LMFDB - Embedded newform 76.8.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,8,Mod(1,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("76.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	76.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$23.7412619368$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8376x^{4} + 135458x^{3} + 16275767x^{2} - 280013424x - 6276171312 $ x^6 - 2x^5 - 8376x^4 + 135458x^3 + 16275767x^2 - 280013424*x - 6276171312
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$52.7878$ of defining polynomial
Character	$\chi$	$=$	76.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-45.7878 q^{3} -327.289 q^{5} -7.23734 q^{7} -90.4805 q^{9} +O(q^{10})$	$q-45.7878 q^{3} -327.289 q^{5} -7.23734 q^{7} -90.4805 q^{9} -7385.22 q^{11} -8089.05 q^{13} +14985.8 q^{15} +20222.5 q^{17} +6859.00 q^{19} +331.382 q^{21} -14848.2 q^{23} +28993.1 q^{25} +104281. q^{27} +91058.4 q^{29} +28778.5 q^{31} +338153. q^{33} +2368.70 q^{35} -25461.9 q^{37} +370380. q^{39} +481699. q^{41} -383681. q^{43} +29613.3 q^{45} +1.39496e6 q^{47} -823491. q^{49} -925942. q^{51} -1.14128e6 q^{53} +2.41710e6 q^{55} -314058. q^{57} +929316. q^{59} -2.58830e6 q^{61} +654.838 q^{63} +2.64746e6 q^{65} +2.48832e6 q^{67} +679867. q^{69} -458373. q^{71} -1.83720e6 q^{73} -1.32753e6 q^{75} +53449.4 q^{77} +70684.8 q^{79} -4.57690e6 q^{81} -7.29853e6 q^{83} -6.61860e6 q^{85} -4.16936e6 q^{87} -7.22382e6 q^{89} +58543.2 q^{91} -1.31770e6 q^{93} -2.24488e6 q^{95} -2.49207e6 q^{97} +668218. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 40 q^{3} + 279 q^{5} - 1565 q^{7} + 3900 q^{9}+O(q^{10}) $ 6 * q + 40 * q^3 + 279 * q^5 - 1565 * q^7 + 3900 * q^9	$ 6 q + 40 q^{3} + 279 q^{5} - 1565 q^{7} + 3900 q^{9} + 7983 q^{11} + 1250 q^{13} + 26790 q^{15} + 21735 q^{17} + 41154 q^{19} - 86346 q^{21} + 100920 q^{23} + 373305 q^{25} + 534790 q^{27} - 58656 q^{29} + 403808 q^{31} + 916430 q^{33} + 463497 q^{35} + 808780 q^{37} + 758704 q^{39} + 556944 q^{41} + 1220735 q^{43} + 3234843 q^{45} + 1915305 q^{47} + 2045883 q^{49} + 908816 q^{51} + 511650 q^{53} + 1813341 q^{55} + 274360 q^{57} + 1300572 q^{59} + 565335 q^{61} - 7170325 q^{63} - 6195012 q^{65} - 45010 q^{67} - 7381528 q^{69} - 1424106 q^{71} - 11153825 q^{73} + 3941974 q^{75} - 17515425 q^{77} - 6392144 q^{79} + 6187530 q^{81} - 3164160 q^{83} - 19479255 q^{85} - 25999500 q^{87} - 14502678 q^{89} - 9736226 q^{91} - 18344300 q^{93} + 1913661 q^{95} - 21377010 q^{97} + 24032935 q^{99}+O(q^{100}) $ 6 * q + 40 * q^3 + 279 * q^5 - 1565 * q^7 + 3900 * q^9 + 7983 * q^11 + 1250 * q^13 + 26790 * q^15 + 21735 * q^17 + 41154 * q^19 - 86346 * q^21 + 100920 * q^23 + 373305 * q^25 + 534790 * q^27 - 58656 * q^29 + 403808 * q^31 + 916430 * q^33 + 463497 * q^35 + 808780 * q^37 + 758704 * q^39 + 556944 * q^41 + 1220735 * q^43 + 3234843 * q^45 + 1915305 * q^47 + 2045883 * q^49 + 908816 * q^51 + 511650 * q^53 + 1813341 * q^55 + 274360 * q^57 + 1300572 * q^59 + 565335 * q^61 - 7170325 * q^63 - 6195012 * q^65 - 45010 * q^67 - 7381528 * q^69 - 1424106 * q^71 - 11153825 * q^73 + 3941974 * q^75 - 17515425 * q^77 - 6392144 * q^79 + 6187530 * q^81 - 3164160 * q^83 - 19479255 * q^85 - 25999500 * q^87 - 14502678 * q^89 - 9736226 * q^91 - 18344300 * q^93 + 1913661 * q^95 - 21377010 * q^97 + 24032935 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−45.7878	−0.979096	−0.489548	−	0.871976i	$-0.662838\pi$
$3$	−45.7878	−0.979096	−0.489548	+	0.871976i	$0.662838\pi$
$4$	0	0
$5$	−327.289	−1.17094	−0.585472	−	0.810692i	$-0.699091\pi$
$5$	−327.289	−1.17094	−0.585472	+	0.810692i	$0.699091\pi$
$6$	0	0
$7$	−7.23734	−0.00797510	−0.00398755	−	0.999992i	$-0.501269\pi$
$7$	−7.23734	−0.00797510	−0.00398755	+	0.999992i	$0.501269\pi$
$8$	0	0
$9$	−90.4805	−0.0413720
$10$	0	0
$11$	−7385.22	−1.67297	−0.836487	−	0.547987i	$-0.815394\pi$
$11$	−7385.22	−1.67297	−0.836487	+	0.547987i	$0.815394\pi$
$12$	0	0
$13$	−8089.05	−1.02116	−0.510582	−	0.859829i	$-0.670570\pi$
$13$	−8089.05	−1.02116	−0.510582	+	0.859829i	$0.670570\pi$
$14$	0	0
$15$	14985.8	1.14647
$16$	0	0
$17$	20222.5	0.998305	0.499153	−	0.866514i	$-0.333645\pi$
$17$	20222.5	0.998305	0.499153	+	0.866514i	$0.333645\pi$
$18$	0	0
$19$	6859.00	0.229416
$19$	6859.00	0.229416
$20$	0	0
$21$	331.382	0.00780838
$22$	0	0
$23$	−14848.2	−0.254464	−0.127232	−	0.991873i	$-0.540609\pi$
$23$	−14848.2	−0.254464	−0.127232	+	0.991873i	$0.540609\pi$
$24$	0	0
$25$	28993.1	0.371112
$26$	0	0
$27$	104281.	1.01960
$28$	0	0
$29$	91058.4	0.693309	0.346655	−	0.937993i	$-0.387318\pi$
$29$	91058.4	0.693309	0.346655	+	0.937993i	$0.387318\pi$
$30$	0	0
$31$	28778.5	0.173501	0.0867505	−	0.996230i	$-0.472352\pi$
$31$	28778.5	0.173501	0.0867505	+	0.996230i	$0.472352\pi$
$32$	0	0
$33$	338153.	1.63800
$34$	0	0
$35$	2368.70	0.00933840
$36$	0	0
$37$	−25461.9	−0.0826388	−0.0413194	−	0.999146i	$-0.513156\pi$
$37$	−25461.9	−0.0826388	−0.0413194	+	0.999146i	$0.513156\pi$
$38$	0	0
$39$	370380.	0.999818
$40$	0	0
$41$	481699.	1.09152	0.545761	−	0.837941i	$-0.316241\pi$
$41$	481699.	1.09152	0.545761	+	0.837941i	$0.316241\pi$
$42$	0	0
