LMFDB - Embedded newform 32.7.d.b.15.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,7,Mod(15,32)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("32.15");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	32.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$7.36173067584$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.3803625.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 6x^{2} - 16x + 256 $ x^4 - x^3 + 6x^2 - 16x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			15.1
Root			$2.81174 - 2.84502i$ of defining polynomial
Character	$\chi$	$=$	32.15
Dual form			32.7.d.b.15.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.49390 q^{3} -59.7107i q^{5} +483.584i q^{7} -656.854 q^{9} +O(q^{10})$	$q-8.49390 q^{3} -59.7107i q^{5} +483.584i q^{7} -656.854 q^{9} -1412.15 q^{11} +3450.70i q^{13} +507.177i q^{15} -3056.78 q^{17} -968.104 q^{19} -4107.51i q^{21} +3314.31i q^{23} +12059.6 q^{25} +11771.3 q^{27} -26351.6i q^{29} -27104.3i q^{31} +11994.7 q^{33} +28875.1 q^{35} +36097.0i q^{37} -29309.9i q^{39} -6860.73 q^{41} -92831.6 q^{43} +39221.2i q^{45} +159323. i q^{47} -116204. q^{49} +25964.0 q^{51} -86612.3i q^{53} +84320.6i q^{55} +8222.98 q^{57} -128806. q^{59} +189486. i q^{61} -317644. i q^{63} +206043. q^{65} +319835. q^{67} -28151.4i q^{69} -196890. i q^{71} -63957.9 q^{73} -102433. q^{75} -682894. i q^{77} +164678. i q^{79} +378862. q^{81} +802946. q^{83} +182522. i q^{85} +223828. i q^{87} -54145.3 q^{89} -1.66870e6 q^{91} +230221. i q^{93} +57806.1i q^{95} -1.10670e6 q^{97} +927577. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 48 q^{3} - 660 q^{9}+O(q^{10}) $ 4 * q + 48 * q^3 - 660 * q^9	$ 4 q + 48 q^{3} - 660 q^{9} - 976 q^{11} + 4168 q^{17} + 1456 q^{19} - 23900 q^{25} - 2592 q^{27} + 84048 q^{33} + 49920 q^{35} - 117944 q^{41} - 197456 q^{43} + 2116 q^{49} + 386016 q^{51} + 126672 q^{57} - 542032 q^{59} - 205440 q^{65} + 790192 q^{67} + 443912 q^{73} - 1765200 q^{75} - 568044 q^{81} + 3465008 q^{83} + 761224 q^{89} - 3398400 q^{91} - 926776 q^{97} + 2459280 q^{99}+O(q^{100}) $ 4 * q + 48 * q^3 - 660 * q^9 - 976 * q^11 + 4168 * q^17 + 1456 * q^19 - 23900 * q^25 - 2592 * q^27 + 84048 * q^33 + 49920 * q^35 - 117944 * q^41 - 197456 * q^43 + 2116 * q^49 + 386016 * q^51 + 126672 * q^57 - 542032 * q^59 - 205440 * q^65 + 790192 * q^67 + 443912 * q^73 - 1765200 * q^75 - 568044 * q^81 + 3465008 * q^83 + 761224 * q^89 - 3398400 * q^91 - 926776 * q^97 + 2459280 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$.

$n$	$5$	$31$
$\chi(n)$	$-1$	$-1$

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
$3$	−8.49390	−0.314589	−0.157294	−	0.987552i	$-0.550277\pi$
$3$	−8.49390	−0.314589	−0.157294	+	0.987552i	$0.550277\pi$
$4$	0	0
$5$	− 59.7107i	− 0.477686i	−0.971058	−	0.238843i	$-0.923232\pi$
$5$	− 59.7107i	− 0.477686i	0.971058	−	0.238843i	$-0.0767681\pi$
$6$	0	0
$7$	483.584i	1.40987i	0.709274	+	0.704933i	$0.249023\pi$
$7$	483.584i	1.40987i	−0.709274	+	0.704933i	$0.750977\pi$
$8$	0	0
$9$	−656.854	−0.901034
$10$	0	0
$11$	−1412.15	−1.06097	−0.530485	−	0.847694i	$-0.677990\pi$
$11$	−1412.15	−1.06097	−0.530485	+	0.847694i	$0.677990\pi$
$12$	0	0
$13$	3450.70i	1.57064i	0.619090	+	0.785320i	$0.287501\pi$
$13$	3450.70i	1.57064i	−0.619090	+	0.785320i	$0.712499\pi$
$14$	0	0
$15$	507.177i	0.150275i
$16$	0	0
$17$	−3056.78	−0.622182	−0.311091	−	0.950380i	$-0.600694\pi$
$17$	−3056.78	−0.622182	−0.311091	+	0.950380i	$0.600694\pi$
$18$	0	0
$19$	−968.104	−0.141144	−0.0705718	−	0.997507i	$-0.522482\pi$
$19$	−968.104	−0.141144	−0.0705718	+	0.997507i	$0.522482\pi$
$20$	0	0
$21$	− 4107.51i	− 0.443528i
$22$	0	0
$23$	3314.31i	0.272401i	0.990681	+	0.136201i	$0.0434892\pi$
$23$	3314.31i	0.272401i	−0.990681	+	0.136201i	$0.956511\pi$
$24$	0	0
$25$	12059.6	0.771817
$26$	0	0
$27$	11771.3	0.598044
$28$	0	0
$29$	− 26351.6i	− 1.08047i	−0.841513	−	0.540236i	$-0.818335\pi$
$29$	− 26351.6i	− 1.08047i	0.841513	−	0.540236i	$-0.181665\pi$
$30$	0	0
$31$	− 27104.3i	− 0.909815i	−0.890539	−	0.454907i	$-0.849672\pi$
$31$	− 27104.3i	− 0.909815i	0.890539	−	0.454907i	$-0.150328\pi$
$32$	0	0
$33$	11994.7	0.333770
$34$	0	0
$35$	28875.1	0.673472
$36$	0	0
$37$	36097.0i	0.712633i	0.934365	+	0.356317i	$0.115968\pi$
$37$	36097.0i	0.712633i	−0.934365	+	0.356317i	$0.884032\pi$
$38$	0	0
$39$	− 29309.9i	− 0.494106i
$40$	0	0
$41$	−6860.73	−0.0995449	−0.0497724	−	0.998761i	$-0.515850\pi$
$41$	−6860.73	−0.0995449	−0.0497724	+	0.998761i	$0.515850\pi$
$42$	0	0
$43$	−92831.6	−1.16759	−0.583795	−	0.811901i	$-0.698433\pi$
$43$	−92831.6	−1.16759	−0.583795	+	0.811901i	$0.698433\pi$
$44$	0	0
$45$	39221.2i	0.430411i
$46$	0	0
$47$	159323.i	1.53456i	0.641309	+	0.767282i	$0.278392\pi$
$47$	159323.i	1.53456i	−0.641309	+	0.767282i	$0.721608\pi$
$48$	0	0
$49$	−116204.	−0.987720
$50$	0	0
$51$	25964.0	0.195732
$52$	0	0
$53$	− 86612.3i	− 0.581771i	−0.956758	−	0.290885i	$-0.906050\pi$
$53$	− 86612.3i	− 0.581771i	0.956758	−	0.290885i	$-0.0939498\pi$
$54$	0	0
$55$	84320.6i	0.506810i
$56$	0	0
$57$	8222.98	0.0444022
$58$	0	0
$59$	−128806.	−0.627165	−0.313582	−	0.949561i	$-0.601529\pi$
