LMFDB - Embedded newform 304.7.e.f.113.2

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:	$30$
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.2
Character	$\chi$	$=$	304.113
Dual form			304.7.e.f.113.29

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-48.7582i q^{3} -200.006 q^{5} +536.711 q^{7} -1648.37 q^{9} +O(q^{10})$	$q-48.7582i q^{3} -200.006 q^{5} +536.711 q^{7} -1648.37 q^{9} +528.705 q^{11} +1938.33i q^{13} +9751.93i q^{15} +6548.79 q^{17} +(2289.83 - 6465.49i) q^{19} -26169.1i q^{21} +18877.3 q^{23} +24377.3 q^{25} +44826.7i q^{27} +21391.9i q^{29} -41792.2i q^{31} -25778.7i q^{33} -107345. q^{35} +25164.2i q^{37} +94509.4 q^{39} -13690.7i q^{41} +11123.7 q^{43} +329683. q^{45} +172525. q^{47} +170410. q^{49} -319308. i q^{51} +120834. i q^{53} -105744. q^{55} +(-315246. - 111648. i) q^{57} -30982.4i q^{59} +96479.8 q^{61} -884697. q^{63} -387676. i q^{65} +393271. i q^{67} -920424. i q^{69} +259029. i q^{71} -24478.8 q^{73} -1.18860e6i q^{75} +283762. q^{77} +429564. i q^{79} +984011. q^{81} -1.05440e6 q^{83} -1.30980e6 q^{85} +1.04303e6 q^{87} -442773. i q^{89} +1.04032e6i q^{91} -2.03771e6 q^{93} +(-457980. + 1.29314e6i) q^{95} -650087. i q^{97} -871500. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 720 q^{7} - 8670 q^{9}+O(q^{10}) $ 30 * q + 720 * q^7 - 8670 * q^9	$ 30 q + 720 q^{7} - 8670 q^{9} + 2524 q^{11} + 9700 q^{17} - 4014 q^{19} + 39376 q^{23} + 110742 q^{25} + 19976 q^{35} + 266500 q^{39} + 106788 q^{43} - 91360 q^{45} - 222756 q^{47} + 593586 q^{49} - 540936 q^{55} - 545972 q^{57} - 242640 q^{61} + 377716 q^{63} + 545964 q^{73} - 272356 q^{77} + 2189926 q^{81} - 1542652 q^{83} - 826908 q^{85} + 2729572 q^{87} - 2139912 q^{93} - 2142716 q^{95} + 293012 q^{99}+O(q^{100}) $ 30 * q + 720 * q^7 - 8670 * q^9 + 2524 * q^11 + 9700 * q^17 - 4014 * q^19 + 39376 * q^23 + 110742 * q^25 + 19976 * q^35 + 266500 * q^39 + 106788 * q^43 - 91360 * q^45 - 222756 * q^47 + 593586 * q^49 - 540936 * q^55 - 545972 * q^57 - 242640 * q^61 + 377716 * q^63 + 545964 * q^73 - 272356 * q^77 + 2189926 * q^81 - 1542652 * q^83 - 826908 * q^85 + 2729572 * q^87 - 2139912 * q^93 - 2142716 * q^95 + 293012 * 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
$3$		−	48.7582i		−	1.80586i	−0.429787	−	0.902930i	$-0.641411\pi$
$3$		−	48.7582i		−	1.80586i	0.429787	−	0.902930i	$-0.358589\pi$
$4$		0			0
$5$	−200.006			−1.60005			−0.800023	−	0.599969i	$-0.795180\pi$
$5$	−200.006			−1.60005			−0.800023	+	0.599969i	$0.795180\pi$
$6$		0			0
$7$	536.711			1.56476			0.782378	−	0.622804i	$-0.214007\pi$
$7$	536.711			1.56476			0.782378	+	0.622804i	$0.214007\pi$
$8$		0			0
$9$	−1648.37			−2.26113
$10$		0			0
$11$	528.705			0.397224			0.198612	−	0.980078i	$-0.436357\pi$
$11$	528.705			0.397224			0.198612	+	0.980078i	$0.436357\pi$
$12$		0			0
$13$			1938.33i			0.882260i	0.897443	+	0.441130i	$0.145422\pi$
$13$			1938.33i			0.882260i	−0.897443	+	0.441130i	$0.854578\pi$
$14$		0			0
$15$			9751.93i			2.88946i
$16$		0			0
$17$	6548.79			1.33295			0.666476	−	0.745527i	$-0.267802\pi$
$17$	6548.79			1.33295			0.666476	+	0.745527i	$0.267802\pi$
$18$		0			0
$19$	2289.83	−	6465.49i	0.333843	−	0.942629i
$19$	2289.83	−	6465.49i	0.333843	−	0.942629i
$20$		0			0
$21$		−	26169.1i		−	2.82573i
$22$		0			0
$23$	18877.3			1.55152			0.775758	−	0.631030i	$-0.217368\pi$
$23$	18877.3			1.55152			0.775758	+	0.631030i	$0.217368\pi$
$24$		0			0
$25$	24377.3			1.56015
$26$		0			0
$27$			44826.7i			2.27743i
$28$		0			0
$29$			21391.9i			0.877113i	0.898704	+	0.438556i	$0.144510\pi$
$29$			21391.9i			0.877113i	−0.898704	+	0.438556i	$0.855490\pi$
$30$		0			0
$31$		−	41792.2i		−	1.40285i	−0.712746	−	0.701423i	$-0.752549\pi$
$31$		−	41792.2i		−	1.40285i	0.712746	−	0.701423i	$-0.247451\pi$
$32$		0			0
$33$		−	25778.7i		−	0.717331i
$34$		0			0
$35$	−107345.			−2.50368
$36$		0			0
$37$			25164.2i			0.496797i	0.968658	+	0.248398i	$0.0799042\pi$
$37$			25164.2i			0.496797i	−0.968658	+	0.248398i	$0.920096\pi$
$38$		0			0
$39$	94509.4			1.59324
$40$		0			0
$41$		−	13690.7i		−	0.198644i	−0.995055	−	0.0993220i	$-0.968333\pi$
$41$		−	13690.7i		−	0.198644i	0.995055	−	0.0993220i	$-0.0316674\pi$
$42$		0			0
$43$	11123.7			0.139908			0.0699540	−	0.997550i	$-0.477715\pi$
$43$	11123.7			0.139908			0.0699540	+	0.997550i	$0.477715\pi$
$44$		0			0
$45$	329683.			3.61792
$46$		0			0
$47$	172525.			1.66172			0.830860	−	0.556482i	$-0.187849\pi$
$47$	172525.			1.66172			0.830860	+	0.556482i	$0.187849\pi$
$48$		0			0
$49$	170410.			1.44846
$50$		0			0
$51$		−	319308.i		−	2.40712i
$52$		0			0
$53$			120834.i			0.811640i	0.913953	+	0.405820i	$0.133014\pi$
$53$			120834.i			0.811640i	−0.913953	+	0.405820i	$0.866986\pi$
$54$		0			0
$55$	−105744.			−0.635577
$56$		0			0
$57$	−315246.	−	111648.i	−1.70226	−	0.602875i
$58$		0			0
$59$		−	30982.4i		−	0.150855i	−0.997151	−	0.0754273i	$-0.975968\pi$
$59$		−	30982.4i		−	0.150855i	0.997151	−	0.0754273i	$-0.0240321\pi$
$60$		0			0
$61$	96479.8			0.425057			0.212528	−	0.977155i	$-0.431830\pi$
$61$	96479.8			0.425057			0.212528	+	0.977155i	$0.431830\pi$
$62$		0			0
$63$	−884697.			−3.53812
$64$		0			0
$65$		−	387676.i		−	1.41166i
$66$		0			0
$67$			393271.i			1.30758i	0.756676	+	0.653790i	$0.226822\pi$
$67$			393271.i			1.30758i	−0.756676	+	0.653790i	$0.773178\pi$
$68$		0			0
$69$		−	920424.i		−	2.80182i
$70$		0			0
$71$			259029.i			0.723726i	0.932231	+	0.361863i	$0.117859\pi$
$71$			259029.i			0.723726i	−0.932231	+	0.361863i	$0.882141\pi$
$72$		0			0
