LMFDB - Embedded newform 192.7.g.a.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,7,Mod(127,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

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

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

chi := DirichletCharacter("192.127");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	192.g (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:	$44.1703840550$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	192.127
Dual form			192.7.g.a.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.5885i q^{3} -150.000 q^{5} +325.626i q^{7} -243.000 q^{9} +O(q^{10})$	$q+15.5885i q^{3} -150.000 q^{5} +325.626i q^{7} -243.000 q^{9} +1475.71i q^{11} -3394.00 q^{13} -2338.27i q^{15} +5178.00 q^{17} +6810.42i q^{19} -5076.00 q^{21} +3990.65i q^{23} +6875.00 q^{25} -3788.00i q^{27} -32142.0 q^{29} -32611.1i q^{31} -23004.0 q^{33} -48843.8i q^{35} +76150.0 q^{37} -52907.2i q^{39} -70038.0 q^{41} -100951. i q^{43} +36450.0 q^{45} +151603. i q^{47} +11617.0 q^{49} +80717.0i q^{51} -66942.0 q^{53} -221356. i q^{55} -106164. q^{57} -390564. i q^{59} +257014. q^{61} -79127.0i q^{63} +509100. q^{65} +321198. i q^{67} -62208.0 q^{69} -343112. i q^{71} +243442. q^{73} +107171. i q^{75} -480528. q^{77} -474963. i q^{79} +59049.0 q^{81} -1.03406e6i q^{83} -776700. q^{85} -501044. i q^{87} -686766. q^{89} -1.10517e6i q^{91} +508356. q^{93} -1.02156e6i q^{95} -942686. q^{97} -358597. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 300 q^{5} - 486 q^{9}+O(q^{10}) $ 2 * q - 300 * q^5 - 486 * q^9	$ 2 q - 300 q^{5} - 486 q^{9} - 6788 q^{13} + 10356 q^{17} - 10152 q^{21} + 13750 q^{25} - 64284 q^{29} - 46008 q^{33} + 152300 q^{37} - 140076 q^{41} + 72900 q^{45} + 23234 q^{49} - 133884 q^{53} - 212328 q^{57} + 514028 q^{61} + 1018200 q^{65} - 124416 q^{69} + 486884 q^{73} - 961056 q^{77} + 118098 q^{81} - 1553400 q^{85} - 1373532 q^{89} + 1016712 q^{93} - 1885372 q^{97}+O(q^{100}) $ 2 * q - 300 * q^5 - 486 * q^9 - 6788 * q^13 + 10356 * q^17 - 10152 * q^21 + 13750 * q^25 - 64284 * q^29 - 46008 * q^33 + 152300 * q^37 - 140076 * q^41 + 72900 * q^45 + 23234 * q^49 - 133884 * q^53 - 212328 * q^57 + 514028 * q^61 + 1018200 * q^65 - 124416 * q^69 + 486884 * q^73 - 961056 * q^77 + 118098 * q^81 - 1553400 * q^85 - 1373532 * q^89 + 1016712 * q^93 - 1885372 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	15.5885i	0.577350i
$3$	15.5885i	0.577350i
$4$	0	0
$5$	−150.000	−1.20000	−0.600000	−	0.800000i	$-0.704833\pi$
$5$	−150.000	−1.20000	−0.600000	+	0.800000i	$0.704833\pi$
$6$	0	0
$7$	325.626i	0.949346i	0.880162	+	0.474673i	$0.157434\pi$
$7$	325.626i	0.949346i	−0.880162	+	0.474673i	$0.842566\pi$
$8$	0	0
$9$	−243.000	−0.333333
$10$	0	0
$11$	1475.71i	1.10872i	0.832277	+	0.554360i	$0.187037\pi$
$11$	1475.71i	1.10872i	−0.832277	+	0.554360i	$0.812963\pi$
$12$	0	0
$13$	−3394.00	−1.54483	−0.772417	−	0.635116i	$-0.780952\pi$
$13$	−3394.00	−1.54483	−0.772417	+	0.635116i	$0.780952\pi$
$14$	0	0
$15$	− 2338.27i	− 0.692820i
$16$	0	0
$17$	5178.00	1.05394	0.526969	−	0.849884i	$-0.323328\pi$
$17$	5178.00	1.05394	0.526969	+	0.849884i	$0.323328\pi$
$18$	0	0
$19$	6810.42i	0.992918i	0.868060	+	0.496459i	$0.165367\pi$
$19$	6810.42i	0.992918i	−0.868060	+	0.496459i	$0.834633\pi$
$20$	0	0
$21$	−5076.00	−0.548105
$22$	0	0
$23$	3990.65i	0.327989i	0.986461	+	0.163995i	$0.0524380\pi$
$23$	3990.65i	0.327989i	−0.986461	+	0.163995i	$0.947562\pi$
$24$	0	0
$25$	6875.00	0.440000
$26$	0	0
$27$	− 3788.00i	− 0.192450i
$28$	0	0
$29$	−32142.0	−1.31789	−0.658945	−	0.752191i	$-0.728997\pi$
$29$	−32142.0	−1.31789	−0.658945	+	0.752191i	$0.728997\pi$
$30$	0	0
$31$	− 32611.1i	− 1.09466i	−0.836916	−	0.547331i	$-0.815644\pi$
$31$	− 32611.1i	− 1.09466i	0.836916	−	0.547331i	$-0.184356\pi$
$32$	0	0
$33$	−23004.0	−0.640120
$34$	0	0
$35$	− 48843.8i	− 1.13921i
$36$	0	0
$37$	76150.0	1.50337	0.751683	−	0.659525i	$-0.229242\pi$
$37$	76150.0	1.50337	0.751683	+	0.659525i	$0.229242\pi$
$38$	0	0
$39$	− 52907.2i	− 0.891910i
$40$	0	0
$41$	−70038.0	−1.01621	−0.508103	−	0.861296i	$-0.669653\pi$
$41$	−70038.0	−1.01621	−0.508103	+	0.861296i	$0.669653\pi$
$42$	0	0
$43$	− 100951.i	− 1.26971i	−0.772631	−	0.634855i	$-0.781060\pi$
$43$	− 100951.i	− 1.26971i	0.772631	−	0.634855i	$-0.218940\pi$
$44$	0	0
$45$	36450.0	0.400000
$46$	0	0
$47$	151603.i	1.46021i	0.683337	+	0.730103i	$0.260528\pi$
$47$	151603.i	1.46021i	−0.683337	+	0.730103i	$0.739472\pi$
$48$	0	0
$49$	11617.0	0.0987429
$50$	0	0
$51$	80717.0i	0.608492i
$52$	0	0
$53$	−66942.0	−0.449646	−0.224823	−	0.974400i	$-0.572180\pi$
$53$	−66942.0	−0.449646	−0.224823	+	0.974400i	$0.572180\pi$
$54$	0	0
$55$	− 221356.i	− 1.33046i
$56$	0	0
$57$	−106164.	−0.573261
$58$	0	0
$59$	− 390564.i	− 1.90167i	−0.309694	−	0.950836i	$-0.600227\pi$
$59$	− 390564.i	− 1.90167i	0.309694	−	0.950836i	$-0.399773\pi$
$60$	0	0
$61$	257014.	1.13232	0.566158	−	0.824297i	$-0.308429\pi$
$61$	257014.	1.13232	0.566158	+	0.824297i	$0.308429\pi$
$62$	0	0
$63$	− 79127.0i	− 0.316449i
$64$	0	0
$65$	509100.	1.85380
$66$	0	0