$43$	−383681.	−0.735921	−0.367960	−	0.929841i	$-0.619944\pi$
$43$	−383681.	−0.735921	−0.367960	+	0.929841i	$0.619944\pi$
$44$	0	0
$45$	29613.3	0.0484443
$46$	0	0
$47$	1.39496e6	1.95984	0.979918	−	0.199400i	$-0.0638993\pi$
$47$	1.39496e6	1.95984	0.979918	+	0.199400i	$0.0638993\pi$
$48$	0	0
$49$	−823491.	−0.999936
$50$	0	0
$51$	−925942.	−0.977436
$52$	0	0
$53$	−1.14128e6	−1.05299	−0.526496	−	0.850178i	$-0.676495\pi$
$53$	−1.14128e6	−1.05299	−0.526496	+	0.850178i	$0.676495\pi$
$54$	0	0
$55$	2.41710e6	1.95896
$56$	0	0
$57$	−314058.	−0.224620
$58$	0	0
$59$	929316.	0.589090	0.294545	−	0.955638i	$-0.404832\pi$
$59$	929316.	0.589090	0.294545	+	0.955638i	$0.404832\pi$
$60$	0	0
$61$	−2.58830e6	−1.46003	−0.730013	−	0.683433i	$-0.760486\pi$
$61$	−2.58830e6	−1.46003	−0.730013	+	0.683433i	$0.760486\pi$
$62$	0	0
$63$	654.838	0.000329946 0
$64$	0	0
$65$	2.64746e6	1.19573
$66$	0	0
$67$	2.48832e6	1.01075	0.505376	−	0.862899i	$-0.331354\pi$
$67$	2.48832e6	1.01075	0.505376	+	0.862899i	$0.331354\pi$
$68$	0	0
$69$	679867.	0.249145
$70$	0	0
$71$	−458373.	−0.151990	−0.0759949	−	0.997108i	$-0.524213\pi$
$71$	−458373.	−0.151990	−0.0759949	+	0.997108i	$0.524213\pi$
$72$	0	0
$73$	−1.83720e6	−0.552748	−0.276374	−	0.961050i	$-0.589133\pi$
$73$	−1.83720e6	−0.552748	−0.276374	+	0.961050i	$0.589133\pi$
$74$	0	0
$75$	−1.32753e6	−0.363354
$76$	0	0
$77$	53449.4	0.0133421
$78$	0	0
$79$	70684.8	0.0161299	0.00806494	−	0.999967i	$-0.497433\pi$
$79$	70684.8	0.0161299	0.00806494	+	0.999967i	$0.497433\pi$
$80$	0	0
$81$	−4.57690e6	−0.956916
$82$	0	0
$83$	−7.29853e6	−1.40108	−0.700539	−	0.713615i	$-0.747057\pi$
$83$	−7.29853e6	−1.40108	−0.700539	+	0.713615i	$0.747057\pi$
$84$	0	0
$85$	−6.61860e6	−1.16896
$86$	0	0
$87$	−4.16936e6	−0.678816
$88$	0	0
$89$	−7.22382e6	−1.08618	−0.543090	−	0.839674i	$-0.682746\pi$
$89$	−7.22382e6	−1.08618	−0.543090	+	0.839674i	$0.682746\pi$
$90$	0	0
$91$	58543.2	0.00814389
$92$	0	0
$93$	−1.31770e6	−0.169874
$94$	0	0
$95$	−2.24488e6	−0.268633
$96$	0	0
$97$	−2.49207e6	−0.277242	−0.138621	−	0.990346i	$-0.544267\pi$
$97$	−2.49207e6	−0.277242	−0.138621	+	0.990346i	$0.544267\pi$
$98$	0	0
$99$	668218.	0.0692142
$100$	0	0
$101$	7.53353e6	0.727568	0.363784	−	0.931483i	$-0.381485\pi$
$101$	7.53353e6	0.727568	0.363784	+	0.931483i	$0.381485\pi$
$102$	0	0
$103$	−3.27164e6	−0.295009	−0.147504	−	0.989061i	$-0.547124\pi$
$103$	−3.27164e6	−0.295009	−0.147504	+	0.989061i	$0.547124\pi$
$104$	0	0
$105$	−108458.	−0.00914318
$106$	0	0
$107$	2.11143e7	1.66623	0.833113	−	0.553103i	$-0.186556\pi$
$107$	2.11143e7	1.66623	0.833113	+	0.553103i	$0.186556\pi$
$108$	0	0
$109$	−1.98753e7	−1.47001	−0.735007	−	0.678060i	$-0.762821\pi$
$109$	−1.98753e7	−1.47001	−0.735007	+	0.678060i	$0.762821\pi$
$110$	0	0
$111$	1.16584e6	0.0809113
$112$	0	0
$113$	2.18494e7	1.42451	0.712253	−	0.701923i	$-0.247675\pi$
$113$	2.18494e7	1.42451	0.712253	+	0.701923i	$0.247675\pi$
$114$	0	0
$115$	4.85966e6	0.297964
$116$	0	0
$117$	731902.	0.0422476
$118$	0	0
$119$	−146357.	−0.00796158
$120$	0	0
$121$	3.50543e7	1.79884
$122$	0	0
$123$	−2.20559e7	−1.06870
$124$	0	0
$125$	1.60803e7	0.736394
$126$	0	0
$127$	2.23070e7	0.966335	0.483168	−	0.875528i	$-0.339486\pi$
$127$	2.23070e7	0.966335	0.483168	+	0.875528i	$0.339486\pi$
$128$	0	0
$129$	1.75679e7	0.720537
$130$	0	0
$131$	−1.66401e7	−0.646704	−0.323352	−	0.946279i	$-0.604810\pi$
$131$	−1.66401e7	−0.646704	−0.323352	+	0.946279i	$0.604810\pi$
$132$	0	0
$133$	−49640.9	−0.00182961
$134$	0	0
$135$	−3.41299e7	−1.19390
$136$	0	0
$137$	−1.73733e7	−0.577244	−0.288622	−	0.957443i	$-0.593197\pi$
$137$	−1.73733e7	−0.577244	−0.288622	+	0.957443i	$0.593197\pi$
$138$	0	0
$139$	3.56278e7	1.12522	0.562610	−	0.826723i	$-0.309797\pi$
$139$	3.56278e7	1.12522	0.562610	+	0.826723i	$0.309797\pi$
$140$	0	0
$141$	−6.38722e7	−1.91887
$142$	0	0
$143$	5.97394e7	1.70838
$144$	0	0
$145$	−2.98024e7	−0.811827
$146$	0	0
$147$	3.77058e7	0.979033
$148$	0	0
$149$	1.85341e7	0.459007	0.229503	−	0.973308i	$-0.426290\pi$
$149$	1.85341e7	0.459007	0.229503	+	0.973308i	$0.426290\pi$
$150$	0	0
$151$	8.13918e7	1.92381	0.961903	−	0.273390i	$-0.0881451\pi$
$151$	8.13918e7	1.92381	0.961903	+	0.273390i	$0.0881451\pi$
$152$	0	0
$153$	−1.82974e6	−0.0413019
$154$	0	0
$155$	−9.41888e6	−0.203160
$156$	0	0
$157$	−1.35918e7	−0.280302	−0.140151	−	0.990130i	$-0.544759\pi$
$157$	−1.35918e7	−0.280302	−0.140151	+	0.990130i	$0.544759\pi$
$158$	0	0
$159$	5.22565e7	1.03098
$160$	0	0
$161$	107462.	0.00202938
$162$	0	0
$163$	4.19247e7	0.758252	0.379126	−	0.925345i	$-0.376225\pi$
$163$	4.19247e7	0.758252	0.379126	+	0.925345i	$0.376225\pi$
$164$	0	0
$165$	−1.10674e8	−1.91801
$166$	0	0
$167$	−1.39042e7	−0.231013	−0.115507	−	0.993307i	$-0.536849\pi$
$167$	−1.39042e7	−0.231013	−0.115507	+	0.993307i	$0.536849\pi$
$168$	0	0
$169$	2.68424e6	0.0427778
$170$	0	0
$171$	−620606.	−0.00949138
$172$	0	0
$173$	−1.29650e8	−1.90376	−0.951879	−	0.306474i	$-0.900851\pi$
$173$	−1.29650e8	−1.90376	−0.951879	+	0.306474i	$0.900851\pi$
$174$	0	0
$175$	−209833.	−0.00295965
$176$	0	0
$177$	−4.25513e7	−0.576775
$178$	0	0
$179$	2.61429e7	0.340697	0.170348	−	0.985384i	$-0.445511\pi$
