LMFDB - Embedded newform 304.7.e.f.113.5

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			113.5
Character	$\chi$	$=$	304.113
Dual form			304.7.e.f.113.26

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-36.8625i q^{3} +9.31317 q^{5} -333.649 q^{7} -629.840 q^{9} +O(q^{10})$	$q-36.8625i q^{3} +9.31317 q^{5} -333.649 q^{7} -629.840 q^{9} +2078.97 q^{11} -3605.38i q^{13} -343.306i q^{15} +990.575 q^{17} +(6858.43 - 88.5667i) q^{19} +12299.1i q^{21} +9880.74 q^{23} -15538.3 q^{25} -3655.26i q^{27} -29664.0i q^{29} -8083.37i q^{31} -76636.0i q^{33} -3107.33 q^{35} -5595.98i q^{37} -132903. q^{39} -50461.2i q^{41} +104427. q^{43} -5865.81 q^{45} -187522. q^{47} -6327.59 q^{49} -36515.0i q^{51} -56414.1i q^{53} +19361.8 q^{55} +(-3264.79 - 252818. i) q^{57} +375732. i q^{59} +326193. q^{61} +210145. q^{63} -33577.5i q^{65} -317128. i q^{67} -364228. i q^{69} +475031. i q^{71} -428704. q^{73} +572779. i q^{75} -693646. q^{77} -418334. i q^{79} -593896. q^{81} +135044. q^{83} +9225.40 q^{85} -1.09349e6 q^{87} +646123. i q^{89} +1.20293e6i q^{91} -297973. q^{93} +(63873.7 - 824.837i) q^{95} +934631. i q^{97} -1.30942e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	36.8625i		−	1.36528i	−0.730757	−	0.682638i	$-0.760833\pi$
$3$		−	36.8625i		−	1.36528i	0.730757	−	0.682638i	$-0.239167\pi$
$4$		0			0
$5$	9.31317			0.0745054			0.0372527	−	0.999306i	$-0.488139\pi$
$5$	9.31317			0.0745054			0.0372527	+	0.999306i	$0.488139\pi$
$6$		0			0
$7$	−333.649			−0.972737			−0.486368	−	0.873754i	$-0.661679\pi$
$7$	−333.649			−0.972737			−0.486368	+	0.873754i	$0.661679\pi$
$8$		0			0
$9$	−629.840			−0.863979
$10$		0			0
$11$	2078.97			1.56196			0.780981	−	0.624555i	$-0.214720\pi$
$11$	2078.97			1.56196			0.780981	+	0.624555i	$0.214720\pi$
$12$		0			0
$13$		−	3605.38i		−	1.64105i	−0.571613	−	0.820524i	$-0.693682\pi$
$13$		−	3605.38i		−	1.64105i	0.571613	−	0.820524i	$-0.306318\pi$
$14$		0			0
$15$		−	343.306i		−	0.101720i
$16$		0			0
$17$	990.575			0.201623			0.100812	−	0.994906i	$-0.467856\pi$
$17$	990.575			0.201623			0.100812	+	0.994906i	$0.467856\pi$
$18$		0			0
$19$	6858.43	−	88.5667i	0.999917	−	0.0129125i
$19$	6858.43	−	88.5667i	0.999917	−	0.0129125i
$20$		0			0
$21$			12299.1i			1.32805i
$22$		0			0
$23$	9880.74			0.812093			0.406047	−	0.913852i	$-0.366907\pi$
$23$	9880.74			0.812093			0.406047	+	0.913852i	$0.366907\pi$
$24$		0			0
$25$	−15538.3			−0.994449
$26$		0			0
$27$		−	3655.26i		−	0.185707i
$28$		0			0
$29$		−	29664.0i		−	1.21629i	−0.793827	−	0.608143i	$-0.791915\pi$
$29$		−	29664.0i		−	1.21629i	0.793827	−	0.608143i	$-0.208085\pi$
$30$		0			0
$31$		−	8083.37i		−	0.271336i	−0.990754	−	0.135668i	$-0.956682\pi$
$31$		−	8083.37i		−	0.271336i	0.990754	−	0.135668i	$-0.0433181\pi$
$32$		0			0
$33$		−	76636.0i		−	2.13251i
$34$		0			0
$35$	−3107.33			−0.0724741
$36$		0			0
$37$		−	5595.98i		−	0.110477i	−0.998473	−	0.0552384i	$-0.982408\pi$
$37$		−	5595.98i		−	0.110477i	0.998473	−	0.0552384i	$-0.0175919\pi$
$38$		0			0
$39$	−132903.			−2.24048
$40$		0			0
$41$		−	50461.2i		−	0.732160i	−0.930583	−	0.366080i	$-0.880700\pi$
$41$		−	50461.2i		−	0.732160i	0.930583	−	0.366080i	$-0.119300\pi$
$42$		0			0
$43$	104427.			1.31343			0.656717	−	0.754137i	$-0.271944\pi$
$43$	104427.			1.31343			0.656717	+	0.754137i	$0.271944\pi$
$44$		0			0
$45$	−5865.81			−0.0643711
$46$		0			0
$47$	−187522.			−1.80617			−0.903085	−	0.429462i	$-0.858703\pi$
$47$	−187522.			−1.80617			−0.903085	+	0.429462i	$0.858703\pi$
$48$		0			0
$49$	−6327.59			−0.0537836
$50$		0			0
$51$		−	36515.0i		−	0.275271i
$52$		0			0
$53$		−	56414.1i		−	0.378931i	−0.981887	−	0.189465i	$-0.939325\pi$
$53$		−	56414.1i		−	0.378931i	0.981887	−	0.189465i	$-0.0606755\pi$
$54$		0			0
$55$	19361.8			0.116375
$56$		0			0
$57$	−3264.79	−	252818.i	−0.0176291	−	1.36516i
$58$		0			0
$59$			375732.i			1.82946i	0.404071	+	0.914728i	$0.367595\pi$
$59$			375732.i			1.82946i	−0.404071	+	0.914728i	$0.632405\pi$
$60$		0			0
$61$	326193.			1.43710			0.718548	−	0.695477i	$-0.244807\pi$
$61$	326193.			1.43710			0.718548	+	0.695477i	$0.244807\pi$
$62$		0			0
$63$	210145.			0.840424
$64$		0			0
$65$		−	33577.5i		−	0.122267i