$67$	321198.i	1.06795i	0.845502	+	0.533973i	$0.179301\pi$
$67$	321198.i	1.06795i	−0.845502	+	0.533973i	$0.820699\pi$
$68$	0	0
$69$	−62208.0	−0.189365
$70$	0	0
$71$	− 343112.i	− 0.958653i	−0.877637	−	0.479326i	$-0.840881\pi$
$71$	− 343112.i	− 0.958653i	0.877637	−	0.479326i	$-0.159119\pi$
$72$	0	0
$73$	243442.	0.625788	0.312894	−	0.949788i	$-0.398702\pi$
$73$	243442.	0.625788	0.312894	+	0.949788i	$0.398702\pi$
$74$	0	0
$75$	107171.i	0.254034i
$76$	0	0
$77$	−480528.	−1.05256
$78$	0	0
$79$	− 474963.i	− 0.963338i	−0.876353	−	0.481669i	$-0.840031\pi$
$79$	− 474963.i	− 0.963338i	0.876353	−	0.481669i	$-0.159969\pi$
$80$	0	0
$81$	59049.0	0.111111
$82$	0	0
$83$	− 1.03406e6i	− 1.80846i	−0.427044	−	0.904231i	$-0.640445\pi$
$83$	− 1.03406e6i	− 1.80846i	0.427044	−	0.904231i	$-0.359555\pi$
$84$	0	0
$85$	−776700.	−1.26473
$86$	0	0
$87$	− 501044.i	− 0.760884i
$88$	0	0
$89$	−686766.	−0.974179	−0.487090	−	0.873352i	$-0.661941\pi$
$89$	−686766.	−0.974179	−0.487090	+	0.873352i	$0.661941\pi$
$90$	0	0
$91$	− 1.10517e6i	− 1.46658i
$92$	0	0
$93$	508356.	0.632003
$94$	0	0
$95$	− 1.02156e6i	− 1.19150i
$96$	0	0
$97$	−942686.	−1.03288	−0.516442	−	0.856322i	$-0.672744\pi$
$97$	−942686.	−1.03288	−0.516442	+	0.856322i	$0.672744\pi$
$98$	0	0
$99$	− 358597.i	− 0.369574i
$100$	0	0
$101$	259650.	0.252014	0.126007	−	0.992029i	$-0.459784\pi$
$101$	259650.	0.252014	0.126007	+	0.992029i	$0.459784\pi$
$102$	0	0
$103$	97971.7i	0.0896580i	0.998995	+	0.0448290i	$0.0142743\pi$
$103$	97971.7i	0.0896580i	−0.998995	+	0.0448290i	$0.985726\pi$
$104$	0	0
$105$	761400.	0.657726
$106$	0	0
$107$	944141.i	0.770700i	0.922770	+	0.385350i	$0.125919\pi$
$107$	944141.i	0.770700i	−0.922770	+	0.385350i	$0.874081\pi$
$108$	0	0
$109$	1.02337e6	0.790232	0.395116	−	0.918631i	$-0.370704\pi$
$109$	1.02337e6	0.790232	0.395116	+	0.918631i	$0.370704\pi$
$110$	0	0
$111$	1.18706e6i	0.867969i
$112$	0	0
$113$	−1.31393e6	−0.910622	−0.455311	−	0.890332i	$-0.650472\pi$
$113$	−1.31393e6	−0.910622	−0.455311	+	0.890332i	$0.650472\pi$
$114$	0	0
$115$	− 598597.i	− 0.393587i
$116$	0	0
$117$	824742.	0.514945
$118$	0	0
$119$	1.68609e6i	1.00055i
$120$	0	0
$121$	−406151.	−0.229262
$122$	0	0
$123$	− 1.09178e6i	− 0.586707i
$124$	0	0
$125$	1.31250e6	0.672000
$126$	0	0
$127$	127139.i	0.0620682i	0.999518	+	0.0310341i	$0.00988005\pi$
$127$	127139.i	0.0620682i	−0.999518	+	0.0310341i	$0.990120\pi$
$128$	0	0
$129$	1.57367e6	0.733068
$130$	0	0
$131$	3.06920e6i	1.36525i	0.730770	+	0.682624i	$0.239161\pi$
$131$	3.06920e6i	1.36525i	−0.730770	+	0.682624i	$0.760839\pi$
$132$	0	0
$133$	−2.21765e6	−0.942622
$134$	0	0
$135$	568199.i	0.230940i
$136$	0	0
$137$	−1.00868e6	−0.392275	−0.196138	−	0.980576i	$-0.562840\pi$
$137$	−1.00868e6	−0.392275	−0.196138	+	0.980576i	$0.562840\pi$
$138$	0	0
$139$	− 606031.i	− 0.225658i	−0.993614	−	0.112829i	$-0.964009\pi$
$139$	− 606031.i	− 0.225658i	0.993614	−	0.112829i	$-0.0359912\pi$
$140$	0	0
$141$	−2.36326e6	−0.843050
$142$	0	0
$143$	− 5.00855e6i	− 1.71279i
$144$	0	0
$145$	4.82130e6	1.58147
$146$	0	0
$147$	181091.i	0.0570092i
$148$	0	0
$149$	−2.39554e6	−0.724177	−0.362089	−	0.932144i	$-0.617936\pi$
$149$	−2.39554e6	−0.724177	−0.362089	+	0.932144i	$0.617936\pi$
$150$	0	0
$151$	3.25467e6i	0.945314i	0.881247	+	0.472657i	$0.156705\pi$
$151$	3.25467e6i	0.945314i	−0.881247	+	0.472657i	$0.843295\pi$
$152$	0	0
$153$	−1.25825e6	−0.351313
$154$	0	0
$155$	4.89166e6i	1.31359i
$156$	0	0
$157$	2.59380e6	0.670251	0.335125	−	0.942174i	$-0.391221\pi$
$157$	2.59380e6	0.670251	0.335125	+	0.942174i	$0.391221\pi$
$158$	0	0
$159$	− 1.04352e6i	− 0.259603i
$160$	0	0
$161$	−1.29946e6	−0.311375
$162$	0	0
$163$	2.97358e6i	0.686620i	0.939222	+	0.343310i	$0.111548\pi$
$163$	2.97358e6i	0.686620i	−0.939222	+	0.343310i	$0.888452\pi$
$164$	0	0
$165$	3.45060e6	0.768144
$166$	0	0
$167$	3.37388e6i	0.724404i	0.932100	+	0.362202i	$0.117975\pi$
$167$	3.37388e6i	0.724404i	−0.932100	+	0.362202i	$0.882025\pi$
$168$	0	0
$169$	6.69243e6	1.38651
$170$	0	0
$171$	− 1.65493e6i	− 0.330973i
$172$	0	0
$173$	−7.20327e6	−1.39121	−0.695603	−	0.718426i	$-0.744863\pi$
$173$	−7.20327e6	−1.39121	−0.695603	+	0.718426i	$0.744863\pi$
$174$	0	0
$175$	2.23868e6i	0.417712i
$176$	0	0
$177$	6.08828e6	1.09793
$178$	0	0
$179$	− 3.11206e6i	− 0.542611i	−0.962493	−	0.271306i	$-0.912545\pi$
$179$	− 3.11206e6i	− 0.542611i	0.962493	−	0.271306i	$-0.0874554\pi$
$180$	0	0
$181$	7.01689e6	1.18334	0.591670	−	0.806181i	$-0.298469\pi$
$181$	7.01689e6	1.18334	0.591670	+	0.806181i	$0.298469\pi$
$182$	0	0
$183$	4.00645e6i	0.653742i
$184$	0	0
$185$	−1.14225e7	−1.80404
$186$	0	0
$187$	7.64121e6i	1.16852i
$188$	0	0
$189$	1.23347e6	0.182702
$190$	0	0
$191$	6.05015e6i	0.868293i	0.900842	+	0.434146i	$0.142950\pi$
$191$	6.05015e6i	0.868293i	−0.900842	+	0.434146i	$0.857050\pi$
$192$	0	0
$193$	−9.37661e6	−1.30429	−0.652144	−	0.758095i	$-0.726130\pi$
$193$	−9.37661e6	−1.30429	−0.652144	+	0.758095i	$0.726130\pi$
$194$	0	0
$195$	7.93608e6i	1.07029i
$196$	0	0
$197$	−4.77205e6	−0.624174	−0.312087	−	0.950053i	$-0.601028\pi$
$197$	−4.77205e6	−0.624174	−0.312087	+	0.950053i	$0.601028\pi$
$198$	0	0
