LMFDB - Embedded newform 304.7.e.f.113.13

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			113.13
Character	$\chi$	$=$	304.113
Dual form			304.7.e.f.113.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.57785i q^{3} +39.7533 q^{5} +24.5800 q^{7} +637.265 q^{9} +O(q^{10})$	$q-9.57785i q^{3} +39.7533 q^{5} +24.5800 q^{7} +637.265 q^{9} +2037.58 q^{11} +2260.99i q^{13} -380.751i q^{15} +3676.03 q^{17} +(-5981.65 + 3356.45i) q^{19} -235.424i q^{21} +2201.31 q^{23} -14044.7 q^{25} -13085.9i q^{27} +26067.2i q^{29} -5905.20i q^{31} -19515.6i q^{33} +977.138 q^{35} +13055.9i q^{37} +21655.4 q^{39} +51499.0i q^{41} -9723.00 q^{43} +25333.4 q^{45} +137312. q^{47} -117045. q^{49} -35208.5i q^{51} +274168. i q^{53} +81000.5 q^{55} +(32147.6 + 57291.4i) q^{57} -231290. i q^{59} -122878. q^{61} +15664.0 q^{63} +89881.8i q^{65} -171624. i q^{67} -21083.8i q^{69} +378499. i q^{71} +76522.4 q^{73} +134518. i q^{75} +50083.8 q^{77} -895511. i q^{79} +339231. q^{81} +896640. q^{83} +146134. q^{85} +249668. q^{87} +1.17938e6i q^{89} +55575.2i q^{91} -56559.1 q^{93} +(-237790. + 133430. i) q^{95} +802702. i q^{97} +1.29848e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	9.57785i		−	0.354735i	−0.984145	−	0.177368i	$-0.943242\pi$
$3$		−	9.57785i		−	0.354735i	0.984145	−	0.177368i	$-0.0567582\pi$
$4$		0			0
$5$	39.7533			0.318026			0.159013	−	0.987276i	$-0.449169\pi$
$5$	39.7533			0.318026			0.159013	+	0.987276i	$0.449169\pi$
$6$		0			0
$7$	24.5800			0.0716619			0.0358310	−	0.999358i	$-0.488592\pi$
$7$	24.5800			0.0716619			0.0358310	+	0.999358i	$0.488592\pi$
$8$		0			0
$9$	637.265			0.874163
$10$		0			0
$11$	2037.58			1.53086			0.765432	−	0.643517i	$-0.222526\pi$
$11$	2037.58			1.53086			0.765432	+	0.643517i	$0.222526\pi$
$12$		0			0
$13$			2260.99i			1.02913i	0.857453	+	0.514563i	$0.172046\pi$
$13$			2260.99i			1.02913i	−0.857453	+	0.514563i	$0.827954\pi$
$14$		0			0
$15$		−	380.751i		−	0.112815i
$16$		0			0
$17$	3676.03			0.748226			0.374113	−	0.927383i	$-0.377947\pi$
$17$	3676.03			0.748226			0.374113	+	0.927383i	$0.377947\pi$
$18$		0			0
$19$	−5981.65	+	3356.45i	−0.872088	+	0.489349i
$19$	−5981.65	+	3356.45i	−0.872088	+	0.489349i
$20$		0			0
$21$		−	235.424i		−	0.0254210i
$22$		0			0
$23$	2201.31			0.180925			0.0904623	−	0.995900i	$-0.471166\pi$
$23$	2201.31			0.180925			0.0904623	+	0.995900i	$0.471166\pi$
$24$		0			0
$25$	−14044.7			−0.898859
$26$		0			0
$27$		−	13085.9i		−	0.664832i
$28$		0			0
$29$			26067.2i			1.06881i	0.845228	+	0.534405i	$0.179464\pi$
$29$			26067.2i			1.06881i	−0.845228	+	0.534405i	$0.820536\pi$
$30$		0			0
$31$		−	5905.20i		−	0.198221i	−0.995076	−	0.0991105i	$-0.968400\pi$
$31$		−	5905.20i		−	0.198221i	0.995076	−	0.0991105i	$-0.0315997\pi$
$32$		0			0
$33$		−	19515.6i		−	0.543051i
$34$		0			0
$35$	977.138			0.0227904
$36$		0			0
$37$			13055.9i			0.257752i	0.991661	+	0.128876i	$0.0411370\pi$
$37$			13055.9i			0.257752i	−0.991661	+	0.128876i	$0.958863\pi$
$38$		0			0
$39$	21655.4			0.365067
$40$		0			0
$41$			51499.0i			0.747218i	0.927586	+	0.373609i	$0.121880\pi$
$41$			51499.0i			0.747218i	−0.927586	+	0.373609i	$0.878120\pi$
$42$		0			0
$43$	−9723.00			−0.122291			−0.0611456	−	0.998129i	$-0.519475\pi$
$43$	−9723.00			−0.122291			−0.0611456	+	0.998129i	$0.519475\pi$
$44$		0			0
$45$	25333.4			0.278007
$46$		0			0
$47$	137312.			1.32256			0.661282	−	0.750138i	$-0.270013\pi$
$47$	137312.			1.32256			0.661282	+	0.750138i	$0.270013\pi$
$48$		0			0
$49$	−117045.			−0.994865
$50$		0			0
$51$		−	35208.5i		−	0.265422i
$52$		0			0
$53$			274168.i			1.84157i	0.390069	+	0.920786i	$0.372451\pi$
$53$			274168.i			1.84157i	−0.390069	+	0.920786i	$0.627549\pi$
$54$		0			0
$55$	81000.5			0.486855
$56$		0			0
$57$	32147.6	+	57291.4i	0.173590	+	0.309360i
$58$		0			0
$59$		−	231290.i		−	1.12616i	−0.826402	−	0.563080i	$-0.809616\pi$
$59$		−	231290.i		−	1.12616i	0.826402	−	0.563080i	$-0.190384\pi$
$60$		0			0
$61$	−122878.			−0.541356			−0.270678	−	0.962670i	$-0.587248\pi$
$61$	−122878.			−0.541356			−0.270678	+	0.962670i	$0.587248\pi$
$62$		0			0
$63$	15664.0			0.0626442
$64$		0			0
$65$			89881.8i			0.327289i
$66$		0			0
$67$		−	171624.i		−	0.570628i	−0.958434	−	0.285314i	$-0.907902\pi$
$67$		−	171624.i		−	0.570628i	0.958434	−	0.285314i	$-0.0920978\pi$
$68$		0			0
$69$		−	21083.8i		−	0.0641804i
$70$		0			0
$71$			378499.i			1.05752i	0.848770	+	0.528762i	$0.177344\pi$
$71$			378499.i			1.05752i	−0.848770	+	0.528762i	$0.822656\pi$
$72$		0			0
$73$	76522.4			0.196707			0.0983536	−	0.995152i	$-0.468642\pi$
$73$	76522.4			0.196707			0.0983536	+	0.995152i	$0.468642\pi$
$74$		0			0
$75$			134518.i			0.318857i
$76$		0			0
$77$	50083.8			0.109705
$78$		0			0
$79$		−	895511.i		−	1.81631i	−0.418635	−	0.908155i	$-0.637491\pi$
$79$		−	895511.i		−	1.81631i	0.418635	−	0.908155i	$-0.362509\pi$
$80$		0			0
$81$	339231.			0.638324
$82$		0			0
$83$	896640.			1.56814			0.784068	−	0.620675i	$-0.213141\pi$
$83$	896640.			1.56814			0.784068	+	0.620675i	$0.213141\pi$
$84$		0			0
$85$	146134.			0.237956
$86$		0			0
$87$	249668.			0.379145
$88$		0			0
$89$			1.17938e6i			1.67296i	0.548000	+	0.836479i	$0.315390\pi$
$89$			1.17938e6i			1.67296i	−0.548000	+	0.836479i	$0.684610\pi$
$90$		0			0
$91$			55575.2i			0.0737492i
$92$		0			0
