	0.698585	−	0.715528i	$-0.253814\pi$
$521$	3.46255e39	1.39717	0.698585	+	0.715528i	$0.253814\pi$
$522$	0	0
$523$	1.15392e39	0.440453	0.220227	−	0.975449i	$-0.429320\pi$
$523$	1.15392e39	0.440453	0.220227	+	0.975449i	$0.429320\pi$
$524$	0	0
$525$	−7.28143e37	−0.0262970
$526$	0	0
$527$	−3.58756e39	−1.22616
$528$	0	0
$529$	−3.08838e39	−0.999133
$530$	0	0
$531$	1.26355e40	3.87011
$532$	0	0
$533$	−6.70055e37	−0.0194342
$534$	0	0
$535$	−1.43854e39	−0.395178
$536$	0	0
$537$	7.59439e39	1.97636
$538$	0	0
$539$	−2.56621e39	−0.632784
$540$	0	0
$541$	−2.74672e39	−0.641879	−0.320940	−	0.947100i	$-0.603999\pi$
$541$	−2.74672e39	−0.641879	−0.320940	+	0.947100i	$0.603999\pi$
$542$	0	0
$543$	−1.09328e40	−2.42177
$544$	0	0
$545$	−4.18776e39	−0.879494
$546$	0	0
$547$	5.50487e39	1.09630	0.548152	−	0.836379i	$-0.315331\pi$
$547$	5.50487e39	1.09630	0.548152	+	0.836379i	$0.315331\pi$
$548$	0	0
$549$	6.50273e39	1.22828
$550$	0	0
$551$	4.67573e39	0.837820
$552$	0	0
$553$	1.11677e38	0.0189866
$554$	0	0
$555$	−7.12518e39	−1.14960
$556$	0	0
$557$	−4.48537e39	−0.686908	−0.343454	−	0.939170i	$-0.611597\pi$
$557$	−4.48537e39	−0.686908	−0.343454	+	0.939170i	$0.611597\pi$
$558$	0	0
$559$	−7.43075e39	−1.08034
$560$	0	0
$561$	−8.63885e39	−1.19260
$562$	0	0
$563$	−6.55822e38	−0.0859834	−0.0429917	−	0.999075i	$-0.513689\pi$
$563$	−6.55822e38	−0.0859834	−0.0429917	+	0.999075i	$0.513689\pi$
$564$	0	0
$565$	−3.17084e38	−0.0394886
$566$	0	0
$567$	1.89795e38	0.0224559
$568$	0	0
$569$	−1.62256e40	−1.82420	−0.912099	−	0.409970i	$-0.865539\pi$
$569$	−1.62256e40	−1.82420	−0.912099	+	0.409970i	$0.865539\pi$
$570$	0	0
$571$	−9.28706e39	−0.992327	−0.496163	−	0.868229i	$-0.665258\pi$
$571$	−9.28706e39	−0.992327	−0.496163	+	0.868229i	$0.665258\pi$
$572$	0	0
$573$	1.56549e40	1.59004
$574$	0	0
$575$	−2.41693e38	−0.0233389
$576$	0	0
$577$	−2.84801e39	−0.261513	−0.130756	−	0.991415i	$-0.541741\pi$
$577$	−2.84801e39	−0.261513	−0.130756	+	0.991415i	$0.541741\pi$
$578$	0	0
$579$	2.77042e40	2.41939
$580$	0	0
$581$	1.49776e38	0.0124420
$582$	0	0
$583$	−1.32026e39	−0.104343
$584$	0	0
$585$	−9.70968e39	−0.730205
$586$	0	0
$587$	5.47078e39	0.391559	0.195779	−	0.980648i	$-0.437276\pi$
$587$	5.47078e39	0.391559	0.195779	+	0.980648i	$0.437276\pi$
$588$	0	0
$589$	1.19494e40	0.814095
$590$	0	0
$591$	−4.09269e40	−2.65455
$592$	0	0
$593$	2.22596e40	1.37475	0.687376	−	0.726302i	$-0.258762\pi$
$593$	2.22596e40	1.37475	0.687376	+	0.726302i	$0.258762\pi$
$594$	0	0
$595$	1.58161e38	0.00930260