$59$	−128806.	−0.627165	−0.313582	+	0.949561i	$0.601529\pi$
$60$	0	0
$61$	189486.i	0.834810i	0.908721	+	0.417405i	$0.137060\pi$
$61$	189486.i	0.834810i	−0.908721	+	0.417405i	$0.862940\pi$
$62$	0	0
$63$	− 317644.i	− 1.27034i
$64$	0	0
$65$	206043.	0.750272
$66$	0	0
$67$	319835.	1.06341	0.531706	−	0.846929i	$-0.321551\pi$
$67$	319835.	1.06341	0.531706	+	0.846929i	$0.321551\pi$
$68$	0	0
$69$	− 28151.4i	− 0.0856945i
$70$	0	0
$71$	− 196890.i	− 0.550108i	−0.961429	−	0.275054i	$-0.911304\pi$
$71$	− 196890.i	− 0.550108i	0.961429	−	0.275054i	$-0.0886957\pi$
$72$	0	0
$73$	−63957.9	−0.164409	−0.0822046	−	0.996615i	$-0.526196\pi$
$73$	−63957.9	−0.164409	−0.0822046	+	0.996615i	$0.526196\pi$
$74$	0	0
$75$	−102433.	−0.242805
$76$	0	0
$77$	− 682894.i	− 1.49583i
$78$	0	0
$79$	164678.i	0.334006i	0.985956	+	0.167003i	$0.0534090\pi$
$79$	164678.i	0.334006i	−0.985956	+	0.167003i	$0.946591\pi$
$80$	0	0
$81$	378862.	0.712896
$82$	0	0
$83$	802946.	1.40428	0.702138	−	0.712041i	$-0.252229\pi$
$83$	802946.	1.40428	0.702138	+	0.712041i	$0.252229\pi$
$84$	0	0
$85$	182522.i	0.297207i
$86$	0	0
$87$	223828.i	0.339905i
$88$	0	0
$89$	−54145.3	−0.0768052	−0.0384026	−	0.999262i	$-0.512227\pi$
$89$	−54145.3	−0.0768052	−0.0384026	+	0.999262i	$0.512227\pi$
$90$	0	0
$91$	−1.66870e6	−2.21439
$92$	0	0
$93$	230221.i	0.286218i
$94$	0	0
$95$	57806.1i	0.0674222i
$96$	0	0
$97$	−1.10670e6	−1.21259	−0.606297	−	0.795238i	$-0.707346\pi$
$97$	−1.10670e6	−1.21259	−0.606297	+	0.795238i	$0.707346\pi$
$98$	0	0
$99$	927577.	0.955971
$100$	0	0
$101$	809583.i	0.785773i	0.919587	+	0.392887i	$0.128524\pi$
$101$	809583.i	0.785773i	−0.919587	+	0.392887i	$0.871476\pi$
$102$	0	0
$103$	619448.i	0.566883i	0.958990	+	0.283442i	$0.0914762\pi$
$103$	619448.i	0.566883i	−0.958990	+	0.283442i	$0.908524\pi$
$104$	0	0
$105$	−245262.	−0.211867
$106$	0	0
$107$	−1.08619e6	−0.886657	−0.443328	−	0.896359i	$-0.646202\pi$
$107$	−1.08619e6	−0.886657	−0.443328	+	0.896359i	$0.646202\pi$
$108$	0	0
$109$	1.71692e6i	1.32578i	0.748717	+	0.662890i	$0.230670\pi$
$109$	1.71692e6i	1.32578i	−0.748717	+	0.662890i	$0.769330\pi$
$110$	0	0
$111$	− 306604.i	− 0.224187i
$112$	0	0
$113$	645423.	0.447311	0.223655	−	0.974668i	$-0.428201\pi$
$113$	645423.	0.447311	0.223655	+	0.974668i	$0.428201\pi$
$114$	0	0
$115$	197900.	0.130122
$116$	0	0
$117$	− 2.26660e6i	− 1.41520i
$118$	0	0
$119$	− 1.47821e6i	− 0.877193i
$120$	0	0
$121$	222613.	0.125659
$122$	0	0
$123$	58274.4	0.0313157
$124$	0	0
$125$	− 1.65307e6i	− 0.846371i
$126$	0	0
$127$	2.51504e6i	1.22781i	0.789378	+	0.613907i	$0.210403\pi$
$127$	2.51504e6i	1.22781i	−0.789378	+	0.613907i	$0.789597\pi$
$128$	0	0
$129$	788502.	0.367311
$130$	0	0
$131$	3.77470e6	1.67907	0.839536	−	0.543305i	$-0.182827\pi$
$131$	3.77470e6	1.67907	0.839536	+	0.543305i	$0.182827\pi$
$132$	0	0
$133$	− 468159.i	− 0.198993i
$134$	0	0
$135$	− 702873.i	− 0.285677i
$136$	0	0
$137$	1.40637e6	0.546939	0.273470	−	0.961881i	$-0.411829\pi$
$137$	1.40637e6	0.546939	0.273470	+	0.961881i	$0.411829\pi$
$138$	0	0
$139$	−2.31561e6	−0.862227	−0.431114	−	0.902298i	$-0.641879\pi$
$139$	−2.31561e6	−0.862227	−0.431114	+	0.902298i	$0.641879\pi$
$140$	0	0
$141$	− 1.35327e6i	− 0.482757i
$142$	0	0
$143$	− 4.87291e6i	− 1.66640i
$144$	0	0
$145$	−1.57347e6	−0.516126
$146$	0	0
$147$	987028.	0.310726
$148$	0	0
$149$	4.74866e6i	1.43553i	0.696285	+	0.717765i	$0.254835\pi$
$149$	4.74866e6i	1.43553i	−0.696285	+	0.717765i	$0.745165\pi$
$150$	0	0
$151$	1.11787e6i	0.324684i	0.986735	+	0.162342i	$0.0519047\pi$
$151$	1.11787e6i	0.324684i	−0.986735	+	0.162342i	$0.948095\pi$
$152$	0	0
$153$	2.00786e6	0.560607
$154$	0	0
$155$	−1.61842e6	−0.434605
$156$	0	0
$157$	1.00407e6i	0.259458i	0.991550	+	0.129729i	$0.0414107\pi$
$157$	1.00407e6i	0.259458i	−0.991550	+	0.129729i	$0.958589\pi$
$158$	0	0
$159$	735676.i	0.183019i
$160$	0	0
$161$	−1.60275e6	−0.384049
$162$	0	0
$163$	−1.47315e6	−0.340161	−0.170080	−	0.985430i	$-0.554403\pi$
$163$	−1.47315e6	−0.340161	−0.170080	+	0.985430i	$0.554403\pi$
$164$	0	0
$165$	− 716211.i	− 0.159437i
$166$	0	0
$167$	5.16350e6i	1.10865i	0.832300	+	0.554326i	$0.187024\pi$
$167$	5.16350e6i	1.10865i	−0.832300	+	0.554326i	$0.812976\pi$
$168$	0	0
$169$	−7.08049e6	−1.46691
$170$	0	0
$171$	635902.	0.127175
$172$	0	0
$173$	− 600245.i	− 0.115928i	−0.998319	−	0.0579642i	$-0.981539\pi$
$173$	− 600245.i	− 0.115928i	0.998319	−	0.0579642i	$-0.0184609\pi$
$174$	0	0
$175$	5.83184e6i	1.08816i
$176$	0	0
$177$	1.09407e6	0.197299
$178$	0	0
$179$	−6.34654e6	−1.10657	−0.553284	−	0.832993i	$-0.686625\pi$
$179$	−6.34654e6	−1.10657	−0.553284	+	0.832993i	$0.686625\pi$
$180$	0	0
$181$	− 1.35031e6i	− 0.227719i	−0.993497	−	0.113859i	$-0.963679\pi$
$181$	− 1.35031e6i	− 0.227719i	0.993497	−	0.113859i	$-0.0363213\pi$
$182$	0	0
$183$	− 1.60947e6i	− 0.262622i
$184$	0	0
$185$	2.15538e6	0.340415
$186$	0	0
$187$	4.31664e6	0.660117
$188$	0	0
$189$	5.69241e6i	0.843162i
$190$	0	0
$191$	− 3.32043e6i	− 0.476534i	−0.971200	−	0.238267i	$-0.923421\pi$
$191$	− 3.32043e6i	− 0.476534i	0.971200	−	0.238267i	$-0.0765794\pi$
$192$	0	0
$193$	5.59202e6	0.777852	0.388926	−	0.921269i	$-0.372846\pi$
