LMFDB - Embedded newform 304.7.e.e.113.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,7,Mod(113,304)]

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, 1]))

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

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

chi := DirichletCharacter("304.113");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	304.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$69.9364414204$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 5050x^{8} + 7354489x^{6} + 2475755792x^{4} + 232626987584x^{2} + 2900002611200 $ x^10 + 5050x^8 + 7354489x^6 + 2475755792x^4 + 232626987584x^2 + 2900002611200
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.1
Root			$-48.7374i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.7.e.e.113.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-51.5658i q^{3} +39.9185 q^{5} +25.1565 q^{7} -1930.03 q^{9} +O(q^{10})$	$q-51.5658i q^{3} +39.9185 q^{5} +25.1565 q^{7} -1930.03 q^{9} +1368.34 q^{11} +157.072i q^{13} -2058.43i q^{15} -2735.86 q^{17} +(3468.77 - 5917.22i) q^{19} -1297.21i q^{21} -17315.8 q^{23} -14031.5 q^{25} +61932.1i q^{27} +26064.3i q^{29} +26833.9i q^{31} -70559.4i q^{33} +1004.21 q^{35} +85898.9i q^{37} +8099.54 q^{39} +53613.3i q^{41} -111934. q^{43} -77044.0 q^{45} -60698.5 q^{47} -117016. q^{49} +141077. i q^{51} -1359.96i q^{53} +54622.0 q^{55} +(-305126. - 178870. i) q^{57} -238887. i q^{59} +429630. q^{61} -48552.7 q^{63} +6270.08i q^{65} -156985. i q^{67} +892901. i q^{69} -365008. i q^{71} -285370. q^{73} +723546. i q^{75} +34422.5 q^{77} +606939. i q^{79} +1.78659e6 q^{81} +875034. q^{83} -109212. q^{85} +1.34403e6 q^{87} +590538. i q^{89} +3951.37i q^{91} +1.38371e6 q^{93} +(138468. - 236207. i) q^{95} -1.00906e6i q^{97} -2.64093e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 112 q^{5} + 224 q^{7} - 2890 q^{9}+O(q^{10}) $ 10 * q - 112 * q^5 + 224 * q^7 - 2890 * q^9	$ 10 q - 112 q^{5} + 224 q^{7} - 2890 q^{9} - 3644 q^{11} - 10420 q^{17} + 17230 q^{19} - 37712 q^{23} - 52078 q^{25} + 161720 q^{35} + 78876 q^{39} - 6308 q^{43} + 309808 q^{45} - 322220 q^{47} - 235770 q^{49} + 377880 q^{55} + 24228 q^{57} + 426304 q^{61} + 517916 q^{63} - 786076 q^{73} + 2303716 q^{77} + 5261090 q^{81} + 101500 q^{83} - 1261380 q^{85} + 2460732 q^{87} - 2827032 q^{93} - 3106292 q^{95} - 1061428 q^{99}+O(q^{100}) $ 10 * q - 112 * q^5 + 224 * q^7 - 2890 * q^9 - 3644 * q^11 - 10420 * q^17 + 17230 * q^19 - 37712 * q^23 - 52078 * q^25 + 161720 * q^35 + 78876 * q^39 - 6308 * q^43 + 309808 * q^45 - 322220 * q^47 - 235770 * q^49 + 377880 * q^55 + 24228 * q^57 + 426304 * q^61 + 517916 * q^63 - 786076 * q^73 + 2303716 * q^77 + 5261090 * q^81 + 101500 * q^83 - 1261380 * q^85 + 2460732 * q^87 - 2827032 * q^93 - 3106292 * q^95 - 1061428 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

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
$3$		−	51.5658i		−	1.90984i	−0.296857	−	0.954922i	$-0.595938\pi$
$3$		−	51.5658i		−	1.90984i	0.296857	−	0.954922i	$-0.404062\pi$
$4$		0			0
$5$	39.9185			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$5$	39.9185			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$6$		0			0
$7$	25.1565			0.0733424			0.0366712	−	0.999327i	$-0.488325\pi$
$7$	25.1565			0.0733424			0.0366712	+	0.999327i	$0.488325\pi$
$8$		0			0
$9$	−1930.03			−2.64750
$10$		0			0
$11$	1368.34			1.02805			0.514026	−	0.857775i	$-0.328154\pi$
$11$	1368.34			1.02805			0.514026	+	0.857775i	$0.328154\pi$
$12$		0			0
$13$			157.072i			0.0714938i	0.999361	+	0.0357469i	$0.0113810\pi$
$13$			157.072i			0.0714938i	−0.999361	+	0.0357469i	$0.988619\pi$
$14$		0			0
$15$		−	2058.43i		−	0.609905i
$16$		0			0
$17$	−2735.86			−0.556862			−0.278431	−	0.960456i	$-0.589814\pi$
$17$	−2735.86			−0.556862			−0.278431	+	0.960456i	$0.589814\pi$
$18$		0			0
$19$	3468.77	−	5917.22i	0.505726	−	0.862694i
$19$	3468.77	−	5917.22i	0.505726	−	0.862694i
$20$		0			0
$21$		−	1297.21i		−	0.140073i
$22$		0			0
$23$	−17315.8			−1.42317			−0.711587	−	0.702598i	$-0.752023\pi$
$23$	−17315.8			−1.42317			−0.711587	+	0.702598i	$0.752023\pi$
$24$		0			0
$25$	−14031.5			−0.898017
$26$		0			0
$27$			61932.1i			3.14648i
$28$		0			0
$29$			26064.3i			1.06869i	0.845266	+	0.534345i	$0.179442\pi$
$29$			26064.3i			1.06869i	−0.845266	+	0.534345i	$0.820558\pi$
$30$		0			0
$31$			26833.9i			0.900739i	0.892842	+	0.450369i	$0.148708\pi$
$31$			26833.9i			0.900739i	−0.892842	+	0.450369i	$0.851292\pi$
$32$		0			0
$33$		−	70559.4i		−	1.96342i
$34$		0			0
$35$	1004.21			0.0234218
$36$		0			0
$37$			85898.9i			1.69583i	0.530132	+	0.847915i	$0.322142\pi$
$37$			85898.9i			1.69583i	−0.530132	+	0.847915i	$0.677858\pi$
$38$		0			0
$39$	8099.54			0.136542
$40$		0			0
$41$			53613.3i			0.777895i	0.921260	+	0.388948i	$0.127161\pi$
$41$			53613.3i			0.777895i	−0.921260	+	0.388948i	$0.872839\pi$
$42$		0			0
$43$	−111934.			−1.40785			−0.703926	−	0.710273i	$-0.748571\pi$
$43$	−111934.			−1.40785			−0.703926	+	0.710273i	$0.748571\pi$
$44$		0			0
$45$	−77044.0			−0.845476
$46$		0			0
$47$	−60698.5			−0.584634			−0.292317	−	0.956321i	$-0.594426\pi$
$47$	−60698.5			−0.584634			−0.292317	+	0.956321i	$0.594426\pi$
$48$		0			0
$49$	−117016.			−0.994621
$50$		0			0
$51$			141077.i			1.06352i
$52$		0			0
$53$		−	1359.96i		−	0.00913479i	−0.999990	−	0.00456739i	$-0.998546\pi$
$53$		−	1359.96i		−	0.00913479i	0.999990	−	0.00456739i	$-0.00145385\pi$
$54$		0			0
$55$	54622.0			0.328306
$56$		0			0
$57$	−305126.	−	178870.i	−1.64761	−	0.965858i
$58$		0			0
$59$		−	238887.i		−	1.16315i	−0.813492	−	0.581575i	$-0.802436\pi$