$73$	−24478.8			−0.0629246			−0.0314623	−	0.999505i	$-0.510016\pi$
$73$	−24478.8			−0.0629246			−0.0314623	+	0.999505i	$0.510016\pi$
$74$		0			0
$75$		−	1.18860e6i		−	2.81741i
$76$		0			0
$77$	283762.			0.621558
$78$		0			0
$79$			429564.i			0.871257i	0.900126	+	0.435629i	$0.143474\pi$
$79$			429564.i			0.871257i	−0.900126	+	0.435629i	$0.856526\pi$
$80$		0			0
$81$	984011.			1.85159
$82$		0			0
$83$	−1.05440e6			−1.84404			−0.922022	−	0.387137i	$-0.873464\pi$
$83$	−1.05440e6			−1.84404			−0.922022	+	0.387137i	$0.873464\pi$
$84$		0			0
$85$	−1.30980e6			−2.13278
$86$		0			0
$87$	1.04303e6			1.58394
$88$		0			0
$89$		−	442773.i		−	0.628074i	−0.949411	−	0.314037i	$-0.898318\pi$
$89$		−	442773.i		−	0.628074i	0.949411	−	0.314037i	$-0.101682\pi$
$90$		0			0
$91$			1.04032e6i			1.38052i
$92$		0			0
$93$	−2.03771e6			−2.53334
$94$		0			0
$95$	−457980.	+	1.29314e6i	−0.534165	+	1.50825i
$96$		0			0
$97$		−	650087.i		−	0.712289i	−0.934431	−	0.356144i	$-0.884091\pi$
$97$		−	650087.i		−	0.712289i	0.934431	−	0.356144i	$-0.115909\pi$
$98$		0			0
$99$	−871500.			−0.898176
$100$		0			0
$101$	833712.			0.809193			0.404596	−	0.914495i	$-0.367412\pi$
$101$	833712.			0.809193			0.404596	+	0.914495i	$0.367412\pi$
$102$		0			0
$103$		−	1.91703e6i		−	1.75436i	−0.480166	−	0.877178i	$-0.659424\pi$
$103$		−	1.91703e6i		−	1.75436i	0.480166	−	0.877178i	$-0.340576\pi$
$104$		0			0
$105$			5.23397e6i			4.52130i
$106$		0			0
$107$		−	635712.i		−	0.518930i	−0.965752	−	0.259465i	$-0.916454\pi$
$107$		−	635712.i		−	0.518930i	0.965752	−	0.259465i	$-0.0835462\pi$
$108$		0			0
$109$		−	213031.i		−	0.164499i	−0.996612	−	0.0822495i	$-0.973790\pi$
$109$		−	213031.i		−	0.164499i	0.996612	−	0.0822495i	$-0.0262104\pi$
$110$		0			0
$111$	1.22696e6			0.897146
$112$		0			0
$113$			1.40999e6i			0.977192i	0.872510	+	0.488596i	$0.162491\pi$
$113$			1.40999e6i			0.977192i	−0.872510	+	0.488596i	$0.837509\pi$
$114$		0			0
$115$	−3.77557e6			−2.48250
$116$		0			0
$117$		−	3.19507e6i		−	1.99491i
$118$		0			0
$119$	3.51481e6			2.08574
$120$		0			0
$121$	−1.49203e6			−0.842213
$122$		0			0
$123$	−667536.			−0.358723
$124$		0			0
$125$	−1.75052e6			−0.896265
$126$		0			0
$127$		−	3.08186e6i		−	1.50453i	−0.658858	−	0.752267i	$-0.728960\pi$
$127$		−	3.08186e6i		−	1.50453i	0.658858	−	0.752267i	$-0.271040\pi$
$128$		0			0
$129$		−	542370.i		−	0.252654i
$130$		0			0
$131$	−1.57928e6			−0.702500			−0.351250	−	0.936282i	$-0.614243\pi$
$131$	−1.57928e6			−0.702500			−0.351250	+	0.936282i	$0.614243\pi$
$132$		0			0
$133$	1.22898e6	−	3.47010e6i	0.522384	−	1.47498i
$134$		0			0
$135$		−	8.96560e6i		−	3.64400i
$136$		0			0
$137$	124358.			0.0483629			0.0241815	−	0.999708i	$-0.492302\pi$
$137$	124358.			0.0483629			0.0241815	+	0.999708i	$0.492302\pi$
$138$		0			0
$139$	3.67389e6			1.36799			0.683994	−	0.729488i	$-0.260242\pi$
$139$	3.67389e6			1.36799			0.683994	+	0.729488i	$0.260242\pi$
$140$		0			0
$141$		−	8.41200e6i		−	3.00083i
$142$		0			0
$143$			1.02480e6i			0.350455i
$144$		0			0
$145$		−	4.27851e6i		−	1.40342i
$146$		0			0
$147$		−	8.30889e6i		−	2.61572i
$148$		0			0
$149$	1.98841e6			0.601099			0.300550	−	0.953766i	$-0.402830\pi$
$149$	1.98841e6			0.601099			0.300550	+	0.953766i	$0.402830\pi$
$150$		0			0
$151$		−	3.64353e6i		−	1.05826i	−0.848541	−	0.529129i	$-0.822519\pi$
$151$		−	3.64353e6i		−	1.05826i	0.848541	−	0.529129i	$-0.177481\pi$
$152$		0			0
$153$	−1.07948e7			−3.01398
$154$		0			0
$155$			8.35868e6i			2.24462i
$156$		0			0
$157$	−5.40292e6			−1.39614			−0.698071	−	0.716029i	$-0.745958\pi$
$157$	−5.40292e6			−1.39614			−0.698071	+	0.716029i	$0.745958\pi$
$158$		0			0
$159$	5.89168e6			1.46571
$160$		0			0
$161$	1.01317e7			2.42775
$162$		0			0
$163$	5.26230e6			1.21510			0.607551	−	0.794281i	$-0.292152\pi$
$163$	5.26230e6			1.21510			0.607551	+	0.794281i	$0.292152\pi$
$164$		0			0
$165$			5.15590e6i			1.14776i
$166$		0			0
$167$			7.23985e6i			1.55446i	0.629216	+	0.777231i	$0.283376\pi$
$167$			7.23985e6i			1.55446i	−0.629216	+	0.777231i	$0.716624\pi$
$168$		0			0
$169$	1.06970e6			0.221617
$170$		0			0
$171$	−3.77448e6	+	1.06575e7i	−0.754865	+	2.13141i
$172$		0			0
$173$			2.04060e6i			0.394113i	0.980392	+	0.197056i	$0.0631382\pi$
$173$			2.04060e6i			0.394113i	−0.980392	+	0.197056i	$0.936862\pi$
$174$		0			0
$175$	1.30836e7			2.44125
$176$		0			0
$177$	−1.51065e6			−0.272422
$178$		0			0
$179$		−	5.60633e6i		−	0.977507i	−0.872422	−	0.488753i	$-0.837452\pi$
$179$		−	5.60633e6i		−	0.977507i	0.872422	−	0.488753i	$-0.162548\pi$
$180$		0			0
$181$		−	6.11913e6i		−	1.03194i	−0.856607	−	0.515970i	$-0.827432\pi$
$181$		−	6.11913e6i		−	1.03194i	0.856607	−	0.515970i	$-0.172568\pi$
$182$		0			0
$183$		−	4.70418e6i		−	0.767593i
$184$		0			0
$185$		−	5.03300e6i		−	0.794898i
$186$		0			0
$187$	3.46238e6			0.529480
$188$		0			0
$189$			2.40590e7i			3.56362i
$190$		0			0
$191$	7.75957e6			1.11362			0.556810	−	0.830640i	$-0.312025\pi$
$191$	7.75957e6			1.11362			0.556810	+	0.830640i	$0.312025\pi$
$192$		0			0
$193$			3.17937e6i			0.442251i	0.975245	+	0.221126i	$0.0709731\pi$
$193$			3.17937e6i			0.442251i	−0.975245	+	0.221126i	$0.929027\pi$
$194$		0			0
$195$	−1.89024e7			−2.54926
$196$		0			0
$197$	4.92471e6			0.644142			0.322071	−	0.946715i	$-0.395621\pi$
$197$	4.92471e6			0.644142			0.322071	+	0.946715i	$0.395621\pi$
$198$		0			0
$199$	−4.04801e6			−0.513668			−0.256834	−	0.966456i	$-0.582679\pi$
$199$	−4.04801e6			−0.513668			−0.256834	+	0.966456i	$0.582679\pi$
$200$		0			0
