LMFDB - Embedded newform 48.7.g.b.31.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,7,Mod(31,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, 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("48.31");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	48.g (of order $2$, degree $1$, 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:	$11.0425960138$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	48.31
Dual form			48.7.g.b.31.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.5885i q^{3} +6.00000 q^{5} -200.918i q^{7} -243.000 q^{9} +O(q^{10})$	$q-15.5885i q^{3} +6.00000 q^{5} -200.918i q^{7} -243.000 q^{9} -644.323i q^{11} -2654.00 q^{13} -93.5307i q^{15} -7206.00 q^{17} -3540.31i q^{19} -3132.00 q^{21} -17459.1i q^{23} -15589.0 q^{25} +3788.00i q^{27} +11550.0 q^{29} +8542.47i q^{31} -10044.0 q^{33} -1205.51i q^{35} +22346.0 q^{37} +41371.8i q^{39} +103626. q^{41} +127389. i q^{43} -1458.00 q^{45} -160831. i q^{47} +77281.0 q^{49} +112330. i q^{51} +168462. q^{53} -3865.94i q^{55} -55188.0 q^{57} -111593. i q^{59} -260470. q^{61} +48823.0i q^{63} -15924.0 q^{65} +317083. i q^{67} -272160. q^{69} -708423. i q^{71} -395918. q^{73} +243008. i q^{75} -129456. q^{77} -556743. i q^{79} +59049.0 q^{81} -89145.2i q^{83} -43236.0 q^{85} -180047. i q^{87} -251886. q^{89} +533236. i q^{91} +133164. q^{93} -21241.9i q^{95} +517474. q^{97} +156570. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 12 q^{5} - 486 q^{9}+O(q^{10}) $ 2 * q + 12 * q^5 - 486 * q^9	$ 2 q + 12 q^{5} - 486 q^{9} - 5308 q^{13} - 14412 q^{17} - 6264 q^{21} - 31178 q^{25} + 23100 q^{29} - 20088 q^{33} + 44692 q^{37} + 207252 q^{41} - 2916 q^{45} + 154562 q^{49} + 336924 q^{53} - 110376 q^{57} - 520940 q^{61} - 31848 q^{65} - 544320 q^{69} - 791836 q^{73} - 258912 q^{77} + 118098 q^{81} - 86472 q^{85} - 503772 q^{89} + 266328 q^{93} + 1034948 q^{97}+O(q^{100}) $ 2 * q + 12 * q^5 - 486 * q^9 - 5308 * q^13 - 14412 * q^17 - 6264 * q^21 - 31178 * q^25 + 23100 * q^29 - 20088 * q^33 + 44692 * q^37 + 207252 * q^41 - 2916 * q^45 + 154562 * q^49 + 336924 * q^53 - 110376 * q^57 - 520940 * q^61 - 31848 * q^65 - 544320 * q^69 - 791836 * q^73 - 258912 * q^77 + 118098 * q^81 - 86472 * q^85 - 503772 * q^89 + 266328 * q^93 + 1034948 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	6.00000	0.0480000	0.0240000	−	0.999712i	$-0.492360\pi$
$5$	6.00000	0.0480000	0.0240000	+	0.999712i	$0.492360\pi$
$6$	0	0
$7$	− 200.918i	− 0.585766i	−0.956148	−	0.292883i	$-0.905385\pi$
$7$	− 200.918i	− 0.585766i	0.956148	−	0.292883i	$-0.0946147\pi$
$8$	0	0
$9$	−243.000	−0.333333
$10$	0	0
$11$	− 644.323i	− 0.484089i	−0.970265	−	0.242045i	$-0.922182\pi$
$11$	− 644.323i	− 0.484089i	0.970265	−	0.242045i	$-0.0778181\pi$
$12$	0	0
$13$	−2654.00	−1.20801	−0.604005	−	0.796980i	$-0.706429\pi$
$13$	−2654.00	−1.20801	−0.604005	+	0.796980i	$0.706429\pi$
$14$	0	0
$15$	− 93.5307i	− 0.0277128i
$16$	0	0
$17$	−7206.00	−1.46672	−0.733360	−	0.679840i	$-0.762049\pi$
$17$	−7206.00	−1.46672	−0.733360	+	0.679840i	$0.762049\pi$
$18$	0	0
$19$	− 3540.31i	− 0.516156i	−0.966124	−	0.258078i	$-0.916911\pi$
$19$	− 3540.31i	− 0.516156i	0.966124	−	0.258078i	$-0.0830891\pi$
$20$	0	0
$21$	−3132.00	−0.338192
$22$	0	0
$23$	− 17459.1i	− 1.43495i	−0.696583	−	0.717476i	$-0.745297\pi$
$23$	− 17459.1i	− 1.43495i	0.696583	−	0.717476i	$-0.254703\pi$
$24$	0	0
$25$	−15589.0	−0.997696
$26$	0	0
$27$	3788.00i	0.192450i
$28$	0	0
$29$	11550.0	0.473574	0.236787	−	0.971562i	$-0.423906\pi$
$29$	11550.0	0.473574	0.236787	+	0.971562i	$0.423906\pi$
$30$	0	0
$31$	8542.47i	0.286747i	0.989669	+	0.143373i	$0.0457950\pi$
$31$	8542.47i	0.286747i	−0.989669	+	0.143373i	$0.954205\pi$
$32$	0	0
$33$	−10044.0	−0.279489
$34$	0	0
$35$	− 1205.51i	− 0.0281168i
$36$	0	0
$37$	22346.0	0.441158	0.220579	−	0.975369i	$-0.429205\pi$
$37$	22346.0	0.441158	0.220579	+	0.975369i	$0.429205\pi$
$38$	0	0
$39$	41371.8i	0.697445i
$40$	0	0
$41$	103626.	1.50355	0.751774	−	0.659421i	$-0.229199\pi$
$41$	103626.	1.50355	0.751774	+	0.659421i	$0.229199\pi$
$42$	0	0
$43$	127389.i	1.60223i	0.598507	+	0.801117i	$0.295761\pi$
$43$	127389.i	1.60223i	−0.598507	+	0.801117i	$0.704239\pi$
$44$	0	0
$45$	−1458.00	−0.0160000
$46$	0	0
$47$	− 160831.i	− 1.54909i	−0.632518	−	0.774546i	$-0.717979\pi$
$47$	− 160831.i	− 1.54909i	0.632518	−	0.774546i	$-0.282021\pi$
$48$	0	0
$49$	77281.0	0.656878
$50$	0	0
$51$	112330.i	0.846812i
$52$	0	0
$53$	168462.	1.13155	0.565776	−	0.824559i	$-0.308577\pi$
$53$	168462.	1.13155	0.565776	+	0.824559i	$0.308577\pi$
$54$	0	0
$55$	− 3865.94i	− 0.0232363i
$56$	0	0
$57$	−55188.0	−0.298003
$58$	0	0
$59$	− 111593.i	− 0.543349i	−0.962389	−	0.271675i	$-0.912423\pi$
$59$	− 111593.i	− 0.543349i	0.962389	−	0.271675i	$-0.0875775\pi$
$60$	0	0
$61$	−260470.	−1.14754	−0.573770	−	0.819016i	$-0.694520\pi$
$61$	−260470.	−1.14754	−0.573770	+	0.819016i	$0.694520\pi$
$62$	0	0
$63$	48823.0i	0.195255i
$64$	0	0
$65$	−15924.0	−0.0579845
$66$	0	0
$67$	317083.i	1.05426i	0.849784	+	0.527131i	$0.176732\pi$