$59$		−	238887.i		−	1.16315i	0.813492	−	0.581575i	$-0.197564\pi$
$60$		0			0
$61$	429630.			1.89280			0.946401	−	0.322995i	$-0.104690\pi$
$61$	429630.			1.89280			0.946401	+	0.322995i	$0.104690\pi$
$62$		0			0
$63$	−48552.7			−0.194174
$64$		0			0
$65$			6270.08i			0.0228314i
$66$		0			0
$67$		−	156985.i		−	0.521957i	−0.965345	−	0.260979i	$-0.915955\pi$
$67$		−	156985.i		−	0.521957i	0.965345	−	0.260979i	$-0.0840452\pi$
$68$		0			0
$69$			892901.i			2.71804i
$70$		0			0
$71$		−	365008.i		−	1.01983i	−0.860225	−	0.509915i	$-0.829677\pi$
$71$		−	365008.i		−	1.01983i	0.860225	−	0.509915i	$-0.170323\pi$
$72$		0			0
$73$	−285370.			−0.733566			−0.366783	−	0.930306i	$-0.619541\pi$
$73$	−285370.			−0.733566			−0.366783	+	0.930306i	$0.619541\pi$
$74$		0			0
$75$			723546.i			1.71507i
$76$		0			0
$77$	34422.5			0.0753998
$78$		0			0
$79$			606939.i			1.23102i	0.788131	+	0.615508i	$0.211049\pi$
$79$			606939.i			1.23102i	−0.788131	+	0.615508i	$0.788951\pi$
$80$		0			0
$81$	1.78659e6			3.36177
$82$		0			0
$83$	875034.			1.53035			0.765175	−	0.643822i	$-0.222652\pi$
$83$	875034.			1.53035			0.765175	+	0.643822i	$0.222652\pi$
$84$		0			0
$85$	−109212.			−0.177833
$86$		0			0
$87$	1.34403e6			2.04103
$88$		0			0
$89$			590538.i			0.837680i	0.908060	+	0.418840i	$0.137563\pi$
$89$			590538.i			0.837680i	−0.908060	+	0.418840i	$0.862437\pi$
$90$		0			0
$91$			3951.37i			0.00524353i
$92$		0			0
$93$	1.38371e6			1.72027
$94$		0			0
$95$	138468.	−	236207.i	0.161503	−	0.275500i
$96$		0			0
$97$		−	1.00906e6i		−	1.10561i	−0.833312	−	0.552803i	$-0.813558\pi$
$97$		−	1.00906e6i		−	1.10561i	0.833312	−	0.552803i	$-0.186442\pi$
$98$		0			0
$99$	−2.64093e6			−2.72177
$100$		0			0
$101$	−534106.			−0.518398			−0.259199	−	0.965824i	$-0.583459\pi$
$101$	−534106.			−0.518398			−0.259199	+	0.965824i	$0.583459\pi$
$102$		0			0
$103$		−	768218.i		−	0.703028i	−0.936183	−	0.351514i	$-0.885667\pi$
$103$		−	768218.i		−	0.703028i	0.936183	−	0.351514i	$-0.114333\pi$
$104$		0			0
$105$		−	51782.8i		−	0.0447319i
$106$		0			0
$107$			1.46546e6i			1.19625i	0.801402	+	0.598126i	$0.204088\pi$
$107$			1.46546e6i			1.19625i	−0.801402	+	0.598126i	$0.795912\pi$
$108$		0			0
$109$		−	1.89935e6i		−	1.46665i	−0.679880	−	0.733323i	$-0.737968\pi$
$109$		−	1.89935e6i		−	1.46665i	0.679880	−	0.733323i	$-0.262032\pi$
$110$		0			0
$111$	4.42944e6			3.23877
$112$		0			0
$113$			1.16465e6i			0.807161i	0.914944	+	0.403581i	$0.132235\pi$
$113$			1.16465e6i			0.807161i	−0.914944	+	0.403581i	$0.867765\pi$
$114$		0			0
$115$	−691220.			−0.454488
$116$		0			0
$117$		−	303154.i		−	0.189280i
$118$		0			0
$119$	−68824.6			−0.0408416
$120$		0			0
$121$	100784.			0.0568900
$122$		0			0
$123$	2.76461e6			1.48566
$124$		0			0
$125$	−1.18384e6			−0.606128
$126$		0			0
$127$			1.04931e6i			0.512264i	0.966642	+	0.256132i	$0.0824482\pi$
$127$			1.04931e6i			0.512264i	−0.966642	+	0.256132i	$0.917552\pi$
$128$		0			0
$129$			5.77197e6i			2.68878i
$130$		0			0
$131$	−265738.			−0.118206			−0.0591030	−	0.998252i	$-0.518824\pi$
$131$	−265738.			−0.118206			−0.0591030	+	0.998252i	$0.518824\pi$
$132$		0			0
$133$	87262.0	−	148856.i	0.0370912	−	0.0632721i
$134$		0			0
$135$			2.47224e6i			1.00482i
$136$		0			0
$137$	−213121.			−0.0828827			−0.0414414	−	0.999141i	$-0.513195\pi$
$137$	−213121.			−0.0828827			−0.0414414	+	0.999141i	$0.513195\pi$
$138$		0			0
$139$	−1.82687e6			−0.680243			−0.340122	−	0.940381i	$-0.610468\pi$
$139$	−1.82687e6			−0.680243			−0.340122	+	0.940381i	$0.610468\pi$
$140$		0			0
$141$			3.12997e6i			1.11656i
$142$		0			0
$143$			214927.i			0.0734993i
$144$		0			0
$145$			1.04045e6i			0.341284i
$146$		0			0
$147$			6.03403e6i			1.89957i
$148$		0			0
$149$	−2.41401e6			−0.729759			−0.364880	−	0.931055i	$-0.618890\pi$
$149$	−2.41401e6			−0.729759			−0.364880	+	0.931055i	$0.618890\pi$
$150$		0			0
$151$			366324.i			0.106398i	0.998584	+	0.0531991i	$0.0169418\pi$
$151$			366324.i			0.106398i	−0.998584	+	0.0531991i	$0.983058\pi$
$152$		0			0
$153$	5.28030e6			1.47429
$154$		0			0
$155$			1.07117e6i			0.287649i
$156$		0			0
$157$	160372.			0.0414409			0.0207204	−	0.999785i	$-0.493404\pi$
$157$	160372.			0.0414409			0.0207204	+	0.999785i	$0.493404\pi$
$158$		0			0
$159$	−70127.4			−0.0174460
$160$		0			0
$161$	−435603.			−0.104379
$162$		0			0
$163$	−7.65828e6			−1.76835			−0.884176	−	0.467154i	$-0.845279\pi$
$163$	−7.65828e6			−1.76835			−0.884176	+	0.467154i	$0.845279\pi$
$164$		0			0
$165$		−	2.81663e6i		−	0.627014i
$166$		0			0
$167$		−	1.10021e6i		−	0.236226i	−0.993000	−	0.118113i	$-0.962316\pi$
$167$		−	1.10021e6i		−	0.236226i	0.993000	−	0.118113i	$-0.0376845\pi$
$168$		0			0
$169$	4.80214e6			0.994889
$170$		0			0
$171$	−6.69484e6	+	1.14204e7i	−1.33891	+	2.28399i
$172$		0			0
$173$			5.74022e6i			1.10864i	0.832304	+	0.554320i	$0.187022\pi$
$173$			5.74022e6i			1.10864i	−0.832304	+	0.554320i	$0.812978\pi$
$174$		0			0
$175$	−352983.			−0.0658627
$176$		0			0
$177$	−1.23184e7			−2.22144
$178$		0			0
$179$		−	5.04968e6i		−	0.880449i	−0.897888	−	0.440225i	$-0.854899\pi$
$179$		−	5.04968e6i		−	0.880449i	0.897888	−	0.440225i	$-0.145101\pi$
$180$		0			0
$181$		−	1.02257e6i		−	0.172447i	−0.996276	−	0.0862234i	$-0.972520\pi$
$181$		−	1.02257e6i		−	0.172447i	0.996276	−	0.0862234i	$-0.0274799\pi$
$182$		0			0
$183$		−	2.21542e7i		−	3.61496i
$184$		0			0
$185$			3.42896e6i			0.541560i
$186$		0			0
$187$	−3.74358e6			−0.572483
$188$		0			0
$189$			1.55799e6i			0.230770i
$190$		0			0
$191$	3.23276e6			0.463952			0.231976	−	0.972722i	$-0.425481\pi$
$191$	3.23276e6			0.463952			0.231976	+	0.972722i	$0.425481\pi$
$192$		0			0
