LMFDB - Embedded newform 304.8.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,8,Mod(1,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("304.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	304.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:	$94.9650477472$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5014x^{3} + 113222x^{2} - 625803x + 567036 $ x^5 - 2x^4 - 5014x^3 + 113222x^2 - 625803x + 567036
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2}\cdot 7 $
Twist minimal:	no (minimal twist has level 76)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$15.8518$ of defining polynomial
Character	$\chi$	$=$	304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.09048 q^{3} +276.339 q^{5} +720.800 q^{7} -2121.54 q^{9} +O(q^{10})$	$q-8.09048 q^{3} +276.339 q^{5} +720.800 q^{7} -2121.54 q^{9} +4974.98 q^{11} +8085.05 q^{13} -2235.72 q^{15} +1375.87 q^{17} +6859.00 q^{19} -5831.62 q^{21} +55033.5 q^{23} -1761.49 q^{25} +34858.2 q^{27} -117756. q^{29} +151040. q^{31} -40250.0 q^{33} +199185. q^{35} -76088.3 q^{37} -65411.9 q^{39} +420662. q^{41} +411546. q^{43} -586266. q^{45} +791764. q^{47} -303991. q^{49} -11131.5 q^{51} -800670. q^{53} +1.37478e6 q^{55} -55492.6 q^{57} -3.09461e6 q^{59} -2.19949e6 q^{61} -1.52921e6 q^{63} +2.23422e6 q^{65} +420260. q^{67} -445247. q^{69} -2.69010e6 q^{71} +598122. q^{73} +14251.3 q^{75} +3.58597e6 q^{77} -2.62187e6 q^{79} +4.35780e6 q^{81} +5.86850e6 q^{83} +380208. q^{85} +952705. q^{87} +6.26993e6 q^{89} +5.82770e6 q^{91} -1.22199e6 q^{93} +1.89541e6 q^{95} +1.21791e7 q^{97} -1.05546e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 14 q^{3} - 280 q^{5} - 414 q^{7} + 3779 q^{9}+O(q^{10}) $ 5 * q + 14 * q^3 - 280 * q^5 - 414 * q^7 + 3779 * q^9	$ 5 q + 14 q^{3} - 280 q^{5} - 414 q^{7} + 3779 q^{9} + 2662 q^{11} - 602 q^{13} + 20800 q^{15} - 27366 q^{17} + 34295 q^{19} - 59964 q^{21} + 67096 q^{23} - 109115 q^{25} + 178778 q^{27} - 372398 q^{29} + 271372 q^{31} - 792700 q^{33} + 608250 q^{35} - 562630 q^{37} + 963904 q^{39} - 956714 q^{41} + 827362 q^{43} - 1165100 q^{45} + 1812982 q^{47} - 862031 q^{49} + 2458254 q^{51} + 486998 q^{53} - 467930 q^{55} + 96026 q^{57} + 367182 q^{59} + 1879732 q^{61} + 1007274 q^{63} + 1790920 q^{65} + 1046394 q^{67} + 7261712 q^{69} + 4664572 q^{71} + 4224942 q^{73} - 8194850 q^{75} + 8611110 q^{77} - 9574024 q^{79} + 11351813 q^{81} - 11754804 q^{83} + 18711750 q^{85} - 3801472 q^{87} + 2782542 q^{89} - 7385214 q^{91} + 29535004 q^{93} - 1920520 q^{95} + 1291574 q^{97} + 9760310 q^{99}+O(q^{100}) $ 5 * q + 14 * q^3 - 280 * q^5 - 414 * q^7 + 3779 * q^9 + 2662 * q^11 - 602 * q^13 + 20800 * q^15 - 27366 * q^17 + 34295 * q^19 - 59964 * q^21 + 67096 * q^23 - 109115 * q^25 + 178778 * q^27 - 372398 * q^29 + 271372 * q^31 - 792700 * q^33 + 608250 * q^35 - 562630 * q^37 + 963904 * q^39 - 956714 * q^41 + 827362 * q^43 - 1165100 * q^45 + 1812982 * q^47 - 862031 * q^49 + 2458254 * q^51 + 486998 * q^53 - 467930 * q^55 + 96026 * q^57 + 367182 * q^59 + 1879732 * q^61 + 1007274 * q^63 + 1790920 * q^65 + 1046394 * q^67 + 7261712 * q^69 + 4664572 * q^71 + 4224942 * q^73 - 8194850 * q^75 + 8611110 * q^77 - 9574024 * q^79 + 11351813 * q^81 - 11754804 * q^83 + 18711750 * q^85 - 3801472 * q^87 + 2782542 * q^89 - 7385214 * q^91 + 29535004 * q^93 - 1920520 * q^95 + 1291574 * q^97 + 9760310 * 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
$3$	−8.09048	−0.173002	−0.0865008	−	0.996252i	$-0.527568\pi$
$3$	−8.09048	−0.173002	−0.0865008	+	0.996252i	$0.527568\pi$
$4$	0	0
$5$	276.339	0.988662	0.494331	−	0.869274i	$-0.335413\pi$
$5$	276.339	0.988662	0.494331	+	0.869274i	$0.335413\pi$
$6$	0	0
$7$	720.800	0.794276	0.397138	−	0.917759i	$-0.370003\pi$
$7$	720.800	0.794276	0.397138	+	0.917759i	$0.370003\pi$
$8$	0	0
$9$	−2121.54	−0.970070
$10$	0	0
$11$	4974.98	1.12698	0.563491	−	0.826122i	$-0.309458\pi$
$11$	4974.98	1.12698	0.563491	+	0.826122i	$0.309458\pi$
$12$	0	0
$13$	8085.05	1.02066	0.510330	−	0.859979i	$-0.329523\pi$
$13$	8085.05	1.02066	0.510330	+	0.859979i	$0.329523\pi$
$14$	0	0
$15$	−2235.72	−0.171040
$16$	0	0
$17$	1375.87	0.0679215	0.0339608	−	0.999423i	$-0.489188\pi$
$17$	1375.87	0.0679215	0.0339608	+	0.999423i	$0.489188\pi$
$18$	0	0
$19$	6859.00	0.229416
$19$	6859.00	0.229416
$20$	0	0
$21$	−5831.62	−0.137411
$22$	0	0
$23$	55033.5	0.943147	0.471574	−	0.881827i	$-0.343686\pi$
$23$	55033.5	0.943147	0.471574	+	0.881827i	$0.343686\pi$
$24$	0	0
$25$	−1761.49	−0.0225471
$26$	0	0
$27$	34858.2	0.340825
$28$	0	0
$29$	−117756.	−0.896585	−0.448292	−	0.893887i	$-0.647968\pi$
$29$	−117756.	−0.896585	−0.448292	+	0.893887i	$0.647968\pi$
$30$	0	0
$31$	151040.	0.910598	0.455299	−	0.890339i	$-0.349532\pi$
$31$	151040.	0.910598	0.455299	+	0.890339i	$0.349532\pi$
$32$	0	0
$33$	−40250.0	−0.194970
$34$	0	0
$35$	199185.	0.785271
$36$	0	0
$37$	−76088.3	−0.246952	−0.123476	−	0.992348i	$-0.539404\pi$
$37$	−76088.3	−0.246952	−0.123476	+	0.992348i	$0.539404\pi$
$38$	0	0
$39$	−65411.9	−0.176576
$40$	0	0
$41$	420662.	0.953212	0.476606	−	0.879117i	$-0.341867\pi$
$41$	420662.	0.953212	0.476606	+	0.879117i	$0.341867\pi$