$67$	317083.i	1.05426i	−0.849784	+	0.527131i	$0.823268\pi$
$68$	0	0
$69$	−272160.	−0.828470
$70$	0	0
$71$	− 708423.i	− 1.97933i	−0.143412	−	0.989663i	$-0.545807\pi$
$71$	− 708423.i	− 1.97933i	0.143412	−	0.989663i	$-0.454193\pi$
$72$	0	0
$73$	−395918.	−1.01774	−0.508870	−	0.860844i	$-0.669937\pi$
$73$	−395918.	−1.01774	−0.508870	+	0.860844i	$0.669937\pi$
$74$	0	0
$75$	243008.i	0.576020i
$76$	0	0
$77$	−129456.	−0.283563
$78$	0	0
$79$	− 556743.i	− 1.12921i	−0.825362	−	0.564604i	$-0.809029\pi$
$79$	− 556743.i	− 1.12921i	0.825362	−	0.564604i	$-0.190971\pi$
$80$	0	0
$81$	59049.0	0.111111
$82$	0	0
$83$	− 89145.2i	− 0.155906i	−0.996957	−	0.0779531i	$-0.975162\pi$
$83$	− 89145.2i	− 0.155906i	0.996957	−	0.0779531i	$-0.0248385\pi$
$84$	0	0
$85$	−43236.0	−0.0704026
$86$	0	0
$87$	− 180047.i	− 0.273418i
$88$	0	0
$89$	−251886.	−0.357301	−0.178650	−	0.983913i	$-0.557173\pi$
$89$	−251886.	−0.357301	−0.178650	+	0.983913i	$0.557173\pi$
$90$	0	0
$91$	533236.i	0.707612i
$92$	0	0
$93$	133164.	0.165553
$94$	0	0
$95$	− 21241.9i	− 0.0247755i
$96$	0	0
$97$	517474.	0.566987	0.283494	−	0.958974i	$-0.408507\pi$
$97$	517474.	0.566987	0.283494	+	0.958974i	$0.408507\pi$
$98$	0	0
$99$	156570.i	0.161363i
$100$	0	0
$101$	1.02382e6	0.993712	0.496856	−	0.867833i	$-0.334488\pi$
$101$	1.02382e6	0.993712	0.496856	+	0.867833i	$0.334488\pi$
$102$	0	0
$103$	90960.4i	0.0832416i	0.999133	+	0.0416208i	$0.0132521\pi$
$103$	90960.4i	0.0832416i	−0.999133	+	0.0416208i	$0.986748\pi$
$104$	0	0
$105$	−18792.0	−0.0162332
$106$	0	0
$107$	2.20282e6i	1.79815i	0.437791	+	0.899077i	$0.355761\pi$
$107$	2.20282e6i	1.79815i	−0.437791	+	0.899077i	$0.644239\pi$
$108$	0	0
$109$	952594.	0.735577	0.367789	−	0.929909i	$-0.380115\pi$
$109$	952594.	0.735577	0.367789	+	0.929909i	$0.380115\pi$
$110$	0	0
$111$	− 348340.i	− 0.254703i
$112$	0	0
$113$	−121038.	−0.0838854	−0.0419427	−	0.999120i	$-0.513355\pi$
$113$	−121038.	−0.0838854	−0.0419427	+	0.999120i	$0.513355\pi$
$114$	0	0
$115$	− 104754.i	− 0.0688777i
$116$	0	0
$117$	644922.	0.402670
$118$	0	0
$119$	1.44781e6i	0.859156i
$120$	0	0
$121$	1.35641e6	0.765658
$122$	0	0
$123$	− 1.61537e6i	− 0.868074i
$124$	0	0
$125$	−187284.	−0.0958894
$126$	0	0
$127$	− 1.44941e6i	− 0.707590i	−0.935323	−	0.353795i	$-0.884891\pi$
$127$	− 1.44941e6i	− 0.707590i	0.935323	−	0.353795i	$-0.115109\pi$
$128$	0	0
$129$	1.98580e6	0.925051
$130$	0	0
$131$	632787.i	0.281478i	0.990047	+	0.140739i	$0.0449478\pi$
$131$	632787.i	0.281478i	−0.990047	+	0.140739i	$0.955052\pi$
$132$	0	0
$133$	−711312.	−0.302347
$134$	0	0
$135$	22728.0i	0.00923760i
$136$	0	0
$137$	−4.47821e6	−1.74158	−0.870789	−	0.491656i	$-0.836392\pi$
$137$	−4.47821e6	−1.74158	−0.870789	+	0.491656i	$0.836392\pi$
$138$	0	0
$139$	− 2.24905e6i	− 0.837443i	−0.908115	−	0.418722i	$-0.862478\pi$
$139$	− 2.24905e6i	− 0.837443i	0.908115	−	0.418722i	$-0.137522\pi$
$140$	0	0
$141$	−2.50711e6	−0.894368
$142$	0	0
$143$	1.71003e6i	0.584785i
$144$	0	0
$145$	69300.0	0.0227316
$146$	0	0
$147$	− 1.20469e6i	− 0.379248i
$148$	0	0
$149$	−1.84051e6	−0.556389	−0.278194	−	0.960525i	$-0.589736\pi$
$149$	−1.84051e6	−0.556389	−0.278194	+	0.960525i	$0.589736\pi$
$150$	0	0
$151$	1.14464e6i	0.332460i	0.986087	+	0.166230i	$0.0531594\pi$
$151$	1.14464e6i	0.332460i	−0.986087	+	0.166230i	$0.946841\pi$
$152$	0	0
$153$	1.75106e6	0.488907
$154$	0	0
$155$	51254.8i	0.0137638i
$156$	0	0
$157$	−5.28775e6	−1.36638	−0.683191	−	0.730240i	$-0.739408\pi$
$157$	−5.28775e6	−1.36638	−0.683191	+	0.730240i	$0.739408\pi$
$158$	0	0
$159$	− 2.62606e6i	− 0.653302i
$160$	0	0
$161$	−3.50784e6	−0.840547
$162$	0	0
$163$	− 6.45869e6i	− 1.49136i	−0.666306	−	0.745678i	$-0.732126\pi$
$163$	− 6.45869e6i	− 1.49136i	0.666306	−	0.745678i	$-0.267874\pi$
$164$	0	0
$165$	−60264.0	−0.0134155
$166$	0	0
$167$	3.97714e6i	0.853927i	0.904269	+	0.426964i	$0.140417\pi$
$167$	3.97714e6i	0.853927i	−0.904269	+	0.426964i	$0.859583\pi$
$168$	0	0
$169$	2.21691e6	0.459290
$170$	0	0
$171$	860296.i	0.172052i
$172$	0	0
$173$	5.73548e6	1.10772	0.553862	−	0.832609i	$-0.313154\pi$
$173$	5.73548e6	1.10772	0.553862	+	0.832609i	$0.313154\pi$
$174$	0	0
$175$	3.13211e6i	0.584417i
$176$	0	0
$177$	−1.73956e6	−0.313703
$178$	0	0
$179$	− 8.16883e6i	− 1.42430i	−0.702029	−	0.712149i	$-0.747722\pi$
$179$	− 8.16883e6i	− 1.42430i	0.702029	−	0.712149i	$-0.252278\pi$
$180$	0	0
$181$	3.24656e6	0.547505	0.273752	−	0.961800i	$-0.411735\pi$
$181$	3.24656e6	0.547505	0.273752	+	0.961800i	$0.411735\pi$
$182$	0	0
$183$	4.06033e6i	0.662533i
$184$	0	0
$185$	134076.	0.0211756
$186$	0	0
$187$	4.64299e6i	0.710024i
$188$	0	0
$189$	761076.	0.112731
$190$	0	0
$191$	1.11705e6i	0.160314i	0.996782	+	0.0801571i	$0.0255422\pi$
$191$	1.11705e6i	0.160314i	−0.996782	+	0.0801571i	$0.974458\pi$
$192$	0	0
$193$	3.12375e6	0.434514	0.217257	−	0.976114i	$-0.430289\pi$
$193$	3.12375e6	0.434514	0.217257	+	0.976114i	$0.430289\pi$
$194$	0	0
$195$	248231.i	0.0334774i
$196$	0	0
$197$	1.08530e7	1.41955	0.709777	−	0.704426i	$-0.248796\pi$
$197$	1.08530e7	1.41955	0.709777	+	0.704426i	$0.248796\pi$