$93$	−56559.1			−0.0703160
$94$		0			0
$95$	−237790.	+	133430.i	−0.277347	+	0.155626i
$96$		0			0
$97$			802702.i			0.879507i	0.898118	+	0.439754i	$0.144934\pi$
$97$			802702.i			0.879507i	−0.898118	+	0.439754i	$0.855066\pi$
$98$		0			0
$99$	1.29848e6			1.33822
$100$		0			0
$101$	1.65702e6			1.60829			0.804145	−	0.594433i	$-0.202623\pi$
$101$	1.65702e6			1.60829			0.804145	+	0.594433i	$0.202623\pi$
$102$		0			0
$103$			596460.i			0.545846i	0.962036	+	0.272923i	$0.0879904\pi$
$103$			596460.i			0.545846i	−0.962036	+	0.272923i	$0.912010\pi$
$104$		0			0
$105$		−	9358.88i		−	0.00808455i
$106$		0			0
$107$			1.73075e6i			1.41281i	0.707809	+	0.706404i	$0.249684\pi$
$107$			1.73075e6i			1.41281i	−0.707809	+	0.706404i	$0.750316\pi$
$108$		0			0
$109$		−	1.01914e6i		−	0.786960i	−0.919333	−	0.393480i	$-0.871271\pi$
$109$		−	1.01914e6i		−	0.786960i	0.919333	−	0.393480i	$-0.128729\pi$
$110$		0			0
$111$	125048.			0.0914338
$112$		0			0
$113$		−	2.78514e6i		−	1.93024i	−0.261807	−	0.965120i	$-0.584318\pi$
$113$		−	2.78514e6i		−	1.93024i	0.261807	−	0.965120i	$-0.415682\pi$
$114$		0			0
$115$	87509.3			0.0575388
$116$		0			0
$117$			1.44085e6i			0.899624i
$118$		0			0
$119$	90357.1			0.0536193
$120$		0			0
$121$	2.38017e6			1.34354
$122$		0			0
$123$	493250.			0.265065
$124$		0			0
$125$	−1.17947e6			−0.603887
$126$		0			0
$127$			764613.i			0.373276i	0.982429	+	0.186638i	$0.0597592\pi$
$127$			764613.i			0.373276i	−0.982429	+	0.186638i	$0.940241\pi$
$128$		0			0
$129$			93125.5i			0.0433810i
$130$		0			0
$131$	−1.01871e6			−0.453143			−0.226571	−	0.973995i	$-0.572752\pi$
$131$	−1.01871e6			−0.453143			−0.226571	+	0.973995i	$0.572752\pi$
$132$		0			0
$133$	−147029.	+	82501.6i	−0.0624955	+	0.0350677i
$134$		0			0
$135$		−	520207.i		−	0.211434i
$136$		0			0
$137$	2.68678e6			1.04489			0.522444	−	0.852673i	$-0.325020\pi$
$137$	2.68678e6			1.04489			0.522444	+	0.852673i	$0.325020\pi$
$138$		0			0
$139$	1.65305e6			0.615518			0.307759	−	0.951464i	$-0.400421\pi$
$139$	1.65305e6			0.615518			0.307759	+	0.951464i	$0.400421\pi$
$140$		0			0
$141$		−	1.31516e6i		−	0.469160i
$142$		0			0
$143$			4.60694e6i			1.57545i
$144$		0			0
$145$			1.03626e6i			0.339910i
$146$		0			0
$147$			1.12104e6i			0.352914i
$148$		0			0
$149$	3.00482e6			0.908364			0.454182	−	0.890909i	$-0.349932\pi$
$149$	3.00482e6			0.908364			0.454182	+	0.890909i	$0.349932\pi$
$150$		0			0
$151$			5.39411e6i			1.56671i	0.621574	+	0.783356i	$0.286494\pi$
$151$			5.39411e6i			1.56671i	−0.621574	+	0.783356i	$0.713506\pi$
$152$		0			0
$153$	2.34261e6			0.654071
$154$		0			0
$155$		−	234751.i		−	0.0630395i
$156$		0			0
$157$	896520.			0.231665			0.115833	−	0.993269i	$-0.463046\pi$
$157$	896520.			0.231665			0.115833	+	0.993269i	$0.463046\pi$
$158$		0			0
$159$	2.62594e6			0.653270
$160$		0			0
$161$	54108.3			0.0129654
$162$		0			0
$163$	−3.44670e6			−0.795867			−0.397933	−	0.917414i	$-0.630273\pi$
$163$	−3.44670e6			−0.795867			−0.397933	+	0.917414i	$0.630273\pi$
$164$		0			0
$165$		−	775811.i		−	0.172705i
$166$		0			0
$167$		−	1.96669e6i		−	0.422266i	−0.977457	−	0.211133i	$-0.932285\pi$
$167$		−	1.96669e6i		−	0.422266i	0.977457	−	0.211133i	$-0.0677152\pi$
$168$		0			0
$169$	−285266.			−0.0591002
$170$		0			0
$171$	−3.81189e6	+	2.13895e6i	−0.762347	+	0.427771i
$172$		0			0
$173$			1.97894e6i			0.382203i	0.981570	+	0.191101i	$0.0612060\pi$
$173$			1.97894e6i			0.382203i	−0.981570	+	0.191101i	$0.938794\pi$
$174$		0			0
$175$	−345219.			−0.0644140
$176$		0			0
$177$	−2.21526e6			−0.399489
$178$		0			0
$179$		−	6.54306e6i		−	1.14083i	−0.821356	−	0.570416i	$-0.806782\pi$
$179$		−	6.54306e6i		−	1.14083i	0.821356	−	0.570416i	$-0.193218\pi$
$180$		0			0
$181$			2.13054e6i			0.359297i	0.983731	+	0.179648i	$0.0574960\pi$
$181$			2.13054e6i			0.359297i	−0.983731	+	0.179648i	$0.942504\pi$
$182$		0			0
$183$			1.17690e6i			0.192038i
$184$		0			0
$185$			519016.i			0.0819720i
$186$		0			0
$187$	7.49021e6			1.14543
$188$		0			0
$189$		−	321652.i		−	0.0476431i
$190$		0			0
$191$	8.61775e6			1.23678			0.618392	−	0.785870i	$-0.287785\pi$
$191$	8.61775e6			1.23678			0.618392	+	0.785870i	$0.287785\pi$
$192$		0			0
$193$		−	6.76454e6i		−	0.940949i	−0.882414	−	0.470475i	$-0.844083\pi$
$193$		−	6.76454e6i		−	0.940949i	0.882414	−	0.470475i	$-0.155917\pi$
$194$		0			0
$195$	860875.			0.116101
$196$		0			0
$197$	8.65041e6			1.13146			0.565728	−	0.824592i	$-0.308595\pi$
$197$	8.65041e6			1.13146			0.565728	+	0.824592i	$0.308595\pi$
$198$		0			0
$199$	−7.87753e6			−0.999611			−0.499806	−	0.866138i	$-0.666595\pi$
$199$	−7.87753e6			−0.999611			−0.499806	+	0.866138i	$0.666595\pi$
$200$		0			0
$201$	−1.64379e6			−0.202422
$202$		0			0
$203$			640733.i			0.0765930i
$204$		0			0
$205$			2.04726e6i			0.237635i
$206$		0			0
$207$	1.40282e6			0.158158
$208$		0			0
$209$	−1.21881e7	+	6.83903e6i	−1.33505	+	0.749127i
$210$		0			0
$211$			9.54073e6i			1.01563i	0.861467	+	0.507814i	$0.169546\pi$
$211$			9.54073e6i			1.01563i	−0.861467	+	0.507814i	$0.830454\pi$
$212$		0			0
$213$	3.62521e6			0.375141
$214$		0			0
$215$	−386521.			−0.0388918
$216$		0			0
$217$		−	145150.i		−	0.0142049i
$218$		0			0
$219$		−	732920.i		−	0.0697790i
$220$		0			0
$221$			8.31147e6i			0.770019i
$222$		0			0