$201$	1.91752e7			2.36131
$202$		0			0
$203$			1.14813e7i			1.37247i
$204$		0			0
$205$			2.73823e6i			0.317840i
$206$		0			0
$207$	−3.11167e7			−3.50819
$208$		0			0
$209$	1.21065e6	−	3.41834e6i	0.132611	−	0.374435i
$210$		0			0
$211$		−	3.50695e6i		−	0.373321i	−0.982424	−	0.186661i	$-0.940234\pi$
$211$		−	3.50695e6i		−	0.373321i	0.982424	−	0.186661i	$-0.0597665\pi$
$212$		0			0
$213$	1.26298e7			1.30695
$214$		0			0
$215$	−2.22480e6			−0.223859
$216$		0			0
$217$		−	2.24303e7i		−	2.19511i
$218$		0			0
$219$			1.19354e6i			0.113633i
$220$		0			0
$221$			1.26937e7i			1.17601i
$222$		0			0
$223$		−	1.26743e7i		−	1.14290i	−0.820637	−	0.571449i	$-0.806381\pi$
$223$		−	1.26743e7i		−	1.14290i	0.820637	−	0.571449i	$-0.193619\pi$
$224$		0			0
$225$	−4.01828e7			−3.52771
$226$		0			0
$227$			3.87026e6i			0.330874i	0.986220	+	0.165437i	$0.0529034\pi$
$227$			3.87026e6i			0.330874i	−0.986220	+	0.165437i	$0.947097\pi$
$228$		0			0
$229$	−9.28607e6			−0.773260			−0.386630	−	0.922235i	$-0.626361\pi$
$229$	−9.28607e6			−0.773260			−0.386630	+	0.922235i	$0.626361\pi$
$230$		0			0
$231$		−	1.38357e7i		−	1.12245i
$232$		0			0
$233$	−497816.			−0.0393551			−0.0196776	−	0.999806i	$-0.506264\pi$
$233$	−497816.			−0.0393551			−0.0196776	+	0.999806i	$0.506264\pi$
$234$		0			0
$235$	−3.45059e7			−2.65883
$236$		0			0
$237$	2.09448e7			1.57337
$238$		0			0
$239$	1.92128e7			1.40733			0.703666	−	0.710531i	$-0.251545\pi$
$239$	1.92128e7			1.40733			0.703666	+	0.710531i	$0.251545\pi$
$240$		0			0
$241$		−	9.98986e6i		−	0.713688i	−0.934164	−	0.356844i	$-0.883853\pi$
$241$		−	9.98986e6i		−	0.713688i	0.934164	−	0.356844i	$-0.116147\pi$
$242$		0			0
$243$		−	1.53000e7i		−	1.06628i
$244$		0			0
$245$	−3.40830e7			−2.31761
$246$		0			0
$247$	1.25322e7	+	4.43844e6i	0.831644	+	0.294537i
$248$		0			0
$249$			5.14107e7i			3.33009i
$250$		0			0
$251$	2.17603e6			0.137608			0.0688039	−	0.997630i	$-0.478082\pi$
$251$	2.17603e6			0.137608			0.0688039	+	0.997630i	$0.478082\pi$
$252$		0			0
$253$	9.98053e6			0.616300
$254$		0			0
$255$			6.38634e7i			3.85151i
$256$		0			0
$257$			679353.i			0.0400217i	0.999800	+	0.0200109i	$0.00637008\pi$
$257$			679353.i			0.0400217i	−0.999800	+	0.0200109i	$0.993630\pi$
$258$		0			0
$259$			1.35059e7i			0.777366i
$260$		0			0
$261$		−	3.52617e7i		−	1.98327i
$262$		0			0
$263$	−4.30125e6			−0.236444			−0.118222	−	0.992987i	$-0.537719\pi$
$263$	−4.30125e6			−0.236444			−0.118222	+	0.992987i	$0.537719\pi$
$264$		0			0
$265$		−	2.41676e7i		−	1.29866i
$266$		0			0
$267$	−2.15888e7			−1.13421
$268$		0			0
$269$		−	2.41857e7i		−	1.24252i	−0.783606	−	0.621258i	$-0.786622\pi$
$269$		−	2.41857e7i		−	1.24252i	0.783606	−	0.621258i	$-0.213378\pi$
$270$		0			0
$271$	1.72186e7			0.865145			0.432572	−	0.901599i	$-0.357606\pi$
$271$	1.72186e7			0.865145			0.432572	+	0.901599i	$0.357606\pi$
$272$		0			0
$273$	5.07242e7			2.49303
$274$		0			0
$275$	1.28884e7			0.619729
$276$		0			0
$277$	−3.88436e6			−0.182760			−0.0913798	−	0.995816i	$-0.529128\pi$
$277$	−3.88436e6			−0.182760			−0.0913798	+	0.995816i	$0.529128\pi$
$278$		0			0
$279$			6.88888e7i			3.17202i
$280$		0			0
$281$			3.08519e7i			1.39047i	0.718781	+	0.695237i	$0.244700\pi$
$281$			3.08519e7i			1.39047i	−0.718781	+	0.695237i	$0.755300\pi$
$282$		0			0
$283$	−3.00137e6			−0.132422			−0.0662109	−	0.997806i	$-0.521091\pi$
$283$	−3.00137e6			−0.132422			−0.0662109	+	0.997806i	$0.521091\pi$
$284$		0			0
$285$	6.30510e7	+	2.23303e7i	2.72369	+	0.964628i
$286$		0			0
$287$		−	7.34797e6i		−	0.310829i
$288$		0			0
$289$	1.87491e7			0.776759
$290$		0			0
$291$	−3.16971e7			−1.28629
$292$		0			0
$293$			4.32041e7i			1.71760i	0.512310	+	0.858801i	$0.328790\pi$
$293$			4.32041e7i			1.71760i	−0.512310	+	0.858801i	$0.671210\pi$
$294$		0			0
$295$			6.19665e6i			0.241374i
$296$		0			0
$297$			2.37001e7i			0.904650i
$298$		0			0
$299$			3.65904e7i			1.36884i
$300$		0			0
$301$	5.97020e6			0.218922
$302$		0			0
$303$		−	4.06503e7i		−	1.46129i
$304$		0			0
$305$	−1.92965e7			−0.680110
$306$		0			0
$307$		−	4.15697e7i		−	1.43669i	−0.695689	−	0.718343i	$-0.744901\pi$
$307$		−	4.15697e7i		−	1.43669i	0.695689	−	0.718343i	$-0.255099\pi$
$308$		0			0
$309$	−9.34711e7			−3.16812
$310$		0			0
$311$	−6.96783e6			−0.231642			−0.115821	−	0.993270i	$-0.536950\pi$
$311$	−6.96783e6			−0.231642			−0.115821	+	0.993270i	$0.536950\pi$
$312$		0			0
$313$	−9.20071e6			−0.300046			−0.150023	−	0.988682i	$-0.547935\pi$
$313$	−9.20071e6			−0.300046			−0.150023	+	0.988682i	$0.547935\pi$
$314$		0			0
$315$	1.76945e8			5.66116
$316$		0			0
$317$		−	5.11085e7i		−	1.60441i	−0.597049	−	0.802204i	$-0.703660\pi$
$317$		−	5.11085e7i		−	1.60441i	0.597049	−	0.802204i	$-0.296340\pi$
$318$		0			0
$319$			1.13100e7i			0.348410i
$320$		0			0
$321$	−3.09962e7			−0.937116
$322$		0			0
$323$	1.49956e7	−	4.23411e7i	0.444997	−	1.25648i
$324$		0			0
$325$			4.72512e7i			1.37646i
$326$		0			0
$327$	−1.03870e7			−0.297062
$328$		0			0
$329$	9.25960e7			2.60019
$330$		0			0
$331$		−	1.44088e6i		−	0.0397323i	−0.999803	−	0.0198661i	$-0.993676\pi$
$331$		−	1.44088e6i		−	0.0397323i	0.999803	−	0.0198661i	$-0.00632401\pi$
$332$		0			0
$333$		−	4.14799e7i		−	1.12332i
$334$		0			0
$335$		−	7.86566e7i		−	2.09219i
$336$		0			0
$337$			6.83356e7i			1.78549i	0.450563	+	0.892745i	$0.351223\pi$
$337$			6.83356e7i			1.78549i	−0.450563	+	0.892745i	$0.648777\pi$
$338$		0			0
$339$	6.87485e7			1.76467
$340$		0			0
$341$		−	2.20957e7i		−	0.557244i
$342$		0			0
$343$	2.83174e7			0.701732