$42$	0	0
$43$	411546.	0.789366	0.394683	−	0.918817i	$-0.370854\pi$
$43$	411546.	0.789366	0.394683	+	0.918817i	$0.370854\pi$
$44$	0	0
$45$	−586266.	−0.959072
$46$	0	0
$47$	791764.	1.11238	0.556190	−	0.831055i	$-0.312263\pi$
$47$	791764.	1.11238	0.556190	+	0.831055i	$0.312263\pi$
$48$	0	0
$49$	−303991.	−0.369126
$50$	0	0
$51$	−11131.5	−0.0117505
$52$	0	0
$53$	−800670.	−0.738734	−0.369367	−	0.929284i	$-0.620425\pi$
$53$	−800670.	−0.738734	−0.369367	+	0.929284i	$0.620425\pi$
$54$	0	0
$55$	1.37478e6	1.11421
$56$	0	0
$57$	−55492.6	−0.0396893
$58$	0	0
$59$	−3.09461e6	−1.96166	−0.980831	−	0.194860i	$-0.937575\pi$
$59$	−3.09461e6	−1.96166	−0.980831	+	0.194860i	$0.937575\pi$
$60$	0	0
$61$	−2.19949e6	−1.24070	−0.620350	−	0.784325i	$-0.713010\pi$
$61$	−2.19949e6	−1.24070	−0.620350	+	0.784325i	$0.713010\pi$
$62$	0	0
$63$	−1.52921e6	−0.770504
$64$	0	0
$65$	2.23422e6	1.00909
$66$	0	0
$67$	420260.	0.170709	0.0853544	−	0.996351i	$-0.472798\pi$
$67$	420260.	0.170709	0.0853544	+	0.996351i	$0.472798\pi$
$68$	0	0
$69$	−445247.	−0.163166
$70$	0	0
$71$	−2.69010e6	−0.891999	−0.446000	−	0.895033i	$-0.647152\pi$
$71$	−2.69010e6	−0.891999	−0.446000	+	0.895033i	$0.647152\pi$
$72$	0	0
$73$	598122.	0.179953	0.0899766	−	0.995944i	$-0.471321\pi$
$73$	598122.	0.179953	0.0899766	+	0.995944i	$0.471321\pi$
$74$	0	0
$75$	14251.3	0.00390069
$76$	0	0
$77$	3.58597e6	0.895136
$78$	0	0
$79$	−2.62187e6	−0.598296	−0.299148	−	0.954207i	$-0.596702\pi$
$79$	−2.62187e6	−0.598296	−0.299148	+	0.954207i	$0.596702\pi$
$80$	0	0
$81$	4.35780e6	0.911107
$82$	0	0
$83$	5.86850e6	1.12656	0.563280	−	0.826266i	$-0.309539\pi$
$83$	5.86850e6	1.12656	0.563280	+	0.826266i	$0.309539\pi$
$84$	0	0
$85$	380208.	0.0671515
$86$	0	0
$87$	952705.	0.155110
$88$	0	0
$89$	6.26993e6	0.942752	0.471376	−	0.881932i	$-0.343757\pi$
$89$	6.26993e6	0.942752	0.471376	+	0.881932i	$0.343757\pi$
$90$	0	0
$91$	5.82770e6	0.810685
$92$	0	0
$93$	−1.22199e6	−0.157535
$94$	0	0
$95$	1.89541e6	0.226815
$96$	0	0
$97$	1.21791e7	1.35492	0.677462	−	0.735558i	$-0.263080\pi$
$97$	1.21791e7	1.35492	0.677462	+	0.735558i	$0.263080\pi$
$98$	0	0
$99$	−1.05546e7	−1.09325
$100$	0	0
$101$	4.80256e6	0.463819	0.231909	−	0.972737i	$-0.425503\pi$
$101$	4.80256e6	0.463819	0.231909	+	0.972737i	$0.425503\pi$
$102$	0	0
$103$	−4.18941e6	−0.377766	−0.188883	−	0.982000i	$-0.560487\pi$
$103$	−4.18941e6	−0.377766	−0.188883	+	0.982000i	$0.560487\pi$
$104$	0	0
$105$	−1.61151e6	−0.135853
$106$	0	0
$107$	1.54409e7	1.21851	0.609255	−	0.792974i	$-0.291469\pi$
$107$	1.54409e7	1.21851	0.609255	+	0.792974i	$0.291469\pi$
$108$	0	0
$109$	2.21155e7	1.63570	0.817850	−	0.575432i	$-0.195166\pi$
$109$	2.21155e7	1.63570	0.817850	+	0.575432i	$0.195166\pi$
$110$	0	0
$111$	615591.	0.0427230
$112$	0	0
$113$	−1.04919e7	−0.684036	−0.342018	−	0.939693i	$-0.611110\pi$
$113$	−1.04919e7	−0.684036	−0.342018	+	0.939693i	$0.611110\pi$
$114$	0	0
$115$	1.52079e7	0.932454
$116$	0	0
$117$	−1.71528e7	−0.990112
$118$	0	0
$119$	991730.	0.0539485
$120$	0	0
$121$	5.26330e6	0.270091
$122$	0	0
$123$	−3.40336e6	−0.164907
$124$	0	0
$125$	−2.20758e7	−1.01095
$126$	0	0
$127$	−1.77716e6	−0.0769863	−0.0384932	−	0.999259i	$-0.512256\pi$
$127$	−1.77716e6	−0.0769863	−0.0384932	+	0.999259i	$0.512256\pi$
$128$	0	0
$129$	−3.32960e6	−0.136562
$130$	0	0
$131$	1.55541e7	0.604498	0.302249	−	0.953229i	$-0.402263\pi$
$131$	1.55541e7	0.604498	0.302249	+	0.953229i	$0.402263\pi$
$132$	0	0
$133$	4.94396e6	0.182219
$134$	0	0
$135$	9.63269e6	0.336961
$136$	0	0
$137$	4.72051e7	1.56844	0.784218	−	0.620485i	$-0.213064\pi$
$137$	4.72051e7	1.56844	0.784218	+	0.620485i	$0.213064\pi$
$138$	0	0
$139$	5.49244e7	1.73466	0.867329	−	0.497735i	$-0.165835\pi$
$139$	5.49244e7	1.73466	0.867329	+	0.497735i	$0.165835\pi$
$140$	0	0
$141$	−6.40575e6	−0.192443
$142$	0	0
$143$	4.02230e7	1.15027
$144$	0	0
$145$	−3.25407e7	−0.886419
$146$	0	0
$147$	2.45943e6	0.0638593
$148$	0	0
$149$	−1.28970e7	−0.319402	−0.159701	−	0.987165i	$-0.551053\pi$
$149$	−1.28970e7	−0.319402	−0.159701	+	0.987165i	$0.551053\pi$
$150$	0	0
$151$	1.86914e7	0.441796	0.220898	−	0.975297i	$-0.429101\pi$
$151$	1.86914e7	0.441796	0.220898	+	0.975297i	$0.429101\pi$
$152$	0	0
$153$	−2.91898e6	−0.0658887
$154$	0	0
$155$	4.17384e7	0.900274
$156$	0	0
$157$	8.68724e7	1.79157	0.895784	−	0.444490i	$-0.146615\pi$
$157$	8.68724e7	1.79157	0.895784	+	0.444490i	$0.146615\pi$
$158$	0	0
$159$	6.47780e6	0.127802
$160$	0	0
$161$	3.96681e7	0.749119
$162$	0	0
$163$	−6.53832e7	−1.18252	−0.591262	−	0.806480i	$-0.701370\pi$
$163$	−6.53832e7	−1.18252	−0.591262	+	0.806480i	$0.701370\pi$
$164$	0	0
$165$	−1.11227e7	−0.192759
$166$	0	0
$167$	1.93667e7	0.321772	0.160886	−	0.986973i	$-0.448565\pi$
$167$	1.93667e7	0.321772	0.160886	+	0.986973i	$0.448565\pi$
$168$	0	0
$169$	2.61950e6	0.0417461
$170$	0	0
$171$	−1.45517e7	−0.222549
$172$	0	0
$173$	1.41324e7	0.207518	0.103759	−	0.994602i	$-0.466913\pi$
$173$	1.41324e7	0.207518	0.103759	+	0.994602i	$0.466913\pi$
$174$	0	0
$175$	−1.26968e6	−0.0179086
$176$	0	0
$177$	2.50369e7	0.339370
$178$	0	0
$179$	−613278.	−0.00799229	−0.00399615	−	0.999992i	$-0.501272\pi$
