LMFDB - Embedded newform 304.7.e.f.113.10

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			113.10
Character	$\chi$	$=$	304.113
Dual form			304.7.e.f.113.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-20.4841i q^{3} -148.208 q^{5} +486.917 q^{7} +309.402 q^{9} +O(q^{10})$	$q-20.4841i q^{3} -148.208 q^{5} +486.917 q^{7} +309.402 q^{9} +1551.05 q^{11} +2321.67i q^{13} +3035.91i q^{15} -4233.52 q^{17} +(-2600.88 + 6346.76i) q^{19} -9974.05i q^{21} -15866.6 q^{23} +6340.66 q^{25} -21270.7i q^{27} -24843.8i q^{29} +11748.4i q^{31} -31771.9i q^{33} -72165.1 q^{35} +61074.3i q^{37} +47557.3 q^{39} +5979.72i q^{41} +37743.1 q^{43} -45855.9 q^{45} -186857. q^{47} +119439. q^{49} +86719.9i q^{51} +190300. i q^{53} -229879. q^{55} +(130008. + 53276.6i) q^{57} +358326. i q^{59} +297880. q^{61} +150653. q^{63} -344090. i q^{65} +67731.9i q^{67} +325013. i q^{69} +201251. i q^{71} -136884. q^{73} -129883. i q^{75} +755234. q^{77} +526034. i q^{79} -210157. q^{81} +24927.7 q^{83} +627443. q^{85} -508904. q^{87} -121816. i q^{89} +1.13046e6i q^{91} +240655. q^{93} +(385471. - 940641. i) q^{95} -1.38573e6i q^{97} +479899. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	20.4841i		−	0.758670i	−0.925259	−	0.379335i	$-0.876153\pi$
$3$		−	20.4841i		−	0.758670i	0.925259	−	0.379335i	$-0.123847\pi$
$4$		0			0
$5$	−148.208			−1.18567			−0.592833	−	0.805326i	$-0.701990\pi$
$5$	−148.208			−1.18567			−0.592833	+	0.805326i	$0.701990\pi$
$6$		0			0
$7$	486.917			1.41958			0.709792	−	0.704412i	$-0.248789\pi$
$7$	486.917			1.41958			0.709792	+	0.704412i	$0.248789\pi$
$8$		0			0
$9$	309.402			0.424420
$10$		0			0
$11$	1551.05			1.16533			0.582665	−	0.812713i	$-0.302010\pi$
$11$	1551.05			1.16533			0.582665	+	0.812713i	$0.302010\pi$
$12$		0			0
$13$			2321.67i			1.05674i	0.849013	+	0.528372i	$0.177197\pi$
$13$			2321.67i			1.05674i	−0.849013	+	0.528372i	$0.822803\pi$
$14$		0			0
$15$			3035.91i			0.899529i
$16$		0			0
$17$	−4233.52			−0.861698			−0.430849	−	0.902424i	$-0.641786\pi$
$17$	−4233.52			−0.861698			−0.430849	+	0.902424i	$0.641786\pi$
$18$		0			0
$19$	−2600.88	+	6346.76i	−0.379192	+	0.925318i
$19$	−2600.88	+	6346.76i	−0.379192	+	0.925318i
$20$		0			0
$21$		−	9974.05i		−	1.07700i
$22$		0			0
$23$	−15866.6			−1.30407			−0.652035	−	0.758189i	$-0.726084\pi$
$23$	−15866.6			−1.30407			−0.652035	+	0.758189i	$0.726084\pi$
$24$		0			0
$25$	6340.66			0.405802
$26$		0			0
$27$		−	21270.7i		−	1.08066i
$28$		0			0
$29$		−	24843.8i		−	1.01865i	−0.860574	−	0.509325i	$-0.829895\pi$
$29$		−	24843.8i		−	1.01865i	0.860574	−	0.509325i	$-0.170105\pi$
$30$		0			0
$31$			11748.4i			0.394360i	0.980367	+	0.197180i	$0.0631784\pi$
$31$			11748.4i			0.394360i	−0.980367	+	0.197180i	$0.936822\pi$
$32$		0			0
$33$		−	31771.9i		−	0.884100i
$34$		0			0
$35$	−72165.1			−1.68315
$36$		0			0
$37$			61074.3i			1.20574i	0.797840	+	0.602870i	$0.205976\pi$
$37$			61074.3i			1.20574i	−0.797840	+	0.602870i	$0.794024\pi$
$38$		0			0
$39$	47557.3			0.801721
$40$		0			0
$41$			5979.72i			0.0867620i	0.999059	+	0.0433810i	$0.0138129\pi$
$41$			5979.72i			0.0867620i	−0.999059	+	0.0433810i	$0.986187\pi$
$42$		0			0
$43$	37743.1			0.474714			0.237357	−	0.971422i	$-0.423719\pi$
$43$	37743.1			0.474714			0.237357	+	0.971422i	$0.423719\pi$
$44$		0			0
$45$	−45855.9			−0.503220
$46$		0			0
$47$	−186857.			−1.79977			−0.899883	−	0.436131i	$-0.856348\pi$
$47$	−186857.			−1.79977			−0.899883	+	0.436131i	$0.856348\pi$
$48$		0			0
$49$	119439.			1.01522
$50$		0			0
$51$			86719.9i			0.653745i
$52$		0			0
$53$			190300.i			1.27824i	0.769109	+	0.639118i	$0.220701\pi$
$53$			190300.i			1.27824i	−0.769109	+	0.639118i	$0.779299\pi$
$54$		0			0
$55$	−229879.			−1.38169
$56$		0			0
$57$	130008.	+	53276.6i	0.702011	+	0.287682i
$58$		0			0
$59$			358326.i			1.74471i	0.488876	+	0.872353i	$0.337407\pi$
$59$			358326.i			1.74471i	−0.488876	+	0.872353i	$0.662593\pi$
$60$		0			0
$61$	297880.			1.31235			0.656177	−	0.754607i	$-0.272172\pi$
$61$	297880.			1.31235			0.656177	+	0.754607i	$0.272172\pi$
$62$		0			0
$63$	150653.			0.602499
$64$		0			0
$65$		−	344090.i		−	1.25295i
$66$		0			0
$67$			67731.9i			0.225200i	0.993640	+	0.112600i	$0.0359179\pi$
$67$			67731.9i			0.225200i	−0.993640	+	0.112600i	$0.964082\pi$
$68$		0			0
$69$			325013.i			0.989359i
$70$		0			0
$71$			201251.i			0.562294i	0.959665	+	0.281147i	$0.0907148\pi$
$71$			201251.i			0.562294i	−0.959665	+	0.281147i	$0.909285\pi$
$72$		0			0
$73$	−136884.			−0.351872			−0.175936	−	0.984402i	$-0.556295\pi$
$73$	−136884.			−0.351872			−0.175936	+	0.984402i	$0.556295\pi$
$74$		0			0
$75$		−	129883.i		−	0.307870i
$76$		0			0
$77$	755234.			1.65428
$78$		0			0
$79$			526034.i			1.06692i	0.845825	+	0.533461i	$0.179109\pi$
$79$			526034.i			1.06692i	−0.845825	+	0.533461i	$0.820891\pi$
$80$		0			0
$81$	−210157.			−0.395448
$82$		0			0
$83$	24927.7			0.0435960			0.0217980	−	0.999762i	$-0.493061\pi$
$83$	24927.7			0.0435960			0.0217980	+	0.999762i	$0.493061\pi$
$84$		0			0
$85$	627443.			1.02169
$86$		0			0
$87$	−508904.			−0.772819
$88$		0			0
$89$		−	121816.i		−	0.172796i	−0.996261	−	0.0863982i	$-0.972464\pi$
$89$		−	121816.i		−	0.172796i	0.996261	−	0.0863982i	$-0.0275357\pi$
$90$		0			0
$91$			1.13046e6i			1.50014i