$199$	5.58137e6i	0.708242i	0.935200	+	0.354121i	$0.115220\pi$
$199$	5.58137e6i	0.708242i	−0.935200	+	0.354121i	$0.884780\pi$
$200$	0	0
$201$	−5.00699e6	−0.616579
$202$	0	0
$203$	− 1.04663e7i	− 1.25113i
$204$	0	0
$205$	1.05057e7	1.21945
$206$	0	0
$207$	− 969727.i	− 0.109330i
$208$	0	0
$209$	−1.00502e7	−1.10087
$210$	0	0
$211$	7.26090e6i	0.772935i	0.922303	+	0.386468i	$0.126305\pi$
$211$	7.26090e6i	0.772935i	−0.922303	+	0.386468i	$0.873695\pi$
$212$	0	0
$213$	5.34859e6	0.553478
$214$	0	0
$215$	1.51426e7i	1.52365i
$216$	0	0
$217$	1.06190e7	1.03921
$218$	0	0
$219$	3.79489e6i	0.361299i
$220$	0	0
$221$	−1.75741e7	−1.62816
$222$	0	0
$223$	− 3.79542e6i	− 0.342251i	−0.985249	−	0.171125i	$-0.945260\pi$
$223$	− 3.79542e6i	− 0.342251i	0.985249	−	0.171125i	$-0.0547404\pi$
$224$	0	0
$225$	−1.67062e6	−0.146667
$226$	0	0
$227$	1.50860e7i	1.28972i	0.764300	+	0.644861i	$0.223085\pi$
$227$	1.50860e7i	1.28972i	−0.764300	+	0.644861i	$0.776915\pi$
$228$	0	0
$229$	−5.90389e6	−0.491623	−0.245811	−	0.969318i	$-0.579054\pi$
$229$	−5.90389e6	−0.491623	−0.245811	+	0.969318i	$0.579054\pi$
$230$	0	0
$231$	− 7.49069e6i	− 0.607695i
$232$	0	0
$233$	1.30595e7	1.03242	0.516212	−	0.856461i	$-0.327342\pi$
$233$	1.30595e7	1.03242	0.516212	+	0.856461i	$0.327342\pi$
$234$	0	0
$235$	− 2.27404e7i	− 1.75225i
$236$	0	0
$237$	7.40394e6	0.556183
$238$	0	0
$239$	− 1.85931e6i	− 0.136194i	−0.997679	−	0.0680969i	$-0.978307\pi$
$239$	− 1.85931e6i	− 0.136194i	0.997679	−	0.0680969i	$-0.0216927\pi$
$240$	0	0
$241$	−5.65134e6	−0.403739	−0.201869	−	0.979412i	$-0.564702\pi$
$241$	−5.65134e6	−0.403739	−0.201869	+	0.979412i	$0.564702\pi$
$242$	0	0
$243$	920483.i	0.0641500i
$244$	0	0
$245$	−1.74255e6	−0.118491
$246$	0	0
$247$	− 2.31146e7i	− 1.53389i
$248$	0	0
$249$	1.61193e7	1.04412
$250$	0	0
$251$	− 2.92421e7i	− 1.84921i	−0.380925	−	0.924606i	$-0.624394\pi$
$251$	− 2.92421e7i	− 1.84921i	0.380925	−	0.924606i	$-0.375606\pi$
$252$	0	0
$253$	−5.88902e6	−0.363648
$254$	0	0
$255$	− 1.21076e7i	− 0.730190i
$256$	0	0
$257$	8.19328e6	0.482679	0.241340	−	0.970441i	$-0.422413\pi$
$257$	8.19328e6	0.482679	0.241340	+	0.970441i	$0.422413\pi$
$258$	0	0
$259$	2.47964e7i	1.42721i
$260$	0	0
$261$	7.81051e6	0.439296
$262$	0	0
$263$	− 6.82637e6i	− 0.375252i	−0.982241	−	0.187626i	$-0.939921\pi$
$263$	− 6.82637e6i	− 0.375252i	0.982241	−	0.187626i	$-0.0600793\pi$
$264$	0	0
$265$	1.00413e7	0.539576
$266$	0	0
$267$	− 1.07056e7i	− 0.562443i
$268$	0	0
$269$	−1.42950e7	−0.734389	−0.367194	−	0.930144i	$-0.619682\pi$
$269$	−1.42950e7	−0.734389	−0.367194	+	0.930144i	$0.619682\pi$
$270$	0	0
$271$	− 2.05403e7i	− 1.03205i	−0.856575	−	0.516023i	$-0.827412\pi$
$271$	− 2.05403e7i	− 1.03205i	0.856575	−	0.516023i	$-0.172588\pi$
$272$	0	0
$273$	1.72279e7	0.846731
$274$	0	0
$275$	1.01455e7i	0.487837i
$276$	0	0
$277$	1.92093e6	0.0903798	0.0451899	−	0.998978i	$-0.485611\pi$
$277$	1.92093e6	0.0903798	0.0451899	+	0.998978i	$0.485611\pi$
$278$	0	0
$279$	7.92449e6i	0.364887i
$280$	0	0
$281$	−2.88921e7	−1.30215	−0.651074	−	0.759015i	$-0.725681\pi$
$281$	−2.88921e7	−1.30215	−0.651074	+	0.759015i	$0.725681\pi$
$282$	0	0
$283$	− 8.82163e6i	− 0.389215i	−0.980881	−	0.194608i	$-0.937657\pi$
$283$	− 8.82163e6i	− 0.389215i	0.980881	−	0.194608i	$-0.0623433\pi$
$284$	0	0
$285$	1.59246e7	0.687914
$286$	0	0
$287$	− 2.28062e7i	− 0.964732i
$288$	0	0
$289$	2.67412e6	0.110786
$290$	0	0
$291$	− 1.46950e7i	− 0.596336i
$292$	0	0
$293$	−3.06685e6	−0.121924	−0.0609620	−	0.998140i	$-0.519417\pi$
$293$	−3.06685e6	−0.121924	−0.0609620	+	0.998140i	$0.519417\pi$
$294$	0	0
$295$	5.85845e7i	2.28201i
$296$	0	0
$297$	5.58997e6	0.213373
$298$	0	0
$299$	− 1.35442e7i	− 0.506689i
$300$	0	0
$301$	3.28722e7	1.20539
$302$	0	0
$303$	4.04754e6i	0.145500i
$304$	0	0
$305$	−3.85521e7	−1.35878
$306$	0	0
$307$	− 5.33706e7i	− 1.84453i	−0.386553	−	0.922267i	$-0.626334\pi$
$307$	− 5.33706e7i	− 1.84453i	0.386553	−	0.922267i	$-0.373666\pi$
$308$	0	0
$309$	−1.52723e6	−0.0517641
$310$	0	0
$311$	− 1.11399e7i	− 0.370340i	−0.982706	−	0.185170i	$-0.940716\pi$
$311$	− 1.11399e7i	− 0.370340i	0.982706	−	0.185170i	$-0.0592837\pi$
$312$	0	0
$313$	−1.26509e7	−0.412561	−0.206280	−	0.978493i	$-0.566136\pi$
$313$	−1.26509e7	−0.412561	−0.206280	+	0.978493i	$0.566136\pi$
$314$	0	0
$315$	1.18691e7i	0.379738i
$316$	0	0
$317$	−3.49712e6	−0.109782	−0.0548912	−	0.998492i	$-0.517481\pi$
$317$	−3.49712e6	−0.109782	−0.0548912	+	0.998492i	$0.517481\pi$
$318$	0	0
$319$	− 4.74322e7i	− 1.46117i
$320$	0	0
$321$	−1.47177e7	−0.444964
$322$	0	0
$323$	3.52644e7i	1.04647i
$324$	0	0
$325$	−2.33338e7	−0.679727
$326$	0	0
$327$	1.59528e7i	0.456241i
$328$	0	0
$329$	−4.93658e7	−1.38624
$330$	0	0
$331$	− 3.50036e7i	− 0.965227i	−0.875834	−	0.482613i	$-0.839688\pi$
$331$	− 3.50036e7i	− 0.965227i	0.875834	−	0.482613i	$-0.160312\pi$
$332$	0	0
$333$	−1.85045e7	−0.501122
$334$	0	0
$335$	− 4.81798e7i	− 1.28153i
$336$	0	0
$337$	3.96237e7	1.03530	0.517649	−	0.855593i	$-0.326807\pi$
$337$	3.96237e7	1.03530	0.517649	+	0.855593i	$0.326807\pi$
$338$	0	0
$339$	− 2.04822e7i	− 0.525748i
$340$	0	0
$341$	4.81244e7	1.21367
$342$	0	0
$343$	4.20923e7i	1.04309i
