LMFDB - Embedded newform 28.8.a.b.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [28,8,Mod(1,28)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("28.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(28, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 28 = 2^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	28.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$8.74678071356$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1009}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 252 $ x^2 - x - 252
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-15.3824$ of defining polynomial
Character	$\chi$	$=$	28.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+38.7648 q^{3} -496.412 q^{5} -343.000 q^{7} -684.293 q^{9} +4035.19 q^{11} -14469.7 q^{13} -19243.3 q^{15} -26750.1 q^{17} +36610.0 q^{19} -13296.3 q^{21} -34920.1 q^{23} +168300. q^{25} -111305. q^{27} +35874.3 q^{29} +198598. q^{31} +156423. q^{33} +170269. q^{35} -114413. q^{37} -560915. q^{39} +245990. q^{41} +30224.0 q^{43} +339692. q^{45} -305233. q^{47} +117649. q^{49} -1.03696e6 q^{51} -1.21053e6 q^{53} -2.00312e6 q^{55} +1.41918e6 q^{57} -1.53557e6 q^{59} -45086.4 q^{61} +234713. q^{63} +7.18295e6 q^{65} +3.08501e6 q^{67} -1.35367e6 q^{69} -1.69051e6 q^{71} -3.86932e6 q^{73} +6.52412e6 q^{75} -1.38407e6 q^{77} +2.00910e6 q^{79} -2.81816e6 q^{81} -8.28720e6 q^{83} +1.32791e7 q^{85} +1.39066e6 q^{87} -6.50624e6 q^{89} +4.96311e6 q^{91} +7.69862e6 q^{93} -1.81737e7 q^{95} -1.14882e7 q^{97} -2.76125e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{3} - 294 q^{5} - 686 q^{7} - 2258 q^{9} - 3492 q^{11} - 16170 q^{13} - 24256 q^{15} - 29232 q^{17} - 3206 q^{19} - 4802 q^{21} - 9360 q^{23} + 131146 q^{25} - 18172 q^{27} + 184704 q^{29} + 165060 q^{31}+ \cdots + 9084332 q^{99}+O(q^{100}) $ 2 * q + 14 * q^3 - 294 * q^5 - 686 * q^7 - 2258 * q^9 - 3492 * q^11 - 16170 * q^13 - 24256 * q^15 - 29232 * q^17 - 3206 * q^19 - 4802 * q^21 - 9360 * q^23 + 131146 * q^25 - 18172 * q^27 + 184704 * q^29 + 165060 * q^31 + 342832 * q^33 + 100842 * q^35 + 286144 * q^37 - 518808 * q^39 - 116760 * q^41 - 294428 * q^43 + 21154 * q^45 - 1014132 * q^47 + 235298 * q^49 - 975500 * q^51 - 1396452 * q^53 - 3526712 * q^55 + 2405212 * q^57 - 2729286 * q^59 + 2466954 * q^61 + 774494 * q^63 + 6838788 * q^65 + 225176 * q^67 - 1986656 * q^69 + 1530312 * q^71 + 1143548 * q^73 + 7444234 * q^75 + 1197756 * q^77 + 7951176 * q^79 - 1682882 * q^81 - 18487854 * q^83 + 12776740 * q^85 - 2295076 * q^87 - 4652508 * q^89 + 5546310 * q^91 + 8529192 * q^93 - 26232912 * q^95 - 26702368 * q^97 + 9084332 * q^99

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
$3$	38.7648	0.828920	0.414460	−	0.910067i	$-0.363970\pi$
$3$	38.7648	0.828920	0.414460	+	0.910067i	$0.363970\pi$
$4$	0	0
$5$	−496.412	−1.77602	−0.888009	−	0.459825i	$-0.847912\pi$
$5$	−496.412	−1.77602	−0.888009	+	0.459825i	$0.847912\pi$
$6$	0	0
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	0	0
$9$	−684.293	−0.312891
$10$	0	0
$11$	4035.19	0.914091	0.457045	−	0.889443i	$-0.348908\pi$
$11$	4035.19	0.914091	0.457045	+	0.889443i	$0.348908\pi$
$12$	0	0
$13$	−14469.7	−1.82666	−0.913331	−	0.407217i	$-0.866499\pi$
$13$	−14469.7	−1.82666	−0.913331	+	0.407217i	$0.866499\pi$
$14$	0	0
$15$	−19243.3	−1.47218
$16$	0	0
$17$	−26750.1	−1.32055	−0.660275	−	0.751024i	$-0.729560\pi$
$17$	−26750.1	−1.32055	−0.660275	+	0.751024i	$0.729560\pi$
$18$	0	0
$19$	36610.0	1.22451	0.612255	−	0.790661i	$-0.290263\pi$
$19$	36610.0	1.22451	0.612255	+	0.790661i	$0.290263\pi$
$20$	0	0
$21$	−13296.3	−0.313302
$22$	0	0
$23$	−34920.1	−0.598449	−0.299225	−	0.954183i	$-0.596728\pi$
$23$	−34920.1	−0.598449	−0.299225	+	0.954183i	$0.596728\pi$
$24$	0	0
$25$	168300.	2.15424
$26$	0	0
$27$	−111305.	−1.08828
$28$	0	0
$29$	35874.3	0.273143	0.136571	−	0.990630i	$-0.456392\pi$
$29$	35874.3	0.273143	0.136571	+	0.990630i	$0.456392\pi$
$30$	0	0
$31$	198598.	1.19732	0.598660	−	0.801004i	$-0.295700\pi$
$31$	198598.	1.19732	0.598660	+	0.801004i	$0.295700\pi$
$32$	0	0
$33$	156423.	0.757708
$34$	0	0
$35$	170269.	0.671272
$36$	0	0
$37$	−114413.	−0.371338	−0.185669	−	0.982612i	$-0.559445\pi$
$37$	−114413.	−0.371338	−0.185669	+	0.982612i	$0.559445\pi$
$38$	0	0
$39$	−560915.	−1.51416
$40$	0	0
$41$	245990.	0.557409	0.278704	−	0.960377i	$-0.410095\pi$
$41$	245990.	0.557409	0.278704	+	0.960377i	$0.410095\pi$
$42$	0	0
$43$	30224.0	0.0579711	0.0289856	−	0.999580i	$-0.490772\pi$
$43$	30224.0	0.0579711	0.0289856	+	0.999580i	$0.490772\pi$
$44$	0	0
$45$	339692.	0.555701
$46$	0	0
$47$	−305233.	−0.428833	−0.214417	−	0.976742i	$-0.568785\pi$
$47$	−305233.	−0.428833	−0.214417	+	0.976742i	$0.568785\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	−1.03696e6	−1.09463
$52$	0	0
$53$	−1.21053e6	−1.11689	−0.558444	−	0.829542i	$-0.688601\pi$
$53$	−1.21053e6	−1.11689	−0.558444	+	0.829542i	$0.688601\pi$
$54$	0	0
$55$	−2.00312e6	−1.62344
$56$	0	0
$57$	1.41918e6	1.01502
$58$	0	0