$92$		0			0
$93$	240655.			0.299189
$94$		0			0
$95$	385471.	−	940641.i	0.449595	−	1.09712i
$96$		0			0
$97$		−	1.38573e6i		−	1.51833i	−0.650901	−	0.759163i	$-0.725609\pi$
$97$		−	1.38573e6i		−	1.51833i	0.650901	−	0.759163i	$-0.274391\pi$
$98$		0			0
$99$	479899.			0.494589
$100$		0			0
$101$	−1.18465e6			−1.14981			−0.574903	−	0.818222i	$-0.694960\pi$
$101$	−1.18465e6			−1.14981			−0.574903	+	0.818222i	$0.694960\pi$
$102$		0			0
$103$		−	1.83902e6i		−	1.68296i	−0.540286	−	0.841482i	$-0.681684\pi$
$103$		−	1.83902e6i		−	1.68296i	0.540286	−	0.841482i	$-0.318316\pi$
$104$		0			0
$105$			1.47824e6i			1.27696i
$106$		0			0
$107$			687154.i			0.560922i	0.959865	+	0.280461i	$0.0904874\pi$
$107$			687154.i			0.560922i	−0.959865	+	0.280461i	$0.909513\pi$
$108$		0			0
$109$			1.03947e6i			0.802659i	0.915934	+	0.401330i	$0.131452\pi$
$109$			1.03947e6i			0.802659i	−0.915934	+	0.401330i	$0.868548\pi$
$110$		0			0
$111$	1.25105e6			0.914758
$112$		0			0
$113$			1.35212e6i			0.937090i	0.883440	+	0.468545i	$0.155222\pi$
$113$			1.35212e6i			0.937090i	−0.883440	+	0.468545i	$0.844778\pi$
$114$		0			0
$115$	2.35156e6			1.54619
$116$		0			0
$117$			718329.i			0.448503i
$118$		0			0
$119$	−2.06137e6			−1.22325
$120$		0			0
$121$	634204.			0.357992
$122$		0			0
$123$	122489.			0.0658237
$124$		0			0
$125$	1.37601e6			0.704520
$126$		0			0
$127$			2.72592e6i			1.33077i	0.746502	+	0.665383i	$0.231732\pi$
$127$			2.72592e6i			1.33077i	−0.746502	+	0.665383i	$0.768268\pi$
$128$		0			0
$129$		−	773134.i		−	0.360152i
$130$		0			0
$131$	1.91708e6			0.852758			0.426379	−	0.904545i	$-0.359789\pi$
$131$	1.91708e6			0.852758			0.426379	+	0.904545i	$0.359789\pi$
$132$		0			0
$133$	−1.26641e6	+	3.09034e6i	−0.538295	+	1.31357i
$134$		0			0
$135$			3.15249e6i			1.28131i
$136$		0			0
$137$	657836.			0.255833			0.127916	−	0.991785i	$-0.459171\pi$
$137$	657836.			0.255833			0.127916	+	0.991785i	$0.459171\pi$
$138$		0			0
$139$	1.35406e6			0.504190			0.252095	−	0.967703i	$-0.418881\pi$
$139$	1.35406e6			0.504190			0.252095	+	0.967703i	$0.418881\pi$
$140$		0			0
$141$			3.82760e6i			1.36543i
$142$		0			0
$143$			3.60103e6i			1.23146i
$144$		0			0
$145$			3.68206e6i			1.20778i
$146$		0			0
$147$		−	2.44660e6i		−	0.770215i
$148$		0			0
$149$	1.72353e6			0.521027			0.260513	−	0.965470i	$-0.416108\pi$
$149$	1.72353e6			0.521027			0.260513	+	0.965470i	$0.416108\pi$
$150$		0			0
$151$			2.98549e6i			0.867132i	0.901122	+	0.433566i	$0.142745\pi$
$151$			2.98549e6i			0.867132i	−0.901122	+	0.433566i	$0.857255\pi$
$152$		0			0
$153$	−1.30986e6			−0.365722
$154$		0			0
$155$		−	1.74121e6i		−	0.467579i
$156$		0			0
$157$	−7.42942e6			−1.91980			−0.959900	−	0.280342i	$-0.909552\pi$
$157$	−7.42942e6			−1.91980			−0.959900	+	0.280342i	$0.909552\pi$
$158$		0			0
$159$	3.89812e6			0.969760
$160$		0			0
$161$	−7.72573e6			−1.85124
$162$		0			0
$163$	4.22280e6			0.975074			0.487537	−	0.873102i	$-0.337895\pi$
$163$	4.22280e6			0.975074			0.487537	+	0.873102i	$0.337895\pi$
$164$		0			0
$165$			4.70886e6i			1.04825i
$166$		0			0
$167$			5.00817e6i			1.07530i	0.843168	+	0.537650i	$0.180688\pi$
$167$			5.00817e6i			1.07530i	−0.843168	+	0.537650i	$0.819312\pi$
$168$		0			0
$169$	−563333.			−0.116709
$170$		0			0
$171$	−804717.	+	1.96370e6i	−0.160937	+	0.392723i
$172$		0			0
$173$		−	9.62022e6i		−	1.85801i	−0.370073	−	0.929003i	$-0.620667\pi$
$173$		−	9.62022e6i		−	1.85801i	0.370073	−	0.929003i	$-0.379333\pi$
$174$		0			0
$175$	3.08738e6			0.576070
$176$		0			0
$177$	7.33998e6			1.32366
$178$		0			0
$179$		−	8.48943e6i		−	1.48020i	−0.672499	−	0.740098i	$-0.734779\pi$
$179$		−	8.48943e6i		−	1.48020i	0.672499	−	0.740098i	$-0.265221\pi$
$180$		0			0
$181$			1.01562e7i			1.71276i	0.516345	+	0.856381i	$0.327292\pi$
$181$			1.01562e7i			1.71276i	−0.516345	+	0.856381i	$0.672708\pi$
$182$		0			0
$183$		−	6.10179e6i		−	0.995644i
$184$		0			0
$185$		−	9.05171e6i		−	1.42960i
$186$		0			0
$187$	−6.56642e6			−1.00416
$188$		0			0
$189$		−	1.03571e7i		−	1.53409i
$190$		0			0
$191$	−7.40248e6			−1.06237			−0.531186	−	0.847255i	$-0.678254\pi$
$191$	−7.40248e6			−1.06237			−0.531186	+	0.847255i	$0.678254\pi$
$192$		0			0
$193$		−	1.08204e6i		−	0.150512i	−0.997164	−	0.0752561i	$-0.976023\pi$
$193$		−	1.08204e6i		−	0.150512i	0.997164	−	0.0752561i	$-0.0239774\pi$
$194$		0			0
$195$	−7.04838e6			−0.950572
$196$		0			0
$197$	7.49418e6			0.980225			0.490112	−	0.871659i	$-0.336956\pi$
$197$	7.49418e6			0.980225			0.490112	+	0.871659i	$0.336956\pi$
$198$		0			0
$199$	1.71954e6			0.218199			0.109100	−	0.994031i	$-0.465203\pi$
$199$	1.71954e6			0.218199			0.109100	+	0.994031i	$0.465203\pi$
$200$		0			0
$201$	1.38743e6			0.170853
$202$		0			0
$203$		−	1.20969e7i		−	1.44606i
$204$		0			0
$205$		−	886244.i		−	0.102871i
$206$		0			0
$207$	−4.90916e6			−0.553473
$208$		0			0
$209$	−4.03410e6	+	9.84416e6i	−0.441884	+	1.07830i
$210$		0			0
$211$			1.62698e7i			1.73195i	0.500086	+	0.865976i	$0.333302\pi$
$211$			1.62698e7i			1.73195i	−0.500086	+	0.865976i	$0.666698\pi$
$212$		0			0
$213$	4.12245e6			0.426595
$214$		0			0
$215$	−5.59384e6			−0.562852
$216$		0			0
$217$			5.72049e6i			0.559827i
$218$		0			0
$219$			2.80395e6i			0.266955i
$220$		0			0
$221$		−	9.82884e6i		−	0.910595i
$222$		0			0