$596$	0	0
$597$	−2.97348e40	−1.66586
$598$	0	0
$599$	−2.08204e39	−0.111122	−0.0555611	−	0.998455i	$-0.517695\pi$
$599$	−2.08204e39	−0.111122	−0.0555611	+	0.998455i	$0.517695\pi$
$600$	0	0
$601$	−1.47384e40	−0.749498	−0.374749	−	0.927126i	$-0.622271\pi$
$601$	−1.47384e40	−0.749498	−0.374749	+	0.927126i	$0.622271\pi$
$602$	0	0
$603$	1.48137e40	0.717895
$604$	0	0
$605$	5.90882e39	0.272928
$606$	0	0
$607$	−1.37809e40	−0.606796	−0.303398	−	0.952864i	$-0.598121\pi$
$607$	−1.37809e40	−0.606796	−0.303398	+	0.952864i	$0.598121\pi$
$608$	0	0
$609$	9.26005e38	0.0388743
$610$	0	0
$611$	2.10335e40	0.842004
$612$	0	0
$613$	1.47390e40	0.562717	0.281358	−	0.959603i	$-0.409215\pi$
$613$	1.47390e40	0.562717	0.281358	+	0.959603i	$0.409215\pi$
$614$	0	0
$615$	5.46244e38	0.0198929
$616$	0	0
$617$	4.06519e39	0.141236	0.0706181	−	0.997503i	$-0.477503\pi$
$617$	4.06519e39	0.141236	0.0706181	+	0.997503i	$0.477503\pi$
$618$	0	0
$619$	−3.64687e40	−1.20894	−0.604472	−	0.796627i	$-0.706616\pi$
$619$	−3.64687e40	−1.20894	−0.604472	+	0.796627i	$0.706616\pi$
$620$	0	0
$621$	1.72616e39	0.0546077
$622$	0	0
$623$	−2.06548e37	−0.000623655 0
$624$	0	0
$625$	1.45997e40	0.420809
$626$	0	0
$627$	2.87742e40	0.791815
$628$	0	0
$629$	−5.91450e40	−1.55412
$630$	0	0
$631$	6.88296e40	1.72723	0.863613	−	0.504155i	$-0.168196\pi$
$631$	6.88296e40	1.72723	0.863613	+	0.504155i	$0.168196\pi$
$632$	0	0
$633$	6.14565e40	1.47304
$634$	0	0
$635$	2.15574e40	0.493602
$636$	0	0
$637$	−3.55640e40	−0.778016
$638$	0	0
$639$	−2.47771e40	−0.517950
$640$	0	0
$641$	5.79208e40	1.15716	0.578580	−	0.815626i	$-0.303607\pi$
$641$	5.79208e40	1.15716	0.578580	+	0.815626i	$0.303607\pi$
$642$	0	0
$643$	6.13559e40	1.17165	0.585824	−	0.810439i	$-0.300771\pi$
$643$	6.13559e40	1.17165	0.585824	+	0.810439i	$0.300771\pi$
$644$	0	0
$645$	6.05772e40	1.10584
$646$	0	0
$647$	1.10620e40	0.193074	0.0965369	−	0.995329i	$-0.469223\pi$
$647$	1.10620e40	0.193074	0.0965369	+	0.995329i	$0.469223\pi$
$648$	0	0
$649$	7.12509e40	1.18917
$650$	0	0
$651$	2.36652e39	0.0377735
$652$	0	0
$653$	9.61992e40	1.46869	0.734346	−	0.678776i	$-0.237489\pi$
$653$	9.61992e40	1.46869	0.734346	+	0.678776i	$0.237489\pi$
$654$	0	0
$655$	−7.22276e39	−0.105488
$656$	0	0
$657$	−6.45677e40	−0.902231
$658$	0	0
$659$	−2.42717e40	−0.324535	−0.162268	−	0.986747i	$-0.551881\pi$
$659$	−2.42717e40	−0.324535	−0.162268	+	0.986747i	$0.551881\pi$
$660$	0	0
$661$	−8.66824e40	−1.10920	−0.554600	−	0.832117i	$-0.687129\pi$