$193$	5.59202e6	0.777852	0.388926	+	0.921269i	$0.372846\pi$
$194$	0	0
$195$	−1.75011e6	−0.236027
$196$	0	0
$197$	− 7.19799e6i	− 0.941483i	−0.882271	−	0.470742i	$-0.843986\pi$
$197$	− 7.19799e6i	− 0.941483i	0.882271	−	0.470742i	$-0.156014\pi$
$198$	0	0
$199$	− 1.15838e7i	− 1.46991i	−0.678116	−	0.734955i	$-0.737203\pi$
$199$	− 1.15838e7i	− 1.46991i	0.678116	−	0.734955i	$-0.262797\pi$
$200$	0	0
$201$	−2.71665e6	−0.334538
$202$	0	0
$203$	1.27432e7	1.52332
$204$	0	0
$205$	409659.i	0.0475511i
$206$	0	0
$207$	− 2.17701e6i	− 0.245443i
$208$	0	0
$209$	1.36711e6	0.149749
$210$	0	0
$211$	639344.	0.0680592	0.0340296	−	0.999421i	$-0.489166\pi$
$211$	639344.	0.0680592	0.0340296	+	0.999421i	$0.489166\pi$
$212$	0	0
$213$	1.67236e6i	0.173058i
$214$	0	0
$215$	5.54304e6i	0.557741i
$216$	0	0
$217$	1.31072e7	1.28272
$218$	0	0
$219$	543252.	0.0517213
$220$	0	0
$221$	− 1.05480e7i	− 0.977224i
$222$	0	0
$223$	− 9.51144e6i	− 0.857693i	−0.903377	−	0.428846i	$-0.858920\pi$
$223$	− 9.51144e6i	− 0.857693i	0.903377	−	0.428846i	$-0.141080\pi$
$224$	0	0
$225$	−7.92141e6	−0.695433
$226$	0	0
$227$	−1.27875e7	−1.09322	−0.546610	−	0.837387i	$-0.684082\pi$
$227$	−1.27875e7	−1.09322	−0.546610	+	0.837387i	$0.684082\pi$
$228$	0	0
$229$	4.61930e6i	0.384653i	0.981331	+	0.192327i	$0.0616033\pi$
$229$	4.61930e6i	0.384653i	−0.981331	+	0.192327i	$0.938397\pi$
$230$	0	0
$231$	5.80043e6i	0.470570i
$232$	0	0
$233$	−9.41375e6	−0.744209	−0.372105	−	0.928191i	$-0.621364\pi$
$233$	−9.41375e6	−0.744209	−0.372105	+	0.928191i	$0.621364\pi$
$234$	0	0
$235$	9.51329e6	0.733039
$236$	0	0
$237$	− 1.39876e6i	− 0.105075i
$238$	0	0
$239$	2.57043e7i	1.88283i	0.337247	+	0.941416i	$0.390504\pi$
$239$	2.57043e7i	1.88283i	−0.337247	+	0.941416i	$0.609496\pi$
$240$	0	0
$241$	8.16838e6	0.583559	0.291779	−	0.956486i	$-0.405753\pi$
$241$	8.16838e6	0.583559	0.291779	+	0.956486i	$0.405753\pi$
$242$	0	0
$243$	−1.17993e7	−0.822313
$244$	0	0
$245$	6.93864e6i	0.471820i
$246$	0	0
$247$	− 3.34063e6i	− 0.221686i
$248$	0	0
$249$	−6.82015e6	−0.441769
$250$	0	0
$251$	8.44540e6	0.534071	0.267035	−	0.963687i	$-0.413956\pi$
$251$	8.44540e6	0.534071	0.267035	+	0.963687i	$0.413956\pi$
$252$	0	0
$253$	− 4.68031e6i	− 0.289010i
$254$	0	0
$255$	− 1.55033e6i	− 0.0934981i
$256$	0	0
$257$	−1.84481e7	−1.08681	−0.543403	−	0.839472i	$-0.682864\pi$
$257$	−1.84481e7	−1.08681	−0.543403	+	0.839472i	$0.682864\pi$
$258$	0	0
$259$	−1.74559e7	−1.00472
$260$	0	0
$261$	1.73092e7i	0.973542i
$262$	0	0
$263$	− 3.84095e6i	− 0.211140i	−0.994412	−	0.105570i	$-0.966333\pi$
$263$	− 3.84095e6i	− 0.211140i	0.994412	−	0.105570i	$-0.0336668\pi$
$264$	0	0
$265$	−5.17168e6	−0.277903
$266$	0	0
$267$	459904.	0.0241621
$268$	0	0
$269$	3.10430e7i	1.59480i	0.603451	+	0.797400i	$0.293792\pi$
$269$	3.10430e7i	1.59480i	−0.603451	+	0.797400i	$0.706208\pi$
$270$	0	0
$271$	− 2.35560e7i	− 1.18357i	−0.806097	−	0.591784i	$-0.798424\pi$
$271$	− 2.35560e7i	− 1.18357i	0.806097	−	0.591784i	$-0.201576\pi$
$272$	0	0
$273$	1.41738e7	0.696623
$274$	0	0
$275$	−1.70300e7	−0.818875
$276$	0	0
$277$	− 2.32189e7i	− 1.09245i	−0.837638	−	0.546225i	$-0.816064\pi$
$277$	− 2.32189e7i	− 1.09245i	0.837638	−	0.546225i	$-0.183936\pi$
$278$	0	0
$279$	1.78035e7i	0.819774i
$280$	0	0
$281$	2.99519e7	1.34991	0.674955	−	0.737859i	$-0.264163\pi$
$281$	2.99519e7	1.34991	0.674955	+	0.737859i	$0.264163\pi$
$282$	0	0
$283$	3.68036e7	1.62380	0.811898	−	0.583800i	$-0.198435\pi$
$283$	3.68036e7	1.62380	0.811898	+	0.583800i	$0.198435\pi$
$284$	0	0
$285$	− 491000.i	− 0.0212103i
$286$	0	0
$287$	− 3.31774e6i	− 0.140345i
$288$	0	0
$289$	−1.47937e7	−0.612890
$290$	0	0
$291$	9.40021e6	0.381469
$292$	0	0
$293$	− 3.36919e7i	− 1.33944i	−0.742615	−	0.669718i	$-0.766415\pi$
$293$	− 3.36919e7i	− 1.33944i	0.742615	−	0.669718i	$-0.233585\pi$
$294$	0	0
$295$	7.69112e6i	0.299588i
$296$	0	0
$297$	−1.66229e7	−0.634508
$298$	0	0
$299$	−1.14367e7	−0.427844
$300$	0	0
$301$	− 4.48918e7i	− 1.64614i
$302$	0	0
$303$	− 6.87652e6i	− 0.247196i
$304$	0	0
$305$	1.13143e7	0.398776
$306$	0	0
$307$	3.69782e7	1.27800	0.639000	−	0.769207i	$-0.279349\pi$
$307$	3.69782e7	1.27800	0.639000	+	0.769207i	$0.279349\pi$
$308$	0	0
$309$	− 5.26153e6i	− 0.178335i
$310$	0	0
$311$	− 3.65356e7i	− 1.21461i	−0.794470	−	0.607303i	$-0.792251\pi$
$311$	− 3.65356e7i	− 1.21461i	0.794470	−	0.607303i	$-0.207749\pi$
$312$	0	0
$313$	5.50169e7	1.79417	0.897083	−	0.441861i	$-0.145682\pi$
$313$	5.50169e7	1.79417	0.897083	+	0.441861i	$0.145682\pi$
$314$	0	0
$315$	−1.89667e7	−0.606821
$316$	0	0
$317$	4.12495e7i	1.29491i	0.762102	+	0.647457i	$0.224168\pi$
$317$	4.12495e7i	1.29491i	−0.762102	+	0.647457i	$0.775832\pi$
$318$	0	0
$319$	3.72125e7i	1.14635i
$320$	0	0
$321$	9.22601e6	0.278932
$322$	0	0
$323$	2.95928e6	0.0878170
$324$	0	0
$325$	4.16141e7i	1.21225i
$326$	0	0
$327$	− 1.45834e7i	− 0.417076i
$328$	0	0
$329$	−7.70461e7	−2.16353
$330$	0	0
$331$	−2.72892e7	−0.752502	−0.376251	−	0.926518i	$-0.622787\pi$
$331$	−2.72892e7	−0.752502	−0.376251	+	0.926518i	$0.622787\pi$
$332$	0	0
$333$	− 2.37104e7i	− 0.642107i
$334$	0	0
$335$	− 1.90976e7i	− 0.507977i
$336$	0	0
$337$	−1.36843e6	−0.0357547	−0.0178774	−	0.999840i	$-0.505691\pi$