$179$	2.61429e7	0.340697	0.170348	+	0.985384i	$0.445511\pi$
$180$	0	0
$181$	8.23876e7	1.03273	0.516365	−	0.856369i	$-0.327285\pi$
$181$	8.23876e7	1.03273	0.516365	+	0.856369i	$0.327285\pi$
$182$	0	0
$183$	1.18513e8	1.42951
$184$	0	0
$185$	8.33339e6	0.0967655
$186$	0	0
$187$	−1.49347e8	−1.67014
$188$	0	0
$189$	−754715.	−0.00813143
$190$	0	0
$191$	7.15752e7	0.743268	0.371634	−	0.928379i	$-0.378798\pi$
$191$	7.15752e7	0.743268	0.371634	+	0.928379i	$0.378798\pi$
$192$	0	0
$193$	−2.00951e7	−0.201206	−0.100603	−	0.994927i	$-0.532077\pi$
$193$	−2.00951e7	−0.201206	−0.100603	+	0.994927i	$0.532077\pi$
$194$	0	0
$195$	−1.21221e8	−1.17073
$196$	0	0
$197$	3.27910e7	0.305578	0.152789	−	0.988259i	$-0.451174\pi$
$197$	3.27910e7	0.305578	0.152789	+	0.988259i	$0.451174\pi$
$198$	0	0
$199$	−5.75954e7	−0.518086	−0.259043	−	0.965866i	$-0.583407\pi$
$199$	−5.75954e7	−0.518086	−0.259043	+	0.965866i	$0.583407\pi$
$200$	0	0
$201$	−1.13935e8	−0.989624
$202$	0	0
$203$	−659021.	−0.00552921
$204$	0	0
$205$	−1.57655e8	−1.27811
$206$	0	0
$207$	1.34348e6	0.0105277
$208$	0	0
$209$	−5.06552e7	−0.383806
$210$	0	0
$211$	−6.53831e7	−0.479156	−0.239578	−	0.970877i	$-0.577009\pi$
$211$	−6.53831e7	−0.479156	−0.239578	+	0.970877i	$0.577009\pi$
$212$	0	0
$213$	2.09879e7	0.148813
$214$	0	0
$215$	1.25575e8	0.861723
$216$	0	0
$217$	−208280.	−0.00138369
$218$	0	0
$219$	8.41214e7	0.541193
$220$	0	0
$221$	−1.63581e8	−1.01943
$222$	0	0
$223$	−1.61991e8	−0.978193	−0.489097	−	0.872230i	$-0.662673\pi$
$223$	−1.61991e8	−0.978193	−0.489097	+	0.872230i	$0.662673\pi$
$224$	0	0
$225$	−2.62331e6	−0.0153536
$226$	0	0
$227$	2.30684e7	0.130896	0.0654481	−	0.997856i	$-0.479152\pi$
$227$	2.30684e7	0.130896	0.0654481	+	0.997856i	$0.479152\pi$
$228$	0	0
$229$	2.24396e8	1.23479	0.617393	−	0.786655i	$-0.288189\pi$
$229$	2.24396e8	1.23479	0.617393	+	0.786655i	$0.288189\pi$
$230$	0	0
$231$	−2.44733e6	−0.0130632
$232$	0	0
$233$	5.21235e7	0.269953	0.134976	−	0.990849i	$-0.456904\pi$
$233$	5.21235e7	0.269953	0.134976	+	0.990849i	$0.456904\pi$
$234$	0	0
$235$	−4.56556e8	−2.29486
$236$	0	0
$237$	−3.23650e6	−0.0157927
$238$	0	0
$239$	2.45843e8	1.16484	0.582418	−	0.812889i	$-0.302107\pi$
$239$	2.45843e8	1.16484	0.582418	+	0.812889i	$0.302107\pi$
$240$	0	0
$241$	−4.23687e8	−1.94978	−0.974890	−	0.222689i	$-0.928517\pi$
$241$	−4.23687e8	−1.94978	−0.974890	+	0.222689i	$0.928517\pi$
$242$	0	0
$243$	−1.84959e7	−0.0826901
$244$	0	0
$245$	2.69519e8	1.17087
$246$	0	0
$247$	−5.54828e7	−0.234271
$248$	0	0
$249$	3.34183e8	1.37179
$250$	0	0
$251$	4.53922e8	1.81185	0.905926	−	0.423435i	$-0.139176\pi$
$251$	4.53922e8	1.81185	0.905926	+	0.423435i	$0.139176\pi$
$252$	0	0
$253$	1.09657e8	0.425712
$254$	0	0
$255$	3.03051e8	1.14452
$256$	0	0
$257$	7.98457e7	0.293417	0.146709	−	0.989180i	$-0.453132\pi$
$257$	7.98457e7	0.293417	0.146709	+	0.989180i	$0.453132\pi$
$258$	0	0
$259$	184276.	0.000659053 0
$260$	0	0
$261$	−8.23901e6	−0.0286836
$262$	0	0
$263$	−2.38715e8	−0.809160	−0.404580	−	0.914503i	$-0.632582\pi$
$263$	−2.38715e8	−0.809160	−0.404580	+	0.914503i	$0.632582\pi$
$264$	0	0
$265$	3.73527e8	1.23300
$266$	0	0
$267$	3.30763e8	1.06347
$268$	0	0
$269$	4.86421e8	1.52363	0.761814	−	0.647796i	$-0.224309\pi$
$269$	4.86421e8	1.52363	0.761814	+	0.647796i	$0.224309\pi$
$270$	0	0
$271$	1.62569e8	0.496187	0.248094	−	0.968736i	$-0.420196\pi$
$271$	1.62569e8	0.496187	0.248094	+	0.968736i	$0.420196\pi$
$272$	0	0
$273$	−2.68056e6	−0.00797365
$274$	0	0
$275$	−2.14120e8	−0.620860
$276$	0	0
$277$	2.76014e8	0.780281	0.390141	−	0.920755i	$-0.372426\pi$
$277$	2.76014e8	0.780281	0.390141	+	0.920755i	$0.372426\pi$
$278$	0	0
$279$	−2.60389e6	−0.00717808
$280$	0	0
$281$	7.76723e7	0.208831	0.104415	−	0.994534i	$-0.466703\pi$
$281$	7.76723e7	0.208831	0.104415	+	0.994534i	$0.466703\pi$
$282$	0	0
$283$	−5.15210e8	−1.35124	−0.675619	−	0.737251i	$-0.736124\pi$
$283$	−5.15210e8	−1.35124	−0.675619	+	0.737251i	$0.736124\pi$
$284$	0	0
$285$	1.02788e8	0.263018
$286$	0	0
$287$	−3.48622e6	−0.00870499
$288$	0	0
$289$	−1.38971e6	−0.00338674
$290$	0	0
$291$	1.14106e8	0.271446
$292$	0	0
$293$	−6.78507e8	−1.57586	−0.787930	−	0.615765i	$-0.788847\pi$
$293$	−6.78507e8	−1.57586	−0.787930	+	0.615765i	$0.788847\pi$
$294$	0	0
$295$	−3.04155e8	−0.689792
$296$	0	0
$297$	−7.70136e8	−1.70577
$298$	0	0
$299$	1.20108e8	0.259850
$300$	0	0
$301$	2.77683e6	0.00586904
$302$	0	0
$303$	−3.44944e8	−0.712359
$304$	0	0
$305$	8.47123e8	1.70961
$306$	0	0
$307$	−5.23062e8	−1.03174	−0.515868	−	0.856668i	$-0.672530\pi$
$307$	−5.23062e8	−1.03174	−0.515868	+	0.856668i	$0.672530\pi$
$308$	0	0
$309$	1.49801e8	0.288842
$310$	0	0
$311$	2.84558e8	0.536426	0.268213	−	0.963360i	$-0.413567\pi$
$311$	2.84558e8	0.536426	0.268213	+	0.963360i	$0.413567\pi$
$312$	0	0
$313$	7.36183e8	1.35700	0.678501	−	0.734599i	$-0.262630\pi$
$313$	7.36183e8	1.35700	0.678501	+	0.734599i	$0.262630\pi$
$314$	0	0
$315$	−214321.	−0.000386348 0
$316$	0	0
$317$	9.77843e7	0.172410	0.0862049	−	0.996277i	$-0.472526\pi$
$317$	9.77843e7	0.172410	0.0862049	+	0.996277i	$0.472526\pi$
$318$	0	0
$319$	−6.72486e8	−1.15989
$320$	0	0
$321$	−9.66777e8	−1.63139
$322$	0	0
$323$	1.38706e8	0.229027
$324$	0	0