$198$	0	0
$199$	8.99733e6i	1.14171i	0.821052	+	0.570853i	$0.193388\pi$
$199$	8.99733e6i	1.14171i	−0.821052	+	0.570853i	$0.806612\pi$
$200$	0	0
$201$	4.94284e6	0.608679
$202$	0	0
$203$	− 2.32060e6i	− 0.277404i
$204$	0	0
$205$	621756.	0.0721703
$206$	0	0
$207$	4.24255e6i	0.478318i
$208$	0	0
$209$	−2.28110e6	−0.249865
$210$	0	0
$211$	3.84129e6i	0.408912i	0.978876	+	0.204456i	$0.0655426\pi$
$211$	3.84129e6i	0.408912i	−0.978876	+	0.204456i	$0.934457\pi$
$212$	0	0
$213$	−1.10432e7	−1.14276
$214$	0	0
$215$	764333.i	0.0769073i
$216$	0	0
$217$	1.71634e6	0.167967
$218$	0	0
$219$	6.17175e6i	0.587592i
$220$	0	0
$221$	1.91247e7	1.77181
$222$	0	0
$223$	− 3.51732e6i	− 0.317174i	−0.987345	−	0.158587i	$-0.949306\pi$
$223$	− 3.51732e6i	− 0.317174i	0.987345	−	0.158587i	$-0.0506938\pi$
$224$	0	0
$225$	3.78813e6	0.332565
$226$	0	0
$227$	− 1.62115e7i	− 1.38595i	−0.720964	−	0.692973i	$-0.756301\pi$
$227$	− 1.62115e7i	− 1.38595i	0.720964	−	0.692973i	$-0.243699\pi$
$228$	0	0
$229$	−1.46861e7	−1.22293	−0.611463	−	0.791273i	$-0.709418\pi$
$229$	−1.46861e7	−1.22293	−0.611463	+	0.791273i	$0.709418\pi$
$230$	0	0
$231$	2.01802e6i	0.163715i
$232$	0	0
$233$	−6.10117e6	−0.482331	−0.241165	−	0.970484i	$-0.577530\pi$
$233$	−6.10117e6	−0.482331	−0.241165	+	0.970484i	$0.577530\pi$
$234$	0	0
$235$	− 964988.i	− 0.0743564i
$236$	0	0
$237$	−8.67877e6	−0.651948
$238$	0	0
$239$	− 2.63536e7i	− 1.93039i	−0.261528	−	0.965196i	$-0.584226\pi$
$239$	− 2.63536e7i	− 1.93039i	0.261528	−	0.965196i	$-0.415774\pi$
$240$	0	0
$241$	−2.15214e6	−0.153752	−0.0768758	−	0.997041i	$-0.524495\pi$
$241$	−2.15214e6	−0.153752	−0.0768758	+	0.997041i	$0.524495\pi$
$242$	0	0
$243$	− 920483.i	− 0.0641500i
$244$	0	0
$245$	463686.	0.0315301
$246$	0	0
$247$	9.39599e6i	0.623522i
$248$	0	0
$249$	−1.38964e6	−0.0900125
$250$	0	0
$251$	7.99802e6i	0.505780i	0.967495	+	0.252890i	$0.0813810\pi$
$251$	7.99802e6i	0.505780i	−0.967495	+	0.252890i	$0.918619\pi$
$252$	0	0
$253$	−1.12493e7	−0.694645
$254$	0	0
$255$	673983.i	0.0406470i
$256$	0	0
$257$	−2.86154e7	−1.68578	−0.842890	−	0.538086i	$-0.819147\pi$
$257$	−2.86154e7	−1.68578	−0.842890	+	0.538086i	$0.819147\pi$
$258$	0	0
$259$	− 4.48971e6i	− 0.258416i
$260$	0	0
$261$	−2.80665e6	−0.157858
$262$	0	0
$263$	− 1.49487e6i	− 0.0821744i	−0.999156	−	0.0410872i	$-0.986918\pi$
$263$	− 1.49487e6i	− 0.0821744i	0.999156	−	0.0410872i	$-0.0130821\pi$
$264$	0	0
$265$	1.01077e6	0.0543145
$266$	0	0
$267$	3.92651e6i	0.206288i
$268$	0	0
$269$	−532290.	−0.0273459	−0.0136729	−	0.999907i	$-0.504352\pi$
$269$	−532290.	−0.0273459	−0.0136729	+	0.999907i	$0.504352\pi$
$270$	0	0
$271$	− 2.88563e7i	− 1.44988i	−0.688811	−	0.724941i	$-0.741867\pi$
$271$	− 2.88563e7i	− 1.44988i	0.688811	−	0.724941i	$-0.258133\pi$
$272$	0	0
$273$	8.31233e6	0.408540
$274$	0	0
$275$	1.00443e7i	0.482974i
$276$	0	0
$277$	−1.82479e7	−0.858567	−0.429284	−	0.903170i	$-0.641234\pi$
$277$	−1.82479e7	−0.858567	−0.429284	+	0.903170i	$0.641234\pi$
$278$	0	0
$279$	− 2.07582e6i	− 0.0955823i
$280$	0	0
$281$	9.52019e6	0.429069	0.214534	−	0.976716i	$-0.431177\pi$
$281$	9.52019e6	0.429069	0.214534	+	0.976716i	$0.431177\pi$
$282$	0	0
$283$	3.81369e7i	1.68262i	0.540552	+	0.841310i	$0.318215\pi$
$283$	3.81369e7i	1.68262i	−0.540552	+	0.841310i	$0.681785\pi$
$284$	0	0
$285$	−331128.	−0.0143041
$286$	0	0
$287$	− 2.08203e7i	− 0.880728i
$288$	0	0
$289$	2.77889e7	1.15127
$290$	0	0
$291$	− 8.06662e6i	− 0.327350i
$292$	0	0
$293$	1.68345e7	0.669266	0.334633	−	0.942349i	$-0.391388\pi$
$293$	1.68345e7	0.669266	0.334633	+	0.942349i	$0.391388\pi$
$294$	0	0
$295$	− 669555.i	− 0.0260808i
$296$	0	0
$297$	2.44069e6	0.0931630
$298$	0	0
$299$	4.63364e7i	1.73344i
$300$	0	0
$301$	2.55947e7	0.938535
$302$	0	0
$303$	− 1.59598e7i	− 0.573720i
$304$	0	0
$305$	−1.56282e6	−0.0550820
$306$	0	0
$307$	− 4.39154e7i	− 1.51775i	−0.651234	−	0.758877i	$-0.725748\pi$
$307$	− 4.39154e7i	− 1.51775i	0.651234	−	0.758877i	$-0.274252\pi$
$308$	0	0
$309$	1.41793e6	0.0480596
$310$	0	0
$311$	5.08371e7i	1.69005i	0.534726	+	0.845025i	$0.320415\pi$
$311$	5.08371e7i	1.69005i	−0.534726	+	0.845025i	$0.679585\pi$
$312$	0	0
$313$	1.30721e7	0.426298	0.213149	−	0.977020i	$-0.431628\pi$
$313$	1.30721e7	0.426298	0.213149	+	0.977020i	$0.431628\pi$
$314$	0	0
$315$	292938.i	0.00937226i
$316$	0	0
$317$	5.07723e6	0.159386	0.0796928	−	0.996819i	$-0.474606\pi$
$317$	5.07723e6	0.159386	0.0796928	+	0.996819i	$0.474606\pi$
$318$	0	0
$319$	− 7.44193e6i	− 0.229252i
$320$	0	0
$321$	3.43385e7	1.03816
$322$	0	0
$323$	2.55115e7i	0.757056i
$324$	0	0
$325$	4.13732e7	1.20523
$326$	0	0
$327$	− 1.48495e7i	− 0.424686i
$328$	0	0
$329$	−3.23139e7	−0.907406
$330$	0	0
$331$	5.59418e7i	1.54260i	0.636474	+	0.771298i	$0.280392\pi$
$331$	5.59418e7i	1.54260i	−0.636474	+	0.771298i	$0.719608\pi$
$332$	0	0
$333$	−5.43008e6	−0.147053
$334$	0	0
$335$	1.90250e6i	0.0506046i
$336$	0	0
$337$	−2.55737e7	−0.668197	−0.334099	−	0.942538i	$-0.608432\pi$
$337$	−2.55737e7	−0.668197	−0.334099	+	0.942538i	$0.608432\pi$
$338$	0	0
$339$	1.88680e6i	0.0484313i
$340$	0	0