$193$		−	338219.i		−	0.0470464i	−0.999723	−	0.0235232i	$-0.992512\pi$
$193$		−	338219.i		−	0.0470464i	0.999723	−	0.0235232i	$-0.00748836\pi$
$194$		0			0
$195$	323322.			0.0436045
$196$		0			0
$197$	1.05750e7			1.38319			0.691593	−	0.722287i	$-0.256909\pi$
$197$	1.05750e7			1.38319			0.691593	+	0.722287i	$0.256909\pi$
$198$		0			0
$199$	6.64038e6			0.842624			0.421312	−	0.906916i	$-0.361570\pi$
$199$	6.64038e6			0.842624			0.421312	+	0.906916i	$0.361570\pi$
$200$		0			0
$201$	−8.09507e6			−0.996857
$202$		0			0
$203$			655685.i			0.0783803i
$204$		0			0
$205$			2.14017e6i			0.248420i
$206$		0			0
$207$	3.34200e7			3.76786
$208$		0			0
$209$	4.74645e6	−	8.09675e6i	0.519912	−	0.886894i
$210$		0			0
$211$			5.89794e6i			0.627846i	0.949448	+	0.313923i	$0.101643\pi$
$211$			5.89794e6i			0.627846i	−0.949448	+	0.313923i	$0.898357\pi$
$212$		0			0
$213$	−1.88219e7			−1.94771
$214$		0			0
$215$	−4.46825e6			−0.449595
$216$		0			0
$217$			675046.i			0.0660624i
$218$		0			0
$219$			1.47153e7i			1.40100i
$220$		0			0
$221$		−	429727.i		−	0.0398122i
$222$		0			0
$223$			1.43507e7i			1.29407i	0.762460	+	0.647036i	$0.223992\pi$
$223$			1.43507e7i			1.29407i	−0.762460	+	0.647036i	$0.776008\pi$
$224$		0			0
$225$	2.70812e7			2.37750
$226$		0			0
$227$		−	9.39539e6i		−	0.803225i	−0.915810	−	0.401613i	$-0.868450\pi$
$227$		−	9.39539e6i		−	0.803225i	0.915810	−	0.401613i	$-0.131550\pi$
$228$		0			0
$229$	1.31738e7			1.09699			0.548497	−	0.836152i	$-0.315200\pi$
$229$	1.31738e7			1.09699			0.548497	+	0.836152i	$0.315200\pi$
$230$		0			0
$231$		−	1.77502e6i		−	0.144002i
$232$		0			0
$233$	−1.34530e7			−1.06353			−0.531767	−	0.846890i	$-0.678472\pi$
$233$	−1.34530e7			−1.06353			−0.531767	+	0.846890i	$0.678472\pi$
$234$		0			0
$235$	−2.42299e6			−0.186702
$236$		0			0
$237$	3.12973e7			2.35105
$238$		0			0
$239$	−2.23712e7			−1.63869			−0.819344	−	0.573302i	$-0.805662\pi$
$239$	−2.23712e7			−1.63869			−0.819344	+	0.573302i	$0.805662\pi$
$240$		0			0
$241$		−	5.17298e6i		−	0.369564i	−0.982780	−	0.184782i	$-0.940842\pi$
$241$		−	5.17298e6i		−	0.369564i	0.982780	−	0.184782i	$-0.0591579\pi$
$242$		0			0
$243$		−	4.69782e7i		−	3.27399i
$244$		0			0
$245$	−4.67111e6			−0.317630
$246$		0			0
$247$	929429.	+	544847.i	0.0616773	+	0.0361563i
$248$		0			0
$249$		−	4.51218e7i		−	2.92273i
$250$		0			0
$251$	−1.08729e6			−0.0687582			−0.0343791	−	0.999409i	$-0.510945\pi$
$251$	−1.08729e6			−0.0687582			−0.0343791	+	0.999409i	$0.510945\pi$
$252$		0			0
$253$	−2.36938e7			−1.46310
$254$		0			0
$255$			5.63158e6i			0.339633i
$256$		0			0
$257$			1.37915e7i			0.812480i	0.913766	+	0.406240i	$0.133160\pi$
$257$			1.37915e7i			0.812480i	−0.913766	+	0.406240i	$0.866840\pi$
$258$		0			0
$259$			2.16091e6i			0.124376i
$260$		0			0
$261$		−	5.03049e7i		−	2.82936i
$262$		0			0
$263$	−1.18420e7			−0.650967			−0.325484	−	0.945548i	$-0.605527\pi$
$263$	−1.18420e7			−0.650967			−0.325484	+	0.945548i	$0.605527\pi$
$264$		0			0
$265$		−	54287.6i		−	0.00291718i
$266$		0			0
$267$	3.04516e7			1.59984
$268$		0			0
$269$		−	2.84915e7i		−	1.46372i	−0.681453	−	0.731862i	$-0.738652\pi$
$269$		−	2.84915e7i		−	1.46372i	0.681453	−	0.731862i	$-0.261348\pi$
$270$		0			0
$271$	−242754.			−0.0121971			−0.00609857	−	0.999981i	$-0.501941\pi$
$271$	−242754.			−0.0121971			−0.00609857	+	0.999981i	$0.501941\pi$
$272$		0			0
$273$	203756.			0.0100143
$274$		0			0
$275$	−1.91998e7			−0.923207
$276$		0			0
$277$	−3.67776e7			−1.73039			−0.865195	−	0.501436i	$-0.832805\pi$
$277$	−3.67776e7			−1.73039			−0.865195	+	0.501436i	$0.832805\pi$
$278$		0			0
$279$		−	5.17903e7i		−	2.38471i
$280$		0			0
$281$			2.78086e7i			1.25331i	0.779295	+	0.626657i	$0.215577\pi$
$281$			2.78086e7i			1.25331i	−0.779295	+	0.626657i	$0.784423\pi$
$282$		0			0
$283$	2.09382e7			0.923803			0.461901	−	0.886931i	$-0.347167\pi$
$283$	2.09382e7			0.923803			0.461901	+	0.886931i	$0.347167\pi$
$284$		0			0
$285$	−1.21802e7	−	7.14023e6i	−0.526162	−	0.308445i
$286$		0			0
$287$			1.34872e6i			0.0570527i
$288$		0			0
$289$	−1.66526e7			−0.689905
$290$		0			0
$291$	−5.20328e7			−2.11154
$292$		0			0
$293$			1.12853e7i			0.448651i	0.974514	+	0.224325i	$0.0720179\pi$
$293$			1.12853e7i			0.448651i	−0.974514	+	0.224325i	$0.927982\pi$
$294$		0			0
$295$		−	9.53601e6i		−	0.371450i
$296$		0			0
$297$			8.47440e7i			3.23474i
$298$		0			0
$299$		−	2.71982e6i		−	0.101748i
$300$		0			0
$301$	−2.81587e6			−0.103255
$302$		0			0
$303$			2.75416e7i			0.990059i
$304$		0			0
$305$	1.71502e7			0.604463
$306$		0			0
$307$		−	762535.i		−	0.0263539i	−0.999913	−	0.0131769i	$-0.995806\pi$
$307$		−	762535.i		−	0.0263539i	0.999913	−	0.0131769i	$-0.00419447\pi$
$308$		0			0
$309$	−3.96138e7			−1.34267
$310$		0			0
$311$	1.78594e7			0.593727			0.296864	−	0.954920i	$-0.404059\pi$
$311$	1.78594e7			0.593727			0.296864	+	0.954920i	$0.404059\pi$
$312$		0			0
$313$	−2.41177e7			−0.786506			−0.393253	−	0.919430i	$-0.628650\pi$
$313$	−2.41177e7			−0.786506			−0.393253	+	0.919430i	$0.628650\pi$
$314$		0			0
$315$	−1.93815e6			−0.0620092
$316$		0			0
$317$			3.51314e7i			1.10285i	0.834223	+	0.551427i	$0.185916\pi$
$317$			3.51314e7i			1.10285i	−0.834223	+	0.551427i	$0.814084\pi$
$318$		0			0
$319$			3.56647e7i			1.09867i
$320$		0			0
$321$	7.55676e7			2.28466
$322$		0			0
$323$	−9.49009e6	+	1.61887e7i	−0.281620	+	0.480402i
$324$		0			0
$325$		−	2.20396e6i		−	0.0642027i
$326$		0			0
$327$	−9.79415e7			−2.80107
$328$		0			0
$329$	−1.52696e6			−0.0428785
$330$		0			0
$331$		−	2.71772e7i		−	0.749412i	−0.927144	−	0.374706i	$-0.877744\pi$
$331$		−	2.71772e7i		−	0.749412i	0.927144	−	0.374706i	$-0.122256\pi$
$332$		0			0