$66$		0			0
$67$		−	317128.i		−	1.05441i	−0.849738	−	0.527206i	$-0.823240\pi$
$67$		−	317128.i		−	1.05441i	0.849738	−	0.527206i	$-0.176760\pi$
$68$		0			0
$69$		−	364228.i		−	1.10873i
$70$		0			0
$71$			475031.i			1.32723i	0.748073	+	0.663616i	$0.230979\pi$
$71$			475031.i			1.32723i	−0.748073	+	0.663616i	$0.769021\pi$
$72$		0			0
$73$	−428704.			−1.10202			−0.551010	−	0.834499i	$-0.685757\pi$
$73$	−428704.			−1.10202			−0.551010	+	0.834499i	$0.685757\pi$
$74$		0			0
$75$			572779.i			1.35770i
$76$		0			0
$77$	−693646.			−1.51938
$78$		0			0
$79$		−	418334.i		−	0.848482i	−0.905549	−	0.424241i	$-0.860541\pi$
$79$		−	418334.i		−	0.848482i	0.905549	−	0.424241i	$-0.139459\pi$
$80$		0			0
$81$	−593896.			−1.11752
$82$		0			0
$83$	135044.			0.236179			0.118089	−	0.993003i	$-0.462323\pi$
$83$	135044.			0.236179			0.118089	+	0.993003i	$0.462323\pi$
$84$		0			0
$85$	9225.40			0.0150220
$86$		0			0
$87$	−1.09349e6			−1.66057
$88$		0			0
$89$			646123.i			0.916526i	0.888817	+	0.458263i	$0.151528\pi$
$89$			646123.i			0.916526i	−0.888817	+	0.458263i	$0.848472\pi$
$90$		0			0
$91$			1.20293e6i			1.59631i
$92$		0			0
$93$	−297973.			−0.370449
$94$		0			0
$95$	63873.7	−	824.837i	0.0744992	−	0.000962049i
$96$		0			0
$97$			934631.i			1.02406i	0.858968	+	0.512029i	$0.171106\pi$
$97$			934631.i			1.02406i	−0.858968	+	0.512029i	$0.828894\pi$
$98$		0			0
$99$	−1.30942e6			−1.34950
$100$		0			0
$101$	−634023.			−0.615376			−0.307688	−	0.951487i	$-0.599555\pi$
$101$	−634023.			−0.615376			−0.307688	+	0.951487i	$0.599555\pi$
$102$		0			0
$103$			36422.2i			0.0333315i	0.999861	+	0.0166657i	$0.00530511\pi$
$103$			36422.2i			0.0333315i	−0.999861	+	0.0166657i	$0.994695\pi$
$104$		0			0
$105$			114544.i			0.0989472i
$106$		0			0
$107$			106022.i			0.0865452i	0.999063	+	0.0432726i	$0.0137784\pi$
$107$			106022.i			0.0865452i	−0.999063	+	0.0432726i	$0.986222\pi$
$108$		0			0
$109$		−	1.49410e6i		−	1.15372i	−0.816842	−	0.576861i	$-0.804277\pi$
$109$		−	1.49410e6i		−	1.15372i	0.816842	−	0.576861i	$-0.195723\pi$
$110$		0			0
$111$	−206281.			−0.150831
$112$		0			0
$113$			1.27498e6i			0.883626i	0.897107	+	0.441813i	$0.145665\pi$
$113$			1.27498e6i			0.883626i	−0.897107	+	0.441813i	$0.854335\pi$
$114$		0			0
$115$	92021.0			0.0605053
$116$		0			0
$117$			2.27081e6i			1.41783i
$118$		0			0
$119$	−330504.			−0.196126
$120$		0			0
$121$	2.55056e6			1.43973
$122$		0			0
$123$	−1.86012e6			−0.999600
$124$		0			0
$125$	−290229.			−0.148597
$126$		0			0
$127$		−	3.68654e6i		−	1.79973i	−0.436164	−	0.899867i	$-0.643663\pi$
$127$		−	3.68654e6i		−	1.79973i	0.436164	−	0.899867i	$-0.356337\pi$
$128$		0			0
$129$		−	3.84944e6i		−	1.79320i
$130$		0			0
$131$	−3.81130e6			−1.69535			−0.847674	−	0.530518i	$-0.821997\pi$
$131$	−3.81130e6			−1.69535			−0.847674	+	0.530518i	$0.821997\pi$
$132$		0			0
$133$	−2.28831e6	+	29550.2i	−0.972655	+	0.0125604i
$134$		0			0
$135$		−	34042.1i		−	0.0138361i
$136$		0			0
$137$	−3.19748e6			−1.24350			−0.621750	−	0.783216i	$-0.713578\pi$
$137$	−3.19748e6			−1.24350			−0.621750	+	0.783216i	$0.713578\pi$
$138$		0			0
$139$	2.09594e6			0.780431			0.390216	−	0.920724i	$-0.372400\pi$
$139$	2.09594e6			0.780431			0.390216	+	0.920724i	$0.372400\pi$
$140$		0			0
$141$			6.91252e6i			2.46592i
$142$		0			0
$143$		−	7.49548e6i		−	2.56325i
$144$		0			0
$145$		−	276266.i		−	0.0906199i
$146$		0			0
$147$			233250.i			0.0734295i
$148$		0			0
$149$	−3.15486e6			−0.953722			−0.476861	−	0.878979i	$-0.658225\pi$
$149$	−3.15486e6			−0.953722			−0.476861	+	0.878979i	$0.658225\pi$
$150$		0			0
$151$			854912.i			0.248308i	0.992263	+	0.124154i	$0.0396217\pi$
$151$			854912.i			0.248308i	−0.992263	+	0.124154i	$0.960378\pi$
$152$		0			0
$153$	−623904.			−0.174198
$154$		0			0
$155$		−	75281.9i		−	0.0202160i
$156$		0			0
$157$	−11774.2			−0.00304252			−0.00152126	−	0.999999i	$-0.500484\pi$
$157$	−11774.2			−0.00304252			−0.00152126	+	0.999999i	$0.500484\pi$
$158$		0			0
$159$	−2.07956e6			−0.517345
$160$		0			0
$161$	−3.29670e6			−0.789953
$162$		0			0
$163$	2.74604e6			0.634079			0.317040	−	0.948412i	$-0.397311\pi$
$163$	2.74604e6			0.634079			0.317040	+	0.948412i	$0.397311\pi$
$164$		0			0
