LMFDB - Embedded newform 304.7.e.d.113.3

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 483x^{6} + 75582x^{4} + 4242376x^{2} + 71047680 $ x^8 + 483x^6 + 75582x^4 + 4242376*x^2 + 71047680
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{11}\cdot 29 $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.3
Root			$-8.08057i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.7.e.d.113.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.2115i q^{3} +162.986 q^{5} +68.0674 q^{7} -11.4646 q^{9} +O(q^{10})$	$q-27.2115i q^{3} +162.986 q^{5} +68.0674 q^{7} -11.4646 q^{9} -838.499 q^{11} +2034.15i q^{13} -4435.09i q^{15} +1267.58 q^{17} +(6046.63 + 3237.92i) q^{19} -1852.21i q^{21} +22322.8 q^{23} +10939.4 q^{25} -19525.2i q^{27} +2151.43i q^{29} +42551.1i q^{31} +22816.8i q^{33} +11094.0 q^{35} -36019.4i q^{37} +55352.1 q^{39} +78361.9i q^{41} -19470.6 q^{43} -1868.57 q^{45} +37502.9 q^{47} -113016. q^{49} -34492.7i q^{51} +166388. i q^{53} -136663. q^{55} +(88108.6 - 164538. i) q^{57} +281997. i q^{59} +373512. q^{61} -780.368 q^{63} +331537. i q^{65} +273335. i q^{67} -607437. i q^{69} -438455. i q^{71} +248225. q^{73} -297677. i q^{75} -57074.4 q^{77} -574454. i q^{79} -539667. q^{81} +344904. q^{83} +206598. q^{85} +58543.5 q^{87} -1.32774e6i q^{89} +138459. i q^{91} +1.15788e6 q^{93} +(985516. + 527735. i) q^{95} -142280. i q^{97} +9613.08 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 108 q^{5} + 140 q^{7} - 1052 q^{9}+O(q^{10}) $ 8 * q + 108 * q^5 + 140 * q^7 - 1052 * q^9	$ 8 q + 108 q^{5} + 140 q^{7} - 1052 q^{9} + 2024 q^{11} + 6008 q^{17} - 20552 q^{19} + 50252 q^{23} + 78492 q^{25} - 210800 q^{35} - 43724 q^{39} - 260800 q^{43} - 191012 q^{45} + 100248 q^{47} - 301872 q^{49} + 52480 q^{55} - 186860 q^{57} - 54548 q^{61} + 137408 q^{63} + 479968 q^{73} - 1755300 q^{77} - 4279648 q^{81} - 483040 q^{83} + 2111780 q^{85} - 2802652 q^{87} + 1507528 q^{93} + 2383888 q^{95} - 528224 q^{99}+O(q^{100}) $ 8 * q + 108 * q^5 + 140 * q^7 - 1052 * q^9 + 2024 * q^11 + 6008 * q^17 - 20552 * q^19 + 50252 * q^23 + 78492 * q^25 - 210800 * q^35 - 43724 * q^39 - 260800 * q^43 - 191012 * q^45 + 100248 * q^47 - 301872 * q^49 + 52480 * q^55 - 186860 * q^57 - 54548 * q^61 + 137408 * q^63 + 479968 * q^73 - 1755300 * q^77 - 4279648 * q^81 - 483040 * q^83 + 2111780 * q^85 - 2802652 * q^87 + 1507528 * q^93 + 2383888 * q^95 - 528224 * 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
$3$		−	27.2115i		−	1.00783i	−0.863752	−	0.503916i	$-0.831892\pi$
$3$		−	27.2115i		−	1.00783i	0.863752	−	0.503916i	$-0.168108\pi$
$4$		0			0
$5$	162.986			1.30389			0.651944	−	0.758267i	$-0.273954\pi$
$5$	162.986			1.30389			0.651944	+	0.758267i	$0.273954\pi$
$6$		0			0
$7$	68.0674			0.198447			0.0992236	−	0.995065i	$-0.468364\pi$
$7$	68.0674			0.198447			0.0992236	+	0.995065i	$0.468364\pi$
$8$		0			0
$9$	−11.4646			−0.0157265
$10$		0			0
$11$	−838.499			−0.629976			−0.314988	−	0.949096i	$-0.602001\pi$
$11$	−838.499			−0.629976			−0.314988	+	0.949096i	$0.602001\pi$
$12$		0			0
$13$			2034.15i			0.925874i	0.886391	+	0.462937i	$0.153204\pi$
$13$			2034.15i			0.925874i	−0.886391	+	0.462937i	$0.846796\pi$
$14$		0			0
$15$		−	4435.09i		−	1.31410i
$16$		0			0
$17$	1267.58			0.258005			0.129003	−	0.991644i	$-0.458822\pi$
$17$	1267.58			0.258005			0.129003	+	0.991644i	$0.458822\pi$
$18$		0			0
$19$	6046.63	+	3237.92i	0.881562	+	0.472069i
$19$	6046.63	+	3237.92i	0.881562	+	0.472069i
$20$		0			0
$21$		−	1852.21i		−	0.200002i
$22$		0			0
$23$	22322.8			1.83470			0.917350	−	0.398081i	$-0.130324\pi$
$23$	22322.8			1.83470			0.917350	+	0.398081i	$0.130324\pi$
$24$		0			0
$25$	10939.4			0.700122
$26$		0			0
$27$		−	19525.2i		−	0.991983i
$28$		0			0
$29$			2151.43i			0.0882130i	0.999027	+	0.0441065i	$0.0140441\pi$
$29$			2151.43i			0.0882130i	−0.999027	+	0.0441065i	$0.985956\pi$
$30$		0			0
$31$			42551.1i			1.42832i	0.699982	+	0.714161i	$0.253191\pi$
$31$			42551.1i			1.42832i	−0.699982	+	0.714161i	$0.746809\pi$
$32$		0			0
$33$			22816.8i			0.634911i
$34$		0			0
$35$	11094.0			0.258753
$36$		0			0
$37$		−	36019.4i		−	0.711101i	−0.934657	−	0.355551i	$-0.884293\pi$
$37$		−	36019.4i		−	0.711101i	0.934657	−	0.355551i	$-0.115707\pi$
$38$		0			0
$39$	55352.1			0.933126
$40$		0			0
$41$			78361.9i			1.13698i	0.822690	+	0.568491i	$0.192472\pi$
$41$			78361.9i			1.13698i	−0.822690	+	0.568491i	$0.807528\pi$
$42$		0			0
$43$	−19470.6			−0.244892			−0.122446	−	0.992475i	$-0.539074\pi$
$43$	−19470.6			−0.244892			−0.122446	+	0.992475i	$0.539074\pi$
$44$		0			0
$45$	−1868.57			−0.0205056
$46$		0			0
$47$	37502.9			0.361220			0.180610	−	0.983555i	$-0.442193\pi$
$47$	37502.9			0.361220			0.180610	+	0.983555i	$0.442193\pi$
$48$		0			0
$49$	−113016.			−0.960619
$50$		0			0
$51$		−	34492.7i		−	0.260026i
$52$		0			0
$53$			166388.i			1.11762i	0.829295	+	0.558811i	$0.188742\pi$
$53$			166388.i			1.11762i	−0.829295	+	0.558811i	$0.811258\pi$
$54$		0			0
$55$	−136663.			−0.821418
$56$		0			0
$57$	88108.6	−	164538.i	0.475766	−	0.888467i
$58$		0			0
$59$			281997.i			1.37306i	0.727104	+	0.686528i	$0.240866\pi$
$59$			281997.i			1.37306i	−0.727104	+	0.686528i	$0.759134\pi$
$60$		0			0
$61$	373512.			1.64556			0.822782	−	0.568357i	$-0.192421\pi$
$61$	373512.			1.64556			0.822782	+	0.568357i	$0.192421\pi$
$62$		0			0
$63$	−780.368			−0.00312088
$64$		0			0
$65$			331537.i			1.20724i
$66$		0			0
$67$			273335.i			0.908805i	0.890796	+	0.454403i	$0.150147\pi$
$67$			273335.i			0.908805i	−0.890796	+	0.454403i	$0.849853\pi$
$68$		0			0
$69$		−	607437.i		−	1.84907i