$344$		0			0
$345$			1.84090e8i			4.48305i
$346$		0			0
$347$	2.59872e7			0.621973			0.310987	−	0.950414i	$-0.399341\pi$
$347$	2.59872e7			0.621973			0.310987	+	0.950414i	$0.399341\pi$
$348$		0			0
$349$	1.61563e7			0.380071			0.190035	−	0.981777i	$-0.439140\pi$
$349$	1.61563e7			0.380071			0.190035	+	0.981777i	$0.439140\pi$
$350$		0			0
$351$	−8.68887e7			−2.00929
$352$		0			0
$353$	5.13046e7			1.16636			0.583179	−	0.812343i	$-0.301808\pi$
$353$	5.13046e7			1.16636			0.583179	+	0.812343i	$0.301808\pi$
$354$		0			0
$355$		−	5.18074e7i		−	1.15799i
$356$		0			0
$357$		−	1.71376e8i		−	3.76656i
$358$		0			0
$359$	−1.41259e7			−0.305305			−0.152652	−	0.988280i	$-0.548781\pi$
$359$	−1.41259e7			−0.305305			−0.152652	+	0.988280i	$0.548781\pi$
$360$		0			0
$361$	−3.65592e7	−	2.96098e7i	−0.777097	−	0.629381i
$362$		0			0
$363$			7.27489e7i			1.52092i
$364$		0			0
$365$	4.89589e6			0.100682
$366$		0			0
$367$	−7.40272e7			−1.49759			−0.748796	−	0.662801i	$-0.769368\pi$
$367$	−7.40272e7			−1.49759			−0.748796	+	0.662801i	$0.769368\pi$
$368$		0			0
$369$			2.25674e7i			0.449160i
$370$		0			0
$371$			6.48532e7i			1.27002i
$372$		0			0
$373$		−	2.98987e6i		−	0.0576137i	−0.999585	−	0.0288068i	$-0.990829\pi$
$373$		−	2.98987e6i		−	0.0576137i	0.999585	−	0.0288068i	$-0.00917077\pi$
$374$		0			0
$375$			8.53522e7i			1.61853i
$376$		0			0
$377$	−4.14645e7			−0.773842
$378$		0			0
$379$			1.44308e7i			0.265078i	0.991178	+	0.132539i	$0.0423130\pi$
$379$			1.44308e7i			0.265078i	−0.991178	+	0.132539i	$0.957687\pi$
$380$		0			0
$381$	−1.50266e8			−2.71698
$382$		0			0
$383$			1.74178e7i			0.310024i	0.987913	+	0.155012i	$0.0495417\pi$
$383$			1.74178e7i			0.310024i	−0.987913	+	0.155012i	$0.950458\pi$
$384$		0			0
$385$	−5.67540e7			−0.994523
$386$		0			0
$387$	−1.83359e7			−0.316351
$388$		0			0
$389$	6.88799e7			1.17016			0.585078	−	0.810977i	$-0.301064\pi$
$389$	6.88799e7			1.17016			0.585078	+	0.810977i	$0.301064\pi$
$390$		0			0
$391$	1.23624e8			2.06810
$392$		0			0
$393$			7.70031e7i			1.26862i
$394$		0			0
$395$		−	8.59153e7i		−	1.39405i
$396$		0			0
$397$	4.93049e7			0.787986			0.393993	−	0.919113i	$-0.371093\pi$
$397$	4.93049e7			0.787986			0.393993	+	0.919113i	$0.371093\pi$
$398$		0			0
$399$	−1.69196e8	−	5.99228e7i	−2.66362	−	0.943352i
$400$		0			0
$401$		−	9.12174e7i		−	1.41463i	−0.706896	−	0.707317i	$-0.749905\pi$
$401$		−	9.12174e7i		−	1.41463i	0.706896	−	0.707317i	$-0.250095\pi$
$402$		0			0
$403$	8.10068e7			1.23767
$404$		0			0
$405$	−1.96808e8			−2.96263
$406$		0			0
$407$			1.33045e7i			0.197340i
$408$		0			0
$409$			2.61534e7i			0.382260i	0.981565	+	0.191130i	$0.0612152\pi$
$409$			2.61534e7i			0.382260i	−0.981565	+	0.191130i	$0.938785\pi$
$410$		0			0
$411$		−	6.06348e6i		−	0.0873367i
$412$		0			0
$413$		−	1.66286e7i		−	0.236051i
$414$		0			0
$415$	2.10886e8			2.95056
$416$		0			0
$417$		−	1.79133e8i		−	2.47039i
$418$		0			0
$419$	1.37278e7			0.186621			0.0933103	−	0.995637i	$-0.470255\pi$
$419$	1.37278e7			0.186621			0.0933103	+	0.995637i	$0.470255\pi$
$420$		0			0
$421$		−	1.13850e8i		−	1.52577i	−0.646537	−	0.762883i	$-0.723783\pi$
$421$		−	1.13850e8i		−	1.52577i	0.646537	−	0.762883i	$-0.276217\pi$
$422$		0			0
$423$	−2.84384e8			−3.75737
$424$		0			0
$425$	1.59642e8			2.07960
$426$		0			0
$427$	5.17818e7			0.665110
$428$		0			0
$429$	4.99676e7			0.632873
$430$		0			0
$431$			3.26676e7i			0.408024i	0.978968	+	0.204012i	$0.0653982\pi$
$431$			3.26676e7i			0.408024i	−0.978968	+	0.204012i	$0.934602\pi$
$432$		0			0
$433$			4.83622e7i			0.595720i	0.954610	+	0.297860i	$0.0962729\pi$
$433$			4.83622e7i			0.595720i	−0.954610	+	0.297860i	$0.903727\pi$
$434$		0			0
$435$	−2.08612e8			−2.53438
$436$		0			0
$437$	4.32259e7	−	1.22051e8i	0.517964	−	1.46250i
$438$		0			0
$439$		−	9.35805e7i		−	1.10609i	−0.833150	−	0.553047i	$-0.813465\pi$
$439$		−	9.35805e7i		−	1.10609i	0.833150	−	0.553047i	$-0.186535\pi$
$440$		0			0
$441$	−2.80898e8			−3.27516
$442$		0			0
$443$	−1.44279e8			−1.65956			−0.829780	−	0.558090i	$-0.811534\pi$
$443$	−1.44279e8			−1.65956			−0.829780	+	0.558090i	$0.811534\pi$
$444$		0			0
$445$			8.85571e7i			1.00495i
$446$		0			0
$447$		−	9.69512e7i		−	1.08550i
$448$		0			0
$449$			1.26362e8i			1.39597i	0.716112	+	0.697986i	$0.245920\pi$
$449$			1.26362e8i			1.39597i	−0.716112	+	0.697986i	$0.754080\pi$
$450$		0			0
$451$		−	7.23836e6i		−	0.0789061i
$452$		0			0
$453$	−1.77652e8			−1.91107
$454$		0			0
$455$		−	2.08070e8i		−	2.20890i
$456$		0			0
$457$	−4.45899e7			−0.467184			−0.233592	−	0.972335i	$-0.575048\pi$
$457$	−4.45899e7			−0.467184			−0.233592	+	0.972335i	$0.575048\pi$
$458$		0			0
$459$			2.93561e8i			3.03571i
$460$		0			0
$461$	6.77217e7			0.691234			0.345617	−	0.938376i	$-0.387670\pi$
$461$	6.77217e7			0.691234			0.345617	+	0.938376i	$0.387670\pi$
$462$		0			0
$463$	1.21686e6			0.0122602			0.00613009	−	0.999981i	$-0.498049\pi$
$463$	1.21686e6			0.0122602			0.00613009	+	0.999981i	$0.498049\pi$
$464$		0			0
$465$	4.07554e8			4.05347
$466$		0			0
$467$	2.06509e6			0.0202763			0.0101382	−	0.999949i	$-0.496773\pi$
$467$	2.06509e6			0.0202763			0.0101382	+	0.999949i	$0.496773\pi$
$468$		0			0
$469$			2.11073e8i			2.04604i
$470$		0			0
$471$			2.63437e8i			2.52124i
$472$		0			0
$473$	5.88114e6			0.0555748
$474$		0			0
$475$	5.58200e7	−	1.57611e8i	0.520846	−	1.47064i
$476$		0			0
$477$		−	1.99180e8i		−	1.83523i
$478$		0			0
$479$	−2.15904e8			−1.96451			−0.982253	−	0.187560i	$-0.939942\pi$