$337$	−1.36843e6	−0.0357547	−0.0178774	+	0.999840i	$0.505691\pi$
$338$	0	0
$339$	−5.48216e6	−0.140719
$340$	0	0
$341$	3.82754e7i	0.965287i
$342$	0	0
$343$	698651.i	0.0173132i
$344$	0	0
$345$	−1.68094e6	−0.0409350
$346$	0	0
$347$	−1.07803e7	−0.258013	−0.129006	−	0.991644i	$-0.541179\pi$
$347$	−1.07803e7	−0.258013	−0.129006	+	0.991644i	$0.541179\pi$
$348$	0	0
$349$	− 1.33186e6i	− 0.0313316i	−0.999877	−	0.0156658i	$-0.995013\pi$
$349$	− 1.33186e6i	− 0.0313316i	0.999877	−	0.0156658i	$-0.00498678\pi$
$350$	0	0
$351$	4.06192e7i	0.939312i
$352$	0	0
$353$	798852.	0.0181611	0.00908055	−	0.999959i	$-0.497110\pi$
$353$	798852.	0.0181611	0.00908055	+	0.999959i	$0.497110\pi$
$354$	0	0
$355$	−1.17564e7	−0.262779
$356$	0	0
$357$	1.25558e7i	0.275955i
$358$	0	0
$359$	5.67611e7i	1.22678i	0.789779	+	0.613392i	$0.210195\pi$
$359$	5.67611e7i	1.22678i	−0.789779	+	0.613392i	$0.789805\pi$
$360$	0	0
$361$	−4.61087e7	−0.980078
$362$	0	0
$363$	−1.89086e6	−0.0395311
$364$	0	0
$365$	3.81897e6i	0.0785359i
$366$	0	0
$367$	− 1.80379e7i	− 0.364911i	−0.983214	−	0.182456i	$-0.941595\pi$
$367$	− 1.80379e7i	− 0.364911i	0.983214	−	0.182456i	$-0.0584046\pi$
$368$	0	0
$369$	4.50650e6	0.0896933
$370$	0	0
$371$	4.18843e7	0.820218
$372$	0	0
$373$	− 3.58446e7i	− 0.690712i	−0.938472	−	0.345356i	$-0.887758\pi$
$373$	− 3.58446e7i	− 0.690712i	0.938472	−	0.345356i	$-0.112242\pi$
$374$	0	0
$375$	1.40410e7i	0.266259i
$376$	0	0
$377$	9.09315e7	1.69703
$378$	0	0
$379$	−9.70544e7	−1.78278	−0.891390	−	0.453238i	$-0.850269\pi$
$379$	−9.70544e7	−1.78278	−0.891390	+	0.453238i	$0.850269\pi$
$380$	0	0
$381$	− 2.13625e7i	− 0.386257i
$382$	0	0
$383$	− 3.16741e7i	− 0.563779i	−0.959447	−	0.281889i	$-0.909039\pi$
$383$	− 3.16741e7i	− 0.563779i	0.959447	−	0.281889i	$-0.0909611\pi$
$384$	0	0
$385$	−4.07761e7	−0.714534
$386$	0	0
$387$	6.09768e7	1.05204
$388$	0	0
$389$	− 5.50526e7i	− 0.935252i	−0.883926	−	0.467626i	$-0.845109\pi$
$389$	− 5.50526e7i	− 0.935252i	0.883926	−	0.467626i	$-0.154891\pi$
$390$	0	0
$391$	− 1.01311e7i	− 0.169483i
$392$	0	0
$393$	−3.20620e7	−0.528217
$394$	0	0
$395$	9.83305e6	0.159550
$396$	0	0
$397$	8.33093e7i	1.33144i	0.746201	+	0.665721i	$0.231876\pi$
$397$	8.33093e7i	1.33144i	−0.746201	+	0.665721i	$0.768124\pi$
$398$	0	0
$399$	3.97650e6i	0.0626011i
$400$	0	0
$401$	4.24726e7	0.658682	0.329341	−	0.944211i	$-0.393174\pi$
$401$	4.24726e7	0.658682	0.329341	+	0.944211i	$0.393174\pi$
$402$	0	0
$403$	9.35286e7	1.42899
$404$	0	0
$405$	− 2.26221e7i	− 0.340540i
$406$	0	0
$407$	− 5.09745e7i	− 0.756083i
$408$	0	0
$409$	4.61036e7	0.673853	0.336926	−	0.941531i	$-0.390613\pi$
$409$	4.61036e7	0.673853	0.336926	+	0.941531i	$0.390613\pi$
$410$	0	0
$411$	−1.19456e7	−0.172061
$412$	0	0
$413$	− 6.22887e7i	− 0.884218i
$414$	0	0
$415$	− 4.79445e7i	− 0.670802i
$416$	0	0
$417$	1.96686e7	0.271247
$418$	0	0
$419$	3.86614e7	0.525575	0.262788	−	0.964854i	$-0.415358\pi$
$419$	3.86614e7	0.525575	0.262788	+	0.964854i	$0.415358\pi$
$420$	0	0
$421$	2.58211e7i	0.346042i	0.984918	+	0.173021i	$0.0553528\pi$
$421$	2.58211e7i	0.346042i	−0.984918	+	0.173021i	$0.944647\pi$
$422$	0	0
$423$	− 1.04652e8i	− 1.38269i
$424$	0	0
$425$	−3.68636e7	−0.480210
$426$	0	0
$427$	−9.16323e7	−1.17697
$428$	0	0
$429$	4.13900e7i	0.524232i
$430$	0	0
$431$	3.09589e7i	0.386681i	0.981132	+	0.193341i	$0.0619322\pi$
$431$	3.09589e7i	0.386681i	−0.981132	+	0.193341i	$0.938068\pi$
$432$	0	0
$433$	4.64302e7	0.571921	0.285961	−	0.958241i	$-0.407687\pi$
$433$	4.64302e7	0.571921	0.285961	+	0.958241i	$0.407687\pi$
$434$	0	0
$435$	1.33649e7	0.162368
$436$	0	0
$437$	− 3.20859e6i	− 0.0384477i
$438$	0	0
$439$	1.33692e8i	1.58020i	0.612979	+	0.790099i	$0.289971\pi$
$439$	1.33692e8i	1.58020i	−0.612979	+	0.790099i	$0.710029\pi$
$440$	0	0
$441$	7.63292e7	0.889969
$442$	0	0
$443$	5.29536e6	0.0609094	0.0304547	−	0.999536i	$-0.490304\pi$
$443$	5.29536e6	0.0609094	0.0304547	+	0.999536i	$0.490304\pi$
$444$	0	0
$445$	3.23305e6i	0.0366887i
$446$	0	0
$447$	− 4.03347e7i	− 0.451602i
$448$	0	0
$449$	−1.20011e8	−1.32582	−0.662908	−	0.748701i	$-0.730678\pi$
$449$	−1.20011e8	−1.32582	−0.662908	+	0.748701i	$0.730678\pi$
$450$	0	0
$451$	9.68840e6	0.105614
$452$	0	0
$453$	− 9.49508e6i	− 0.102142i
$454$	0	0
$455$	9.96392e7i	1.05778i
$456$	0	0
$457$	−1.84689e8	−1.93505	−0.967527	−	0.252767i	$-0.918660\pi$
$457$	−1.84689e8	−1.93505	−0.967527	+	0.252767i	$0.918660\pi$
$458$	0	0
$459$	−3.59823e7	−0.372092
$460$	0	0
$461$	− 1.46730e8i	− 1.49767i	−0.662755	−	0.748836i	$-0.730613\pi$
$461$	− 1.46730e8i	− 1.49767i	0.662755	−	0.748836i	$-0.269387\pi$
$462$	0	0
$463$	1.55124e8i	1.56292i	0.623956	+	0.781459i	$0.285524\pi$
$463$	1.55124e8i	1.56292i	−0.623956	+	0.781459i	$0.714476\pi$
$464$	0	0
$465$	1.37467e7	0.136722
$466$	0	0
$467$	−1.07539e8	−1.05589	−0.527943	−	0.849280i	$-0.677037\pi$
$467$	−1.07539e8	−1.05589	−0.527943	+	0.849280i	$0.677037\pi$
$468$	0	0
$469$	1.54667e8i	1.49927i
$470$	0	0
$471$	− 8.52850e6i	− 0.0816225i
$472$	0	0
$473$	1.31092e8	1.23878
$474$	0	0
$475$	−1.16750e7	−0.108937
$476$	0	0
$477$	5.68916e7i	0.524195i
$478$	0	0
$479$	− 1.48881e7i	− 0.135467i	−0.997703	−	0.0677333i	$-0.978423\pi$