$344$	0	0
$345$	9.33120e6	0.227238
$346$	0	0
$347$	− 1.49793e7i	− 0.358512i	−0.983802	−	0.179256i	$-0.942631\pi$
$347$	− 1.49793e7i	− 0.358512i	0.983802	−	0.179256i	$-0.0573690\pi$
$348$	0	0
$349$	−3.40951e6	−0.0802077	−0.0401039	−	0.999196i	$-0.512769\pi$
$349$	−3.40951e6	−0.0802077	−0.0401039	+	0.999196i	$0.512769\pi$
$350$	0	0
$351$	1.28565e7i	0.297303i
$352$	0	0
$353$	8.55995e6	0.194602	0.0973010	−	0.995255i	$-0.468979\pi$
$353$	8.55995e6	0.194602	0.0973010	+	0.995255i	$0.468979\pi$
$354$	0	0
$355$	5.14669e7i	1.15038i
$356$	0	0
$357$	−2.62835e7	−0.577669
$358$	0	0
$359$	− 2.57477e7i	− 0.556488i	−0.960510	−	0.278244i	$-0.910248\pi$
$359$	− 2.57477e7i	− 0.556488i	0.960510	−	0.278244i	$-0.0897523\pi$
$360$	0	0
$361$	664009.	0.0141141
$362$	0	0
$363$	− 6.33127e6i	− 0.132364i
$364$	0	0
$365$	−3.65163e7	−0.750945
$366$	0	0
$367$	− 7.22681e7i	− 1.46200i	−0.682375	−	0.731002i	$-0.739053\pi$
$367$	− 7.22681e7i	− 1.46200i	0.682375	−	0.731002i	$-0.260947\pi$
$368$	0	0
$369$	1.70192e7	0.338736
$370$	0	0
$371$	− 2.17980e7i	− 0.426870i
$372$	0	0
$373$	5.64405e7	1.08759	0.543794	−	0.839219i	$-0.316987\pi$
$373$	5.64405e7	1.08759	0.543794	+	0.839219i	$0.316987\pi$
$374$	0	0
$375$	2.04599e7i	0.387979i
$376$	0	0
$377$	1.09090e8	2.03592
$378$	0	0
$379$	− 3.92134e7i	− 0.720305i	−0.932893	−	0.360153i	$-0.882725\pi$
$379$	− 3.92134e7i	− 0.720305i	0.932893	−	0.360153i	$-0.117275\pi$
$380$	0	0
$381$	−1.98191e6	−0.0358351
$382$	0	0
$383$	− 2.32420e7i	− 0.413692i	−0.978373	−	0.206846i	$-0.933680\pi$
$383$	− 2.32420e7i	− 0.413692i	0.978373	−	0.206846i	$-0.0663200\pi$
$384$	0	0
$385$	7.20792e7	1.26307
$386$	0	0
$387$	2.45311e7i	0.423237i
$388$	0	0
$389$	−1.01266e8	−1.72034	−0.860169	−	0.510009i	$-0.829642\pi$
$389$	−1.01266e8	−1.72034	−0.860169	+	0.510009i	$0.829642\pi$
$390$	0	0
$391$	2.06636e7i	0.345680i
$392$	0	0
$393$	−4.78441e7	−0.788226
$394$	0	0
$395$	7.12444e7i	1.15601i
$396$	0	0
$397$	−1.04857e8	−1.67581	−0.837906	−	0.545815i	$-0.816220\pi$
$397$	−1.04857e8	−1.67581	−0.837906	+	0.545815i	$0.816220\pi$
$398$	0	0
$399$	− 3.45697e7i	− 0.544223i
$400$	0	0
$401$	−9.68500e7	−1.50199	−0.750994	−	0.660309i	$-0.770425\pi$
$401$	−9.68500e7	−1.50199	−0.750994	+	0.660309i	$0.770425\pi$
$402$	0	0
$403$	1.10682e8i	1.69107i
$404$	0	0
$405$	−8.85735e6	−0.133333
$406$	0	0
$407$	1.12375e8i	1.66681i
$408$	0	0
$409$	−8.23021e7	−1.20293	−0.601466	−	0.798898i	$-0.705416\pi$
$409$	−8.23021e7	−1.20293	−0.601466	+	0.798898i	$0.705416\pi$
$410$	0	0
$411$	− 1.57237e7i	− 0.226480i
$412$	0	0
$413$	1.27177e8	1.80534
$414$	0	0
$415$	1.55108e8i	2.17015i
$416$	0	0
$417$	9.44708e6	0.130284
$418$	0	0
$419$	5.87082e7i	0.798099i	0.916929	+	0.399050i	$0.130660\pi$
$419$	5.87082e7i	0.798099i	−0.916929	+	0.399050i	$0.869340\pi$
$420$	0	0
$421$	7.33609e7	0.983146	0.491573	−	0.870836i	$-0.336422\pi$
$421$	7.33609e7	0.983146	0.491573	+	0.870836i	$0.336422\pi$
$422$	0	0
$423$	− 3.68395e7i	− 0.486735i
$424$	0	0
$425$	3.55988e7	0.463733
$426$	0	0
$427$	8.36903e7i	1.07496i
$428$	0	0
$429$	7.80756e7	0.988879
$430$	0	0
$431$	1.26607e8i	1.58134i	0.612240	+	0.790672i	$0.290268\pi$
$431$	1.26607e8i	1.58134i	−0.612240	+	0.790672i	$0.709732\pi$
$432$	0	0
$433$	5.59750e7	0.689493	0.344747	−	0.938696i	$-0.387965\pi$
$433$	5.59750e7	0.689493	0.344747	+	0.938696i	$0.387965\pi$
$434$	0	0
$435$	7.51566e7i	0.913060i
$436$	0	0
$437$	−2.71780e7	−0.325666
$438$	0	0
$439$	− 1.59949e7i	− 0.189055i	−0.995522	−	0.0945277i	$-0.969866\pi$
$439$	− 1.59949e7i	− 0.189055i	0.995522	−	0.0945277i	$-0.0301341\pi$
$440$	0	0
$441$	−2.82293e6	−0.0329143
$442$	0	0
$443$	− 1.15233e8i	− 1.32545i	−0.748861	−	0.662727i	$-0.769399\pi$
$443$	− 1.15233e8i	− 1.32545i	0.748861	−	0.662727i	$-0.230601\pi$
$444$	0	0
$445$	1.03015e8	1.16901
$446$	0	0
$447$	− 3.73428e7i	− 0.418104i
$448$	0	0
$449$	9.55375e7	1.05544	0.527721	−	0.849418i	$-0.323047\pi$
$449$	9.55375e7	1.05544	0.527721	+	0.849418i	$0.323047\pi$
$450$	0	0
$451$	− 1.03356e8i	− 1.12669i
$452$	0	0
$453$	−5.07353e7	−0.545777
$454$	0	0
$455$	1.65776e8i	1.75990i
$456$	0	0
$457$	−1.29504e8	−1.35686	−0.678428	−	0.734667i	$-0.737339\pi$
$457$	−1.29504e8	−1.35686	−0.678428	+	0.734667i	$0.737339\pi$
$458$	0	0
$459$	− 1.96142e7i	− 0.202831i
$460$	0	0
$461$	1.05039e8	1.07213	0.536066	−	0.844176i	$-0.319910\pi$
$461$	1.05039e8	1.07213	0.536066	+	0.844176i	$0.319910\pi$
$462$	0	0
$463$	− 1.04436e8i	− 1.05223i	−0.850415	−	0.526113i	$-0.823649\pi$
$463$	− 1.04436e8i	− 1.05223i	0.850415	−	0.526113i	$-0.176351\pi$
$464$	0	0
$465$	−7.62534e7	−0.758404
$466$	0	0
$467$	1.54569e7i	0.151765i	0.997117	+	0.0758824i	$0.0241774\pi$
$467$	1.54569e7i	0.151765i	−0.997117	+	0.0758824i	$0.975823\pi$
$468$	0	0
$469$	−1.04590e8	−1.01385
$470$	0	0
$471$	4.04333e7i	0.386969i
$472$	0	0
$473$	1.48974e8	1.40775
$474$	0	0
$475$	4.68217e7i	0.436884i
$476$	0	0
$477$	1.62669e7	0.149882
$478$	0	0
$479$	− 1.25737e7i	− 0.114408i	−0.998363	−	0.0572042i	$-0.981781\pi$
$479$	− 1.25737e7i	− 0.114408i	0.998363	−	0.0572042i	$-0.0182186\pi$
$480$	0	0
$481$	−2.58453e8	−2.32245
$482$	0	0
$483$	− 2.02565e7i	− 0.179773i
$484$	0	0