$325$	−2.34527e8	−0.378966
$326$	0	0
$327$	9.10046e8	1.43928
$328$	0	0
$329$	−1.00958e7	−0.0156299
$330$	0	0
$331$	3.27931e8	0.497032	0.248516	−	0.968628i	$-0.420057\pi$
$331$	3.27931e8	0.497032	0.248516	+	0.968628i	$0.420057\pi$
$332$	0	0
$333$	2.30380e6	0.00341893
$334$	0	0
$335$	−8.14401e8	−1.18354
$336$	0	0
$337$	5.85723e8	0.833657	0.416828	−	0.908985i	$-0.363142\pi$
$337$	5.85723e8	0.833657	0.416828	+	0.908985i	$0.363142\pi$
$338$	0	0
$339$	−1.00043e9	−1.39473
$340$	0	0
$341$	−2.12535e8	−0.290263
$342$	0	0
$343$	1.19201e7	0.0159497
$344$	0	0
$345$	−2.22513e8	−0.291735
$346$	0	0
$347$	6.26384e8	0.804800	0.402400	−	0.915464i	$-0.368176\pi$
$347$	6.26384e8	0.804800	0.402400	+	0.915464i	$0.368176\pi$
$348$	0	0
$349$	3.77516e6	0.00475386	0.00237693	−	0.999997i	$-0.499243\pi$
$349$	3.77516e6	0.00475386	0.00237693	+	0.999997i	$0.499243\pi$
$350$	0	0
$351$	−8.43532e8	−1.04118
$352$	0	0
$353$	9.82282e8	1.18857	0.594285	−	0.804255i	$-0.297435\pi$
$353$	9.82282e8	1.18857	0.594285	+	0.804255i	$0.297435\pi$
$354$	0	0
$355$	1.50020e8	0.177972
$356$	0	0
$357$	6.70136e6	0.00779515
$358$	0	0
$359$	1.29362e9	1.47562	0.737810	−	0.675009i	$-0.235860\pi$
$359$	1.29362e9	1.47562	0.737810	+	0.675009i	$0.235860\pi$
$360$	0	0
$361$	4.70459e7	0.0526316
$362$	0	0
$363$	−1.60506e9	−1.76124
$364$	0	0
$365$	6.01296e8	0.647237
$366$	0	0
$367$	−1.61562e9	−1.70611	−0.853057	−	0.521818i	$-0.825254\pi$
$367$	−1.61562e9	−1.70611	−0.853057	+	0.521818i	$0.825254\pi$
$368$	0	0
$369$	−4.35844e7	−0.0451584
$370$	0	0
$371$	8.25981e6	0.00839772
$372$	0	0
$373$	−1.44636e9	−1.44310	−0.721549	−	0.692364i	$-0.756569\pi$
$373$	−1.44636e9	−1.44310	−0.721549	+	0.692364i	$0.756569\pi$
$374$	0	0
$375$	−7.36282e8	−0.721000
$376$	0	0
$377$	−7.36576e8	−0.707983
$378$	0	0
$379$	7.88497e6	0.00743982	0.00371991	−	0.999993i	$-0.498816\pi$
$379$	7.88497e6	0.00743982	0.00371991	+	0.999993i	$0.498816\pi$
$380$	0	0
$381$	−1.02139e9	−0.946134
$382$	0	0
$383$	−9.41181e8	−0.856007	−0.428004	−	0.903777i	$-0.640783\pi$
$383$	−9.41181e8	−0.856007	−0.428004	+	0.903777i	$0.640783\pi$
$384$	0	0
$385$	−1.74934e7	−0.0156229
$386$	0	0
$387$	3.47157e7	0.0304465
$388$	0	0
$389$	5.55056e8	0.478094	0.239047	−	0.971008i	$-0.423165\pi$
$389$	5.55056e8	0.478094	0.239047	+	0.971008i	$0.423165\pi$
$390$	0	0
$391$	−3.00268e8	−0.254033
$392$	0	0
$393$	7.61912e8	0.633185
$394$	0	0
$395$	−2.31344e7	−0.0188872
$396$	0	0
$397$	−3.18980e8	−0.255857	−0.127928	−	0.991783i	$-0.540833\pi$
$397$	−3.18980e8	−0.255857	−0.127928	+	0.991783i	$0.540833\pi$
$398$	0	0
$399$	2.27295e6	0.00179137
$400$	0	0
$401$	2.09411e9	1.62179	0.810896	−	0.585190i	$-0.198980\pi$
$401$	2.09411e9	1.62179	0.810896	+	0.585190i	$0.198980\pi$
$402$	0	0
$403$	−2.32791e8	−0.177173
$404$	0	0
$405$	1.49797e9	1.12050
$406$	0	0
$407$	1.88041e8	0.138253
$408$	0	0
$409$	1.89588e9	1.37019	0.685094	−	0.728455i	$-0.259761\pi$
$409$	1.89588e9	1.37019	0.685094	+	0.728455i	$0.259761\pi$
$410$	0	0
$411$	7.95484e8	0.565177
$412$	0	0
$413$	−6.72578e6	−0.00469805
$414$	0	0
$415$	2.38873e9	1.64058
$416$	0	0
$417$	−1.63132e9	−1.10170
$418$	0	0
$419$	2.01272e9	1.33670	0.668351	−	0.743846i	$-0.267001\pi$
$419$	2.01272e9	1.33670	0.668351	+	0.743846i	$0.267001\pi$
$420$	0	0
$421$	−2.55849e9	−1.67108	−0.835539	−	0.549432i	$-0.814844\pi$
$421$	−2.55849e9	−1.67108	−0.835539	+	0.549432i	$0.814844\pi$
$422$	0	0
$423$	−1.26217e8	−0.0810823
$424$	0	0
$425$	5.86312e8	0.370483
$426$	0	0
$427$	1.87324e7	0.0116439
$428$	0	0
$429$	−2.73533e9	−1.67267
$430$	0	0
$431$	1.98218e8	0.119254	0.0596268	−	0.998221i	$-0.481009\pi$
$431$	1.98218e8	0.119254	0.0596268	+	0.998221i	$0.481009\pi$
$432$	0	0
$433$	1.30895e9	0.774848	0.387424	−	0.921902i	$-0.373365\pi$
$433$	1.30895e9	0.774848	0.387424	+	0.921902i	$0.373365\pi$
$434$	0	0
$435$	1.36459e9	0.794856
$436$	0	0
$437$	−1.01844e8	−0.0583782
$438$	0	0
$439$	2.59080e9	1.46153	0.730766	−	0.682628i	$-0.239163\pi$
$439$	2.59080e9	1.46153	0.730766	+	0.682628i	$0.239163\pi$
$440$	0	0
$441$	7.45099e7	0.0413693
$442$	0	0
$443$	2.65506e9	1.45098	0.725489	−	0.688233i	$-0.241614\pi$
$443$	2.65506e9	1.45098	0.725489	+	0.688233i	$0.241614\pi$
$444$	0	0
$445$	2.36428e9	1.27186
$446$	0	0
$447$	−8.48634e8	−0.449412
$448$	0	0
$449$	2.83397e9	1.47752	0.738760	−	0.673969i	$-0.235412\pi$
$449$	2.83397e9	1.47752	0.738760	+	0.673969i	$0.235412\pi$
$450$	0	0
$451$	−3.55745e9	−1.82609
$452$	0	0
$453$	−3.72675e9	−1.88359
$454$	0	0
$455$	−1.91606e7	−0.00953604
$456$	0	0
$457$	1.80187e9	0.883112	0.441556	−	0.897234i	$-0.354427\pi$
$457$	1.80187e9	0.883112	0.441556	+	0.897234i	$0.354427\pi$
$458$	0	0
$459$	2.10882e9	1.01787
$460$	0	0
$461$	6.77819e8	0.322226	0.161113	−	0.986936i	$-0.448492\pi$
$461$	6.77819e8	0.322226	0.161113	+	0.986936i	$0.448492\pi$
$462$	0	0
$463$	4.64787e8	0.217631	0.108815	−	0.994062i	$-0.465294\pi$
$463$	4.64787e8	0.217631	0.108815	+	0.994062i	$0.465294\pi$
$464$	0	0
$465$	4.31270e8	0.198913
$466$	0	0
$467$	−2.49504e9	−1.13362	−0.566811	−	0.823848i	$-0.691823\pi$
$467$	−2.49504e9	−1.13362	−0.566811	+	0.823848i	$0.691823\pi$
$468$	0	0
$469$	−1.80089e7	−0.00806085
$470$	0	0