$70$		0			0
$71$		−	438455.i		−	1.22504i	−0.790455	−	0.612520i	$-0.790156\pi$
$71$		−	438455.i		−	1.22504i	0.790455	−	0.612520i	$-0.209844\pi$
$72$		0			0
$73$	248225.			0.638083			0.319042	−	0.947741i	$-0.396639\pi$
$73$	248225.			0.638083			0.319042	+	0.947741i	$0.396639\pi$
$74$		0			0
$75$		−	297677.i		−	0.705605i
$76$		0			0
$77$	−57074.4			−0.125017
$78$		0			0
$79$		−	574454.i		−	1.16513i	−0.812785	−	0.582564i	$-0.802049\pi$
$79$		−	574454.i		−	1.16513i	0.812785	−	0.582564i	$-0.197951\pi$
$80$		0			0
$81$	−539667.			−1.01548
$82$		0			0
$83$	344904.			0.603204			0.301602	−	0.953434i	$-0.402479\pi$
$83$	344904.			0.603204			0.301602	+	0.953434i	$0.402479\pi$
$84$		0			0
$85$	206598.			0.336410
$86$		0			0
$87$	58543.5			0.0889040
$88$		0			0
$89$		−	1.32774e6i		−	1.88340i	−0.336450	−	0.941701i	$-0.609226\pi$
$89$		−	1.32774e6i		−	1.88340i	0.336450	−	0.941701i	$-0.390774\pi$
$90$		0			0
$91$			138459.i			0.183737i
$92$		0			0
$93$	1.15788e6			1.43951
$94$		0			0
$95$	985516.	+	527735.i	1.14946	+	0.615525i
$96$		0			0
$97$		−	142280.i		−	0.155894i	−0.996958	−	0.0779469i	$-0.975164\pi$
$97$		−	142280.i		−	0.155894i	0.996958	−	0.0779469i	$-0.0248365\pi$
$98$		0			0
$99$	9613.08			0.00990734
$100$		0			0
$101$	987059.			0.958030			0.479015	−	0.877807i	$-0.340994\pi$
$101$	987059.			0.958030			0.479015	+	0.877807i	$0.340994\pi$
$102$		0			0
$103$		−	645029.i		−	0.590293i	−0.955452	−	0.295147i	$-0.904632\pi$
$103$		−	645029.i		−	0.590293i	0.955452	−	0.295147i	$-0.0953685\pi$
$104$		0			0
$105$		−	301885.i		−	0.260780i
$106$		0			0
$107$		−	1.86047e6i		−	1.51869i	−0.650686	−	0.759347i	$-0.725519\pi$
$107$		−	1.86047e6i		−	1.51869i	0.650686	−	0.759347i	$-0.274481\pi$
$108$		0			0
$109$		−	1.46049e6i		−	1.12776i	−0.825856	−	0.563881i	$-0.809307\pi$
$109$		−	1.46049e6i		−	1.12776i	0.825856	−	0.563881i	$-0.190693\pi$
$110$		0			0
$111$	−980141.			−0.716671
$112$		0			0
$113$			1.81890e6i			1.26059i	0.776357	+	0.630293i	$0.217065\pi$
$113$			1.81890e6i			1.26059i	−0.776357	+	0.630293i	$0.782935\pi$
$114$		0			0
$115$	3.63830e6			2.39224
$116$		0			0
$117$		−	23320.7i		−	0.0145608i
$118$		0			0
$119$	86280.9			0.0512004
$120$		0			0
$121$	−1.06848e6			−0.603130
$122$		0			0
$123$	2.13234e6			1.14589
$124$		0			0
$125$	−763687.			−0.391008
$126$		0			0
$127$		−	1.08950e6i		−	0.531883i	−0.963989	−	0.265942i	$-0.914317\pi$
$127$		−	1.08950e6i		−	0.531883i	0.963989	−	0.265942i	$-0.0856829\pi$
$128$		0			0
$129$			529824.i			0.246810i
$130$		0			0
$131$	−2.62081e6			−1.16579			−0.582897	−	0.812546i	$-0.698081\pi$
$131$	−2.62081e6			−1.16579			−0.582897	+	0.812546i	$0.698081\pi$
$132$		0			0
$133$	411579.	+	220397.i	0.174943	+	0.0936808i
$134$		0			0
$135$		−	3.18233e6i		−	1.29343i
$136$		0			0
$137$	2.48390e6			0.965989			0.482995	−	0.875623i	$-0.339549\pi$
$137$	2.48390e6			0.965989			0.482995	+	0.875623i	$0.339549\pi$
$138$		0			0
$139$	119484.			0.0444904			0.0222452	−	0.999753i	$-0.492919\pi$
$139$	119484.			0.0444904			0.0222452	+	0.999753i	$0.492919\pi$
$140$		0			0
$141$		−	1.02051e6i		−	0.364049i
$142$		0			0
$143$		−	1.70563e6i		−	0.583279i
$144$		0			0
$145$			350652.i			0.115020i
$146$		0			0
$147$			3.07533e6i			0.968143i
$148$		0			0
$149$	5.90602e6			1.78540			0.892702	−	0.450648i	$-0.148807\pi$
$149$	5.90602e6			1.78540			0.892702	+	0.450648i	$0.148807\pi$
$150$		0			0
$151$			5.87343e6i			1.70593i	0.521969	+	0.852964i	$0.325198\pi$
$151$			5.87343e6i			1.70593i	−0.521969	+	0.852964i	$0.674802\pi$
$152$		0			0
$153$	−14532.3			−0.00405753
$154$		0			0
$155$			6.93523e6i			1.86237i
$156$		0			0
$157$	−4.87470e6			−1.25965			−0.629823	−	0.776738i	$-0.716873\pi$
$157$	−4.87470e6			−1.25965			−0.629823	+	0.776738i	$0.716873\pi$
$158$		0			0
$159$	4.52767e6			1.12638
$160$		0			0
$161$	1.51946e6			0.364091
$162$		0			0
$163$	1.81131e6			0.418244			0.209122	−	0.977890i	$-0.432939\pi$
$163$	1.81131e6			0.418244			0.209122	+	0.977890i	$0.432939\pi$
$164$		0			0
$165$			3.71881e6i			0.827852i
$166$		0			0
$167$		−	3.80324e6i		−	0.816590i	−0.912850	−	0.408295i	$-0.866123\pi$
$167$		−	3.80324e6i		−	0.816590i	0.912850	−	0.408295i	$-0.133877\pi$
$168$		0			0
$169$	689058.			0.142757
$170$		0			0
$171$	−69322.4	−	37121.6i	−0.0138639	−	0.00742400i
$172$		0			0
$173$		−	4.95029e6i		−	0.956076i	−0.878339	−	0.478038i	$-0.841348\pi$
$173$		−	4.95029e6i		−	0.956076i	0.878339	−	0.478038i	$-0.158652\pi$
$174$		0			0
$175$	744617.			0.138937
$176$		0			0
$177$	7.67355e6			1.38381
$178$		0			0
$179$		−	408437.i		−	0.0712141i	−0.999366	−	0.0356071i	$-0.988664\pi$
$179$		−	408437.i		−	0.0712141i	0.999366	−	0.0356071i	$-0.0113365\pi$
$180$		0			0
$181$			969863.i			0.163559i	0.996650	+	0.0817795i	$0.0260603\pi$
$181$			969863.i			0.163559i	−0.996650	+	0.0817795i	$0.973940\pi$
$182$		0			0
$183$		−	1.01638e7i		−	1.65845i
$184$		0			0
$185$		−	5.87066e6i		−	0.927196i
$186$		0			0
$187$	−1.06286e6			−0.162537
$188$		0			0
$189$		−	1.32903e6i		−	0.196856i
$190$		0			0
$191$	942772.			0.135303			0.0676514	−	0.997709i	$-0.478449\pi$
$191$	942772.			0.135303			0.0676514	+	0.997709i	$0.478449\pi$
$192$		0			0
$193$		−	3.46837e6i		−	0.482452i	−0.970469	−	0.241226i	$-0.922451\pi$
$193$		−	3.46837e6i		−	0.482452i	0.970469	−	0.241226i	$-0.0775495\pi$
$194$		0			0
$195$	9.02162e6			1.21669
$196$		0			0
$197$	6.68728e6			0.874683			0.437341	−	0.899296i	$-0.355920\pi$
$197$	6.68728e6			0.874683			0.437341	+	0.899296i	$0.355920\pi$
$198$		0			0
$199$	−1.46573e7			−1.85992			−0.929958	−	0.367665i	$-0.880157\pi$