$223$			9.60067e6i			0.865739i	0.901457	+	0.432870i	$0.142499\pi$
$223$			9.60067e6i			0.865739i	−0.901457	+	0.432870i	$0.857501\pi$
$224$		0			0
$225$	1.96181e6			0.172231
$226$		0			0
$227$			3.85025e6i			0.329163i	0.986363	+	0.164582i	$0.0526274\pi$
$227$			3.85025e6i			0.329163i	−0.986363	+	0.164582i	$0.947373\pi$
$228$		0			0
$229$	2.04424e7			1.70226			0.851128	−	0.524959i	$-0.175919\pi$
$229$	2.04424e7			1.70226			0.851128	+	0.524959i	$0.175919\pi$
$230$		0			0
$231$		−	1.54703e7i		−	1.25505i
$232$		0			0
$233$	1.10376e7			0.872580			0.436290	−	0.899806i	$-0.356292\pi$
$233$	1.10376e7			0.872580			0.436290	+	0.899806i	$0.356292\pi$
$234$		0			0
$235$	2.76938e7			2.13392
$236$		0			0
$237$	1.07753e7			0.809442
$238$		0			0
$239$	2.09731e6			0.153628			0.0768139	−	0.997045i	$-0.475525\pi$
$239$	2.09731e6			0.153628			0.0768139	+	0.997045i	$0.475525\pi$
$240$		0			0
$241$			5.18077e6i			0.370120i	0.982727	+	0.185060i	$0.0592480\pi$
$241$			5.18077e6i			0.370120i	−0.982727	+	0.185060i	$0.940752\pi$
$242$		0			0
$243$		−	1.12015e7i		−	0.780650i
$244$		0			0
$245$	−1.77019e7			−1.20371
$246$		0			0
$247$	−1.47351e7	−	6.03838e6i	−0.977825	−	0.400709i
$248$		0			0
$249$		−	510620.i		−	0.0330750i
$250$		0			0
$251$	−4.74820e6			−0.300267			−0.150134	−	0.988666i	$-0.547970\pi$
$251$	−4.74820e6			−0.300267			−0.150134	+	0.988666i	$0.547970\pi$
$252$		0			0
$253$	−2.46100e7			−1.51967
$254$		0			0
$255$		−	1.28526e7i		−	0.775122i
$256$		0			0
$257$		−	2.64932e7i		−	1.56076i	−0.625308	−	0.780378i	$-0.715027\pi$
$257$		−	2.64932e7i		−	1.56076i	0.625308	−	0.780378i	$-0.284973\pi$
$258$		0			0
$259$			2.97381e7i			1.71165i
$260$		0			0
$261$		−	7.68673e6i		−	0.432335i
$262$		0			0
$263$	−1.29631e7			−0.712595			−0.356297	−	0.934373i	$-0.615961\pi$
$263$	−1.29631e7			−0.712595			−0.356297	+	0.934373i	$0.615961\pi$
$264$		0			0
$265$		−	2.82040e7i		−	1.51556i
$266$		0			0
$267$	−2.49529e6			−0.131095
$268$		0			0
$269$			4.88068e6i			0.250740i	0.992110	+	0.125370i	$0.0400118\pi$
$269$			4.88068e6i			0.250740i	−0.992110	+	0.125370i	$0.959988\pi$
$270$		0			0
$271$	−1.97961e7			−0.994654			−0.497327	−	0.867563i	$-0.665685\pi$
$271$	−1.97961e7			−0.994654			−0.497327	+	0.867563i	$0.665685\pi$
$272$		0			0
$273$	2.31564e7			1.13811
$274$		0			0
$275$	9.83470e6			0.472893
$276$		0			0
$277$	2.91795e7			1.37290			0.686450	−	0.727177i	$-0.259168\pi$
$277$	2.91795e7			1.37290			0.686450	+	0.727177i	$0.259168\pi$
$278$		0			0
$279$			3.63497e6i			0.167374i
$280$		0			0
$281$			1.23817e6i			0.0558036i	0.999611	+	0.0279018i	$0.00888256\pi$
$281$			1.23817e6i			0.0558036i	−0.999611	+	0.0279018i	$0.991117\pi$
$282$		0			0
$283$	−1.20000e7			−0.529444			−0.264722	−	0.964325i	$-0.585280\pi$
$283$	−1.20000e7			−0.529444			−0.264722	+	0.964325i	$0.585280\pi$
$284$		0			0
$285$	−1.92682e7	−	7.89603e6i	−0.832350	−	0.341094i
$286$		0			0
$287$			2.91163e6i			0.123166i
$288$		0			0
$289$	−6.21485e6			−0.257476
$290$		0			0
$291$	−2.83855e7			−1.15191
$292$		0			0
$293$		−	5.44360e6i		−	0.216413i	−0.994128	−	0.108206i	$-0.965489\pi$
$293$		−	5.44360e6i		−	0.216413i	0.994128	−	0.108206i	$-0.0345108\pi$
$294$		0			0
$295$		−	5.31068e7i		−	2.06864i
$296$		0			0
$297$		−	3.29920e7i		−	1.25933i
$298$		0			0
$299$		−	3.68370e7i		−	1.37807i
$300$		0			0
$301$	1.83778e7			0.673897
$302$		0			0
$303$			2.42664e7i			0.872323i
$304$		0			0
$305$	−4.41482e7			−1.55601
$306$		0			0
$307$			5.51649e6i			0.190655i	0.995446	+	0.0953275i	$0.0303898\pi$
$307$			5.51649e6i			0.190655i	−0.995446	+	0.0953275i	$0.969610\pi$
$308$		0			0
$309$	−3.76706e7			−1.27681
$310$		0			0
$311$	4.16573e7			1.38487			0.692436	−	0.721479i	$-0.256538\pi$
$311$	4.16573e7			1.38487			0.692436	+	0.721479i	$0.256538\pi$
$312$		0			0
$313$	−2.17487e7			−0.709251			−0.354625	−	0.935008i	$-0.615392\pi$
$313$	−2.17487e7			−0.709251			−0.354625	+	0.935008i	$0.615392\pi$
$314$		0			0
$315$	−2.23280e7			−0.714362
$316$		0			0
$317$			2.68938e7i			0.844255i	0.906536	+	0.422128i	$0.138717\pi$
$317$			2.68938e7i			0.844255i	−0.906536	+	0.422128i	$0.861283\pi$
$318$		0			0
$319$		−	3.85341e7i		−	1.18706i
$320$		0			0
$321$	1.40757e7			0.425555
$322$		0			0
$323$	1.10109e7	−	2.68691e7i	0.326749	−	0.797345i
$324$		0			0
$325$			1.47209e7i			0.428830i
$326$		0			0
$327$	2.12925e7			0.608953
$328$		0			0
$329$	−9.09839e7			−2.55492
$330$		0			0
$331$			6.67899e6i			0.184173i	0.995751	+	0.0920866i	$0.0293537\pi$
$331$			6.67899e6i			0.184173i	−0.995751	+	0.0920866i	$0.970646\pi$
$332$		0			0
$333$			1.88965e7i			0.511739i
$334$		0			0
$335$		−	1.00384e7i		−	0.267012i
$336$		0			0
$337$			1.22339e7i			0.319650i	0.987145	+	0.159825i	$0.0510930\pi$
$337$			1.22339e7i			0.319650i	−0.987145	+	0.159825i	$0.948907\pi$
$338$		0			0
$339$	2.76970e7			0.710942
$340$		0			0
$341$			1.82224e7i			0.459559i
$342$		0			0
$343$	871691.			0.0216013
$344$		0			0
$345$		−	4.81696e7i		−	1.17305i
$346$		0			0
$347$	−4.95049e7			−1.18484			−0.592421	−	0.805629i	$-0.701828\pi$
$347$	−4.95049e7			−1.18484			−0.592421	+	0.805629i	$0.701828\pi$
$348$		0			0
$349$	−3.20492e7			−0.753948			−0.376974	−	0.926224i	$-0.623035\pi$
$349$	−3.20492e7			−0.753948			−0.376974	+	0.926224i	$0.623035\pi$
$350$		0			0