$165$		−	713724.i		−	0.158883i
$166$		0			0
$167$		−	891541.i		−	0.191422i	−0.995409	−	0.0957110i	$-0.969488\pi$
$167$		−	891541.i		−	0.191422i	0.995409	−	0.0957110i	$-0.0305125\pi$
$168$		0			0
$169$	−8.17196e6			−1.69304
$170$		0			0
$171$	−4.31972e6	+	55782.9i	−0.863907	+	0.0111561i
$172$		0			0
$173$		−	9.29788e6i		−	1.79575i	−0.440251	−	0.897875i	$-0.645111\pi$
$173$		−	9.29788e6i		−	1.79575i	0.440251	−	0.897875i	$-0.354889\pi$
$174$		0			0
$175$	5.18432e6			0.967337
$176$		0			0
$177$	1.38504e7			2.49771
$178$		0			0
$179$			3.43520e6i			0.598953i	0.954104	+	0.299476i	$0.0968120\pi$
$179$			3.43520e6i			0.598953i	−0.954104	+	0.299476i	$0.903188\pi$
$180$		0			0
$181$		−	2.19465e6i		−	0.370109i	−0.982728	−	0.185055i	$-0.940754\pi$
$181$		−	2.19465e6i		−	0.370109i	0.982728	−	0.185055i	$-0.0592462\pi$
$182$		0			0
$183$		−	1.20243e7i		−	1.96203i
$184$		0			0
$185$		−	52116.3i		−	0.00823111i
$186$		0			0
$187$	2.05938e6			0.314928
$188$		0			0
$189$			1.21957e6i			0.180644i
$190$		0			0
$191$	220212.			0.0316039			0.0158020	−	0.999875i	$-0.494970\pi$
$191$	220212.			0.0316039			0.0158020	+	0.999875i	$0.494970\pi$
$192$		0			0
$193$			7.93817e6i			1.10420i	0.833777	+	0.552101i	$0.186174\pi$
$193$			7.93817e6i			1.10420i	−0.833777	+	0.552101i	$0.813826\pi$
$194$		0			0
$195$	−1.23775e6			−0.166928
$196$		0			0
$197$	−6.76666e6			−0.885066			−0.442533	−	0.896752i	$-0.645920\pi$
$197$	−6.76666e6			−0.885066			−0.442533	+	0.896752i	$0.645920\pi$
$198$		0			0
$199$	−9.87122e6			−1.25260			−0.626299	−	0.779583i	$-0.715431\pi$
$199$	−9.87122e6			−1.25260			−0.626299	+	0.779583i	$0.715431\pi$
$200$		0			0
$201$	−1.16901e7			−1.43956
$202$		0			0
$203$			9.89736e6i			1.18313i
$204$		0			0
$205$		−	469954.i		−	0.0545498i
$206$		0			0
$207$	−6.22329e6			−0.701631
$208$		0			0
$209$	1.42585e7	−	184128.i	1.56183	−	0.0201688i
$210$		0			0
$211$			3.53026e6i			0.375802i	0.982188	+	0.187901i	$0.0601684\pi$
$211$			3.53026e6i			0.375802i	−0.982188	+	0.187901i	$0.939832\pi$
$212$		0			0
$213$	1.75108e7			1.81204
$214$		0			0
$215$	972549.			0.0978580
$216$		0			0
$217$			2.69701e6i			0.263939i
$218$		0			0
$219$			1.58031e7i			1.50456i
$220$		0			0
$221$		−	3.57140e6i		−	0.330873i
$222$		0			0
$223$			1.15307e7i			1.03978i	0.854233	+	0.519890i	$0.174027\pi$
$223$			1.15307e7i			1.03978i	−0.854233	+	0.519890i	$0.825973\pi$
$224$		0			0
$225$	9.78663e6			0.859183
$226$		0			0
$227$		−	7.61735e6i		−	0.651218i	−0.945504	−	0.325609i	$-0.894431\pi$
$227$		−	7.61735e6i		−	0.651218i	0.945504	−	0.325609i	$-0.105569\pi$
$228$		0			0
$229$	−1.99152e6			−0.165836			−0.0829178	−	0.996556i	$-0.526424\pi$
$229$	−1.99152e6			−0.165836			−0.0829178	+	0.996556i	$0.526424\pi$
$230$		0			0
$231$			2.55695e7i			2.07437i
$232$		0			0
$233$	−230939.			−0.0182570			−0.00912850	−	0.999958i	$-0.502906\pi$
$233$	−230939.			−0.0182570			−0.00912850	+	0.999958i	$0.502906\pi$
$234$		0			0
$235$	−1.74642e6			−0.134569
$236$		0			0
$237$	−1.54208e7			−1.15841
$238$		0			0
$239$	1.56786e7			1.14845			0.574226	−	0.818697i	$-0.305303\pi$
$239$	1.56786e7			1.14845			0.574226	+	0.818697i	$0.305303\pi$
$240$		0			0
$241$		−	1.48946e7i		−	1.06409i	−0.846717	−	0.532044i	$-0.821424\pi$
$241$		−	1.48946e7i		−	1.06409i	0.846717	−	0.532044i	$-0.178576\pi$
$242$		0			0
$243$			1.92278e7i			1.34002i
$244$		0			0
$245$	−58929.9			−0.00400717
$246$		0			0
$247$	−319317.	−	2.47272e7i	−0.0211900	−	1.64091i
$248$		0			0
$249$		−	4.97805e6i		−	0.322449i
$250$		0			0
$251$	2.36471e7			1.49540			0.747699	−	0.664038i	$-0.231159\pi$
$251$	2.36471e7			1.49540			0.747699	+	0.664038i	$0.231159\pi$
$252$		0			0
$253$	2.05418e7			1.26846
$254$		0			0
$255$		−	340071.i		−	0.0205092i
$256$		0			0
$257$		−	7.20182e6i		−	0.424270i	−0.977240	−	0.212135i	$-0.931958\pi$
$257$		−	7.20182e6i		−	0.424270i	0.977240	−	0.212135i	$-0.0680417\pi$
$258$		0			0
$259$			1.86709e6i			0.107465i
$260$		0			0
$261$			1.86836e7i			1.05085i
$262$		0			0
$263$	2.97999e7			1.63813			0.819063	−	0.573704i	$-0.194494\pi$
$263$	2.97999e7			1.63813			0.819063	+	0.573704i	$0.194494\pi$
$264$		0			0
$265$		−	525394.i		−	0.0282324i
$266$		0			0