$59$	−1.53557e6	−0.973391	−0.486696	−	0.873572i	$-0.661798\pi$
$59$	−1.53557e6	−0.973391	−0.486696	+	0.873572i	$0.661798\pi$
$60$	0	0
$61$	−45086.4	−0.0254326	−0.0127163	−	0.999919i	$-0.504048\pi$
$61$	−45086.4	−0.0254326	−0.0127163	+	0.999919i	$0.504048\pi$
$62$	0	0
$63$	234713.	0.118262
$64$	0	0
$65$	7.18295e6	3.24419
$66$	0	0
$67$	3.08501e6	1.25312	0.626562	−	0.779371i	$-0.284461\pi$
$67$	3.08501e6	1.25312	0.626562	+	0.779371i	$0.284461\pi$
$68$	0	0
$69$	−1.35367e6	−0.496067
$70$	0	0
$71$	−1.69051e6	−0.560550	−0.280275	−	0.959920i	$-0.590426\pi$
$71$	−1.69051e6	−0.560550	−0.280275	+	0.959920i	$0.590426\pi$
$72$	0	0
$73$	−3.86932e6	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$73$	−3.86932e6	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$74$	0	0
$75$	6.52412e6	1.78570
$76$	0	0
$77$	−1.38407e6	−0.345494
$78$	0	0
$79$	2.00910e6	0.458464	0.229232	−	0.973372i	$-0.426378\pi$
$79$	2.00910e6	0.458464	0.229232	+	0.973372i	$0.426378\pi$
$80$	0	0
$81$	−2.81816e6	−0.589208
$82$	0	0
$83$	−8.28720e6	−1.59087	−0.795435	−	0.606039i	$-0.792758\pi$
$83$	−8.28720e6	−1.59087	−0.795435	+	0.606039i	$0.792758\pi$
$84$	0	0
$85$	1.32791e7	2.34532
$86$	0	0
$87$	1.39066e6	0.226414
$88$	0	0
$89$	−6.50624e6	−0.978285	−0.489142	−	0.872204i	$-0.662690\pi$
$89$	−6.50624e6	−0.978285	−0.489142	+	0.872204i	$0.662690\pi$
$90$	0	0
$91$	4.96311e6	0.690414
$92$	0	0
$93$	7.69862e6	0.992482
$94$	0	0
$95$	−1.81737e7	−2.17475
$96$	0	0
$97$	−1.14882e7	−1.27806	−0.639028	−	0.769183i	$-0.720663\pi$
$97$	−1.14882e7	−1.27806	−0.639028	+	0.769183i	$0.720663\pi$
$98$	0	0
$99$	−2.76125e6	−0.286011
$100$	0	0
$101$	−6.90453e6	−0.666821	−0.333410	−	0.942782i	$-0.608199\pi$
$101$	−6.90453e6	−0.666821	−0.333410	+	0.942782i	$0.608199\pi$
$102$	0	0
$103$	4.17396e6	0.376372	0.188186	−	0.982133i	$-0.439739\pi$
$103$	4.17396e6	0.376372	0.188186	+	0.982133i	$0.439739\pi$
$104$	0	0
$105$	6.60045e6	0.556431
$106$	0	0
$107$	−6.65685e6	−0.525322	−0.262661	−	0.964888i	$-0.584600\pi$
$107$	−6.65685e6	−0.525322	−0.262661	+	0.964888i	$0.584600\pi$
$108$	0	0
$109$	−1.11394e7	−0.823893	−0.411946	−	0.911208i	$-0.635151\pi$
$109$	−1.11394e7	−0.823893	−0.411946	+	0.911208i	$0.635151\pi$
$110$	0	0
$111$	−4.43520e6	−0.307810
$112$	0	0
$113$	4.18354e6	0.272753	0.136377	−	0.990657i	$-0.456454\pi$
$113$	4.18354e6	0.272753	0.136377	+	0.990657i	$0.456454\pi$
$114$	0	0
$115$	1.73347e7	1.06286
$116$	0	0
$117$	9.90153e6	0.571547
$118$	0	0
$119$	9.17530e6	0.499121
$120$	0	0
$121$	−3.20444e6	−0.164439
$122$	0	0
$123$	9.53574e6	0.462047
$124$	0	0
$125$	−4.47641e7	−2.04996
$126$	0	0
$127$	3.65300e7	1.58247	0.791237	−	0.611510i	$-0.209438\pi$
$127$	3.65300e7	1.58247	0.791237	+	0.611510i	$0.209438\pi$
$128$	0	0
$129$	1.17162e6	0.0480534
$130$	0	0
$131$	5.08168e7	1.97496	0.987479	−	0.157748i	$-0.0504234\pi$
$131$	5.08168e7	1.97496	0.987479	+	0.157748i	$0.0504234\pi$
$132$	0	0
$133$	−1.25572e7	−0.462821
$134$	0	0
$135$	5.52532e7	1.93281
$136$	0	0
$137$	3.03952e7	1.00991	0.504955	−	0.863146i	$-0.331509\pi$
$137$	3.03952e7	1.00991	0.504955	+	0.863146i	$0.331509\pi$
$138$	0	0
$139$	−2.92052e7	−0.922377	−0.461189	−	0.887302i	$-0.652577\pi$
$139$	−2.92052e7	−0.922377	−0.461189	+	0.887302i	$0.652577\pi$
$140$	0	0
$141$	−1.18323e7	−0.355469
$142$	0	0
$143$	−5.83880e7	−1.66973
$144$	0	0
$145$	−1.78084e7	−0.485107
$146$	0	0
$147$	4.56064e6	0.118417
$148$	0	0
$149$	−3.70950e7	−0.918677	−0.459339	−	0.888261i	$-0.651914\pi$
$149$	−3.70950e7	−0.918677	−0.459339	+	0.888261i	$0.651914\pi$
$150$	0	0
$151$	1.82394e7	0.431114	0.215557	−	0.976491i	$-0.430843\pi$
$151$	1.82394e7	0.431114	0.215557	+	0.976491i	$0.430843\pi$
$152$	0	0
$153$	1.83049e7	0.413189
$154$	0	0
$155$	−9.85867e7	−2.12646
$156$	0	0
$157$	2.12064e7	0.437340	0.218670	−	0.975799i	$-0.429828\pi$
$157$	2.12064e7	0.437340	0.218670	+	0.975799i	$0.429828\pi$
$158$	0	0
$159$	−4.69258e7	−0.925810
$160$	0	0
$161$	1.19776e7	0.226193
$162$	0	0
$163$	4.50953e7	0.815595	0.407797	−	0.913072i	$-0.366297\pi$
$163$	4.50953e7	0.815595	0.407797	+	0.913072i	$0.366297\pi$
$164$	0	0
$165$	−7.76503e7	−1.34570
$166$	0	0
$167$	6.32823e7	1.05142	0.525708	−	0.850665i	$-0.323801\pi$
$167$	6.32823e7	1.05142	0.525708	+	0.850665i	$0.323801\pi$
$168$	0	0
$169$	1.46624e8	2.33670
$170$	0	0
$171$	−2.50520e7	−0.383138
$172$	0	0
$173$	−3.34193e7	−0.490722	−0.245361	−	0.969432i	$-0.578907\pi$
$173$	−3.34193e7	−0.490722	−0.245361	+	0.969432i	$0.578907\pi$
$174$	0	0
$175$	−5.77270e7	−0.814227
$176$	0	0
$177$	−5.95260e7	−0.806863
$178$	0	0
$179$	−1.29965e8	−1.69372	−0.846861	−	0.531814i	$-0.821511\pi$
$179$	−1.29965e8	−1.69372	−0.846861	+	0.531814i	$0.821511\pi$
$180$	0	0
$181$	1.91607e7	0.240180	0.120090	−	0.992763i	$-0.461682\pi$
$181$	1.91607e7	0.240180	0.120090	+	0.992763i	$0.461682\pi$
$182$	0	0
$183$	−1.74776e6	−0.0210816
$184$	0	0
$185$	5.67961e7	0.659504
$186$	0	0
$187$	−1.07942e8	−1.20710
$188$	0	0
$189$	3.81776e7	0.411332
$190$	0	0
$191$	3.54364e7	0.367987	0.183994	−	0.982927i	$-0.441097\pi$