$199$	−1.46573e7			−1.85992			−0.929958	+	0.367665i	$0.880157\pi$
$200$		0			0
$201$	7.43785e6			0.915924
$202$		0			0
$203$			146442.i			0.0175056i
$204$		0			0
$205$			1.27719e7i			1.48250i
$206$		0			0
$207$	−255923.			−0.0288535
$208$		0			0
$209$	−5.07009e6	−	2.71499e6i	−0.555363	−	0.297392i
$210$		0			0
$211$			6.77046e6i			0.720727i	0.932812	+	0.360363i	$0.117347\pi$
$211$			6.77046e6i			0.720727i	−0.932812	+	0.360363i	$0.882653\pi$
$212$		0			0
$213$	−1.19310e7			−1.23463
$214$		0			0
$215$	−3.17343e6			−0.319311
$216$		0			0
$217$			2.89635e6i			0.283447i
$218$		0			0
$219$		−	6.75458e6i		−	0.643081i
$220$		0			0
$221$			2.57844e6i			0.238881i
$222$		0			0
$223$		−	4.08252e6i		−	0.368141i	−0.982913	−	0.184070i	$-0.941073\pi$
$223$		−	4.08252e6i		−	0.368141i	0.982913	−	0.184070i	$-0.0589274\pi$
$224$		0			0
$225$	−125416.			−0.0110105
$226$		0			0
$227$			504687.i			0.0431464i	0.999767	+	0.0215732i	$0.00686750\pi$
$227$			504687.i			0.0431464i	−0.999767	+	0.0215732i	$0.993133\pi$
$228$		0			0
$229$	−1.03411e7			−0.861114			−0.430557	−	0.902563i	$-0.641683\pi$
$229$	−1.03411e7			−0.861114			−0.430557	+	0.902563i	$0.641683\pi$
$230$		0			0
$231$			1.55308e6i			0.125996i
$232$		0			0
$233$	1.69778e6			0.134219			0.0671093	−	0.997746i	$-0.478622\pi$
$233$	1.69778e6			0.134219			0.0671093	+	0.997746i	$0.478622\pi$
$234$		0			0
$235$	6.11245e6			0.470990
$236$		0			0
$237$	−1.56317e7			−1.17425
$238$		0			0
$239$	874360.			0.0640467			0.0320233	−	0.999487i	$-0.489805\pi$
$239$	874360.			0.0640467			0.0320233	+	0.999487i	$0.489805\pi$
$240$		0			0
$241$		−	1.03438e7i		−	0.738970i	−0.929237	−	0.369485i	$-0.879534\pi$
$241$		−	1.03438e7i		−	0.738970i	0.929237	−	0.369485i	$-0.120466\pi$
$242$		0			0
$243$			451275.i			0.0314501i
$244$		0			0
$245$	−1.84200e7			−1.25254
$246$		0			0
$247$	−6.58640e6	+	1.22997e7i	−0.437077	+	0.816215i
$248$		0			0
$249$		−	9.38536e6i		−	0.607929i
$250$		0			0
$251$	−9.79922e6			−0.619684			−0.309842	−	0.950788i	$-0.600276\pi$
$251$	−9.79922e6			−0.619684			−0.309842	+	0.950788i	$0.600276\pi$
$252$		0			0
$253$	−1.87176e7			−1.15582
$254$		0			0
$255$		−	5.62183e6i		−	0.339045i
$256$		0			0
$257$		−	1.70359e7i		−	1.00361i	−0.864980	−	0.501807i	$-0.832669\pi$
$257$		−	1.70359e7i		−	1.00361i	0.864980	−	0.501807i	$-0.167331\pi$
$258$		0			0
$259$		−	2.45175e6i		−	0.141116i
$260$		0			0
$261$		−	24665.3i		−	0.00138728i
$262$		0			0
$263$	1.17852e7			0.647844			0.323922	−	0.946084i	$-0.394999\pi$
$263$	1.17852e7			0.647844			0.323922	+	0.946084i	$0.394999\pi$
$264$		0			0
$265$			2.71189e7i			1.45725i
$266$		0			0
$267$	−3.61298e7			−1.89815
$268$		0			0
$269$			3.33034e7i			1.71093i	0.517861	+	0.855465i	$0.326728\pi$
$269$			3.33034e7i			1.71093i	−0.517861	+	0.855465i	$0.673272\pi$
$270$		0			0
$271$	−1.41907e7			−0.713011			−0.356505	−	0.934293i	$-0.616032\pi$
$271$	−1.41907e7			−0.713011			−0.356505	+	0.934293i	$0.616032\pi$
$272$		0			0
$273$	3.76768e6			0.185176
$274$		0			0
$275$	−9.17267e6			−0.441060
$276$		0			0
$277$	−1.04534e7			−0.491833			−0.245916	−	0.969291i	$-0.579089\pi$
$277$	−1.04534e7			−0.491833			−0.245916	+	0.969291i	$0.579089\pi$
$278$		0			0
$279$		−	487833.i		−	0.0224625i
$280$		0			0
$281$			8.53758e6i			0.384783i	0.981318	+	0.192392i	$0.0616244\pi$
$281$			8.53758e6i			0.384783i	−0.981318	+	0.192392i	$0.938376\pi$
$282$		0			0
$283$	85032.8			0.00375169			0.00187584	−	0.999998i	$-0.499403\pi$
$283$	85032.8			0.00375169			0.00187584	+	0.999998i	$0.499403\pi$
$284$		0			0
$285$	1.43605e7	−	2.68173e7i	0.620346	−	1.15846i
$286$		0			0
$287$			5.33389e6i			0.225631i
$288$		0			0
$289$	−2.25308e7			−0.933433
$290$		0			0
$291$	−3.87165e6			−0.157115
$292$		0			0
$293$			2.14342e7i			0.852127i	0.904693	+	0.426063i	$0.140100\pi$
$293$			2.14342e7i			0.852127i	−0.904693	+	0.426063i	$0.859900\pi$
$294$		0			0
$295$			4.59615e7i			1.79031i
$296$		0			0
$297$			1.63719e7i			0.624926i
$298$		0			0
$299$			4.54078e7i			1.69870i
$300$		0			0
$301$	−1.32531e6			−0.0485981
$302$		0			0
$303$		−	2.68593e7i		−	0.965534i
$304$		0			0
$305$	6.08771e7			2.14563
$306$		0			0
$307$		−	3.85363e7i		−	1.33185i	−0.746019	−	0.665925i	$-0.768037\pi$
$307$		−	3.85363e7i		−	1.33185i	0.746019	−	0.665925i	$-0.231963\pi$
$308$		0			0
$309$	−1.75522e7			−0.594917
$310$		0			0
$311$	1.89191e7			0.628955			0.314478	−	0.949265i	$-0.398171\pi$
$311$	1.89191e7			0.628955			0.314478	+	0.949265i	$0.398171\pi$
$312$		0			0
$313$	4.03438e7			1.31566			0.657830	−	0.753166i	$-0.271474\pi$
$313$	4.03438e7			1.31566			0.657830	+	0.753166i	$0.271474\pi$
$314$		0			0
$315$	−127189.			−0.00406928
$316$		0			0
$317$		−	54381.7i		−	0.00170716i	−1.00000		0.000853581i	$-0.999728\pi$
$317$		−	54381.7i		−	0.00170716i	1.00000		0.000853581i	$-0.000271703\pi$
$318$		0			0
$319$		−	1.80397e6i		−	0.0555721i
$320$		0			0
$321$	−5.06260e7			−1.53059
$322$		0			0
$323$	7.66459e6	+	4.10432e6i	0.227448	+	0.121796i
$324$		0			0
$325$			2.22523e7i			0.648225i
$326$		0			0
$327$	−3.97420e7			−1.13660
$328$		0			0
$329$	2.55273e6			0.0716831
$330$		0			0
$331$			2.30790e7i			0.636403i	0.948023	+	0.318202i	$0.103079\pi$
$331$			2.30790e7i			0.636403i	−0.948023	+	0.318202i	$0.896921\pi$
$332$		0			0
$333$			412949.i			0.0111831i
$334$		0			0
$335$			4.45497e7i			1.18498i
$336$		0			0
$337$			3.42608e7i			0.895175i	0.894240	+	0.447588i	$0.147717\pi$
$337$			3.42608e7i			0.895175i	−0.894240	+	0.447588i	$0.852283\pi$
$338$		0			0
$339$	4.94948e7			1.27046
$340$		0			0
$341$		−	3.56791e7i		−	0.899809i
$342$		0			0
$343$	−1.57008e7			−0.389079