$267$	2.38177e7			1.25131
$268$		0			0
$269$		−	6.51885e6i		−	0.334899i	−0.985881	−	0.167450i	$-0.946447\pi$
$269$		−	6.51885e6i		−	0.334899i	0.985881	−	0.167450i	$-0.0535532\pi$
$270$		0			0
$271$	1.80842e7			0.908639			0.454320	−	0.890839i	$-0.349882\pi$
$271$	1.80842e7			0.908639			0.454320	+	0.890839i	$0.349882\pi$
$272$		0			0
$273$	4.43430e7			2.17940
$274$		0			0
$275$	−3.23036e7			−1.55329
$276$		0			0
$277$	−2.02268e7			−0.951675			−0.475837	−	0.879533i	$-0.657855\pi$
$277$	−2.02268e7			−0.951675			−0.475837	+	0.879533i	$0.657855\pi$
$278$		0			0
$279$			5.09124e6i			0.234429i
$280$		0			0
$281$			1.94438e7i			0.876319i	0.898897	+	0.438160i	$0.144370\pi$
$281$			1.94438e7i			0.876319i	−0.898897	+	0.438160i	$0.855630\pi$
$282$		0			0
$283$	−2.02701e7			−0.894326			−0.447163	−	0.894453i	$-0.647566\pi$
$283$	−2.02701e7			−0.894326			−0.447163	+	0.894453i	$0.647566\pi$
$284$		0			0
$285$	−30405.5	−	2.35454e6i	−0.00131346	−	0.101712i
$286$		0			0
$287$			1.68363e7i			0.712198i
$288$		0			0
$289$	−2.31563e7			−0.959348
$290$		0			0
$291$	3.44528e7			1.39812
$292$		0			0
$293$		−	3.28208e7i		−	1.30481i	−0.757872	−	0.652404i	$-0.773761\pi$
$293$		−	3.28208e7i		−	1.30481i	0.757872	−	0.652404i	$-0.226239\pi$
$294$		0			0
$295$			3.49925e6i			0.136304i
$296$		0			0
$297$		−	7.59919e6i		−	0.290067i
$298$		0			0
$299$		−	3.56238e7i		−	1.33268i
$300$		0			0
$301$	−3.48420e7			−1.27763
$302$		0			0
$303$			2.33716e7i			0.840159i
$304$		0			0
$305$	3.03790e6			0.107071
$306$		0			0
$307$		−	4.34323e7i		−	1.50106i	−0.660837	−	0.750529i	$-0.729799\pi$
$307$		−	4.34323e7i		−	1.50106i	0.660837	−	0.750529i	$-0.270201\pi$
$308$		0			0
$309$	1.34261e6			0.0455066
$310$		0			0
$311$	−3.90880e7			−1.29946			−0.649729	−	0.760166i	$-0.725118\pi$
$311$	−3.90880e7			−1.29946			−0.649729	+	0.760166i	$0.725118\pi$
$312$		0			0
$313$	5.74162e7			1.87241			0.936206	−	0.351453i	$-0.114312\pi$
$313$	5.74162e7			1.87241			0.936206	+	0.351453i	$0.114312\pi$
$314$		0			0
$315$	1.95712e6			0.0626161
$316$		0			0
$317$			2.58961e6i			0.0812936i	0.999174	+	0.0406468i	$0.0129418\pi$
$317$			2.58961e6i			0.0812936i	−0.999174	+	0.0406468i	$0.987058\pi$
$318$		0			0
$319$		−	6.16706e7i		−	1.89979i
$320$		0			0
$321$	3.90822e6			0.118158
$322$		0			0
$323$	6.79379e6	−	87732.0i	0.201606	−	0.00260346i
$324$		0			0
$325$			5.60214e7i			1.63194i
$326$		0			0
$327$	−5.50764e7			−1.57515
$328$		0			0
$329$	6.25665e7			1.75693
$330$		0			0
$331$		−	2.42549e7i		−	0.668831i	−0.942426	−	0.334415i	$-0.891461\pi$
$331$		−	2.42549e7i		−	0.668831i	0.942426	−	0.334415i	$-0.108539\pi$
$332$		0			0
$333$			3.52457e6i			0.0954495i
$334$		0			0
$335$		−	2.95347e6i		−	0.0785593i
$336$		0			0
$337$		−	2.96882e7i		−	0.775702i	−0.921722	−	0.387851i	$-0.873218\pi$
$337$		−	2.96882e7i		−	0.775702i	0.921722	−	0.387851i	$-0.126782\pi$
$338$		0			0
$339$	4.69990e7			1.20639
$340$		0			0
$341$		−	1.68051e7i		−	0.423817i
$342$		0			0
$343$	4.13646e7			1.02505
$344$		0			0
$345$		−	3.39212e6i		−	0.0826065i
$346$		0			0
$347$	3.82196e7			0.914739			0.457370	−	0.889277i	$-0.348792\pi$
$347$	3.82196e7			0.914739			0.457370	+	0.889277i	$0.348792\pi$
$348$		0			0
$349$	2.35960e7			0.555088			0.277544	−	0.960713i	$-0.410480\pi$
$349$	2.35960e7			0.555088			0.277544	+	0.960713i	$0.410480\pi$
$350$		0			0
$351$	−1.31786e7			−0.304753
$352$		0			0
$353$	4.38762e7			0.997481			0.498740	−	0.866751i	$-0.333796\pi$
$353$	4.38762e7			0.997481			0.498740	+	0.866751i	$0.333796\pi$
$354$		0			0
$355$			4.42405e6i			0.0988860i
$356$		0			0
$357$			1.21832e7i			0.267767i
$358$		0			0
$359$	−1.87184e6			−0.0404562			−0.0202281	−	0.999795i	$-0.506439\pi$
$359$	−1.87184e6			−0.0404562			−0.0202281	+	0.999795i	$0.506439\pi$
$360$		0			0
$361$	4.70302e7	−	1.21486e6i	0.999667	−	0.0258228i
$362$		0			0
$363$		−	9.40200e7i		−	1.96562i
$364$		0			0
$365$	−3.99260e6			−0.0821063
$366$		0			0
$367$	2.51394e7			0.508577			0.254289	−	0.967128i	$-0.418159\pi$
$367$	2.51394e7			0.508577			0.254289	+	0.967128i	$0.418159\pi$
$368$		0			0
$369$			3.17825e7i			0.632570i
$370$		0			0
$371$			1.88225e7i			0.368600i
$372$		0			0