$661$	−8.66824e40	−1.10920	−0.554600	+	0.832117i	$0.687129\pi$
$662$	0	0
$663$	−1.19722e41	−1.46631
$664$	0	0
$665$	−5.26802e38	−0.00617638
$666$	0	0
$667$	3.07370e39	0.0345014
$668$	0	0
$669$	−2.83791e41	−3.05014
$670$	0	0
$671$	3.66684e40	0.377412
$672$	0	0
$673$	−8.20694e40	−0.809027	−0.404513	−	0.914532i	$-0.632559\pi$
$673$	−8.20694e40	−0.809027	−0.404513	+	0.914532i	$0.632559\pi$
$674$	0	0
$675$	−1.55662e41	−1.46987
$676$	0	0
$677$	1.82761e41	1.65328	0.826640	−	0.562730i	$-0.190249\pi$
$677$	1.82761e41	1.65328	0.826640	+	0.562730i	$0.190249\pi$
$678$	0	0
$679$	4.91685e38	0.00426163
$680$	0	0
$681$	−1.77416e41	−1.47353
$682$	0	0
$683$	3.66780e39	0.0291948	0.0145974	−	0.999893i	$-0.495353\pi$
$683$	3.66780e39	0.0291948	0.0145974	+	0.999893i	$0.495353\pi$
$684$	0	0
$685$	6.97460e40	0.532115
$686$	0	0
$687$	7.35141e40	0.537648
$688$	0	0
$689$	−1.82968e40	−0.128291
$690$	0	0
$691$	−5.62493e40	−0.378169	−0.189084	−	0.981961i	$-0.560552\pi$
$691$	−5.62493e40	−0.378169	−0.189084	+	0.981961i	$0.560552\pi$
$692$	0	0
$693$	3.83638e39	0.0247337
$694$	0	0
$695$	−1.30676e40	−0.0808012
$696$	0	0
$697$	4.53429e39	0.0268927
$698$	0	0
$699$	−3.38324e41	−1.92493
$700$	0	0
$701$	4.31831e40	0.235724	0.117862	−	0.993030i	$-0.462396\pi$
$701$	4.31831e40	0.235724	0.117862	+	0.993030i	$0.462396\pi$
$702$	0	0
$703$	1.97000e41	1.03184
$704$	0	0
$705$	−1.71470e41	−0.861878
$706$	0	0
$707$	−1.66844e39	−0.00804878
$708$	0	0
$709$	3.70730e41	1.71667	0.858336	−	0.513088i	$-0.171499\pi$
$709$	3.70730e41	1.71667	0.858336	+	0.513088i	$0.171499\pi$
$710$	0	0
$711$	4.63944e41	2.06232
$712$	0	0
$713$	7.85524e39	0.0335244
$714$	0	0
$715$	−5.47522e40	−0.224369
$716$	0	0
$717$	2.45311e41	0.965361
$718$	0	0
$719$	3.14942e40	0.119031	0.0595157	−	0.998227i	$-0.481044\pi$
$719$	3.14942e40	0.119031	0.0595157	+	0.998227i	$0.481044\pi$
$720$	0	0
$721$	−4.49128e39	−0.0163045
$722$	0	0
$723$	−3.28800e41	−1.14664
$724$	0	0
$725$	−2.77180e41	−0.928670
$726$	0	0
$727$	−2.91745e41	−0.939192	−0.469596	−	0.882881i	$-0.655600\pi$
$727$	−2.91745e41	−0.939192	−0.469596	+	0.882881i	$0.655600\pi$
$728$	0	0
$729$	−2.60209e41	−0.804957
$730$	0	0
$731$	5.02842e41	1.49496
$732$	0	0
$733$	−3.81345e41	−1.08971	−0.544854	−	0.838531i	$-0.683415\pi$
$733$	−3.81345e41	−1.08971	−0.544854	+	0.838531i	$0.683415\pi$
$734$	0	0
$735$	2.89926e41	0.796379
$736$	0	0
$737$	8.35331e40	0.220587
$738$	0	0
$739$	−5.69450e41	−1.44581	−0.722905	−	0.690947i	$-0.757194\pi$