$223$		−	1.30556e7i		−	1.17729i	−0.808393	−	0.588643i	$-0.799662\pi$
$223$		−	1.30556e7i		−	1.17729i	0.808393	−	0.588643i	$-0.200338\pi$
$224$		0			0
$225$	−8.95018e6			−0.785749
$226$		0			0
$227$		−	2.04795e7i		−	1.75082i	−0.483381	−	0.875410i	$-0.660591\pi$
$227$		−	2.04795e7i		−	1.75082i	0.483381	−	0.875410i	$-0.339409\pi$
$228$		0			0
$229$	6.74824e6			0.561932			0.280966	−	0.959718i	$-0.409345\pi$
$229$	6.74824e6			0.561932			0.280966	+	0.959718i	$0.409345\pi$
$230$		0			0
$231$		−	479695.i		−	0.0389161i
$232$		0			0
$233$	−2.05770e7			−1.62673			−0.813363	−	0.581756i	$-0.802366\pi$
$233$	−2.05770e7			−1.62673			−0.813363	+	0.581756i	$0.802366\pi$
$234$		0			0
$235$	5.45862e6			0.420610
$236$		0			0
$237$	−8.57708e6			−0.644309
$238$		0			0
$239$	−6.04676e6			−0.442924			−0.221462	−	0.975169i	$-0.571083\pi$
$239$	−6.04676e6			−0.442924			−0.221462	+	0.975169i	$0.571083\pi$
$240$		0			0
$241$		−	2.10356e6i		−	0.150281i	−0.997173	−	0.0751406i	$-0.976059\pi$
$241$		−	2.10356e6i		−	0.150281i	0.997173	−	0.0751406i	$-0.0239405\pi$
$242$		0			0
$243$		−	1.27887e7i		−	0.891268i
$244$		0			0
$245$	−4.65292e6			−0.316393
$246$		0			0
$247$	−7.58889e6	−	1.35245e7i	−0.503602	−	0.897488i
$248$		0			0
$249$		−	8.58789e6i		−	0.556273i
$250$		0			0
$251$	5.40271e6			0.341657			0.170829	−	0.985301i	$-0.445356\pi$
$251$	5.40271e6			0.341657			0.170829	+	0.985301i	$0.445356\pi$
$252$		0			0
$253$	4.48534e6			0.276971
$254$		0			0
$255$		−	1.39965e6i		−	0.0844112i
$256$		0			0
$257$		−	3.06890e6i		−	0.180794i	−0.995906	−	0.0903969i	$-0.971186\pi$
$257$		−	3.06890e6i		−	0.180794i	0.995906	−	0.0903969i	$-0.0288136\pi$
$258$		0			0
$259$			320915.i			0.0184710i
$260$		0			0
$261$			1.66117e7i			0.934314i
$262$		0			0
$263$	−3.44896e6			−0.189593			−0.0947963	−	0.995497i	$-0.530220\pi$
$263$	−3.44896e6			−0.189593			−0.0947963	+	0.995497i	$0.530220\pi$
$264$		0			0
$265$			1.08991e7i			0.585668i
$266$		0			0
$267$	1.12960e7			0.593457
$268$		0			0
$269$		−	1.79669e7i		−	0.923029i	−0.887133	−	0.461514i	$-0.847306\pi$
$269$		−	1.79669e7i		−	0.923029i	0.887133	−	0.461514i	$-0.152694\pi$
$270$		0			0
$271$	−1.37060e7			−0.688658			−0.344329	−	0.938849i	$-0.611894\pi$
$271$	−1.37060e7			−0.688658			−0.344329	+	0.938849i	$0.611894\pi$
$272$		0			0
$273$	532291.			0.0261614
$274$		0			0
$275$	−2.86171e7			−1.37603
$276$		0			0
$277$	−2.02798e7			−0.954166			−0.477083	−	0.878858i	$-0.658306\pi$
$277$	−2.02798e7			−0.954166			−0.477083	+	0.878858i	$0.658306\pi$
$278$		0			0
$279$		−	3.76318e6i		−	0.173277i
$280$		0			0
$281$		−	5.85077e6i		−	0.263690i	−0.991270	−	0.131845i	$-0.957910\pi$
$281$		−	5.85077e6i		−	0.263690i	0.991270	−	0.131845i	$-0.0420902\pi$
$282$		0			0
$283$	−1.54149e7			−0.680112			−0.340056	−	0.940405i	$-0.610446\pi$
$283$	−1.54149e7			−0.680112			−0.340056	+	0.940405i	$0.610446\pi$
$284$		0			0
$285$	1.27797e6	+	2.27752e6i	0.0552060	+	0.0983847i
$286$		0			0
$287$			1.26585e6i			0.0535471i
$288$		0			0
$289$	−1.06243e7			−0.440158
$290$		0			0
$291$	7.68817e6			0.311992
$292$		0			0
$293$			4.41322e6i			0.175450i	0.996145	+	0.0877250i	$0.0279597\pi$
$293$			4.41322e6i			0.175450i	−0.996145	+	0.0877250i	$0.972040\pi$
$294$		0			0
$295$		−	9.19453e6i		−	0.358149i
$296$		0			0
$297$		−	2.66635e7i		−	1.01777i
$298$		0			0
$299$			4.97714e6i			0.186194i
$300$		0			0
$301$	−238992.			−0.00876362
$302$		0			0
$303$		−	1.58707e7i		−	0.570517i
$304$		0			0
$305$	−4.88479e6			−0.172165
$306$		0			0
$307$			1.99040e7i			0.687900i	0.938988	+	0.343950i	$0.111765\pi$
$307$			1.99040e7i			0.687900i	−0.938988	+	0.343950i	$0.888235\pi$
$308$		0			0
$309$	5.71281e6			0.193631
$310$		0			0
$311$	−6.97619e6			−0.231919			−0.115960	−	0.993254i	$-0.536994\pi$
$311$	−6.97619e6			−0.231919			−0.115960	+	0.993254i	$0.536994\pi$
$312$		0			0
$313$	−2.89181e7			−0.943055			−0.471528	−	0.881851i	$-0.656297\pi$
$313$	−2.89181e7			−0.943055			−0.471528	+	0.881851i	$0.656297\pi$
$314$		0			0
$315$	622695.			0.0199225
$316$		0			0
$317$			2.10910e7i			0.662092i	0.943615	+	0.331046i	$0.107402\pi$
$317$			2.10910e7i			0.662092i	−0.943615	+	0.331046i	$0.892598\pi$
$318$		0			0
$319$			5.31140e7i			1.63620i
$320$		0			0
$321$	1.65769e7			0.501173
$322$		0			0
$323$	−2.19887e7	+	1.23384e7i	−0.652519	+	0.366144i
$324$		0			0
$325$		−	3.17549e7i		−	0.925039i
$326$		0			0
$327$	−9.76114e6			−0.279163
$328$		0			0
$329$	3.37515e6			0.0947774
$330$		0			0
$331$		−	1.16821e7i		−	0.322133i	−0.986944	−	0.161066i	$-0.948507\pi$
$331$		−	1.16821e7i		−	0.322133i	0.986944	−	0.161066i	$-0.0514934\pi$
$332$		0			0
$333$			8.32008e6i			0.225317i
$334$		0			0
$335$		−	6.82261e6i		−	0.181475i
$336$		0			0
$337$		−	5.89357e7i		−	1.53989i	−0.638112	−	0.769944i	$-0.720284\pi$
$337$		−	5.89357e7i		−	1.53989i	0.638112	−	0.769944i	$-0.279716\pi$
$338$		0			0
$339$	−2.66756e7			−0.684724
$340$		0			0
$341$		−	1.20323e7i		−	0.303449i
$342$		0			0
$343$	−5.76878e6			−0.142956
$344$		0			0
$345$		−	838152.i		−	0.0204110i
$346$		0			0
$347$	−2.96423e7			−0.709452			−0.354726	−	0.934970i	$-0.615426\pi$
$347$	−2.96423e7			−0.709452			−0.354726	+	0.934970i	$0.615426\pi$
$348$		0			0
$349$	3.30243e7			0.776885			0.388443	−	0.921473i	$-0.373013\pi$
$349$	3.30243e7			0.776885			0.388443	+	0.921473i	$0.373013\pi$