$351$	4.93836e7			1.14199
$352$		0			0
$353$	5.71516e7			1.29929			0.649643	−	0.760240i	$-0.274918\pi$
$353$	5.71516e7			1.29929			0.649643	+	0.760240i	$0.274918\pi$
$354$		0			0
$355$		−	2.98271e7i		−	0.666692i
$356$		0			0
$357$			4.22254e7i			0.928045i
$358$		0			0
$359$	−3.66916e7			−0.793018			−0.396509	−	0.918031i	$-0.629778\pi$
$359$	−3.66916e7			−0.793018			−0.396509	+	0.918031i	$0.629778\pi$
$360$		0			0
$361$	−3.35167e7	−	3.30143e7i	−0.712427	−	0.701746i
$362$		0			0
$363$		−	1.29911e7i		−	0.271598i
$364$		0			0
$365$	2.02874e7			0.417203
$366$		0			0
$367$	4.06215e7			0.821784			0.410892	−	0.911684i	$-0.365217\pi$
$367$	4.06215e7			0.821784			0.410892	+	0.911684i	$0.365217\pi$
$368$		0			0
$369$			1.85014e6i			0.0368235i
$370$		0			0
$371$			9.26603e7i			1.81456i
$372$		0			0
$373$			5.30138e7i			1.02156i	0.859713	+	0.510778i	$0.170643\pi$
$373$			5.30138e7i			1.02156i	−0.859713	+	0.510778i	$0.829357\pi$
$374$		0			0
$375$		−	2.81864e7i		−	0.534498i
$376$		0			0
$377$	5.76792e7			1.07645
$378$		0			0
$379$		−	1.70291e7i		−	0.312806i	−0.987693	−	0.156403i	$-0.950010\pi$
$379$		−	1.70291e7i		−	0.312806i	0.987693	−	0.156403i	$-0.0499898\pi$
$380$		0			0
$381$	5.58380e7			1.00961
$382$		0			0
$383$			1.06812e8i			1.90118i	0.310457	+	0.950588i	$0.399518\pi$
$383$			1.06812e8i			1.90118i	−0.310457	+	0.950588i	$0.600482\pi$
$384$		0			0
$385$	−1.11932e8			−1.96142
$386$		0			0
$387$	1.16778e7			0.201478
$388$		0			0
$389$	1.01951e8			1.73197			0.865987	−	0.500067i	$-0.166691\pi$
$389$	1.01951e8			1.73197			0.865987	+	0.500067i	$0.166691\pi$
$390$		0			0
$391$	6.71717e7			1.12371
$392$		0			0
$393$		−	3.92696e7i		−	0.646962i
$394$		0			0
$395$		−	7.79626e7i		−	1.26501i
$396$		0			0
$397$	−1.20566e8			−1.92687			−0.963434	−	0.267947i	$-0.913655\pi$
$397$	−1.20566e8			−1.92687			−0.963434	+	0.267947i	$0.913655\pi$
$398$		0			0
$399$	6.33029e7	+	2.59413e7i	0.996563	+	0.408388i
$400$		0			0
$401$		−	3.05690e7i		−	0.474075i	−0.971500	−	0.237038i	$-0.923823\pi$
$401$		−	3.05690e7i		−	0.474075i	0.971500	−	0.237038i	$-0.0761765\pi$
$402$		0			0
$403$	−2.72758e7			−0.416738
$404$		0			0
$405$	3.11471e7			0.468869
$406$		0			0
$407$			9.47295e7i			1.40508i
$408$		0			0
$409$		−	6.61146e7i		−	0.966335i	−0.875528	−	0.483168i	$-0.839486\pi$
$409$		−	6.61146e7i		−	0.966335i	0.875528	−	0.483168i	$-0.160514\pi$
$410$		0			0
$411$		−	1.34752e7i		−	0.194092i
$412$		0			0
$413$			1.74475e8i			2.47676i
$414$		0			0
$415$	−3.69448e6			−0.0516903
$416$		0			0
$417$		−	2.77367e7i		−	0.382514i
$418$		0			0
$419$	−7.87196e7			−1.07014			−0.535070	−	0.844808i	$-0.679715\pi$
$419$	−7.87196e7			−1.07014			−0.535070	+	0.844808i	$0.679715\pi$
$420$		0			0
$421$			8.52736e7i			1.14279i	0.820674	+	0.571397i	$0.193598\pi$
$421$			8.52736e7i			1.14279i	−0.820674	+	0.571397i	$0.806402\pi$
$422$		0			0
$423$	−5.78140e7			−0.763856
$424$		0			0
$425$	−2.68433e7			−0.349679
$426$		0			0
$427$	1.45043e8			1.86300
$428$		0			0
$429$	7.37638e7			0.934268
$430$		0			0
$431$			1.10604e8i			1.38146i	0.723113	+	0.690730i	$0.242711\pi$
$431$			1.10604e8i			1.38146i	−0.723113	+	0.690730i	$0.757289\pi$
$432$		0			0
$433$			1.02466e7i			0.126216i	0.998007	+	0.0631082i	$0.0201013\pi$
$433$			1.02466e7i			0.126216i	−0.998007	+	0.0631082i	$0.979899\pi$
$434$		0			0
$435$	7.54237e7			0.916305
$436$		0			0
$437$	4.12671e7	−	1.00702e8i	0.494493	−	1.20668i
$438$		0			0
$439$		−	8.46843e7i		−	1.00094i	−0.865753	−	0.500471i	$-0.833160\pi$
$439$		−	8.46843e7i		−	1.00094i	0.865753	−	0.500471i	$-0.166840\pi$
$440$		0			0
$441$	3.69547e7			0.430878
$442$		0			0
$443$	−1.17625e8			−1.35297			−0.676486	−	0.736455i	$-0.736498\pi$
$443$	−1.17625e8			−1.35297			−0.676486	+	0.736455i	$0.736498\pi$
$444$		0			0
$445$			1.80541e7i			0.204879i
$446$		0			0
$447$		−	3.53050e7i		−	0.395288i
$448$		0			0
$449$		−	6.96739e7i		−	0.769717i	−0.922976	−	0.384859i	$-0.874250\pi$
$449$		−	6.96739e7i		−	0.769717i	0.922976	−	0.384859i	$-0.125750\pi$
$450$		0			0
$451$			9.27486e6i			0.101106i
$452$		0			0
$453$	6.11551e7			0.657867
$454$		0			0
$455$		−	1.67543e8i		−	1.77866i
$456$		0			0
$457$	6.74264e7			0.706450			0.353225	−	0.935538i	$-0.385085\pi$
$457$	6.74264e7			0.706450			0.353225	+	0.935538i	$0.385085\pi$
$458$		0			0
$459$			9.00501e7i			0.931207i
$460$		0			0
$461$	1.66919e8			1.70374			0.851869	−	0.523755i	$-0.175469\pi$
$461$	1.66919e8			1.70374			0.851869	+	0.523755i	$0.175469\pi$
$462$		0			0
$463$	3.31076e7			0.333568			0.166784	−	0.985993i	$-0.446662\pi$
$463$	3.31076e7			0.333568			0.166784	+	0.985993i	$0.446662\pi$
$464$		0			0
$465$	−3.56670e7			−0.354738
$466$		0			0
$467$	1.18305e6			0.0116159			0.00580793	−	0.999983i	$-0.498151\pi$
$467$	1.18305e6			0.0116159			0.00580793	+	0.999983i	$0.498151\pi$
$468$		0			0
$469$			3.29798e7i			0.319691i
$470$		0			0
$471$			1.52185e8i			1.45650i
$472$		0			0
$473$	5.85416e7			0.553198
$474$		0			0
$475$	−1.64913e7	+	4.02426e7i	−0.153877	+	0.375496i
$476$		0			0
$477$			5.88792e7i			0.542509i
$478$		0			0
$479$	−1.14067e8			−1.03790			−0.518948	−	0.854806i	$-0.673676\pi$
$479$	−1.14067e8			−1.03790			−0.518948	+	0.854806i	$0.673676\pi$
$480$		0			0
$481$	−1.41794e8			−1.27416
$482$		0			0
$483$			1.58254e8i			1.40448i