$739$	−5.69450e41	−1.44581	−0.722905	+	0.690947i	$0.757194\pi$
$740$	0	0
$741$	3.98768e41	0.973546
$742$	0	0
$743$	−2.24951e41	−0.528140	−0.264070	−	0.964503i	$-0.585065\pi$
$743$	−2.24951e41	−0.528140	−0.264070	+	0.964503i	$0.585065\pi$
$744$	0	0
$745$	2.66039e40	0.0600731
$746$	0	0
$747$	6.22224e41	1.35144
$748$	0	0
$749$	7.87693e39	0.0164578
$750$	0	0
$751$	6.45261e41	1.29705	0.648524	−	0.761194i	$-0.275386\pi$
$751$	6.45261e41	1.29705	0.648524	+	0.761194i	$0.275386\pi$
$752$	0	0
$753$	1.42432e42	2.75474
$754$	0	0
$755$	−5.79734e40	−0.107894
$756$	0	0
$757$	−8.68640e41	−1.55579	−0.777893	−	0.628397i	$-0.783711\pi$
$757$	−8.68640e41	−1.55579	−0.777893	+	0.628397i	$0.783711\pi$
$758$	0	0
$759$	1.89154e40	0.0326069
$760$	0	0
$761$	4.39407e41	0.729104	0.364552	−	0.931183i	$-0.381222\pi$
$761$	4.39407e41	0.729104	0.364552	+	0.931183i	$0.381222\pi$
$762$	0	0
$763$	2.29306e40	0.0366278
$764$	0	0
$765$	6.57058e41	1.01045
$766$	0	0
$767$	9.87434e41	1.46209
$768$	0	0
$769$	−3.86450e41	−0.551012	−0.275506	−	0.961299i	$-0.588845\pi$
$769$	−3.86450e41	−0.551012	−0.275506	+	0.961299i	$0.588845\pi$
$770$	0	0
$771$	−1.23698e42	−1.69853
$772$	0	0
$773$	3.89700e41	0.515380	0.257690	−	0.966228i	$-0.417039\pi$
$773$	3.89700e41	0.515380	0.257690	+	0.966228i	$0.417039\pi$
$774$	0	0
$775$	−7.08370e41	−0.902372
$776$	0	0
$777$	3.90148e40	0.0478768
$778$	0	0
$779$	−1.51028e40	−0.0178552
$780$	0	0
$781$	−1.39716e41	−0.159150
$782$	0	0
$783$	1.97961e42	2.17287
$784$	0	0
$785$	−1.60870e41	−0.170163
$786$	0	0
$787$	−8.84928e41	−0.902140	−0.451070	−	0.892488i	$-0.648958\pi$
$787$	−8.84928e41	−0.902140	−0.451070	+	0.892488i	$0.648958\pi$
$788$	0	0
$789$	−3.01297e42	−2.96059
$790$	0	0
$791$	1.73624e39	0.00164456
$792$	0	0
$793$	5.08171e41	0.464032
$794$	0	0
$795$	1.49160e41	0.131319
$796$	0	0
$797$	−6.26660e41	−0.531969	−0.265985	−	0.963977i	$-0.585697\pi$
$797$	−6.26660e41	−0.531969	−0.265985	+	0.963977i	$0.585697\pi$
$798$	0	0
$799$	−1.42335e42	−1.16515
$800$	0	0
$801$	−8.58073e40	−0.0677412
$802$	0	0
$803$	−3.64093e41	−0.277228
$804$	0	0
$805$	−3.46306e38	−0.000254343 0
$806$	0	0
$807$	−3.40392e42	−2.41164
$808$	0	0
$809$	−8.71861e41	−0.595928	−0.297964	−	0.954577i	$-0.596308\pi$
$809$	−8.71861e41	−0.595928	−0.297964	+	0.954577i	$0.596308\pi$
$810$	0	0
$811$	−9.09697e41	−0.599922	−0.299961	−	0.953952i	$-0.596974\pi$
$811$	−9.09697e41	−0.599922	−0.299961	+	0.953952i	$0.596974\pi$
$812$	0	0
$813$	1.27958e42	0.814245
$814$	0	0
$815$	7.43865e40	0.0456782