$350$		0			0
$351$	2.95870e7			0.684196
$352$		0			0
$353$	−4.25129e7			−0.966487			−0.483244	−	0.875486i	$-0.660542\pi$
$353$	−4.25129e7			−0.966487			−0.483244	+	0.875486i	$0.660542\pi$
$354$		0			0
$355$			1.50466e7i			0.336320i
$356$		0			0
$357$		−	865427.i		−	0.0190207i
$358$		0			0
$359$	6.57363e7			1.42076			0.710382	−	0.703817i	$-0.248522\pi$
$359$	6.57363e7			1.42076			0.710382	+	0.703817i	$0.248522\pi$
$360$		0			0
$361$	2.45144e7	−	4.01542e7i	0.521074	−	0.853511i
$362$		0			0
$363$		−	2.27969e7i		−	0.476602i
$364$		0			0
$365$	3.04202e6			0.0625581
$366$		0			0
$367$	−8.47087e7			−1.71368			−0.856841	−	0.515581i	$-0.827576\pi$
$367$	−8.47087e7			−1.71368			−0.856841	+	0.515581i	$0.827576\pi$
$368$		0			0
$369$			3.28185e7i			0.653190i
$370$		0			0
$371$			6.73905e6i			0.131971i
$372$		0			0
$373$		−	2.47658e7i		−	0.477229i	−0.971114	−	0.238614i	$-0.923307\pi$
$373$		−	2.47658e7i		−	0.477229i	0.971114	−	0.238614i	$-0.0766932\pi$
$374$		0			0
$375$			1.12968e7i			0.214220i
$376$		0			0
$377$	−5.89377e7			−1.09994
$378$		0			0
$379$			8.03807e7i			1.47650i	0.674526	+	0.738251i	$0.264348\pi$
$379$			8.03807e7i			1.47650i	−0.674526	+	0.738251i	$0.735652\pi$
$380$		0			0
$381$	7.32335e6			0.132414
$382$		0			0
$383$			2.89276e7i			0.514891i	0.966293	+	0.257446i	$0.0828808\pi$
$383$			2.89276e7i			0.514891i	−0.966293	+	0.257446i	$0.917119\pi$
$384$		0			0
$385$	1.99099e6			0.0348890
$386$		0			0
$387$	−6.19612e6			−0.106902
$388$		0			0
$389$	9.89845e7			1.68158			0.840792	−	0.541359i	$-0.182090\pi$
$389$	9.89845e7			1.68158			0.840792	+	0.541359i	$0.182090\pi$
$390$		0			0
$391$	8.09209e6			0.135373
$392$		0			0
$393$			9.75702e6i			0.160746i
$394$		0			0
$395$		−	3.55995e7i		−	0.577634i
$396$		0			0
$397$	7.98371e7			1.27595			0.637974	−	0.770058i	$-0.279773\pi$
$397$	7.98371e7			1.27595			0.637974	+	0.770058i	$0.279773\pi$
$398$		0			0
$399$	790188.	+	1.40822e6i	0.0124398	+	0.0221694i
$400$		0			0
$401$			6.92918e7i			1.07460i	0.843390	+	0.537302i	$0.180557\pi$
$401$			6.92918e7i			1.07460i	−0.843390	+	0.537302i	$0.819443\pi$
$402$		0			0
$403$	1.33516e7			0.203994
$404$		0			0
$405$	1.34856e7			0.203004
$406$		0			0
$407$			2.66025e7i			0.394583i
$408$		0			0
$409$		−	6.56471e7i		−	0.959502i	−0.877405	−	0.479751i	$-0.840727\pi$
$409$		−	6.56471e7i		−	0.959502i	0.877405	−	0.479751i	$-0.159273\pi$
$410$		0			0
$411$		−	2.57336e7i		−	0.370659i
$412$		0			0
$413$		−	5.68511e6i		−	0.0807028i
$414$		0			0
$415$	3.56444e7			0.498709
$416$		0			0
$417$		−	1.58326e7i		−	0.218346i
$418$		0			0
$419$	1.08833e7			0.147951			0.0739756	−	0.997260i	$-0.476431\pi$
$419$	1.08833e7			0.147951			0.0739756	+	0.997260i	$0.476431\pi$
$420$		0			0
$421$			1.30849e8i			1.75357i	0.480879	+	0.876787i	$0.340318\pi$
$421$			1.30849e8i			1.75357i	−0.480879	+	0.876787i	$0.659682\pi$
$422$		0			0
$423$	8.75044e7			1.15614
$424$		0			0
$425$	−5.16287e7			−0.672550
$426$		0			0
$427$	−3.02034e6			−0.0387946
$428$		0			0
$429$	4.41246e7			0.558868
$430$		0			0
$431$		−	9.85958e7i		−	1.23148i	−0.787950	−	0.615739i	$-0.788858\pi$
$431$		−	9.85958e7i		−	1.23148i	0.787950	−	0.615739i	$-0.211142\pi$
$432$		0			0
$433$		−	7.55457e7i		−	0.930564i	−0.885163	−	0.465282i	$-0.845953\pi$
$433$		−	7.55457e7i		−	0.930564i	0.885163	−	0.465282i	$-0.154047\pi$
$434$		0			0
$435$	9.92512e6			0.120578
$436$		0			0
$437$	−1.31675e7	+	7.38858e6i	−0.157782	+	0.0885354i
$438$		0			0
$439$		−	5.81343e7i		−	0.687130i	−0.939129	−	0.343565i	$-0.888365\pi$
$439$		−	5.81343e7i		−	0.687130i	0.939129	−	0.343565i	$-0.111635\pi$
$440$		0			0
$441$	−7.45885e7			−0.869674
$442$		0			0
$443$	7.18816e7			0.826811			0.413406	−	0.910547i	$-0.364339\pi$
$443$	7.18816e7			0.826811			0.413406	+	0.910547i	$0.364339\pi$
$444$		0			0
$445$			4.68844e7i			0.532044i
$446$		0			0
$447$		−	2.87797e7i		−	0.322229i
$448$		0			0
$449$			3.66408e7i			0.404787i	0.979304	+	0.202393i	$0.0648720\pi$
$449$			3.66408e7i			0.404787i	−0.979304	+	0.202393i	$0.935128\pi$
$450$		0			0
$451$			1.04933e8i			1.14389i
$452$		0			0
$453$	5.16640e7			0.555768
$454$		0			0
$455$			2.20930e6i			0.0234542i
$456$		0			0
$457$	−2.83569e7			−0.297105			−0.148553	−	0.988905i	$-0.547461\pi$
$457$	−2.83569e7			−0.297105			−0.148553	+	0.988905i	$0.547461\pi$
$458$		0			0
$459$		−	4.81041e7i		−	0.497444i
$460$		0			0
$461$	8.22265e7			0.839284			0.419642	−	0.907690i	$-0.362156\pi$
$461$	8.22265e7			0.839284			0.419642	+	0.907690i	$0.362156\pi$
$462$		0			0
$463$	−7.69296e7			−0.775087			−0.387544	−	0.921851i	$-0.626676\pi$
$463$	−7.69296e7			−0.775087			−0.387544	+	0.921851i	$0.626676\pi$
$464$		0			0
$465$	−2.24841e6			−0.0223623
$466$		0			0
$467$	8.31699e7			0.816612			0.408306	−	0.912845i	$-0.366120\pi$
$467$	8.31699e7			0.816612			0.408306	+	0.912845i	$0.366120\pi$
$468$		0			0
$469$		−	4.21852e6i		−	0.0408923i
$470$		0			0
$471$		−	8.58674e6i		−	0.0821799i
$472$		0			0
$473$	−1.98114e7			−0.187211
$474$		0			0
$475$	8.40103e7	−	4.71402e7i	0.783884	−	0.439856i
$476$		0			0
$477$			1.74717e8i			1.60983i
$478$		0			0
$479$	1.29133e6			0.0117499			0.00587493	−	0.999983i	$-0.498130\pi$
$479$	1.29133e6			0.0117499			0.00587493	+	0.999983i	$0.498130\pi$
$480$		0			0
$481$	−2.95193e7			−0.265260
$482$		0			0