$341$	5.50411e6	0.138811
$342$	0	0
$343$	− 3.91649e7i	− 0.970543i
$344$	0	0
$345$	−1.63296e6	−0.0397666
$346$	0	0
$347$	4.96031e6i	0.118719i	0.998237	+	0.0593595i	$0.0189058\pi$
$347$	4.96031e6i	0.118719i	−0.998237	+	0.0593595i	$0.981094\pi$
$348$	0	0
$349$	−3.06563e7	−0.721179	−0.360590	−	0.932725i	$-0.617424\pi$
$349$	−3.06563e7	−0.721179	−0.360590	+	0.932725i	$0.617424\pi$
$350$	0	0
$351$	− 1.00533e7i	− 0.232482i
$352$	0	0
$353$	−5.33353e7	−1.21252	−0.606262	−	0.795265i	$-0.707332\pi$
$353$	−5.33353e7	−1.21252	−0.606262	+	0.795265i	$0.707332\pi$
$354$	0	0
$355$	− 4.25054e6i	− 0.0950077i
$356$	0	0
$357$	2.25692e7	0.496034
$358$	0	0
$359$	2.83885e7i	0.613564i	0.951780	+	0.306782i	$0.0992522\pi$
$359$	2.83885e7i	0.613564i	−0.951780	+	0.306782i	$0.900748\pi$
$360$	0	0
$361$	3.45121e7	0.733583
$362$	0	0
$363$	− 2.11443e7i	− 0.442053i
$364$	0	0
$365$	−2.37551e6	−0.0488515
$366$	0	0
$367$	− 5.75671e7i	− 1.16460i	−0.812974	−	0.582299i	$-0.802153\pi$
$367$	− 5.75671e7i	− 1.16460i	0.812974	−	0.582299i	$-0.197847\pi$
$368$	0	0
$369$	−2.51811e7	−0.501183
$370$	0	0
$371$	− 3.38470e7i	− 0.662825i
$372$	0	0
$373$	−4.68328e7	−0.902451	−0.451226	−	0.892410i	$-0.649013\pi$
$373$	−4.68328e7	−0.902451	−0.451226	+	0.892410i	$0.649013\pi$
$374$	0	0
$375$	2.91947e6i	0.0553618i
$376$	0	0
$377$	−3.06537e7	−0.572083
$378$	0	0
$379$	− 7.92476e7i	− 1.45569i	−0.685742	−	0.727845i	$-0.740522\pi$
$379$	− 7.92476e7i	− 1.45569i	0.685742	−	0.727845i	$-0.259478\pi$
$380$	0	0
$381$	−2.25941e7	−0.408527
$382$	0	0
$383$	− 2.00575e7i	− 0.357010i	−0.983939	−	0.178505i	$-0.942874\pi$
$383$	− 2.00575e7i	− 0.357010i	0.983939	−	0.178505i	$-0.0571260\pi$
$384$	0	0
$385$	−776736.	−0.0136110
$386$	0	0
$387$	− 3.09555e7i	− 0.534078i
$388$	0	0
$389$	−4.79345e7	−0.814328	−0.407164	−	0.913355i	$-0.633482\pi$
$389$	−4.79345e7	−0.814328	−0.407164	+	0.913355i	$0.633482\pi$
$390$	0	0
$391$	1.25810e8i	2.10468i
$392$	0	0
$393$	9.86418e6	0.162511
$394$	0	0
$395$	− 3.34046e6i	− 0.0542020i
$396$	0	0
$397$	−3.43785e7	−0.549434	−0.274717	−	0.961525i	$-0.588584\pi$
$397$	−3.43785e7	−0.549434	−0.274717	+	0.961525i	$0.588584\pi$
$398$	0	0
$399$	1.10883e7i	0.174560i
$400$	0	0
$401$	9.09234e7	1.41008	0.705038	−	0.709170i	$-0.250930\pi$
$401$	9.09234e7	1.41008	0.705038	+	0.709170i	$0.250930\pi$
$402$	0	0
$403$	− 2.26717e7i	− 0.346393i
$404$	0	0
$405$	354294.	0.00533333
$406$	0	0
$407$	− 1.43980e7i	− 0.213560i
$408$	0	0
$409$	−1.15454e8	−1.68748	−0.843739	−	0.536753i	$-0.819651\pi$
$409$	−1.15454e8	−1.68748	−0.843739	+	0.536753i	$0.819651\pi$
$410$	0	0
$411$	6.98084e7i	1.00550i
$412$	0	0
$413$	−2.24209e7	−0.318276
$414$	0	0
$415$	− 534871.i	− 0.00748350i
$416$	0	0
$417$	−3.50593e7	−0.483498
$418$	0	0
$419$	− 7.40305e7i	− 1.00640i	−0.864171	−	0.503198i	$-0.832157\pi$
$419$	− 7.40305e7i	− 1.00640i	0.864171	−	0.503198i	$-0.167843\pi$
$420$	0	0
$421$	1.14811e8	1.53863	0.769317	−	0.638867i	$-0.220596\pi$
$421$	1.14811e8	1.53863	0.769317	+	0.638867i	$0.220596\pi$
$422$	0	0
$423$	3.90820e7i	0.516364i
$424$	0	0
$425$	1.12334e8	1.46334
$426$	0	0
$427$	5.23331e7i	0.672191i
$428$	0	0
$429$	2.66568e7	0.337626
$430$	0	0
$431$	− 3.42559e6i	− 0.0427862i	−0.999771	−	0.0213931i	$-0.993190\pi$
$431$	− 3.42559e6i	− 0.0427862i	0.999771	−	0.0213931i	$-0.00681016\pi$
$432$	0	0
$433$	1.08786e8	1.34002	0.670008	−	0.742354i	$-0.266291\pi$
$433$	1.08786e8	1.34002	0.670008	+	0.742354i	$0.266291\pi$
$434$	0	0
$435$	− 1.08028e6i	− 0.0131241i
$436$	0	0
$437$	−6.18106e7	−0.740659
$438$	0	0
$439$	1.13400e8i	1.34036i	0.742199	+	0.670180i	$0.233783\pi$
$439$	1.13400e8i	1.34036i	−0.742199	+	0.670180i	$0.766217\pi$
$440$	0	0
$441$	−1.87793e7	−0.218959
$442$	0	0
$443$	− 2.44463e7i	− 0.281191i	−0.990067	−	0.140595i	$-0.955098\pi$
$443$	− 2.44463e7i	− 0.281191i	0.990067	−	0.140595i	$-0.0449017\pi$
$444$	0	0
$445$	−1.51132e6	−0.0171504
$446$	0	0
$447$	2.86906e7i	0.321231i
$448$	0	0
$449$	9.03699e7	0.998354	0.499177	−	0.866500i	$-0.333636\pi$
$449$	9.03699e7	0.998354	0.499177	+	0.866500i	$0.333636\pi$
$450$	0	0
$451$	− 6.67686e7i	− 0.727851i
$452$	0	0
$453$	1.78432e7	0.191946
$454$	0	0
$455$	3.19942e6i	0.0339654i
$456$	0	0
$457$	−5.82048e7	−0.609832	−0.304916	−	0.952379i	$-0.598628\pi$
$457$	−5.82048e7	−0.609832	−0.304916	+	0.952379i	$0.598628\pi$
$458$	0	0
$459$	− 2.72963e7i	− 0.282271i
$460$	0	0
$461$	1.73271e8	1.76858	0.884288	−	0.466942i	$-0.154644\pi$
$461$	1.73271e8	1.76858	0.884288	+	0.466942i	$0.154644\pi$
$462$	0	0
$463$	1.25400e8i	1.26344i	0.775197	+	0.631719i	$0.217650\pi$
$463$	1.25400e8i	1.26344i	−0.775197	+	0.631719i	$0.782350\pi$
$464$	0	0
$465$	798984.	0.00794656
$466$	0	0
$467$	− 1.45905e8i	− 1.43258i	−0.697800	−	0.716292i	$-0.745838\pi$
$467$	− 1.45905e8i	− 1.43258i	0.697800	−	0.716292i	$-0.254162\pi$
$468$	0	0
$469$	6.37077e7	0.617551
$470$	0	0
$471$	8.24279e7i	0.788881i
$472$	0	0
$473$	8.20796e7	0.775625
$474$	0	0
$475$	5.51899e7i	0.514966i
$476$	0	0
$477$	−4.09363e7	−0.377184
$478$	0	0
$479$	− 7.62417e6i	− 0.0693723i	−0.999398	−	0.0346861i	$-0.988957\pi$