$816$	0	0
$817$	−1.67486e42	−0.992565
$818$	0	0
$819$	5.31666e40	0.0304104
$820$	0	0
$821$	1.52548e42	0.842231	0.421115	−	0.907007i	$-0.361639\pi$
$821$	1.52548e42	0.842231	0.421115	+	0.907007i	$0.361639\pi$
$822$	0	0
$823$	1.67746e42	0.894037	0.447019	−	0.894525i	$-0.352486\pi$
$823$	1.67746e42	0.894037	0.447019	+	0.894525i	$0.352486\pi$
$824$	0	0
$825$	−1.70575e42	−0.877676
$826$	0	0
$827$	3.12102e42	1.55048	0.775242	−	0.631665i	$-0.217628\pi$
$827$	3.12102e42	1.55048	0.775242	+	0.631665i	$0.217628\pi$
$828$	0	0
$829$	−1.01143e42	−0.485174	−0.242587	−	0.970130i	$-0.577996\pi$
$829$	−1.01143e42	−0.485174	−0.242587	+	0.970130i	$0.577996\pi$
$830$	0	0
$831$	−1.08501e42	−0.502595
$832$	0	0
$833$	2.40663e42	1.07661
$834$	0	0
$835$	6.70050e41	0.289503
$836$	0	0
$837$	5.05915e42	2.11134
$838$	0	0
$839$	−1.10356e42	−0.444887	−0.222443	−	0.974946i	$-0.571403\pi$
$839$	−1.10356e42	−0.444887	−0.222443	+	0.974946i	$0.571403\pi$
$840$	0	0
$841$	9.57314e41	0.372832
$842$	0	0
$843$	5.49860e42	2.06896
$844$	0	0
$845$	4.93863e41	0.179550
$846$	0	0
$847$	−3.23545e40	−0.0113665
$848$	0	0
$849$	−7.59106e42	−2.57716
$850$	0	0
$851$	1.29502e41	0.0424912
$852$	0	0
$853$	1.80886e42	0.573646	0.286823	−	0.957984i	$-0.407401\pi$
$853$	1.80886e42	0.573646	0.286823	+	0.957984i	$0.407401\pi$
$854$	0	0
$855$	−2.18852e42	−0.670876
$856$	0	0
$857$	3.58273e42	1.06168	0.530838	−	0.847473i	$-0.321877\pi$
$857$	3.58273e42	1.06168	0.530838	+	0.847473i	$0.321877\pi$
$858$	0	0
$859$	−3.55534e42	−1.01855	−0.509273	−	0.860605i	$-0.670086\pi$
$859$	−3.55534e42	−1.01855	−0.509273	+	0.860605i	$0.670086\pi$
$860$	0	0
$861$	−2.99103e39	−0.000828467 0
$862$	0	0
$863$	5.78959e42	1.55057	0.775285	−	0.631612i	$-0.217606\pi$
$863$	5.78959e42	1.55057	0.775285	+	0.631612i	$0.217606\pi$
$864$	0	0
$865$	−2.37553e42	−0.615216
$866$	0	0
$867$	1.11695e42	0.279740
$868$	0	0
$869$	2.61614e42	0.633687
$870$	0	0
$871$	1.15765e42	0.271215
$872$	0	0
$873$	2.04263e42	0.462897
$874$	0	0
$875$	7.06303e40	0.0154837
$876$	0	0
$877$	−1.74620e42	−0.370340	−0.185170	−	0.982707i	$-0.559284\pi$
$877$	−1.74620e42	−0.370340	−0.185170	+	0.982707i	$0.559284\pi$
$878$	0	0
$879$	−1.29265e43	−2.65243
$880$	0	0
$881$	6.11305e42	1.21369	0.606844	−	0.794821i	$-0.292435\pi$
$881$	6.11305e42	1.21369	0.606844	+	0.794821i	$0.292435\pi$
$882$	0	0
$883$	5.67511e42	1.09029	0.545147	−	0.838340i	$-0.316474\pi$
$883$	5.67511e42	1.09029	0.545147	+	0.838340i	$0.316474\pi$
$884$	0	0
$885$	−8.04978e42	−1.49660