$479$	− 1.48881e7i	− 0.135467i	0.997703	−	0.0677333i	$-0.0215767\pi$
$480$	0	0
$481$	−1.24560e8	−1.11929
$482$	0	0
$483$	1.36136e7	0.120818
$484$	0	0
$485$	6.60819e7i	0.579239i
$486$	0	0
$487$	− 9.69346e7i	− 0.839251i	−0.907697	−	0.419625i	$-0.862161\pi$
$487$	− 9.69346e7i	− 0.839251i	0.907697	−	0.419625i	$-0.137839\pi$
$488$	0	0
$489$	1.25128e7	0.107011
$490$	0	0
$491$	6.45564e7	0.545374	0.272687	−	0.962103i	$-0.412088\pi$
$491$	6.45564e7	0.545374	0.272687	+	0.962103i	$0.412088\pi$
$492$	0	0
$493$	8.05512e7i	0.672250i
$494$	0	0
$495$	− 5.53863e7i	− 0.456653i
$496$	0	0
$497$	9.52126e7	0.775578
$498$	0	0
$499$	−8.09040e7	−0.651131	−0.325565	−	0.945520i	$-0.605555\pi$
$499$	−8.09040e7	−0.651131	−0.325565	+	0.945520i	$0.605555\pi$
$500$	0	0
$501$	− 4.38583e7i	− 0.348770i
$502$	0	0
$503$	− 1.32342e8i	− 1.03990i	−0.854196	−	0.519951i	$-0.825950\pi$
$503$	− 1.32342e8i	− 1.03990i	0.854196	−	0.519951i	$-0.174050\pi$
$504$	0	0
$505$	4.83408e7	0.375352
$506$	0	0
$507$	6.01410e7	0.461473
$508$	0	0
$509$	1.39233e7i	0.105582i	0.998606	+	0.0527909i	$0.0168117\pi$
$509$	1.39233e7i	0.105582i	−0.998606	+	0.0527909i	$0.983188\pi$
$510$	0	0
$511$	− 3.09290e7i	− 0.231795i
$512$	0	0
$513$	−1.13958e7	−0.0844101
$514$	0	0
$515$	3.69877e7	0.270792
$516$	0	0
$517$	− 2.24989e8i	− 1.62813i
$518$	0	0
$519$	5.09842e6i	0.0364698i
$520$	0	0
$521$	−9.25151e7	−0.654183	−0.327092	−	0.944993i	$-0.606069\pi$
$521$	−9.25151e7	−0.654183	−0.327092	+	0.944993i	$0.606069\pi$
$522$	0	0
$523$	5.17905e7	0.362030	0.181015	−	0.983480i	$-0.442062\pi$
$523$	5.17905e7	0.362030	0.181015	+	0.983480i	$0.442062\pi$
$524$	0	0
$525$	− 4.95351e7i	− 0.342322i
$526$	0	0
$527$	8.28518e7i	0.566070i
$528$	0	0
$529$	1.37051e8	0.925797
$530$	0	0
$531$	8.46070e7	0.565097
$532$	0	0
$533$	− 2.36743e7i	− 0.156349i
$534$	0	0
$535$	6.48573e7i	0.423543i
$536$	0	0
$537$	5.39069e7	0.348114
$538$	0	0
$539$	1.64098e8	1.04794
$540$	0	0
$541$	2.48317e7i	0.156825i	0.996921	+	0.0784125i	$0.0249851\pi$
$541$	2.48317e7i	0.156825i	−0.996921	+	0.0784125i	$0.975015\pi$
$542$	0	0
$543$	1.14694e7i	0.0716378i
$544$	0	0
$545$	1.02519e8	0.633306
$546$	0	0
$547$	−8.23167e7	−0.502951	−0.251476	−	0.967864i	$-0.580916\pi$
$547$	−8.23167e7	−0.502951	−0.251476	+	0.967864i	$0.580916\pi$
$548$	0	0
$549$	− 1.24465e8i	− 0.752192i
$550$	0	0
$551$	2.55111e7i	0.152502i
$552$	0	0
$553$	−7.96357e7	−0.470904
$554$	0	0
$555$	−1.83076e7	−0.107091
$556$	0	0
$557$	− 1.99169e7i	− 0.115254i	−0.998338	−	0.0576271i	$-0.981647\pi$
$557$	− 1.99169e7i	− 0.115254i	0.998338	−	0.0576271i	$-0.0183534\pi$
$558$	0	0
$559$	− 3.20333e8i	− 1.83386i
$560$	0	0
$561$	−3.66651e7	−0.207666
$562$	0	0
$563$	−3.05819e8	−1.71372	−0.856859	−	0.515550i	$-0.827588\pi$
$563$	−3.05819e8	−1.71372	−0.856859	+	0.515550i	$0.827588\pi$
$564$	0	0
$565$	− 3.85387e7i	− 0.213674i
$566$	0	0
$567$	1.83212e8i	1.00509i
$568$	0	0
$569$	−2.25936e6	−0.0122644	−0.00613222	−	0.999981i	$-0.501952\pi$
$569$	−2.25936e6	−0.0122644	−0.00613222	+	0.999981i	$0.501952\pi$
$570$	0	0
$571$	6.55464e7	0.352079	0.176040	−	0.984383i	$-0.443671\pi$
$571$	6.55464e7	0.352079	0.176040	+	0.984383i	$0.443671\pi$
$572$	0	0
$573$	2.82034e7i	0.149912i
$574$	0	0
$575$	3.99693e7i	0.210244i
$576$	0	0
$577$	5.43876e7	0.283121	0.141561	−	0.989930i	$-0.454788\pi$
$577$	5.43876e7	0.283121	0.141561	+	0.989930i	$0.454788\pi$
$578$	0	0
$579$	−4.74981e7	−0.244703
$580$	0	0
$581$	3.88292e8i	1.97984i
$582$	0	0
$583$	1.22310e8i	0.617242i
$584$	0	0
$585$	−1.35340e8	−0.676020
$586$	0	0
$587$	3.85369e8	1.90529	0.952647	−	0.304078i	$-0.0983482\pi$
$587$	3.85369e8	1.90529	0.952647	+	0.304078i	$0.0983482\pi$
$588$	0	0
$589$	2.62398e7i	0.128414i
$590$	0	0
$591$	6.11390e7i	0.296180i
$592$	0	0
$593$	−3.00090e8	−1.43909	−0.719545	−	0.694446i	$-0.755650\pi$
$593$	−3.00090e8	−1.43909	−0.719545	+	0.694446i	$0.755650\pi$
$594$	0	0
$595$	−8.82649e7	−0.419022
$596$	0	0
$597$	9.83914e7i	0.462417i
$598$	0	0
$599$	− 3.11371e8i	− 1.44876i	−0.689400	−	0.724381i	$-0.742126\pi$
$599$	− 3.11371e8i	− 1.44876i	0.689400	−	0.724381i	$-0.257874\pi$
$600$	0	0
$601$	2.10925e8	0.971637	0.485818	−	0.874060i	$-0.338522\pi$
$601$	2.10925e8	0.971637	0.485818	+	0.874060i	$0.338522\pi$
$602$	0	0
$603$	−2.10085e8	−0.958171
$604$	0	0
$605$	− 1.32924e7i	− 0.0600257i
$606$	0	0
$607$	3.28208e8i	1.46752i	0.679410	+	0.733759i	$0.262236\pi$
$607$	3.28208e8i	1.46752i	−0.679410	+	0.733759i	$0.737764\pi$
$608$	0	0
$609$	−1.08240e8	−0.479220
$610$	0	0
$611$	−5.49776e8	−2.41025
$612$	0	0
$613$	1.43677e8i	0.623743i	0.950124	+	0.311872i	$0.100956\pi$
$613$	1.43677e8i	0.623743i	−0.950124	+	0.311872i	$0.899044\pi$
$614$	0	0
$615$	− 3.47960e6i	− 0.0149591i
$616$	0	0
$617$	−1.50375e8	−0.640206	−0.320103	−	0.947383i	$-0.603717\pi$
$617$	−1.50375e8	−0.640206	−0.320103	+	0.947383i	$0.603717\pi$
$618$	0	0
$619$	2.07480e8	0.874793	0.437396	−	0.899269i	$-0.355901\pi$
$619$	2.07480e8	0.874793	0.437396	+	0.899269i	$0.355901\pi$
$620$	0	0
$621$	3.90137e7i	0.162908i
$622$	0	0
$623$	− 2.61838e7i	− 0.108285i
$624$	0	0
$625$	8.97259e7	0.367517
$626$	0	0
$627$	−1.16121e7	−0.0471094
$628$	0	0
$629$	− 1.10341e8i	− 0.443388i