$191$	3.54364e7	0.367987	0.183994	+	0.982927i	$0.441097\pi$
$192$	0	0
$193$	1.80339e8	1.80567	0.902837	−	0.429983i	$-0.141481\pi$
$193$	1.80339e8	1.80567	0.902837	+	0.429983i	$0.141481\pi$
$194$	0	0
$195$	2.78445e8	2.68917
$196$	0	0
$197$	−1.39463e8	−1.29965	−0.649826	−	0.760083i	$-0.725158\pi$
$197$	−1.39463e8	−1.29965	−0.649826	+	0.760083i	$0.725158\pi$
$198$	0	0
$199$	1.46312e8	1.31612	0.658060	−	0.752965i	$-0.271377\pi$
$199$	1.46312e8	1.31612	0.658060	+	0.752965i	$0.271377\pi$
$200$	0	0
$201$	1.19590e8	1.03874
$202$	0	0
$203$	−1.23049e7	−0.103238
$204$	0	0
$205$	−1.22112e8	−0.989969
$206$	0	0
$207$	2.38956e7	0.187250
$208$	0	0
$209$	1.47728e8	1.11931
$210$	0	0
$211$	−1.02935e8	−0.754351	−0.377176	−	0.926142i	$-0.623105\pi$
$211$	−1.02935e8	−0.754351	−0.377176	+	0.926142i	$0.623105\pi$
$212$	0	0
$213$	−6.55324e7	−0.464651
$214$	0	0
$215$	−1.50035e7	−0.102958
$216$	0	0
$217$	−6.81193e7	−0.452544
$218$	0	0
$219$	−1.49993e8	−0.964978
$220$	0	0
$221$	3.87067e8	2.41220
$222$	0	0
$223$	−3.26446e8	−1.97126	−0.985632	−	0.168906i	$-0.945977\pi$
$223$	−3.26446e8	−1.97126	−0.985632	+	0.168906i	$0.945977\pi$
$224$	0	0
$225$	−1.15167e8	−0.674044
$226$	0	0
$227$	−8.99538e7	−0.510422	−0.255211	−	0.966885i	$-0.582145\pi$
$227$	−8.99538e7	−0.510422	−0.255211	+	0.966885i	$0.582145\pi$
$228$	0	0
$229$	−1.01929e8	−0.560886	−0.280443	−	0.959871i	$-0.590481\pi$
$229$	−1.01929e8	−0.560886	−0.280443	+	0.959871i	$0.590481\pi$
$230$	0	0
$231$	−5.36531e7	−0.286387
$232$	0	0
$233$	−4.26915e7	−0.221103	−0.110552	−	0.993870i	$-0.535262\pi$
$233$	−4.26915e7	−0.221103	−0.110552	+	0.993870i	$0.535262\pi$
$234$	0	0
$235$	1.51521e8	0.761616
$236$	0	0
$237$	7.78821e7	0.380030
$238$	0	0
$239$	−6.02163e7	−0.285313	−0.142656	−	0.989772i	$-0.545564\pi$
$239$	−6.02163e7	−0.285313	−0.142656	+	0.989772i	$0.545564\pi$
$240$	0	0
$241$	−4.71536e7	−0.216998	−0.108499	−	0.994097i	$-0.534604\pi$
$241$	−4.71536e7	−0.216998	−0.108499	+	0.994097i	$0.534604\pi$
$242$	0	0
$243$	1.34179e8	0.599876
$244$	0	0
$245$	−5.84024e7	−0.253717
$246$	0	0
$247$	−5.29736e8	−2.23677
$248$	0	0
$249$	−3.21251e8	−1.31870
$250$	0	0
$251$	1.94014e8	0.774416	0.387208	−	0.921992i	$-0.373440\pi$
$251$	1.94014e8	0.774416	0.387208	+	0.921992i	$0.373440\pi$
$252$	0	0
$253$	−1.40909e8	−0.547037
$254$	0	0
$255$	5.14761e8	1.94408
$256$	0	0
$257$	−6.78743e7	−0.249425	−0.124712	−	0.992193i	$-0.539801\pi$
$257$	−6.78743e7	−0.249425	−0.124712	+	0.992193i	$0.539801\pi$
$258$	0	0
$259$	3.92437e7	0.140353
$260$	0	0
$261$	−2.45485e7	−0.0854640
$262$	0	0
$263$	1.61200e8	0.546410	0.273205	−	0.961956i	$-0.411916\pi$
$263$	1.61200e8	0.546410	0.273205	+	0.961956i	$0.411916\pi$
$264$	0	0
$265$	6.00921e8	1.98361
$266$	0	0
$267$	−2.52213e8	−0.810920
$268$	0	0
$269$	−2.64104e8	−0.827261	−0.413631	−	0.910445i	$-0.635739\pi$
$269$	−2.64104e8	−0.827261	−0.413631	+	0.910445i	$0.635739\pi$
$270$	0	0
$271$	−4.92433e7	−0.150299	−0.0751493	−	0.997172i	$-0.523943\pi$
$271$	−4.92433e7	−0.150299	−0.0751493	+	0.997172i	$0.523943\pi$
$272$	0	0
$273$	1.92394e8	0.572298
$274$	0	0
$275$	6.79123e8	1.96917
$276$	0	0
$277$	−2.85803e8	−0.807957	−0.403978	−	0.914769i	$-0.632373\pi$
$277$	−2.85803e8	−0.807957	−0.403978	+	0.914769i	$0.632373\pi$
$278$	0	0
$279$	−1.35900e8	−0.374631
$280$	0	0
$281$	5.50944e8	1.48127	0.740637	−	0.671905i	$-0.234524\pi$
$281$	5.50944e8	1.48127	0.740637	+	0.671905i	$0.234524\pi$
$282$	0	0
$283$	2.84145e8	0.745226	0.372613	−	0.927987i	$-0.378462\pi$
$283$	2.84145e8	0.745226	0.372613	+	0.927987i	$0.378462\pi$
$284$	0	0
$285$	−7.04498e8	−1.80270
$286$	0	0
$287$	−8.43745e7	−0.210681
$288$	0	0
$289$	3.05231e8	0.743852
$290$	0	0
$291$	−4.45337e8	−1.05941
$292$	0	0
$293$	4.33632e7	0.100713	0.0503564	−	0.998731i	$-0.483964\pi$
$293$	4.33632e7	0.100713	0.0503564	+	0.998731i	$0.483964\pi$
$294$	0	0
$295$	7.62276e8	1.72876
$296$	0	0
$297$	−4.49136e8	−0.994788
$298$	0	0
$299$	5.05283e8	1.09317
$300$	0	0
$301$	−1.03668e7	−0.0219110
$302$	0	0
$303$	−2.67652e8	−0.552741
$304$	0	0
$305$	2.23814e7	0.0451688
$306$	0	0
$307$	−4.07582e8	−0.803954	−0.401977	−	0.915650i	$-0.631677\pi$
$307$	−4.07582e8	−0.803954	−0.401977	+	0.915650i	$0.631677\pi$
$308$	0	0
$309$	1.61802e8	0.311983
$310$	0	0
$311$	7.68264e8	1.44827	0.724134	−	0.689659i	$-0.242240\pi$
$311$	7.68264e8	1.44827	0.724134	+	0.689659i	$0.242240\pi$
$312$	0	0
$313$	4.31310e8	0.795031	0.397515	−	0.917596i	$-0.369873\pi$
$313$	4.31310e8	0.795031	0.397515	+	0.917596i	$0.369873\pi$
$314$	0	0
$315$	−1.16514e8	−0.210035
$316$	0	0
$317$	1.57984e8	0.278551	0.139276	−	0.990254i	$-0.455523\pi$
$317$	1.57984e8	0.278551	0.139276	+	0.990254i	$0.455523\pi$
$318$	0	0
$319$	1.44759e8	0.249677
$320$	0	0
$321$	−2.58051e8	−0.435450
$322$	0	0
$323$	−9.79323e8	−1.61703
$324$	0	0
$325$	−2.43526e9	−3.93507
$326$	0	0
$327$	−4.31818e8	−0.682941
$328$	0	0
$329$	1.04695e8	0.162084
$330$	0	0
$331$	9.75366e8	1.47832	0.739162	−	0.673527i	$-0.235222\pi$
$331$	9.75366e8	1.47832	0.739162	+	0.673527i	$0.235222\pi$
$332$	0	0
$333$	7.82922e7	0.116189