$886$	0	0
$887$	−1.28948e42	−0.232019	−0.116009	−	0.993248i	$-0.537010\pi$
$887$	−1.28948e42	−0.232019	−0.116009	+	0.993248i	$0.537010\pi$
$888$	0	0
$889$	−1.18040e41	−0.0205568
$890$	0	0
$891$	4.44615e42	0.749475
$892$	0	0
$893$	4.74087e42	0.773592
$894$	0	0
$895$	−3.25715e42	−0.514521
$896$	0	0
$897$	2.62140e41	0.0400906
$898$	0	0
$899$	9.00859e42	1.33396
$900$	0	0
$901$	1.23815e42	0.177527
$902$	0	0
$903$	−3.31698e41	−0.0460543
$904$	0	0
$905$	4.68894e42	0.630478
$906$	0	0
$907$	6.85671e42	0.892915	0.446457	−	0.894805i	$-0.352685\pi$
$907$	6.85671e42	0.892915	0.446457	+	0.894805i	$0.352685\pi$
$908$	0	0
$909$	−6.93130e42	−0.874255
$910$	0	0
$911$	6.18262e42	0.755363	0.377682	−	0.925936i	$-0.376721\pi$
$911$	6.18262e42	0.755363	0.377682	+	0.925936i	$0.376721\pi$
$912$	0	0
$913$	3.50868e42	0.415257
$914$	0	0
$915$	−4.14272e42	−0.474985
$916$	0	0
$917$	3.95491e40	0.00439321
$918$	0	0
$919$	−7.15141e42	−0.769691	−0.384846	−	0.922981i	$-0.625745\pi$
$919$	−7.15141e42	−0.769691	−0.384846	+	0.922981i	$0.625745\pi$
$920$	0	0
$921$	−9.19938e42	−0.959387
$922$	0	0
$923$	−1.93626e42	−0.195677
$924$	0	0
$925$	−1.16783e43	−1.14373
$926$	0	0
$927$	−1.86583e43	−1.77099
$928$	0	0
$929$	1.19347e43	1.09795	0.548975	−	0.835839i	$-0.315018\pi$
$929$	1.19347e43	1.09795	0.548975	+	0.835839i	$0.315018\pi$
$930$	0	0
$931$	−8.01598e42	−0.714803
$932$	0	0
$933$	−2.60363e43	−2.25058
$934$	0	0
$935$	3.70510e42	0.310479
$936$	0	0
$937$	−4.02931e42	−0.327345	−0.163673	−	0.986515i	$-0.552334\pi$
$937$	−4.02931e42	−0.327345	−0.163673	+	0.986515i	$0.552334\pi$
$938$	0	0
$939$	−3.89442e43	−3.06755
$940$	0	0
$941$	9.40592e42	0.718376	0.359188	−	0.933265i	$-0.383054\pi$
$941$	9.40592e42	0.718376	0.359188	+	0.933265i	$0.383054\pi$
$942$	0	0
$943$	−9.92816e39	−0.000735274 0
$944$	0	0
$945$	−2.23038e41	−0.0160183
$946$	0	0
$947$	1.85611e43	1.29279	0.646397	−	0.763001i	$-0.276275\pi$
$947$	1.85611e43	1.29279	0.646397	+	0.763001i	$0.276275\pi$
$948$	0	0
$949$	−5.04579e42	−0.340855
$950$	0	0
$951$	1.49276e43	0.978078
$952$	0	0
$953$	−3.20332e42	−0.203588	−0.101794	−	0.994805i	$-0.532458\pi$
$953$	−3.20332e42	−0.203588	−0.101794	+	0.994805i	$0.532458\pi$
$954$	0	0
$955$	−6.71420e42	−0.413948
$956$	0	0
$957$	2.16927e43	1.29745
$958$	0	0
$959$	−3.81903e41	−0.0221607
$960$	0	0
$961$	5.26074e42	0.296181
$962$	0	0
$963$	3.27235e43	1.78764
$964$	0	0
$965$	−1.18820e43	−0.629860
$966$	0	0
$967$	−5.23025e42	−0.269053	−0.134527	−	0.990910i	$-0.542951\pi$