$483$		−	518241.i		−	0.00459929i
$484$		0			0
$485$			3.19101e7i			0.279706i
$486$		0			0
$487$			7.68906e7i			0.665712i	0.942978	+	0.332856i	$0.108012\pi$
$487$			7.68906e7i			0.665712i	−0.942978	+	0.332856i	$0.891988\pi$
$488$		0			0
$489$			3.30120e7i			0.282322i
$490$		0			0
$491$	4.30715e7			0.363869			0.181934	−	0.983311i	$-0.441764\pi$
$491$	4.30715e7			0.363869			0.181934	+	0.983311i	$0.441764\pi$
$492$		0			0
$493$			9.58240e7i			0.799711i
$494$		0			0
$495$	5.16187e7			0.425590
$496$		0			0
$497$			9.30353e6i			0.0757842i
$498$		0			0
$499$	1.81528e8			1.46097			0.730485	−	0.682929i	$-0.239294\pi$
$499$	1.81528e8			1.46097			0.730485	+	0.682929i	$0.239294\pi$
$500$		0			0
$501$	−1.88366e7			−0.149793
$502$		0			0
$503$	1.30674e8			1.02680			0.513401	−	0.858149i	$-0.328385\pi$
$503$	1.30674e8			1.02680			0.513401	+	0.858149i	$0.328385\pi$
$504$		0			0
$505$	6.58721e7			0.511479
$506$		0			0
$507$			2.73223e6i			0.0209649i
$508$		0			0
$509$			1.32034e8i			1.00122i	0.865672	+	0.500611i	$0.166891\pi$
$509$			1.32034e8i			1.00122i	−0.865672	+	0.500611i	$0.833109\pi$
$510$		0			0
$511$	1.88092e6			0.0140964
$512$		0			0
$513$	4.39221e7	+	7.82752e7i	0.325335	+	0.579792i
$514$		0			0
$515$			2.37113e7i			0.173593i
$516$		0			0
$517$	2.79785e8			2.02466
$518$		0			0
$519$	1.89540e7			0.135581
$520$		0			0
$521$		−	7.76766e7i		−	0.549259i	−0.961550	−	0.274629i	$-0.911445\pi$
$521$		−	7.76766e7i		−	0.549259i	0.961550	−	0.274629i	$-0.0885552\pi$
$522$		0			0
$523$			2.29556e8i			1.60466i	0.596878	+	0.802332i	$0.296408\pi$
$523$			2.29556e8i			1.60466i	−0.596878	+	0.802332i	$0.703592\pi$
$524$		0			0
$525$			3.30645e6i			0.0228499i
$526$		0			0
$527$		−	2.17077e7i		−	0.148314i
$528$		0			0
$529$	−1.43190e8			−0.967266
$530$		0			0
$531$		−	1.47393e8i		−	0.984448i
$532$		0			0
$533$	−1.16439e8			−0.768982
$534$		0			0
$535$			6.88031e7i			0.449310i
$536$		0			0
$537$	−6.26684e7			−0.404693
$538$		0			0
$539$	−2.38488e8			−1.52300
$540$		0			0
$541$	2.48772e8			1.57112			0.785561	−	0.618785i	$-0.212375\pi$
$541$	2.48772e8			1.57112			0.785561	+	0.618785i	$0.212375\pi$
$542$		0			0
$543$	2.04060e7			0.127455
$544$		0			0
$545$		−	4.05140e7i		−	0.250274i
$546$		0			0
$547$			9.76194e6i			0.0596450i	0.999555	+	0.0298225i	$0.00949421\pi$
$547$			9.76194e6i			0.0596450i	−0.999555	+	0.0298225i	$0.990506\pi$
$548$		0			0
$549$	−7.83055e7			−0.473233
$550$		0			0
$551$	−8.74932e7	−	1.55925e8i	−0.523022	−	0.932096i
$552$		0			0
$553$		−	2.20117e7i		−	0.130160i
$554$		0			0
$555$	4.97106e6			0.0290784
$556$		0			0
$557$	−6.00727e7			−0.347625			−0.173813	−	0.984779i	$-0.555609\pi$
$557$	−6.00727e7			−0.347625			−0.173813	+	0.984779i	$0.555609\pi$
$558$		0			0
$559$		−	2.19836e7i		−	0.125853i
$560$		0			0
$561$		−	7.17401e7i		−	0.406325i
$562$		0			0
$563$		−	3.10421e8i		−	1.73950i	−0.493489	−	0.869752i	$-0.664279\pi$
$563$		−	3.10421e8i		−	1.73950i	0.493489	−	0.869752i	$-0.335721\pi$
$564$		0			0
$565$		−	1.10718e8i		−	0.613867i
$566$		0			0
$567$	8.33832e6			0.0457435
$568$		0			0
$569$		−	9.88441e6i		−	0.0536555i	−0.999640	−	0.0268277i	$-0.991459\pi$
$569$		−	9.88441e6i		−	0.0536555i	0.999640	−	0.0268277i	$-0.00854056\pi$
$570$		0			0
$571$	−1.10963e8			−0.596032			−0.298016	−	0.954561i	$-0.596325\pi$
$571$	−1.10963e8			−0.596032			−0.298016	+	0.954561i	$0.596325\pi$
$572$		0			0
$573$		−	8.25395e7i		−	0.438731i
$574$		0			0
$575$	−3.09167e7			−0.162626
$576$		0			0
$577$	−2.09358e8			−1.08984			−0.544920	−	0.838488i	$-0.683440\pi$
$577$	−2.09358e8			−1.08984			−0.544920	+	0.838488i	$0.683440\pi$
$578$		0			0
$579$	−6.47897e7			−0.333788
$580$		0			0
$581$	2.20395e7			0.112376
$582$		0			0
$583$			5.58638e8i			2.81919i
$584$		0			0
$585$			5.72785e7i			0.286104i
$586$		0			0
$587$	−1.11290e8			−0.550227			−0.275113	−	0.961412i	$-0.588715\pi$
$587$	−1.11290e8			−0.550227			−0.275113	+	0.961412i	$0.588715\pi$
$588$		0			0
$589$	1.98205e7	+	3.53228e7i	0.0969993	+	0.172866i
$590$		0			0
$591$		−	8.28523e7i		−	0.401368i
$592$		0			0
$593$	−2.06430e8			−0.989939			−0.494970	−	0.868910i	$-0.664821\pi$
$593$	−2.06430e8			−0.989939			−0.494970	+	0.868910i	$0.664821\pi$
$594$		0			0
$595$	3.59199e6			0.0170524
$596$		0			0
$597$			7.54499e7i			0.354597i
$598$		0			0
$599$			5.95076e7i			0.276880i	0.990371	+	0.138440i	$0.0442088\pi$
$599$			5.95076e7i			0.276880i	−0.990371	+	0.138440i	$0.955791\pi$
$600$		0			0
$601$		−	1.64566e8i		−	0.758085i	−0.925379	−	0.379043i	$-0.876253\pi$
$601$		−	1.64566e8i		−	0.758085i	0.925379	−	0.379043i	$-0.123747\pi$
$602$		0			0
$603$		−	1.09370e8i		−	0.498822i
$604$		0			0
$605$	9.46194e7			0.427282
$606$		0			0
$607$		−	1.22663e8i		−	0.548461i	−0.961664	−	0.274231i	$-0.911577\pi$
$607$		−	1.22663e8i		−	0.548461i	0.961664	−	0.274231i	$-0.0884232\pi$
$608$		0			0
$609$	6.13685e6			0.0271702
$610$		0			0
$611$			3.10462e8i			1.36108i
$612$		0			0
$613$	4.19746e8			1.82224			0.911119	−	0.412144i	$-0.135220\pi$
$613$	4.19746e8			1.82224			0.911119	+	0.412144i	$0.135220\pi$
$614$		0			0
$615$	1.96083e7			0.0842976
$616$		0			0
$617$	−2.03358e8			−0.865779			−0.432889	−	0.901447i	$-0.642506\pi$
$617$	−2.03358e8			−0.865779			−0.432889	+	0.901447i	$0.642506\pi$
$618$		0			0
$619$	−5.67160e7			−0.239130			−0.119565	−	0.992826i	$-0.538150\pi$