$333$		−	1.65787e8i		−	4.48972i
$334$		0			0
$335$		−	6.26663e6i		−	0.166686i
$336$		0			0
$337$			6.30310e7i			1.64689i	0.567397	+	0.823444i	$0.307951\pi$
$337$			6.30310e7i			1.64689i	−0.567397	+	0.823444i	$0.692049\pi$
$338$		0			0
$339$	6.00561e7			1.54155
$340$		0			0
$341$			3.67178e7i			0.926006i
$342$		0			0
$343$	−5.90334e6			−0.146290
$344$		0			0
$345$			3.56433e7i			0.868002i
$346$		0			0
$347$	−3.67501e6			−0.0879571			−0.0439785	−	0.999032i	$-0.514003\pi$
$347$	−3.67501e6			−0.0879571			−0.0439785	+	0.999032i	$0.514003\pi$
$348$		0			0
$349$	−4.34206e7			−1.02146			−0.510728	−	0.859742i	$-0.670624\pi$
$349$	−4.34206e7			−1.02146			−0.510728	+	0.859742i	$0.670624\pi$
$350$		0			0
$351$	−9.72779e6			−0.224954
$352$		0			0
$353$	−1.09059e7			−0.247934			−0.123967	−	0.992286i	$-0.539562\pi$
$353$	−1.09059e7			−0.247934			−0.123967	+	0.992286i	$0.539562\pi$
$354$		0			0
$355$		−	1.45706e7i		−	0.325681i
$356$		0			0
$357$			3.54900e6i			0.0780011i
$358$		0			0
$359$	−8.16663e7			−1.76506			−0.882530	−	0.470256i	$-0.844161\pi$
$359$	−8.16663e7			−1.76506			−0.882530	+	0.470256i	$0.844161\pi$
$360$		0			0
$361$	−2.29811e7	−	4.10510e7i	−0.488483	−	0.872574i
$362$		0			0
$363$		−	5.19701e6i		−	0.108651i
$364$		0			0
$365$	−1.13915e7			−0.234263
$366$		0			0
$367$	−5.31200e7			−1.07463			−0.537316	−	0.843381i	$-0.680562\pi$
$367$	−5.31200e7			−1.07463			−0.537316	+	0.843381i	$0.680562\pi$
$368$		0			0
$369$		−	1.03475e8i		−	2.05948i
$370$		0			0
$371$		−	34211.8i		−	0.000669967i
$372$		0			0
$373$			2.35968e7i			0.454703i	0.973813	+	0.227351i	$0.0730066\pi$
$373$			2.35968e7i			0.454703i	−0.973813	+	0.227351i	$0.926993\pi$
$374$		0			0
$375$			6.10459e7i			1.15761i
$376$		0			0
$377$	−4.09397e6			−0.0764048
$378$		0			0
$379$		−	6.20101e7i		−	1.13906i	−0.821972	−	0.569528i	$-0.807126\pi$
$379$		−	6.20101e7i		−	1.13906i	0.821972	−	0.569528i	$-0.192874\pi$
$380$		0			0
$381$	5.41087e7			0.978345
$382$		0			0
$383$		−	6.21448e7i		−	1.10614i	−0.833136	−	0.553068i	$-0.813457\pi$
$383$		−	6.21448e7i		−	1.10614i	0.833136	−	0.553068i	$-0.186543\pi$
$384$		0			0
$385$	1.37410e6			0.0240788
$386$		0			0
$387$	2.16036e8			3.72730
$388$		0			0
$389$	−4.71506e7			−0.801011			−0.400506	−	0.916294i	$-0.631165\pi$
$389$	−4.71506e7			−0.801011			−0.400506	+	0.916294i	$0.631165\pi$
$390$		0			0
$391$	4.73736e7			0.792512
$392$		0			0
$393$			1.37030e7i			0.225755i
$394$		0			0
$395$			2.42281e7i			0.393123i
$396$		0			0
$397$	5.12175e7			0.818553			0.409277	−	0.912410i	$-0.365781\pi$
$397$	5.12175e7			0.818553			0.409277	+	0.912410i	$0.365781\pi$
$398$		0			0
$399$	−7.67589e6	−	4.49974e6i	−0.120840	−	0.0708383i
$400$		0			0
$401$		−	6.30653e7i		−	0.978041i	−0.872272	−	0.489021i	$-0.837354\pi$
$401$		−	6.30653e7i		−	0.978041i	0.872272	−	0.489021i	$-0.162646\pi$
$402$		0			0
$403$	−4.21485e6			−0.0643973
$404$		0			0
$405$	7.13178e7			1.07358
$406$		0			0
$407$			1.17539e8i			1.74340i
$408$		0			0
$409$			6.35675e7i			0.929107i	0.885545	+	0.464553i	$0.153785\pi$
$409$			6.35675e7i			0.929107i	−0.885545	+	0.464553i	$0.846215\pi$
$410$		0			0
$411$			1.09897e7i			0.158293i
$412$		0			0
$413$		−	6.00954e6i		−	0.0853083i
$414$		0			0
$415$	3.49301e7			0.488715
$416$		0			0
$417$			9.42042e7i			1.29916i
$418$		0			0
$419$	−3.77509e7			−0.513198			−0.256599	−	0.966518i	$-0.582602\pi$
$419$	−3.77509e7			−0.513198			−0.256599	+	0.966518i	$0.582602\pi$
$420$		0			0
$421$		−	6.28689e7i		−	0.842537i	−0.906936	−	0.421269i	$-0.861585\pi$
$421$		−	6.28689e7i		−	0.842537i	0.906936	−	0.421269i	$-0.138415\pi$
$422$		0			0
$423$	1.17150e8			1.54782
$424$		0			0
$425$	3.83883e7			0.500071
$426$		0			0
$427$	1.08080e7			0.138823
$428$		0			0
$429$	1.10829e7			0.140372
$430$		0			0
$431$		−	7.64637e7i		−	0.955045i	−0.878620	−	0.477522i	$-0.841535\pi$
$431$		−	7.64637e7i		−	0.955045i	0.878620	−	0.477522i	$-0.158465\pi$
$432$		0			0
$433$			3.83030e7i			0.471812i	0.971776	+	0.235906i	$0.0758058\pi$
$433$			3.83030e7i			0.471812i	−0.971776	+	0.235906i	$0.924194\pi$
$434$		0			0
$435$	5.36515e7			0.651800
$436$		0			0
$437$	−6.00645e7	+	1.02461e8i	−0.719736	+	1.22776i
$438$		0			0
$439$			5.64059e7i			0.666701i	0.942803	+	0.333350i	$0.108179\pi$
$439$			5.64059e7i			0.666701i	−0.942803	+	0.333350i	$0.891821\pi$
$440$		0			0
$441$	2.25845e8			2.63326
$442$		0			0
$443$	1.05839e8			1.21740			0.608701	−	0.793400i	$-0.291691\pi$
$443$	1.05839e8			1.21740			0.608701	+	0.793400i	$0.291691\pi$
$444$		0			0
$445$			2.35734e7i			0.267512i
$446$		0			0
$447$			1.24480e8i			1.39373i
$448$		0			0
$449$		−	8.94076e7i		−	0.987724i	−0.869540	−	0.493862i	$-0.835585\pi$
$449$		−	8.94076e7i		−	0.987724i	0.869540	−	0.493862i	$-0.164415\pi$
$450$		0			0
$451$			7.33611e7i			0.799717i
$452$		0			0
$453$	1.88898e7			0.203204
$454$		0			0
$455$			157733.i			0.00167451i
$456$		0			0
$457$	−3.60536e7			−0.377747			−0.188873	−	0.982001i	$-0.560484\pi$
$457$	−3.60536e7			−0.377747			−0.188873	+	0.982001i	$0.560484\pi$
$458$		0			0
$459$		−	1.69438e8i		−	1.75215i
$460$		0			0
$461$	4.22650e7			0.431398			0.215699	−	0.976460i	$-0.430797\pi$
$461$	4.22650e7			0.431398			0.215699	+	0.976460i	$0.430797\pi$
$462$		0			0
$463$	−1.05402e8			−1.06195			−0.530977	−	0.847386i	$-0.678175\pi$
$463$	−1.05402e8			−1.06195			−0.530977	+	0.847386i	$0.678175\pi$
$464$		0			0
$465$	5.52357e7			0.549365
$466$		0			0
$467$	1.16729e8			1.14611			0.573055	−	0.819517i	$-0.305758\pi$
$467$	1.16729e8			1.14611			0.573055	+	0.819517i	$0.305758\pi$
$468$		0			0
$469$		−	3.94919e6i		−	0.0382816i
$470$		0			0
$471$		−	8.26970e6i		−	0.0791456i
$472$		0			0