$630$	0	0
$631$	− 9.08069e7i	− 0.361436i	−0.983535	−	0.180718i	$-0.942158\pi$
$631$	− 9.08069e7i	− 0.361436i	0.983535	−	0.180718i	$-0.0578421\pi$
$632$	0	0
$633$	−5.43052e6	−0.0214107
$634$	0	0
$635$	1.50174e8	0.586509
$636$	0	0
$637$	− 4.00986e8i	− 1.55135i
$638$	0	0
$639$	1.29328e8i	0.495666i
$640$	0	0
$641$	1.95456e8	0.742121	0.371061	−	0.928609i	$-0.378994\pi$
$641$	1.95456e8	0.742121	0.371061	+	0.928609i	$0.378994\pi$
$642$	0	0
$643$	3.72026e8	1.39939	0.699697	−	0.714440i	$-0.253318\pi$
$643$	3.72026e8	1.39939	0.699697	+	0.714440i	$0.253318\pi$
$644$	0	0
$645$	− 4.70820e7i	− 0.175459i
$646$	0	0
$647$	2.47447e8i	0.913627i	0.889563	+	0.456813i	$0.151009\pi$
$647$	2.47447e8i	0.913627i	−0.889563	+	0.456813i	$0.848991\pi$
$648$	0	0
$649$	1.81894e8	0.665404
$650$	0	0
$651$	−1.11331e8	−0.403528
$652$	0	0
$653$	1.98187e8i	0.711764i	0.934531	+	0.355882i	$0.115820\pi$
$653$	1.98187e8i	0.711764i	−0.934531	+	0.355882i	$0.884180\pi$
$654$	0	0
$655$	− 2.25390e8i	− 0.802068i
$656$	0	0
$657$	4.20110e7	0.148138
$658$	0	0
$659$	5.71625e7	0.199735	0.0998677	−	0.995001i	$-0.468158\pi$
$659$	5.71625e7	0.199735	0.0998677	+	0.995001i	$0.468158\pi$
$660$	0	0
$661$	− 5.04948e8i	− 1.74841i	−0.485561	−	0.874203i	$-0.661385\pi$
$661$	− 5.04948e8i	− 1.74841i	0.485561	−	0.874203i	$-0.338615\pi$
$662$	0	0
$663$	8.95938e7i	0.307424i
$664$	0	0
$665$	−2.79541e7	−0.0950563
$666$	0	0
$667$	8.73374e7	0.294322
$668$	0	0
$669$	8.07893e7i	0.269821i
$670$	0	0
$671$	− 2.67583e8i	− 0.885709i
$672$	0	0
$673$	−1.59423e8	−0.523004	−0.261502	−	0.965203i	$-0.584218\pi$
$673$	−1.59423e8	−0.523004	−0.261502	+	0.965203i	$0.584218\pi$
$674$	0	0
$675$	1.41958e8	0.461580
$676$	0	0
$677$	− 1.89358e8i	− 0.610265i	−0.952310	−	0.305132i	$-0.901299\pi$
$677$	− 1.89358e8i	− 0.610265i	0.952310	−	0.305132i	$-0.0987007\pi$
$678$	0	0
$679$	− 5.35183e8i	− 1.70959i
$680$	0	0
$681$	1.08616e8	0.343915
$682$	0	0
$683$	−6.31662e7	−0.198254	−0.0991272	−	0.995075i	$-0.531605\pi$
$683$	−6.31662e7	−0.198254	−0.0991272	+	0.995075i	$0.531605\pi$
$684$	0	0
$685$	− 8.39755e7i	− 0.261265i
$686$	0	0
$687$	− 3.92358e7i	− 0.121008i
$688$	0	0
$689$	2.98872e8	0.913752
$690$	0	0
$691$	1.37266e8	0.416033	0.208016	−	0.978125i	$-0.433299\pi$
$691$	1.37266e8	0.416033	0.208016	+	0.978125i	$0.433299\pi$
$692$	0	0
$693$	4.48561e8i	1.34779i
$694$	0	0
$695$	1.38267e8i	0.411873i
$696$	0	0
$697$	2.09718e7	0.0619350
$698$	0	0
$699$	7.99595e7	0.234120
$700$	0	0
$701$	6.02955e8i	1.75037i	0.483784	+	0.875187i	$0.339262\pi$
$701$	6.02955e8i	1.75037i	−0.483784	+	0.875187i	$0.660738\pi$
$702$	0	0
$703$	− 3.49456e7i	− 0.100584i
$704$	0	0
$705$	−8.08050e7	−0.230606
$706$	0	0
$707$	−3.91501e8	−1.10783
$708$	0	0
$709$	4.41370e7i	0.123841i	0.998081	+	0.0619205i	$0.0197225\pi$
$709$	4.41370e7i	0.123841i	−0.998081	+	0.0619205i	$0.980277\pi$
$710$	0	0
$711$	− 1.08169e8i	− 0.300951i
$712$	0	0
$713$	8.98319e7	0.247835
$714$	0	0
$715$	−2.90965e8	−0.796017
$716$	0	0
$717$	− 2.18330e8i	− 0.592318i
$718$	0	0
$719$	3.59817e8i	0.968045i	0.875056	+	0.484022i	$0.160825\pi$
$719$	3.59817e8i	0.968045i	−0.875056	+	0.484022i	$0.839175\pi$
$720$	0	0
$721$	−2.99555e8	−0.799229
$722$	0	0
$723$	−6.93814e7	−0.183581
$724$	0	0
$725$	− 3.17791e8i	− 0.833926i
$726$	0	0
$727$	1.23355e7i	0.0321035i	0.999871	+	0.0160518i	$0.00510965\pi$
$727$	1.23355e7i	0.0321035i	−0.999871	+	0.0160518i	$0.994890\pi$
$728$	0	0
$729$	−1.75968e8	−0.454205
$730$	0	0
$731$	2.83766e8	0.726453
$732$	0	0
$733$	3.54832e7i	0.0900972i	0.998985	+	0.0450486i	$0.0143443\pi$
$733$	3.54832e7i	0.0900972i	−0.998985	+	0.0450486i	$0.985656\pi$
$734$	0	0
$735$	− 5.89361e7i	− 0.148429i
$736$	0	0
$737$	−4.51656e8	−1.12825
$738$	0	0
$739$	4.32139e8	1.07076	0.535378	−	0.844612i	$-0.320169\pi$
$739$	4.32139e8	1.07076	0.535378	+	0.844612i	$0.320169\pi$
$740$	0	0
$741$	2.83750e7i	0.0697399i
$742$	0	0
$743$	7.42345e8i	1.80984i	0.425586	+	0.904918i	$0.360068\pi$
$743$	7.42345e8i	1.80984i	−0.425586	+	0.904918i	$0.639932\pi$
$744$	0	0
$745$	2.83546e8	0.685732
$746$	0	0
$747$	−5.27418e8	−1.26530
$748$	0	0
$749$	− 5.25265e8i	− 1.25007i
$750$	0	0
$751$	6.22745e8i	1.47025i	0.677933	+	0.735124i	$0.262876\pi$
$751$	6.22745e8i	1.47025i	−0.677933	+	0.735124i	$0.737124\pi$
$752$	0	0
$753$	−7.17344e7	−0.168013
$754$	0	0
$755$	6.67488e7	0.155097
$756$	0	0
$757$	8.38267e8i	1.93239i	0.257815	+	0.966194i	$0.416998\pi$
$757$	8.38267e8i	1.93239i	−0.257815	+	0.966194i	$0.583002\pi$
$758$	0	0
$759$	3.97541e7i	0.0909193i
$760$	0	0
$761$	−6.44538e8	−1.46250	−0.731248	−	0.682112i	$-0.761062\pi$
$761$	−6.44538e8	−1.46250	−0.731248	+	0.682112i	$0.761062\pi$
$762$	0	0
$763$	−8.30277e8	−1.86917
$764$	0	0
$765$	− 1.19891e8i	− 0.267794i
$766$	0	0
$767$	− 4.44472e8i	− 0.985050i
$768$	0	0
$769$	7.83458e8	1.72281	0.861404	−	0.507921i	$-0.169586\pi$
$769$	7.83458e8	1.72281	0.861404	+	0.507921i	$0.169586\pi$
$770$	0	0
$771$	1.56696e8	0.341897
$772$	0	0
$773$	− 7.79311e7i	− 0.168722i	−0.996435	−	0.0843611i	$-0.973115\pi$
$773$	− 7.79311e7i	− 0.168722i	0.996435	−	0.0843611i	$-0.0268849\pi$
$774$	0	0
$775$	− 3.26868e8i	− 0.702210i
$776$	0	0
$777$	1.48269e8	0.316073