$334$	0	0
$335$	−1.53144e9	−2.22557
$336$	0	0
$337$	−9.07880e8	−1.29218	−0.646091	−	0.763260i	$-0.723597\pi$
$337$	−9.07880e8	−1.29218	−0.646091	+	0.763260i	$0.723597\pi$
$338$	0	0
$339$	1.62174e8	0.226090
$340$	0	0
$341$	8.01382e8	1.09446
$342$	0	0
$343$	−4.03536e7	−0.0539949
$344$	0	0
$345$	6.71977e8	0.881024
$346$	0	0
$347$	−5.79720e7	−0.0744843	−0.0372422	−	0.999306i	$-0.511857\pi$
$347$	−5.79720e7	−0.0744843	−0.0372422	+	0.999306i	$0.511857\pi$
$348$	0	0
$349$	−1.33879e9	−1.68587	−0.842935	−	0.538016i	$-0.819174\pi$
$349$	−1.33879e9	−1.68587	−0.842935	+	0.538016i	$0.819174\pi$
$350$	0	0
$351$	1.61055e9	1.98792
$352$	0	0
$353$	5.82234e8	0.704509	0.352254	−	0.935904i	$-0.385415\pi$
$353$	5.82234e8	0.704509	0.352254	+	0.935904i	$0.385415\pi$
$354$	0	0
$355$	8.39192e8	0.995548
$356$	0	0
$357$	3.55678e8	0.413731
$358$	0	0
$359$	−3.20435e8	−0.365518	−0.182759	−	0.983158i	$-0.558503\pi$
$359$	−3.20435e8	−0.365518	−0.182759	+	0.983158i	$0.558503\pi$
$360$	0	0
$361$	4.46421e8	0.499424
$362$	0	0
$363$	−1.24219e8	−0.136306
$364$	0	0
$365$	1.92078e9	2.06753
$366$	0	0
$367$	6.55277e8	0.691981	0.345990	−	0.938238i	$-0.387543\pi$
$367$	6.55277e8	0.691981	0.345990	+	0.938238i	$0.387543\pi$
$368$	0	0
$369$	−1.68329e8	−0.174408
$370$	0	0
$371$	4.15211e8	0.422144
$372$	0	0
$373$	−5.42928e8	−0.541703	−0.270851	−	0.962621i	$-0.587305\pi$
$373$	−5.42928e8	−0.541703	−0.270851	+	0.962621i	$0.587305\pi$
$374$	0	0
$375$	−1.73527e9	−1.69925
$376$	0	0
$377$	−5.19090e8	−0.498940
$378$	0	0
$379$	−1.57480e9	−1.48590	−0.742949	−	0.669348i	$-0.766573\pi$
$379$	−1.57480e9	−1.48590	−0.742949	+	0.669348i	$0.766573\pi$
$380$	0	0
$381$	1.41608e9	1.31174
$382$	0	0
$383$	−8.88822e8	−0.808386	−0.404193	−	0.914674i	$-0.632448\pi$
$383$	−8.88822e8	−0.808386	−0.404193	+	0.914674i	$0.632448\pi$
$384$	0	0
$385$	6.87069e8	0.613603
$386$	0	0
$387$	−2.06820e7	−0.0181387
$388$	0	0
$389$	1.34673e9	1.15999	0.579997	−	0.814619i	$-0.303054\pi$
$389$	1.34673e9	1.15999	0.579997	+	0.814619i	$0.303054\pi$
$390$	0	0
$391$	9.34116e8	0.790282
$392$	0	0
$393$	1.96990e9	1.63708
$394$	0	0
$395$	−9.97340e8	−0.814242
$396$	0	0
$397$	2.15927e9	1.73197	0.865985	−	0.500070i	$-0.166693\pi$
$397$	2.15927e9	1.73197	0.865985	+	0.500070i	$0.166693\pi$
$398$	0	0
$399$	−4.86778e8	−0.383642
$400$	0	0
$401$	−1.87617e9	−1.45300	−0.726500	−	0.687166i	$-0.758854\pi$
$401$	−1.87617e9	−1.45300	−0.726500	+	0.687166i	$0.758854\pi$
$402$	0	0
$403$	−2.87366e9	−2.18710
$404$	0	0
$405$	1.39897e9	1.04644
$406$	0	0
$407$	−4.61678e8	−0.339437
$408$	0	0
$409$	2.27449e6	0.00164381	0.000821906	−	1.00000i	$-0.499738\pi$
$409$	2.27449e6	0.00164381	0.000821906		1.00000i	$0.499738\pi$
$410$	0	0
$411$	1.17826e9	0.837135
$412$	0	0
$413$	5.26700e8	0.367907
$414$	0	0
$415$	4.11387e9	2.82542
$416$	0	0
$417$	−1.13213e9	−0.764577
$418$	0	0
$419$	−2.40944e9	−1.60017	−0.800087	−	0.599884i	$-0.795213\pi$
$419$	−2.40944e9	−1.60017	−0.800087	+	0.599884i	$0.795213\pi$
$420$	0	0
$421$	−1.08307e9	−0.707404	−0.353702	−	0.935358i	$-0.615077\pi$
$421$	−1.08307e9	−0.707404	−0.353702	+	0.935358i	$0.615077\pi$
$422$	0	0
$423$	2.08869e8	0.134178
$424$	0	0
$425$	−4.50205e9	−2.84479
$426$	0	0
$427$	1.54646e7	0.00961262
$428$	0	0
$429$	−2.26340e9	−1.38408
$430$	0	0
$431$	5.05987e8	0.304417	0.152208	−	0.988348i	$-0.451361\pi$
$431$	5.05987e8	0.304417	0.152208	+	0.988348i	$0.451361\pi$
$432$	0	0
$433$	3.73323e8	0.220992	0.110496	−	0.993877i	$-0.464756\pi$
$433$	3.73323e8	0.220992	0.110496	+	0.993877i	$0.464756\pi$
$434$	0	0
$435$	−6.90339e8	−0.402115
$436$	0	0
$437$	−1.27842e9	−0.732807
$438$	0	0
$439$	−8.88930e6	−0.00501466	−0.00250733	−	0.999997i	$-0.500798\pi$
$439$	−8.88930e6	−0.00501466	−0.00250733	+	0.999997i	$0.500798\pi$
$440$	0	0
$441$	−8.05064e7	−0.0446988
$442$	0	0
$443$	1.02405e9	0.559638	0.279819	−	0.960053i	$-0.409726\pi$
$443$	1.02405e9	0.559638	0.279819	+	0.960053i	$0.409726\pi$
$444$	0	0
$445$	3.22978e9	1.73745
$446$	0	0
$447$	−1.43798e9	−0.761510
$448$	0	0
$449$	8.32711e8	0.434142	0.217071	−	0.976156i	$-0.430350\pi$
$449$	8.32711e8	0.434142	0.217071	+	0.976156i	$0.430350\pi$
$450$	0	0
$451$	9.92615e8	0.509522
$452$	0	0
$453$	7.07047e8	0.357359
$454$	0	0
$455$	−2.46375e9	−1.22619
$456$	0	0
$457$	−2.28781e9	−1.12128	−0.560639	−	0.828060i	$-0.689444\pi$
$457$	−2.28781e9	−1.12128	−0.560639	+	0.828060i	$0.689444\pi$
$458$	0	0
$459$	2.97742e9	1.43713
$460$	0	0
$461$	7.84459e8	0.372921	0.186461	−	0.982462i	$-0.440298\pi$
$461$	7.84459e8	0.372921	0.186461	+	0.982462i	$0.440298\pi$
$462$	0	0
$463$	−2.31590e9	−1.08439	−0.542196	−	0.840252i	$-0.682407\pi$
$463$	−2.31590e9	−1.08439	−0.542196	+	0.840252i	$0.682407\pi$
$464$	0	0
$465$	−3.82169e9	−1.76267
$466$	0	0
$467$	−6.12659e8	−0.278362	−0.139181	−	0.990267i	$-0.544447\pi$
$467$	−6.12659e8	−0.278362	−0.139181	+	0.990267i	$0.544447\pi$
$468$	0	0
$469$	−1.05816e9	−0.473637
$470$	0	0
$471$	8.22063e8	0.362520
$472$	0	0
$473$	1.21959e8	0.0529908
$474$	0	0
$475$	6.16147e9	2.63789
$476$	0	0