$179$	−613278.	−0.00799229	−0.00399615	+	0.999992i	$0.501272\pi$
$180$	0	0
$181$	−7.33331e7	−0.919232	−0.459616	−	0.888118i	$-0.652013\pi$
$181$	−7.33331e7	−0.919232	−0.459616	+	0.888118i	$0.652013\pi$
$182$	0	0
$183$	1.77949e7	0.214643
$184$	0	0
$185$	−2.10262e7	−0.244152
$186$	0	0
$187$	6.84495e6	0.0765464
$188$	0	0
$189$	2.51258e7	0.270709
$190$	0	0
$191$	−1.08428e7	−0.112597	−0.0562984	−	0.998414i	$-0.517930\pi$
$191$	−1.08428e7	−0.112597	−0.0562984	+	0.998414i	$0.517930\pi$
$192$	0	0
$193$	1.57865e8	1.58065	0.790326	−	0.612687i	$-0.209911\pi$
$193$	1.57865e8	1.58065	0.790326	+	0.612687i	$0.209911\pi$
$194$	0	0
$195$	−1.80759e7	−0.174574
$196$	0	0
$197$	3.95231e7	0.368315	0.184157	−	0.982897i	$-0.441044\pi$
$197$	3.95231e7	0.368315	0.184157	+	0.982897i	$0.441044\pi$
$198$	0	0
$199$	−6.52880e6	−0.0587283	−0.0293642	−	0.999569i	$-0.509348\pi$
$199$	−6.52880e6	−0.0587283	−0.0293642	+	0.999569i	$0.509348\pi$
$200$	0	0
$201$	−3.40010e6	−0.0295329
$202$	0	0
$203$	−8.48787e7	−0.712136
$204$	0	0
$205$	1.16245e8	0.942405
$206$	0	0
$207$	−1.16756e8	−0.914919
$208$	0	0
$209$	3.41234e7	0.258548
$210$	0	0
$211$	−1.25402e8	−0.918999	−0.459500	−	0.888178i	$-0.651971\pi$
$211$	−1.25402e8	−0.918999	−0.459500	+	0.888178i	$0.651971\pi$
$212$	0	0
$213$	2.17642e7	0.154317
$214$	0	0
$215$	1.13726e8	0.780416
$216$	0	0
$217$	1.08870e8	0.723266
$218$	0	0
$219$	−4.83909e6	−0.0311322
$220$	0	0
$221$	1.11240e7	0.0693248
$222$	0	0
$223$	2.73524e8	1.65169	0.825844	−	0.563899i	$-0.190699\pi$
$223$	2.73524e8	1.65169	0.825844	+	0.563899i	$0.190699\pi$
$224$	0	0
$225$	3.73709e6	0.0218723
$226$	0	0
$227$	1.45717e8	0.826835	0.413418	−	0.910542i	$-0.364335\pi$
$227$	1.45717e8	0.826835	0.413418	+	0.910542i	$0.364335\pi$
$228$	0	0
$229$	1.70169e8	0.936391	0.468196	−	0.883625i	$-0.344904\pi$
$229$	1.70169e8	0.936391	0.468196	+	0.883625i	$0.344904\pi$
$230$	0	0
$231$	−2.90122e7	−0.154860
$232$	0	0
$233$	−2.20408e8	−1.14151	−0.570757	−	0.821119i	$-0.693350\pi$
$233$	−2.20408e8	−1.14151	−0.570757	+	0.821119i	$0.693350\pi$
$234$	0	0
$235$	2.18796e8	1.09977
$236$	0	0
$237$	2.12122e7	0.103506
$238$	0	0
$239$	−8.50266e7	−0.402867	−0.201434	−	0.979502i	$-0.564560\pi$
$239$	−8.50266e7	−0.402867	−0.201434	+	0.979502i	$0.564560\pi$
$240$	0	0
$241$	−1.65575e8	−0.761967	−0.380983	−	0.924582i	$-0.624414\pi$
$241$	−1.65575e8	−0.761967	−0.380983	+	0.924582i	$0.624414\pi$
$242$	0	0
$243$	−1.11492e8	−0.498448
$244$	0	0
$245$	−8.40047e7	−0.364941
$246$	0	0
$247$	5.54554e7	0.234155
$248$	0	0
$249$	−4.74790e7	−0.194896
$250$	0	0
$251$	2.70250e8	1.07872	0.539359	−	0.842076i	$-0.318667\pi$
$251$	2.70250e8	1.07872	0.539359	+	0.842076i	$0.318667\pi$
$252$	0	0
$253$	2.73791e8	1.06291
$254$	0	0
$255$	−3.07607e6	−0.0116173
$256$	0	0
$257$	5.67907e7	0.208695	0.104347	−	0.994541i	$-0.466725\pi$
$257$	5.67907e7	0.208695	0.104347	+	0.994541i	$0.466725\pi$
$258$	0	0
$259$	−5.48444e7	−0.196148
$260$	0	0
$261$	2.49825e8	0.869750
$262$	0	0
$263$	2.98253e6	0.0101097	0.00505487	−	0.999987i	$-0.498391\pi$
$263$	2.98253e6	0.0101097	0.00505487	+	0.999987i	$0.498391\pi$
$264$	0	0
$265$	−2.21257e8	−0.730358
$266$	0	0
$267$	−5.07267e7	−0.163098
$268$	0	0
$269$	−2.74367e8	−0.859407	−0.429703	−	0.902970i	$-0.641382\pi$
$269$	−2.74367e8	−0.859407	−0.429703	+	0.902970i	$0.641382\pi$
$270$	0	0
$271$	1.28592e8	0.392483	0.196241	−	0.980556i	$-0.437126\pi$
$271$	1.28592e8	0.392483	0.196241	+	0.980556i	$0.437126\pi$
$272$	0	0
$273$	−4.71489e7	−0.140250
$274$	0	0
$275$	−8.76340e6	−0.0254102
$276$	0	0
$277$	−5.63559e8	−1.59316	−0.796582	−	0.604531i	$-0.793361\pi$
$277$	−5.63559e8	−1.59316	−0.796582	+	0.604531i	$0.793361\pi$
$278$	0	0
$279$	−3.20439e8	−0.883344
$280$	0	0
$281$	−4.69126e8	−1.26130	−0.630648	−	0.776069i	$-0.717211\pi$
$281$	−4.69126e8	−1.26130	−0.630648	+	0.776069i	$0.717211\pi$
$282$	0	0
$283$	−2.16530e8	−0.567891	−0.283946	−	0.958840i	$-0.591644\pi$
$283$	−2.16530e8	−0.567891	−0.283946	+	0.958840i	$0.591644\pi$
$284$	0	0
$285$	−1.53348e7	−0.0392393
$286$	0	0
$287$	3.03213e8	0.757114
$288$	0	0
$289$	−4.08446e8	−0.995387
$290$	0	0
$291$	−9.85349e7	−0.234404
$292$	0	0
$293$	−3.53977e8	−0.822127	−0.411063	−	0.911607i	$-0.634843\pi$
$293$	−3.53977e8	−0.822127	−0.411063	+	0.911607i	$0.634843\pi$
$294$	0	0
$295$	−8.55163e8	−1.93942
$296$	0	0
$297$	1.73419e8	0.384104
$298$	0	0
$299$	4.44948e8	0.962632
$300$	0	0
$301$	2.96642e8	0.626975
$302$	0	0
$303$	−3.88550e7	−0.0802414
$304$	0	0
$305$	−6.07805e8	−1.22663
$306$	0	0
$307$	6.23436e8	1.22972	0.614862	−	0.788635i	$-0.289212\pi$
$307$	6.23436e8	1.22972	0.614862	+	0.788635i	$0.289212\pi$
$308$	0	0
$309$	3.38943e7	0.0653540
$310$	0	0
$311$	−5.17438e8	−0.975432	−0.487716	−	0.873002i	$-0.662170\pi$
$311$	−5.17438e8	−0.975432	−0.487716	+	0.873002i	$0.662170\pi$
$312$	0	0
$313$	−2.03327e8	−0.374791	−0.187395	−	0.982285i	$-0.560005\pi$
$313$	−2.03327e8	−0.374791	−0.187395	+	0.982285i	$0.560005\pi$
$314$	0	0
$315$	−4.22581e8	−0.761768
$316$	0	0
$317$	−2.01407e6	−0.00355114	−0.00177557	−	0.999998i	$-0.500565\pi$
$317$	−2.01407e6	−0.00355114	−0.00177557	+	0.999998i	$0.500565\pi$
$318$	0	0
$319$	−5.85836e8	−1.01044
$320$	0	0