$473$	−1.53164e8			−1.44734
$474$		0			0
$475$	−4.86721e7	+	8.30275e7i	−0.454150	+	0.774714i
$476$		0			0
$477$			2.62476e6i			0.0241844i
$478$		0			0
$479$	2.50394e7			0.227833			0.113916	−	0.993490i	$-0.463660\pi$
$479$	2.50394e7			0.227833			0.113916	+	0.993490i	$0.463660\pi$
$480$		0			0
$481$	−1.34923e7			−0.121241
$482$		0			0
$483$			2.24622e7i			0.199348i
$484$		0			0
$485$		−	4.02801e7i		−	0.353074i
$486$		0			0
$487$			7.51465e7i			0.650612i	0.945609	+	0.325306i	$0.105467\pi$
$487$			7.51465e7i			0.650612i	−0.945609	+	0.325306i	$0.894533\pi$
$488$		0			0
$489$			3.94905e8i			3.37728i
$490$		0			0
$491$	−1.74282e8			−1.47234			−0.736170	−	0.676797i	$-0.763367\pi$
$491$	−1.74282e8			−1.47234			−0.736170	+	0.676797i	$0.763367\pi$
$492$		0			0
$493$		−	7.13083e7i		−	0.595113i
$494$		0			0
$495$	−1.05422e8			−0.869193
$496$		0			0
$497$		−	9.18231e6i		−	0.0747967i
$498$		0			0
$499$	−1.36562e8			−1.09908			−0.549538	−	0.835469i	$-0.685196\pi$
$499$	−1.36562e8			−1.09908			−0.549538	+	0.835469i	$0.685196\pi$
$500$		0			0
$501$	−5.67334e7			−0.451155
$502$		0			0
$503$	−1.51696e8			−1.19198			−0.595991	−	0.802991i	$-0.703241\pi$
$503$	−1.51696e8			−1.19198			−0.595991	+	0.802991i	$0.703241\pi$
$504$		0			0
$505$	−2.13207e7			−0.165549
$506$		0			0
$507$		−	2.47626e8i		−	1.90008i
$508$		0			0
$509$		−	2.19152e8i		−	1.66185i	−0.556386	−	0.830924i	$-0.687812\pi$
$509$		−	2.19152e8i		−	1.66185i	0.556386	−	0.830924i	$-0.312188\pi$
$510$		0			0
$511$	−7.17889e6			−0.0538015
$512$		0			0
$513$	3.66466e8	+	2.14828e8i	2.71445	+	1.59125i
$514$		0			0
$515$		−	3.06661e7i		−	0.224511i
$516$		0			0
$517$	−8.30560e7			−0.601034
$518$		0			0
$519$	2.95999e8			2.11733
$520$		0			0
$521$			1.55803e8i			1.10170i	0.834605	+	0.550849i	$0.185696\pi$
$521$			1.55803e8i			1.10170i	−0.834605	+	0.550849i	$0.814304\pi$
$522$		0			0
$523$			3.83749e7i			0.268251i	0.990964	+	0.134126i	$0.0428226\pi$
$523$			3.83749e7i			0.268251i	−0.990964	+	0.134126i	$0.957177\pi$
$524$		0			0
$525$			1.82018e7i			0.125788i
$526$		0			0
$527$		−	7.34139e7i		−	0.501587i
$528$		0			0
$529$	1.51800e8			1.02543
$530$		0			0
$531$			4.61059e8i			3.07945i
$532$		0			0
$533$	−8.42115e6			−0.0556147
$534$		0			0
$535$			5.84990e7i			0.382021i
$536$		0			0
$537$	−2.60391e8			−1.68152
$538$		0			0
$539$	−1.60117e8			−1.02252
$540$		0			0
$541$	2.36964e8			1.49655			0.748274	−	0.663390i	$-0.230883\pi$
$541$	2.36964e8			1.49655			0.748274	+	0.663390i	$0.230883\pi$
$542$		0			0
$543$	−5.27294e7			−0.329347
$544$		0			0
$545$		−	7.58193e7i		−	0.468371i
$546$		0			0
$547$			1.85456e8i			1.13313i	0.824018	+	0.566563i	$0.191727\pi$
$547$			1.85456e8i			1.13313i	−0.824018	+	0.566563i	$0.808273\pi$
$548$		0			0
$549$	−8.29199e8			−5.01120
$550$		0			0
$551$	1.54228e8	+	9.04111e7i	0.921953	+	0.540464i
$552$		0			0
$553$			1.52684e7i			0.0902857i
$554$		0			0
$555$	1.76817e8			1.03430
$556$		0			0
$557$	−4.79023e7			−0.277198			−0.138599	−	0.990349i	$-0.544260\pi$
$557$	−4.79023e7			−0.277198			−0.138599	+	0.990349i	$0.544260\pi$
$558$		0			0
$559$		−	1.75817e7i		−	0.100653i
$560$		0			0
$561$			1.93041e8i			1.09335i
$562$		0			0
$563$			6.64627e7i			0.372437i	0.982508	+	0.186219i	$0.0596233\pi$
$563$			6.64627e7i			0.372437i	−0.982508	+	0.186219i	$0.940377\pi$
$564$		0			0
$565$			4.64911e7i			0.257766i
$566$		0			0
$567$	4.49441e7			0.246561
$568$		0			0
$569$			2.63899e8i			1.43252i	0.697834	+	0.716259i	$0.254147\pi$
$569$			2.63899e8i			1.43252i	−0.697834	+	0.716259i	$0.745853\pi$
$570$		0			0
$571$	−2.72456e8			−1.46348			−0.731742	−	0.681581i	$-0.761293\pi$
$571$	−2.72456e8			−1.46348			−0.731742	+	0.681581i	$0.761293\pi$
$572$		0			0
$573$		−	1.66700e8i		−	0.886076i
$574$		0			0
$575$	2.42966e8			1.27803
$576$		0			0
$577$	1.43860e8			0.748879			0.374439	−	0.927251i	$-0.377835\pi$
$577$	1.43860e8			0.748879			0.374439	+	0.927251i	$0.377835\pi$
$578$		0			0
$579$	−1.74405e7			−0.0898513
$580$		0			0
$581$	2.20128e7			0.112240
$582$		0			0
$583$		−	1.86088e6i		−	0.00939103i
$584$		0			0
$585$		−	1.21014e7i		−	0.0604463i
$586$		0			0
$587$	−2.03952e8			−1.00836			−0.504179	−	0.863599i	$-0.668205\pi$
$587$	−2.03952e8			−1.00836			−0.504179	+	0.863599i	$0.668205\pi$
$588$		0			0
$589$	1.58782e8	+	9.30807e7i	0.777062	+	0.455527i
$590$		0			0
$591$		−	5.45307e8i		−	2.64167i
$592$		0			0
$593$	−1.29678e7			−0.0621872			−0.0310936	−	0.999516i	$-0.509899\pi$
$593$	−1.29678e7			−0.0621872			−0.0310936	+	0.999516i	$0.509899\pi$
$594$		0			0
$595$	−2.74738e6			−0.0130427
$596$		0			0
$597$		−	3.42417e8i		−	1.60928i
$598$		0			0
$599$		−	3.81320e8i		−	1.77423i	−0.461549	−	0.887114i	$-0.652706\pi$
$599$		−	3.81320e8i		−	1.77423i	0.461549	−	0.887114i	$-0.347294\pi$
$600$		0			0
$601$			1.47262e8i			0.678370i	0.940720	+	0.339185i	$0.110151\pi$
$601$			1.47262e8i			0.678370i	−0.940720	+	0.339185i	$0.889849\pi$
$602$		0			0
$603$			3.02987e8i			1.38188i
$604$		0			0
$605$	4.02315e6			0.0181677
$606$		0			0
$607$		−	9.07363e7i		−	0.405709i	−0.979209	−	0.202855i	$-0.934978\pi$
$607$		−	9.07363e7i		−	0.405709i	0.979209	−	0.202855i	$-0.0650219\pi$
$608$		0			0
$609$	3.38109e7			0.149694
$610$		0			0
$611$		−	9.53403e6i		−	0.0417977i
$612$		0			0
$613$	−1.02413e8			−0.444603			−0.222302	−	0.974978i	$-0.571357\pi$
$613$	−1.02413e8			−0.444603			−0.222302	+	0.974978i	$0.571357\pi$
$614$		0			0
$615$	1.10359e8			0.474443
$616$		0			0
$617$	−2.66718e8			−1.13552			−0.567762	−	0.823193i	$-0.692191\pi$
$617$	−2.66718e8			−1.13552			−0.567762	+	0.823193i	$0.692191\pi$
$618$		0			0
$619$	4.23626e8			1.78612			0.893061	−	0.449936i	$-0.148553\pi$