$484$		0			0
$485$			2.05377e8i			1.80023i
$486$		0			0
$487$		−	2.07228e7i		−	0.179416i	−0.995968	−	0.0897080i	$-0.971407\pi$
$487$		−	2.07228e7i		−	0.179416i	0.995968	−	0.0897080i	$-0.0285934\pi$
$488$		0			0
$489$		−	8.65002e7i		−	0.739759i
$490$		0			0
$491$	−1.02709e8			−0.867685			−0.433843	−	0.900989i	$-0.642843\pi$
$491$	−1.02709e8			−0.867685			−0.433843	+	0.900989i	$0.642843\pi$
$492$		0			0
$493$			1.05177e8i			0.877769i
$494$		0			0
$495$	−7.11249e7			−0.586417
$496$		0			0
$497$			9.79926e7i			0.798223i
$498$		0			0
$499$	−2.53668e7			−0.204157			−0.102078	−	0.994776i	$-0.532549\pi$
$499$	−2.53668e7			−0.204157			−0.102078	+	0.994776i	$0.532549\pi$
$500$		0			0
$501$	1.02588e8			0.815798
$502$		0			0
$503$	5.81598e7			0.457003			0.228502	−	0.973544i	$-0.426617\pi$
$503$	5.81598e7			0.457003			0.228502	+	0.973544i	$0.426617\pi$
$504$		0			0
$505$	1.75574e8			1.36328
$506$		0			0
$507$			1.15394e7i			0.0885438i
$508$		0			0
$509$		−	1.40926e7i		−	0.106866i	−0.998571	−	0.0534330i	$-0.982984\pi$
$509$		−	1.40926e7i		−	0.106866i	0.998571	−	0.0534330i	$-0.0170163\pi$
$510$		0			0
$511$	−6.66513e7			−0.499512
$512$		0			0
$513$	1.35000e8	+	5.53226e7i	0.999958	+	0.409779i
$514$		0			0
$515$			2.72558e8i			1.99543i
$516$		0			0
$517$	−2.89825e8			−2.09732
$518$		0			0
$519$	−1.97062e8			−1.40961
$520$		0			0
$521$		−	1.98404e8i		−	1.40293i	−0.712702	−	0.701467i	$-0.752529\pi$
$521$		−	1.98404e8i		−	1.40293i	0.712702	−	0.701467i	$-0.247471\pi$
$522$		0			0
$523$		−	1.38487e8i		−	0.968065i	−0.875050	−	0.484032i	$-0.839172\pi$
$523$		−	1.38487e8i		−	0.968065i	0.875050	−	0.484032i	$-0.160828\pi$
$524$		0			0
$525$		−	6.32421e7i		−	0.437047i
$526$		0			0
$527$		−	4.97370e7i		−	0.339819i
$528$		0			0
$529$	1.03714e8			0.700598
$530$		0			0
$531$			1.10867e8i			0.740488i
$532$		0			0
$533$	−1.38829e7			−0.0916852
$534$		0			0
$535$		−	1.01842e8i		−	0.665066i
$536$		0			0
$537$	−1.73898e8			−1.12298
$538$		0			0
$539$	1.85257e8			1.18306
$540$		0			0
$541$	−3.19552e7			−0.201814			−0.100907	−	0.994896i	$-0.532174\pi$
$541$	−3.19552e7			−0.201814			−0.100907	+	0.994896i	$0.532174\pi$
$542$		0			0
$543$	2.08041e8			1.29942
$544$		0			0
$545$		−	1.54057e8i		−	0.951685i
$546$		0			0
$547$			1.14740e8i			0.701054i	0.936553	+	0.350527i	$0.113998\pi$
$547$			1.14740e8i			0.701054i	−0.936553	+	0.350527i	$0.886002\pi$
$548$		0			0
$549$	9.21645e7			0.556989
$550$		0			0
$551$	1.57678e8	+	6.46158e7i	0.942575	+	0.386264i
$552$		0			0
$553$			2.56135e8i			1.51458i
$554$		0			0
$555$	−1.85416e8			−1.08460
$556$		0			0
$557$	−6.01870e7			−0.348287			−0.174144	−	0.984720i	$-0.555716\pi$
$557$	−6.01870e7			−0.348287			−0.174144	+	0.984720i	$0.555716\pi$
$558$		0			0
$559$			8.76270e7i			0.501652i
$560$		0			0
$561$			1.34507e8i			0.761828i
$562$		0			0
$563$		−	2.87414e8i		−	1.61058i	−0.592880	−	0.805291i	$-0.702009\pi$
$563$		−	2.87414e8i		−	1.61058i	0.592880	−	0.805291i	$-0.297991\pi$
$564$		0			0
$565$		−	2.00396e8i		−	1.11108i
$566$		0			0
$567$	−1.02329e8			−0.561372
$568$		0			0
$569$		−	8.07865e7i		−	0.438533i	−0.975665	−	0.219266i	$-0.929634\pi$
$569$		−	8.07865e7i		−	0.438533i	0.975665	−	0.219266i	$-0.0703664\pi$
$570$		0			0
$571$	1.81823e8			0.976652			0.488326	−	0.872661i	$-0.337608\pi$
$571$	1.81823e8			0.976652			0.488326	+	0.872661i	$0.337608\pi$
$572$		0			0
$573$			1.51633e8i			0.805990i
$574$		0			0
$575$	−1.00605e8			−0.529195
$576$		0			0
$577$	−8.48366e7			−0.441627			−0.220814	−	0.975316i	$-0.570871\pi$
$577$	−8.48366e7			−0.441627			−0.220814	+	0.975316i	$0.570871\pi$
$578$		0			0
$579$	−2.21646e7			−0.114189
$580$		0			0
$581$	1.21377e7			0.0618882
$582$		0			0
$583$			2.95165e8i			1.48957i
$584$		0			0
$585$		−	1.06462e8i		−	0.531775i
$586$		0			0
$587$	−8.19063e7			−0.404951			−0.202476	−	0.979287i	$-0.564899\pi$
$587$	−8.19063e7			−0.404951			−0.202476	+	0.979287i	$0.564899\pi$
$588$		0			0
$589$	−7.45641e7	−	3.05561e7i	−0.364908	−	0.149538i
$590$		0			0
$591$		−	1.53512e8i		−	0.743667i
$592$		0			0
$593$	−2.50741e8			−1.20243			−0.601217	−	0.799086i	$-0.705317\pi$
$593$	−2.50741e8			−1.20243			−0.601217	+	0.799086i	$0.705317\pi$
$594$		0			0
$595$	3.05513e8			1.45037
$596$		0			0
$597$		−	3.52233e7i		−	0.165541i
$598$		0			0
$599$			3.66718e6i			0.0170629i	0.999964	+	0.00853144i	$0.00271567\pi$
$599$			3.66718e6i			0.0170629i	−0.999964	+	0.00853144i	$0.997284\pi$
$600$		0			0
$601$			1.46673e8i			0.675657i	0.941208	+	0.337828i	$0.109692\pi$
$601$			1.46673e8i			0.675657i	−0.941208	+	0.337828i	$0.890308\pi$
$602$		0			0
$603$			2.09564e7i			0.0955794i
$604$		0			0
$605$	−9.39943e7			−0.424459
$606$		0			0
$607$			1.23484e7i			0.0552134i	0.999619	+	0.0276067i	$0.00878860\pi$
$607$			1.23484e7i			0.0552134i	−0.999619	+	0.0276067i	$0.991211\pi$
$608$		0			0
$609$	−2.47794e8			−1.09708
$610$		0			0
$611$		−	4.33820e8i		−	1.90189i
$612$		0			0
$613$	3.22952e8			1.40203			0.701014	−	0.713147i	$-0.252731\pi$
$613$	3.22952e8			1.40203			0.701014	+	0.713147i	$0.252731\pi$
$614$		0			0
$615$	−1.81539e7			−0.0780449
$616$		0			0
$617$	−9.53671e7			−0.406016			−0.203008	−	0.979177i	$-0.565072\pi$
$617$	−9.53671e7			−0.406016			−0.203008	+	0.979177i	$0.565072\pi$
$618$		0			0