$373$			3.07434e7i			0.592414i	0.955124	+	0.296207i	$0.0957219\pi$
$373$			3.07434e7i			0.592414i	−0.955124	+	0.296207i	$0.904278\pi$
$374$		0			0
$375$			1.06985e7i			0.202876i
$376$		0			0
$377$	−1.06950e8			−1.99598
$378$		0			0
$379$		−	7.69260e7i		−	1.41304i	−0.707692	−	0.706521i	$-0.750263\pi$
$379$		−	7.69260e7i		−	1.41304i	0.707692	−	0.706521i	$-0.249737\pi$
$380$		0			0
$381$	−1.35895e8			−2.45713
$382$		0			0
$383$			7.21580e7i			1.28436i	0.766552	+	0.642182i	$0.221971\pi$
$383$			7.21580e7i			1.28436i	−0.766552	+	0.642182i	$0.778029\pi$
$384$		0			0
$385$	−6.46004e6			−0.113202
$386$		0			0
$387$	−6.57725e7			−1.13478
$388$		0			0
$389$	2.93341e7			0.498337			0.249169	−	0.968460i	$-0.419843\pi$
$389$	2.93341e7			0.498337			0.249169	+	0.968460i	$0.419843\pi$
$390$		0			0
$391$	9.78762e6			0.163737
$392$		0			0
$393$			1.40494e8i			2.31462i
$394$		0			0
$395$		−	3.89602e6i		−	0.0632164i
$396$		0			0
$397$	−1.61834e7			−0.258641			−0.129321	−	0.991603i	$-0.541280\pi$
$397$	−1.61834e7			−0.258641			−0.129321	+	0.991603i	$0.541280\pi$
$398$		0			0
$399$	1.08929e6	+	8.43525e7i	0.0171485	+	1.32794i
$400$		0			0
$401$			6.59267e7i			1.02242i	0.859457	+	0.511209i	$0.170802\pi$
$401$			6.59267e7i			1.02242i	−0.859457	+	0.511209i	$0.829198\pi$
$402$		0			0
$403$	−2.91436e7			−0.445275
$404$		0			0
$405$	−5.53105e6			−0.0832612
$406$		0			0
$407$		−	1.16339e7i		−	0.172560i
$408$		0			0
$409$			3.82877e7i			0.559614i	0.960056	+	0.279807i	$0.0902706\pi$
$409$			3.82877e7i			0.559614i	−0.960056	+	0.279807i	$0.909729\pi$
$410$		0			0
$411$			1.17867e8i			1.69772i
$412$		0			0
$413$		−	1.25362e8i		−	1.77958i
$414$		0			0
$415$	1.25769e6			0.0175966
$416$		0			0
$417$		−	7.72615e7i		−	1.06550i
$418$		0			0
$419$	−4.03454e7			−0.548469			−0.274235	−	0.961663i	$-0.588425\pi$
$419$	−4.03454e7			−0.548469			−0.274235	+	0.961663i	$0.588425\pi$
$420$		0			0
$421$			2.50228e7i			0.335343i	0.985843	+	0.167672i	$0.0536248\pi$
$421$			2.50228e7i			0.335343i	−0.985843	+	0.167672i	$0.946375\pi$
$422$		0			0
$423$	1.18109e8			1.56049
$424$		0			0
$425$	−1.53918e7			−0.200504
$426$		0			0
$427$	−1.08834e8			−1.39792
$428$		0			0
$429$	−2.76302e8			−3.49955
$430$		0			0
$431$		−	4.41217e7i		−	0.551087i	−0.961289	−	0.275544i	$-0.911142\pi$
$431$		−	4.41217e7i		−	0.551087i	0.961289	−	0.275544i	$-0.0888579\pi$
$432$		0			0
$433$		−	6.46505e7i		−	0.796357i	−0.917308	−	0.398179i	$-0.869642\pi$
$433$		−	6.46505e7i		−	0.796357i	0.917308	−	0.398179i	$-0.130358\pi$
$434$		0			0
$435$	−1.01838e7			−0.123721
$436$		0			0
$437$	6.77664e7	−	875105.i	0.812026	−	0.0104861i
$438$		0			0
$439$			9.76909e7i			1.15468i	0.816505	+	0.577339i	$0.195909\pi$
$439$			9.76909e7i			1.15468i	−0.816505	+	0.577339i	$0.804091\pi$
$440$		0			0
$441$	3.98537e6			0.0464679
$442$		0			0
$443$	−1.16199e8			−1.33656			−0.668282	−	0.743908i	$-0.732970\pi$
$443$	−1.16199e8			−1.33656			−0.668282	+	0.743908i	$0.732970\pi$
$444$		0			0
$445$			6.01745e6i			0.0682861i
$446$		0			0
$447$			1.16296e8i			1.30209i
$448$		0			0
$449$		−	6.32693e7i		−	0.698962i	−0.936943	−	0.349481i	$-0.886358\pi$
$449$		−	6.32693e7i		−	0.698962i	0.936943	−	0.349481i	$-0.113642\pi$
$450$		0			0
$451$		−	1.04907e8i		−	1.14361i
$452$		0			0
$453$	3.15142e7			0.339009
$454$		0			0
$455$			1.12031e7i			0.118933i
$456$		0			0
$457$	7.55248e7			0.791300			0.395650	−	0.918401i	$-0.370519\pi$
$457$	7.55248e7			0.791300			0.395650	+	0.918401i	$0.370519\pi$
$458$		0			0
$459$		−	3.62081e6i		−	0.0374428i
$460$		0			0
$461$	−1.12105e7			−0.114425			−0.0572124	−	0.998362i	$-0.518221\pi$
$461$	−1.12105e7			−0.114425			−0.0572124	+	0.998362i	$0.518221\pi$
$462$		0			0
$463$	−1.81606e8			−1.82973			−0.914865	−	0.403760i	$-0.867703\pi$
$463$	−1.81606e8			−1.82973			−0.914865	+	0.403760i	$0.867703\pi$
$464$		0			0
$465$	−2.77507e6			−0.0276004
$466$		0			0
$467$	1.31691e8			1.29302			0.646512	−	0.762904i	$-0.276227\pi$
$467$	1.31691e8			1.29302			0.646512	+	0.762904i	$0.276227\pi$
$468$		0			0
$469$			1.05809e8i			1.02566i
$470$		0			0
$471$			434027.i			0.00415388i
$472$		0			0
$473$	2.17101e8			2.05154
$474$		0			0
$475$	−1.06568e8	+	1.37617e6i	−0.994366	+	0.0128408i