$619$	4.23626e8			1.78612			0.893061	+	0.449936i	$0.148553\pi$
$620$		0			0
$621$		−	1.07240e9i		−	4.47799i
$622$		0			0
$623$			1.48558e7i			0.0614375i
$624$		0			0
$625$	1.71985e8			0.704451
$626$		0			0
$627$	−4.17515e8	−	2.44754e8i	−1.69383	−	0.992951i
$628$		0			0
$629$		−	2.35008e8i		−	0.944344i
$630$		0			0
$631$	1.14563e8			0.455991			0.227996	−	0.973662i	$-0.426783\pi$
$631$	1.14563e8			0.455991			0.227996	+	0.973662i	$0.426783\pi$
$632$		0			0
$633$	3.04132e8			1.19909
$634$		0			0
$635$			4.18871e7i			0.163591i
$636$		0			0
$637$		−	1.83800e7i		−	0.0711093i
$638$		0			0
$639$			7.04477e8i			2.70000i
$640$		0			0
$641$		−	3.79080e8i		−	1.43932i	−0.694328	−	0.719658i	$-0.744298\pi$
$641$		−	3.79080e8i		−	1.43932i	0.694328	−	0.719658i	$-0.255702\pi$
$642$		0			0
$643$	−1.05304e8			−0.396107			−0.198054	−	0.980191i	$-0.563462\pi$
$643$	−1.05304e8			−0.396107			−0.198054	+	0.980191i	$0.563462\pi$
$644$		0			0
$645$			2.30409e8i			0.858657i
$646$		0			0
$647$	−4.50756e8			−1.66429			−0.832144	−	0.554560i	$-0.812887\pi$
$647$	−4.50756e8			−1.66429			−0.832144	+	0.554560i	$0.812887\pi$
$648$		0			0
$649$		−	3.26878e8i		−	1.19578i
$650$		0			0
$651$	3.48093e7			0.126169
$652$		0			0
$653$	−3.55740e8			−1.27759			−0.638797	−	0.769376i	$-0.720568\pi$
$653$	−3.55740e8			−1.27759			−0.638797	+	0.769376i	$0.720568\pi$
$654$		0			0
$655$	−1.06079e7			−0.0377489
$656$		0			0
$657$	5.50772e8			1.94212
$658$		0			0
$659$		−	1.16578e6i		−	0.00407344i	−0.999998	−	0.00203672i	$-0.999352\pi$
$659$		−	1.16578e6i		−	0.00407344i	0.999998	−	0.00203672i	$-0.000648309\pi$
$660$		0			0
$661$			5.43258e8i			1.88106i	0.339715	+	0.940528i	$0.389669\pi$
$661$			5.43258e8i			1.88106i	−0.339715	+	0.940528i	$0.610331\pi$
$662$		0			0
$663$	−2.21592e7			−0.0760351
$664$		0			0
$665$	3.48337e6	−	5.94212e6i	0.0118450	−	0.0202058i
$666$		0			0
$667$		−	4.51323e8i		−	1.52093i
$668$		0			0
$669$	7.40005e8			2.47148
$670$		0			0
$671$	5.87878e8			1.94590
$672$		0			0
$673$			1.90275e8i			0.624217i	0.950046	+	0.312109i	$0.101035\pi$
$673$			1.90275e8i			0.624217i	−0.950046	+	0.312109i	$0.898965\pi$
$674$		0			0
$675$		−	8.69001e8i		−	2.82559i
$676$		0			0
$677$		−	4.81730e7i		−	0.155252i	−0.996983	−	0.0776261i	$-0.975266\pi$
$677$		−	4.81730e7i		−	0.155252i	0.996983	−	0.0776261i	$-0.0247340\pi$
$678$		0			0
$679$		−	2.53843e7i		−	0.0810879i
$680$		0			0
$681$	−4.84481e8			−1.53403
$682$		0			0
$683$			1.89512e8i			0.594806i	0.954752	+	0.297403i	$0.0961205\pi$
$683$			1.89512e8i			0.594806i	−0.954752	+	0.297403i	$0.903879\pi$
$684$		0			0
$685$	−8.50747e6			−0.0264684
$686$		0			0
$687$		−	6.79317e8i		−	2.09509i
$688$		0			0
$689$	213612.			0.000653081
$690$		0			0
$691$	−6.88902e7			−0.208796			−0.104398	−	0.994536i	$-0.533292\pi$
$691$	−6.88902e7			−0.208796			−0.104398	+	0.994536i	$0.533292\pi$
$692$		0			0
$693$	−6.64365e7			−0.199621
$694$		0			0
$695$	−7.29261e7			−0.217234
$696$		0			0
$697$		−	1.46679e8i		−	0.433180i
$698$		0			0
$699$			6.93715e8i			2.03119i
$700$		0			0
$701$	3.42895e8			0.995420			0.497710	−	0.867343i	$-0.334174\pi$
$701$	3.42895e8			0.995420			0.497710	+	0.867343i	$0.334174\pi$
$702$		0			0
$703$	5.08283e8	+	2.97964e8i	1.46298	+	0.857625i
$704$		0			0
$705$			1.24944e8i			0.356572i
$706$		0			0
$707$	−1.34362e7			−0.0380206
$708$		0			0
$709$	−1.95474e8			−0.548468			−0.274234	−	0.961663i	$-0.588424\pi$
$709$	−1.95474e8			−0.548468			−0.274234	+	0.961663i	$0.588424\pi$
$710$		0			0
$711$		−	1.17141e9i		−	3.25912i
$712$		0			0
$713$		−	4.64650e8i		−	1.28191i
$714$		0			0
$715$			8.57958e6i			0.0234719i
$716$		0			0
$717$			1.15359e9i			3.12964i
$718$		0			0
$719$	2.31962e8			0.624066			0.312033	−	0.950071i	$-0.398990\pi$
$719$	2.31962e8			0.624066			0.312033	+	0.950071i	$0.398990\pi$
$720$		0			0
$721$		−	1.93256e7i		−	0.0515618i
$722$		0			0
$723$	−2.66749e8			−0.705810
$724$		0			0
$725$		−	3.65721e8i		−	0.959702i
$726$		0			0
$727$	3.54964e8			0.923806			0.461903	−	0.886930i	$-0.347167\pi$
$727$	3.54964e8			0.923806			0.461903	+	0.886930i	$0.347167\pi$
$728$		0			0
$729$	−1.12005e9			−2.89103
$730$		0			0
$731$	3.06236e8			0.783980
$732$		0			0
$733$	−4.30151e8			−1.09222			−0.546108	−	0.837715i	$-0.683891\pi$
$733$	−4.30151e8			−1.09222			−0.546108	+	0.837715i	$0.683891\pi$
$734$		0			0
$735$			2.40870e8i			0.606625i
$736$		0			0
$737$		−	2.14809e8i		−	0.536599i
$738$		0			0
$739$	−2.90982e8			−0.720995			−0.360497	−	0.932760i	$-0.617393\pi$
$739$	−2.90982e8			−0.720995			−0.360497	+	0.932760i	$0.617393\pi$
$740$		0			0
$741$	2.80955e7	−	4.79267e7i	0.0690529	−	0.117794i
$742$		0			0
$743$		−	3.06553e8i		−	0.747375i	−0.927555	−	0.373688i	$-0.878093\pi$
$743$		−	3.06553e8i		−	0.747375i	0.927555	−	0.373688i	$-0.121907\pi$
$744$		0			0
$745$	−9.63636e7			−0.233047
$746$		0			0
$747$	−1.68884e9			−4.05161
$748$		0			0
$749$			3.68658e7i			0.0877361i
$750$		0			0
$751$		−	4.56545e8i		−	1.07786i	−0.842349	−	0.538932i	$-0.818828\pi$
$751$		−	4.56545e8i		−	1.07786i	0.842349	−	0.538932i	$-0.181172\pi$
$752$		0			0
$753$			5.60670e7i			0.131317i
$754$		0			0
$755$			1.46231e7i			0.0339781i
$756$		0			0
$757$	−2.12640e7			−0.0490182			−0.0245091	−	0.999700i	$-0.507802\pi$
$757$	−2.12640e7			−0.0490182			−0.0245091	+	0.999700i	$0.507802\pi$
$758$		0			0
$759$			1.22179e9i			2.79429i
$760$		0			0
$761$	2.83195e8			0.642586			0.321293	−	0.946980i	$-0.395883\pi$
$761$	2.83195e8			0.642586			0.321293	+	0.946980i	$0.395883\pi$
$762$		0			0
$763$		−	4.77809e7i		−	0.107567i
$764$		0			0
$765$	2.10782e8			0.470813
$766$		0			0
$767$	3.75224e7			0.0831581
$768$		0			0