$477$	8.28356e8	0.349464
$478$	0	0
$479$	7.28950e8	0.303056	0.151528	−	0.988453i	$-0.451581\pi$
$479$	7.28950e8	0.303056	0.151528	+	0.988453i	$0.451581\pi$
$480$	0	0
$481$	1.65553e9	0.678310
$482$	0	0
$483$	4.64308e8	0.187496
$484$	0	0
$485$	5.70287e9	2.26985
$486$	0	0
$487$	−8.68384e8	−0.340691	−0.170346	−	0.985384i	$-0.554488\pi$
$487$	−8.68384e8	−0.340691	−0.170346	+	0.985384i	$0.554488\pi$
$488$	0	0
$489$	1.74811e9	0.676063
$490$	0	0
$491$	−3.47503e9	−1.32487	−0.662435	−	0.749119i	$-0.730477\pi$
$491$	−3.47503e9	−1.32487	−0.662435	+	0.749119i	$0.730477\pi$
$492$	0	0
$493$	−9.59641e8	−0.360699
$494$	0	0
$495$	1.37072e9	0.507961
$496$	0	0
$497$	5.79846e8	0.211868
$498$	0	0
$499$	−3.68171e9	−1.32647	−0.663236	−	0.748410i	$-0.730817\pi$
$499$	−3.68171e9	−1.32647	−0.663236	+	0.748410i	$0.730817\pi$
$500$	0	0
$501$	2.45312e9	0.871540
$502$	0	0
$503$	3.79099e9	1.32820	0.664102	−	0.747642i	$-0.268814\pi$
$503$	3.79099e9	1.32820	0.664102	+	0.747642i	$0.268814\pi$
$504$	0	0
$505$	3.42749e9	1.18429
$506$	0	0
$507$	5.68385e9	1.93693
$508$	0	0
$509$	−4.09313e9	−1.37576	−0.687880	−	0.725824i	$-0.741459\pi$
$509$	−4.09313e9	−1.37576	−0.687880	+	0.725824i	$0.741459\pi$
$510$	0	0
$511$	1.32718e9	0.440003
$512$	0	0
$513$	−4.07488e9	−1.33261
$514$	0	0
$515$	−2.07200e9	−0.668445
$516$	0	0
$517$	−1.23167e9	−0.391993
$518$	0	0
$519$	−1.29549e9	−0.406770
$520$	0	0
$521$	1.69546e9	0.525237	0.262618	−	0.964900i	$-0.415414\pi$
$521$	1.69546e9	0.525237	0.262618	+	0.964900i	$0.415414\pi$
$522$	0	0
$523$	−1.53714e9	−0.469849	−0.234924	−	0.972014i	$-0.575484\pi$
$523$	−1.53714e9	−0.469849	−0.234924	+	0.972014i	$0.575484\pi$
$524$	0	0
$525$	−2.23777e9	−0.674929
$526$	0	0
$527$	−5.31254e9	−1.58112
$528$	0	0
$529$	−2.18542e9	−0.641858
$530$	0	0
$531$	1.05078e9	0.304566
$532$	0	0
$533$	−3.55940e9	−1.01820
$534$	0	0
$535$	3.30454e9	0.932982
$536$	0	0
$537$	−5.03808e9	−1.40396
$538$	0	0
$539$	4.74736e8	0.130584
$540$	0	0
$541$	−4.34509e9	−1.17980	−0.589901	−	0.807476i	$-0.700833\pi$
$541$	−4.34509e9	−1.17980	−0.589901	+	0.807476i	$0.700833\pi$
$542$	0	0
$543$	7.42760e8	0.199090
$544$	0	0
$545$	5.52976e9	1.46325
$546$	0	0
$547$	2.30195e9	0.601367	0.300683	−	0.953724i	$-0.402785\pi$
$547$	2.30195e9	0.601367	0.300683	+	0.953724i	$0.402785\pi$
$548$	0	0
$549$	3.08523e7	0.00795764
$550$	0	0
$551$	1.31336e9	0.334466
$552$	0	0
$553$	−6.89120e8	−0.173283
$554$	0	0
$555$	2.20169e9	0.546676
$556$	0	0
$557$	1.71107e9	0.419541	0.209770	−	0.977751i	$-0.432728\pi$
$557$	1.71107e9	0.419541	0.209770	+	0.977751i	$0.432728\pi$
$558$	0	0
$559$	−4.37332e8	−0.105894
$560$	0	0
$561$	−4.18434e9	−1.00059
$562$	0	0
$563$	2.16633e9	0.511617	0.255809	−	0.966727i	$-0.417658\pi$
$563$	2.16633e9	0.511617	0.255809	+	0.966727i	$0.417658\pi$
$564$	0	0
$565$	−2.07676e9	−0.484415
$566$	0	0
$567$	9.66630e8	0.222700
$568$	0	0
$569$	−2.76063e9	−0.628225	−0.314112	−	0.949386i	$-0.601707\pi$
$569$	−2.76063e9	−0.628225	−0.314112	+	0.949386i	$0.601707\pi$
$570$	0	0
$571$	7.25907e8	0.163175	0.0815877	−	0.996666i	$-0.474001\pi$
$571$	7.25907e8	0.163175	0.0815877	+	0.996666i	$0.474001\pi$
$572$	0	0
$573$	1.37368e9	0.305032
$574$	0	0
$575$	−5.87705e9	−1.28921
$576$	0	0
$577$	−5.37035e9	−1.16382	−0.581911	−	0.813252i	$-0.697695\pi$
$577$	−5.37035e9	−1.16382	−0.581911	+	0.813252i	$0.697695\pi$
$578$	0	0
$579$	6.99080e9	1.49676
$580$	0	0
$581$	2.84251e9	0.601292
$582$	0	0
$583$	−4.88471e9	−1.02094
$584$	0	0
$585$	−4.91524e9	−1.01508
$586$	0	0
$587$	−7.86145e9	−1.60424	−0.802120	−	0.597163i	$-0.796295\pi$
$587$	−7.86145e9	−1.60424	−0.802120	+	0.597163i	$0.796295\pi$
$588$	0	0
$589$	7.27069e9	1.46613
$590$	0	0
$591$	−5.40625e9	−1.07731
$592$	0	0
$593$	5.44412e9	1.07210	0.536051	−	0.844185i	$-0.319915\pi$
$593$	5.44412e9	1.07210	0.536051	+	0.844185i	$0.319915\pi$
$594$	0	0
$595$	−4.55473e9	−0.886448
$596$	0	0
$597$	5.67177e9	1.09096
$598$	0	0
$599$	7.97408e9	1.51596	0.757978	−	0.652280i	$-0.226188\pi$
$599$	7.97408e9	1.51596	0.757978	+	0.652280i	$0.226188\pi$
$600$	0	0
$601$	−6.27378e8	−0.117888	−0.0589439	−	0.998261i	$-0.518773\pi$
$601$	−6.27378e8	−0.117888	−0.0589439	+	0.998261i	$0.518773\pi$
$602$	0	0
$603$	−2.11105e9	−0.392092
$604$	0	0
$605$	1.59072e9	0.292046
$606$	0	0
$607$	3.79758e9	0.689202	0.344601	−	0.938749i	$-0.388014\pi$
$607$	3.79758e9	0.689202	0.344601	+	0.938749i	$0.388014\pi$
$608$	0	0
$609$	−4.76995e8	−0.0855763
$610$	0	0
$611$	4.41663e9	0.783334
$612$	0	0
$613$	1.06120e10	1.86074	0.930369	−	0.366625i	$-0.119487\pi$
$613$	1.06120e10	1.86074	0.930369	+	0.366625i	$0.119487\pi$
$614$	0	0
$615$	−4.73366e9	−0.820605
$616$	0	0
$617$	2.26828e8	0.0388776	0.0194388	−	0.999811i	$-0.493812\pi$
$617$	2.26828e8	0.0388776	0.0194388	+	0.999811i	$0.493812\pi$
$618$	0	0
$619$	4.91801e9	0.833435	0.416718	−	0.909036i	$-0.363180\pi$
$619$	4.91801e9	0.833435	0.416718	+	0.909036i	$0.363180\pi$
$620$	0	0
$621$	3.88678e9	0.651282
$622$	0	0
$623$	2.23164e9	0.369757
$624$	0	0
$625$	9.07300e9	1.48652
$626$	0	0
$627$	5.72665e9	0.927821