$485$	1.41403e8	1.23946
$486$	0	0
$487$	− 5.25365e7i	− 0.454856i	−0.973795	−	0.227428i	$-0.926968\pi$
$487$	− 5.25365e7i	− 0.454856i	0.973795	−	0.227428i	$-0.0730317\pi$
$488$	0	0
$489$	−4.63535e7	−0.396420
$490$	0	0
$491$	− 4.04753e7i	− 0.341936i	−0.985277	−	0.170968i	$-0.945310\pi$
$491$	− 4.04753e7i	− 0.341936i	0.985277	−	0.170968i	$-0.0546895\pi$
$492$	0	0
$493$	−1.66431e8	−1.38897
$494$	0	0
$495$	5.37895e7i	0.443488i
$496$	0	0
$497$	1.11726e8	0.910093
$498$	0	0
$499$	1.91720e8i	1.54300i	0.636230	+	0.771499i	$0.280493\pi$
$499$	1.91720e8i	1.54300i	−0.636230	+	0.771499i	$0.719507\pi$
$500$	0	0
$501$	−5.25936e7	−0.418235
$502$	0	0
$503$	− 3.32464e7i	− 0.261241i	−0.991432	−	0.130620i	$-0.958303\pi$
$503$	− 3.32464e7i	− 0.261241i	0.991432	−	0.130620i	$-0.0416969\pi$
$504$	0	0
$505$	−3.89475e7	−0.302416
$506$	0	0
$507$	1.04325e8i	0.800503i
$508$	0	0
$509$	−1.87407e8	−1.42112	−0.710562	−	0.703634i	$-0.751559\pi$
$509$	−1.87407e8	−1.42112	−0.710562	+	0.703634i	$0.751559\pi$
$510$	0	0
$511$	7.92709e7i	0.594089i
$512$	0	0
$513$	2.57979e7	0.191087
$514$	0	0
$515$	− 1.46958e7i	− 0.107590i
$516$	0	0
$517$	−2.23722e8	−1.61896
$518$	0	0
$519$	− 1.12288e8i	− 0.803213i
$520$	0	0
$521$	1.24378e8	0.879486	0.439743	−	0.898124i	$-0.355069\pi$
$521$	1.24378e8	0.879486	0.439743	+	0.898124i	$0.355069\pi$
$522$	0	0
$523$	− 1.67205e8i	− 1.16881i	−0.811463	−	0.584404i	$-0.801328\pi$
$523$	− 1.67205e8i	− 1.16881i	0.811463	−	0.584404i	$-0.198672\pi$
$524$	0	0
$525$	−3.48975e7	−0.241166
$526$	0	0
$527$	− 1.68860e8i	− 1.15371i
$528$	0	0
$529$	1.32111e8	0.892423
$530$	0	0
$531$	9.49070e7i	0.633891i
$532$	0	0
$533$	2.37709e8	1.56987
$534$	0	0
$535$	− 1.41621e8i	− 0.924840i
$536$	0	0
$537$	4.85122e7	0.313277
$538$	0	0
$539$	1.71433e7i	0.109478i
$540$	0	0
$541$	−9.84248e6	−0.0621603	−0.0310801	−	0.999517i	$-0.509895\pi$
$541$	−9.84248e6	−0.0621603	−0.0310801	+	0.999517i	$0.509895\pi$
$542$	0	0
$543$	1.09383e8i	0.683201i
$544$	0	0
$545$	−1.53506e8	−0.948279
$546$	0	0
$547$	− 1.24398e8i	− 0.760064i	−0.924973	−	0.380032i	$-0.875913\pi$
$547$	− 1.24398e8i	− 0.760064i	0.924973	−	0.380032i	$-0.124087\pi$
$548$	0	0
$549$	−6.24544e7	−0.377438
$550$	0	0
$551$	− 2.18901e8i	− 1.30856i
$552$	0	0
$553$	1.54660e8	0.914540
$554$	0	0
$555$	− 1.78059e8i	− 1.04156i
$556$	0	0
$557$	−1.33942e8	−0.775091	−0.387545	−	0.921851i	$-0.626677\pi$
$557$	−1.33942e8	−0.775091	−0.387545	+	0.921851i	$0.626677\pi$
$558$	0	0
$559$	3.42627e8i	1.96149i
$560$	0	0
$561$	−1.19115e8	−0.674647
$562$	0	0
$563$	− 1.20675e7i	− 0.0676228i	−0.999428	−	0.0338114i	$-0.989235\pi$
$563$	− 1.20675e7i	− 0.0676228i	0.999428	−	0.0338114i	$-0.0107646\pi$
$564$	0	0
$565$	1.97090e8	1.09275
$566$	0	0
$567$	1.92279e7i	0.105483i
$568$	0	0
$569$	3.50088e7	0.190038	0.0950191	−	0.995475i	$-0.469709\pi$
$569$	3.50088e7	0.190038	0.0950191	+	0.995475i	$0.469709\pi$
$570$	0	0
$571$	1.91768e8i	1.03007i	0.857168	+	0.515037i	$0.172222\pi$
$571$	1.91768e8i	1.03007i	−0.857168	+	0.515037i	$0.827778\pi$
$572$	0	0
$573$	−9.43125e7	−0.501309
$574$	0	0
$575$	2.74357e7i	0.144315i
$576$	0	0
$577$	2.40733e8	1.25317	0.626583	−	0.779354i	$-0.284453\pi$
$577$	2.40733e8	1.25317	0.626583	+	0.779354i	$0.284453\pi$
$578$	0	0
$579$	− 1.46167e8i	− 0.753031i
$580$	0	0
$581$	3.36715e8	1.71686
$582$	0	0
$583$	− 9.87868e7i	− 0.498532i
$584$	0	0
$585$	−1.23711e8	−0.617934
$586$	0	0
$587$	3.66145e8i	1.81025i	0.425147	+	0.905124i	$0.360222\pi$
$587$	3.66145e8i	1.81025i	−0.425147	+	0.905124i	$0.639778\pi$
$588$	0	0
$589$	2.22095e8	1.08691
$590$	0	0
$591$	− 7.43888e7i	− 0.360367i
$592$	0	0
$593$	2.91532e7	0.139805	0.0699024	−	0.997554i	$-0.477731\pi$
$593$	2.91532e7	0.139805	0.0699024	+	0.997554i	$0.477731\pi$
$594$	0	0
$595$	− 2.52913e8i	− 1.20066i
$596$	0	0
$597$	−8.70049e7	−0.408903
$598$	0	0
$599$	− 1.30694e8i	− 0.608100i	−0.952656	−	0.304050i	$-0.901661\pi$
$599$	− 1.30694e8i	− 0.608100i	0.952656	−	0.304050i	$-0.0983389\pi$
$600$	0	0
$601$	−3.80928e8	−1.75477	−0.877383	−	0.479791i	$-0.840712\pi$
$601$	−3.80928e8	−1.75477	−0.877383	+	0.479791i	$0.840712\pi$
$602$	0	0
$603$	− 7.80512e7i	− 0.355982i
$604$	0	0
$605$	6.09226e7	0.275114
$606$	0	0
$607$	− 3.04928e8i	− 1.36342i	−0.731620	−	0.681712i	$-0.761236\pi$
$607$	− 3.04928e8i	− 1.36342i	0.731620	−	0.681712i	$-0.238764\pi$
$608$	0	0
$609$	1.63153e8	0.722342
$610$	0	0
$611$	− 5.14540e8i	− 2.25578i
$612$	0	0
$613$	−6.34479e7	−0.275445	−0.137723	−	0.990471i	$-0.543978\pi$
$613$	−6.34479e7	−0.275445	−0.137723	+	0.990471i	$0.543978\pi$
$614$	0	0
$615$	1.63768e8i	0.704049i
$616$	0	0
$617$	−3.34561e7	−0.142436	−0.0712181	−	0.997461i	$-0.522689\pi$
$617$	−3.34561e7	−0.142436	−0.0712181	+	0.997461i	$0.522689\pi$
$618$	0	0
$619$	− 1.89984e8i	− 0.801022i	−0.916292	−	0.400511i	$-0.868833\pi$
$619$	− 1.89984e8i	− 0.801022i	0.916292	−	0.400511i	$-0.131167\pi$
$620$	0	0
$621$	1.51165e7	0.0631216
$622$	0	0
$623$	− 2.23629e8i	− 0.924833i
$624$	0	0
$625$	−3.04297e8	−1.24640
$626$	0	0
$627$	− 1.56667e8i	− 0.635587i
$628$	0	0
$629$	3.94305e8	1.58446
$630$	0	0
$631$	3.35903e8i	1.33698i	0.743719	+	0.668492i	$0.233060\pi$