$619$	−5.67160e7			−0.239130			−0.119565	+	0.992826i	$0.538150\pi$
$620$		0			0
$621$		−	2.88061e7i		−	0.120284i
$622$		0			0
$623$			2.89893e7i			0.119887i
$624$		0			0
$625$	1.72560e8			0.706807
$626$		0			0
$627$	6.55032e7	+	1.16736e8i	0.265742	+	0.473588i
$628$		0			0
$629$			4.79940e7i			0.192857i
$630$		0			0
$631$	−1.70660e7			−0.0679272			−0.0339636	−	0.999423i	$-0.510813\pi$
$631$	−1.70660e7			−0.0679272			−0.0339636	+	0.999423i	$0.510813\pi$
$632$		0			0
$633$	9.13797e7			0.360279
$634$		0			0
$635$			3.03959e7i			0.118712i
$636$		0			0
$637$		−	2.64637e8i		−	1.02384i
$638$		0			0
$639$			2.41204e8i			0.924448i
$640$		0			0
$641$		−	3.32488e8i		−	1.26242i	−0.775614	−	0.631208i	$-0.782560\pi$
$641$		−	3.32488e8i		−	1.26242i	0.775614	−	0.631208i	$-0.217440\pi$
$642$		0			0
$643$	−8.53612e7			−0.321091			−0.160545	−	0.987028i	$-0.551325\pi$
$643$	−8.53612e7			−0.321091			−0.160545	+	0.987028i	$0.551325\pi$
$644$		0			0
$645$			3.70204e6i			0.0137963i
$646$		0			0
$647$	−1.48441e8			−0.548077			−0.274038	−	0.961719i	$-0.588360\pi$
$647$	−1.48441e8			−0.548077			−0.274038	+	0.961719i	$0.588360\pi$
$648$		0			0
$649$		−	4.71271e8i		−	1.72400i
$650$		0			0
$651$	−1.39023e6			−0.00503898
$652$		0			0
$653$	3.79406e6			0.0136259			0.00681293	−	0.999977i	$-0.497831\pi$
$653$	3.79406e6			0.0136259			0.00681293	+	0.999977i	$0.497831\pi$
$654$		0			0
$655$	−4.04969e7			−0.144111
$656$		0			0
$657$	4.87650e7			0.171954
$658$		0			0
$659$		−	3.30971e8i		−	1.15647i	−0.815871	−	0.578233i	$-0.803742\pi$
$659$		−	3.30971e8i		−	1.15647i	0.815871	−	0.578233i	$-0.196258\pi$
$660$		0			0
$661$		−	4.18510e8i		−	1.44911i	−0.689217	−	0.724555i	$-0.742045\pi$
$661$		−	4.18510e8i		−	1.44911i	0.689217	−	0.724555i	$-0.257955\pi$
$662$		0			0
$663$	7.96061e7			0.273153
$664$		0			0
$665$	−5.84490e6	+	3.27971e6i	−0.0198752	+	0.0111525i
$666$		0			0
$667$			5.73820e7i			0.193374i
$668$		0			0
$669$	−1.25045e8			−0.417625
$670$		0			0
$671$	−2.50373e8			−0.828742
$672$		0			0
$673$			1.49735e8i			0.491224i	0.969368	+	0.245612i	$0.0789889\pi$
$673$			1.49735e8i			0.491224i	−0.969368	+	0.245612i	$0.921011\pi$
$674$		0			0
$675$			1.83787e8i			0.597590i
$676$		0			0
$677$			4.14666e8i			1.33639i	0.743987	+	0.668194i	$0.232933\pi$
$677$			4.14666e8i			1.33639i	−0.743987	+	0.668194i	$0.767067\pi$
$678$		0			0
$679$			1.97305e7i			0.0630272i
$680$		0			0
$681$	−1.96150e8			−0.621078
$682$		0			0
$683$		−	4.96037e8i		−	1.55687i	−0.627725	−	0.778435i	$-0.716014\pi$
$683$		−	4.96037e8i		−	1.55687i	0.627725	−	0.778435i	$-0.283986\pi$
$684$		0			0
$685$	1.06808e8			0.332302
$686$		0			0
$687$		−	6.46336e7i		−	0.199337i
$688$		0			0
$689$	−6.19890e8			−1.89521
$690$		0			0
$691$	4.16636e7			0.126277			0.0631383	−	0.998005i	$-0.479889\pi$
$691$	4.16636e7			0.126277			0.0631383	+	0.998005i	$0.479889\pi$
$692$		0			0
$693$	3.19166e7			0.0958997
$694$		0			0
$695$	6.57141e7			0.195751
$696$		0			0
$697$			1.89312e8i			0.559088i
$698$		0			0
$699$			1.97084e8i			0.577057i
$700$		0			0
$701$	−4.52831e8			−1.31457			−0.657283	−	0.753644i	$-0.728294\pi$
$701$	−4.52831e8			−1.31457			−0.657283	+	0.753644i	$0.728294\pi$
$702$		0			0
$703$	−4.38215e7	−	7.80960e7i	−0.126131	−	0.224783i
$704$		0			0
$705$		−	5.22819e7i		−	0.149205i
$706$		0			0
$707$	4.07297e7			0.115253
$708$		0			0
$709$	3.93524e8			1.10416			0.552080	−	0.833791i	$-0.313834\pi$
$709$	3.93524e8			1.10416			0.552080	+	0.833791i	$0.313834\pi$
$710$		0			0
$711$		−	5.70678e8i		−	1.58775i
$712$		0			0
$713$		−	1.29992e7i		−	0.0358631i
$714$		0			0
$715$			1.83141e8i			0.501035i
$716$		0			0
$717$			5.79150e7i			0.157121i
$718$		0			0
$719$	−6.60835e8			−1.77790			−0.888949	−	0.458007i	$-0.848563\pi$
$719$	−6.60835e8			−1.77790			−0.888949	+	0.458007i	$0.848563\pi$
$720$		0			0
$721$			1.46610e7i			0.0391164i
$722$		0			0
$723$	−2.01476e7			−0.0533100
$724$		0			0
$725$		−	3.66106e8i		−	0.960710i
$726$		0			0
$727$	−6.20490e8			−1.61485			−0.807424	−	0.589971i	$-0.799139\pi$
$727$	−6.20490e8			−1.61485			−0.807424	+	0.589971i	$0.799139\pi$
$728$		0			0
$729$	1.24811e8			0.322159
$730$		0			0
$731$	−3.57421e7			−0.0915014
$732$		0			0
$733$	−5.26030e8			−1.33567			−0.667835	−	0.744310i	$-0.732779\pi$
$733$	−5.26030e8			−1.33567			−0.667835	+	0.744310i	$0.732779\pi$
$734$		0			0
$735$			4.45650e7i			0.112236i
$736$		0			0
$737$		−	3.49697e8i		−	0.873553i
$738$		0			0
$739$	−4.27969e8			−1.06042			−0.530211	−	0.847865i	$-0.677887\pi$
$739$	−4.27969e8			−1.06042			−0.530211	+	0.847865i	$0.677887\pi$
$740$		0			0
$741$	−1.29535e8	+	7.26853e7i	−0.318371	+	0.178645i
$742$		0			0
$743$		−	5.98413e8i		−	1.45893i	−0.684018	−	0.729465i	$-0.739769\pi$
$743$		−	5.98413e8i		−	1.45893i	0.684018	−	0.729465i	$-0.260231\pi$
$744$		0			0
$745$	1.19452e8			0.288884
$746$		0			0
$747$	5.71397e8			1.37081
$748$		0			0
$749$			4.25419e7i			0.101245i
$750$		0			0
$751$			1.99975e8i			0.472125i	0.971738	+	0.236062i	$0.0758570\pi$
$751$			1.99975e8i			0.472125i	−0.971738	+	0.236062i	$0.924143\pi$
$752$		0			0
$753$		−	5.17464e7i		−	0.121198i
$754$		0			0
$755$			2.14434e8i			0.498256i
$756$		0			0
$757$	3.41631e7			0.0787535			0.0393768	−	0.999224i	$-0.487463\pi$