$619$	−6.49271e7			−0.273750			−0.136875	−	0.990588i	$-0.543706\pi$
$619$	−6.49271e7			−0.273750			−0.136875	+	0.990588i	$0.543706\pi$
$620$		0			0
$621$			3.37494e8i			1.40926i
$622$		0			0
$623$		−	5.93143e7i		−	0.245299i
$624$		0			0
$625$	−3.03009e8			−1.24113
$626$		0			0
$627$	2.01649e8	+	8.26349e7i	0.818074	+	0.335244i
$628$		0			0
$629$		−	2.58559e8i		−	1.03898i
$630$		0			0
$631$	2.32451e8			0.925217			0.462608	−	0.886563i	$-0.346914\pi$
$631$	2.32451e8			0.925217			0.462608	+	0.886563i	$0.346914\pi$
$632$		0			0
$633$	3.33273e8			1.31398
$634$		0			0
$635$		−	4.04004e8i		−	1.57784i
$636$		0			0
$637$			2.77298e8i			1.07282i
$638$		0			0
$639$			6.22675e7i			0.238648i
$640$		0			0
$641$		−	4.11723e8i		−	1.56326i	−0.623744	−	0.781629i	$-0.714389\pi$
$641$		−	4.11723e8i		−	1.56326i	0.623744	−	0.781629i	$-0.285611\pi$
$642$		0			0
$643$	4.62395e8			1.73932			0.869661	−	0.493649i	$-0.164337\pi$
$643$	4.62395e8			1.73932			0.869661	+	0.493649i	$0.164337\pi$
$644$		0			0
$645$			1.14585e8i			0.427019i
$646$		0			0
$647$	−3.00194e8			−1.10838			−0.554191	−	0.832390i	$-0.686972\pi$
$647$	−3.00194e8			−1.10838			−0.554191	+	0.832390i	$0.686972\pi$
$648$		0			0
$649$			5.55783e8i			2.03316i
$650$		0			0
$651$	1.17179e8			0.424724
$652$		0			0
$653$	716158.			0.00257199			0.00128599	−	0.999999i	$-0.499591\pi$
$653$	716158.			0.00257199			0.00128599	+	0.999999i	$0.499591\pi$
$654$		0			0
$655$	−2.84127e8			−1.01109
$656$		0			0
$657$	−4.23523e7			−0.149342
$658$		0			0
$659$			6.34068e6i			0.0221554i	0.999939	+	0.0110777i	$0.00352622\pi$
$659$			6.34068e6i			0.0221554i	−0.999939	+	0.0110777i	$0.996474\pi$
$660$		0			0
$661$		−	3.42226e8i		−	1.18497i	−0.805581	−	0.592486i	$-0.798147\pi$
$661$		−	3.42226e8i		−	1.18497i	0.805581	−	0.592486i	$-0.201853\pi$
$662$		0			0
$663$	−2.01335e8			−0.690841
$664$		0			0
$665$	1.87693e8	−	4.58014e8i	0.638237	−	1.55745i
$666$		0			0
$667$			3.94188e8i			1.32839i
$668$		0			0
$669$	1.96661e8			0.656810
$670$		0			0
$671$	4.62027e8			1.52933
$672$		0			0
$673$		−	4.48332e8i		−	1.47080i	−0.677632	−	0.735401i	$-0.736994\pi$
$673$		−	4.48332e8i		−	1.47080i	0.677632	−	0.735401i	$-0.263006\pi$
$674$		0			0
$675$		−	1.34870e8i		−	0.438536i
$676$		0			0
$677$			2.73027e8i			0.879912i	0.898019	+	0.439956i	$0.145006\pi$
$677$			2.73027e8i			0.879912i	−0.898019	+	0.439956i	$0.854994\pi$
$678$		0			0
$679$		−	6.74738e8i		−	2.15539i
$680$		0			0
$681$	7.88689e7			0.249726
$682$		0			0
$683$		−	1.58980e8i		−	0.498976i	−0.968378	−	0.249488i	$-0.919738\pi$
$683$		−	1.58980e8i		−	0.498976i	0.968378	−	0.249488i	$-0.0802623\pi$
$684$		0			0
$685$	−9.74966e7			−0.303332
$686$		0			0
$687$		−	4.18743e8i		−	1.29145i
$688$		0			0
$689$	−4.41813e8			−1.35077
$690$		0			0
$691$	−6.99279e7			−0.211942			−0.105971	−	0.994369i	$-0.533795\pi$
$691$	−6.99279e7			−0.211942			−0.105971	+	0.994369i	$0.533795\pi$
$692$		0			0
$693$	2.33671e8			0.702110
$694$		0			0
$695$	−2.00683e8			−0.597800
$696$		0			0
$697$		−	2.53153e7i		−	0.0747626i
$698$		0			0
$699$		−	2.26094e8i		−	0.662000i
$700$		0			0
$701$	−4.04905e8			−1.17544			−0.587718	−	0.809066i	$-0.699974\pi$
$701$	−4.04905e8			−1.17544			−0.587718	+	0.809066i	$0.699974\pi$
$702$		0			0
$703$	−3.87624e8	−	1.58847e8i	−1.11569	−	0.457207i
$704$		0			0
$705$		−	5.67281e8i		−	1.61894i
$706$		0			0
$707$	−5.76824e8			−1.63224
$708$		0			0
$709$	−2.24142e8			−0.628903			−0.314452	−	0.949274i	$-0.601821\pi$
$709$	−2.24142e8			−0.628903			−0.314452	+	0.949274i	$0.601821\pi$
$710$		0			0
$711$			1.62756e8i			0.452823i
$712$		0			0
$713$		−	1.86407e8i		−	0.514273i
$714$		0			0
$715$		−	5.33702e8i		−	1.46009i
$716$		0			0
$717$		−	4.29616e7i		−	0.116553i
$718$		0			0
$719$	5.24823e8			1.41197			0.705987	−	0.708225i	$-0.250504\pi$
$719$	5.24823e8			1.41197			0.705987	+	0.708225i	$0.250504\pi$
$720$		0			0
$721$		−	8.95450e8i		−	2.38911i
$722$		0			0
$723$	1.06123e8			0.280799
$724$		0			0
$725$		−	1.57526e8i		−	0.413370i
$726$		0			0
$727$	5.49414e8			1.42987			0.714935	−	0.699191i	$-0.246456\pi$
$727$	5.49414e8			1.42987			0.714935	+	0.699191i	$0.246456\pi$
$728$		0			0
$729$	−3.82657e8			−0.987704
$730$		0			0
$731$	−1.59786e8			−0.409061
$732$		0			0
$733$	−3.20871e8			−0.814738			−0.407369	−	0.913264i	$-0.633554\pi$
$733$	−3.20871e8			−0.814738			−0.407369	+	0.913264i	$0.633554\pi$
$734$		0			0
$735$			3.62607e8i			0.913217i
$736$		0			0
$737$			1.05056e8i			0.262432i
$738$		0			0
$739$	3.46153e7			0.0857699			0.0428850	−	0.999080i	$-0.486345\pi$
$739$	3.46153e7			0.0857699			0.0428850	+	0.999080i	$0.486345\pi$
$740$		0			0
$741$	−1.23691e8	+	3.01834e8i	−0.304006	+	0.741846i
$742$		0			0
$743$			4.70807e8i			1.14783i	0.818916	+	0.573913i	$0.194575\pi$
$743$			4.70807e8i			1.14783i	−0.818916	+	0.573913i	$0.805425\pi$
$744$		0			0
$745$	−2.55441e8			−0.617764
$746$		0			0
$747$	7.71266e6			0.0185030
$748$		0			0
$749$			3.34587e8i			0.796276i
$750$		0			0
$751$		−	3.60786e8i		−	0.851786i	−0.904774	−	0.425893i	$-0.859960\pi$
$751$		−	3.60786e8i		−	0.851786i	0.904774	−	0.425893i	$-0.140040\pi$
$752$		0			0
$753$			9.72626e7i			0.227804i
$754$		0			0
$755$		−	4.42474e8i		−	1.02813i
$756$		0			0
$757$	2.57653e8			0.593947			0.296974	−	0.954886i	$-0.404023\pi$