$479$	−2.15904e8			−1.96451			−0.982253	+	0.187560i	$0.939942\pi$
$480$		0			0
$481$	−4.87765e7			−0.438304
$482$		0			0
$483$		−	4.94002e8i		−	4.38417i
$484$		0			0
$485$			1.30021e8i			1.13969i
$486$		0			0
$487$			1.33297e8i			1.15407i	0.816720	+	0.577035i	$0.195790\pi$
$487$			1.33297e8i			1.15407i	−0.816720	+	0.577035i	$0.804210\pi$
$488$		0			0
$489$		−	2.56580e8i		−	2.19430i
$490$		0			0
$491$	−1.16823e8			−0.986926			−0.493463	−	0.869767i	$-0.664269\pi$
$491$	−1.16823e8			−0.986926			−0.493463	+	0.869767i	$0.664269\pi$
$492$		0			0
$493$			1.40091e8i			1.16915i
$494$		0			0
$495$	1.74305e8			1.43712
$496$		0			0
$497$			1.39024e8i			1.13245i
$498$		0			0
$499$	2.10771e8			1.69632			0.848161	−	0.529738i	$-0.177710\pi$
$499$	2.10771e8			1.69632			0.848161	+	0.529738i	$0.177710\pi$
$500$		0			0
$501$	3.53002e8			2.80714
$502$		0			0
$503$	−1.05595e8			−0.829734			−0.414867	−	0.909882i	$-0.636172\pi$
$503$	−1.05595e8			−0.829734			−0.414867	+	0.909882i	$0.636172\pi$
$504$		0			0
$505$	−1.66747e8			−1.29475
$506$		0			0
$507$		−	5.21568e7i		−	0.400209i
$508$		0			0
$509$		−	1.87764e8i		−	1.42384i	−0.702263	−	0.711918i	$-0.747827\pi$
$509$		−	1.87764e8i		−	1.42384i	0.702263	−	0.711918i	$-0.252173\pi$
$510$		0			0
$511$	−1.31380e7			−0.0984617
$512$		0			0
$513$	2.89826e8	+	1.02646e8i	2.14677	+	0.760306i
$514$		0			0
$515$			3.83418e8i			2.80705i
$516$		0			0
$517$	9.12147e7			0.660075
$518$		0			0
$519$	9.94962e7			0.711712
$520$		0			0
$521$		−	6.60813e6i		−	0.0467267i	−0.999727	−	0.0233634i	$-0.992563\pi$
$521$		−	6.60813e6i		−	0.0467267i	0.999727	−	0.0233634i	$-0.00743746\pi$
$522$		0			0
$523$		−	9.32891e7i		−	0.652118i	−0.945349	−	0.326059i	$-0.894279\pi$
$523$		−	9.32891e7i		−	0.652118i	0.945349	−	0.326059i	$-0.105721\pi$
$524$		0			0
$525$		−	6.37933e8i		−	4.40856i
$526$		0			0
$527$		−	2.73688e8i		−	1.86992i
$528$		0			0
$529$	2.08317e8			1.40720
$530$		0			0
$531$			5.10703e7i			0.341102i
$532$		0			0
$533$	2.65371e7			0.175256
$534$		0			0
$535$			1.27146e8i			0.830312i
$536$		0			0
$537$	−2.73355e8			−1.76524
$538$		0			0
$539$	9.00966e7			0.575363
$540$		0			0
$541$	−3.22147e7			−0.203452			−0.101726	−	0.994812i	$-0.532437\pi$
$541$	−3.22147e7			−0.203452			−0.101726	+	0.994812i	$0.532437\pi$
$542$		0			0
$543$	−2.98358e8			−1.86354
$544$		0			0
$545$			4.26074e7i			0.263206i
$546$		0			0
$547$		−	1.39680e8i		−	0.853436i	−0.904385	−	0.426718i	$-0.859670\pi$
$547$		−	1.39680e8i		−	0.853436i	0.904385	−	0.426718i	$-0.140330\pi$
$548$		0			0
$549$	−1.59034e8			−0.961110
$550$		0			0
$551$	1.38309e8	+	4.89839e7i	0.826791	+	0.292818i
$552$		0			0
$553$			2.30552e8i			1.36330i
$554$		0			0
$555$	−2.45400e8			−1.43547
$556$		0			0
$557$	1.37604e8			0.796282			0.398141	−	0.917324i	$-0.369655\pi$
$557$	1.37604e8			0.796282			0.398141	+	0.917324i	$0.369655\pi$
$558$		0			0
$559$			2.15613e7i			0.123435i
$560$		0			0
$561$		−	1.68819e8i		−	0.956168i
$562$		0			0
$563$		−	2.37756e7i		−	0.133231i	−0.997779	−	0.0666156i	$-0.978780\pi$
$563$		−	2.37756e7i		−	0.133231i	0.997779	−	0.0666156i	$-0.0212201\pi$
$564$		0			0
$565$		−	2.82006e8i		−	1.56355i
$566$		0			0
$567$	5.28130e8			2.89729
$568$		0			0
$569$			2.31155e7i			0.125478i	0.998030	+	0.0627389i	$0.0199835\pi$
$569$			2.31155e7i			0.125478i	−0.998030	+	0.0627389i	$0.980016\pi$
$570$		0			0
$571$	7.45006e7			0.400177			0.200088	−	0.979778i	$-0.435877\pi$
$571$	7.45006e7			0.400177			0.200088	+	0.979778i	$0.435877\pi$
$572$		0			0
$573$		−	3.78343e8i		−	2.01104i
$574$		0			0
$575$	4.60178e8			2.42060
$576$		0			0
$577$	−1.89442e8			−0.986164			−0.493082	−	0.869983i	$-0.664130\pi$
$577$	−1.89442e8			−0.986164			−0.493082	+	0.869983i	$0.664130\pi$
$578$		0			0
$579$	1.55021e8			0.798645
$580$		0			0
$581$	−5.65909e8			−2.88548
$582$		0			0
$583$			6.38858e7i			0.322403i
$584$		0			0
$585$			6.39033e8i			3.19195i
$586$		0			0
$587$	4.84968e7			0.239772			0.119886	−	0.992788i	$-0.461747\pi$
$587$	4.84968e7			0.239772			0.119886	+	0.992788i	$0.461747\pi$
$588$		0			0
$589$	−2.70207e8	−	9.56971e7i	−1.32236	−	0.468331i
$590$		0			0
$591$		−	2.40120e8i		−	1.16323i
$592$		0			0
$593$	2.12533e7			0.101921			0.0509604	−	0.998701i	$-0.483772\pi$
$593$	2.12533e7			0.101921			0.0509604	+	0.998701i	$0.483772\pi$
$594$		0			0
$595$	−7.02982e8			−3.33729
$596$		0			0
$597$			1.97374e8i			0.927613i
$598$		0			0
$599$			3.83752e8i			1.78554i	0.450511	+	0.892771i	$0.351242\pi$
$599$			3.83752e8i			1.78554i	−0.450511	+	0.892771i	$0.648758\pi$
$600$		0			0
$601$		−	4.20417e8i		−	1.93668i	−0.249643	−	0.968338i	$-0.580313\pi$
$601$		−	4.20417e8i		−	1.93668i	0.249643	−	0.968338i	$-0.419687\pi$
$602$		0			0
$603$		−	6.48255e8i		−	2.95661i
$604$		0			0
$605$	2.98415e8			1.34758
$606$		0			0
$607$			9.10423e7i			0.407078i	0.979067	+	0.203539i	$0.0652443\pi$
$607$			9.10423e7i			0.407078i	−0.979067	+	0.203539i	$0.934756\pi$
$608$		0			0
$609$	5.59807e8			2.47848
$610$		0			0
$611$			3.34409e8i			1.46607i
$612$		0			0
$613$	−2.54414e8			−1.10449			−0.552243	−	0.833683i	$-0.686228\pi$
$613$	−2.54414e8			−1.10449			−0.552243	+	0.833683i	$0.686228\pi$
$614$		0			0
$615$	1.33511e8			0.573974
$616$		0			0
$617$	−1.53267e8			−0.652521			−0.326260	−	0.945280i	$-0.605789\pi$
$617$	−1.53267e8			−0.652521			−0.326260	+	0.945280i	$0.605789\pi$
$618$		0			0
$619$	3.38743e8			1.42823			0.714116	−	0.700028i	$-0.246829\pi$
$619$	3.38743e8			1.42823			0.714116	+	0.700028i	$0.246829\pi$
$620$		0			0
$621$			8.46207e8i			3.53347i
$622$		0			0