$476$		0			0
$477$			3.55319e7i			0.327388i
$478$		0			0
$479$	−1.61518e7			−0.146965			−0.0734824	−	0.997297i	$-0.523411\pi$
$479$	−1.61518e7			−0.146965			−0.0734824	+	0.997297i	$0.523411\pi$
$480$		0			0
$481$	−2.01756e7			−0.181298
$482$		0			0
$483$			1.21524e8i			1.07850i
$484$		0			0
$485$			8.70438e6i			0.0762979i
$486$		0			0
$487$		−	4.24942e7i		−	0.367911i	−0.982935	−	0.183956i	$-0.941110\pi$
$487$		−	4.24942e7i		−	0.367911i	0.982935	−	0.183956i	$-0.0588902\pi$
$488$		0			0
$489$		−	1.01226e8i		−	0.865693i
$490$		0			0
$491$	9.28537e7			0.784431			0.392216	−	0.919873i	$-0.371709\pi$
$491$	9.28537e7			0.784431			0.392216	+	0.919873i	$0.371709\pi$
$492$		0			0
$493$		−	2.93844e7i		−	0.245232i
$494$		0			0
$495$	−1.21949e7			−0.100545
$496$		0			0
$497$		−	1.58493e8i		−	1.29105i
$498$		0			0
$499$	−6.15919e7			−0.495703			−0.247852	−	0.968798i	$-0.579725\pi$
$499$	−6.15919e7			−0.495703			−0.247852	+	0.968798i	$0.579725\pi$
$500$		0			0
$501$	−3.28644e7			−0.261344
$502$		0			0
$503$	−601690.			−0.00472791			−0.00236395	−	0.999997i	$-0.500752\pi$
$503$	−601690.			−0.00472791			−0.00236395	+	0.999997i	$0.500752\pi$
$504$		0			0
$505$	−5.90476e6			−0.0458489
$506$		0			0
$507$			3.01239e8i			2.31146i
$508$		0			0
$509$		−	1.22352e8i		−	0.927810i	−0.885885	−	0.463905i	$-0.846448\pi$
$509$		−	1.22352e8i		−	0.927810i	0.885885	−	0.463905i	$-0.153552\pi$
$510$		0			0
$511$	1.43037e8			1.07197
$512$		0			0
$513$	−323735.	−	2.50694e7i	−0.00239793	−	0.185691i
$514$		0			0
$515$			339206.i			0.00248337i
$516$		0			0
$517$	−3.89853e8			−2.82117
$518$		0			0
$519$	−3.42743e8			−2.45169
$520$		0			0
$521$		−	8.43784e7i		−	0.596648i	−0.954465	−	0.298324i	$-0.903572\pi$
$521$		−	8.43784e7i		−	0.596648i	0.954465	−	0.298324i	$-0.0964275\pi$
$522$		0			0
$523$		−	1.49701e8i		−	1.04646i	−0.852193	−	0.523228i	$-0.824728\pi$
$523$		−	1.49701e8i		−	1.04646i	0.852193	−	0.523228i	$-0.175272\pi$
$524$		0			0
$525$		−	1.91107e8i		−	1.32068i
$526$		0			0
$527$		−	8.00719e6i		−	0.0547077i
$528$		0			0
$529$	−5.04068e7			−0.340504
$530$		0			0
$531$		−	2.36651e8i		−	1.58061i
$532$		0			0
$533$	−1.81932e8			−1.20151
$534$		0			0
$535$			987397.i			0.00644808i
$536$		0			0
$537$	1.26630e8			0.817736
$538$		0			0
$539$	−1.31549e7			−0.0840079
$540$		0			0
$541$	1.27860e8			0.807500			0.403750	−	0.914869i	$-0.367707\pi$
$541$	1.27860e8			0.807500			0.403750	+	0.914869i	$0.367707\pi$
$542$		0			0
$543$	−8.09003e7			−0.505301
$544$		0			0
$545$		−	1.39149e7i		−	0.0859586i
$546$		0			0
$547$			2.63042e8i			1.60718i	0.595186	+	0.803588i	$0.297078\pi$
$547$			2.63042e8i			1.60718i	−0.595186	+	0.803588i	$0.702922\pi$
$548$		0			0
$549$	−2.05450e8			−1.24162
$550$		0			0
$551$	−2.62724e6	−	2.03449e8i	−0.0157053	−	1.21619i
$552$		0			0
$553$			1.39577e8i			0.825349i
$554$		0			0
$555$	−1.92113e6			−0.0112377
$556$		0			0
$557$	−9.24005e7			−0.534698			−0.267349	−	0.963600i	$-0.586148\pi$
$557$	−9.24005e7			−0.534698			−0.267349	+	0.963600i	$0.586148\pi$
$558$		0			0
$559$		−	3.76500e8i		−	2.15541i
$560$		0			0
$561$		−	7.59137e7i		−	0.429964i
$562$		0			0
$563$		−	2.60607e8i		−	1.46037i	−0.683252	−	0.730183i	$-0.739435\pi$
$563$		−	2.60607e8i		−	1.46037i	0.683252	−	0.730183i	$-0.260565\pi$
$564$		0			0
$565$			1.18741e7i			0.0658349i
$566$		0			0
$567$	1.98152e8			1.08705
$568$		0			0
$569$			1.57246e8i			0.853576i	0.904352	+	0.426788i	$0.140355\pi$
$569$			1.57246e8i			0.853576i	−0.904352	+	0.426788i	$0.859645\pi$
$570$		0			0
$571$	2.97511e8			1.59806			0.799032	−	0.601288i	$-0.205346\pi$
$571$	2.97511e8			1.59806			0.799032	+	0.601288i	$0.205346\pi$
$572$		0			0
$573$		−	8.11756e6i		−	0.0431481i
$574$		0			0
$575$	−1.53530e8			−0.807585
$576$		0			0
$577$	1.47265e7			0.0766606			0.0383303	−	0.999265i	$-0.487796\pi$
$577$	1.47265e7			0.0766606			0.0383303	+	0.999265i	$0.487796\pi$
$578$		0			0
$579$	2.92621e8			1.50754
$580$		0			0
$581$	−4.50572e7			−0.229740
$582$		0			0
$583$		−	1.17283e8i		−	0.591876i
$584$		0			0
$585$			2.11485e7i			0.105636i
$586$		0			0
$587$	−7.64950e7			−0.378198			−0.189099	−	0.981958i	$-0.560557\pi$
$587$	−7.64950e7			−0.378198			−0.189099	+	0.981958i	$0.560557\pi$