$479$	− 7.62417e6i	− 0.0693723i	0.999398	−	0.0346861i	$-0.0110432\pi$
$480$	0	0
$481$	−5.93063e7	−0.532924
$482$	0	0
$483$	5.46818e7i	0.485290i
$484$	0	0
$485$	3.10484e6	0.0272154
$486$	0	0
$487$	− 1.71349e8i	− 1.48352i	−0.670665	−	0.741760i	$-0.733991\pi$
$487$	− 1.71349e8i	− 1.48352i	0.670665	−	0.741760i	$-0.266009\pi$
$488$	0	0
$489$	−1.00681e8	−0.861035
$490$	0	0
$491$	3.30473e6i	0.0279185i	0.999903	+	0.0139592i	$0.00444351\pi$
$491$	3.30473e6i	0.0279185i	−0.999903	+	0.0139592i	$0.995556\pi$
$492$	0	0
$493$	−8.32293e7	−0.694601
$494$	0	0
$495$	939423.i	0.00774543i
$496$	0	0
$497$	−1.42335e8	−1.15942
$498$	0	0
$499$	1.26207e8i	1.01574i	0.861435	+	0.507868i	$0.169566\pi$
$499$	1.26207e8i	1.01574i	−0.861435	+	0.507868i	$0.830434\pi$
$500$	0	0
$501$	6.19974e7	0.493015
$502$	0	0
$503$	212336.i	0.00166847i	1.00000		0.000834236i	$0.000265546\pi$
$503$	212336.i	0.00166847i	−1.00000		0.000834236i	$0.999734\pi$
$504$	0	0
$505$	6.14293e6	0.0476982
$506$	0	0
$507$	− 3.45582e7i	− 0.265171i
$508$	0	0
$509$	−1.49792e8	−1.13589	−0.567944	−	0.823067i	$-0.692261\pi$
$509$	−1.49792e8	−1.13589	−0.567944	+	0.823067i	$0.692261\pi$
$510$	0	0
$511$	7.95470e7i	0.596158i
$512$	0	0
$513$	1.34107e7	0.0993342
$514$	0	0
$515$	545762.i	0.00399560i
$516$	0	0
$517$	−1.03627e8	−0.749899
$518$	0	0
$519$	− 8.94073e7i	− 0.639544i
$520$	0	0
$521$	1.11422e8	0.787876	0.393938	−	0.919137i	$-0.371113\pi$
$521$	1.11422e8	0.787876	0.393938	+	0.919137i	$0.371113\pi$
$522$	0	0
$523$	− 2.69486e8i	− 1.88379i	−0.335911	−	0.941894i	$-0.609044\pi$
$523$	− 2.69486e8i	− 1.88379i	0.335911	−	0.941894i	$-0.390956\pi$
$524$	0	0
$525$	4.88247e7	0.337413
$526$	0	0
$527$	− 6.15571e7i	− 0.420578i
$528$	0	0
$529$	−1.56783e8	−1.05909
$530$	0	0
$531$	2.71170e7i	0.181116i
$532$	0	0
$533$	−2.75023e8	−1.81630
$534$	0	0
$535$	1.32169e7i	0.0863114i
$536$	0	0
$537$	−1.27339e8	−0.822319
$538$	0	0
$539$	− 4.97939e7i	− 0.317987i
$540$	0	0
$541$	3.03425e8	1.91628	0.958142	−	0.286292i	$-0.0924227\pi$
$541$	3.03425e8	1.91628	0.958142	+	0.286292i	$0.0924227\pi$
$542$	0	0
$543$	− 5.06089e7i	− 0.316102i
$544$	0	0
$545$	5.71556e6	0.0353077
$546$	0	0
$547$	1.92937e7i	0.117884i	0.998261	+	0.0589418i	$0.0187726\pi$
$547$	1.92937e7i	0.117884i	−0.998261	+	0.0589418i	$0.981227\pi$
$548$	0	0
$549$	6.32942e7	0.382514
$550$	0	0
$551$	− 4.08906e7i	− 0.244438i
$552$	0	0
$553$	−1.11860e8	−0.661452
$554$	0	0
$555$	− 2.09004e6i	− 0.0122257i
$556$	0	0
$557$	−1.71693e8	−0.993544	−0.496772	−	0.867881i	$-0.665482\pi$
$557$	−1.71693e8	−0.993544	−0.496772	+	0.867881i	$0.665482\pi$
$558$	0	0
$559$	− 3.38090e8i	− 1.93552i
$560$	0	0
$561$	7.23771e7	0.409933
$562$	0	0
$563$	2.21392e8i	1.24062i	0.784358	+	0.620308i	$0.212992\pi$
$563$	2.21392e8i	1.24062i	−0.784358	+	0.620308i	$0.787008\pi$
$564$	0	0
$565$	−726228.	−0.00402650
$566$	0	0
$567$	− 1.18640e7i	− 0.0650852i
$568$	0	0
$569$	2.56991e8	1.39502	0.697511	−	0.716574i	$-0.254291\pi$
$569$	2.56991e8	1.39502	0.697511	+	0.716574i	$0.254291\pi$
$570$	0	0
$571$	2.71249e8i	1.45700i	0.685045	+	0.728501i	$0.259782\pi$
$571$	2.71249e8i	1.45700i	−0.685045	+	0.728501i	$0.740218\pi$
$572$	0	0
$573$	1.74131e7	0.0925574
$574$	0	0
$575$	2.72169e8i	1.43165i
$576$	0	0
$577$	−3.41573e6	−0.0177810	−0.00889049	−	0.999960i	$-0.502830\pi$
$577$	−3.41573e6	−0.0177810	−0.00889049	+	0.999960i	$0.502830\pi$
$578$	0	0
$579$	− 4.86944e7i	− 0.250867i
$580$	0	0
$581$	−1.79109e7	−0.0913247
$582$	0	0
$583$	− 1.08544e8i	− 0.547772i
$584$	0	0
$585$	3.86953e6	0.0193282
$586$	0	0
$587$	2.32848e8i	1.15122i	0.817725	+	0.575609i	$0.195235\pi$
$587$	2.32848e8i	1.15122i	−0.817725	+	0.575609i	$0.804765\pi$
$588$	0	0
$589$	3.02430e7	0.148006
$590$	0	0
$591$	− 1.69182e8i	− 0.819580i
$592$	0	0
$593$	−8.86693e7	−0.425216	−0.212608	−	0.977138i	$-0.568196\pi$
$593$	−8.86693e7	−0.425216	−0.212608	+	0.977138i	$0.568196\pi$
$594$	0	0
$595$	8.68689e6i	0.0412395i
$596$	0	0
$597$	1.40255e8	0.659165
$598$	0	0
$599$	− 2.37591e7i	− 0.110547i	−0.998471	−	0.0552737i	$-0.982397\pi$
$599$	− 2.37591e7i	− 0.110547i	0.998471	−	0.0552737i	$-0.0176031\pi$
$600$	0	0
$601$	2.56421e8	1.18122	0.590609	−	0.806958i	$-0.298888\pi$
$601$	2.56421e8	1.18122	0.590609	+	0.806958i	$0.298888\pi$
$602$	0	0
$603$	− 7.70512e7i	− 0.351421i
$604$	0	0
$605$	8.13845e6	0.0367516
$606$	0	0
$607$	2.08193e8i	0.930895i	0.885076	+	0.465447i	$0.154107\pi$
$607$	2.08193e8i	0.930895i	−0.885076	+	0.465447i	$0.845893\pi$
$608$	0	0
$609$	−3.61746e7	−0.160159
$610$	0	0
$611$	4.26846e8i	1.87132i
$612$	0	0
$613$	2.59662e8	1.12727	0.563634	−	0.826024i	$-0.309403\pi$
$613$	2.59662e8	1.12727	0.563634	+	0.826024i	$0.309403\pi$
$614$	0	0
$615$	− 9.69222e6i	− 0.0416675i
$616$	0	0
$617$	1.40534e8	0.598309	0.299155	−	0.954205i	$-0.403295\pi$
$617$	1.40534e8	0.598309	0.299155	+	0.954205i	$0.403295\pi$
$618$	0	0
$619$	− 1.58033e8i	− 0.666307i	−0.942873	−	0.333154i	$-0.891887\pi$
$619$	− 1.58033e8i	− 0.666307i	0.942873	−	0.333154i	$-0.108113\pi$
$620$	0	0
$621$	6.61349e7	0.276157
$622$	0	0
$623$	5.06084e7i	0.209295i