$967$	−5.23025e42	−0.269053	−0.134527	+	0.990910i	$0.542951\pi$
$968$	0	0
$969$	−2.69848e43	−1.34718
$970$	0	0
$971$	2.58906e43	1.25448	0.627239	−	0.778827i	$-0.284185\pi$
$971$	2.58906e43	1.25448	0.627239	+	0.778827i	$0.284185\pi$
$972$	0	0
$973$	7.15536e40	0.00336508
$974$	0	0
$975$	−2.36393e43	−1.07911
$976$	0	0
$977$	6.80879e42	0.301716	0.150858	−	0.988555i	$-0.451796\pi$
$977$	6.80879e42	0.301716	0.150858	+	0.988555i	$0.451796\pi$
$978$	0	0
$979$	−4.83861e41	−0.0208148
$980$	0	0
$981$	9.52620e43	3.97850
$982$	0	0
$983$	4.18545e43	1.69713	0.848566	−	0.529089i	$-0.177466\pi$
$983$	4.18545e43	1.69713	0.848566	+	0.529089i	$0.177466\pi$
$984$	0	0
$985$	1.75531e43	0.691080
$986$	0	0
$987$	9.38906e41	0.0358941
$988$	0	0
$989$	−1.10101e42	−0.0408738
$990$	0	0
$991$	−2.63339e43	−0.949397	−0.474699	−	0.880148i	$-0.657443\pi$
$991$	−2.63339e43	−0.949397	−0.474699	+	0.880148i	$0.657443\pi$
$992$	0	0
$993$	−2.28237e43	−0.799138
$994$	0	0
$995$	1.27529e43	0.433686
$996$	0	0
$997$	−1.61216e43	−0.532511	−0.266255	−	0.963903i	$-0.585786\pi$
$997$	−1.61216e43	−0.532511	−0.266255	+	0.963903i	$0.585786\pi$
$998$	0	0
$999$	8.34058e43	2.67606

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.30.a.c.1.2		2
4.3	odd	2		1.30.a.a.1.2	✓	2
12.11	even	2		9.30.a.a.1.1		2
20.3	even	4		25.30.b.a.24.1		4
20.7	even	4		25.30.b.a.24.4		4
20.19	odd	2		25.30.a.a.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.30.a.a.1.2	✓	2	4.3	odd	2
9.30.a.a.1.1		2	12.11	even	2
16.30.a.c.1.2		2	1.1	even	1	trivial
25.30.a.a.1.1		2	20.19	odd	2
25.30.b.a.24.1		4	20.3	even	4
25.30.b.a.24.4		4	20.7	even	4

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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 30 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q+1.44920e7 q^{3} -6.21544e9 q^{5} +3.40335e10 q^{7} +1.41387e14 q^{9} +O(q^{10})\)	\(q+1.44920e7 q^{3} -6.21544e9 q^{5} +3.40335e10 q^{7} +1.41387e14 q^{9} +7.97271e14 q^{11} +1.10490e16 q^{13} -9.00740e16 q^{15} -7.47691e17 q^{17} +2.49040e18 q^{19} +4.93212e17 q^{21} +1.63712e18 q^{23} -1.47633e20 q^{25} +1.05439e21 q^{27} +1.87750e21 q^{29} +4.79819e21 q^{31} +1.15540e22 q^{33} -2.11533e20 q^{35} +7.91036e22 q^{37} +1.60122e23 q^{39} -6.06439e21 q^{41} -6.72526e23 q^{43} -8.78782e23 q^{45} +1.90366e24 q^{47} -3.21875e24 q^{49} -1.08355e25 q^{51} -1.65597e24 q^{53} -4.95539e24 q^{55} +3.60909e25 q^{57} +8.93685e25 q^{59} +4.59924e25 q^{61} +4.81189e24 q^{63} -6.86745e25 q^{65} +1.04774e26 