$588$		0			0
$589$	−715918.	−	5.54392e7i	−0.00350362	−	0.271313i
$590$		0			0
$591$			2.49436e8i			1.20836i
$592$		0			0
$593$	1.94688e8			0.933630			0.466815	−	0.884355i	$-0.345401\pi$
$593$	1.94688e8			0.933630			0.466815	+	0.884355i	$0.345401\pi$
$594$		0			0
$595$	−3.07804e6			−0.0146125
$596$		0			0
$597$			3.63877e8i			1.71014i
$598$		0			0
$599$			3.86043e8i			1.79620i	0.439787	+	0.898102i	$0.355054\pi$
$599$			3.86043e8i			1.79620i	−0.439787	+	0.898102i	$0.644946\pi$
$600$		0			0
$601$			1.94757e8i			0.897157i	0.893743	+	0.448579i	$0.148070\pi$
$601$			1.94757e8i			0.897157i	−0.893743	+	0.448579i	$0.851930\pi$
$602$		0			0
$603$			1.99740e8i			0.910989i
$604$		0			0
$605$	2.37538e7			0.107267
$606$		0			0
$607$		−	4.26582e8i		−	1.90738i	−0.300795	−	0.953689i	$-0.597252\pi$
$607$		−	4.26582e8i		−	1.90738i	0.300795	−	0.953689i	$-0.402748\pi$
$608$		0			0
$609$	3.64841e8			1.61529
$610$		0			0
$611$			6.76088e8i			2.96401i
$612$		0			0
$613$	−1.49126e8			−0.647398			−0.323699	−	0.946160i	$-0.604927\pi$
$613$	−1.49126e8			−0.647398			−0.323699	+	0.946160i	$0.604927\pi$
$614$		0			0
$615$	−1.73236e7			−0.0744756
$616$		0			0
$617$	8.34867e7			0.355436			0.177718	−	0.984081i	$-0.443128\pi$
$617$	8.34867e7			0.355436			0.177718	+	0.984081i	$0.443128\pi$
$618$		0			0
$619$	4.48909e7			0.189272			0.0946360	−	0.995512i	$-0.469831\pi$
$619$	4.48909e7			0.189272			0.0946360	+	0.995512i	$0.469831\pi$
$620$		0			0
$621$		−	3.61167e7i		−	0.150811i
$622$		0			0
$623$		−	2.15578e8i		−	0.891539i
$624$		0			0
$625$	2.40082e8			0.983378
$626$		0			0
$627$	−6.78740e6	−	5.25602e8i	−0.0275360	−	2.13233i
$628$		0			0
$629$		−	5.54324e6i		−	0.0222747i
$630$		0			0
$631$	−7.40984e7			−0.294931			−0.147466	−	0.989067i	$-0.547112\pi$
$631$	−7.40984e7			−0.294931			−0.147466	+	0.989067i	$0.547112\pi$
$632$		0			0
$633$	1.30134e8			0.513074
$634$		0			0
$635$		−	3.43334e7i		−	0.134090i
$636$		0			0
$637$			2.28134e7i			0.0882614i
$638$		0			0
$639$		−	2.99194e8i		−	1.14670i
$640$		0			0
$641$		−	1.18864e8i		−	0.451310i	−0.974207	−	0.225655i	$-0.927548\pi$
$641$		−	1.18864e8i		−	0.451310i	0.974207	−	0.225655i	$-0.0724523\pi$
$642$		0			0
$643$	−4.19389e8			−1.57756			−0.788778	−	0.614679i	$-0.789286\pi$
$643$	−4.19389e8			−1.57756			−0.788778	+	0.614679i	$0.789286\pi$
$644$		0			0
$645$		−	3.58505e7i		−	0.133603i
$646$		0			0
$647$	2.10442e8			0.776999			0.388499	−	0.921449i	$-0.372994\pi$
$647$	2.10442e8			0.776999			0.388499	+	0.921449i	$0.372994\pi$
$648$		0			0
$649$			7.81135e8i			2.85754i
$650$		0			0
$651$	9.94183e7			0.360349
$652$		0			0
$653$	9.63966e7			0.346196			0.173098	−	0.984905i	$-0.444622\pi$
$653$	9.63966e7			0.346196			0.173098	+	0.984905i	$0.444622\pi$
$654$		0			0
$655$	−3.54952e7			−0.126312
$656$		0			0
$657$	2.70015e8			0.952121
$658$		0			0
$659$			3.19745e8i			1.11724i	0.829423	+	0.558621i	$0.188669\pi$
$659$			3.19745e8i			1.11724i	−0.829423	+	0.558621i	$0.811331\pi$
$660$		0			0
$661$			1.19591e8i			0.414089i	0.978332	+	0.207044i	$0.0663845\pi$
$661$			1.19591e8i			0.414089i	−0.978332	+	0.207044i	$0.933616\pi$
$662$		0			0
$663$	−1.31651e8			−0.451733
$664$		0			0
$665$	−2.13114e7	+	275206.i	−0.0724681	+	0.000935821i
$666$		0			0
$667$		−	2.93102e8i		−	0.987739i
$668$		0			0
$669$	4.25051e8			1.41959
$670$		0			0
$671$	6.78147e8			2.24469
$672$		0			0
$673$			1.69048e8i			0.554581i	0.960786	+	0.277291i	$0.0894365\pi$
$673$			1.69048e8i			0.554581i	−0.960786	+	0.277291i	$0.910564\pi$
$674$		0			0
$675$			5.67964e7i			0.184676i
$676$		0			0
$677$			2.40878e8i			0.776303i	0.921596	+	0.388151i	$0.126886\pi$
$677$			2.40878e8i			0.776303i	−0.921596	+	0.388151i	$0.873114\pi$
$678$		0			0
$679$		−	3.11838e8i		−	0.996139i
$680$		0			0
$681$	−2.80794e8			−0.889093
$682$		0			0
$683$			5.35144e8i			1.67961i	0.542887	+	0.839805i	$0.317331\pi$
$683$			5.35144e8i			1.67961i	−0.542887	+	0.839805i	$0.682669\pi$
$684$		0			0
$685$	−2.97787e7			−0.0926474
$686$		0			0
$687$			7.34122e7i			0.226411i
$688$		0			0
$689$	−2.03394e8			−0.621843
$690$		0			0
$691$	1.08573e8			0.329070			0.164535	−	0.986371i	$-0.447388\pi$