$628$	0	0
$629$	3.06057e9	0.490371
$630$	0	0
$631$	6.22228e9	0.985931	0.492966	−	0.870049i	$-0.335913\pi$
$631$	6.22228e9	0.985931	0.492966	+	0.870049i	$0.335913\pi$
$632$	0	0
$633$	−3.99024e9	−0.625297
$634$	0	0
$635$	−1.81339e10	−2.81050
$636$	0	0
$637$	−1.70235e9	−0.260952
$638$	0	0
$639$	1.15681e9	0.175391
$640$	0	0
$641$	−2.71355e9	−0.406944	−0.203472	−	0.979081i	$-0.565223\pi$
$641$	−2.71355e9	−0.406944	−0.203472	+	0.979081i	$0.565223\pi$
$642$	0	0
$643$	−5.96086e9	−0.884240	−0.442120	−	0.896956i	$-0.645774\pi$
$643$	−5.96086e9	−0.884240	−0.442120	+	0.896956i	$0.645774\pi$
$644$	0	0
$645$	−5.81609e8	−0.0853438
$646$	0	0
$647$	1.87137e9	0.271640	0.135820	−	0.990734i	$-0.456633\pi$
$647$	1.87137e9	0.271640	0.135820	+	0.990734i	$0.456633\pi$
$648$	0	0
$649$	−6.19631e9	−0.889768
$650$	0	0
$651$	−2.64063e9	−0.375123
$652$	0	0
$653$	8.68243e9	1.22024	0.610120	−	0.792309i	$-0.291121\pi$
$653$	8.68243e9	1.22024	0.610120	+	0.792309i	$0.291121\pi$
$654$	0	0
$655$	−2.52261e10	−3.50756
$656$	0	0
$657$	2.64775e9	0.364249
$658$	0	0
$659$	1.17028e10	1.59291	0.796457	−	0.604696i	$-0.206705\pi$
$659$	1.17028e10	1.59291	0.796457	+	0.604696i	$0.206705\pi$
$660$	0	0
$661$	−4.23524e9	−0.570391	−0.285195	−	0.958469i	$-0.592059\pi$
$661$	−4.23524e9	−0.570391	−0.285195	+	0.958469i	$0.592059\pi$
$662$	0	0
$663$	1.50046e10	1.99952
$664$	0	0
$665$	6.23357e9	0.821979
$666$	0	0
$667$	−1.25273e9	−0.163462
$668$	0	0
$669$	−1.26546e10	−1.63402
$670$	0	0
$671$	−1.81932e8	−0.0232477
$672$	0	0
$673$	−1.64366e9	−0.207854	−0.103927	−	0.994585i	$-0.533141\pi$
$673$	−1.64366e9	−0.207854	−0.103927	+	0.994585i	$0.533141\pi$
$674$	0	0
$675$	−1.87327e10	−2.34442
$676$	0	0
$677$	1.08838e10	1.34810	0.674048	−	0.738688i	$-0.264554\pi$
$677$	1.08838e10	1.34810	0.674048	+	0.738688i	$0.264554\pi$
$678$	0	0
$679$	3.94045e9	0.483060
$680$	0	0
$681$	−3.48704e9	−0.423099
$682$	0	0
$683$	−3.85277e9	−0.462701	−0.231351	−	0.972870i	$-0.574314\pi$
$683$	−3.85277e9	−0.462701	−0.231351	+	0.972870i	$0.574314\pi$
$684$	0	0
$685$	−1.50885e10	−1.79362
$686$	0	0
$687$	−3.95126e9	−0.464930
$688$	0	0
$689$	1.75160e10	2.04018
$690$	0	0
$691$	−1.55439e10	−1.79220	−0.896100	−	0.443852i	$-0.853612\pi$
$691$	−1.55439e10	−1.79220	−0.896100	+	0.443852i	$0.853612\pi$
$692$	0	0
$693$	9.47109e8	0.108102
$694$	0	0
$695$	1.44978e10	1.63816
$696$	0	0
$697$	−6.58026e9	−0.736086
$698$	0	0
$699$	−1.65493e9	−0.183277
$700$	0	0
$701$	−7.75099e9	−0.849854	−0.424927	−	0.905228i	$-0.639700\pi$
$701$	−7.75099e9	−0.849854	−0.424927	+	0.905228i	$0.639700\pi$
$702$	0	0
$703$	−4.18867e9	−0.454707
$704$	0	0
$705$	5.87369e9	0.631319
$706$	0	0
$707$	2.36825e9	0.252035
$708$	0	0
$709$	−1.14664e10	−1.20827	−0.604137	−	0.796881i	$-0.706482\pi$
$709$	−1.14664e10	−1.20827	−0.604137	+	0.796881i	$0.706482\pi$
$710$	0	0
$711$	−1.37481e9	−0.143450
$712$	0	0
$713$	−6.93507e9	−0.716535
$714$	0	0
$715$	2.89845e10	2.96548
$716$	0	0
$717$	−2.33427e9	−0.236502
$718$	0	0
$719$	−1.63852e10	−1.64399	−0.821997	−	0.569492i	$-0.807140\pi$
$719$	−1.63852e10	−1.64399	−0.821997	+	0.569492i	$0.807140\pi$
$720$	0	0
$721$	−1.43167e9	−0.142255
$722$	0	0
$723$	−1.82790e9	−0.179874
$724$	0	0
$725$	6.03765e9	0.588416
$726$	0	0
$727$	1.66277e10	1.60495	0.802474	−	0.596687i	$-0.203517\pi$
$727$	1.66277e10	1.60495	0.802474	+	0.596687i	$0.203517\pi$
$728$	0	0
$729$	1.13647e10	1.08646
$730$	0	0
$731$	−8.08495e8	−0.0765537
$732$	0	0
$733$	1.45773e10	1.36714	0.683568	−	0.729886i	$-0.260427\pi$
$733$	1.45773e10	1.36714	0.683568	+	0.729886i	$0.260427\pi$
$734$	0	0
$735$	−2.26396e9	−0.210311
$736$	0	0
$737$	1.24486e10	1.14547
$738$	0	0
$739$	4.30003e9	0.391937	0.195968	−	0.980610i	$-0.437215\pi$
$739$	4.30003e9	0.391937	0.195968	+	0.980610i	$0.437215\pi$
$740$	0	0
$741$	−2.05351e10	−1.85410
$742$	0	0
$743$	−9.14731e9	−0.818150	−0.409075	−	0.912501i	$-0.634148\pi$
$743$	−9.14731e9	−0.818150	−0.409075	+	0.912501i	$0.634148\pi$
$744$	0	0
$745$	1.84144e10	1.63159
$746$	0	0
$747$	5.67088e9	0.497769
$748$	0	0
$749$	2.28330e9	0.198553
$750$	0	0
$751$	1.71874e9	0.148071	0.0740354	−	0.997256i	$-0.476412\pi$
$751$	1.71874e9	0.148071	0.0740354	+	0.997256i	$0.476412\pi$
$752$	0	0
$753$	7.52089e9	0.641929
$754$	0	0
$755$	−9.05428e9	−0.765666
$756$	0	0
$757$	−1.58386e10	−1.32703	−0.663517	−	0.748161i	$-0.730937\pi$
$757$	−1.58386e10	−1.32703	−0.663517	+	0.748161i	$0.730937\pi$
$758$	0	0
$759$	−5.46230e9	−0.453450
$760$	0	0
$761$	3.29329e9	0.270884	0.135442	−	0.990785i	$-0.456755\pi$
$761$	3.29329e9	0.270884	0.135442	+	0.990785i	$0.456755\pi$
$762$	0	0
$763$	3.82083e9	0.311402
$764$	0	0
$765$	−9.08680e9	−0.733831
$766$	0	0
$767$	2.22193e10	1.77806
$768$	0	0
$769$	−1.80581e10	−1.43196	−0.715978	−	0.698123i	$-0.754019\pi$
$769$	−1.80581e10	−1.43196	−0.715978	+	0.698123i	$0.754019\pi$
$770$	0	0
$771$	−2.63113e9	−0.206753
$772$	0	0
$773$	−1.57529e10	−1.22668	−0.613342	−	0.789817i	$-0.710175\pi$
$773$	−1.57529e10	−1.22668	−0.613342	+	0.789817i	$0.710175\pi$
$774$	0	0
$775$	3.34242e10	2.57932