$321$	−1.24924e8	−0.210804
$322$	0	0
$323$	9.43712e6	0.0155823
$324$	0	0
$325$	−1.42418e7	−0.0230129
$326$	0	0
$327$	−1.78925e8	−0.282978
$328$	0	0
$329$	5.70703e8	0.883537
$330$	0	0
$331$	−4.89819e8	−0.742399	−0.371200	−	0.928553i	$-0.621053\pi$
$331$	−4.89819e8	−0.742399	−0.371200	+	0.928553i	$0.621053\pi$
$332$	0	0
$333$	1.61425e8	0.239560
$334$	0	0
$335$	1.16134e8	0.168773
$336$	0	0
$337$	7.87197e8	1.12041	0.560207	−	0.828352i	$-0.310721\pi$
$337$	7.87197e8	1.12041	0.560207	+	0.828352i	$0.310721\pi$
$338$	0	0
$339$	8.48844e7	0.118339
$340$	0	0
$341$	7.51423e8	1.02623
$342$	0	0
$343$	−8.12726e8	−1.08746
$344$	0	0
$345$	−1.23039e8	−0.161316
$346$	0	0
$347$	2.92885e8	0.376308	0.188154	−	0.982140i	$-0.439750\pi$
$347$	2.92885e8	0.376308	0.188154	+	0.982140i	$0.439750\pi$
$348$	0	0
$349$	1.32289e9	1.66584	0.832921	−	0.553392i	$-0.186667\pi$
$349$	1.32289e9	1.66584	0.832921	+	0.553392i	$0.186667\pi$
$350$	0	0
$351$	2.81830e8	0.347866
$352$	0	0
$353$	−1.07075e9	−1.29562	−0.647811	−	0.761801i	$-0.724315\pi$
$353$	−1.07075e9	−1.29562	−0.647811	+	0.761801i	$0.724315\pi$
$354$	0	0
$355$	−7.43381e8	−0.881886
$356$	0	0
$357$	−8.02357e6	−0.00933316
$358$	0	0
$359$	5.11066e8	0.582970	0.291485	−	0.956575i	$-0.405851\pi$
$359$	5.11066e8	0.582970	0.291485	+	0.956575i	$0.405851\pi$
$360$	0	0
$361$	4.70459e7	0.0526316
$362$	0	0
$363$	−4.25826e7	−0.0467261
$364$	0	0
$365$	1.65285e8	0.177913
$366$	0	0
$367$	3.57375e8	0.377392	0.188696	−	0.982036i	$-0.439574\pi$
$367$	3.57375e8	0.377392	0.188696	+	0.982036i	$0.439574\pi$
$368$	0	0
$369$	−8.92453e8	−0.924683
$370$	0	0
$371$	−5.77123e8	−0.586759
$372$	0	0
$373$	−1.92494e8	−0.192060	−0.0960300	−	0.995378i	$-0.530614\pi$
$373$	−1.92494e8	−0.192060	−0.0960300	+	0.995378i	$0.530614\pi$
$374$	0	0
$375$	1.78604e8	0.174897
$376$	0	0
$377$	−9.52066e8	−0.915108
$378$	0	0
$379$	−1.22141e9	−1.15245	−0.576226	−	0.817291i	$-0.695475\pi$
$379$	−1.22141e9	−1.15245	−0.576226	+	0.817291i	$0.695475\pi$
$380$	0	0
$381$	1.43781e7	0.0133188
$382$	0	0
$383$	−6.37917e7	−0.0580188	−0.0290094	−	0.999579i	$-0.509235\pi$
$383$	−6.37917e7	−0.0580188	−0.0290094	+	0.999579i	$0.509235\pi$
$384$	0	0
$385$	9.90944e8	0.884987
$386$	0	0
$387$	−8.73112e8	−0.765741
$388$	0	0
$389$	−2.07082e9	−1.78369	−0.891845	−	0.452340i	$-0.850589\pi$
$389$	−2.07082e9	−1.78369	−0.891845	+	0.452340i	$0.850589\pi$
$390$	0	0
$391$	7.57192e7	0.0640600
$392$	0	0
$393$	−1.25840e8	−0.104579
$394$	0	0
$395$	−7.24526e8	−0.591513
$396$	0	0
$397$	1.48707e9	1.19280	0.596398	−	0.802689i	$-0.296598\pi$
$397$	1.48707e9	1.19280	0.596398	+	0.802689i	$0.296598\pi$
$398$	0	0
$399$	−3.99991e7	−0.0315242
$400$	0	0
$401$	4.94594e8	0.383039	0.191520	−	0.981489i	$-0.438658\pi$
$401$	4.94594e8	0.383039	0.191520	+	0.981489i	$0.438658\pi$
$402$	0	0
$403$	1.22117e9	0.929411
$404$	0	0
$405$	1.20423e9	0.900777
$406$	0	0
$407$	−3.78538e8	−0.278310
$408$	0	0
$409$	−1.77741e9	−1.28457	−0.642284	−	0.766467i	$-0.722013\pi$
$409$	−1.77741e9	−1.28457	−0.642284	+	0.766467i	$0.722013\pi$
$410$	0	0
$411$	−3.81912e8	−0.271342
$412$	0	0
$413$	−2.23060e9	−1.55810
$414$	0	0
$415$	1.62170e9	1.11379
$416$	0	0
$417$	−4.44365e8	−0.300098
$418$	0	0
$419$	5.25911e8	0.349271	0.174636	−	0.984633i	$-0.444125\pi$
$419$	5.25911e8	0.349271	0.174636	+	0.984633i	$0.444125\pi$
$420$	0	0
$421$	−1.27157e9	−0.830527	−0.415264	−	0.909701i	$-0.636311\pi$
$421$	−1.27157e9	−0.830527	−0.415264	+	0.909701i	$0.636311\pi$
$422$	0	0
$423$	−1.67976e9	−1.07909
$424$	0	0
$425$	−2.42359e6	−0.00153144
$426$	0	0
$427$	−1.58539e9	−0.985459
$428$	0	0
$429$	−3.25423e8	−0.198998
$430$	0	0
$431$	2.07412e9	1.24786	0.623928	−	0.781482i	$-0.285536\pi$
$431$	2.07412e9	1.24786	0.623928	+	0.781482i	$0.285536\pi$
$432$	0	0
$433$	−2.00301e9	−1.18570	−0.592851	−	0.805312i	$-0.701998\pi$
$433$	−2.00301e9	−1.18570	−0.592851	+	0.805312i	$0.701998\pi$
$434$	0	0
$435$	2.63270e8	0.153352
$436$	0	0
$437$	3.77475e8	0.216373
$438$	0	0
$439$	−7.57162e8	−0.427133	−0.213566	−	0.976929i	$-0.568508\pi$
$439$	−7.57162e8	−0.427133	−0.213566	+	0.976929i	$0.568508\pi$
$440$	0	0
$441$	6.44930e8	0.358078
$442$	0	0
$443$	4.44853e7	0.0243110	0.0121555	−	0.999926i	$-0.496131\pi$
$443$	4.44853e7	0.0243110	0.0121555	+	0.999926i	$0.496131\pi$
$444$	0	0
$445$	1.73263e9	0.932063
$446$	0	0
$447$	1.04343e8	0.0552570
$448$	0	0
$449$	−3.40918e9	−1.77741	−0.888706	−	0.458477i	$-0.848395\pi$
$449$	−3.40918e9	−1.77741	−0.888706	+	0.458477i	$0.848395\pi$
$450$	0	0
$451$	2.09279e9	1.07425
$452$	0	0
$453$	−1.51222e8	−0.0764314
$454$	0	0
$455$	1.61042e9	0.801494
$456$	0	0
$457$	3.41561e9	1.67403	0.837013	−	0.547183i	$-0.184300\pi$
$457$	3.41561e9	1.67403	0.837013	+	0.547183i	$0.184300\pi$
$458$	0	0
$459$	4.79605e7	0.0231494
$460$	0	0
$461$	−1.71518e9	−0.815371	−0.407686	−	0.913122i	$-0.633664\pi$
$461$	−1.71518e9	−0.815371	−0.407686	+	0.913122i	$0.633664\pi$
$462$	0	0
$463$	2.34891e9	1.09985	0.549924	−	0.835215i	$-0.314657\pi$
$463$	2.34891e9	1.09985	0.549924	+	0.835215i	$0.314657\pi$
$464$	0	0
$465$	−3.37684e8	−0.155749
$466$	0	0
$467$	1.93983e9	0.881364	0.440682	−	0.897663i	$-0.354737\pi$