$769$	1.63966e8			0.360557			0.180278	−	0.983616i	$-0.442300\pi$
$769$	1.63966e8			0.360557			0.180278	+	0.983616i	$0.442300\pi$
$770$		0			0
$771$	7.11170e8			1.55171
$772$		0			0
$773$		−	2.80416e8i		−	0.607106i	−0.952815	−	0.303553i	$-0.901827\pi$
$773$		−	2.80416e8i		−	0.607106i	0.952815	−	0.303553i	$-0.0981730\pi$
$774$		0			0
$775$		−	3.76520e8i		−	0.808878i
$776$		0			0
$777$	1.11429e8			0.237539
$778$		0			0
$779$	3.17242e8	+	1.85972e8i	0.671086	+	0.393402i
$780$		0			0
$781$		−	4.99454e8i		−	1.04844i
$782$		0			0
$783$	−1.61422e9			−3.36261
$784$		0			0
$785$	6.40181e6			0.0132341
$786$		0			0
$787$		−	1.78603e8i		−	0.366407i	−0.983075	−	0.183203i	$-0.941353\pi$
$787$		−	1.78603e8i		−	0.366407i	0.983075	−	0.183203i	$-0.0586467\pi$
$788$		0			0
$789$			6.10644e8i			1.24325i
$790$		0			0
$791$			2.92985e7i			0.0591992i
$792$		0			0
$793$			6.74828e7i			0.135324i
$794$		0			0
$795$	−2.79938e6			−0.00557136
$796$		0			0
$797$		−	5.39911e8i		−	1.06647i	−0.845968	−	0.533233i	$-0.820977\pi$
$797$		−	5.39911e8i		−	1.06647i	0.845968	−	0.533233i	$-0.179023\pi$
$798$		0			0
$799$	1.66063e8			0.325561
$800$		0			0
$801$		−	1.13976e9i		−	2.21776i
$802$		0			0
$803$	−3.90482e8			−0.754144
$804$		0			0
$805$	−1.73886e7			−0.0333333
$806$		0			0
$807$	−1.46919e9			−2.79548
$808$		0			0
$809$	−4.24059e8			−0.800904			−0.400452	−	0.916318i	$-0.631147\pi$
$809$	−4.24059e8			−0.800904			−0.400452	+	0.916318i	$0.631147\pi$
$810$		0			0
$811$			1.50159e8i			0.281507i	0.990045	+	0.140754i	$0.0449525\pi$
$811$			1.50159e8i			0.281507i	−0.990045	+	0.140754i	$0.955047\pi$
$812$		0			0
$813$			1.25178e7i			0.0232946i
$814$		0			0
$815$	−3.05707e8			−0.564720
$816$		0			0
$817$	−3.88274e8	+	6.62339e8i	−0.711987	+	1.21455i
$818$		0			0
$819$		−	7.62627e6i		−	0.0138823i
$820$		0			0
$821$	−1.77116e8			−0.320057			−0.160029	−	0.987112i	$-0.551159\pi$
$821$	−1.77116e8			−0.320057			−0.160029	+	0.987112i	$0.551159\pi$
$822$		0			0
$823$	5.42120e8			0.972514			0.486257	−	0.873816i	$-0.338362\pi$
$823$	5.42120e8			0.972514			0.486257	+	0.873816i	$0.338362\pi$
$824$		0			0
$825$			9.90054e8i			1.76318i
$826$		0			0
$827$			2.17728e8i			0.384945i	0.981302	+	0.192472i	$0.0616506\pi$
$827$			2.17728e8i			0.384945i	−0.981302	+	0.192472i	$0.938349\pi$
$828$		0			0
$829$		−	2.27590e8i		−	0.399474i	−0.979850	−	0.199737i	$-0.935991\pi$
$829$		−	2.27590e8i		−	0.399474i	0.979850	−	0.199737i	$-0.0640088\pi$
$830$		0			0
$831$			1.89647e9i			3.30477i
$832$		0			0
$833$	3.20140e8			0.553867
$834$		0			0
$835$		−	4.39189e7i		−	0.0754383i
$836$		0			0
$837$	−1.66188e9			−2.83415
$838$		0			0
$839$		−	3.59559e8i		−	0.608814i	−0.952542	−	0.304407i	$-0.901542\pi$
$839$		−	3.59559e8i		−	0.608814i	0.952542	−	0.304407i	$-0.0984582\pi$
$840$		0			0
$841$	−8.45237e7			−0.142099
$842$		0			0
$843$	1.43397e9			2.39364
$844$		0			0
$845$	1.91694e8			0.317716
$846$		0			0
$847$	2.53537e6			0.00417245
$848$		0			0
$849$		−	1.07969e9i		−	1.76432i
$850$		0			0
$851$		−	1.48741e9i		−	2.41346i
$852$		0			0
$853$	8.96407e8			1.44430			0.722151	−	0.691736i	$-0.243154\pi$
$853$	8.96407e8			1.44430			0.722151	+	0.691736i	$0.243154\pi$
$854$		0			0
$855$	−2.67248e8	+	4.55886e8i	−0.427579	+	0.729387i
$856$		0			0
$857$			2.39428e7i			0.0380394i	0.999819	+	0.0190197i	$0.00605452\pi$
$857$			2.39428e7i			0.0380394i	−0.999819	+	0.0190197i	$0.993945\pi$
$858$		0			0
$859$	5.73924e8			0.905471			0.452736	−	0.891645i	$-0.350448\pi$
$859$	5.73924e8			0.905471			0.452736	+	0.891645i	$0.350448\pi$
$860$		0			0
$861$	6.95479e7			0.108962
$862$		0			0
$863$			9.59089e8i			1.49220i	0.665835	+	0.746099i	$0.268076\pi$
$863$			9.59089e8i			1.49220i	−0.665835	+	0.746099i	$0.731924\pi$
$864$		0			0
$865$			2.29141e8i			0.354042i
$866$		0			0
$867$			8.58706e8i			1.31761i
$868$		0			0
$869$			8.30497e8i			1.26555i
$870$		0			0
$871$	2.46580e7			0.0373167
$872$		0			0
$873$			1.94751e9i			2.92710i
$874$		0			0
$875$	−2.97813e7			−0.0444549
$876$		0			0
$877$			2.35518e8i			0.349160i	0.984643	+	0.174580i	$0.0558568\pi$
$877$			2.35518e8i			0.349160i	−0.984643	+	0.174580i	$0.944143\pi$
$878$		0			0
$879$	5.81933e8			0.856853
$880$		0			0
$881$	6.93176e8			1.01372			0.506858	−	0.862030i	$-0.330807\pi$
$881$	6.93176e8			1.01372			0.506858	+	0.862030i	$0.330807\pi$
$882$		0			0
$883$	9.08954e8			1.32026			0.660130	−	0.751151i	$-0.270501\pi$
$883$	9.08954e8			1.32026			0.660130	+	0.751151i	$0.270501\pi$
$884$		0			0
$885$	−4.91732e8			−0.709412
$886$		0			0
$887$		−	1.21284e9i		−	1.73794i	−0.494868	−	0.868968i	$-0.664783\pi$
$887$		−	1.21284e9i		−	1.73794i	0.494868	−	0.868968i	$-0.335217\pi$
$888$		0			0
$889$			2.63970e7i			0.0375707i
$890$		0			0
$891$	2.44465e9			3.45608
$892$		0			0
$893$	−2.10549e8	+	3.59166e8i	−0.295665	+	0.504361i
$894$		0			0
$895$		−	2.01576e8i		−	0.281170i
$896$		0			0
$897$	−1.40250e8			−0.194323
$898$		0			0
$899$	−6.99407e8			−0.962611
$900$		0			0
$901$			3.72066e6i			0.00508682i
$902$		0			0
$903$			1.45202e8i			0.197202i
$904$		0			0
$905$		−	4.08193e7i		−	0.0550706i
$906$		0			0
$907$			6.70720e8i			0.898916i	0.893301	+	0.449458i	$0.148383\pi$
$907$			6.70720e8i			0.898916i	−0.893301	+	0.449458i	$0.851617\pi$
$908$		0			0
$909$	1.03084e9			1.37246
$910$		0			0
$911$		−	4.11499e8i		−	0.544269i	−0.962259	−	0.272134i	$-0.912270\pi$
$911$		−	4.11499e8i		−	0.544269i	0.962259	−	0.272134i	$-0.0877296\pi$
$912$		0			0
$913$	1.19734e9			1.57328
$914$		0			0
$915$		−	8.84363e8i		−	1.15443i
$916$		0			0
$917$	−6.68502e6			−0.00866952
$918$		0			0
$919$	−9.76999e8			−1.25877			−0.629387	−	0.777092i	$-0.716694\pi$