$471$	6.22336e8	0.274443
$472$	0	0
$473$	2.83357e9	1.23118
$474$	0	0
$475$	1.98864e8	0.0851388
$476$	0	0
$477$	1.03263e8	0.0435644
$478$	0	0
$479$	−4.18007e9	−1.73784	−0.868919	−	0.494954i	$-0.835185\pi$
$479$	−4.18007e9	−1.73784	−0.868919	+	0.494954i	$0.835185\pi$
$480$	0	0
$481$	2.05962e8	0.0843879
$482$	0	0
$483$	−4.92043e6	−0.00198696
$484$	0	0
$485$	8.15627e8	0.324635
$486$	0	0
$487$	−3.27170e9	−1.28358	−0.641788	−	0.766882i	$-0.721807\pi$
$487$	−3.27170e9	−1.28358	−0.641788	+	0.766882i	$0.721807\pi$
$488$	0	0
$489$	−1.91964e9	−0.742401
$490$	0	0
$491$	3.53343e9	1.34714	0.673568	−	0.739125i	$-0.264761\pi$
$491$	3.53343e9	1.34714	0.673568	+	0.739125i	$0.264761\pi$
$492$	0	0
$493$	1.84143e9	0.692134
$494$	0	0
$495$	−2.18701e8	−0.0810460
$496$	0	0
$497$	3.31740e6	0.00121213
$498$	0	0
$499$	−1.89202e9	−0.681670	−0.340835	−	0.940123i	$-0.610710\pi$
$499$	−1.89202e9	−0.681670	−0.340835	+	0.940123i	$0.610710\pi$
$500$	0	0
$501$	6.36641e8	0.226184
$502$	0	0
$503$	−3.57591e9	−1.25285	−0.626424	−	0.779482i	$-0.715482\pi$
$503$	−3.57591e9	−1.25285	−0.626424	+	0.779482i	$0.715482\pi$
$504$	0	0
$505$	−2.46564e9	−0.851942
$506$	0	0
$507$	−1.22906e8	−0.0418836
$508$	0	0
$509$	1.56522e9	0.526095	0.263047	−	0.964783i	$-0.415272\pi$
$509$	1.56522e9	0.526095	0.263047	+	0.964783i	$0.415272\pi$
$510$	0	0
$511$	1.32965e7	0.00440822
$512$	0	0
$513$	7.15262e8	0.233913
$514$	0	0
$515$	1.07077e9	0.345439
$516$	0	0
$517$	−1.03021e10	−3.27875
$518$	0	0
$519$	5.93639e9	1.86396
$520$	0	0
$521$	−3.00555e9	−0.931091	−0.465546	−	0.885024i	$-0.654142\pi$
$521$	−3.00555e9	−0.931091	−0.465546	+	0.885024i	$0.654142\pi$
$522$	0	0
$523$	7.78224e8	0.237875	0.118938	−	0.992902i	$-0.462051\pi$
$523$	7.78224e8	0.237875	0.118938	+	0.992902i	$0.462051\pi$
$524$	0	0
$525$	9.60778e6	0.00289778
$526$	0	0
$527$	5.81973e8	0.173207
$528$	0	0
$529$	−3.18436e9	−0.935248
$530$	0	0
$531$	−8.40850e7	−0.0243718
$532$	0	0
$533$	−3.89649e9	−1.11462
$534$	0	0
$535$	−6.91048e9	−1.95106
$536$	0	0
$537$	−1.19702e9	−0.333575
$538$	0	0
$539$	6.08166e9	1.67287
$540$	0	0
$541$	−2.49739e9	−0.678103	−0.339052	−	0.940768i	$-0.610106\pi$
$541$	−2.49739e9	−0.678103	−0.339052	+	0.940768i	$0.610106\pi$
$542$	0	0
$543$	−3.77234e9	−1.01114
$544$	0	0
$545$	6.50497e9	1.72130
$546$	0	0
$547$	1.74933e9	0.457000	0.228500	−	0.973544i	$-0.426618\pi$
$547$	1.74933e9	0.457000	0.228500	+	0.973544i	$0.426618\pi$
$548$	0	0
$549$	2.34191e8	0.0604042
$550$	0	0
$551$	6.24569e8	0.159056
$552$	0	0
$553$	−511570.	−0.000128637 0
$554$	0	0
$555$	−3.81567e8	−0.0947427
$556$	0	0
$557$	−5.03357e9	−1.23419	−0.617096	−	0.786888i	$-0.711691\pi$
$557$	−5.03357e9	−1.23419	−0.617096	+	0.786888i	$0.711691\pi$
$558$	0	0
$559$	3.10362e9	0.751497
$560$	0	0
$561$	6.83829e9	1.63522
$562$	0	0
$563$	−1.96338e9	−0.463687	−0.231843	−	0.972753i	$-0.574476\pi$
$563$	−1.96338e9	−0.463687	−0.231843	+	0.972753i	$0.574476\pi$
$564$	0	0
$565$	−7.15106e9	−1.66802
$566$	0	0
$567$	3.31246e7	0.00763150
$568$	0	0
$569$	−7.52794e9	−1.71310	−0.856551	−	0.516063i	$-0.827397\pi$
$569$	−7.52794e9	−1.71310	−0.856551	+	0.516063i	$0.827397\pi$
$570$	0	0
$571$	4.51273e9	1.01441	0.507204	−	0.861826i	$-0.330679\pi$
$571$	4.51273e9	1.01441	0.507204	+	0.861826i	$0.330679\pi$
$572$	0	0
$573$	−3.27727e9	−0.727731
$574$	0	0
$575$	−4.30496e8	−0.0944347
$576$	0	0
$577$	7.68161e9	1.66470	0.832351	−	0.554248i	$-0.186994\pi$
$577$	7.68161e9	1.66470	0.832351	+	0.554248i	$0.186994\pi$
$578$	0	0
$579$	9.20110e8	0.196999
$580$	0	0
$581$	5.28220e7	0.0111737
$582$	0	0
$583$	8.42857e9	1.76163
$584$	0	0
$585$	−2.39543e8	−0.0494696
$586$	0	0
$587$	6.20148e9	1.26550	0.632750	−	0.774356i	$-0.281926\pi$
$587$	6.20148e9	1.26550	0.632750	+	0.774356i	$0.281926\pi$
$588$	0	0
$589$	1.97392e8	0.0398039
$590$	0	0
$591$	−1.50142e9	−0.299190
$592$	0	0
$593$	9.45103e8	0.186118	0.0930588	−	0.995661i	$-0.470336\pi$
$593$	9.45103e8	0.186118	0.0930588	+	0.995661i	$0.470336\pi$
$594$	0	0
$595$	4.79011e7	0.00932257
$596$	0	0
$597$	2.63716e9	0.507256
$598$	0	0
$599$	−1.02414e9	−0.194700	−0.0973498	−	0.995250i	$-0.531037\pi$
$599$	−1.02414e9	−0.194700	−0.0973498	+	0.995250i	$0.531037\pi$
$600$	0	0
$601$	2.91585e8	0.0547904	0.0273952	−	0.999625i	$-0.491279\pi$
$601$	2.91585e8	0.0547904	0.0273952	+	0.999625i	$0.491279\pi$
$602$	0	0
$603$	−2.25145e8	−0.0418168
$604$	0	0
$605$	−1.14729e10	−2.10634
$606$	0	0
$607$	4.37431e9	0.793870	0.396935	−	0.917847i	$-0.370074\pi$
$607$	4.37431e9	0.793870	0.396935	+	0.917847i	$0.370074\pi$
$608$	0	0
$609$	3.01751e7	0.00541362
$610$	0	0
$611$	−1.12839e10	−2.00132
$612$	0	0
$613$	8.10621e9	1.42137	0.710683	−	0.703512i	$-0.248386\pi$
$613$	8.10621e9	1.42137	0.710683	+	0.703512i	$0.248386\pi$
$614$	0	0
$615$	7.21866e9	1.25139
$616$	0	0
$617$	−2.56707e9	−0.439987	−0.219993	−	0.975501i	$-0.570604\pi$
$617$	−2.56707e9	−0.439987	−0.219993	+	0.975501i	$0.570604\pi$
$618$	0	0
$619$	−4.52539e9	−0.766900	−0.383450	−	0.923562i	$-0.625264\pi$
$619$	−4.52539e9	−0.766900	−0.383450	+	0.923562i	$0.625264\pi$
$620$	0	0
$621$	−1.54838e9	−0.259453
$622$	0	0
$623$	5.22813e7	0.00866240
$624$	0	0
$625$	−7.52800e9	−1.23339
$626$	0	0