$467$	1.93983e9	0.881364	0.440682	+	0.897663i	$0.354737\pi$
$468$	0	0
$469$	3.02923e8	0.135590
$470$	0	0
$471$	−7.02840e8	−0.309944
$472$	0	0
$473$	2.04743e9	0.889602
$474$	0	0
$475$	−1.20821e7	−0.00517266
$476$	0	0
$477$	1.69866e9	0.716624
$478$	0	0
$479$	−4.99335e8	−0.207595	−0.103798	−	0.994598i	$-0.533099\pi$
$479$	−4.99335e8	−0.207595	−0.103798	+	0.994598i	$0.533099\pi$
$480$	0	0
$481$	−6.15177e8	−0.252053
$482$	0	0
$483$	−3.20934e8	−0.129599
$484$	0	0
$485$	3.36557e9	1.33956
$486$	0	0
$487$	2.33245e9	0.915084	0.457542	−	0.889188i	$-0.348730\pi$
$487$	2.33245e9	0.915084	0.457542	+	0.889188i	$0.348730\pi$
$488$	0	0
$489$	5.28982e8	0.204578
$490$	0	0
$491$	1.75314e9	0.668391	0.334196	−	0.942504i	$-0.391535\pi$
$491$	1.75314e9	0.668391	0.334196	+	0.942504i	$0.391535\pi$
$492$	0	0
$493$	−1.62018e8	−0.0608974
$494$	0	0
$495$	−2.91667e9	−1.08086
$496$	0	0
$497$	−1.93902e9	−0.708493
$498$	0	0
$499$	−1.10432e9	−0.397872	−0.198936	−	0.980012i	$-0.563749\pi$
$499$	−1.10432e9	−0.397872	−0.198936	+	0.980012i	$0.563749\pi$
$500$	0	0
$501$	−1.56686e8	−0.0556671
$502$	0	0
$503$	−2.39626e9	−0.839548	−0.419774	−	0.907629i	$-0.637891\pi$
$503$	−2.39626e9	−0.839548	−0.419774	+	0.907629i	$0.637891\pi$
$504$	0	0
$505$	1.32714e9	0.458560
$506$	0	0
$507$	−2.11930e7	−0.00722213
$508$	0	0
$509$	−1.57893e9	−0.530702	−0.265351	−	0.964152i	$-0.585488\pi$
$509$	−1.57893e9	−0.530702	−0.265351	+	0.964152i	$0.585488\pi$
$510$	0	0
$511$	4.31126e8	0.142933
$512$	0	0
$513$	2.39092e8	0.0781907
$514$	0	0
$515$	−1.15770e9	−0.373483
$516$	0	0
$517$	3.93901e9	1.25363
$518$	0	0
$519$	−1.14338e8	−0.0359009
$520$	0	0
$521$	2.97045e9	0.920215	0.460108	−	0.887863i	$-0.347811\pi$
$521$	2.97045e9	0.920215	0.460108	+	0.887863i	$0.347811\pi$
$522$	0	0
$523$	−8.32922e8	−0.254594	−0.127297	−	0.991865i	$-0.540630\pi$
$523$	−8.32922e8	−0.254594	−0.127297	+	0.991865i	$0.540630\pi$
$524$	0	0
$525$	1.02724e7	0.00309822
$526$	0	0
$527$	2.07812e8	0.0618492
$528$	0	0
$529$	−3.76141e8	−0.110473
$530$	0	0
$531$	6.56536e9	1.90295
$532$	0	0
$533$	3.40107e9	0.972905
$534$	0	0
$535$	4.26693e9	1.20469
$536$	0	0
$537$	4.96171e6	0.00138268
$538$	0	0
$539$	−1.51235e9	−0.415998
$540$	0	0
$541$	8.68340e8	0.235776	0.117888	−	0.993027i	$-0.462388\pi$
$541$	8.68340e8	0.235776	0.117888	+	0.993027i	$0.462388\pi$
$542$	0	0
$543$	5.93300e8	0.159028
$544$	0	0
$545$	6.11138e9	1.61715
$546$	0	0
$547$	−6.78914e9	−1.77362	−0.886808	−	0.462139i	$-0.847082\pi$
$547$	−6.78914e9	−1.77362	−0.886808	+	0.462139i	$0.847082\pi$
$548$	0	0
$549$	4.66631e9	1.20357
$550$	0	0
$551$	−8.07691e8	−0.205691
$552$	0	0
$553$	−1.88984e9	−0.475212
$554$	0	0
$555$	1.70112e8	0.0422386
$556$	0	0
$557$	7.20950e9	1.76771	0.883857	−	0.467757i	$-0.154938\pi$
$557$	7.20950e9	1.76771	0.883857	+	0.467757i	$0.154938\pi$
$558$	0	0
$559$	3.32737e9	0.805674
$560$	0	0
$561$	−5.53790e7	−0.0132426
$562$	0	0
$563$	7.85174e9	1.85433	0.927164	−	0.374655i	$-0.122239\pi$
$563$	7.85174e9	1.85433	0.927164	+	0.374655i	$0.122239\pi$
$564$	0	0
$565$	−2.89932e9	−0.676281
$566$	0	0
$567$	3.14110e9	0.723671
$568$	0	0
$569$	−4.52870e9	−1.03058	−0.515289	−	0.857017i	$-0.672315\pi$
$569$	−4.52870e9	−1.03058	−0.515289	+	0.857017i	$0.672315\pi$
$570$	0	0
$571$	−7.34549e9	−1.65118	−0.825590	−	0.564271i	$-0.809157\pi$
$571$	−7.34549e9	−1.65118	−0.825590	+	0.564271i	$0.809157\pi$
$572$	0	0
$573$	8.77237e7	0.0194794
$574$	0	0
$575$	−9.69411e7	−0.0212653
$576$	0	0
$577$	3.37956e8	0.0732393	0.0366197	−	0.999329i	$-0.488341\pi$
$577$	3.37956e8	0.0732393	0.0366197	+	0.999329i	$0.488341\pi$
$578$	0	0
$579$	−1.27721e9	−0.273455
$580$	0	0
$581$	4.23002e9	0.894799
$582$	0	0
$583$	−3.98332e9	−0.832540
$584$	0	0
$585$	−4.73999e9	−0.978886
$586$	0	0
$587$	5.74946e9	1.17326	0.586629	−	0.809856i	$-0.300455\pi$
$587$	5.74946e9	1.17326	0.586629	+	0.809856i	$0.300455\pi$
$588$	0	0
$589$	1.03599e9	0.208906
$590$	0	0
$591$	−3.19761e8	−0.0637190
$592$	0	0
$593$	1.26534e9	0.249181	0.124590	−	0.992208i	$-0.460238\pi$
$593$	1.26534e9	0.249181	0.124590	+	0.992208i	$0.460238\pi$
$594$	0	0
$595$	2.74054e8	0.0533368
$596$	0	0
$597$	5.28211e7	0.0101601
$598$	0	0
$599$	−9.89338e9	−1.88084	−0.940418	−	0.340021i	$-0.889566\pi$
$599$	−9.89338e9	−1.88084	−0.940418	+	0.340021i	$0.889566\pi$
$600$	0	0
$601$	−7.60429e9	−1.42889	−0.714443	−	0.699693i	$-0.753320\pi$
$601$	−7.60429e9	−1.42889	−0.714443	+	0.699693i	$0.753320\pi$
$602$	0	0
$603$	−8.91599e8	−0.165600
$604$	0	0
$605$	1.45446e9	0.267028
$606$	0	0
$607$	−4.96147e9	−0.900430	−0.450215	−	0.892920i	$-0.648653\pi$
$607$	−4.96147e9	−0.900430	−0.450215	+	0.892920i	$0.648653\pi$
$608$	0	0
$609$	6.86710e8	0.123201
$610$	0	0
$611$	6.40145e9	1.13536
$612$	0	0
$613$	5.04021e9	0.883765	0.441883	−	0.897073i	$-0.354311\pi$
$613$	5.04021e9	0.883765	0.441883	+	0.897073i	$0.354311\pi$
$614$	0	0
$615$	−9.40482e8	−0.163037
$616$	0	0
$617$	−7.84954e9	−1.34538	−0.672692	−	0.739922i	$-0.734862\pi$
$617$	−7.84954e9	−1.34538	−0.672692	+	0.739922i	$0.734862\pi$
$618$	0	0
$619$	2.69511e9	0.456729	0.228365	−	0.973576i	$-0.426662\pi$
$619$	2.69511e9	0.456729	0.228365	+	0.973576i	$0.426662\pi$
$620$	0	0
$621$	1.91837e9	0.321448