$344$		0			0
$345$		−	9.90036e7i		−	2.41098i
$346$		0			0
$347$	−4.85838e7			−1.16280			−0.581398	−	0.813619i	$-0.697494\pi$
$347$	−4.85838e7			−1.16280			−0.581398	+	0.813619i	$0.697494\pi$
$348$		0			0
$349$	1.56896e6			0.0369092			0.0184546	−	0.999830i	$-0.494125\pi$
$349$	1.56896e6			0.0369092			0.0184546	+	0.999830i	$0.494125\pi$
$350$		0			0
$351$	3.97171e7			0.918452
$352$		0			0
$353$	−1.75807e7			−0.399679			−0.199840	−	0.979829i	$-0.564042\pi$
$353$	−1.75807e7			−0.399679			−0.199840	+	0.979829i	$0.564042\pi$
$354$		0			0
$355$		−	7.14620e7i		−	1.59731i
$356$		0			0
$357$		−	2.34783e6i		−	0.0516015i
$358$		0			0
$359$	−4.45139e7			−0.962082			−0.481041	−	0.876698i	$-0.659741\pi$
$359$	−4.45139e7			−0.962082			−0.481041	+	0.876698i	$0.659741\pi$
$360$		0			0
$361$	2.60776e7	+	3.91570e7i	0.554302	+	0.832316i
$362$		0			0
$363$			2.90749e7i			0.607854i
$364$		0			0
$365$	4.04572e7			0.831989
$366$		0			0
$367$	5.49654e7			1.11197			0.555983	−	0.831194i	$-0.312342\pi$
$367$	5.49654e7			1.11197			0.555983	+	0.831194i	$0.312342\pi$
$368$		0			0
$369$		−	898390.i		−	0.0178808i
$370$		0			0
$371$			1.13256e7i			0.221789i
$372$		0			0
$373$			6.59390e7i			1.27062i	0.772257	+	0.635310i	$0.219128\pi$
$373$			6.59390e7i			1.27062i	−0.772257	+	0.635310i	$0.780872\pi$
$374$		0			0
$375$			2.07810e7i			0.394070i
$376$		0			0
$377$	−4.37632e6			−0.0816742
$378$		0			0
$379$			6.89315e6i			0.126619i	0.997994	+	0.0633096i	$0.0201656\pi$
$379$			6.89315e6i			0.126619i	−0.997994	+	0.0633096i	$0.979834\pi$
$380$		0			0
$381$	−2.96469e7			−0.536049
$382$		0			0
$383$			1.15806e7i			0.206126i	0.994675	+	0.103063i	$0.0328644\pi$
$383$			1.15806e7i			0.206126i	−0.994675	+	0.103063i	$0.967136\pi$
$384$		0			0
$385$	−9.30233e6			−0.163008
$386$		0			0
$387$	223223.			0.00385129
$388$		0			0
$389$	−4.64344e7			−0.788843			−0.394422	−	0.918930i	$-0.629055\pi$
$389$	−4.64344e7			−0.788843			−0.394422	+	0.918930i	$0.629055\pi$
$390$		0			0
$391$	2.82959e7			0.473363
$392$		0			0
$393$			7.13162e7i			1.17493i
$394$		0			0
$395$		−	9.36279e7i		−	1.51920i
$396$		0			0
$397$	−2.07096e7			−0.330979			−0.165490	−	0.986212i	$-0.552920\pi$
$397$	−2.07096e7			−0.330979			−0.165490	+	0.986212i	$0.552920\pi$
$398$		0			0
$399$	5.99732e6	−	1.11997e7i	0.0944145	−	0.176314i
$400$		0			0
$401$		−	7.17700e7i		−	1.11304i	−0.830835	−	0.556519i	$-0.812137\pi$
$401$		−	7.17700e7i		−	1.11304i	0.830835	−	0.556519i	$-0.187863\pi$
$402$		0			0
$403$	−8.65552e7			−1.32245
$404$		0			0
$405$	−8.79582e7			−1.32407
$406$		0			0
$407$			3.02022e7i			0.447977i
$408$		0			0
$409$		−	4.28155e7i		−	0.625793i	−0.949787	−	0.312897i	$-0.898701\pi$
$409$		−	4.28155e7i		−	0.625793i	0.949787	−	0.312897i	$-0.101299\pi$
$410$		0			0
$411$		−	6.75906e7i		−	0.973555i
$412$		0			0
$413$			1.91948e7i			0.272479i
$414$		0			0
$415$	5.62145e7			0.786510
$416$		0			0
$417$		−	3.25134e6i		−	0.0448389i
$418$		0			0
$419$	7.35781e6			0.100024			0.0500122	−	0.998749i	$-0.484074\pi$
$419$	7.35781e6			0.100024			0.0500122	+	0.998749i	$0.484074\pi$
$420$		0			0
$421$			2.90256e7i			0.388987i	0.980904	+	0.194493i	$0.0623063\pi$
$421$			2.90256e7i			0.388987i	−0.980904	+	0.194493i	$0.937694\pi$
$422$		0			0
$423$	−429957.			−0.00568073
$424$		0			0
$425$	1.38666e7			0.180635
$426$		0			0
$427$	2.54240e7			0.326558
$428$		0			0
$429$	−4.64127e7			−0.587848
$430$		0			0
$431$			3.68503e6i			0.0460266i	0.999735	+	0.0230133i	$0.00732600\pi$
$431$			3.68503e6i			0.0460266i	−0.999735	+	0.0230133i	$0.992674\pi$
$432$		0			0
$433$		−	1.43578e8i		−	1.76858i	−0.466940	−	0.884289i	$-0.654643\pi$
$433$		−	1.43578e8i		−	1.76858i	0.466940	−	0.884289i	$-0.345357\pi$
$434$		0			0
$435$	9.54177e6			0.115921
$436$		0			0
$437$	1.34978e8	+	7.22795e7i	1.61740	+	0.866105i
$438$		0			0
$439$		−	2.84502e7i		−	0.336273i	−0.985764	−	0.168137i	$-0.946225\pi$
$439$		−	2.84502e7i		−	0.336273i	0.985764	−	0.168137i	$-0.0537749\pi$
$440$		0			0
$441$	1.29569e6			0.0151072
$442$		0			0
$443$	−6.56999e7			−0.755707			−0.377853	−	0.925865i	$-0.623338\pi$
$443$	−6.56999e7			−0.755707			−0.377853	+	0.925865i	$0.623338\pi$
$444$		0			0
$445$		−	2.16403e8i		−	2.45574i
$446$		0			0
$447$		−	1.60712e8i		−	1.79939i
$448$		0			0
$449$			3.86254e7i			0.426711i	0.976975	+	0.213356i	$0.0684393\pi$
$449$			3.86254e7i			0.426711i	−0.976975	+	0.213356i	$0.931561\pi$
$450$		0			0
$451$		−	6.57063e7i		−	0.716271i
$452$		0			0
$453$	1.59825e8			1.71929
$454$		0			0
$455$			2.25669e7i			0.239573i
$456$		0			0
$457$	1.14170e8			1.19620			0.598098	−	0.801423i	$-0.295923\pi$
$457$	1.14170e8			1.19620			0.598098	+	0.801423i	$0.295923\pi$
$458$		0			0
$459$		−	2.47498e7i		−	0.255937i
$460$		0			0
$461$	−1.69044e8			−1.72543			−0.862713	−	0.505694i	$-0.831236\pi$
$461$	−1.69044e8			−1.72543			−0.862713	+	0.505694i	$0.831236\pi$
$462$		0			0
$463$	1.44410e8			1.45497			0.727484	−	0.686125i	$-0.240690\pi$
$463$	1.44410e8			1.45497			0.727484	+	0.686125i	$0.240690\pi$
$464$		0			0
$465$	1.88718e8			1.87696
$466$		0			0
$467$	1.64334e8			1.61353			0.806763	−	0.590875i	$-0.201217\pi$
$467$	1.64334e8			1.61353			0.806763	+	0.590875i	$0.201217\pi$
$468$		0			0
$469$			1.86052e7i			0.180350i
$470$		0			0
$471$			1.32648e8i			1.26951i
$472$		0			0
$473$	1.63261e7			0.154276
$474$		0			0
$475$	6.61465e7	+	3.54209e7i	0.617200	+	0.330506i
$476$		0			0
$477$		−	1.90758e6i		−	0.0175763i
$478$		0			0
$479$	−7.81340e7			−0.710941			−0.355470	−	0.934688i	$-0.615679\pi$
$479$	−7.81340e7			−0.710941			−0.355470	+	0.934688i	$0.615679\pi$