$627$	2.31939e9	0.375783
$628$	0	0
$629$	−5.14902e8	−0.0824988
$630$	0	0
$631$	3.73157e9	0.591273	0.295637	−	0.955300i	$-0.404468\pi$
$631$	3.73157e9	0.591273	0.295637	+	0.955300i	$0.404468\pi$
$632$	0	0
$633$	2.99375e9	0.469140
$634$	0	0
$635$	−7.30083e9	−1.13152
$636$	0	0
$637$	6.66126e9	1.02110
$638$	0	0
$639$	4.14738e7	0.00628812
$640$	0	0
$641$	−3.11544e9	−0.467215	−0.233607	−	0.972331i	$-0.575053\pi$
$641$	−3.11544e9	−0.467215	−0.233607	+	0.972331i	$0.575053\pi$
$642$	0	0
$643$	1.15291e10	1.71023	0.855117	−	0.518435i	$-0.173485\pi$
$643$	1.15291e10	1.71023	0.855117	+	0.518435i	$0.173485\pi$
$644$	0	0
$645$	−5.74978e9	−0.843709
$646$	0	0
$647$	3.27557e9	0.475468	0.237734	−	0.971330i	$-0.423595\pi$
$647$	3.27557e9	0.475468	0.237734	+	0.971330i	$0.423595\pi$
$648$	0	0
$649$	−6.86321e9	−0.985532
$650$	0	0
$651$	9.53667e6	0.00135476
$652$	0	0
$653$	1.84365e9	0.259108	0.129554	−	0.991572i	$-0.458645\pi$
$653$	1.84365e9	0.259108	0.129554	+	0.991572i	$0.458645\pi$
$654$	0	0
$655$	5.44611e9	0.757255
$656$	0	0
$657$	1.66231e8	0.0228683
$658$	0	0
$659$	2.52876e9	0.344199	0.172099	−	0.985080i	$-0.444945\pi$
$659$	2.52876e9	0.344199	0.172099	+	0.985080i	$0.444945\pi$
$660$	0	0
$661$	1.22655e10	1.65189	0.825945	−	0.563751i	$-0.190642\pi$
$661$	1.22655e10	1.65189	0.825945	+	0.563751i	$0.190642\pi$
$662$	0	0
$663$	7.49000e9	0.998123
$664$	0	0
$665$	1.62469e7	0.00214238
$666$	0	0
$667$	−1.35206e9	−0.176423
$668$	0	0
$669$	7.41722e9	0.957744
$670$	0	0
$671$	1.91152e10	2.44259
$672$	0	0
$673$	1.20903e10	1.52891	0.764457	−	0.644675i	$-0.223007\pi$
$673$	1.20903e10	1.52891	0.764457	+	0.644675i	$0.223007\pi$
$674$	0	0
$675$	3.02342e9	0.378386
$676$	0	0
$677$	−1.40974e10	−1.74614	−0.873068	−	0.487599i	$-0.837873\pi$
$677$	−1.40974e10	−1.74614	−0.873068	+	0.487599i	$0.837873\pi$
$678$	0	0
$679$	1.80360e7	0.00221103
$680$	0	0
$681$	−1.05625e9	−0.128160
$682$	0	0
$683$	−6.18013e9	−0.742208	−0.371104	−	0.928591i	$-0.621021\pi$
$683$	−6.18013e9	−0.742208	−0.371104	+	0.928591i	$0.621021\pi$
$684$	0	0
$685$	5.68608e9	0.675921
$686$	0	0
$687$	−1.02746e10	−1.20897
$688$	0	0
$689$	9.23184e9	1.07528
$690$	0	0
$691$	−1.64234e10	−1.89361	−0.946805	−	0.321807i	$-0.895710\pi$
$691$	−1.64234e10	−1.89361	−0.946805	+	0.321807i	$0.895710\pi$
$692$	0	0
$693$	−4.83613e6	−0.000551990 0
$694$	0	0
$695$	−1.16606e10	−1.31757
$696$	0	0
$697$	9.74116e9	1.08967
$698$	0	0
$699$	−2.38662e9	−0.264310
$700$	0	0
$701$	1.54352e10	1.69238	0.846192	−	0.532878i	$-0.178890\pi$
$701$	1.54352e10	1.69238	0.846192	+	0.532878i	$0.178890\pi$
$702$	0	0
$703$	−1.74643e8	−0.0189587
$704$	0	0
$705$	2.09047e10	2.24689
$706$	0	0
$707$	−5.45227e7	−0.00580243
$708$	0	0
$709$	−6.39261e9	−0.673622	−0.336811	−	0.941572i	$-0.609348\pi$
$709$	−6.39261e9	−0.673622	−0.336811	+	0.941572i	$0.609348\pi$
$710$	0	0
$711$	−6.39560e6	−0.000667325 0
$712$	0	0
$713$	−4.27309e8	−0.0441498
$714$	0	0
$715$	−1.95521e10	−2.00042
$716$	0	0
$717$	−1.12566e10	−1.14049
$718$	0	0
$719$	−1.36461e10	−1.36917	−0.684583	−	0.728935i	$-0.740015\pi$
$719$	−1.36461e10	−1.36917	−0.684583	+	0.728935i	$0.740015\pi$
$720$	0	0
$721$	2.36780e7	0.00235272
$722$	0	0
$723$	1.93997e10	1.90902
$724$	0	0
$725$	2.64006e9	0.257295
$726$	0	0
$727$	−6.23152e9	−0.601483	−0.300742	−	0.953706i	$-0.597234\pi$
$727$	−6.23152e9	−0.601483	−0.300742	+	0.953706i	$0.597234\pi$
$728$	0	0
$729$	1.08566e10	1.03788
$730$	0	0
$731$	−7.75899e9	−0.734674
$732$	0	0
$733$	−4.27101e8	−0.0400559	−0.0200280	−	0.999799i	$-0.506376\pi$
$733$	−4.27101e8	−0.0400559	−0.0200280	+	0.999799i	$0.506376\pi$
$734$	0	0
$735$	−1.23407e10	−1.14639
$736$	0	0
$737$	−1.83768e10	−1.69096
$738$	0	0
$739$	−8.40020e9	−0.765656	−0.382828	−	0.923820i	$-0.625050\pi$
$739$	−8.40020e9	−0.765656	−0.382828	+	0.923820i	$0.625050\pi$
$740$	0	0
$741$	2.54043e9	0.229374
$742$	0	0
$743$	1.15259e10	1.03089	0.515447	−	0.856922i	$-0.327626\pi$
$743$	1.15259e10	1.03089	0.515447	+	0.856922i	$0.327626\pi$
$744$	0	0
$745$	−6.06600e9	−0.537472
$746$	0	0
$747$	6.60375e8	0.0579653
$748$	0	0
$749$	−1.52812e8	−0.0132883
$750$	0	0
$751$	−1.23152e10	−1.06097	−0.530484	−	0.847695i	$-0.677990\pi$
$751$	−1.23152e10	−1.06097	−0.530484	+	0.847695i	$0.677990\pi$
$752$	0	0
$753$	−2.07841e10	−1.77398
$754$	0	0
$755$	−2.66386e10	−2.25267
$756$	0	0
$757$	−2.01829e10	−1.69101	−0.845506	−	0.533965i	$-0.820701\pi$
$757$	−2.01829e10	−1.69101	−0.845506	+	0.533965i	$0.820701\pi$
$758$	0	0
$759$	−5.02097e9	−0.416813
$760$	0	0
$761$	2.38786e10	1.96410	0.982050	−	0.188623i	$-0.0604025\pi$
$761$	2.38786e10	1.96410	0.982050	+	0.188623i	$0.0604025\pi$
$762$	0	0
$763$	1.43844e8	0.0117235
$764$	0	0
$765$	5.98854e8	0.0483622
$766$	0	0
$767$	−7.51729e9	−0.601558
$768$	0	0
$769$	1.43150e10	1.13514	0.567570	−	0.823325i	$-0.307884\pi$
$769$	1.43150e10	1.13514	0.567570	+	0.823325i	$0.307884\pi$
$770$	0	0
$771$	−3.65596e9	−0.287284
$772$	0	0
$773$	−2.31981e10	−1.80644	−0.903220	−	0.429177i	$-0.858804\pi$
$773$	−2.31981e10	−1.80644	−0.903220	+	0.429177i	$0.858804\pi$
$774$	0	0
$775$	8.34377e8	0.0643882
$776$	0	0
$777$	−8.43760e6	−0.000645276 0
$778$	0	0
$779$	3.30398e9	0.250412
$780$	0	0