$622$	0	0
$623$	4.51936e9	0.748805
$624$	0	0
$625$	−5.96280e9	−0.976945
$626$	0	0
$627$	−2.76075e8	−0.0447291
$628$	0	0
$629$	−1.04688e8	−0.0167733
$630$	0	0
$631$	5.44546e9	0.862843	0.431422	−	0.902150i	$-0.358012\pi$
$631$	5.44546e9	0.862843	0.431422	+	0.902150i	$0.358012\pi$
$632$	0	0
$633$	1.01456e9	0.158988
$634$	0	0
$635$	−4.91100e8	−0.0761135
$636$	0	0
$637$	−2.45778e9	−0.376752
$638$	0	0
$639$	5.70717e9	0.865302
$640$	0	0
$641$	−1.19511e10	−1.79228	−0.896139	−	0.443773i	$-0.853640\pi$
$641$	−1.19511e10	−1.79228	−0.896139	+	0.443773i	$0.853640\pi$
$642$	0	0
$643$	1.08936e10	1.61597	0.807987	−	0.589200i	$-0.200557\pi$
$643$	1.08936e10	1.61597	0.807987	+	0.589200i	$0.200557\pi$
$644$	0	0
$645$	−9.20101e8	−0.135013
$646$	0	0
$647$	−1.48672e9	−0.215807	−0.107903	−	0.994161i	$-0.534414\pi$
$647$	−1.48672e9	−0.215807	−0.107903	+	0.994161i	$0.534414\pi$
$648$	0	0
$649$	−1.53956e10	−2.21076
$650$	0	0
$651$	−8.80809e8	−0.125126
$652$	0	0
$653$	8.66204e9	1.21737	0.608687	−	0.793411i	$-0.291697\pi$
$653$	8.66204e9	1.21737	0.608687	+	0.793411i	$0.291697\pi$
$654$	0	0
$655$	4.29821e9	0.597644
$656$	0	0
$657$	−1.26894e9	−0.174567
$658$	0	0
$659$	4.47037e9	0.608477	0.304239	−	0.952596i	$-0.401598\pi$
$659$	4.47037e9	0.608477	0.304239	+	0.952596i	$0.401598\pi$
$660$	0	0
$661$	8.62221e9	1.16122	0.580608	−	0.814183i	$-0.302815\pi$
$661$	8.62221e9	1.16122	0.580608	+	0.814183i	$0.302815\pi$
$662$	0	0
$663$	−8.99986e7	−0.0119933
$664$	0	0
$665$	1.36621e9	0.180153
$666$	0	0
$667$	−6.48054e9	−0.845611
$668$	0	0
$669$	−2.21294e9	−0.285744
$670$	0	0
$671$	−1.09424e10	−1.39825
$672$	0	0
$673$	2.08380e9	0.263513	0.131757	−	0.991282i	$-0.457938\pi$
$673$	2.08380e9	0.263513	0.131757	+	0.991282i	$0.457938\pi$
$674$	0	0
$675$	−6.14025e7	−0.00768463
$676$	0	0
$677$	7.68414e9	0.951777	0.475888	−	0.879506i	$-0.342127\pi$
$677$	7.68414e9	0.951777	0.475888	+	0.879506i	$0.342127\pi$
$678$	0	0
$679$	8.77871e9	1.07618
$680$	0	0
$681$	−1.17892e9	−0.143044
$682$	0	0
$683$	−7.90692e9	−0.949587	−0.474794	−	0.880097i	$-0.657477\pi$
$683$	−7.90692e9	−0.949587	−0.474794	+	0.880097i	$0.657477\pi$
$684$	0	0
$685$	1.30446e10	1.55065
$686$	0	0
$687$	−1.37675e9	−0.161997
$688$	0	0
$689$	−6.47346e9	−0.753996
$690$	0	0
$691$	−1.47299e10	−1.69835	−0.849175	−	0.528112i	$-0.822900\pi$
$691$	−1.47299e10	−1.69835	−0.849175	+	0.528112i	$0.822900\pi$
$692$	0	0
$693$	−7.60779e9	−0.868345
$694$	0	0
$695$	1.51778e10	1.71499
$696$	0	0
$697$	5.78778e8	0.0647437
$698$	0	0
$699$	1.78320e9	0.197484
$700$	0	0
$701$	1.76814e10	1.93867	0.969335	−	0.245744i	$-0.0790323\pi$
$701$	1.76814e10	1.93867	0.969335	+	0.245744i	$0.0790323\pi$
$702$	0	0
$703$	−5.21889e8	−0.0566546
$704$	0	0
$705$	−1.77016e9	−0.190261
$706$	0	0
$707$	3.46169e9	0.368400
$708$	0	0
$709$	−3.52758e9	−0.371719	−0.185859	−	0.982576i	$-0.559507\pi$
$709$	−3.52758e9	−0.371719	−0.185859	+	0.982576i	$0.559507\pi$
$710$	0	0
$711$	5.56241e9	0.580390
$712$	0	0
$713$	8.31227e9	0.858828
$714$	0	0
$715$	1.11152e10	1.13722
$716$	0	0
$717$	6.87906e8	0.0696967
$718$	0	0
$719$	1.24573e10	1.24990	0.624949	−	0.780666i	$-0.285120\pi$
$719$	1.24573e10	1.24990	0.624949	+	0.780666i	$0.285120\pi$
$720$	0	0
$721$	−3.01972e9	−0.300050
$722$	0	0
$723$	1.33958e9	0.131821
$724$	0	0
$725$	2.07427e8	0.0202154
$726$	0	0
$727$	−1.43981e10	−1.38975	−0.694873	−	0.719132i	$-0.744540\pi$
$727$	−1.43981e10	−1.38975	−0.694873	+	0.719132i	$0.744540\pi$
$728$	0	0
$729$	−8.62848e9	−0.824875
$730$	0	0
$731$	5.66235e8	0.0536150
$732$	0	0
$733$	−1.94697e10	−1.82598	−0.912990	−	0.407983i	$-0.866232\pi$
$733$	−1.94697e10	−1.82598	−0.912990	+	0.407983i	$0.866232\pi$
$734$	0	0
$735$	6.79638e8	0.0631353
$736$	0	0
$737$	2.09078e9	0.192386
$738$	0	0
$739$	−1.90895e10	−1.73996	−0.869979	−	0.493089i	$-0.835868\pi$
$739$	−1.90895e10	−1.73996	−0.869979	+	0.493089i	$0.835868\pi$
$740$	0	0
$741$	−4.48660e8	−0.0405092
$742$	0	0
$743$	−1.60614e10	−1.43655	−0.718277	−	0.695758i	$-0.755069\pi$
$743$	−1.60614e10	−1.43655	−0.718277	+	0.695758i	$0.755069\pi$
$744$	0	0
$745$	−3.56395e9	−0.315780
$746$	0	0
$747$	−1.24503e10	−1.09284
$748$	0	0
$749$	1.11298e10	0.967833
$750$	0	0
$751$	−1.54515e10	−1.33116	−0.665579	−	0.746327i	$-0.731815\pi$
$751$	−1.54515e10	−1.33116	−0.665579	+	0.746327i	$0.731815\pi$
$752$	0	0
$753$	−2.18645e9	−0.186620
$754$	0	0
$755$	5.16516e9	0.436787
$756$	0	0
$757$	−1.26828e10	−1.06262	−0.531312	−	0.847176i	$-0.678301\pi$
$757$	−1.26828e10	−1.06262	−0.531312	+	0.847176i	$0.678301\pi$
$758$	0	0
$759$	−2.21510e9	−0.183885
$760$	0	0
$761$	1.77060e10	1.45638	0.728190	−	0.685375i	$-0.240362\pi$
$761$	1.77060e10	1.45638	0.728190	+	0.685375i	$0.240362\pi$
$762$	0	0
$763$	1.59408e10	1.29920
$764$	0	0
$765$	−8.06629e8	−0.0651416
$766$	0	0
$767$	−2.50201e10	−2.00219
$768$	0	0
$769$	1.88518e9	0.149489	0.0747446	−	0.997203i	$-0.476186\pi$
$769$	1.88518e9	0.149489	0.0747446	+	0.997203i	$0.476186\pi$
$770$	0	0
$771$	−4.59464e8	−0.0361045
$772$	0	0
$773$	6.99272e8	0.0544525	0.0272262	−	0.999629i	$-0.491333\pi$
$773$	6.99272e8	0.0544525	0.0272262	+	0.999629i	$0.491333\pi$
$774$	0	0
$775$	−2.66056e8	−0.0205314
$776$	0	0