$631$	3.35903e8i	1.33698i	−0.743719	+	0.668492i	$0.766940\pi$
$632$	0	0
$633$	−1.13186e8	−0.446255
$634$	0	0
$635$	− 1.90709e7i	− 0.0744818i
$636$	0	0
$637$	−3.94281e7	−0.152541
$638$	0	0
$639$	8.33763e7i	0.319551i
$640$	0	0
$641$	−2.60654e8	−0.989671	−0.494835	−	0.868987i	$-0.664772\pi$
$641$	−2.60654e8	−0.989671	−0.494835	+	0.868987i	$0.664772\pi$
$642$	0	0
$643$	− 3.73815e8i	− 1.40612i	−0.711128	−	0.703062i	$-0.751815\pi$
$643$	− 3.73815e8i	− 1.40612i	0.711128	−	0.703062i	$-0.248185\pi$
$644$	0	0
$645$	−2.36050e8	−0.879681
$646$	0	0
$647$	− 2.43406e8i	− 0.898708i	−0.893354	−	0.449354i	$-0.851654\pi$
$647$	− 2.43406e8i	− 0.898708i	0.893354	−	0.449354i	$-0.148346\pi$
$648$	0	0
$649$	5.76358e8	2.10842
$650$	0	0
$651$	1.65534e8i	0.599989i
$652$	0	0
$653$	4.12306e7	0.148074	0.0740372	−	0.997255i	$-0.476412\pi$
$653$	4.12306e7	0.148074	0.0740372	+	0.997255i	$0.476412\pi$
$654$	0	0
$655$	− 4.60380e8i	− 1.63830i
$656$	0	0
$657$	−5.91564e7	−0.208596
$658$	0	0
$659$	9.91383e7i	0.346406i	0.984886	+	0.173203i	$0.0554117\pi$
$659$	9.91383e7i	0.346406i	−0.984886	+	0.173203i	$0.944588\pi$
$660$	0	0
$661$	−3.85360e8	−1.33433	−0.667163	−	0.744912i	$-0.732492\pi$
$661$	−3.85360e8	−1.33433	−0.667163	+	0.744912i	$0.732492\pi$
$662$	0	0
$663$	− 2.73954e8i	− 0.940019i
$664$	0	0
$665$	3.32647e8	1.13115
$666$	0	0
$667$	− 1.28267e8i	− 0.432253i
$668$	0	0
$669$	5.91647e7	0.197599
$670$	0	0
$671$	3.79277e8i	1.25542i
$672$	0	0
$673$	4.80231e7	0.157545	0.0787726	−	0.996893i	$-0.474900\pi$
$673$	4.80231e7	0.157545	0.0787726	+	0.996893i	$0.474900\pi$
$674$	0	0
$675$	− 2.60425e7i	− 0.0846780i
$676$	0	0
$677$	−2.63635e8	−0.849644	−0.424822	−	0.905277i	$-0.639663\pi$
$677$	−2.63635e8	−0.849644	−0.424822	+	0.905277i	$0.639663\pi$
$678$	0	0
$679$	− 3.06963e8i	− 0.980565i
$680$	0	0
$681$	−2.35167e8	−0.744622
$682$	0	0
$683$	5.05574e8i	1.58680i	0.608699	+	0.793401i	$0.291692\pi$
$683$	5.05574e8i	1.58680i	−0.608699	+	0.793401i	$0.708308\pi$
$684$	0	0
$685$	1.51302e8	0.470730
$686$	0	0
$687$	− 9.20325e7i	− 0.283838i
$688$	0	0
$689$	2.27201e8	0.694629
$690$	0	0
$691$	7.79953e7i	0.236393i	0.992990	+	0.118196i	$0.0377113\pi$
$691$	7.79953e7i	0.236393i	−0.992990	+	0.118196i	$0.962289\pi$
$692$	0	0
$693$	1.16768e8	0.350853
$694$	0	0
$695$	9.09046e7i	0.270789i
$696$	0	0
$697$	−3.62657e8	−1.07102
$698$	0	0
$699$	2.03577e8i	0.596070i
$700$	0	0
$701$	−4.66810e8	−1.35515	−0.677573	−	0.735456i	$-0.736968\pi$
$701$	−4.66810e8	−1.35515	−0.677573	+	0.735456i	$0.736968\pi$
$702$	0	0
$703$	5.18614e8i	1.49272i
$704$	0	0
$705$	3.54488e8	1.01166
$706$	0	0
$707$	8.45487e7i	0.239248i
$708$	0	0
$709$	−1.77560e8	−0.498203	−0.249101	−	0.968477i	$-0.580135\pi$
$709$	−1.77560e8	−0.498203	−0.249101	+	0.968477i	$0.580135\pi$
$710$	0	0
$711$	1.15416e8i	0.321113i
$712$	0	0
$713$	1.30139e8	0.359037
$714$	0	0
$715$	7.51283e8i	2.05535i
$716$	0	0
$717$	2.89837e7	0.0786316
$718$	0	0
$719$	2.87726e8i	0.774093i	0.922060	+	0.387046i	$0.126505\pi$
$719$	2.87726e8i	0.774093i	−0.922060	+	0.387046i	$0.873495\pi$
$720$	0	0
$721$	−3.19021e7	−0.0851164
$722$	0	0
$723$	− 8.80957e7i	− 0.233099i
$724$	0	0
$725$	−2.20976e8	−0.579871
$726$	0	0
$727$	− 1.20038e8i	− 0.312404i	−0.987725	−	0.156202i	$-0.950075\pi$
$727$	− 1.20038e8i	− 0.312404i	0.987725	−	0.156202i	$-0.0499251\pi$
$728$	0	0
$729$	−1.43489e7	−0.0370370
$730$	0	0
$731$	− 5.22723e8i	− 1.33820i
$732$	0	0
$733$	2.05697e8	0.522295	0.261147	−	0.965299i	$-0.415899\pi$
$733$	2.05697e8	0.522295	0.261147	+	0.965299i	$0.415899\pi$
$734$	0	0
$735$	− 2.71637e7i	− 0.0684111i
$736$	0	0
$737$	−4.73995e8	−1.18405
$738$	0	0
$739$	4.80558e8i	1.19073i	0.803456	+	0.595364i	$0.202992\pi$
$739$	4.80558e8i	1.19073i	−0.803456	+	0.595364i	$0.797008\pi$
$740$	0	0
$741$	3.60321e8	0.885594
$742$	0	0
$743$	− 1.96089e8i	− 0.478065i	−0.971012	−	0.239032i	$-0.923170\pi$
$743$	− 1.96089e8i	− 0.478065i	0.971012	−	0.239032i	$-0.0768302\pi$
$744$	0	0
$745$	3.59331e8	0.869013
$746$	0	0
$747$	2.51275e8i	0.602821i
$748$	0	0
$749$	−3.07436e8	−0.731661
$750$	0	0
$751$	1.78956e8i	0.422501i	0.977432	+	0.211250i	$0.0677535\pi$
$751$	1.78956e8i	0.422501i	−0.977432	+	0.211250i	$0.932246\pi$
$752$	0	0
$753$	4.55839e8	1.06764
$754$	0	0
$755$	− 4.88200e8i	− 1.13438i
$756$	0	0
$757$	−2.54288e8	−0.586190	−0.293095	−	0.956083i	$-0.594685\pi$
$757$	−2.54288e8	−0.586190	−0.293095	+	0.956083i	$0.594685\pi$
$758$	0	0
$759$	− 9.18008e7i	− 0.209953i
$760$	0	0
$761$	4.52852e8	1.02755	0.513775	−	0.857925i	$-0.328247\pi$
$761$	4.52852e8	1.02755	0.513775	+	0.857925i	$0.328247\pi$
$762$	0	0
$763$	3.33237e8i	0.750204i
$764$	0	0
$765$	1.88738e8	0.421575
$766$	0	0
$767$	1.32557e9i	2.93777i
$768$	0	0
$769$	−5.75007e8	−1.26443	−0.632214	−	0.774794i	$-0.717854\pi$
$769$	−5.75007e8	−1.26443	−0.632214	+	0.774794i	$0.717854\pi$
$770$	0	0
$771$	1.27721e8i	0.278675i
$772$	0	0
$773$	2.85200e8	0.617464	0.308732	−	0.951149i	$-0.400095\pi$
$773$	2.85200e8	0.617464	0.308732	+	0.951149i	$0.400095\pi$
$774$	0	0
$775$	− 2.24201e8i	− 0.481651i
$776$	0	0
$777$	−3.86537e8	−0.824002
$778$	0	0
$779$	− 4.76988e8i	− 1.00901i
$780$	0	0