$778$	0	0
$779$	6.64190e6	0.0140501
$780$	0	0
$781$	2.78038e8i	0.583649i
$782$	0	0
$783$	− 3.10193e8i	− 0.646170i
$784$	0	0
$785$	5.99539e7	0.123939
$786$	0	0
$787$	4.76683e8	0.977924	0.488962	−	0.872305i	$-0.337376\pi$
$787$	4.76683e8	0.977924	0.488962	+	0.872305i	$0.337376\pi$
$788$	0	0
$789$	3.26246e7i	0.0664224i
$790$	0	0
$791$	3.12116e8i	0.630648i
$792$	0	0
$793$	−6.53858e8	−1.31118
$794$	0	0
$795$	4.39277e7	0.0874253
$796$	0	0
$797$	− 2.49479e7i	− 0.0492788i	−0.999696	−	0.0246394i	$-0.992156\pi$
$797$	− 2.49479e7i	− 0.0492788i	0.999696	−	0.0246394i	$-0.00784375\pi$
$798$	0	0
$799$	− 4.87016e8i	− 0.954779i
$800$	0	0
$801$	3.55655e7	0.0692040
$802$	0	0
$803$	9.03184e7	0.174433
$804$	0	0
$805$	9.57010e7i	0.183455i
$806$	0	0
$807$	− 2.63676e8i	− 0.501707i
$808$	0	0
$809$	6.82458e8	1.28893	0.644467	−	0.764632i	$-0.277079\pi$
$809$	6.82458e8	1.28893	0.644467	+	0.764632i	$0.277079\pi$
$810$	0	0
$811$	−3.51060e8	−0.658140	−0.329070	−	0.944306i	$-0.606735\pi$
$811$	−3.51060e8	−0.658140	−0.329070	+	0.944306i	$0.606735\pi$
$812$	0	0
$813$	2.00082e8i	0.372337i
$814$	0	0
$815$	8.79628e7i	0.162490i
$816$	0	0
$817$	8.98706e7	0.164798
$818$	0	0
$819$	1.09609e9	1.99524
$820$	0	0
$821$	− 6.38664e8i	− 1.15410i	−0.816709	−	0.577049i	$-0.804204\pi$
$821$	− 6.38664e8i	− 1.15410i	0.816709	−	0.577049i	$-0.195796\pi$
$822$	0	0
$823$	− 1.77607e8i	− 0.318611i	−0.987229	−	0.159305i	$-0.949075\pi$
$823$	− 1.77607e8i	− 0.318611i	0.987229	−	0.159305i	$-0.0509254\pi$
$824$	0	0
$825$	1.44651e8	0.257609
$826$	0	0
$827$	−2.12047e7	−0.0374900	−0.0187450	−	0.999824i	$-0.505967\pi$
$827$	−2.12047e7	−0.0374900	−0.0187450	+	0.999824i	$0.505967\pi$
$828$	0	0
$829$	− 6.30661e8i	− 1.10696i	−0.832862	−	0.553480i	$-0.813299\pi$
$829$	− 6.30661e8i	− 1.10696i	0.832862	−	0.553480i	$-0.186701\pi$
$830$	0	0
$831$	1.97219e8i	0.343673i
$832$	0	0
$833$	3.55211e8	0.614542
$834$	0	0
$835$	3.08316e8	0.529587
$836$	0	0
$837$	− 3.19053e8i	− 0.544109i
$838$	0	0
$839$	− 9.45472e8i	− 1.60089i	−0.599403	−	0.800447i	$-0.704595\pi$
$839$	− 9.45472e8i	− 1.60089i	0.599403	−	0.800447i	$-0.295405\pi$
$840$	0	0
$841$	−9.95855e7	−0.167420
$842$	0	0
$843$	−2.54408e8	−0.424667
$844$	0	0
$845$	4.22781e8i	0.700721i
$846$	0	0
$847$	1.07652e8i	0.177163i
$848$	0	0
$849$	−3.12606e8	−0.510828
$850$	0	0
$851$	−1.19637e8	−0.194122
$852$	0	0
$853$	− 3.08597e8i	− 0.497216i	−0.968604	−	0.248608i	$-0.920027\pi$
$853$	− 3.08597e8i	− 0.497216i	0.968604	−	0.248608i	$-0.0799731\pi$
$854$	0	0
$855$	− 3.79702e7i	− 0.0607497i
$856$	0	0
$857$	2.45043e8	0.389314	0.194657	−	0.980871i	$-0.437641\pi$
$857$	2.45043e8	0.389314	0.194657	+	0.980871i	$0.437641\pi$
$858$	0	0
$859$	−6.90100e8	−1.08876	−0.544380	−	0.838839i	$-0.683235\pi$
$859$	−6.90100e8	−1.08876	−0.544380	+	0.838839i	$0.683235\pi$
$860$	0	0
$861$	2.81805e7i	0.0441509i
$862$	0	0
$863$	6.89145e8i	1.07221i	0.844153	+	0.536103i	$0.180104\pi$
$863$	6.89145e8i	1.07221i	−0.844153	+	0.536103i	$0.819896\pi$
$864$	0	0
$865$	−3.58410e7	−0.0553773
$866$	0	0
$867$	1.25656e8	0.192808
$868$	0	0
$869$	− 2.32551e8i	− 0.354371i
$870$	0	0
$871$	1.10365e9i	1.67024i
$872$	0	0
$873$	7.26941e8	1.09259
$874$	0	0
$875$	7.99397e8	1.19327
$876$	0	0
$877$	8.03085e8i	1.19059i	0.803507	+	0.595296i	$0.202965\pi$
$877$	8.03085e8i	1.19059i	−0.803507	+	0.595296i	$0.797035\pi$
$878$	0	0
$879$	2.86175e8i	0.421372i
$880$	0	0
$881$	−4.58820e8	−0.670988	−0.335494	−	0.942042i	$-0.608903\pi$
$881$	−4.58820e8	−0.670988	−0.335494	+	0.942042i	$0.608903\pi$
$882$	0	0
$883$	−1.80291e8	−0.261874	−0.130937	−	0.991391i	$-0.541799\pi$
$883$	−1.80291e8	−0.261874	−0.130937	+	0.991391i	$0.541799\pi$
$884$	0	0
$885$	− 6.53277e7i	− 0.0942469i
$886$	0	0
$887$	− 6.63542e8i	− 0.950818i	−0.879765	−	0.475409i	$-0.842300\pi$
$887$	− 6.63542e8i	− 0.950818i	0.879765	−	0.475409i	$-0.157700\pi$
$888$	0	0
$889$	−1.21623e9	−1.73105
$890$	0	0
$891$	−5.35011e8	−0.756362
$892$	0	0
$893$	− 1.54241e8i	− 0.216594i
$894$	0	0
$895$	3.78956e8i	0.528591i
$896$	0	0
$897$	9.71419e7	0.134595
$898$	0	0
$899$	−7.14242e8	−0.983029
$900$	0	0
$901$	2.64755e8i	0.361967i
$902$	0	0
$903$	3.81307e8i	0.517859i
$904$	0	0
$905$	−8.06281e7	−0.108778
$906$	0	0
$907$	−2.51825e8	−0.337503	−0.168751	−	0.985659i	$-0.553973\pi$
$907$	−2.51825e8	−0.337503	−0.168751	+	0.985659i	$0.553973\pi$
$908$	0	0
$909$	− 5.31777e8i	− 0.708008i
$910$	0	0
$911$	− 1.33414e9i	− 1.76459i	−0.470694	−	0.882297i	$-0.655996\pi$
$911$	− 1.33414e9i	− 1.76459i	0.470694	−	0.882297i	$-0.344004\pi$
$912$	0	0
$913$	−1.13388e9	−1.48990
$914$	0	0
$915$	−9.61028e7	−0.125451
$916$	0	0
$917$	1.82539e9i	2.36726i
$918$	0	0
$919$	9.68344e8i	1.24762i	0.781575	+	0.623811i	$0.214417\pi$
$919$	9.68344e8i	1.24762i	−0.781575	+	0.623811i	$0.785583\pi$
$920$	0	0
$921$	−3.14089e8	−0.402044
$922$	0	0
$923$	6.79406e8	0.864021
$924$	0	0
$925$	4.35317e8i	0.550022i
$926$	0	0
$927$	− 4.06887e8i	− 0.510781i
$928$	0	0
$929$	9.60464e8	1.19794	0.598968	−	0.800773i	$-0.295577\pi$
$929$	9.60464e8	1.19794	0.598968	+	0.800773i	$0.295577\pi$
$930$	0	0
$931$	1.12498e8	0.139410
$932$	0	0
$933$	3.10330e8i	0.382102i
$934$	0	0