$776$	0	0
$777$	1.52127e9	0.116341
$778$	0	0
$779$	9.00569e9	0.682552
$780$	0	0
$781$	−6.82154e9	−0.512394
$782$	0	0
$783$	−3.99298e9	−0.297257
$784$	0	0
$785$	−1.05271e10	−0.776724
$786$	0	0
$787$	1.02224e10	0.747551	0.373776	−	0.927519i	$-0.378063\pi$
$787$	1.02224e10	0.747551	0.373776	+	0.927519i	$0.378063\pi$
$788$	0	0
$789$	6.24886e9	0.452930
$790$	0	0
$791$	−1.43495e9	−0.103091
$792$	0	0
$793$	6.52387e8	0.0464568
$794$	0	0
$795$	2.32946e10	1.64426
$796$	0	0
$797$	1.44391e10	1.01027	0.505134	−	0.863041i	$-0.331443\pi$
$797$	1.44391e10	1.01027	0.505134	+	0.863041i	$0.331443\pi$
$798$	0	0
$799$	8.16502e9	0.566296
$800$	0	0
$801$	4.45218e9	0.306097
$802$	0	0
$803$	−1.56134e10	−1.06413
$804$	0	0
$805$	−5.94582e9	−0.401722
$806$	0	0
$807$	−1.02379e10	−0.685733
$808$	0	0
$809$	−1.00885e10	−0.669897	−0.334949	−	0.942236i	$-0.608719\pi$
$809$	−1.00885e10	−0.669897	−0.334949	+	0.942236i	$0.608719\pi$
$810$	0	0
$811$	1.35981e9	0.0895172	0.0447586	−	0.998998i	$-0.485748\pi$
$811$	1.35981e9	0.0895172	0.0447586	+	0.998998i	$0.485748\pi$
$812$	0	0
$813$	−1.90890e9	−0.124585
$814$	0	0
$815$	−2.23859e10	−1.44851
$816$	0	0
$817$	1.10650e9	0.0709862
$818$	0	0
$819$	−3.39623e9	−0.216024
$820$	0	0
$821$	−1.49802e10	−0.944751	−0.472375	−	0.881397i	$-0.656603\pi$
$821$	−1.49802e10	−0.944751	−0.472375	+	0.881397i	$0.656603\pi$
$822$	0	0
$823$	−1.96016e10	−1.22572	−0.612861	−	0.790191i	$-0.709981\pi$
$823$	−1.96016e10	−1.22572	−0.612861	+	0.790191i	$0.709981\pi$
$824$	0	0
$825$	2.63260e10	1.63229
$826$	0	0
$827$	1.68768e9	0.103758	0.0518789	−	0.998653i	$-0.483479\pi$
$827$	1.68768e9	0.103758	0.0518789	+	0.998653i	$0.483479\pi$
$828$	0	0
$829$	−1.64541e9	−0.100307	−0.0501536	−	0.998742i	$-0.515971\pi$
$829$	−1.64541e9	−0.100307	−0.0501536	+	0.998742i	$0.515971\pi$
$830$	0	0
$831$	−1.10791e10	−0.669732
$832$	0	0
$833$	−3.14713e9	−0.188650
$834$	0	0
$835$	−3.14141e10	−1.86733
$836$	0	0
$837$	−2.21050e10	−1.30302
$838$	0	0
$839$	−3.49725e9	−0.204437	−0.102219	−	0.994762i	$-0.532594\pi$
$839$	−3.49725e9	−0.204437	−0.102219	+	0.994762i	$0.532594\pi$
$840$	0	0
$841$	−1.59629e10	−0.925393
$842$	0	0
$843$	2.13572e10	1.22786
$844$	0	0
$845$	−7.27861e10	−4.15002
$846$	0	0
$847$	1.09912e9	0.0621519
$848$	0	0
$849$	1.10148e10	0.617733
$850$	0	0
$851$	3.99531e9	0.222227
$852$	0	0
$853$	2.86798e10	1.58218	0.791088	−	0.611702i	$-0.209515\pi$
$853$	2.86798e10	1.58218	0.791088	+	0.611702i	$0.209515\pi$
$854$	0	0
$855$	1.24361e10	0.680461
$856$	0	0
$857$	3.67333e10	1.99355	0.996776	−	0.0802366i	$-0.0255676\pi$
$857$	3.67333e10	1.99355	0.996776	+	0.0802366i	$0.0255676\pi$
$858$	0	0
$859$	5.54734e9	0.298613	0.149306	−	0.988791i	$-0.452296\pi$
$859$	5.54734e9	0.298613	0.149306	+	0.988791i	$0.452296\pi$
$860$	0	0
$861$	−3.27076e9	−0.174638
$862$	0	0
$863$	5.26057e9	0.278609	0.139305	−	0.990250i	$-0.455513\pi$
$863$	5.26057e9	0.278609	0.139305	+	0.990250i	$0.455513\pi$
$864$	0	0
$865$	1.65897e10	0.871532
$866$	0	0
$867$	1.18322e10	0.616594
$868$	0	0
$869$	8.10707e9	0.419078
$870$	0	0
$871$	−4.46392e10	−2.28904
$872$	0	0
$873$	7.86129e9	0.399893
$874$	0	0
$875$	1.53541e10	0.774811
$876$	0	0
$877$	2.83160e10	1.41753	0.708767	−	0.705443i	$-0.249252\pi$
$877$	2.83160e10	1.41753	0.708767	+	0.705443i	$0.249252\pi$
$878$	0	0
$879$	1.68096e9	0.0834828
$880$	0	0
$881$	−5.58205e9	−0.275029	−0.137514	−	0.990500i	$-0.543911\pi$
$881$	−5.58205e9	−0.275029	−0.137514	+	0.990500i	$0.543911\pi$
$882$	0	0
$883$	−2.81316e10	−1.37509	−0.687547	−	0.726140i	$-0.741312\pi$
$883$	−2.81316e10	−1.37509	−0.687547	+	0.726140i	$0.741312\pi$
$884$	0	0
$885$	2.95494e10	1.43300
$886$	0	0
$887$	−9.92105e6	−0.000477337 0	−0.000238668	−	1.00000i	$-0.500076\pi$
$887$	−9.92105e6	−0.000477337 0	−0.000238668		1.00000i	$0.500076\pi$
$888$	0	0
$889$	−1.25298e10	−0.598119
$890$	0	0
$891$	−1.13718e10	−0.538589
$892$	0	0
$893$	−1.11746e10	−0.525111
$894$	0	0
$895$	6.45164e10	3.00808
$896$	0	0
$897$	1.95872e10	0.906147
$898$	0	0
$899$	7.12457e9	0.327039
$900$	0	0
$901$	3.23818e10	1.47491
$902$	0	0
$903$	−4.01867e8	−0.0181625
$904$	0	0
$905$	−9.51161e9	−0.426564
$906$	0	0
$907$	−1.61173e9	−0.0717242	−0.0358621	−	0.999357i	$-0.511418\pi$
$907$	−1.61173e9	−0.0717242	−0.0358621	+	0.999357i	$0.511418\pi$
$908$	0	0
$909$	4.72472e9	0.208642
$910$	0	0
$911$	−2.31768e10	−1.01564	−0.507818	−	0.861464i	$-0.669548\pi$
$911$	−2.31768e10	−1.01564	−0.507818	+	0.861464i	$0.669548\pi$
$912$	0	0
$913$	−3.34404e10	−1.45420
$914$	0	0
$915$	8.67611e8	0.0374413
$916$	0	0
$917$	−1.74302e10	−0.746464
$918$	0	0
$919$	−1.36641e10	−0.580733	−0.290367	−	0.956916i	$-0.593777\pi$
$919$	−1.36641e10	−0.580733	−0.290367	+	0.956916i	$0.593777\pi$
$920$	0	0
$921$	−1.57998e10	−0.666414
$922$	0	0
$923$	2.44613e10	1.02394
$924$	0	0
$925$	−1.92558e10	−0.799953
$926$	0	0
$927$	−2.85621e9	−0.117764
$928$	0	0
$929$	−3.34282e10	−1.36791	−0.683956	−	0.729523i	$-0.739742\pi$
$929$	−3.34282e10	−1.36791	−0.683956	+	0.729523i	$0.739742\pi$
$930$	0	0