$623$		−	2.37641e8i		−	0.982782i
$624$		0			0
$625$	−3.07820e7			−0.126083
$626$		0			0
$627$	−1.66672e8	−	5.90290e7i	−0.676177	−	0.239476i
$628$		0			0
$629$			1.64795e8i			0.662206i
$630$		0			0
$631$	−3.40484e8			−1.35522			−0.677609	−	0.735423i	$-0.736984\pi$
$631$	−3.40484e8			−1.35522			−0.677609	+	0.735423i	$0.736984\pi$
$632$		0			0
$633$	−1.70993e8			−0.674166
$634$		0			0
$635$			6.16390e8i			2.40732i
$636$		0			0
$637$			3.30310e8i			1.27792i
$638$		0			0
$639$		−	4.26975e8i		−	1.63644i
$640$		0			0
$641$		−	1.82902e8i		−	0.694457i	−0.937781	−	0.347229i	$-0.887123\pi$
$641$		−	1.82902e8i		−	0.694457i	0.937781	−	0.347229i	$-0.112877\pi$
$642$		0			0
$643$	1.80050e8			0.677268			0.338634	−	0.940918i	$-0.390035\pi$
$643$	1.80050e8			0.677268			0.338634	+	0.940918i	$0.390035\pi$
$644$		0			0
$645$			1.08477e8i			0.404259i
$646$		0			0
$647$	2.45601e8			0.906811			0.453406	−	0.891304i	$-0.350209\pi$
$647$	2.45601e8			0.906811			0.453406	+	0.891304i	$0.350209\pi$
$648$		0			0
$649$		−	1.63805e7i		−	0.0599230i
$650$		0			0
$651$	−1.09366e9			−3.96406
$652$		0			0
$653$	4.72102e8			1.69549			0.847747	−	0.530401i	$-0.177959\pi$
$653$	4.72102e8			1.69549			0.847747	+	0.530401i	$0.177959\pi$
$654$		0			0
$655$	3.15866e8			1.12403
$656$		0			0
$657$	4.03500e7			0.142281
$658$		0			0
$659$		−	2.43351e8i		−	0.850308i	−0.905121	−	0.425154i	$-0.860220\pi$
$659$		−	2.43351e8i		−	0.850308i	0.905121	−	0.425154i	$-0.139780\pi$
$660$		0			0
$661$		−	2.16910e8i		−	0.751060i	−0.926810	−	0.375530i	$-0.877461\pi$
$661$		−	2.16910e8i		−	0.751060i	0.926810	−	0.375530i	$-0.122539\pi$
$662$		0			0
$663$	6.18922e8			2.12371
$664$		0			0
$665$	−2.45803e8	+	6.94040e8i	−0.835838	+	2.36004i
$666$		0			0
$667$			4.03821e8i			1.36086i
$668$		0			0
$669$	−6.17974e8			−2.06392
$670$		0			0
$671$	5.10093e7			0.168843
$672$		0			0
$673$			3.67917e8i			1.20699i	0.797366	+	0.603496i	$0.206226\pi$
$673$			3.67917e8i			1.20699i	−0.797366	+	0.603496i	$0.793774\pi$
$674$		0			0
$675$			1.09276e9i			3.55313i
$676$		0			0
$677$			3.47209e8i			1.11899i	0.828834	+	0.559494i	$0.189005\pi$
$677$			3.47209e8i			1.11899i	−0.828834	+	0.559494i	$0.810995\pi$
$678$		0			0
$679$		−	3.48909e8i		−	1.11456i
$680$		0			0
$681$	1.88707e8			0.597512
$682$		0			0
$683$		−	1.46683e7i		−	0.0460382i	−0.999735	−	0.0230191i	$-0.992672\pi$
$683$		−	1.46683e7i		−	0.0460382i	0.999735	−	0.0230191i	$-0.00732785\pi$
$684$		0			0
$685$	−2.48723e7			−0.0773829
$686$		0			0
$687$			4.52773e8i			1.39640i
$688$		0			0
$689$	−2.34217e8			−0.716078
$690$		0			0
$691$	−1.15880e7			−0.0351216			−0.0175608	−	0.999846i	$-0.505590\pi$
$691$	−1.15880e7			−0.0351216			−0.0175608	+	0.999846i	$0.505590\pi$
$692$		0			0
$693$	−4.67744e8			−1.40543
$694$		0			0
$695$	−7.34800e8			−2.18884
$696$		0			0
$697$		−	8.96578e7i		−	0.264783i
$698$		0			0
$699$			2.42726e7i			0.0710699i
$700$		0			0
$701$	1.56644e8			0.454737			0.227368	−	0.973809i	$-0.426988\pi$
$701$	1.56644e8			0.454737			0.227368	+	0.973809i	$0.426988\pi$
$702$		0			0
$703$	1.62699e8	+	5.76219e7i	0.468295	+	0.165852i
$704$		0			0
$705$			1.68245e9i			4.80148i
$706$		0			0
$707$	4.47463e8			1.26619
$708$		0			0
$709$	−2.53434e8			−0.711092			−0.355546	−	0.934659i	$-0.615705\pi$
$709$	−2.53434e8			−0.711092			−0.355546	+	0.934659i	$0.615705\pi$
$710$		0			0
$711$		−	7.08078e8i		−	1.97003i
$712$		0			0
$713$		−	7.88924e8i		−	2.17654i
$714$		0			0
$715$		−	2.04966e8i		−	0.560744i
$716$		0			0
$717$		−	9.36781e8i		−	2.54145i
$718$		0			0
$719$	−6.92521e7			−0.186314			−0.0931572	−	0.995651i	$-0.529696\pi$
$719$	−6.92521e7			−0.186314			−0.0931572	+	0.995651i	$0.529696\pi$
$720$		0			0
$721$		−	1.02889e9i		−	2.74514i
$722$		0			0
$723$	−4.87088e8			−1.28882
$724$		0			0
$725$			5.21478e8i			1.36843i
$726$		0			0
$727$	2.61429e8			0.680377			0.340189	−	0.940357i	$-0.389509\pi$
$727$	2.61429e8			0.680377			0.340189	+	0.940357i	$0.389509\pi$
$728$		0			0
$729$	−2.86576e7			−0.0739702
$730$		0			0
$731$	7.28465e7			0.186491
$732$		0			0
$733$	5.79496e8			1.47143			0.735714	−	0.677293i	$-0.236847\pi$
$733$	5.79496e8			1.47143			0.735714	+	0.677293i	$0.236847\pi$
$734$		0			0
$735$			1.66183e9i			4.18527i
$736$		0			0
$737$			2.07925e8i			0.519402i
$738$		0			0
$739$	−1.04122e8			−0.257993			−0.128997	−	0.991645i	$-0.541176\pi$
$739$	−1.04122e8			−0.257993			−0.128997	+	0.991645i	$0.541176\pi$
$740$		0			0
$741$	2.16411e8	−	6.11049e8i	0.531892	−	1.50183i
$742$		0			0
$743$		−	1.31519e8i		−	0.320643i	−0.987065	−	0.160321i	$-0.948747\pi$
$743$		−	1.31519e8i		−	0.320643i	0.987065	−	0.160321i	$-0.0512531\pi$
$744$		0			0
$745$	−3.97693e8			−0.961787
$746$		0			0
$747$	1.73804e9			4.16963
$748$		0			0
$749$		−	3.41194e8i		−	0.811999i
$750$		0			0
$751$			2.65619e8i			0.627105i	0.949571	+	0.313552i	$0.101519\pi$
$751$			2.65619e8i			0.627105i	−0.949571	+	0.313552i	$0.898481\pi$
$752$		0			0
$753$		−	1.06099e8i		−	0.248501i
$754$		0			0
$755$			7.28728e8i			1.69326i
$756$		0			0
$757$	2.34318e8			0.540154			0.270077	−	0.962839i	$-0.412951\pi$
$757$	2.34318e8			0.540154			0.270077	+	0.962839i	$0.412951\pi$
$758$		0			0
$759$		−	4.86633e8i		−	1.11295i
$760$		0			0
$761$	−2.90283e8			−0.658670			−0.329335	−	0.944213i	$-0.606825\pi$
$761$	−2.90283e8			−0.658670			−0.329335	+	0.944213i	$0.606825\pi$
$762$		0			0
$763$		−	1.14336e8i		−	0.257401i
$764$		0			0
$765$	2.15902e9			4.82251
$766$		0			0
$767$	6.00539e7			0.133093
$768$		0			0