q^{67} +2.37252e25 q^{69} -1.75243e26 q^{71} -4.56674e26 q^{73} -2.13949e27 q^{75} +2.71339e25 q^{77} +3.28137e27 q^{79} +5.57671e27 q^{81} +4.40086e27 q^{83} +4.64723e27 q^{85} +2.72087e28 q^{87} -6.06897e26 q^{89} +3.76036e26 q^{91} +6.95352e28 q^{93} -1.54790e28 q^{95} +1.44471e28 q^{97} +1.12724e29 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4967640 q^{3} - 17477788500 q^{5} + 3020312682800 q^{7} + 163469569523706 q^{9}+O(q^{10}) \) 2 * q + 4967640 * q^3 - 17477788500 * q^5 + 3020312682800 * q^7 + 163469569523706 * q^9	\( 2 q + 4967640 q^{3} - 17477788500 q^{5} + 3020312682800 q^{7} + 163469569523706 q^{9} + 20\!\cdots\!16 q^{11}+ \cdots + 14\!\cdots\!48 q^{99}+O(q^{100}) \) 2 * q + 4967640 * q^3 - 17477788500 * q^5 + 3020312682800 * q^7 + 163469569523706 * q^9 + 2055380519588616 * q^11 + 17139371179332220 * q^13 + 17192351392506000 * q^15 - 664795927907749980 * q^17 - 1232169445452155080 * q^19 - 27949113476248821696 * q^21 + 18588430504374379920 * q^23 - 207056854502617281250 * q^25 + 1497723629507205824880 * q^27 + 996096417765761595420 * q^29 + 1087258664181924030656 * q^31 - 428627571739162757280 * q^33 - 33844046536902649068000 * q^35 + 98878127759336138311660 * q^37 + 102115380027539490485328 * q^39 - 106699837259610025757676 * q^41 - 510640433114974031366200 * q^43 - 1127484267514124881450500 * q^45 + 4521107671109825795039520 * q^47 + 2479210387392854586312114 * q^49 - 11625042659032475755122384 * q^51 - 16141544629666790250663540 * q^53 - 19124657798421120971058000 * q^55 + 71545878452212360967003040 * q^57 + 83497291515660150276062760 * q^59 - 15876762943230179624113316 * q^61 + 70756669640265175617870960 * q^63 - 137266226551726798673511000 * q^65 - 24468754434128844890782120 * q^67 - 137724750477330905859409728 * q^69 + 188428428253360067209577136 * q^71 + 995872721297568533052080980 * q^73 - 1573516410323456985690375000 * q^75 + 3784200478484579386131729600 * q^77 + 7218460852587735797823642080 * q^79 - 161317068789001769884865358 * q^81 - 2210846225248155523694669640 * q^83 + 3713635892367734904614943000 * q^85 + 35603430680717182696869516240 * q^87 + 5872964746600969363962031860 * q^89 + 18563550073842319069958420896 * q^91 + 104879397265778648689061310720 * q^93 + 26445947767121183571013410000 * q^95 + 109649982219471997093227005380 * q^97 + 140506044809020243496724809448 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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