$624$	0	0
$625$	2.42454e8	0.993093
$626$	0	0
$627$	3.55589e7i	0.144260i
$628$	0	0
$629$	−1.61025e8	−0.647056
$630$	0	0
$631$	− 2.49902e8i	− 0.994677i	−0.867556	−	0.497339i	$-0.834311\pi$
$631$	− 2.49902e8i	− 0.994677i	0.867556	−	0.497339i	$-0.165689\pi$
$632$	0	0
$633$	5.98798e7	0.236086
$634$	0	0
$635$	− 8.69649e6i	− 0.0339643i
$636$	0	0
$637$	−2.05104e8	−0.793515
$638$	0	0
$639$	1.72147e8i	0.659775i
$640$	0	0
$641$	−2.94095e7	−0.111664	−0.0558321	−	0.998440i	$-0.517781\pi$
$641$	−2.94095e7	−0.111664	−0.0558321	+	0.998440i	$0.517781\pi$
$642$	0	0
$643$	2.55925e8i	0.962675i	0.876535	+	0.481337i	$0.159849\pi$
$643$	2.55925e8i	0.962675i	−0.876535	+	0.481337i	$0.840151\pi$
$644$	0	0
$645$	1.19148e7	0.0444024
$646$	0	0
$647$	− 4.08342e8i	− 1.50769i	−0.657054	−	0.753844i	$-0.728197\pi$
$647$	− 4.08342e8i	− 1.50769i	0.657054	−	0.753844i	$-0.271803\pi$
$648$	0	0
$649$	−7.19016e7	−0.263030
$650$	0	0
$651$	− 2.67550e7i	− 0.0969756i
$652$	0	0
$653$	2.06095e8	0.740164	0.370082	−	0.928999i	$-0.379330\pi$
$653$	2.06095e8	0.740164	0.370082	+	0.928999i	$0.379330\pi$
$654$	0	0
$655$	3.79672e6i	0.0135109i
$656$	0	0
$657$	9.62081e7	0.339247
$658$	0	0
$659$	7.85056e7i	0.274312i	0.990549	+	0.137156i	$0.0437961\pi$
$659$	7.85056e7i	0.274312i	−0.990549	+	0.137156i	$0.956204\pi$
$660$	0	0
$661$	−5.50505e7	−0.190615	−0.0953074	−	0.995448i	$-0.530383\pi$
$661$	−5.50505e7	−0.190615	−0.0953074	+	0.995448i	$0.530383\pi$
$662$	0	0
$663$	− 2.98125e8i	− 1.02296i
$664$	0	0
$665$	−4.26787e6	−0.0145126
$666$	0	0
$667$	− 2.01652e8i	− 0.679557i
$668$	0	0
$669$	−5.48295e7	−0.183120
$670$	0	0
$671$	1.67827e8i	0.555512i
$672$	0	0
$673$	−4.82818e7	−0.158394	−0.0791969	−	0.996859i	$-0.525236\pi$
$673$	−4.82818e7	−0.158394	−0.0791969	+	0.996859i	$0.525236\pi$
$674$	0	0
$675$	− 5.90511e7i	− 0.192007i
$676$	0	0
$677$	−4.02892e8	−1.29844	−0.649221	−	0.760600i	$-0.724905\pi$
$677$	−4.02892e8	−1.29844	−0.649221	+	0.760600i	$0.724905\pi$
$678$	0	0
$679$	− 1.03970e8i	− 0.332122i
$680$	0	0
$681$	−2.52713e8	−0.800176
$682$	0	0
$683$	− 1.26946e8i	− 0.398434i	−0.979955	−	0.199217i	$-0.936160\pi$
$683$	− 1.26946e8i	− 0.398434i	0.979955	−	0.199217i	$-0.0638398\pi$
$684$	0	0
$685$	−2.68693e7	−0.0835958
$686$	0	0
$687$	2.28934e8i	0.706056i
$688$	0	0
$689$	−4.47098e8	−1.36693
$690$	0	0
$691$	1.39216e8i	0.421944i	0.977492	+	0.210972i	$0.0676629\pi$
$691$	1.39216e8i	0.421944i	−0.977492	+	0.210972i	$0.932337\pi$
$692$	0	0
$693$	3.14578e7	0.0945211
$694$	0	0
$695$	− 1.34943e7i	− 0.0401973i
$696$	0	0
$697$	−7.46729e8	−2.20528
$698$	0	0
$699$	9.51078e7i	0.278474i
$700$	0	0
$701$	4.45981e8	1.29468	0.647340	−	0.762201i	$-0.275881\pi$
$701$	4.45981e8	1.29468	0.647340	+	0.762201i	$0.275881\pi$
$702$	0	0
$703$	− 7.91118e7i	− 0.227706i
$704$	0	0
$705$	−1.50427e7	−0.0429297
$706$	0	0
$707$	− 2.05704e8i	− 0.582083i
$708$	0	0
$709$	1.59057e8	0.446288	0.223144	−	0.974786i	$-0.428368\pi$
$709$	1.59057e8	0.446288	0.223144	+	0.974786i	$0.428368\pi$
$710$	0	0
$711$	1.35289e8i	0.376403i
$712$	0	0
$713$	1.49144e8	0.411468
$714$	0	0
$715$	1.02602e7i	0.0280697i
$716$	0	0
$717$	−4.10811e8	−1.11451
$718$	0	0
$719$	− 3.34049e8i	− 0.898718i	−0.893351	−	0.449359i	$-0.851652\pi$
$719$	− 3.34049e8i	− 0.898718i	0.893351	−	0.449359i	$-0.148348\pi$
$720$	0	0
$721$	1.82756e7	0.0487602
$722$	0	0
$723$	3.35486e7i	0.0887686i
$724$	0	0
$725$	−1.80053e8	−0.472483
$726$	0	0
$727$	2.60960e8i	0.679157i	0.940578	+	0.339579i	$0.110284\pi$
$727$	2.60960e8i	0.679157i	−0.940578	+	0.339579i	$0.889716\pi$
$728$	0	0
$729$	−1.43489e7	−0.0370370
$730$	0	0
$731$	− 9.17964e8i	− 2.35003i
$732$	0	0
$733$	−3.56906e8	−0.906238	−0.453119	−	0.891450i	$-0.649689\pi$
$733$	−3.56906e8	−0.906238	−0.453119	+	0.891450i	$0.649689\pi$
$734$	0	0
$735$	− 7.22815e6i	− 0.0182039i
$736$	0	0
$737$	2.04304e8	0.510357
$738$	0	0
$739$	− 8.61653e7i	− 0.213501i	−0.994286	−	0.106750i	$-0.965955\pi$
$739$	− 8.61653e7i	− 0.213501i	0.994286	−	0.106750i	$-0.0340446\pi$
$740$	0	0
$741$	1.46469e8	0.359990
$742$	0	0
$743$	− 2.61364e7i	− 0.0637204i	−0.999492	−	0.0318602i	$-0.989857\pi$
$743$	− 2.61364e7i	− 0.0637204i	0.999492	−	0.0318602i	$-0.0101431\pi$
$744$	0	0
$745$	−1.10430e7	−0.0267067
$746$	0	0
$747$	2.16623e7i	0.0519688i
$748$	0	0
$749$	4.42585e8	1.05330
$750$	0	0
$751$	5.80052e8i	1.36945i	0.728800	+	0.684727i	$0.240078\pi$
$751$	5.80052e8i	1.36945i	−0.728800	+	0.684727i	$0.759922\pi$
$752$	0	0
$753$	1.24677e8	0.292012
$754$	0	0
$755$	6.86786e6i	0.0159581i
$756$	0	0
$757$	3.35602e8	0.773636	0.386818	−	0.922156i	$-0.373574\pi$
$757$	3.35602e8	0.773636	0.386818	+	0.922156i	$0.373574\pi$
$758$	0	0
$759$	1.75359e8i	0.401054i
$760$	0	0
$761$	2.34765e8	0.532696	0.266348	−	0.963877i	$-0.414183\pi$
$761$	2.34765e8	0.532696	0.266348	+	0.963877i	$0.414183\pi$
$762$	0	0
$763$	− 1.91393e8i	− 0.430877i
$764$	0	0
$765$	1.05063e7	0.0234675
$766$	0	0
$767$	2.96167e8i	0.656372i
$768$	0	0
$769$	3.78738e8	0.832837	0.416419	−	0.909173i	$-0.363285\pi$
$769$	3.78738e8	0.832837	0.416419	+	0.909173i	$0.363285\pi$