$781$	5.06333e8	1.06288
$782$	0	0
$783$	1.21754e8i	0.253628i
$784$	0	0
$785$	−3.89070e8	−0.804301
$786$	0	0
$787$	− 2.03737e8i	− 0.417971i	−0.977919	−	0.208986i	$-0.932984\pi$
$787$	− 2.03737e8i	− 0.417971i	0.977919	−	0.208986i	$-0.0670162\pi$
$788$	0	0
$789$	1.06413e8	0.216652
$790$	0	0
$791$	− 4.27850e8i	− 0.864495i
$792$	0	0
$793$	−8.72306e8	−1.74924
$794$	0	0
$795$	1.56528e8i	0.311524i
$796$	0	0
$797$	1.73958e8	0.343612	0.171806	−	0.985131i	$-0.445040\pi$
$797$	1.73958e8	0.343612	0.171806	+	0.985131i	$0.445040\pi$
$798$	0	0
$799$	7.85000e8i	1.53897i
$800$	0	0
$801$	1.66884e8	0.324726
$802$	0	0
$803$	3.59249e8i	0.693824i
$804$	0	0
$805$	1.94918e8	0.373650
$806$	0	0
$807$	− 2.22836e8i	− 0.424000i
$808$	0	0
$809$	−4.47671e8	−0.845500	−0.422750	−	0.906246i	$-0.638935\pi$
$809$	−4.47671e8	−0.845500	−0.422750	+	0.906246i	$0.638935\pi$
$810$	0	0
$811$	5.93222e7i	0.111213i	0.998453	+	0.0556064i	$0.0177092\pi$
$811$	5.93222e7i	0.111213i	−0.998453	+	0.0556064i	$0.982291\pi$
$812$	0	0
$813$	3.20192e8	0.595852
$814$	0	0
$815$	− 4.46037e8i	− 0.823944i
$816$	0	0
$817$	6.87518e8	1.26072
$818$	0	0
$819$	2.68557e8i	0.488860i
$820$	0	0
$821$	3.86250e7	0.0697974	0.0348987	−	0.999391i	$-0.488889\pi$
$821$	3.86250e7	0.0697974	0.0348987	+	0.999391i	$0.488889\pi$
$822$	0	0
$823$	4.03810e8i	0.724399i	0.932101	+	0.362200i	$0.117974\pi$
$823$	4.03810e8i	0.724399i	−0.932101	+	0.362200i	$0.882026\pi$
$824$	0	0
$825$	−1.58152e8	−0.281653
$826$	0	0
$827$	3.21027e8i	0.567578i	0.958887	+	0.283789i	$0.0915915\pi$
$827$	3.21027e8i	0.567578i	−0.958887	+	0.283789i	$0.908408\pi$
$828$	0	0
$829$	6.98016e7	0.122519	0.0612593	−	0.998122i	$-0.480488\pi$
$829$	6.98016e7	0.122519	0.0612593	+	0.998122i	$0.480488\pi$
$830$	0	0
$831$	2.99443e7i	0.0521808i
$832$	0	0
$833$	6.01528e7	0.104069
$834$	0	0
$835$	− 5.06082e8i	− 0.869284i
$836$	0	0
$837$	−1.23531e8	−0.210668
$838$	0	0
$839$	5.16728e8i	0.874936i	0.899234	+	0.437468i	$0.144125\pi$
$839$	5.16728e8i	0.874936i	−0.899234	+	0.437468i	$0.855875\pi$
$840$	0	0
$841$	4.38285e8	0.736832
$842$	0	0
$843$	− 4.50383e8i	− 0.751795i
$844$	0	0
$845$	−1.00386e9	−1.66381
$846$	0	0
$847$	− 1.32253e8i	− 0.217649i
$848$	0	0
$849$	1.37516e8	0.224713
$850$	0	0
$851$	3.03888e8i	0.493088i
$852$	0	0
$853$	3.41054e7	0.0549510	0.0274755	−	0.999622i	$-0.491253\pi$
$853$	3.41054e7	0.0549510	0.0274755	+	0.999622i	$0.491253\pi$
$854$	0	0
$855$	2.48240e8i	0.397167i
$856$	0	0
$857$	5.96601e8	0.947854	0.473927	−	0.880564i	$-0.342836\pi$
$857$	5.96601e8	0.947854	0.473927	+	0.880564i	$0.342836\pi$
$858$	0	0
$859$	4.96999e8i	0.784109i	0.919942	+	0.392054i	$0.128236\pi$
$859$	4.96999e8i	0.784109i	−0.919942	+	0.392054i	$0.871764\pi$
$860$	0	0
$861$	3.55513e8	0.556988
$862$	0	0
$863$	3.28718e8i	0.511435i	0.966752	+	0.255718i	$0.0823117\pi$
$863$	3.28718e8i	0.511435i	−0.966752	+	0.255718i	$0.917688\pi$
$864$	0	0
$865$	1.08049e9	1.66945
$866$	0	0
$867$	4.16853e7i	0.0639626i
$868$	0	0
$869$	7.00906e8	1.06807
$870$	0	0
$871$	− 1.09015e9i	− 1.64980i
$872$	0	0
$873$	2.29073e8	0.344295
$874$	0	0
$875$	4.27384e8i	0.637960i
$876$	0	0
$877$	7.37191e8	1.09290	0.546451	−	0.837491i	$-0.315979\pi$
$877$	7.37191e8	1.09290	0.546451	+	0.837491i	$0.315979\pi$
$878$	0	0
$879$	− 4.78074e7i	− 0.0703928i
$880$	0	0
$881$	−1.05804e9	−1.54730	−0.773649	−	0.633614i	$-0.781571\pi$
$881$	−1.05804e9	−1.54730	−0.773649	+	0.633614i	$0.781571\pi$
$882$	0	0
$883$	− 9.53856e7i	− 0.138548i	−0.997598	−	0.0692741i	$-0.977932\pi$
$883$	− 9.53856e7i	− 0.138548i	0.997598	−	0.0692741i	$-0.0220683\pi$
$884$	0	0
$885$	−9.13243e8	−1.31752
$886$	0	0
$887$	6.48931e8i	0.929882i	0.885342	+	0.464941i	$0.153925\pi$
$887$	6.48931e8i	0.929882i	−0.885342	+	0.464941i	$0.846075\pi$
$888$	0	0
$889$	−4.13999e7	−0.0589242
$890$	0	0
$891$	8.71390e7i	0.123191i
$892$	0	0
$893$	−1.03248e9	−1.44986
$894$	0	0
$895$	4.66809e8i	0.651133i
$896$	0	0
$897$	2.11134e8	0.292537
$898$	0	0
$899$	1.04818e9i	1.44264i
$900$	0	0
$901$	−3.46626e8	−0.473900
$902$	0	0
$903$	5.12427e8i	0.695934i
$904$	0	0
$905$	−1.05253e9	−1.42001
$906$	0	0
$907$	6.09067e6i	0.00816288i	0.999992	+	0.00408144i	$0.00129917\pi$
$907$	6.09067e6i	0.00816288i	−0.999992	+	0.00408144i	$0.998701\pi$
$908$	0	0
$909$	−6.30950e7	−0.0840046
$910$	0	0
$911$	− 1.17870e9i	− 1.55901i	−0.626397	−	0.779504i	$-0.715471\pi$
$911$	− 1.17870e9i	− 1.55901i	0.626397	−	0.779504i	$-0.284529\pi$
$912$	0	0
$913$	1.52596e9	2.00508
$914$	0	0
$915$	− 6.00968e8i	− 0.784491i
$916$	0	0
$917$	−9.99410e8	−1.29609
$918$	0	0
$919$	− 4.51537e7i	− 0.0581764i	−0.999577	−	0.0290882i	$-0.990740\pi$
$919$	− 4.51537e7i	− 0.0581764i	0.999577	−	0.0290882i	$-0.00926037\pi$
$920$	0	0
$921$	8.31965e8	1.06494
$922$	0	0
$923$	1.16452e9i	1.48096i
$924$	0	0
$925$	5.23531e8	0.661481
$926$	0	0
$927$	− 2.38071e7i	− 0.0298860i
$928$	0	0
$929$	−7.05122e8	−0.879463	−0.439731	−	0.898129i	$-0.644926\pi$
$929$	−7.05122e8	−0.879463	−0.439731	+	0.898129i	$0.644926\pi$
$930$	0	0
$931$	7.91167e7i	0.0980436i
$932$	0	0
$933$	1.73654e8	0.213816
$934$	0	0
$935$	− 1.14618e9i	− 1.40223i
$936$	0	0