$935$	− 2.57750e8i	− 0.315328i
$936$	0	0
$937$	9.88309e8	1.20136	0.600681	−	0.799489i	$-0.294896\pi$
$937$	9.88309e8	1.20136	0.600681	+	0.799489i	$0.294896\pi$
$938$	0	0
$939$	−4.67308e8	−0.564425
$940$	0	0
$941$	− 1.34529e8i	− 0.161454i	−0.996736	−	0.0807269i	$-0.974276\pi$
$941$	− 1.34529e8i	− 0.161454i	0.996736	−	0.0807269i	$-0.0257242\pi$
$942$	0	0
$943$	− 2.27386e7i	− 0.0271162i
$944$	0	0
$945$	3.39898e8	0.402766
$946$	0	0
$947$	−4.96808e8	−0.584976	−0.292488	−	0.956269i	$-0.594483\pi$
$947$	−4.96808e8	−0.584976	−0.292488	+	0.956269i	$0.594483\pi$
$948$	0	0
$949$	− 2.20699e8i	− 0.258227i
$950$	0	0
$951$	− 3.50369e8i	− 0.407366i
$952$	0	0
$953$	−4.73453e8	−0.547014	−0.273507	−	0.961870i	$-0.588184\pi$
$953$	−4.73453e8	−0.547014	−0.273507	+	0.961870i	$0.588184\pi$
$954$	0	0
$955$	−1.98265e8	−0.227633
$956$	0	0
$957$	− 3.16080e8i	− 0.360629i
$958$	0	0
$959$	6.80099e8i	0.771110i
$960$	0	0
$961$	1.52861e8	0.172238
$962$	0	0
$963$	7.13470e8	0.798908
$964$	0	0
$965$	− 3.33903e8i	− 0.371568i
$966$	0	0
$967$	3.26806e8i	0.361418i	0.983537	+	0.180709i	$0.0578393\pi$
$967$	3.26806e8i	0.361418i	−0.983537	+	0.180709i	$0.942161\pi$
$968$	0	0
$969$	−2.51358e7	−0.0276263
$970$	0	0
$971$	7.38503e8	0.806667	0.403333	−	0.915053i	$-0.367851\pi$
$971$	7.38503e8	0.806667	0.403333	+	0.915053i	$0.367851\pi$
$972$	0	0
$973$	− 1.11979e9i	− 1.21562i
$974$	0	0
$975$	− 3.53466e8i	− 0.381359i
$976$	0	0
$977$	−8.98069e8	−0.962999	−0.481500	−	0.876446i	$-0.659908\pi$
$977$	−8.98069e8	−0.962999	−0.481500	+	0.876446i	$0.659908\pi$
$978$	0	0
$979$	7.64614e7	0.0814880
$980$	0	0
$981$	− 1.12777e9i	− 1.19457i
$982$	0	0
$983$	− 5.63791e8i	− 0.593551i	−0.954947	−	0.296775i	$-0.904089\pi$
$983$	− 5.63791e8i	− 0.593551i	0.954947	−	0.296775i	$-0.0959112\pi$
$984$	0	0
$985$	−4.29797e8	−0.449733
$986$	0	0
$987$	6.54422e8	0.680622
$988$	0	0
$989$	− 3.07672e8i	− 0.318053i
$990$	0	0
$991$	4.94832e8i	0.508437i	0.967147	+	0.254218i	$0.0818182\pi$
$991$	4.94832e8i	0.508437i	−0.967147	+	0.254218i	$0.918182\pi$
$992$	0	0
$993$	2.31792e8	0.236729
$994$	0	0
$995$	−6.91675e8	−0.702155
$996$	0	0
$997$	− 2.35279e8i	− 0.237409i	−0.992930	−	0.118705i	$-0.962126\pi$
$997$	− 2.35279e8i	− 0.237409i	0.992930	−	0.118705i	$-0.0378742\pi$
$998$	0	0
$999$	4.24909e8i	0.426186i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.7.d.b.15.1		4
3.2	odd	2		288.7.b.b.271.3		4
4.3	odd	2		8.7.d.b.3.4	yes	4
8.3	odd	2	inner	32.7.d.b.15.2		4
8.5	even	2		8.7.d.b.3.3	✓	4
12.11	even	2		72.7.b.b.19.1		4
16.3	odd	4		256.7.c.l.255.5		8
16.5	even	4		256.7.c.l.255.6		8
16.11	odd	4		256.7.c.l.255.4		8
16.13	even	4		256.7.c.l.255.3		8
24.5	odd	2		72.7.b.b.19.2		4
24.11	even	2		288.7.b.b.271.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.7.d.b.3.3	✓	4	8.5	even	2
8.7.d.b.3.4	yes	4	4.3	odd	2
32.7.d.b.15.1		4	1.1	even	1	trivial
32.7.d.b.15.2		4	8.3	odd	2	inner
72.7.b.b.19.1		4	12.11	even	2
72.7.b.b.19.2		4	24.5	odd	2
256.7.c.l.255.3		8	16.13	even	4
256.7.c.l.255.4		8	16.11	odd	4
256.7.c.l.255.5		8	16.3	odd	4
256.7.c.l.255.6		8	16.5	even	4
288.7.b.b.271.2		4	24.11	even	2
288.7.b.b.271.3		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	32.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-8.49390 q^{3} -59.7107i q^{5} +483.584i q^{7} -656.854 q^{9} +O(q^{10})\)	\(q-8.49390 q^{3} -59.7107i q^{5} +483.584i q^{7} -656.854 q^{9} -1412.15 q^{11} +3450.70i q^{13} +507.177i q^{15} -3056.78 q^{17} -968.104 q^{19} -4107.51i q^{21} +3314.31i q^{23} +12059.6 q^{25} +11771.3 q^{27} -26351.6i q^{29} -27104.3i q^{31} +11994.7 q^{33} +28875.1 q^{35} +36097.0i q^{37} -29309.9i q^{39} -6860.73 q^{41} -92831.6 q^{43} +39221.2i q^{45} +159323. i q^{47} -116204. q^{49} +25964.0 q^{51} -86612.3i q^{53} +84320.6i q^{55} +8222.98 q^{57} -128806. q^{59} +189486. i q^{61} -317644. i q^{63} +206043. q^{65} +319835. q^{67} -28151.4i q^{69} -196890. i q^{71} -63957.9 q^{73} -102433. q^{75} -682894. i q^{77} +164678. i q^{79} +378862. q^{81} +802946. q^{83} +182522. i q^{85} +223828. i q^{87} -54145.3 q^{89} -1.66870e6 q^{91} +230221. i q^{93} +57806.1i q^{95} -1.10670e6 q^{97} +927577. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 48 q^{3} - 660 q^{9}+O(q^{10}) \) 4 * q + 48 * q^3 - 660 * q^9	\( 4 q + 48 q^{3} - 660 q^{9} - 976 q^{11} + 4168 q^{17} + 1456 q^{19} - 23900 q^{25} - 2592 q^{27} + 84048 q^{33} + 49920 q^{35} - 117944 q^{41} - 197456 q^{43} + 2116 q^{49} + 386016 q^{51} + 126672 q^{57} - 542032 q^{59} - 205440 q^{65} + 790192 q^{67} + 443912 q^{73} - 1765200 q^{75} - 568044 q^{81} + 3465008 q^{83} + 761224 q^{89} - 3398400 q^{91} - 926776 q^{97} + 2459280 q^{99}+O(q^{100}) \) 4 * q + 48 * q^3 - 660 * q^9 - 976 * q^11 + 4168 * q^17 + 1456 * q^19 - 23900 * q^25 - 2592 * q^27 + 84048 * q^33 + 49920 * q^35 - 117944 * q^41 - 197456 * q^43 + 2116 * q^49 + 386016 * q^51 + 126672 * q^57 - 542032 * q^59 - 205440 * q^65 + 790192 * q^67 + 443912 * q^73 - 1765200 * q^75 - 568044 * q^81 + 3465008 * q^83 + 761224 * q^89 - 3398400 * q^91 - 926776 * q^97 + 2459280 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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