$770$	0	0
$771$	4.46070e8i	0.973285i
$772$	0	0
$773$	−1.34904e8	−0.292070	−0.146035	−	0.989279i	$-0.546651\pi$
$773$	−1.34904e8	−0.292070	−0.146035	+	0.989279i	$0.546651\pi$
$774$	0	0
$775$	− 1.33169e8i	− 0.286086i
$776$	0	0
$777$	−6.99877e7	−0.149196
$778$	0	0
$779$	− 3.66868e8i	− 0.776065i
$780$	0	0
$781$	−4.56453e8	−0.958171
$782$	0	0
$783$	4.37513e7i	0.0911394i
$784$	0	0
$785$	−3.17265e7	−0.0655863
$786$	0	0
$787$	8.58679e7i	0.176160i	0.996113	+	0.0880799i	$0.0280731\pi$
$787$	8.58679e7i	0.176160i	−0.996113	+	0.0880799i	$0.971927\pi$
$788$	0	0
$789$	−2.33027e7	−0.0474434
$790$	0	0
$791$	2.43187e7i	0.0491373i
$792$	0	0
$793$	6.91287e8	1.38624
$794$	0	0
$795$	− 1.57564e7i	− 0.0313585i
$796$	0	0
$797$	8.82735e8	1.74363	0.871817	−	0.489832i	$-0.162942\pi$
$797$	8.82735e8	1.74363	0.871817	+	0.489832i	$0.162942\pi$
$798$	0	0
$799$	1.15895e9i	2.27208i
$800$	0	0
$801$	6.12083e7	0.119100
$802$	0	0
$803$	2.55099e8i	0.492677i
$804$	0	0
$805$	−2.10470e7	−0.0403463
$806$	0	0
$807$	8.29758e6i	0.0157881i
$808$	0	0
$809$	3.19181e7	0.0602825	0.0301412	−	0.999546i	$-0.490404\pi$
$809$	3.19181e7	0.0602825	0.0301412	+	0.999546i	$0.490404\pi$
$810$	0	0
$811$	8.59397e7i	0.161113i	0.996750	+	0.0805567i	$0.0256698\pi$
$811$	8.59397e7i	0.161113i	−0.996750	+	0.0805567i	$0.974330\pi$
$812$	0	0
$813$	−4.49825e8	−0.837089
$814$	0	0
$815$	− 3.87521e7i	− 0.0715851i
$816$	0	0
$817$	4.50996e8	0.827003
$818$	0	0
$819$	− 1.29576e8i	− 0.235871i
$820$	0	0
$821$	7.71027e8	1.39328	0.696642	−	0.717419i	$-0.254676\pi$
$821$	7.71027e8	1.39328	0.696642	+	0.717419i	$0.254676\pi$
$822$	0	0
$823$	− 2.04669e8i	− 0.367158i	−0.983005	−	0.183579i	$-0.941232\pi$
$823$	− 2.04669e8i	− 0.367158i	0.983005	−	0.183579i	$-0.0587683\pi$
$824$	0	0
$825$	1.56576e8	0.278845
$826$	0	0
$827$	− 2.55414e8i	− 0.451574i	−0.974177	−	0.225787i	$-0.927505\pi$
$827$	− 2.55414e8i	− 0.451574i	0.974177	−	0.225787i	$-0.0724953\pi$
$828$	0	0
$829$	−6.29211e8	−1.10442	−0.552208	−	0.833706i	$-0.686215\pi$
$829$	−6.29211e8	−1.10442	−0.552208	+	0.833706i	$0.686215\pi$
$830$	0	0
$831$	2.84457e8i	0.495694i
$832$	0	0
$833$	−5.56887e8	−0.963456
$834$	0	0
$835$	2.38628e7i	0.0409885i
$836$	0	0
$837$	−3.23589e7	−0.0551845
$838$	0	0
$839$	2.24118e8i	0.379481i	0.981834	+	0.189741i	$0.0607647\pi$
$839$	2.24118e8i	0.379481i	−0.981834	+	0.189741i	$0.939235\pi$
$840$	0	0
$841$	−4.61421e8	−0.775728
$842$	0	0
$843$	− 1.48405e8i	− 0.247723i
$844$	0	0
$845$	1.33014e7	0.0220459
$846$	0	0
$847$	− 2.72527e8i	− 0.448496i
$848$	0	0
$849$	5.94496e8	0.971461
$850$	0	0
$851$	− 3.90140e8i	− 0.633042i
$852$	0	0
$853$	2.09382e8	0.337359	0.168679	−	0.985671i	$-0.446050\pi$
$853$	2.09382e8	0.337359	0.168679	+	0.985671i	$0.446050\pi$
$854$	0	0
$855$	5.16177e6i	0.00825849i
$856$	0	0
$857$	−3.32297e8	−0.527940	−0.263970	−	0.964531i	$-0.585032\pi$
$857$	−3.32297e8	−0.527940	−0.263970	+	0.964531i	$0.585032\pi$
$858$	0	0
$859$	− 6.58920e8i	− 1.03957i	−0.854297	−	0.519784i	$-0.826012\pi$
$859$	− 6.58920e8i	− 1.03957i	0.854297	−	0.519784i	$-0.173988\pi$
$860$	0	0
$861$	−3.24557e8	−0.508488
$862$	0	0
$863$	1.00694e9i	1.56665i	0.621610	+	0.783327i	$0.286479\pi$
$863$	1.00694e9i	1.56665i	−0.621610	+	0.783327i	$0.713521\pi$
$864$	0	0
$865$	3.44129e7	0.0531707
$866$	0	0
$867$	− 4.33186e8i	− 0.664686i
$868$	0	0
$869$	−3.58723e8	−0.546637
$870$	0	0
$871$	− 8.41538e8i	− 1.27356i
$872$	0	0
$873$	−1.25746e8	−0.188996
$874$	0	0
$875$	3.76287e7i	0.0561688i
$876$	0	0
$877$	7.31416e8	1.08434	0.542170	−	0.840269i	$-0.317603\pi$
$877$	7.31416e8	1.08434	0.542170	+	0.840269i	$0.317603\pi$
$878$	0	0
$879$	− 2.62425e8i	− 0.386401i
$880$	0	0
$881$	−1.16606e9	−1.70528	−0.852638	−	0.522502i	$-0.824999\pi$
$881$	−1.16606e9	−1.70528	−0.852638	+	0.522502i	$0.824999\pi$
$882$	0	0
$883$	7.42439e7i	0.107840i	0.998545	+	0.0539198i	$0.0171715\pi$
$883$	7.42439e7i	0.107840i	−0.998545	+	0.0539198i	$0.982828\pi$
$884$	0	0
$885$	−1.04373e7	−0.0150577
$886$	0	0
$887$	− 2.35897e8i	− 0.338027i	−0.985614	−	0.169014i	$-0.945942\pi$
$887$	− 2.35897e8i	− 0.338027i	0.985614	−	0.169014i	$-0.0540582\pi$
$888$	0	0
$889$	−2.91213e8	−0.414482
$890$	0	0
$891$	− 3.80466e7i	− 0.0537877i
$892$	0	0
$893$	−5.69393e8	−0.799572
$894$	0	0
$895$	− 4.90130e7i	− 0.0683663i
$896$	0	0
$897$	7.22313e8	1.00080
$898$	0	0
$899$	9.86656e7i	0.135796i
$900$	0	0
$901$	−1.21394e9	−1.65967
$902$	0	0
$903$	− 3.98982e8i	− 0.541864i
$904$	0	0
$905$	1.94794e7	0.0262802
$906$	0	0
$907$	− 2.54036e8i	− 0.340466i	−0.985404	−	0.170233i	$-0.945548\pi$
$907$	− 2.54036e8i	− 0.340466i	0.985404	−	0.170233i	$-0.0544520\pi$
$908$	0	0
$909$	−2.48789e8	−0.331237
$910$	0	0
$911$	− 1.03209e9i	− 1.36509i	−0.730844	−	0.682544i	$-0.760874\pi$
$911$	− 1.03209e9i	− 1.36509i	0.730844	−	0.682544i	$-0.239126\pi$
$912$	0	0
$913$	−5.74383e7	−0.0754726
$914$	0	0
$915$	2.43620e7i	0.0318016i
$916$	0	0
$917$	1.27138e8	0.164880
$918$	0	0
$919$	1.13004e9i	1.45595i	0.685602	+	0.727976i	$0.259539\pi$
$919$	1.13004e9i	1.45595i	−0.685602	+	0.727976i	$0.740461\pi$
$920$	0	0