$931$	4.30713e9	0.174930
$932$	0	0
$933$	2.97816e10	1.20050
$934$	0	0
$935$	5.35836e10	2.14384
$936$	0	0
$937$	1.04114e10	0.413450	0.206725	−	0.978399i	$-0.433720\pi$
$937$	1.04114e10	0.413450	0.206725	+	0.978399i	$0.433720\pi$
$938$	0	0
$939$	1.67196e10	0.659017
$940$	0	0
$941$	5.00017e10	1.95623	0.978117	−	0.208055i	$-0.0667134\pi$
$941$	5.00017e10	1.95623	0.978117	+	0.208055i	$0.0667134\pi$
$942$	0	0
$943$	−8.58998e9	−0.333581
$944$	0	0
$945$	−1.89518e10	−0.730533
$946$	0	0
$947$	−1.82730e10	−0.699172	−0.349586	−	0.936904i	$-0.613678\pi$
$947$	−1.82730e10	−0.699172	−0.349586	+	0.936904i	$0.613678\pi$
$948$	0	0
$949$	5.59880e10	2.12649
$950$	0	0
$951$	6.12420e9	0.230897
$952$	0	0
$953$	−1.24663e10	−0.466565	−0.233282	−	0.972409i	$-0.574947\pi$
$953$	−1.24663e10	−0.466565	−0.233282	+	0.972409i	$0.574947\pi$
$954$	0	0
$955$	−1.75911e10	−0.653552
$956$	0	0
$957$	5.61156e9	0.206963
$958$	0	0
$959$	−1.04256e10	−0.381710
$960$	0	0
$961$	1.19287e10	0.433573
$962$	0	0
$963$	4.55524e9	0.164369
$964$	0	0
$965$	−8.95225e10	−3.20691
$966$	0	0
$967$	4.12045e9	0.146539	0.0732693	−	0.997312i	$-0.476657\pi$
$967$	4.12045e9	0.146539	0.0732693	+	0.997312i	$0.476657\pi$
$968$	0	0
$969$	−3.79632e10	−1.34039
$970$	0	0
$971$	−1.24238e10	−0.435499	−0.217749	−	0.976005i	$-0.569872\pi$
$971$	−1.24238e10	−0.435499	−0.217749	+	0.976005i	$0.569872\pi$
$972$	0	0
$973$	1.00174e10	0.348626
$974$	0	0
$975$	−9.44021e10	−3.26186
$976$	0	0
$977$	4.51303e10	1.54824	0.774119	−	0.633041i	$-0.218193\pi$
$977$	4.51303e10	1.54824	0.774119	+	0.633041i	$0.218193\pi$
$978$	0	0
$979$	−2.62539e10	−0.894241
$980$	0	0
$981$	7.62265e9	0.257789
$982$	0	0
$983$	5.75558e9	0.193264	0.0966321	−	0.995320i	$-0.469193\pi$
$983$	5.75558e9	0.193264	0.0966321	+	0.995320i	$0.469193\pi$
$984$	0	0
$985$	6.92312e10	2.30821
$986$	0	0
$987$	4.05847e9	0.134355
$988$	0	0
$989$	−1.05542e9	−0.0346928
$990$	0	0
$991$	−3.64108e10	−1.18843	−0.594214	−	0.804307i	$-0.702537\pi$
$991$	−3.64108e10	−1.18843	−0.594214	+	0.804307i	$0.702537\pi$
$992$	0	0
$993$	3.78098e10	1.22541
$994$	0	0
$995$	−7.26313e10	−2.33745
$996$	0	0
$997$	1.13033e10	0.361220	0.180610	−	0.983555i	$-0.442193\pi$
$997$	1.13033e10	0.361220	0.180610	+	0.983555i	$0.442193\pi$
$998$	0	0
$999$	1.27348e10	0.404121

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.8.a.b.1.2	✓	2
3.2	odd	2		252.8.a.f.1.2		2
4.3	odd	2		112.8.a.h.1.1		2
7.2	even	3		196.8.e.b.165.1		4
7.3	odd	6		196.8.e.c.177.2		4
7.4	even	3		196.8.e.b.177.1		4
7.5	odd	6		196.8.e.c.165.2		4
7.6	odd	2		196.8.a.a.1.1		2
8.3	odd	2		448.8.a.q.1.2		2
8.5	even	2		448.8.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.8.a.b.1.2	✓	2	1.1	even	1	trivial
112.8.a.h.1.1		2	4.3	odd	2
196.8.a.a.1.1		2	7.6	odd	2
196.8.e.b.165.1		4	7.2	even	3
196.8.e.b.177.1		4	7.4	even	3
196.8.e.c.165.2		4	7.5	odd	6
196.8.e.c.177.2		4	7.3	odd	6
252.8.a.f.1.2		2	3.2	odd	2
448.8.a.o.1.1		2	8.5	even	2
448.8.a.q.1.2		2	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	28.a (trivial)

\(f(q)\)	\(=\)	\(q+38.7648 q^{3} -496.412 q^{5} -343.000 q^{7} -684.293 q^{9} +4035.19 q^{11} -14469.7 q^{13} -19243.3 q^{15} -26750.1 q^{17} +36610.0 q^{19} -13296.3 q^{21} -34920.1 q^{23} +168300. q^{25} -111305. q^{27} +35874.3 q^{29} +198598. q^{31} +156423. q^{33} +170269. q^{35} -114413. q^{37} -560915. q^{39} +245990. q^{41} +30224.0 q^{43} +339692. q^{45} -305233. q^{47} +117649. q^{49} -1.03696e6 q^{51} -1.21053e6 q^{53} -2.00312e6 q^{55} +1.41918e6 q^{57} -1.53557e6 q^{59} -45086.4 q^{61} +234713. q^{63} +7.18295e6 q^{65} +3.08501e6 q^{67} -1.35367e6 q^{69} -1.69051e6 q^{71} -3.86932e6 q^{73} +6.52412e6 q^{75} -1.38407e6 q^{77} +2.00910e6 q^{79} -2.81816e6 q^{81} -8.28720e6 q^{83} +1.32791e7 q^{85} +1.39066e6 q^{87} -6.50624e6 q^{89} +4.96311e6 q^{91} +7.69862e6 q^{93} -1.81737e7 q^{95} -1.14882e7 q^{97} -2.76125e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{3} - 294 q^{5} - 686 q^{7} - 2258 q^{9} - 3492 q^{11} - 16170 q^{13} - 24256 q^{15} - 29232 q^{17} - 3206 q^{19} - 4802 q^{21} - 9360 q^{23} + 131146 q^{25} - 18172 q^{27} + 184704 q^{29} + 165060 q^{31}+ \cdots + 9084332 q^{99}+O(q^{100}) \) 2 * q + 14 * q^3 - 294 * q^5 - 686 * q^7 - 2258 * q^9 - 3492 * q^11 - 16170 * q^13 - 24256 * q^15 - 29232 * q^17 - 3206 * q^19 - 4802 * q^21 - 9360 * q^23 + 131146 * q^25 - 18172 * q^27 + 184704 * q^29 + 165060 * q^31 + 342832 * q^33 + 100842 * q^35 + 286144 * q^37 - 518808 * q^39 - 116760 * q^41 - 294428 * q^43 + 21154 * q^45 - 1014132 * q^47 + 235298 * q^49 - 975500 * q^51 - 1396452 * q^53 - 3526712 * q^55 + 2405212 * q^57 - 2729286 * q^59 + 2466954 * q^61 + 774494 * q^63 + 6838788 * q^65 + 225176 * q^67 - 1986656 * q^69 + 1530312 * q^71 + 1143548 * q^73 + 7444234 * q^75 + 1197756 * q^77 + 7951176 * q^79 - 1682882 * q^81 - 18487854 * q^83 + 12776740 * q^85 - 2295076 * q^87 - 4652508 * q^89 + 5546310 * q^91 + 8529192 * q^93 - 26232912 * q^95 - 26702368 * q^97 + 9084332 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more