$480$		0			0
$481$	7.32687e7			0.658390
$482$		0			0
$483$		−	4.13466e7i		−	0.366943i
$484$		0			0
$485$		−	2.31896e7i		−	0.203268i
$486$		0			0
$487$			9.85344e7i			0.853102i	0.904463	+	0.426551i	$0.140272\pi$
$487$			9.85344e7i			0.853102i	−0.904463	+	0.426551i	$0.859728\pi$
$488$		0			0
$489$		−	4.92884e7i		−	0.421520i
$490$		0			0
$491$	−1.14313e7			−0.0965718			−0.0482859	−	0.998834i	$-0.515376\pi$
$491$	−1.14313e7			−0.0965718			−0.0482859	+	0.998834i	$0.515376\pi$
$492$		0			0
$493$			2.72711e6i			0.0227594i
$494$		0			0
$495$	1.56680e6			0.0129181
$496$		0			0
$497$		−	2.98445e7i		−	0.243106i
$498$		0			0
$499$	1.65611e8			1.33287			0.666435	−	0.745563i	$-0.267819\pi$
$499$	1.65611e8			1.33287			0.666435	+	0.745563i	$0.267819\pi$
$500$		0			0
$501$	−1.03492e8			−0.822986
$502$		0			0
$503$	−2.52135e7			−0.198120			−0.0990602	−	0.995081i	$-0.531584\pi$
$503$	−2.52135e7			−0.198120			−0.0990602	+	0.995081i	$0.531584\pi$
$504$		0			0
$505$	1.60877e8			1.24916
$506$		0			0
$507$		−	1.87503e7i		−	0.143875i
$508$		0			0
$509$			2.06343e8i			1.56472i	0.622825	+	0.782361i	$0.285985\pi$
$509$			2.06343e8i			1.56472i	−0.622825	+	0.782361i	$0.714015\pi$
$510$		0			0
$511$	1.68961e7			0.126626
$512$		0			0
$513$	6.32210e7	−	1.18062e8i	0.468284	−	0.874494i
$514$		0			0
$515$		−	1.05131e8i		−	0.769676i
$516$		0			0
$517$	−3.14462e7			−0.227560
$518$		0			0
$519$	−1.34705e8			−0.963565
$520$		0			0
$521$			3.79045e7i			0.268026i	0.990980	+	0.134013i	$0.0427864\pi$
$521$			3.79045e7i			0.268026i	−0.990980	+	0.134013i	$0.957214\pi$
$522$		0			0
$523$			2.05573e8i			1.43701i	0.695521	+	0.718506i	$0.255174\pi$
$523$			2.05573e8i			1.43701i	−0.695521	+	0.718506i	$0.744826\pi$
$524$		0			0
$525$		−	2.02621e7i		−	0.140025i
$526$		0			0
$527$			5.39370e7i			0.368515i
$528$		0			0
$529$	3.50272e8			2.36613
$530$		0			0
$531$		−	3.23299e6i		−	0.0215934i
$532$		0			0
$533$	−1.59400e8			−1.05270
$534$		0			0
$535$		−	3.03230e8i		−	1.98021i
$536$		0			0
$537$	−1.11142e7			−0.0717719
$538$		0			0
$539$	9.47636e7			0.605167
$540$		0			0
$541$	−1.15918e8			−0.732078			−0.366039	−	0.930600i	$-0.619286\pi$
$541$	−1.15918e8			−0.732078			−0.366039	+	0.930600i	$0.619286\pi$
$542$		0			0
$543$	2.63914e7			0.164840
$544$		0			0
$545$		−	2.38039e8i		−	1.47048i
$546$		0			0
$547$			2.18070e8i			1.33240i	0.745773	+	0.666200i	$0.232080\pi$
$547$			2.18070e8i			1.33240i	−0.745773	+	0.666200i	$0.767920\pi$
$548$		0			0
$549$	−4.28218e6			−0.0258790
$550$		0			0
$551$	−6.96615e6	+	1.30089e7i	−0.0416426	+	0.0777652i
$552$		0			0
$553$		−	3.91016e7i		−	0.231217i
$554$		0			0
$555$	−1.59749e8			−0.934458
$556$		0			0
$557$	−1.32552e8			−0.767045			−0.383522	−	0.923532i	$-0.625289\pi$
$557$	−1.32552e8			−0.767045			−0.383522	+	0.923532i	$0.625289\pi$
$558$		0			0
$559$		−	3.96060e7i		−	0.226739i
$560$		0			0
$561$			2.89221e7i			0.163810i
$562$		0			0
$563$			1.90793e8i			1.06914i	0.845123	+	0.534572i	$0.179527\pi$
$563$			1.90793e8i			1.06914i	−0.845123	+	0.534572i	$0.820473\pi$
$564$		0			0
$565$			2.96454e8i			1.64366i
$566$		0			0
$567$	−3.67338e7			−0.201519
$568$		0			0
$569$			3.28510e8i			1.78325i	0.452776	+	0.891624i	$0.350434\pi$
$569$			3.28510e8i			1.78325i	−0.452776	+	0.891624i	$0.649566\pi$
$570$		0			0
$571$	−7.87374e7			−0.422934			−0.211467	−	0.977385i	$-0.567824\pi$
$571$	−7.87374e7			−0.422934			−0.211467	+	0.977385i	$0.567824\pi$
$572$		0			0
$573$		−	2.56542e7i		−	0.136363i
$574$		0			0
$575$	2.44198e8			1.28451
$576$		0			0
$577$	1.70970e8			0.890003			0.445002	−	0.895530i	$-0.353203\pi$
$577$	1.70970e8			0.890003			0.445002	+	0.895530i	$0.353203\pi$
$578$		0			0
$579$	−9.43796e7			−0.486231
$580$		0			0
$581$	2.34767e7			0.119704
$582$		0			0
$583$		−	1.39516e8i		−	0.704076i
$584$		0			0
$585$		−	3.80095e6i		−	0.0189856i
$586$		0			0
$587$	−1.21563e8			−0.601017			−0.300509	−	0.953779i	$-0.597156\pi$
$587$	−1.21563e8			−0.601017			−0.300509	+	0.953779i	$0.597156\pi$
$588$		0			0
$589$	−1.37777e8	+	2.57291e8i	−0.674266	+	1.25915i
$590$		0			0
$591$		−	1.81971e8i		−	0.881534i
$592$		0			0
$593$	−2.74861e8			−1.31810			−0.659052	−	0.752097i	$-0.729042\pi$
$593$	−2.74861e8			−1.31810			−0.659052	+	0.752097i	$0.729042\pi$
$594$		0			0
$595$	1.40626e7			0.0667596
$596$		0			0
$597$			3.98846e8i			1.87448i
$598$		0			0
$599$		−	6.76259e7i		−	0.314653i	−0.987547	−	0.157327i	$-0.949712\pi$
$599$		−	6.76259e7i		−	0.314653i	0.987547	−	0.157327i	$-0.0502876\pi$
$600$		0			0
$601$		−	3.84218e8i		−	1.76992i	−0.465666	−	0.884960i	$-0.654185\pi$
$601$		−	3.84218e8i		−	1.76992i	0.465666	−	0.884960i	$-0.345815\pi$
$602$		0			0
$603$		−	3.13369e6i		−	0.0142923i
$604$		0			0
$605$	−1.74147e8			−0.786413
$606$		0			0
$607$			2.23120e8i			0.997637i	0.866706	+	0.498819i	$0.166233\pi$
$607$			2.23120e8i			0.997637i	−0.866706	+	0.498819i	$0.833767\pi$
$608$		0			0
$609$	3.98491e6			0.0176427
$610$		0			0
$611$			7.62864e7i			0.334444i
$612$		0			0
$613$	−3.10738e8			−1.34900			−0.674501	−	0.738274i	$-0.735641\pi$
$613$	−3.10738e8			−1.34900			−0.674501	+	0.738274i	$0.735641\pi$
$614$		0			0
$615$	3.47542e8			1.49411
$616$		0			0
$617$	2.92870e8			1.24687			0.623433	−	0.781877i	$-0.285738\pi$
$617$	2.92870e8			1.24687			0.623433	+	0.781877i	$0.285738\pi$
$618$		0			0
$619$	2.18192e8			0.919957			0.459979	−	0.887930i	$-0.347857\pi$
$619$	2.18192e8			0.919957			0.459979	+	0.887930i	$0.347857\pi$
$620$		0			0
$621$		−	4.35857e8i		−	1.81999i
$622$		0			0
$623$		−	9.03758e7i		−	0.373756i
$624$		0			0
$625$	−2.95398e8			−1.20995
$626$		0			0