$757$	2.57653e8			0.593947			0.296974	+	0.954886i	$0.404023\pi$
$758$		0			0
$759$			5.04113e8i			1.15293i
$760$		0			0
$761$	6.18479e8			1.40337			0.701683	−	0.712490i	$-0.252432\pi$
$761$	6.18479e8			1.40337			0.701683	+	0.712490i	$0.252432\pi$
$762$		0			0
$763$			5.06134e8i			1.13944i
$764$		0			0
$765$	1.94132e8			0.433624
$766$		0			0
$767$	−8.31914e8			−1.84371
$768$		0			0
$769$	−2.28868e8			−0.503275			−0.251637	−	0.967822i	$-0.580969\pi$
$769$	−2.28868e8			−0.503275			−0.251637	+	0.967822i	$0.580969\pi$
$770$		0			0
$771$	−5.42689e8			−1.18410
$772$		0			0
$773$		−	2.02792e8i		−	0.439048i	−0.975607	−	0.219524i	$-0.929550\pi$
$773$		−	2.02792e8i		−	0.439048i	0.975607	−	0.219524i	$-0.0704504\pi$
$774$		0			0
$775$			7.44925e7i			0.160032i
$776$		0			0
$777$	6.09158e8			1.29858
$778$		0			0
$779$	−3.79518e7	−	1.55525e7i	−0.0802824	−	0.0328994i
$780$		0			0
$781$			3.12151e8i			0.655257i
$782$		0			0
$783$	−5.28447e8			−1.10082
$784$		0			0
$785$	1.10110e9			2.27624
$786$		0			0
$787$			7.62647e8i			1.56459i	0.622910	+	0.782293i	$0.285950\pi$
$787$			7.62647e8i			1.56459i	−0.622910	+	0.782293i	$0.714050\pi$
$788$		0			0
$789$			2.65538e8i			0.540624i
$790$		0			0
$791$			6.58373e8i			1.33028i
$792$		0			0
$793$			6.91577e8i			1.38682i
$794$		0			0
$795$	−5.77734e8			−1.14981
$796$		0			0
$797$		−	6.92506e8i		−	1.36788i	−0.729537	−	0.683941i	$-0.760264\pi$
$797$		−	6.92506e8i		−	1.36788i	0.729537	−	0.683941i	$-0.239736\pi$
$798$		0			0
$799$	7.91064e8			1.55086
$800$		0			0
$801$		−	3.76901e7i		−	0.0733382i
$802$		0			0
$803$	−2.12315e8			−0.410047
$804$		0			0
$805$	1.14502e9			2.19495
$806$		0			0
$807$	9.99763e7			0.190229
$808$		0			0
$809$	−7.47780e8			−1.41230			−0.706152	−	0.708060i	$-0.749570\pi$
$809$	−7.47780e8			−1.41230			−0.706152	+	0.708060i	$0.749570\pi$
$810$		0			0
$811$			3.61841e8i			0.678352i	0.940723	+	0.339176i	$0.110148\pi$
$811$			3.61841e8i			0.678352i	−0.940723	+	0.339176i	$0.889852\pi$
$812$		0			0
$813$			4.05505e8i			0.754614i
$814$		0			0
$815$	−6.25853e8			−1.15611
$816$		0			0
$817$	−9.81653e7	+	2.39546e8i	−0.180008	+	0.439262i
$818$		0			0
$819$			3.49766e8i			0.636688i
$820$		0			0
$821$	4.85911e8			0.878066			0.439033	−	0.898471i	$-0.355321\pi$
$821$	4.85911e8			0.878066			0.439033	+	0.898471i	$0.355321\pi$
$822$		0			0
$823$	−3.19986e8			−0.574027			−0.287013	−	0.957927i	$-0.592662\pi$
$823$	−3.19986e8			−0.574027			−0.287013	+	0.957927i	$0.592662\pi$
$824$		0			0
$825$		−	2.01455e8i		−	0.358770i
$826$		0			0
$827$			3.25171e8i			0.574904i	0.957795	+	0.287452i	$0.0928082\pi$
$827$			3.25171e8i			0.574904i	−0.957795	+	0.287452i	$0.907192\pi$
$828$		0			0
$829$		−	5.92117e8i		−	1.03931i	−0.854377	−	0.519654i	$-0.826061\pi$
$829$		−	5.92117e8i		−	1.03931i	0.854377	−	0.519654i	$-0.173939\pi$
$830$		0			0
$831$		−	5.97716e8i		−	1.04158i
$832$		0			0
$833$	−5.05649e8			−0.874810
$834$		0			0
$835$		−	7.42252e8i		−	1.27495i
$836$		0			0
$837$	2.49897e8			0.426171
$838$		0			0
$839$			1.25885e8i			0.213151i	0.994305	+	0.106575i	$0.0339885\pi$
$839$			1.25885e8i			0.213151i	−0.994305	+	0.106575i	$0.966011\pi$
$840$		0			0
$841$	−2.23933e7			−0.0376470
$842$		0			0
$843$	2.53628e7			0.0423365
$844$		0			0
$845$	8.34906e7			0.138378
$846$		0			0
$847$	3.08805e8			0.508199
$848$		0			0
$849$			2.45808e8i			0.401674i
$850$		0			0
$851$		−	9.69042e8i		−	1.57237i
$852$		0			0
$853$	−1.05928e9			−1.70672			−0.853361	−	0.521320i	$-0.825440\pi$
$853$	−1.05928e9			−1.70672			−0.853361	+	0.521320i	$0.825440\pi$
$854$		0			0
$855$	1.19266e8	−	2.91036e8i	0.190817	−	0.465638i
$856$		0			0
$857$		−	8.95883e8i		−	1.42334i	−0.702514	−	0.711670i	$-0.747939\pi$
$857$		−	8.95883e8i		−	1.42334i	0.702514	−	0.711670i	$-0.252061\pi$
$858$		0			0
$859$	8.18846e8			1.29188			0.645941	−	0.763388i	$-0.276465\pi$
$859$	8.18846e8			1.29188			0.645941	+	0.763388i	$0.276465\pi$
$860$		0			0
$861$	5.96421e7			0.0934422
$862$		0			0
$863$		−	4.25321e8i		−	0.661735i	−0.943677	−	0.330867i	$-0.892659\pi$
$863$		−	4.25321e8i		−	0.661735i	0.943677	−	0.330867i	$-0.107341\pi$
$864$		0			0
$865$			1.42580e9i			2.20297i
$866$		0			0
$867$			1.27306e8i			0.195339i
$868$		0			0
$869$			8.15907e8i			1.24332i
$870$		0			0
$871$	−1.57251e8			−0.237979
$872$		0			0
$873$		−	4.28749e8i		−	0.644407i
$874$		0			0
$875$	6.70005e8			1.00012
$876$		0			0
$877$			3.21386e8i			0.476463i	0.971208	+	0.238231i	$0.0765676\pi$
$877$			3.21386e8i			0.476463i	−0.971208	+	0.238231i	$0.923432\pi$
$878$		0			0
$879$	−1.11507e8			−0.164186
$880$		0			0
$881$	−2.83890e7			−0.0415166			−0.0207583	−	0.999785i	$-0.506608\pi$
$881$	−2.83890e7			−0.0415166			−0.0207583	+	0.999785i	$0.506608\pi$
$882$		0			0
$883$	−9.67813e8			−1.40575			−0.702877	−	0.711311i	$-0.748102\pi$
$883$	−9.67813e8			−1.40575			−0.702877	+	0.711311i	$0.748102\pi$
$884$		0			0
$885$	−1.08785e9			−1.56941
$886$		0			0
$887$		−	3.08094e8i		−	0.441482i	−0.975332	−	0.220741i	$-0.929152\pi$
$887$		−	3.08094e8i		−	0.441482i	0.975332	−	0.220741i	$-0.0708476\pi$
$888$		0			0
$889$			1.32730e9i			1.88913i
$890$		0			0
$891$	−3.25965e8			−0.460827
$892$		0			0
$893$	4.85993e8	−	1.18594e9i	0.682457	−	1.66536i
$894$		0			0
$895$			1.25820e9i			1.75502i
$896$		0			0
$897$	−7.54573e8			−1.04550