$937$	−7.32052e8	−0.889864	−0.444932	−	0.895564i	$-0.646772\pi$
$937$	−7.32052e8	−0.889864	−0.444932	+	0.895564i	$0.646772\pi$
$938$	0	0
$939$	− 1.97208e8i	− 0.238192i
$940$	0	0
$941$	−1.64058e9	−1.96892	−0.984461	−	0.175603i	$-0.943813\pi$
$941$	−1.64058e9	−1.96892	−0.984461	+	0.175603i	$0.943813\pi$
$942$	0	0
$943$	− 2.79497e8i	− 0.333305i
$944$	0	0
$945$	−1.85020e8	−0.219242
$946$	0	0
$947$	− 1.04106e9i	− 1.22581i	−0.790156	−	0.612906i	$-0.790000\pi$
$947$	− 1.04106e9i	− 1.22581i	0.790156	−	0.612906i	$-0.210000\pi$
$948$	0	0
$949$	−8.26242e8	−0.966738
$950$	0	0
$951$	− 5.45147e7i	− 0.0633829i
$952$	0	0
$953$	−1.01652e9	−1.17446	−0.587231	−	0.809419i	$-0.699782\pi$
$953$	−1.01652e9	−1.17446	−0.587231	+	0.809419i	$0.699782\pi$
$954$	0	0
$955$	− 9.07523e8i	− 1.04195i
$956$	0	0
$957$	7.39395e8	0.843608
$958$	0	0
$959$	− 3.28451e8i	− 0.372405i
$960$	0	0
$961$	−1.75977e8	−0.198283
$962$	0	0
$963$	− 2.29426e8i	− 0.256900i
$964$	0	0
$965$	1.40649e9	1.56515
$966$	0	0
$967$	9.48126e8i	1.04854i	0.851551	+	0.524272i	$0.175662\pi$
$967$	9.48126e8i	1.04854i	−0.851551	+	0.524272i	$0.824338\pi$
$968$	0	0
$969$	−5.49717e8	−0.604182
$970$	0	0
$971$	− 1.06242e9i	− 1.16048i	−0.814446	−	0.580240i	$-0.802959\pi$
$971$	− 1.06242e9i	− 1.16048i	0.814446	−	0.580240i	$-0.197041\pi$
$972$	0	0
$973$	1.97339e8	0.214227
$974$	0	0
$975$	− 3.63737e8i	− 0.392441i
$976$	0	0
$977$	−1.10671e9	−1.18673	−0.593363	−	0.804935i	$-0.702200\pi$
$977$	−1.10671e9	−1.18673	−0.593363	+	0.804935i	$0.702200\pi$
$978$	0	0
$979$	− 1.01347e9i	− 1.08009i
$980$	0	0
$981$	−2.48680e8	−0.263411
$982$	0	0
$983$	− 1.37448e9i	− 1.44703i	−0.690310	−	0.723513i	$-0.742526\pi$
$983$	− 1.37448e9i	− 1.44703i	0.690310	−	0.723513i	$-0.257474\pi$
$984$	0	0
$985$	7.15807e8	0.749009
$986$	0	0
$987$	− 7.69537e8i	− 0.800346i
$988$	0	0
$989$	4.02859e8	0.416451
$990$	0	0
$991$	− 6.25564e8i	− 0.642762i	−0.946950	−	0.321381i	$-0.895853\pi$
$991$	− 6.25564e8i	− 0.642762i	0.946950	−	0.321381i	$-0.104147\pi$
$992$	0	0
$993$	5.45653e8	0.557274
$994$	0	0
$995$	− 8.37205e8i	− 0.849890i
$996$	0	0
$997$	1.20853e9	1.21947	0.609736	−	0.792604i	$-0.291275\pi$
$997$	1.20853e9	1.21947	0.609736	+	0.792604i	$0.291275\pi$
$998$	0	0
$999$	− 2.88456e8i	− 0.289323i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.7.g.a.127.2		2
3.2	odd	2		576.7.g.j.127.2		2
4.3	odd	2	inner	192.7.g.a.127.1		2
8.3	odd	2		48.7.g.c.31.2	yes	2
8.5	even	2		48.7.g.c.31.1	✓	2
12.11	even	2		576.7.g.j.127.1		2
16.3	odd	4		768.7.b.d.127.4		4
16.5	even	4		768.7.b.d.127.3		4
16.11	odd	4		768.7.b.d.127.1		4
16.13	even	4		768.7.b.d.127.2		4
24.5	odd	2		144.7.g.b.127.2		2
24.11	even	2		144.7.g.b.127.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.7.g.c.31.1	✓	2	8.5	even	2
48.7.g.c.31.2	yes	2	8.3	odd	2
144.7.g.b.127.1		2	24.11	even	2
144.7.g.b.127.2		2	24.5	odd	2
192.7.g.a.127.1		2	4.3	odd	2	inner
192.7.g.a.127.2		2	1.1	even	1	trivial
576.7.g.j.127.1		2	12.11	even	2
576.7.g.j.127.2		2	3.2	odd	2
768.7.b.d.127.1		4	16.11	odd	4
768.7.b.d.127.2		4	16.13	even	4
768.7.b.d.127.3		4	16.5	even	4
768.7.b.d.127.4		4	16.3	odd	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	192.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+15.5885i q^{3} -150.000 q^{5} +325.626i q^{7} -243.000 q^{9} +O(q^{10})\)	\(q+15.5885i q^{3} -150.000 q^{5} +325.626i q^{7} -243.000 q^{9} +1475.71i q^{11} -3394.00 q^{13} -2338.27i q^{15} +5178.00 q^{17} +6810.42i q^{19} -5076.00 q^{21} +3990.65i q^{23} +6875.00 q^{25} -3788.00i q^{27} -32142.0 q^{29} -32611.1i q^{31} -23004.0 q^{33} -48843.8i q^{35} +76150.0 q^{37} -52907.2i q^{39} -70038.0 q^{41} -100951. i q^{43} +36450.0 q^{45} +151603. i q^{47} +11617.0 q^{49} +80717.0i q^{51} -66942.0 q^{53} -221356. i q^{55} -106164. q^{57} -390564. i q^{59} +257014. q^{61} -79127.0i q^{63} +509100. q^{65} +321198. i q^{67} -62208.0 q^{69} -343112. i q^{71} +243442. q^{73} +107171. i q^{75} -480528. q^{77} -474963. i q^{79} +59049.0 q^{81} -1.03406e6i q^{83} -776700. q^{85} -501044. i q^{87} -686766. q^{89} -1.10517e6i q^{91} +508356. q^{93} -1.02156e6i q^{95} -942686. q^{97} -358597. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 300 q^{5} - 486 q^{9}+O(q^{10}) \) 2 * q - 300 * q^5 - 486 * q^9	\( 2 q - 300 q^{5} - 486 q^{9} - 6788 q^{13} + 10356 q^{17} - 10152 q^{21} + 13750 q^{25} - 64284 q^{29} - 46008 q^{33} + 152300 q^{37} - 140076 q^{41} + 72900 q^{45} + 23234 q^{49} - 133884 q^{53} - 212328 q^{57} + 514028 q^{61} + 1018200 q^{65} - 124416 q^{69} + 486884 q^{73} - 961056 q^{77} + 118098 q^{81} - 1553400 q^{85} - 1373532 q^{89} + 1016712 q^{93} - 1885372 q^{97}+O(q^{100}) \) 2 * q - 300 * q^5 - 486 * q^9 - 6788 * q^13 + 10356 * q^17 - 10152 * q^21 + 13750 * q^25 - 64284 * q^29 - 46008 * q^33 + 152300 * q^37 - 140076 * q^41 + 72900 * q^45 + 23234 * q^49 - 133884 * q^53 - 212328 * q^57 + 514028 * q^61 + 1018200 * q^65 - 124416 * q^69 + 486884 * q^73 - 961056 * q^77 + 118098 * q^81 - 1553400 * q^85 - 1373532 * q^89 + 1016712 * q^93 - 1885372 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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