$757$	3.41631e7			0.0787535			0.0393768	+	0.999224i	$0.487463\pi$
$758$		0			0
$759$		−	4.29600e7i		−	0.0982514i
$760$		0			0
$761$	−1.12757e7			−0.0255853			−0.0127927	−	0.999918i	$-0.504072\pi$
$761$	−1.12757e7			−0.0255853			−0.0127927	+	0.999918i	$0.504072\pi$
$762$		0			0
$763$		−	2.50504e7i		−	0.0563951i
$764$		0			0
$765$	9.31263e7			0.208012
$766$		0			0
$767$	5.22944e8			1.15896
$768$		0			0
$769$	−7.58702e8			−1.66837			−0.834184	−	0.551486i	$-0.814061\pi$
$769$	−7.58702e8			−1.66837			−0.834184	+	0.551486i	$0.814061\pi$
$770$		0			0
$771$	−2.93935e7			−0.0641339
$772$		0			0
$773$			7.99380e8i			1.73067i	0.501191	+	0.865337i	$0.332895\pi$
$773$			7.99380e8i			1.73067i	−0.501191	+	0.865337i	$0.667105\pi$
$774$		0			0
$775$			8.29366e7i			0.178173i
$776$		0			0
$777$	3.07368e6			0.00655233
$778$		0			0
$779$	−1.72854e8	−	3.08049e8i	−0.365651	−	0.651640i
$780$		0			0
$781$			7.71222e8i			1.61892i
$782$		0			0
$783$	3.41113e8			0.710579
$784$		0			0
$785$	3.56396e7			0.0736757
$786$		0			0
$787$		−	4.00150e8i		−	0.820915i	−0.911880	−	0.410457i	$-0.865369\pi$
$787$		−	4.00150e8i		−	0.820915i	0.911880	−	0.410457i	$-0.134631\pi$
$788$		0			0
$789$			3.30337e7i			0.0672552i
$790$		0			0
$791$		−	6.84588e7i		−	0.138325i
$792$		0			0
$793$		−	2.77825e8i		−	0.557124i
$794$		0			0
$795$	1.04390e8			0.207757
$796$		0			0
$797$		−	5.45772e7i		−	0.107804i	−0.998546	−	0.0539022i	$-0.982834\pi$
$797$		−	5.45772e7i		−	0.107804i	0.998546	−	0.0539022i	$-0.0171659\pi$
$798$		0			0
$799$	5.04765e8			0.989576
$800$		0			0
$801$			7.51579e8i			1.46244i
$802$		0			0
$803$	1.55920e8			0.301132
$804$		0			0
$805$	2.15098e6			0.00412334
$806$		0			0
$807$	−1.72084e8			−0.327431
$808$		0			0
$809$	6.85746e8			1.29514			0.647571	−	0.762005i	$-0.275785\pi$
$809$	6.85746e8			1.29514			0.647571	+	0.762005i	$0.275785\pi$
$810$		0			0
$811$		−	7.18103e8i		−	1.34625i	−0.739530	−	0.673123i	$-0.764952\pi$
$811$		−	7.18103e8i		−	1.34625i	0.739530	−	0.673123i	$-0.235048\pi$
$812$		0			0
$813$			1.31274e8i			0.244291i
$814$		0			0
$815$	−1.37018e8			−0.253107
$816$		0			0
$817$	5.81596e7	−	3.26347e7i	0.106649	−	0.0598431i
$818$		0			0
$819$			3.54161e7i			0.0644688i
$820$		0			0
$821$	−5.74972e7			−0.103900			−0.0519502	−	0.998650i	$-0.516544\pi$
$821$	−5.74972e7			−0.103900			−0.0519502	+	0.998650i	$0.516544\pi$
$822$		0			0
$823$	5.30659e8			0.951954			0.475977	−	0.879458i	$-0.342095\pi$
$823$	5.30659e8			0.951954			0.475977	+	0.879458i	$0.342095\pi$
$824$		0			0
$825$			2.74091e8i			0.488127i
$826$		0			0
$827$			2.50522e8i			0.442924i	0.975169	+	0.221462i	$0.0710829\pi$
$827$			2.50522e8i			0.442924i	−0.975169	+	0.221462i	$0.928917\pi$
$828$		0			0
$829$		−	8.38543e8i		−	1.47184i	−0.677067	−	0.735922i	$-0.736749\pi$
$829$		−	8.38543e8i		−	1.47184i	0.677067	−	0.735922i	$-0.263251\pi$
$830$		0			0
$831$			1.94237e8i			0.338476i
$832$		0			0
$833$	−4.30261e8			−0.744383
$834$		0			0
$835$		−	7.81823e7i		−	0.134292i
$836$		0			0
$837$	−7.72748e7			−0.131784
$838$		0			0
$839$			7.20856e8i			1.22057i	0.792182	+	0.610285i	$0.208945\pi$
$839$			7.20856e8i			1.22057i	−0.792182	+	0.610285i	$0.791055\pi$
$840$		0			0
$841$	−8.46764e7			−0.142356
$842$		0			0
$843$	−5.60378e7			−0.0935402
$844$		0			0
$845$	−1.13402e7			−0.0187954
$846$		0			0
$847$	5.85046e7			0.0962808
$848$		0			0
$849$			1.47641e8i			0.241260i
$850$		0			0
$851$			2.87401e7i			0.0466338i
$852$		0			0
$853$	−6.93429e8			−1.11726			−0.558631	−	0.829417i	$-0.688673\pi$
$853$	−6.93429e8			−1.11726			−0.558631	+	0.829417i	$0.688673\pi$
$854$		0			0
$855$	−1.51535e8	+	8.50301e7i	−0.242446	+	0.136042i
$856$		0			0
$857$			8.66899e8i			1.37729i	0.725098	+	0.688646i	$0.241795\pi$
$857$			8.66899e8i			1.37729i	−0.725098	+	0.688646i	$0.758205\pi$
$858$		0			0
$859$	−9.01517e8			−1.42231			−0.711155	−	0.703035i	$-0.751828\pi$
$859$	−9.01517e8			−1.42231			−0.711155	+	0.703035i	$0.751828\pi$
$860$		0			0
$861$	1.21241e7			0.0189950
$862$		0			0
$863$		−	6.19623e8i		−	0.964040i	−0.876160	−	0.482020i	$-0.839903\pi$
$863$		−	6.19623e8i		−	0.964040i	0.876160	−	0.482020i	$-0.160097\pi$
$864$		0			0
$865$			7.86693e7i			0.121551i
$866$		0			0
$867$			1.01758e8i			0.156140i
$868$		0			0
$869$		−	1.82467e9i		−	2.78052i
$870$		0			0
$871$	3.88039e8			0.587248
$872$		0			0
$873$			5.11534e8i			0.768832i
$874$		0			0
$875$	−2.89914e7			−0.0432757
$876$		0			0
$877$			6.20572e7i			0.0920012i	0.998941	+	0.0460006i	$0.0146476\pi$
$877$			6.20572e7i			0.0920012i	−0.998941	+	0.0460006i	$0.985352\pi$
$878$		0			0
$879$	4.22692e7			0.0622383
$880$		0			0
$881$	3.32637e8			0.486455			0.243227	−	0.969969i	$-0.421794\pi$
$881$	3.32637e8			0.486455			0.243227	+	0.969969i	$0.421794\pi$
$882$		0			0
$883$	1.25059e8			0.181649			0.0908245	−	0.995867i	$-0.471050\pi$
$883$	1.25059e8			0.181649			0.0908245	+	0.995867i	$0.471050\pi$
$884$		0			0
$885$	−8.80638e7			−0.127048
$886$		0			0
$887$		−	4.10218e8i		−	0.587820i	−0.955833	−	0.293910i	$-0.905043\pi$
$887$		−	4.10218e8i		−	0.587820i	0.955833	−	0.293910i	$-0.0949566\pi$
$888$		0			0
$889$			1.87942e7i			0.0267497i
$890$		0			0
$891$	6.91211e8			0.977186
$892$		0			0
$893$	−8.21355e8	+	4.60882e8i	−1.15339	+	0.647196i
$894$		0			0
$895$		−	2.60108e8i		−	0.362815i
$896$		0			0
$897$	4.76703e7			0.0660497