$769$	−2.11713e8			−0.465552			−0.232776	−	0.972530i	$-0.574781\pi$
$769$	−2.11713e8			−0.465552			−0.232776	+	0.972530i	$0.574781\pi$
$770$		0			0
$771$	3.31241e7			0.0722737
$772$		0			0
$773$			1.33497e8i			0.289024i	0.989503	+	0.144512i	$0.0461613\pi$
$773$			1.33497e8i			0.289024i	−0.989503	+	0.144512i	$0.953839\pi$
$774$		0			0
$775$		−	1.01878e9i		−	2.18865i
$776$		0			0
$777$	6.58526e8			1.40381
$778$		0			0
$779$	−8.85173e7	−	3.13495e7i	−0.187247	−	0.0663160i
$780$		0			0
$781$			1.36950e8i			0.287481i
$782$		0			0
$783$	−9.58928e8			−1.99756
$784$		0			0
$785$	1.08061e9			2.23389
$786$		0			0
$787$		−	3.03802e8i		−	0.623257i	−0.950204	−	0.311628i	$-0.899126\pi$
$787$		−	3.03802e8i		−	0.623257i	0.950204	−	0.311628i	$-0.100874\pi$
$788$		0			0
$789$			2.09722e8i			0.426984i
$790$		0			0
$791$			7.56756e8i			1.52907i
$792$		0			0
$793$			1.87009e8i			0.375011i
$794$		0			0
$795$	−1.17837e9			−2.34520
$796$		0			0
$797$		−	6.29670e8i		−	1.24376i	−0.783111	−	0.621882i	$-0.786368\pi$
$797$		−	6.29670e8i		−	1.24376i	0.783111	−	0.621882i	$-0.213632\pi$
$798$		0			0
$799$	1.12983e9			2.21499
$800$		0			0
$801$			7.29851e8i			1.42016i
$802$		0			0
$803$	−1.29420e7			−0.0249952
$804$		0			0
$805$	−2.02639e9			−3.88451
$806$		0			0
$807$	−1.17925e9			−2.24381
$808$		0			0
$809$	7.12103e8			1.34492			0.672461	−	0.740133i	$-0.265237\pi$
$809$	7.12103e8			1.34492			0.672461	+	0.740133i	$0.265237\pi$
$810$		0			0
$811$			7.04655e8i			1.32103i	0.750811	+	0.660517i	$0.229663\pi$
$811$			7.04655e8i			1.32103i	−0.750811	+	0.660517i	$0.770337\pi$
$812$		0			0
$813$		−	8.39546e8i		−	1.56233i
$814$		0			0
$815$	−1.05249e9			−1.94422
$816$		0			0
$817$	2.54713e7	−	7.19199e7i	0.0467074	−	0.131881i
$818$		0			0
$819$		−	1.71483e9i		−	3.12154i
$820$		0			0
$821$	−8.77996e8			−1.58658			−0.793292	−	0.608842i	$-0.791634\pi$
$821$	−8.77996e8			−1.58658			−0.793292	+	0.608842i	$0.791634\pi$
$822$		0			0
$823$	4.48972e6			0.00805414			0.00402707	−	0.999992i	$-0.498718\pi$
$823$	4.48972e6			0.00805414			0.00402707	+	0.999992i	$0.498718\pi$
$824$		0			0
$825$		−	6.28417e8i		−	1.11914i
$826$		0			0
$827$			3.93291e8i			0.695341i	0.937617	+	0.347671i	$0.113027\pi$
$827$			3.93291e8i			0.695341i	−0.937617	+	0.347671i	$0.886973\pi$
$828$		0			0
$829$		−	4.43942e7i		−	0.0779224i	−0.999241	−	0.0389612i	$-0.987595\pi$
$829$		−	4.43942e7i		−	0.0779224i	0.999241	−	0.0389612i	$-0.0124049\pi$
$830$		0			0
$831$			1.89395e8i			0.330038i
$832$		0			0
$833$	1.11598e9			1.93073
$834$		0			0
$835$		−	1.44801e9i		−	2.48721i
$836$		0			0
$837$	1.87340e9			3.19488
$838$		0			0
$839$			7.12436e8i			1.20631i	0.797623	+	0.603157i	$0.206091\pi$
$839$			7.12436e8i			1.20631i	−0.797623	+	0.603157i	$0.793909\pi$
$840$		0			0
$841$	1.37210e8			0.230673
$842$		0			0
$843$	1.50428e9			2.51100
$844$		0			0
$845$	−2.13947e8			−0.354597
$846$		0			0
$847$	−8.00790e8			−1.31786
$848$		0			0
$849$			1.46341e8i			0.239136i
$850$		0			0
$851$			4.75033e8i			0.770788i
$852$		0			0
$853$	−5.78089e8			−0.931425			−0.465712	−	0.884936i	$-0.654202\pi$
$853$	−5.78089e8			−0.931425			−0.465712	+	0.884936i	$0.654202\pi$
$854$		0			0
$855$	7.54918e8	−	2.13156e9i	1.20782	−	3.41035i
$856$		0			0
$857$			8.48564e8i			1.34816i	0.738657	+	0.674081i	$0.235460\pi$
$857$			8.48564e8i			1.34816i	−0.738657	+	0.674081i	$0.764540\pi$
$858$		0			0
$859$	1.64312e8			0.259233			0.129617	−	0.991564i	$-0.458625\pi$
$859$	1.64312e8			0.259233			0.129617	+	0.991564i	$0.458625\pi$
$860$		0			0
$861$	−3.58274e8			−0.561314
$862$		0			0
$863$			7.18996e8i			1.11865i	0.828948	+	0.559325i	$0.188940\pi$
$863$			7.18996e8i			1.11865i	−0.828948	+	0.559325i	$0.811060\pi$
$864$		0			0
$865$		−	4.08133e8i		−	0.630599i
$866$		0			0
$867$		−	9.14172e8i		−	1.40272i
$868$		0			0
$869$			2.27113e8i			0.346084i
$870$		0			0
$871$	−7.62288e8			−1.15363
$872$		0			0
$873$			1.07158e9i			1.61058i
$874$		0			0
$875$	−9.39523e8			−1.40244
$876$		0			0
$877$			6.53997e8i			0.969566i	0.874635	+	0.484783i	$0.161101\pi$
$877$			6.53997e8i			0.969566i	−0.874635	+	0.484783i	$0.838899\pi$
$878$		0			0
$879$	2.10656e9			3.10175
$880$		0			0
$881$	−1.17967e9			−1.72517			−0.862587	−	0.505908i	$-0.831158\pi$
$881$	−1.17967e9			−1.72517			−0.862587	+	0.505908i	$0.831158\pi$
$882$		0			0
$883$	5.21584e7			0.0757603			0.0378802	−	0.999282i	$-0.487939\pi$
$883$	5.21584e7			0.0757603			0.0378802	+	0.999282i	$0.487939\pi$
$884$		0			0
$885$	3.02138e8			0.435889
$886$		0			0
$887$		−	5.01621e8i		−	0.718795i	−0.933185	−	0.359397i	$-0.882982\pi$
$887$		−	5.01621e8i		−	0.718795i	0.933185	−	0.359397i	$-0.117018\pi$
$888$		0			0
$889$		−	1.65407e9i		−	2.35423i
$890$		0			0
$891$	5.20252e8			0.735496
$892$		0			0
$893$	3.95053e8	−	1.11546e9i	0.554754	−	1.56638i
$894$		0			0
$895$			1.12130e9i			1.56406i
$896$		0			0
$897$	1.78408e9			2.47194
$898$		0			0
$899$	8.94014e8			1.23045
$900$		0			0
$901$			7.91320e8i			1.08188i
$902$		0			0
$903$		−	2.91096e8i		−	0.395342i
$904$		0			0
$905$			1.22386e9i			1.65115i
$906$		0			0
$907$			1.42613e9i			1.91133i	0.294456	+	0.955665i	$0.404862\pi$
$907$			1.42613e9i			1.91133i	−0.294456	+	0.955665i	$0.595138\pi$
$908$		0			0
$909$	−1.37426e9			−1.82969
$910$		0			0
$911$		−	1.19743e9i		−	1.58378i	−0.610661	−	0.791892i	$-0.709096\pi$
$911$		−	1.19743e9i		−	1.58378i	0.610661	−	0.791892i	$-0.290904\pi$
$912$		0			0
$913$	−5.57467e8			−0.732499
$914$		0			0
$915$			9.40864e8i			1.22818i
$916$		0			0
$917$	−8.47620e8			−1.09924
$918$		0			0