$781$	3.38518e9	0.254275
$782$	0	0
$783$	9.49563e9	0.706900
$784$	0	0
$785$	4.44843e9	0.328219
$786$	0	0
$787$	5.01002e9	0.366377	0.183188	−	0.983078i	$-0.441358\pi$
$787$	5.01002e9	0.366377	0.183188	+	0.983078i	$0.441358\pi$
$788$	0	0
$789$	1.09302e10	0.792245
$790$	0	0
$791$	−1.58131e8	−0.0113606
$792$	0	0
$793$	2.09369e10	1.49093
$794$	0	0
$795$	−1.71030e10	−1.20722
$796$	0	0
$797$	−6.37017e9	−0.445705	−0.222852	−	0.974852i	$-0.571537\pi$
$797$	−6.37017e9	−0.445705	−0.222852	+	0.974852i	$0.571537\pi$
$798$	0	0
$799$	2.82096e10	1.95651
$800$	0	0
$801$	6.53615e8	0.0449374
$802$	0	0
$803$	1.35681e10	0.924732
$804$	0	0
$805$	−3.51710e7	−0.00237629
$806$	0	0
$807$	−2.22721e10	−1.49178
$808$	0	0
$809$	1.20280e10	0.798679	0.399339	−	0.916803i	$-0.369239\pi$
$809$	1.20280e10	0.798679	0.399339	+	0.916803i	$0.369239\pi$
$810$	0	0
$811$	1.94493e10	1.28035	0.640177	−	0.768227i	$-0.278861\pi$
$811$	1.94493e10	1.28035	0.640177	+	0.768227i	$0.278861\pi$
$812$	0	0
$813$	−7.44368e9	−0.485815
$814$	0	0
$815$	−1.37215e10	−0.887871
$816$	0	0
$817$	−2.63167e9	−0.168832
$818$	0	0
$819$	−5.29702e6	−0.000336929 0
$820$	0	0
$821$	−8.42483e9	−0.531325	−0.265663	−	0.964066i	$-0.585591\pi$
$821$	−8.42483e9	−0.531325	−0.265663	+	0.964066i	$0.585591\pi$
$822$	0	0
$823$	1.62536e10	1.01637	0.508183	−	0.861249i	$-0.330317\pi$
$823$	1.62536e10	1.01637	0.508183	+	0.861249i	$0.330317\pi$
$824$	0	0
$825$	9.80409e9	0.607881
$826$	0	0
$827$	7.08282e9	0.435448	0.217724	−	0.976010i	$-0.430137\pi$
$827$	7.08282e9	0.435448	0.217724	+	0.976010i	$0.430137\pi$
$828$	0	0
$829$	−2.36936e10	−1.44441	−0.722204	−	0.691680i	$-0.756871\pi$
$829$	−2.36936e10	−1.44441	−0.722204	+	0.691680i	$0.756871\pi$
$830$	0	0
$831$	−1.26380e10	−0.763970
$832$	0	0
$833$	−1.66530e10	−0.998242
$834$	0	0
$835$	4.55068e9	0.270504
$836$	0	0
$837$	3.00104e9	0.176902
$838$	0	0
$839$	−1.00516e10	−0.587582	−0.293791	−	0.955870i	$-0.594917\pi$
$839$	−1.00516e10	−0.587582	−0.293791	+	0.955870i	$0.594917\pi$
$840$	0	0
$841$	−8.95825e9	−0.519323
$842$	0	0
$843$	−3.55644e9	−0.204465
$844$	0	0
$845$	−8.78523e8	−0.0500904
$846$	0	0
$847$	−2.53700e8	−0.0143459
$848$	0	0
$849$	2.35903e10	1.32299
$850$	0	0
$851$	3.78064e8	0.0210286
$852$	0	0
$853$	1.15829e10	0.638991	0.319495	−	0.947588i	$-0.396487\pi$
$853$	1.15829e10	0.638991	0.319495	+	0.947588i	$0.396487\pi$
$854$	0	0
$855$	2.03117e8	0.0111139
$856$	0	0
$857$	−1.74319e10	−0.946045	−0.473023	−	0.881050i	$-0.656837\pi$
$857$	−1.74319e10	−0.946045	−0.473023	+	0.881050i	$0.656837\pi$
$858$	0	0
$859$	2.73062e10	1.46989	0.734947	−	0.678125i	$-0.237207\pi$
$859$	2.73062e10	1.46989	0.734947	+	0.678125i	$0.237207\pi$
$860$	0	0
$861$	1.59626e8	0.00852302
$862$	0	0
$863$	2.79663e10	1.48114	0.740571	−	0.671978i	$-0.234555\pi$
$863$	2.79663e10	1.48114	0.740571	+	0.671978i	$0.234555\pi$
$864$	0	0
$865$	4.24331e10	2.22920
$866$	0	0
$867$	6.36317e7	0.00331594
$868$	0	0
$869$	−5.22023e8	−0.0269849
$870$	0	0
$871$	−2.01282e10	−1.03215
$872$	0	0
$873$	2.25484e8	0.0114701
$874$	0	0
$875$	−1.16379e8	−0.00587281
$876$	0	0
$877$	−1.89430e10	−0.948312	−0.474156	−	0.880441i	$-0.657247\pi$
$877$	−1.89430e10	−0.948312	−0.474156	+	0.880441i	$0.657247\pi$
$878$	0	0
$879$	3.10673e10	1.54292
$880$	0	0
$881$	1.79935e10	0.886544	0.443272	−	0.896387i	$-0.353817\pi$
$881$	1.79935e10	0.886544	0.443272	+	0.896387i	$0.353817\pi$
$882$	0	0
$883$	−2.16412e10	−1.05784	−0.528918	−	0.848673i	$-0.677402\pi$
$883$	−2.16412e10	−1.05784	−0.528918	+	0.848673i	$0.677402\pi$
$884$	0	0
$885$	1.39266e10	0.675372
$886$	0	0
$887$	−2.55519e10	−1.22939	−0.614696	−	0.788764i	$-0.710721\pi$
$887$	−2.55519e10	−1.22939	−0.614696	+	0.788764i	$0.710721\pi$
$888$	0	0
$889$	−1.61443e8	−0.00770662
$890$	0	0
$891$	3.38014e10	1.60090
$892$	0	0
$893$	9.56804e9	0.449617
$894$	0	0
$895$	−8.55628e9	−0.398937
$896$	0	0
$897$	−5.49948e9	−0.254418
$898$	0	0
$899$	2.62052e9	0.120290
$900$	0	0
$901$	−2.30794e10	−1.05121
$902$	0	0
$903$	−1.27145e8	−0.00574635
$904$	0	0
$905$	−2.69645e10	−1.20927
$906$	0	0
$907$	−2.10770e10	−0.937959	−0.468980	−	0.883209i	$-0.655378\pi$
$907$	−2.10770e10	−0.937959	−0.468980	+	0.883209i	$0.655378\pi$
$908$	0	0
$909$	−6.81638e8	−0.0301009
$910$	0	0
$911$	2.21981e10	0.972751	0.486376	−	0.873750i	$-0.338319\pi$
$911$	2.21981e10	0.972751	0.486376	+	0.873750i	$0.338319\pi$
$912$	0	0
$913$	5.39012e10	2.34396
$914$	0	0
$915$	−3.87879e10	−1.67387
$916$	0	0
$917$	1.20430e8	0.00515753
$918$	0	0
$919$	3.06495e10	1.30262	0.651311	−	0.758811i	$-0.274219\pi$
$919$	3.06495e10	1.30262	0.651311	+	0.758811i	$0.274219\pi$
$920$	0	0
$921$	2.39498e10	1.01017
$922$	0	0
$923$	3.70780e9	0.155207
$924$	0	0
$925$	−7.38218e8	−0.0306682
$926$	0	0
$927$	2.96019e8	0.0122051
$928$	0	0
$929$	−2.86598e10	−1.17279	−0.586393	−	0.810027i	$-0.699453\pi$
$929$	−2.86598e10	−1.17279	−0.586393	+	0.810027i	$0.699453\pi$
$930$	0	0
$931$	−5.64832e9	−0.229401
$932$	0	0
$933$	−1.30293e10	−0.525213
$934$	0	0
$935$	4.88798e10	1.95564
$936$	0	0
$937$	3.86611e10	1.53527	0.767637	−	0.640885i	$-0.221432\pi$
$937$	3.86611e10	1.53527	0.767637	+	0.640885i	$0.221432\pi$
$938$	0	0