$627$	−7.38790e7	+	1.37965e8i	−0.299722	+	0.559713i
$628$		0			0
$629$		−	4.56575e7i		−	0.183468i
$630$		0			0
$631$	−1.59902e8			−0.636452			−0.318226	−	0.948015i	$-0.603087\pi$
$631$	−1.59902e8			−0.636452			−0.318226	+	0.948015i	$0.603087\pi$
$632$		0			0
$633$	1.84234e8			0.726372
$634$		0			0
$635$		−	1.77573e8i		−	0.693516i
$636$		0			0
$637$		−	2.29891e8i		−	0.889412i
$638$		0			0
$639$			5.02673e6i			0.0192656i
$640$		0			0
$641$		−	3.07834e7i		−	0.116881i	−0.998291	−	0.0584403i	$-0.981387\pi$
$641$		−	3.07834e7i		−	0.116881i	0.998291	−	0.0584403i	$-0.0186127\pi$
$642$		0			0
$643$	−4.42423e7			−0.166420			−0.0832098	−	0.996532i	$-0.526517\pi$
$643$	−4.42423e7			−0.166420			−0.0832098	+	0.996532i	$0.526517\pi$
$644$		0			0
$645$			8.63538e7i			0.321812i
$646$		0			0
$647$	−4.20721e8			−1.55339			−0.776697	−	0.629874i	$-0.783106\pi$
$647$	−4.20721e8			−1.55339			−0.776697	+	0.629874i	$0.783106\pi$
$648$		0			0
$649$		−	2.36454e8i		−	0.864993i
$650$		0			0
$651$	7.88138e7			0.285667
$652$		0			0
$653$	−1.79550e8			−0.644832			−0.322416	−	0.946598i	$-0.604495\pi$
$653$	−1.79550e8			−0.644832			−0.322416	+	0.946598i	$0.604495\pi$
$654$		0			0
$655$	−4.27155e8			−1.52006
$656$		0			0
$657$	−2.84581e6			−0.0100348
$658$		0			0
$659$		−	2.06027e8i		−	0.719892i	−0.932973	−	0.359946i	$-0.882795\pi$
$659$		−	2.06027e8i		−	0.719892i	0.932973	−	0.359946i	$-0.117205\pi$
$660$		0			0
$661$			7.13718e7i			0.247128i	0.992337	+	0.123564i	$0.0394325\pi$
$661$			7.13718e7i			0.247128i	−0.992337	+	0.123564i	$0.960568\pi$
$662$		0			0
$663$	7.01633e7			0.240752
$664$		0			0
$665$	6.70815e7	+	3.59216e7i	0.228107	+	0.122149i
$666$		0			0
$667$			4.80259e7i			0.161845i
$668$		0			0
$669$	−1.11091e8			−0.371024
$670$		0			0
$671$	−3.13189e8			−1.03667
$672$		0			0
$673$		−	1.69751e8i		−	0.556886i	−0.960453	−	0.278443i	$-0.910182\pi$
$673$		−	1.69751e8i		−	0.556886i	0.960453	−	0.278443i	$-0.0898184\pi$
$674$		0			0
$675$		−	2.13594e8i		−	0.694509i
$676$		0			0
$677$			1.37841e8i			0.444234i	0.975020	+	0.222117i	$0.0712966\pi$
$677$			1.37841e8i			0.444234i	−0.975020	+	0.222117i	$0.928703\pi$
$678$		0			0
$679$		−	9.68464e6i		−	0.0309367i
$680$		0			0
$681$	1.37333e7			0.0434843
$682$		0			0
$683$			4.63188e8i			1.45377i	0.686759	+	0.726885i	$0.259033\pi$
$683$			4.63188e8i			1.45377i	−0.686759	+	0.726885i	$0.740967\pi$
$684$		0			0
$685$	4.04841e8			1.25954
$686$		0			0
$687$			2.81397e8i			0.867859i
$688$		0			0
$689$	−3.38458e8			−1.03478
$690$		0			0
$691$	1.90281e8			0.576715			0.288357	−	0.957523i	$-0.406891\pi$
$691$	1.90281e8			0.576715			0.288357	+	0.957523i	$0.406891\pi$
$692$		0			0
$693$	654337.			0.00196608
$694$		0			0
$695$	1.94743e7			0.0580105
$696$		0			0
$697$			9.93300e7i			0.293347i
$698$		0			0
$699$		−	4.61990e7i		−	0.135270i
$700$		0			0
$701$	−1.02507e7			−0.0297576			−0.0148788	−	0.999889i	$-0.504736\pi$
$701$	−1.02507e7			−0.0297576			−0.0148788	+	0.999889i	$0.504736\pi$
$702$		0			0
$703$	1.16628e8	−	2.17796e8i	0.335689	−	0.626880i
$704$		0			0
$705$		−	1.66329e8i		−	0.474679i
$706$		0			0
$707$	6.71866e7			0.190118
$708$		0			0
$709$	3.16806e7			0.0888905			0.0444452	−	0.999012i	$-0.485848\pi$
$709$	3.16806e7			0.0888905			0.0444452	+	0.999012i	$0.485848\pi$
$710$		0			0
$711$			6.58590e6i			0.0183234i
$712$		0			0
$713$			9.49861e8i			2.62054i
$714$		0			0
$715$		−	2.77993e8i		−	0.760530i
$716$		0			0
$717$		−	2.37926e7i		−	0.0645483i
$718$		0			0
$719$	7.46352e7			0.200797			0.100398	−	0.994947i	$-0.467988\pi$
$719$	7.46352e7			0.200797			0.100398	+	0.994947i	$0.467988\pi$
$720$		0			0
$721$		−	4.39055e7i		−	0.117142i
$722$		0			0
$723$	−2.81469e8			−0.744758
$724$		0			0
$725$			2.35353e7i			0.0617599i
$726$		0			0
$727$	−6.64160e8			−1.72850			−0.864250	−	0.503063i	$-0.832206\pi$
$727$	−6.64160e8			−1.72850			−0.864250	+	0.503063i	$0.832206\pi$
$728$		0			0
$729$	−3.81138e8			−0.983783
$730$		0			0
$731$	−2.46805e7			−0.0631833
$732$		0			0
$733$	−5.29085e8			−1.34343			−0.671713	−	0.740811i	$-0.734441\pi$
$733$	−5.29085e8			−1.34343			−0.671713	+	0.740811i	$0.734441\pi$
$734$		0			0
$735$			5.01235e8i			1.26235i
$736$		0			0
$737$		−	2.29191e8i		−	0.572526i
$738$		0			0
$739$	−1.46711e8			−0.363521			−0.181761	−	0.983343i	$-0.558180\pi$
$739$	−1.46711e8			−0.363521			−0.181761	+	0.983343i	$0.558180\pi$
$740$		0			0
$741$	3.34694e8	+	1.79226e8i	0.822608	+	0.440500i
$742$		0			0
$743$		−	5.65796e8i		−	1.37941i	−0.724091	−	0.689705i	$-0.757740\pi$
$743$		−	5.65796e8i		−	1.37941i	0.724091	−	0.689705i	$-0.242260\pi$
$744$		0			0
$745$	9.62598e8			2.32796
$746$		0			0
$747$	−3.95420e6			−0.00948630
$748$		0			0
$749$		−	1.26637e8i		−	0.301381i
$750$		0			0
$751$		−	6.51395e8i		−	1.53789i	−0.639317	−	0.768943i	$-0.720783\pi$
$751$		−	6.51395e8i		−	1.53789i	0.639317	−	0.768943i	$-0.279217\pi$
$752$		0			0
$753$			2.66651e8i			0.624538i
$754$		0			0
$755$			9.57286e8i			2.22434i
$756$		0			0
$757$	−2.49239e8			−0.574550			−0.287275	−	0.957848i	$-0.592749\pi$
$757$	−2.49239e8			−0.574550			−0.287275	+	0.957848i	$0.592749\pi$
$758$		0			0
$759$			5.09335e8i			1.16487i
$760$		0			0
$761$	−2.33143e8			−0.529015			−0.264508	−	0.964384i	$-0.585209\pi$
$761$	−2.33143e8			−0.529015			−0.264508	+	0.964384i	$0.585209\pi$
$762$		0			0
$763$		−	9.94115e7i		−	0.223801i
$764$		0			0
$765$	−2.36857e6			−0.00529056
$766$		0			0
$767$	−5.73623e8			−1.27128
$768$		0			0
$769$	−2.23195e8			−0.490801			−0.245400	−	0.969422i	$-0.578919\pi$
$769$	−2.23195e8			−0.490801			−0.245400	+	0.969422i	$0.578919\pi$