$898$		0			0
$899$	2.91875e8			0.401715
$900$		0			0
$901$		−	8.05639e8i		−	1.10145i
$902$		0			0
$903$		−	3.76452e8i		−	0.511265i
$904$		0			0
$905$		−	1.50524e9i		−	2.03076i
$906$		0			0
$907$			3.39287e7i			0.0454721i	0.999742	+	0.0227360i	$0.00723773\pi$
$907$			3.39287e7i			0.0454721i	−0.999742	+	0.0227360i	$0.992762\pi$
$908$		0			0
$909$	−3.66532e8			−0.488000
$910$		0			0
$911$		−	3.10107e8i		−	0.410163i	−0.978745	−	0.205082i	$-0.934254\pi$
$911$		−	3.10107e8i		−	0.410163i	0.978745	−	0.205082i	$-0.0657459\pi$
$912$		0			0
$913$	3.86641e7			0.0508037
$914$		0			0
$915$			9.04336e8i			1.18050i
$916$		0			0
$917$	9.33458e8			1.21056
$918$		0			0
$919$	2.65408e8			0.341954			0.170977	−	0.985275i	$-0.445308\pi$
$919$	2.65408e8			0.341954			0.170977	+	0.985275i	$0.445308\pi$
$920$		0			0
$921$	1.13000e8			0.144644
$922$		0			0
$923$	−4.67238e8			−0.594201
$924$		0			0
$925$			3.87252e8i			0.489292i
$926$		0			0
$927$		−	5.68996e8i		−	0.714283i
$928$		0			0
$929$	1.10783e9			1.38174			0.690868	−	0.722981i	$-0.257229\pi$
$929$	1.10783e9			1.38174			0.690868	+	0.722981i	$0.257229\pi$
$930$		0			0
$931$	−3.10647e8	+	7.58052e8i	−0.384962	+	0.939398i
$932$		0			0
$933$		−	8.53311e8i		−	1.05066i
$934$		0			0
$935$	9.73197e8			1.19060
$936$		0			0
$937$	1.54052e9			1.87262			0.936309	−	0.351177i	$-0.114218\pi$
$937$	1.54052e9			1.87262			0.936309	+	0.351177i	$0.114218\pi$
$938$		0			0
$939$			4.45502e8i			0.538087i
$940$		0			0
$941$			1.62276e8i			0.194753i	0.995248	+	0.0973766i	$0.0310451\pi$
$941$			1.62276e8i			0.194753i	−0.995248	+	0.0973766i	$0.968955\pi$
$942$		0			0
$943$		−	9.48779e7i		−	0.113144i
$944$		0			0
$945$			1.53500e9i			1.81892i
$946$		0			0
$947$	8.28066e8			0.975024			0.487512	−	0.873116i	$-0.337905\pi$
$947$	8.28066e8			0.975024			0.487512	+	0.873116i	$0.337905\pi$
$948$		0			0
$949$		−	3.17800e8i		−	0.371839i
$950$		0			0
$951$	5.50894e8			0.640511
$952$		0			0
$953$			9.44682e8i			1.09146i	0.837962	+	0.545729i	$0.183747\pi$
$953$			9.44682e8i			1.09146i	−0.837962	+	0.545729i	$0.816253\pi$
$954$		0			0
$955$	1.09711e9			1.25962
$956$		0			0
$957$	−7.89337e8			−0.900588
$958$		0			0
$959$	3.20311e8			0.363176
$960$		0			0
$961$	7.49479e8			0.844480
$962$		0			0
$963$			2.12607e8i			0.238066i
$964$		0			0
$965$			1.60367e8i			0.178457i
$966$		0			0
$967$	1.35521e9			1.49874			0.749372	−	0.662149i	$-0.230355\pi$
$967$	1.35521e9			1.49874			0.749372	+	0.662149i	$0.230355\pi$
$968$		0			0
$969$	−5.50390e8	−	2.25548e8i	−0.604922	−	0.247895i
$970$		0			0
$971$			9.89835e8i			1.08120i	0.841280	+	0.540599i	$0.181802\pi$
$971$			9.89835e8i			1.08120i	−0.841280	+	0.540599i	$0.818198\pi$
$972$		0			0
$973$	6.59316e8			0.715739
$974$		0			0
$975$	3.01545e8			0.325340
$976$		0			0
$977$			9.87205e8i			1.05858i	0.848441	+	0.529290i	$0.177542\pi$
$977$			9.87205e8i			1.05858i	−0.848441	+	0.529290i	$0.822458\pi$
$978$		0			0
$979$		−	1.88943e8i		−	0.201365i
$980$		0			0
$981$			3.21613e8i			0.340664i
$982$		0			0
$983$		−	9.10074e8i		−	0.958112i	−0.877784	−	0.479056i	$-0.840979\pi$
$983$		−	9.10074e8i		−	0.958112i	0.877784	−	0.479056i	$-0.159021\pi$
$984$		0			0
$985$	−1.11070e9			−1.16222
$986$		0			0
$987$			1.86372e9i			1.93834i
$988$		0			0
$989$	−5.98856e8			−0.619061
$990$		0			0
$991$			8.61627e8i			0.885316i	0.896691	+	0.442658i	$0.145964\pi$
$991$			8.61627e8i			0.885316i	−0.896691	+	0.442658i	$0.854036\pi$
$992$		0			0
$993$	1.36813e8			0.139727
$994$		0			0
$995$	−2.54850e8			−0.258712
$996$		0			0
$997$	−7.43177e8			−0.749906			−0.374953	−	0.927044i	$-0.622341\pi$
$997$	−7.43177e8			−0.749906			−0.374953	+	0.927044i	$0.622341\pi$
$998$		0			0
$999$	1.29909e9			1.30300

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.7.e.f.113.10		30
4.3	odd	2		152.7.e.a.113.21	yes	30
19.18	odd	2	inner	304.7.e.f.113.21		30
76.75	even	2		152.7.e.a.113.10	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.7.e.a.113.10	✓	30	76.75	even	2
152.7.e.a.113.21	yes	30	4.3	odd	2
304.7.e.f.113.10		30	1.1	even	1	trivial
304.7.e.f.113.21		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)

\(f(q)\)	\(=\)	\(q-20.4841i q^{3} -148.208 q^{5} +486.917 q^{7} +309.402 q^{9} +O(q^{10})\)	\(q-20.4841i q^{3} -148.208 q^{5} +486.917 q^{7} +309.402 q^{9} +1551.05 q^{11} +2321.67i q^{13} +3035.91i q^{15} -4233.52 q^{17} +(-2600.88 + 6346.76i) q^{19} -9974.05i q^{21} -15866.6 q^{23} +6340.66 q^{25} -21270.7i q^{27} -24843.8i q^{29} +11748.4i q^{31} -31771.9i q^{33} -72165.1 q^{35} +61074.3i q^{37} +47557.3 q^{39} +5979.72i q^{41} +37743.1 q^{43} -45855.9 q^{45} -186857. q^{47} +119439. q^{49} +86719.9i q^{51} +190300. i q^{53} -229879. q^{55} +(130008. + 53276.6i) q^{57} +358326. i q^{59} +297880. q^{61} +150653. q^{63} -344090. i q^{65} +67731.9i q^{67} +325013. i q^{69} +201251. i q^{71} -136884. q^{73} -129883. i q^{75} +755234. q^{77} +526034. i q^{79} -210157. q^{81} +24927.7 q^{83} +627443. q^{85} -508904. q^{87} -121816. i q^{89} +1.13046e6i q^{91} +240655. q^{93} +(385471. - 940641. i) q^{95} -1.38573e6i q^{97} +479899. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 720 q^{7} - 8670 q^{9}+O(q^{10}) \) 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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