$898$		0			0
$899$	1.53932e8			0.211861
$900$		0			0
$901$			1.00785e9i			1.37791i
$902$		0			0
$903$			2.28903e6i			0.00310876i
$904$		0			0
$905$			8.46959e7i			0.114266i
$906$		0			0
$907$			1.03301e9i			1.38447i	0.721673	+	0.692234i	$0.243373\pi$
$907$			1.03301e9i			1.38447i	−0.721673	+	0.692234i	$0.756627\pi$
$908$		0			0
$909$	1.05596e9			1.40591
$910$		0			0
$911$			2.56887e8i			0.339772i	0.985464	+	0.169886i	$0.0543400\pi$
$911$			2.56887e8i			0.339772i	−0.985464	+	0.169886i	$0.945660\pi$
$912$		0			0
$913$	1.82697e9			2.40060
$914$		0			0
$915$			4.67858e7i			0.0610732i
$916$		0			0
$917$	−2.50398e7			−0.0324731
$918$		0			0
$919$	9.16840e8			1.18126			0.590632	−	0.806941i	$-0.298879\pi$
$919$	9.16840e8			1.18126			0.590632	+	0.806941i	$0.298879\pi$
$920$		0			0
$921$	1.90638e8			0.244022
$922$		0			0
$923$	−8.55783e8			−1.08833
$924$		0			0
$925$		−	1.83366e8i		−	0.231683i
$926$		0			0
$927$			3.80103e8i			0.477158i
$928$		0			0
$929$	1.07063e9			1.33534			0.667671	−	0.744456i	$-0.267291\pi$
$929$	1.07063e9			1.33534			0.667671	+	0.744456i	$0.267291\pi$
$930$		0			0
$931$	7.00121e8	−	3.92855e8i	0.867609	−	0.486836i
$932$		0			0
$933$			6.68169e7i			0.0822700i
$934$		0			0
$935$	2.97760e8			0.364277
$936$		0			0
$937$	1.33730e8			0.162558			0.0812791	−	0.996691i	$-0.474100\pi$
$937$	1.33730e8			0.162558			0.0812791	+	0.996691i	$0.474100\pi$
$938$		0			0
$939$			2.76974e8i			0.334535i
$940$		0			0
$941$		−	1.06260e9i		−	1.27526i	−0.770342	−	0.637631i	$-0.779914\pi$
$941$		−	1.06260e9i		−	1.27526i	0.770342	−	0.637631i	$-0.220086\pi$
$942$		0			0
$943$			1.13365e8i			0.135190i
$944$		0			0
$945$		−	1.27867e7i		−	0.0151518i
$946$		0			0
$947$	2.77740e8			0.327031			0.163516	−	0.986541i	$-0.447717\pi$
$947$	2.77740e8			0.327031			0.163516	+	0.986541i	$0.447717\pi$
$948$		0			0
$949$			1.73016e8i			0.202436i
$950$		0			0
$951$	2.02006e8			0.234868
$952$		0			0
$953$		−	1.07558e9i		−	1.24269i	−0.783538	−	0.621344i	$-0.786587\pi$
$953$		−	1.07558e9i		−	1.24269i	0.783538	−	0.621344i	$-0.213413\pi$
$954$		0			0
$955$	3.42584e8			0.393330
$956$		0			0
$957$	5.08718e8			0.580419
$958$		0			0
$959$	6.60411e7			0.0748787
$960$		0			0
$961$	8.52632e8			0.960708
$962$		0			0
$963$			1.10295e9i			1.23502i
$964$		0			0
$965$		−	2.68913e8i		−	0.299247i
$966$		0			0
$967$	−4.21410e8			−0.466042			−0.233021	−	0.972472i	$-0.574861\pi$
$967$	−4.21410e8			−0.466042			−0.233021	+	0.972472i	$0.574861\pi$
$968$		0			0
$969$	1.18176e8	+	2.10605e8i	0.129884	+	0.231471i
$970$		0			0
$971$		−	3.14911e8i		−	0.343978i	−0.985099	−	0.171989i	$-0.944981\pi$
$971$		−	3.14911e8i		−	0.343978i	0.985099	−	0.171989i	$-0.0550194\pi$
$972$		0			0
$973$	4.06320e7			0.0441092
$974$		0			0
$975$	−3.04143e8			−0.328144
$976$		0			0
$977$			4.24295e8i			0.454972i	0.973781	+	0.227486i	$0.0730505\pi$
$977$			4.24295e8i			0.454972i	−0.973781	+	0.227486i	$0.926949\pi$
$978$		0			0
$979$			2.40309e9i			2.56107i
$980$		0			0
$981$		−	6.49460e8i		−	0.687931i
$982$		0			0
$983$			7.58248e8i			0.798271i	0.916892	+	0.399136i	$0.130690\pi$
$983$			7.58248e8i			0.798271i	−0.916892	+	0.399136i	$0.869310\pi$
$984$		0			0
$985$	3.43882e8			0.359833
$986$		0			0
$987$		−	3.23267e7i		−	0.0336209i
$988$		0			0
$989$	−2.14033e7			−0.0221255
$990$		0			0
$991$			1.06634e9i			1.09566i	0.836590	+	0.547830i	$0.184546\pi$
$991$			1.06634e9i			1.09566i	−0.836590	+	0.547830i	$0.815454\pi$
$992$		0			0
$993$	−1.11889e8			−0.114272
$994$		0			0
$995$	−3.13158e8			−0.317903
$996$		0			0
$997$	1.55268e9			1.56674			0.783369	−	0.621557i	$-0.213500\pi$
$997$	1.55268e9			1.56674			0.783369	+	0.621557i	$0.213500\pi$
$998$		0			0
$999$	1.70848e8			0.171362

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.7.e.f.113.13		30
4.3	odd	2		152.7.e.a.113.18	yes	30
19.18	odd	2	inner	304.7.e.f.113.18		30
76.75	even	2		152.7.e.a.113.13	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.7.e.a.113.13	✓	30	76.75	even	2
152.7.e.a.113.18	yes	30	4.3	odd	2
304.7.e.f.113.13		30	1.1	even	1	trivial
304.7.e.f.113.18		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)

\(f(q)\)	\(=\)	\(q-9.57785i q^{3} +39.7533 q^{5} +24.5800 q^{7} +637.265 q^{9} +O(q^{10})\)	\(q-9.57785i q^{3} +39.7533 q^{5} +24.5800 q^{7} +637.265 q^{9} +2037.58 q^{11} +2260.99i q^{13} -380.751i q^{15} +3676.03 q^{17} +(-5981.65 + 3356.45i) q^{19} -235.424i q^{21} +2201.31 q^{23} -14044.7 q^{25} -13085.9i q^{27} +26067.2i q^{29} -5905.20i q^{31} -19515.6i q^{33} +977.138 q^{35} +13055.9i q^{37} +21655.4 q^{39} +51499.0i q^{41} -9723.00 q^{43} +25333.4 q^{45} +137312. q^{47} -117045. q^{49} -35208.5i q^{51} +274168. i q^{53} +81000.5 q^{55} +(32147.6 + 57291.4i) q^{57} -231290. i q^{59} -122878. q^{61} +15664.0 q^{63} +89881.8i q^{65} -171624. i q^{67} -21083.8i q^{69} +378499. i q^{71} +76522.4 q^{73} +134518. i q^{75} +50083.8 q^{77} -895511. i q^{79} +339231. q^{81} +896640. q^{83} +146134. q^{85} +249668. q^{87} +1.17938e6i q^{89} +55575.2i q^{91} -56559.1 q^{93} +(-237790. + 133430. i) q^{95} +802702. i q^{97} +1.29848e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 720 q^{7} - 8670 q^{9}+O(q^{10}) \) 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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