$921$	−6.84573e8	−0.876276
$922$	0	0
$923$	1.88015e9i	2.39105i
$924$	0	0
$925$	−3.48352e8	−0.440142
$926$	0	0
$927$	− 2.21034e7i	− 0.0277472i
$928$	0	0
$929$	−1.40049e9	−1.74676	−0.873382	−	0.487036i	$-0.838078\pi$
$929$	−1.40049e9	−1.74676	−0.873382	+	0.487036i	$0.838078\pi$
$930$	0	0
$931$	− 2.73599e8i	− 0.339051i
$932$	0	0
$933$	7.92472e8	0.975751
$934$	0	0
$935$	2.78579e7i	0.0340812i
$936$	0	0
$937$	3.55654e8	0.432324	0.216162	−	0.976357i	$-0.430646\pi$
$937$	3.55654e8	0.432324	0.216162	+	0.976357i	$0.430646\pi$
$938$	0	0
$939$	− 2.03774e8i	− 0.246123i
$940$	0	0
$941$	1.04392e8	0.125285	0.0626427	−	0.998036i	$-0.480047\pi$
$941$	1.04392e8	0.125285	0.0626427	+	0.998036i	$0.480047\pi$
$942$	0	0
$943$	− 1.80921e9i	− 2.15752i
$944$	0	0
$945$	4.56646e6	0.00541108
$946$	0	0
$947$	9.83576e8i	1.15813i	0.815281	+	0.579066i	$0.196583\pi$
$947$	9.83576e8i	1.15813i	−0.815281	+	0.579066i	$0.803417\pi$
$948$	0	0
$949$	1.05077e9	1.22944
$950$	0	0
$951$	− 7.91462e7i	− 0.0920213i
$952$	0	0
$953$	3.13478e8	0.362184	0.181092	−	0.983466i	$-0.442037\pi$
$953$	3.13478e8	0.362184	0.181092	+	0.983466i	$0.442037\pi$
$954$	0	0
$955$	6.70229e6i	0.00769508i
$956$	0	0
$957$	−1.16008e8	−0.132359
$958$	0	0
$959$	8.99753e8i	1.02016i
$960$	0	0
$961$	8.14530e8	0.917776
$962$	0	0
$963$	− 5.35284e8i	− 0.599384i
$964$	0	0
$965$	1.87425e7	0.0208567
$966$	0	0
$967$	− 3.95187e8i	− 0.437042i	−0.975832	−	0.218521i	$-0.929877\pi$
$967$	− 3.95187e8i	− 0.437042i	0.975832	−	0.218521i	$-0.0701232\pi$
$968$	0	0
$969$	3.97685e8	0.437087
$970$	0	0
$971$	− 2.04334e8i	− 0.223194i	−0.993754	−	0.111597i	$-0.964403\pi$
$971$	− 2.04334e8i	− 0.223194i	0.993754	−	0.111597i	$-0.0355966\pi$
$972$	0	0
$973$	−4.51875e8	−0.490546
$974$	0	0
$975$	− 6.44944e8i	− 0.695839i
$976$	0	0
$977$	−3.87988e7	−0.0416040	−0.0208020	−	0.999784i	$-0.506622\pi$
$977$	−3.87988e7	−0.0416040	−0.0208020	+	0.999784i	$0.506622\pi$
$978$	0	0
$979$	1.62296e8i	0.172966i
$980$	0	0
$981$	−2.31480e8	−0.245192
$982$	0	0
$983$	7.59099e8i	0.799168i	0.916697	+	0.399584i	$0.130845\pi$
$983$	7.59099e8i	0.799168i	−0.916697	+	0.399584i	$0.869155\pi$
$984$	0	0
$985$	6.51181e7	0.0681386
$986$	0	0
$987$	5.03724e8i	0.523891i
$988$	0	0
$989$	2.22409e9	2.29913
$990$	0	0
$991$	5.08413e8i	0.522391i	0.965286	+	0.261195i	$0.0841167\pi$
$991$	5.08413e8i	0.522391i	−0.965286	+	0.261195i	$0.915883\pi$
$992$	0	0
$993$	8.72046e8	0.890618
$994$	0	0
$995$	5.39840e7i	0.0548019i
$996$	0	0
$997$	1.50556e8	0.151919	0.0759594	−	0.997111i	$-0.475798\pi$
$997$	1.50556e8	0.151919	0.0759594	+	0.997111i	$0.475798\pi$
$998$	0	0
$999$	8.46465e7i	0.0849010i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.7.g.b.31.1	✓	2	1.1	even	1	trivial
48.7.g.b.31.2	yes	2	4.3	odd	2	inner
144.7.g.c.127.1		2	3.2	odd	2
144.7.g.c.127.2		2	12.11	even	2
192.7.g.b.127.1		2	8.3	odd	2
192.7.g.b.127.2		2	8.5	even	2
576.7.g.g.127.1		2	24.5	odd	2
576.7.g.g.127.2		2	24.11	even	2
768.7.b.b.127.1		4	16.3	odd	4
768.7.b.b.127.2		4	16.5	even	4
768.7.b.b.127.3		4	16.13	even	4
768.7.b.b.127.4		4	16.11	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	48.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-15.5885i q^{3} +6.00000 q^{5} -200.918i q^{7} -243.000 q^{9} +O(q^{10})\)	\(q-15.5885i q^{3} +6.00000 q^{5} -200.918i q^{7} -243.000 q^{9} -644.323i q^{11} -2654.00 q^{13} -93.5307i q^{15} -7206.00 q^{17} -3540.31i q^{19} -3132.00 q^{21} -17459.1i q^{23} -15589.0 q^{25} +3788.00i q^{27} +11550.0 q^{29} +8542.47i q^{31} -10044.0 q^{33} -1205.51i q^{35} +22346.0 q^{37} +41371.8i q^{39} +103626. q^{41} +127389. i q^{43} -1458.00 q^{45} -160831. i q^{47} +77281.0 q^{49} +112330. i q^{51} +168462. q^{53} -3865.94i q^{55} -55188.0 q^{57} -111593. i q^{59} -260470. q^{61} +48823.0i q^{63} -15924.0 q^{65} +317083. i q^{67} -272160. q^{69} -708423. i q^{71} -395918. q^{73} +243008. i q^{75} -129456. q^{77} -556743. i q^{79} +59049.0 q^{81} -89145.2i q^{83} -43236.0 q^{85} -180047. i q^{87} -251886. q^{89} +533236. i q^{91} +133164. q^{93} -21241.9i q^{95} +517474. q^{97} +156570. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12 q^{5} - 486 q^{9}+O(q^{10}) \) 2 * q + 12 * q^5 - 486 * q^9	\( 2 q + 12 q^{5} - 486 q^{9} - 5308 q^{13} - 14412 q^{17} - 6264 q^{21} - 31178 q^{25} + 23100 q^{29} - 20088 q^{33} + 44692 q^{37} + 207252 q^{41} - 2916 q^{45} + 154562 q^{49} + 336924 q^{53} - 110376 q^{57} - 520940 q^{61} - 31848 q^{65} - 544320 q^{69} - 791836 q^{73} - 258912 q^{77} + 118098 q^{81} - 86472 q^{85} - 503772 q^{89} + 266328 q^{93} + 1034948 q^{97}+O(q^{100}) \) 2 * q + 12 * q^5 - 486 * q^9 - 5308 * q^13 - 14412 * q^17 - 6264 * q^21 - 31178 * q^25 + 23100 * q^29 - 20088 * q^33 + 44692 * q^37 + 207252 * q^41 - 2916 * q^45 + 154562 * q^49 + 336924 * q^53 - 110376 * q^57 - 520940 * q^61 - 31848 * q^65 - 544320 * q^69 - 791836 * q^73 - 258912 * q^77 + 118098 * q^81 - 86472 * q^85 - 503772 * q^89 + 266328 * q^93 + 1034948 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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