$919$	−8.78298e8			−1.13161			−0.565803	−	0.824541i	$-0.691434\pi$
$919$	−8.78298e8			−1.13161			−0.565803	+	0.824541i	$0.691434\pi$
$920$		0			0
$921$	−2.02687e9			−2.59446
$922$		0			0
$923$	−5.02083e8			−0.638514
$924$		0			0
$925$			6.13437e8i			0.775077i
$926$		0			0
$927$			3.15997e9i			3.96683i
$928$		0			0
$929$	−4.54253e8			−0.566567			−0.283283	−	0.959036i	$-0.591424\pi$
$929$	−4.54253e8			−0.566567			−0.283283	+	0.959036i	$0.591424\pi$
$930$		0			0
$931$	3.90210e8	−	1.10178e9i	0.483559	−	1.36536i
$932$		0			0
$933$			3.39739e8i			0.418312i
$934$		0			0
$935$	−6.92496e8			−0.847193
$936$		0			0
$937$	−1.42133e8			−0.172773			−0.0863863	−	0.996262i	$-0.527532\pi$
$937$	−1.42133e8			−0.172773			−0.0863863	+	0.996262i	$0.527532\pi$
$938$		0			0
$939$			4.48610e8i			0.541842i
$940$		0			0
$941$			6.11671e8i			0.734090i	0.930203	+	0.367045i	$0.119630\pi$
$941$			6.11671e8i			0.734090i	−0.930203	+	0.367045i	$0.880370\pi$
$942$		0			0
$943$		−	2.58444e8i		−	0.308199i
$944$		0			0
$945$		−	4.81194e9i		−	5.70197i
$946$		0			0
$947$	−1.46500e9			−1.72499			−0.862496	−	0.506065i	$-0.831100\pi$
$947$	−1.46500e9			−1.72499			−0.862496	+	0.506065i	$0.831100\pi$
$948$		0			0
$949$		−	4.74478e7i		−	0.0555159i
$950$		0			0
$951$	−2.49196e9			−2.89734
$952$		0			0
$953$			6.21057e8i			0.717551i	0.933424	+	0.358776i	$0.116806\pi$
$953$			6.21057e8i			0.717551i	−0.933424	+	0.358776i	$0.883194\pi$
$954$		0			0
$955$	−1.55196e9			−1.78185
$956$		0			0
$957$	5.51456e8			0.629180
$958$		0			0
$959$	6.67444e7			0.0756761
$960$		0			0
$961$	−8.59082e8			−0.967975
$962$		0			0
$963$			1.04789e9i			1.17337i
$964$		0			0
$965$		−	6.35893e8i		−	0.707623i
$966$		0			0
$967$	2.85908e8			0.316189			0.158095	−	0.987424i	$-0.449465\pi$
$967$	2.85908e8			0.316189			0.158095	+	0.987424i	$0.449465\pi$
$968$		0			0
$969$	−2.06448e9	−	7.31161e8i	−2.26902	−	0.803603i
$970$		0			0
$971$		−	6.35318e8i		−	0.693958i	−0.937873	−	0.346979i	$-0.887208\pi$
$971$		−	6.35318e8i		−	0.693958i	0.937873	−	0.346979i	$-0.112792\pi$
$972$		0			0
$973$	1.97182e9			2.14057
$974$		0			0
$975$	2.30389e9			2.48569
$976$		0			0
$977$			1.25172e9i			1.34222i	0.741358	+	0.671109i	$0.234182\pi$
$977$			1.25172e9i			1.34222i	−0.741358	+	0.671109i	$0.765818\pi$
$978$		0			0
$979$		−	2.34096e8i		−	0.249486i
$980$		0			0
$981$			3.51153e8i			0.371954i
$982$		0			0
$983$			1.11393e9i			1.17273i	0.810047	+	0.586365i	$0.199441\pi$
$983$			1.11393e9i			1.17273i	−0.810047	+	0.586365i	$0.800559\pi$
$984$		0			0
$985$	−9.84970e8			−1.03066
$986$		0			0
$987$		−	4.51482e9i		−	4.69557i
$988$		0			0
$989$	2.09985e8			0.217070
$990$		0			0
$991$			5.45307e7i			0.0560299i	0.999608	+	0.0280150i	$0.00891860\pi$
$991$			5.45307e7i			0.0560299i	−0.999608	+	0.0280150i	$0.991081\pi$
$992$		0			0
$993$	−7.02547e7			−0.0717510
$994$		0			0
$995$	8.09626e8			0.821893
$996$		0			0
$997$	1.27388e9			1.28541			0.642706	−	0.766113i	$-0.277812\pi$
$997$	1.27388e9			1.28541			0.642706	+	0.766113i	$0.277812\pi$
$998$		0			0
$999$	−1.12803e9			−1.13142

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.7.e.f.113.2		30
4.3	odd	2		152.7.e.a.113.29	yes	30
19.18	odd	2	inner	304.7.e.f.113.29		30
76.75	even	2		152.7.e.a.113.2	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.7.e.a.113.2	✓	30	76.75	even	2
152.7.e.a.113.29	yes	30	4.3	odd	2
304.7.e.f.113.2		30	1.1	even	1	trivial
304.7.e.f.113.29		30	19.18	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-48.7582i q^{3} -200.006 q^{5} +536.711 q^{7} -1648.37 q^{9} +O(q^{10})\)	\(q-48.7582i q^{3} -200.006 q^{5} +536.711 q^{7} -1648.37 q^{9} +528.705 q^{11} +1938.33i q^{13} +9751.93i q^{15} +6548.79 q^{17} +(2289.83 - 6465.49i) q^{19} -26169.1i q^{21} +18877.3 q^{23} +24377.3 q^{25} +44826.7i q^{27} +21391.9i q^{29} -41792.2i q^{31} -25778.7i q^{33} -107345. q^{35} +25164.2i q^{37} +94509.4 q^{39} -13690.7i q^{41} +11123.7 q^{43} +329683. q^{45} +172525. q^{47} +170410. q^{49} -319308. i q^{51} +120834. i q^{53} -105744. q^{55} +(-315246. - 111648. i) q^{57} -30982.4i q^{59} +96479.8 q^{61} -884697. q^{63} -387676. i q^{65} +393271. i q^{67} -920424. i q^{69} +259029. i q^{71} -24478.8 q^{73} -1.18860e6i q^{75} +283762. q^{77} +429564. i q^{79} +984011. q^{81} -1.05440e6 q^{83} -1.30980e6 q^{85} +1.04303e6 q^{87} -442773. i q^{89} +1.04032e6i q^{91} -2.03771e6 q^{93} +(-457980. + 1.29314e6i) q^{95} -650087. i q^{97} -871500. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 720 q^{7} - 8670 q^{9}+O(q^{10}) \) 30 * q + 720 * q^7 - 8670 * q^9	\( 30 q + 720 q^{7} - 8670 q^{9} + 2524 q^{11} + 9700 q^{17} - 4014 q^{19} + 39376 q^{23} + 110742 q^{25} + 19976 q^{35} + 266500 q^{39} + 106788 q^{43} - 91360 q^{45} - 222756 q^{47} + 593586 q^{49} - 540936 q^{55} - 545972 q^{57} - 242640 q^{61} + 377716 q^{63} + 545964 q^{73} - 272356 q^{77} + 2189926 q^{81} - 1542652 q^{83} - 826908 q^{85} + 2729572 q^{87} - 2139912 q^{93} - 2142716 q^{95} + 293012 q^{99}+O(q^{100}) \) 30 * q + 720 * q^7 - 8670 * q^9 + 2524 * q^11 + 9700 * q^17 - 4014 * q^19 + 39376 * q^23 + 110742 * q^25 + 19976 * q^35 + 266500 * q^39 + 106788 * q^43 - 91360 * q^45 - 222756 * q^47 + 593586 * q^49 - 540936 * q^55 - 545972 * q^57 - 242640 * q^61 + 377716 * q^63 + 545964 * q^73 - 272356 * q^77 + 2189926 * q^81 - 1542652 * q^83 - 826908 * q^85 + 2729572 * q^87 - 2139912 * q^93 - 2142716 * q^95 + 293012 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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