$770$		0			0
$771$	−4.63573e8			−1.01147
$772$		0			0
$773$		−	6.80275e8i		−	1.47281i	−0.676542	−	0.736404i	$-0.736522\pi$
$773$		−	6.80275e8i		−	1.47281i	0.676542	−	0.736404i	$-0.263478\pi$
$774$		0			0
$775$			4.65484e8i			0.999999i
$776$		0			0
$777$	−6.67157e7			−0.142221
$778$		0			0
$779$	−2.53730e8	+	4.73825e8i	−0.536733	+	1.00232i
$780$		0			0
$781$			3.67644e8i			0.771746i
$782$		0			0
$783$	4.20071e7			0.0875058
$784$		0			0
$785$	−7.94507e8			−1.64244
$786$		0			0
$787$			6.88578e8i			1.41263i	0.707897	+	0.706316i	$0.249644\pi$
$787$			6.88578e8i			1.41263i	−0.707897	+	0.706316i	$0.750356\pi$
$788$		0			0
$789$		−	3.20693e8i		−	0.652918i
$790$		0			0
$791$			1.23807e8i			0.250160i
$792$		0			0
$793$			7.59778e8i			1.52359i
$794$		0			0
$795$	7.37947e8			1.46867
$796$		0			0
$797$		−	1.81534e7i		−	0.0358577i	−0.999839	−	0.0179288i	$-0.994293\pi$
$797$		−	1.81534e7i		−	0.0358577i	0.999839	−	0.0179288i	$-0.00570723\pi$
$798$		0			0
$799$	4.75380e7			0.0931966
$800$		0			0
$801$			1.52221e7i			0.0296194i
$802$		0			0
$803$	−2.08137e8			−0.401978
$804$		0			0
$805$	2.47650e8			0.474734
$806$		0			0
$807$	9.06235e8			1.72433
$808$		0			0
$809$	−7.93848e8			−1.49931			−0.749655	−	0.661829i	$-0.769781\pi$
$809$	−7.93848e8			−1.49931			−0.749655	+	0.661829i	$0.769781\pi$
$810$		0			0
$811$		−	6.10611e8i		−	1.14473i	−0.820000	−	0.572364i	$-0.806027\pi$
$811$		−	6.10611e8i		−	1.14473i	0.820000	−	0.572364i	$-0.193973\pi$
$812$		0			0
$813$			3.86150e8i			0.718595i
$814$		0			0
$815$	2.95218e8			0.545343
$816$		0			0
$817$	−1.17732e8	−	6.30442e7i	−0.215887	−	0.115606i
$818$		0			0
$819$		−	1.58738e6i		−	0.00288955i
$820$		0			0
$821$	2.18871e8			0.395512			0.197756	−	0.980251i	$-0.436635\pi$
$821$	2.18871e8			0.395512			0.197756	+	0.980251i	$0.436635\pi$
$822$		0			0
$823$	8.66944e8			1.55522			0.777609	−	0.628748i	$-0.216432\pi$
$823$	8.66944e8			1.55522			0.777609	+	0.628748i	$0.216432\pi$
$824$		0			0
$825$			2.49602e8i			0.444515i
$826$		0			0
$827$		−	4.78521e8i		−	0.846027i	−0.906123	−	0.423014i	$-0.860972\pi$
$827$		−	4.78521e8i		−	0.846027i	0.906123	−	0.423014i	$-0.139028\pi$
$828$		0			0
$829$			4.93510e8i			0.866229i	0.901339	+	0.433114i	$0.142585\pi$
$829$			4.93510e8i			0.866229i	−0.901339	+	0.433114i	$0.857415\pi$
$830$		0			0
$831$			2.84452e8i			0.495685i
$832$		0			0
$833$	−1.43257e8			−0.247845
$834$		0			0
$835$		−	6.19874e8i		−	1.06474i
$836$		0			0
$837$	8.30819e8			1.41687
$838$		0			0
$839$			1.79399e8i			0.303762i	0.988399	+	0.151881i	$0.0485331\pi$
$839$			1.79399e8i			0.303762i	−0.988399	+	0.151881i	$0.951467\pi$
$840$		0			0
$841$	5.90195e8			0.992218
$842$		0			0
$843$	2.32320e8			0.387797
$844$		0			0
$845$	1.12307e8			0.186138
$846$		0			0
$847$	−7.27287e7			−0.119689
$848$		0			0
$849$		−	2.31387e6i		−	0.00378108i
$850$		0			0
$851$		−	8.04054e8i		−	1.30466i
$852$		0			0
$853$	−4.99419e8			−0.804670			−0.402335	−	0.915493i	$-0.631801\pi$
$853$	−4.99419e8			−0.804670			−0.402335	+	0.915493i	$0.631801\pi$
$854$		0			0
$855$	−1.12986e7	−	6.05029e6i	−0.0180770	−	0.00968006i
$856$		0			0
$857$		−	2.91259e8i		−	0.462740i	−0.972866	−	0.231370i	$-0.925679\pi$
$857$		−	2.91259e8i		−	0.462740i	0.972866	−	0.231370i	$-0.0743208\pi$
$858$		0			0
$859$	5.79375e8			0.914072			0.457036	−	0.889448i	$-0.348911\pi$
$859$	5.79375e8			0.914072			0.457036	+	0.889448i	$0.348911\pi$
$860$		0			0
$861$	1.45143e8			0.227398
$862$		0			0
$863$		−	4.85743e8i		−	0.755743i	−0.925858	−	0.377872i	$-0.876656\pi$
$863$		−	4.85743e8i		−	0.755743i	0.925858	−	0.377872i	$-0.123344\pi$
$864$		0			0
$865$		−	8.06828e8i		−	1.24662i
$866$		0			0
$867$			6.13097e8i			0.940744i
$868$		0			0
$869$			4.81679e8i			0.734003i
$870$		0			0
$871$	−5.56003e8			−0.841440
$872$		0			0
$873$			1.63119e6i			0.00245167i
$874$		0			0
$875$	−5.19822e7			−0.0775944
$876$		0			0
$877$		−	1.04191e9i		−	1.54465i	−0.635228	−	0.772324i	$-0.719094\pi$
$877$		−	1.04191e9i		−	1.54465i	0.635228	−	0.772324i	$-0.280906\pi$
$878$		0			0
$879$	5.83256e8			0.858801
$880$		0			0
$881$	1.61613e8			0.236346			0.118173	−	0.992993i	$-0.462296\pi$
$881$	1.61613e8			0.236346			0.118173	+	0.992993i	$0.462296\pi$
$882$		0			0
$883$	−1.08770e9			−1.57989			−0.789943	−	0.613180i	$-0.789890\pi$
$883$	−1.08770e9			−1.57989			−0.789943	+	0.613180i	$0.789890\pi$
$884$		0			0
$885$	1.25068e9			1.80433
$886$		0			0
$887$		−	4.42391e7i		−	0.0633922i	−0.999498	−	0.0316961i	$-0.989909\pi$
$887$		−	4.42391e7i		−	0.0633922i	0.999498	−	0.0316961i	$-0.0100909\pi$
$888$		0			0
$889$		−	7.41595e7i		−	0.105551i
$890$		0			0
$891$	4.52510e8			0.639728
$892$		0			0
$893$	2.26766e8	+	1.21431e8i	0.318438	+	0.170521i
$894$		0			0
$895$		−	6.65695e7i		−	0.0928552i
$896$		0			0
$897$	1.23561e9			1.71201
$898$		0			0
$899$	−9.15457e7			−0.125997
$900$		0			0
$901$			2.10911e8i			0.288353i
$902$		0			0
$903$			3.60637e7i			0.0489787i
$904$		0			0
$905$			1.58074e8i			0.213262i
$906$		0			0
$907$		−	8.05692e7i		−	0.107981i	−0.998541	−	0.0539904i	$-0.982806\pi$
$907$		−	8.05692e7i		−	0.107981i	0.998541	−	0.0539904i	$-0.0171941\pi$
$908$		0			0
$909$	−1.13163e7			−0.0150665
$910$		0			0
$911$		−	2.55068e7i		−	0.0337366i	−0.999858	−	0.0168683i	$-0.994630\pi$
$911$		−	2.55068e7i		−	0.0337366i	0.999858	−	0.0168683i	$-0.00536960\pi$
$912$		0			0
$913$	−2.89202e8			−0.380004
$914$		0			0
$915$		−	1.65656e9i		−	2.16244i
$916$		0			0
$917$	−1.78392e8			−0.231349
$918$		0			0
$919$	9.10218e8			1.17273			0.586366	−	0.810046i	$-0.300558\pi$
$919$	9.10218e8			1.17273			0.586366	+	0.810046i	$0.300558\pi$