$777$	4.43718e8	0.0339338
$778$	0	0
$779$	2.88532e9	0.218682
$780$	0	0
$781$	−1.33832e10	−1.00527
$782$	0	0
$783$	−4.10477e9	−0.305579
$784$	0	0
$785$	2.40063e10	1.77126
$786$	0	0
$787$	−2.19911e10	−1.60818	−0.804092	−	0.594505i	$-0.797348\pi$
$787$	−2.19911e10	−1.60818	−0.804092	+	0.594505i	$0.797348\pi$
$788$	0	0
$789$	−2.41301e7	−0.00174900
$790$	0	0
$791$	−7.56255e9	−0.543314
$792$	0	0
$793$	−1.77830e10	−1.26633
$794$	0	0
$795$	1.79007e9	0.126353
$796$	0	0
$797$	1.41708e10	0.991496	0.495748	−	0.868466i	$-0.334894\pi$
$797$	1.41708e10	0.991496	0.495748	+	0.868466i	$0.334894\pi$
$798$	0	0
$799$	1.08937e9	0.0755546
$800$	0	0
$801$	−1.33019e10	−0.914536
$802$	0	0
$803$	2.97565e9	0.202804
$804$	0	0
$805$	1.09619e10	0.740626
$806$	0	0
$807$	2.21976e9	0.148679
$808$	0	0
$809$	−2.00198e10	−1.32935	−0.664677	−	0.747131i	$-0.731431\pi$
$809$	−2.00198e10	−1.32935	−0.664677	+	0.747131i	$0.731431\pi$
$810$	0	0
$811$	6.19672e9	0.407933	0.203967	−	0.978978i	$-0.434617\pi$
$811$	6.19672e9	0.407933	0.203967	+	0.978978i	$0.434617\pi$
$812$	0	0
$813$	−1.04037e9	−0.0679001
$814$	0	0
$815$	−1.80680e10	−1.16912
$816$	0	0
$817$	2.82279e9	0.181093
$818$	0	0
$819$	−1.23637e10	−0.786422
$820$	0	0
$821$	−1.36180e10	−0.858839	−0.429419	−	0.903105i	$-0.641282\pi$
$821$	−1.36180e10	−0.858839	−0.429419	+	0.903105i	$0.641282\pi$
$822$	0	0
$823$	9.96103e9	0.622880	0.311440	−	0.950266i	$-0.399189\pi$
$823$	9.96103e9	0.622880	0.311440	+	0.950266i	$0.399189\pi$
$824$	0	0
$825$	7.09001e7	0.00439601
$826$	0	0
$827$	−2.53194e10	−1.55663	−0.778313	−	0.627876i	$-0.783924\pi$
$827$	−2.53194e10	−1.55663	−0.778313	+	0.627876i	$0.783924\pi$
$828$	0	0
$829$	−2.10063e10	−1.28059	−0.640294	−	0.768130i	$-0.721187\pi$
$829$	−2.10063e10	−1.28059	−0.640294	+	0.768130i	$0.721187\pi$
$830$	0	0
$831$	4.55946e9	0.275620
$832$	0	0
$833$	−4.18253e8	−0.0250716
$834$	0	0
$835$	5.35179e9	0.318124
$836$	0	0
$837$	5.26499e9	0.310355
$838$	0	0
$839$	1.76564e10	1.03213	0.516066	−	0.856549i	$-0.327396\pi$
$839$	1.76564e10	1.03213	0.516066	+	0.856549i	$0.327396\pi$
$840$	0	0
$841$	−3.38332e9	−0.196136
$842$	0	0
$843$	3.79545e9	0.218206
$844$	0	0
$845$	7.23872e8	0.0412727
$846$	0	0
$847$	3.79379e9	0.214526
$848$	0	0
$849$	1.75183e9	0.0982461
$850$	0	0
$851$	−4.18740e9	−0.232912
$852$	0	0
$853$	−1.73063e9	−0.0954734	−0.0477367	−	0.998860i	$-0.515201\pi$
$853$	−1.73063e9	−0.0954734	−0.0477367	+	0.998860i	$0.515201\pi$
$854$	0	0
$855$	−4.02120e9	−0.220026
$856$	0	0
$857$	−1.07792e8	−0.00584995	−0.00292498	−	0.999996i	$-0.500931\pi$
$857$	−1.07792e8	−0.00584995	−0.00292498	+	0.999996i	$0.500931\pi$
$858$	0	0
$859$	1.21663e10	0.654911	0.327455	−	0.944867i	$-0.393809\pi$
$859$	1.21663e10	0.654911	0.327455	+	0.944867i	$0.393809\pi$
$860$	0	0
$861$	−2.45314e9	−0.130982
$862$	0	0
$863$	2.32114e10	1.22932	0.614659	−	0.788793i	$-0.289293\pi$
$863$	2.32114e10	1.22932	0.614659	+	0.788793i	$0.289293\pi$
$864$	0	0
$865$	3.90535e9	0.205165
$866$	0	0
$867$	3.30452e9	0.172203
$868$	0	0
$869$	−1.30438e10	−0.674270
$870$	0	0
$871$	3.39782e9	0.174236
$872$	0	0
$873$	−2.58385e10	−1.31437
$874$	0	0
$875$	−1.59122e10	−0.802976
$876$	0	0
$877$	−2.30902e10	−1.15593	−0.577963	−	0.816063i	$-0.696152\pi$
$877$	−2.30902e10	−1.15593	−0.577963	+	0.816063i	$0.696152\pi$
$878$	0	0
$879$	2.86385e9	0.142229
$880$	0	0
$881$	6.17232e9	0.304112	0.152056	−	0.988372i	$-0.451411\pi$
$881$	6.17232e9	0.304112	0.152056	+	0.988372i	$0.451411\pi$
$882$	0	0
$883$	1.18206e10	0.577801	0.288901	−	0.957359i	$-0.406710\pi$
$883$	1.18206e10	0.577801	0.288901	+	0.957359i	$0.406710\pi$
$884$	0	0
$885$	6.91868e9	0.335523
$886$	0	0
$887$	−4.11165e10	−1.97826	−0.989129	−	0.147049i	$-0.953022\pi$
$887$	−4.11165e10	−1.97826	−0.989129	+	0.147049i	$0.953022\pi$
$888$	0	0
$889$	−1.28098e9	−0.0611484
$890$	0	0
$891$	2.16800e10	1.02680
$892$	0	0
$893$	5.43071e9	0.255197
$894$	0	0
$895$	−1.69473e8	−0.00790168
$896$	0	0
$897$	−3.59985e9	−0.166537
$898$	0	0
$899$	−1.77859e10	−0.816428
$900$	0	0
$901$	−1.10162e9	−0.0501759
$902$	0	0
$903$	−2.39998e9	−0.108468
$904$	0	0
$905$	−2.02648e10	−0.908810
$906$	0	0
$907$	1.36671e10	0.608204	0.304102	−	0.952639i	$-0.401644\pi$
$907$	1.36671e10	0.608204	0.304102	+	0.952639i	$0.401644\pi$
$908$	0	0
$909$	−1.01889e10	−0.449937
$910$	0	0
$911$	3.65765e10	1.60283	0.801417	−	0.598106i	$-0.204080\pi$
$911$	3.65765e10	1.60283	0.801417	+	0.598106i	$0.204080\pi$
$912$	0	0
$913$	2.91957e10	1.26961
$914$	0	0
$915$	4.91744e9	0.212210
$916$	0	0
$917$	1.12114e10	0.480138
$918$	0	0
$919$	3.71530e9	0.157903	0.0789514	−	0.996878i	$-0.474843\pi$
$919$	3.71530e9	0.157903	0.0789514	+	0.996878i	$0.474843\pi$
$920$	0	0
$921$	−5.04390e9	−0.212744
$922$	0	0
$923$	−2.17496e10	−0.910427
$924$	0	0
$925$	1.34029e8	0.00556805
$926$	0	0
$927$	8.88801e9	0.366459
$928$	0	0
$929$	−3.28521e10	−1.34434	−0.672168	−	0.740398i	$-0.734637\pi$
$929$	−3.28521e10	−1.34434	−0.672168	+	0.740398i	$0.734637\pi$
$930$	0	0
$931$	−2.08507e9	−0.0846832
$932$	0	0
$933$	4.18632e9	0.168751
$934$	0	0
$935$	1.89153e9	0.0756786
$936$	0	0
$937$	2.29737e10	0.912310	0.456155	−	0.889900i	$-0.349226\pi$