$939$	−3.37082e10	−1.32864
$940$	0	0
$941$	1.23980e10	0.485050	0.242525	−	0.970145i	$-0.422024\pi$
$941$	1.23980e10	0.485050	0.242525	+	0.970145i	$0.422024\pi$
$942$	0	0
$943$	−7.15238e9	−0.277754
$944$	0	0
$945$	2.47010e8	0.00952146
$946$	0	0
$947$	2.09134e10	0.800204	0.400102	−	0.916471i	$-0.368975\pi$
$947$	2.09134e10	0.800204	0.400102	+	0.916471i	$0.368975\pi$
$948$	0	0
$949$	1.48612e10	0.564447
$950$	0	0
$951$	−4.47733e9	−0.168806
$952$	0	0
$953$	−1.91527e10	−0.716810	−0.358405	−	0.933566i	$-0.616679\pi$
$953$	−1.91527e10	−0.716810	−0.358405	+	0.933566i	$0.616679\pi$
$954$	0	0
$955$	−2.34258e10	−0.870326
$956$	0	0
$957$	3.07916e10	1.13564
$958$	0	0
$959$	1.25736e8	0.00460358
$960$	0	0
$961$	−2.66844e10	−0.969897
$962$	0	0
$963$	−1.91043e9	−0.0689350
$964$	0	0
$965$	6.57691e9	0.235601
$966$	0	0
$967$	−1.81505e10	−0.645498	−0.322749	−	0.946485i	$-0.604607\pi$
$967$	−1.81505e10	−0.645498	−0.322749	+	0.946485i	$0.604607\pi$
$968$	0	0
$969$	−6.35104e9	−0.224239
$970$	0	0
$971$	−2.44463e10	−0.856930	−0.428465	−	0.903558i	$-0.640945\pi$
$971$	−2.44463e10	−0.856930	−0.428465	+	0.903558i	$0.640945\pi$
$972$	0	0
$973$	−2.57851e8	−0.00897374
$974$	0	0
$975$	1.07385e10	0.371044
$976$	0	0
$977$	−3.79574e10	−1.30216	−0.651082	−	0.759008i	$-0.725685\pi$
$977$	−3.79574e10	−1.30216	−0.651082	+	0.759008i	$0.725685\pi$
$978$	0	0
$979$	5.33495e10	1.81715
$980$	0	0
$981$	1.79833e9	0.0608173
$982$	0	0
$983$	−3.18473e10	−1.06939	−0.534695	−	0.845045i	$-0.679573\pi$
$983$	−3.18473e10	−1.06939	−0.534695	+	0.845045i	$0.679573\pi$
$984$	0	0
$985$	−1.07321e10	−0.357815
$986$	0	0
$987$	4.62265e8	0.0153032
$988$	0	0
$989$	5.69699e9	0.187266
$990$	0	0
$991$	−2.83369e10	−0.924899	−0.462450	−	0.886646i	$-0.653029\pi$
$991$	−2.83369e10	−0.924899	−0.462450	+	0.886646i	$0.653029\pi$
$992$	0	0
$993$	−1.50152e10	−0.486641
$994$	0	0
$995$	1.88503e10	0.606650
$996$	0	0
$997$	−3.85690e10	−1.23255	−0.616276	−	0.787530i	$-0.711359\pi$
$997$	−3.85690e10	−1.23255	−0.616276	+	0.787530i	$0.711359\pi$
$998$	0	0
$999$	−2.65518e9	−0.0842588

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.8.a.b.1.1	✓	6
4.3	odd	2		304.8.a.i.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.a.b.1.1	✓	6	1.1	even	1	trivial
304.8.a.i.1.6		6	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	76.a (trivial)

\(f(q)\)	\(=\)	\(q-45.7878 q^{3} -327.289 q^{5} -7.23734 q^{7} -90.4805 q^{9} +O(q^{10})\)	\(q-45.7878 q^{3} -327.289 q^{5} -7.23734 q^{7} -90.4805 q^{9} -7385.22 q^{11} -8089.05 q^{13} +14985.8 q^{15} +20222.5 q^{17} +6859.00 q^{19} +331.382 q^{21} -14848.2 q^{23} +28993.1 q^{25} +104281. q^{27} +91058.4 q^{29} +28778.5 q^{31} +338153. q^{33} +2368.70 q^{35} -25461.9 q^{37} +370380. q^{39} +481699. q^{41} -383681. q^{43} +29613.3 q^{45} +1.39496e6 q^{47} -823491. q^{49} -925942. q^{51} -1.14128e6 q^{53} +2.41710e6 q^{55} -314058. q^{57} +929316. q^{59} -2.58830e6 q^{61} +654.838 q^{63} +2.64746e6 q^{65} +2.48832e6 q^{67} +679867. q^{69} -458373. q^{71} -1.83720e6 q^{73} -1.32753e6 q^{75} +53449.4 q^{77} +70684.8 q^{79} -4.57690e6 q^{81} -7.29853e6 q^{83} -6.61860e6 q^{85} -4.16936e6 q^{87} -7.22382e6 q^{89} +58543.2 q^{91} -1.31770e6 q^{93} -2.24488e6 q^{95} -2.49207e6 q^{97} +668218. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 40 q^{3} + 279 q^{5} - 1565 q^{7} + 3900 q^{9}+O(q^{10}) \) 6 * q + 40 * q^3 + 279 * q^5 - 1565 * q^7 + 3900 * q^9	\( 6 q + 40 q^{3} + 279 q^{5} - 1565 q^{7} + 3900 q^{9} + 7983 q^{11} + 1250 q^{13} + 26790 q^{15} + 21735 q^{17} + 41154 q^{19} - 86346 q^{21} + 100920 q^{23} + 373305 q^{25} + 534790 q^{27} - 58656 q^{29} + 403808 q^{31} + 916430 q^{33} + 463497 q^{35} + 808780 q^{37} + 758704 q^{39} + 556944 q^{41} + 1220735 q^{43} + 3234843 q^{45} + 1915305 q^{47} + 2045883 q^{49} + 908816 q^{51} + 511650 q^{53} + 1813341 q^{55} + 274360 q^{57} + 1300572 q^{59} + 565335 q^{61} - 7170325 q^{63} - 6195012 q^{65} - 45010 q^{67} - 7381528 q^{69} - 1424106 q^{71} - 11153825 q^{73} + 3941974 q^{75} - 17515425 q^{77} - 6392144 q^{79} + 6187530 q^{81} - 3164160 q^{83} - 19479255 q^{85} - 25999500 q^{87} - 14502678 q^{89} - 9736226 q^{91} - 18344300 q^{93} + 1913661 q^{95} - 21377010 q^{97} + 24032935 q^{99}+O(q^{100}) \) 6 * q + 40 * q^3 + 279 * q^5 - 1565 * q^7 + 3900 * q^9 + 7983 * q^11 + 1250 * q^13 + 26790 * q^15 + 21735 * q^17 + 41154 * q^19 - 86346 * q^21 + 100920 * q^23 + 373305 * q^25 + 534790 * q^27 - 58656 * q^29 + 403808 * q^31 + 916430 * q^33 + 463497 * q^35 + 808780 * q^37 + 758704 * q^39 + 556944 * q^41 + 1220735 * q^43 + 3234843 * q^45 + 1915305 * q^47 + 2045883 * q^49 + 908816 * q^51 + 511650 * q^53 + 1813341 * q^55 + 274360 * q^57 + 1300572 * q^59 + 565335 * q^61 - 7170325 * q^63 - 6195012 * q^65 - 45010 * q^67 - 7381528 * q^69 - 1424106 * q^71 - 11153825 * q^73 + 3941974 * q^75 - 17515425 * q^77 - 6392144 * q^79 + 6187530 * q^81 - 3164160 * q^83 - 19479255 * q^85 - 25999500 * q^87 - 14502678 * q^89 - 9736226 * q^91 - 18344300 * q^93 + 1913661 * q^95 - 21377010 * q^97 + 24032935 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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