$920$		0			0
$921$	−1.04863e9			−1.34228
$922$		0			0
$923$	8.91882e8			1.13423
$924$		0			0
$925$		−	3.94031e8i		−	0.497857i
$926$		0			0
$927$			7.39502e6i			0.00928326i
$928$		0			0
$929$	8.64268e8			1.07796			0.538978	−	0.842320i	$-0.318810\pi$
$929$	8.64268e8			1.07796			0.538978	+	0.842320i	$0.318810\pi$
$930$		0			0
$931$	−6.83365e8	−	3.65936e8i	−0.846845	−	0.453478i
$932$		0			0
$933$		−	5.14817e8i		−	0.633882i
$934$		0			0
$935$	−1.73232e8			−0.211930
$936$		0			0
$937$	−2.08746e8			−0.253746			−0.126873	−	0.991919i	$-0.540494\pi$
$937$	−2.08746e8			−0.253746			−0.126873	+	0.991919i	$0.540494\pi$
$938$		0			0
$939$		−	1.09781e9i		−	1.32597i
$940$		0			0
$941$		−	1.20142e9i		−	1.44187i	−0.693004	−	0.720933i	$-0.743713\pi$
$941$		−	1.20142e9i		−	1.44187i	0.693004	−	0.720933i	$-0.256287\pi$
$942$		0			0
$943$			1.74926e9i			2.08602i
$944$		0			0
$945$		−	2.16613e8i		−	0.256678i
$946$		0			0
$947$	−1.33535e8			−0.157234			−0.0786170	−	0.996905i	$-0.525050\pi$
$947$	−1.33535e8			−0.157234			−0.0786170	+	0.996905i	$0.525050\pi$
$948$		0			0
$949$			5.04927e8i			0.590785i
$950$		0			0
$951$	−1.47981e6			−0.00172053
$952$		0			0
$953$		−	6.97538e8i		−	0.805915i	−0.915219	−	0.402957i	$-0.867982\pi$
$953$		−	6.97538e8i		−	0.805915i	0.915219	−	0.402957i	$-0.132018\pi$
$954$		0			0
$955$	1.53659e8			0.176420
$956$		0			0
$957$	−4.90887e7			−0.0560074
$958$		0			0
$959$	1.69073e8			0.191698
$960$		0			0
$961$	−9.23095e8			−1.04010
$962$		0			0
$963$			2.13296e7i			0.0238838i
$964$		0			0
$965$		−	5.65296e8i		−	0.629063i
$966$		0			0
$967$	4.93793e8			0.546092			0.273046	−	0.962001i	$-0.411969\pi$
$967$	4.93793e8			0.546092			0.273046	+	0.962001i	$0.411969\pi$
$968$		0			0
$969$	1.11685e8	−	2.08565e8i	0.122750	−	0.229229i
$970$		0			0
$971$			1.98464e8i			0.216783i	0.994108	+	0.108391i	$0.0345700\pi$
$971$			1.98464e8i			0.216783i	−0.994108	+	0.108391i	$0.965430\pi$
$972$		0			0
$973$	8.13299e6			0.00882900
$974$		0			0
$975$	6.05519e8			0.653302
$976$		0			0
$977$			5.38145e8i			0.577052i	0.957472	+	0.288526i	$0.0931652\pi$
$977$			5.38145e8i			0.577052i	−0.957472	+	0.288526i	$0.906835\pi$
$978$		0			0
$979$			1.11331e9i			1.18650i
$980$		0			0
$981$			1.67439e7i			0.0177358i
$982$		0			0
$983$		−	1.73889e8i		−	0.183067i	−0.995802	−	0.0915336i	$-0.970823\pi$
$983$		−	1.73889e8i		−	0.183067i	0.995802	−	0.0915336i	$-0.0291769\pi$
$984$		0			0
$985$	1.08993e9			1.14049
$986$		0			0
$987$		−	6.94635e7i		−	0.0722445i
$988$		0			0
$989$	−4.34638e8			−0.449303
$990$		0			0
$991$		−	1.27730e9i		−	1.31241i	−0.754581	−	0.656207i	$-0.772160\pi$
$991$		−	1.27730e9i		−	1.31241i	0.754581	−	0.656207i	$-0.227840\pi$
$992$		0			0
$993$	6.28013e8			0.641388
$994$		0			0
$995$	−2.38893e9			−2.42512
$996$		0			0
$997$	−2.66502e8			−0.268915			−0.134458	−	0.990919i	$-0.542929\pi$
$997$	−2.66502e8			−0.268915			−0.134458	+	0.990919i	$0.542929\pi$
$998$		0			0
$999$	−7.03286e8			−0.705400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.7.e.d.113.3		8
4.3	odd	2		19.7.b.b.18.3	✓	8
12.11	even	2		171.7.c.d.37.6		8
19.18	odd	2	inner	304.7.e.d.113.6		8
76.75	even	2		19.7.b.b.18.6	yes	8
228.227	odd	2		171.7.c.d.37.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.7.b.b.18.3	✓	8	4.3	odd	2
19.7.b.b.18.6	yes	8	76.75	even	2
171.7.c.d.37.3		8	228.227	odd	2
171.7.c.d.37.6		8	12.11	even	2
304.7.e.d.113.3		8	1.1	even	1	trivial
304.7.e.d.113.6		8	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)

\(f(q)\)	\(=\)	\(q-27.2115i q^{3} +162.986 q^{5} +68.0674 q^{7} -11.4646 q^{9} +O(q^{10})\)	\(q-27.2115i q^{3} +162.986 q^{5} +68.0674 q^{7} -11.4646 q^{9} -838.499 q^{11} +2034.15i q^{13} -4435.09i q^{15} +1267.58 q^{17} +(6046.63 + 3237.92i) q^{19} -1852.21i q^{21} +22322.8 q^{23} +10939.4 q^{25} -19525.2i q^{27} +2151.43i q^{29} +42551.1i q^{31} +22816.8i q^{33} +11094.0 q^{35} -36019.4i q^{37} +55352.1 q^{39} +78361.9i q^{41} -19470.6 q^{43} -1868.57 q^{45} +37502.9 q^{47} -113016. q^{49} -34492.7i q^{51} +166388. i q^{53} -136663. q^{55} +(88108.6 - 164538. i) q^{57} +281997. i q^{59} +373512. q^{61} -780.368 q^{63} +331537. i q^{65} +273335. i q^{67} -607437. i q^{69} -438455. i q^{71} +248225. q^{73} -297677. i q^{75} -57074.4 q^{77} -574454. i q^{79} -539667. q^{81} +344904. q^{83} +206598. q^{85} +58543.5 q^{87} -1.32774e6i q^{89} +138459. i q^{91} +1.15788e6 q^{93} +(985516. + 527735. i) q^{95} -142280. i q^{97} +9613.08 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 108 q^{5} + 140 q^{7} - 1052 q^{9}+O(q^{10}) \) 8 * q + 108 * q^5 + 140 * q^7 - 1052 * q^9	\( 8 q + 108 q^{5} + 140 q^{7} - 1052 q^{9} + 2024 q^{11} + 6008 q^{17} - 20552 q^{19} + 50252 q^{23} + 78492 q^{25} - 210800 q^{35} - 43724 q^{39} - 260800 q^{43} - 191012 q^{45} + 100248 q^{47} - 301872 q^{49} + 52480 q^{55} - 186860 q^{57} - 54548 q^{61} + 137408 q^{63} + 479968 q^{73} - 1755300 q^{77} - 4279648 q^{81} - 483040 q^{83} + 2111780 q^{85} - 2802652 q^{87} + 1507528 q^{93} + 2383888 q^{95} - 528224 q^{99}+O(q^{100}) \) 8 * q + 108 * q^5 + 140 * q^7 - 1052 * q^9 + 2024 * q^11 + 6008 * q^17 - 20552 * q^19 + 50252 * q^23 + 78492 * q^25 - 210800 * q^35 - 43724 * q^39 - 260800 * q^43 - 191012 * q^45 + 100248 * q^47 - 301872 * q^49 + 52480 * q^55 - 186860 * q^57 - 54548 * q^61 + 137408 * q^63 + 479968 * q^73 - 1755300 * q^77 - 4279648 * q^81 - 483040 * q^83 + 2111780 * q^85 - 2802652 * q^87 + 1507528 * q^93 + 2383888 * q^95 - 528224 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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