$937$	2.29737e10	0.912310	0.456155	+	0.889900i	$0.349226\pi$
$938$	0	0
$939$	1.64501e9	0.0648394
$940$	0	0
$941$	1.00811e10	0.394407	0.197204	−	0.980363i	$-0.436814\pi$
$941$	1.00811e10	0.394407	0.197204	+	0.980363i	$0.436814\pi$
$942$	0	0
$943$	2.31505e10	0.899020
$944$	0	0
$945$	6.94324e9	0.267640
$946$	0	0
$947$	1.75260e10	0.670589	0.335295	−	0.942113i	$-0.391164\pi$
$947$	1.75260e10	0.670589	0.335295	+	0.942113i	$0.391164\pi$
$948$	0	0
$949$	4.83584e9	0.183671
$950$	0	0
$951$	1.62948e7	0.000614353 0
$952$	0	0
$953$	−2.66888e10	−0.998859	−0.499429	−	0.866355i	$-0.666457\pi$
$953$	−2.66888e10	−0.998859	−0.499429	+	0.866355i	$0.666457\pi$
$954$	0	0
$955$	−2.99630e9	−0.111320
$956$	0	0
$957$	4.73969e9	0.174807
$958$	0	0
$959$	3.40254e10	1.24577
$960$	0	0
$961$	−4.69946e9	−0.170811
$962$	0	0
$963$	−3.27585e10	−1.18204
$964$	0	0
$965$	4.36244e10	1.56273
$966$	0	0
$967$	−2.40155e10	−0.854080	−0.427040	−	0.904233i	$-0.640444\pi$
$967$	−2.40155e10	−0.854080	−0.427040	+	0.904233i	$0.640444\pi$
$968$	0	0
$969$	−7.63508e7	−0.00269576
$970$	0	0
$971$	3.24355e10	1.13698	0.568491	−	0.822689i	$-0.307527\pi$
$971$	3.24355e10	1.13698	0.568491	+	0.822689i	$0.307527\pi$
$972$	0	0
$973$	3.95895e10	1.37780
$974$	0	0
$975$	1.15223e8	0.00398127
$976$	0	0
$977$	−1.53419e10	−0.526317	−0.263158	−	0.964753i	$-0.584764\pi$
$977$	−1.53419e10	−0.526317	−0.263158	+	0.964753i	$0.584764\pi$
$978$	0	0
$979$	3.11928e10	1.06247
$980$	0	0
$981$	−4.69189e10	−1.58674
$982$	0	0
$983$	6.48494e9	0.217755	0.108878	−	0.994055i	$-0.465274\pi$
$983$	6.48494e9	0.217755	0.108878	+	0.994055i	$0.465274\pi$
$984$	0	0
$985$	1.09218e10	0.364139
$986$	0	0
$987$	−4.61726e9	−0.152853
$988$	0	0
$989$	2.26488e10	0.744489
$990$	0	0
$991$	−3.82059e10	−1.24702	−0.623510	−	0.781816i	$-0.714294\pi$
$991$	−3.82059e10	−1.24702	−0.623510	+	0.781816i	$0.714294\pi$
$992$	0	0
$993$	3.96287e9	0.128436
$994$	0	0
$995$	−1.80416e9	−0.0580625
$996$	0	0
$997$	4.90627e10	1.56790	0.783950	−	0.620824i	$-0.213202\pi$
$997$	4.90627e10	1.56790	0.783950	+	0.620824i	$0.213202\pi$
$998$	0	0
$999$	−2.65230e9	−0.0841673

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.8.a.g.1.3		5
4.3	odd	2		76.8.a.a.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.a.a.1.3	✓	5	4.3	odd	2
304.8.a.g.1.3		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)

\(f(q)\)	\(=\)	\(q-8.09048 q^{3} +276.339 q^{5} +720.800 q^{7} -2121.54 q^{9} +O(q^{10})\)	\(q-8.09048 q^{3} +276.339 q^{5} +720.800 q^{7} -2121.54 q^{9} +4974.98 q^{11} +8085.05 q^{13} -2235.72 q^{15} +1375.87 q^{17} +6859.00 q^{19} -5831.62 q^{21} +55033.5 q^{23} -1761.49 q^{25} +34858.2 q^{27} -117756. q^{29} +151040. q^{31} -40250.0 q^{33} +199185. q^{35} -76088.3 q^{37} -65411.9 q^{39} +420662. q^{41} +411546. q^{43} -586266. q^{45} +791764. q^{47} -303991. q^{49} -11131.5 q^{51} -800670. q^{53} +1.37478e6 q^{55} -55492.6 q^{57} -3.09461e6 q^{59} -2.19949e6 q^{61} -1.52921e6 q^{63} +2.23422e6 q^{65} +420260. q^{67} -445247. q^{69} -2.69010e6 q^{71} +598122. q^{73} +14251.3 q^{75} +3.58597e6 q^{77} -2.62187e6 q^{79} +4.35780e6 q^{81} +5.86850e6 q^{83} +380208. q^{85} +952705. q^{87} +6.26993e6 q^{89} +5.82770e6 q^{91} -1.22199e6 q^{93} +1.89541e6 q^{95} +1.21791e7 q^{97} -1.05546e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 14 q^{3} - 280 q^{5} - 414 q^{7} + 3779 q^{9}+O(q^{10}) \) 5 * q + 14 * q^3 - 280 * q^5 - 414 * q^7 + 3779 * q^9	\( 5 q + 14 q^{3} - 280 q^{5} - 414 q^{7} + 3779 q^{9} + 2662 q^{11} - 602 q^{13} + 20800 q^{15} - 27366 q^{17} + 34295 q^{19} - 59964 q^{21} + 67096 q^{23} - 109115 q^{25} + 178778 q^{27} - 372398 q^{29} + 271372 q^{31} - 792700 q^{33} + 608250 q^{35} - 562630 q^{37} + 963904 q^{39} - 956714 q^{41} + 827362 q^{43} - 1165100 q^{45} + 1812982 q^{47} - 862031 q^{49} + 2458254 q^{51} + 486998 q^{53} - 467930 q^{55} + 96026 q^{57} + 367182 q^{59} + 1879732 q^{61} + 1007274 q^{63} + 1790920 q^{65} + 1046394 q^{67} + 7261712 q^{69} + 4664572 q^{71} + 4224942 q^{73} - 8194850 q^{75} + 8611110 q^{77} - 9574024 q^{79} + 11351813 q^{81} - 11754804 q^{83} + 18711750 q^{85} - 3801472 q^{87} + 2782542 q^{89} - 7385214 q^{91} + 29535004 q^{93} - 1920520 q^{95} + 1291574 q^{97} + 9760310 q^{99}+O(q^{100}) \) 5 * q + 14 * q^3 - 280 * q^5 - 414 * q^7 + 3779 * q^9 + 2662 * q^11 - 602 * q^13 + 20800 * q^15 - 27366 * q^17 + 34295 * q^19 - 59964 * q^21 + 67096 * q^23 - 109115 * q^25 + 178778 * q^27 - 372398 * q^29 + 271372 * q^31 - 792700 * q^33 + 608250 * q^35 - 562630 * q^37 + 963904 * q^39 - 956714 * q^41 + 827362 * q^43 - 1165100 * q^45 + 1812982 * q^47 - 862031 * q^49 + 2458254 * q^51 + 486998 * q^53 - 467930 * q^55 + 96026 * q^57 + 367182 * q^59 + 1879732 * q^61 + 1007274 * q^63 + 1790920 * q^65 + 1046394 * q^67 + 7261712 * q^69 + 4664572 * q^71 + 4224942 * q^73 - 8194850 * q^75 + 8611110 * q^77 - 9574024 * q^79 + 11351813 * q^81 - 11754804 * q^83 + 18711750 * q^85 - 3801472 * q^87 + 2782542 * q^89 - 7385214 * q^91 + 29535004 * q^93 - 1920520 * q^95 + 1291574 * q^97 + 9760310 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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