$919$	−9.76999e8			−1.25877			−0.629387	+	0.777092i	$0.716694\pi$
$920$		0			0
$921$	−3.93207e7			−0.0503318
$922$		0			0
$923$	5.73325e7			0.0729115
$924$		0			0
$925$		−	1.20529e9i		−	1.52288i
$926$		0			0
$927$			1.48268e9i			1.86127i
$928$		0			0
$929$	7.27336e8			0.907168			0.453584	−	0.891213i	$-0.350145\pi$
$929$	7.27336e8			0.907168			0.453584	+	0.891213i	$0.350145\pi$
$930$		0			0
$931$	−4.05903e8	+	6.92410e8i	−0.503005	+	0.858054i
$932$		0			0
$933$		−	9.20936e8i		−	1.13393i
$934$		0			0
$935$	−1.49438e8			−0.182821
$936$		0			0
$937$	−7.48977e8			−0.910437			−0.455218	−	0.890380i	$-0.650439\pi$
$937$	−7.48977e8			−0.910437			−0.455218	+	0.890380i	$0.650439\pi$
$938$		0			0
$939$			1.24365e9i			1.50210i
$940$		0			0
$941$			7.23039e7i			0.0867746i	0.999058	+	0.0433873i	$0.0138149\pi$
$941$			7.23039e7i			0.0867746i	−0.999058	+	0.0433873i	$0.986185\pi$
$942$		0			0
$943$		−	9.28356e8i		−	1.10708i
$944$		0			0
$945$			6.21927e7i			0.0736961i
$946$		0			0
$947$	−1.18089e9			−1.39046			−0.695231	−	0.718786i	$-0.744698\pi$
$947$	−1.18089e9			−1.39046			−0.695231	+	0.718786i	$0.744698\pi$
$948$		0			0
$949$		−	4.48236e7i		−	0.0524455i
$950$		0			0
$951$	1.81158e9			2.10628
$952$		0			0
$953$		−	5.02689e8i		−	0.580792i	−0.956907	−	0.290396i	$-0.906213\pi$
$953$		−	5.02689e8i		−	0.580792i	0.956907	−	0.290396i	$-0.0937870\pi$
$954$		0			0
$955$	1.29047e8			0.148162
$956$		0			0
$957$	1.83908e9			2.09829
$958$		0			0
$959$	−5.36136e6			−0.00607882
$960$		0			0
$961$	1.67445e8			0.188670
$962$		0			0
$963$		−	2.82838e9i		−	3.16708i
$964$		0			0
$965$		−	1.35012e7i		−	0.0150242i
$966$		0			0
$967$	7.99455e8			0.884127			0.442063	−	0.896984i	$-0.354247\pi$
$967$	7.99455e8			0.884127			0.442063	+	0.896984i	$0.354247\pi$
$968$		0			0
$969$	8.34783e8	+	4.89364e8i	0.917492	+	0.537849i
$970$		0			0
$971$		−	1.39572e9i		−	1.52455i	−0.647255	−	0.762274i	$-0.724083\pi$
$971$		−	1.39572e9i		−	1.52455i	0.647255	−	0.762274i	$-0.275917\pi$
$972$		0			0
$973$	−4.59577e7			−0.0498907
$974$		0			0
$975$	−1.13649e8			−0.122617
$976$		0			0
$977$		−	1.42354e9i		−	1.52646i	−0.646129	−	0.763228i	$-0.723613\pi$
$977$		−	1.42354e9i		−	1.52646i	0.646129	−	0.763228i	$-0.276387\pi$
$978$		0			0
$979$			8.08055e8i			0.861178i
$980$		0			0
$981$			3.66580e9i			3.88295i
$982$		0			0
$983$		−	3.32507e8i		−	0.350058i	−0.984563	−	0.175029i	$-0.943998\pi$
$983$		−	3.32507e8i		−	0.350058i	0.984563	−	0.175029i	$-0.0560019\pi$
$984$		0			0
$985$	4.22138e8			0.441718
$986$		0			0
$987$			7.87388e7i			0.0818912i
$988$		0			0
$989$	1.93823e9			2.00362
$990$		0			0
$991$			8.50991e8i			0.874388i	0.899367	+	0.437194i	$0.144028\pi$
$991$			8.50991e8i			0.874388i	−0.899367	+	0.437194i	$0.855972\pi$
$992$		0			0
$993$	−1.40141e9			−1.43126
$994$		0			0
$995$	2.65074e8			0.269091
$996$		0			0
$997$	1.54040e9			1.55435			0.777174	−	0.629286i	$-0.216653\pi$
$997$	1.54040e9			1.55435			0.777174	+	0.629286i	$0.216653\pi$
$998$		0			0
$999$	−5.31990e9			−5.33589

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.7.e.e.113.1		10
4.3	odd	2		38.7.b.a.37.5	✓	10
12.11	even	2		342.7.d.a.37.7		10
19.18	odd	2	inner	304.7.e.e.113.10		10
76.75	even	2		38.7.b.a.37.6	yes	10
228.227	odd	2		342.7.d.a.37.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.7.b.a.37.5	✓	10	4.3	odd	2
38.7.b.a.37.6	yes	10	76.75	even	2
304.7.e.e.113.1		10	1.1	even	1	trivial
304.7.e.e.113.10		10	19.18	odd	2	inner
342.7.d.a.37.2		10	228.227	odd	2
342.7.d.a.37.7		10	12.11	even	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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-51.5658i q^{3} +39.9185 q^{5} +25.1565 q^{7} -1930.03 q^{9} +O(q^{10})\)	\(q-51.5658i q^{3} +39.9185 q^{5} +25.1565 q^{7} -1930.03 q^{9} +1368.34 q^{11} +157.072i q^{13} -2058.43i q^{15} -2735.86 q^{17} +(3468.77 - 5917.22i) q^{19} -1297.21i q^{21} -17315.8 q^{23} -14031.5 q^{25} +61932.1i q^{27} +26064.3i q^{29} +26833.9i q^{31} -70559.4i q^{33} +1004.21 q^{35} +85898.9i q^{37} +8099.54 q^{39} +53613.3i q^{41} -111934. q^{43} -77044.0 q^{45} -60698.5 q^{47} -117016. q^{49} +141077. i q^{51} -1359.96i q^{53} +54622.0 q^{55} +(-305126. - 178870. i) q^{57} -238887. i q^{59} +429630. q^{61} -48552.7 q^{63} +6270.08i q^{65} -156985. i q^{67} +892901. i q^{69} -365008. i q^{71} -285370. q^{73} +723546. i q^{75} +34422.5 q^{77} +606939. i q^{79} +1.78659e6 q^{81} +875034. q^{83} -109212. q^{85} +1.34403e6 q^{87} +590538. i q^{89} +3951.37i q^{91} +1.38371e6 q^{93} +(138468. - 236207. i) q^{95} -1.00906e6i q^{97} -2.64093e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 112 q^{5} + 224 q^{7} - 2890 q^{9}+O(q^{10}) \) 10 * q - 112 * q^5 + 224 * q^7 - 2890 * q^9	\( 10 q - 112 q^{5} + 224 q^{7} - 2890 q^{9} - 3644 q^{11} - 10420 q^{17} + 17230 q^{19} - 37712 q^{23} - 52078 q^{25} + 161720 q^{35} + 78876 q^{39} - 6308 q^{43} + 309808 q^{45} - 322220 q^{47} - 235770 q^{49} + 377880 q^{55} + 24228 q^{57} + 426304 q^{61} + 517916 q^{63} - 786076 q^{73} + 2303716 q^{77} + 5261090 q^{81} + 101500 q^{83} - 1261380 q^{85} + 2460732 q^{87} - 2827032 q^{93} - 3106292 q^{95} - 1061428 q^{99}+O(q^{100}) \) 10 * q - 112 * q^5 + 224 * q^7 - 2890 * q^9 - 3644 * q^11 - 10420 * q^17 + 17230 * q^19 - 37712 * q^23 - 52078 * q^25 + 161720 * q^35 + 78876 * q^39 - 6308 * q^43 + 309808 * q^45 - 322220 * q^47 - 235770 * q^49 + 377880 * q^55 + 24228 * q^57 + 426304 * q^61 + 517916 * q^63 - 786076 * q^73 + 2303716 * q^77 + 5261090 * q^81 + 101500 * q^83 - 1261380 * q^85 + 2460732 * q^87 - 2827032 * q^93 - 3106292 * q^95 - 1061428 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more