$691$	1.08573e8			0.329070			0.164535	+	0.986371i	$0.447388\pi$
$692$		0			0
$693$	4.36886e8			1.31271
$694$		0			0
$695$	1.95199e7			0.0581463
$696$		0			0
$697$		−	4.99856e7i		−	0.147620i
$698$		0			0
$699$			8.51298e6i			0.0249258i
$700$		0			0
$701$	4.00994e7			0.116408			0.0582042	−	0.998305i	$-0.481463\pi$
$701$	4.00994e7			0.116408			0.0582042	+	0.998305i	$0.481463\pi$
$702$		0			0
$703$	−495617.	−	3.83796e7i	−0.00142653	−	0.110468i
$704$		0			0
$705$			6.43775e7i			0.183724i
$706$		0			0
$707$	2.11541e8			0.598599
$708$		0			0
$709$	−4.51118e8			−1.26576			−0.632880	−	0.774250i	$-0.718127\pi$
$709$	−4.51118e8			−1.26576			−0.632880	+	0.774250i	$0.718127\pi$
$710$		0			0
$711$			2.63484e8i			0.733070i
$712$		0			0
$713$		−	7.98697e7i		−	0.220350i
$714$		0			0
$715$		−	6.98067e7i		−	0.190976i
$716$		0			0
$717$		−	5.77951e8i		−	1.56795i
$718$		0			0
$719$	−4.29069e7			−0.115436			−0.0577179	−	0.998333i	$-0.518382\pi$
$719$	−4.29069e7			−0.115436			−0.0577179	+	0.998333i	$0.518382\pi$
$720$		0			0
$721$		−	1.21522e7i		−	0.0324227i
$722$		0			0
$723$	−5.49051e8			−1.45277
$724$		0			0
$725$			4.60927e8i			1.20954i
$726$		0			0
$727$	3.86921e8			1.00697			0.503487	−	0.864003i	$-0.332050\pi$
$727$	3.86921e8			1.00697			0.503487	+	0.864003i	$0.332050\pi$
$728$		0			0
$729$	2.75833e8			0.711972
$730$		0			0
$731$	1.03443e8			0.264819
$732$		0			0
$733$	3.34664e8			0.849761			0.424880	−	0.905250i	$-0.360316\pi$
$733$	3.34664e8			0.849761			0.424880	+	0.905250i	$0.360316\pi$
$734$		0			0
$735$			2.17230e6i			0.00547089i
$736$		0			0
$737$		−	6.59300e8i		−	1.64695i
$738$		0			0
$739$	6.22639e8			1.54278			0.771388	−	0.636365i	$-0.219563\pi$
$739$	6.22639e8			1.54278			0.771388	+	0.636365i	$0.219563\pi$
$740$		0			0
$741$	−9.11507e8	+	1.17708e7i	−2.24030	+	0.0289302i
$742$		0			0
$743$			1.01752e8i			0.248072i	0.992278	+	0.124036i	$0.0395838\pi$
$743$			1.01752e8i			0.248072i	−0.992278	+	0.124036i	$0.960416\pi$
$744$		0			0
$745$	−2.93818e7			−0.0710574
$746$		0			0
$747$	−8.50561e7			−0.204053
$748$		0			0
$749$		−	3.53740e7i		−	0.0841857i
$750$		0			0
$751$			7.91911e8i			1.86963i	0.355131	+	0.934816i	$0.384436\pi$
$751$			7.91911e8i			1.86963i	−0.355131	+	0.934816i	$0.615564\pi$
$752$		0			0
$753$		−	8.71690e8i		−	2.04163i
$754$		0			0
$755$			7.96194e6i			0.0185003i
$756$		0			0
$757$	5.94781e8			1.37110			0.685550	−	0.728025i	$-0.259562\pi$
$757$	5.94781e8			1.37110			0.685550	+	0.728025i	$0.259562\pi$
$758$		0			0
$759$		−	7.57220e8i		−	1.73180i
$760$		0			0
$761$	3.98303e8			0.903773			0.451886	−	0.892076i	$-0.350751\pi$
$761$	3.98303e8			0.903773			0.451886	+	0.892076i	$0.350751\pi$
$762$		0			0
$763$			4.98506e8i			1.12227i
$764$		0			0
$765$	−5.81053e6			−0.0129787
$766$		0			0
$767$	1.35466e9			3.00222
$768$		0			0
$769$	−8.66816e8			−1.90611			−0.953055	−	0.302796i	$-0.902080\pi$
$769$	−8.66816e8			−1.90611			−0.953055	+	0.302796i	$0.902080\pi$
$770$		0			0
$771$	−2.65477e8			−0.579246
$772$		0			0
$773$			9.22212e8i			1.99661i	0.0582351	+	0.998303i	$0.481453\pi$
$773$			9.22212e8i			1.99661i	−0.0582351	+	0.998303i	$0.518547\pi$
$774$		0			0
$775$			1.25602e8i			0.269830i
$776$		0			0
$777$	6.88255e7			0.146719
$778$		0			0
$779$	−4.46918e6	−	3.46084e8i	−0.00945400	−	0.732099i
$780$		0			0
$781$			9.87576e8i			2.07309i
$782$		0			0
$783$	−1.08430e8			−0.225872
$784$		0			0
$785$	−109655.			−0.000226684
$786$		0			0
$787$		−	2.11475e8i		−	0.433846i	−0.976189	−	0.216923i	$-0.930398\pi$
$787$		−	2.11475e8i		−	0.433846i	0.976189	−	0.216923i	$-0.0696020\pi$
$788$		0			0
$789$		−	1.09850e9i		−	2.23649i
$790$		0			0
$791$		−	4.25396e8i		−	0.859536i
$792$		0			0
$793$		−	1.17605e9i		−	2.35834i
$794$		0			0
$795$	−1.93673e7			−0.0385450
$796$		0			0
$797$		−	7.79060e8i		−	1.53885i	−0.638737	−	0.769425i	$-0.720543\pi$
$797$		−	7.79060e8i		−	1.53885i	0.638737	−	0.769425i	$-0.279457\pi$
$798$		0			0
$799$	−1.85755e8			−0.364166
$800$		0			0
$801$		−	4.06954e8i		−	0.791859i
$802$		0			0
$803$	−8.91264e8			−1.72131
$804$		0			0
$805$	−3.07027e7			−0.0588557
$806$		0			0
$807$	−2.40301e8			−0.457230
$808$		0			0
$809$	5.89102e8			1.11262			0.556308	−	0.830976i	$-0.312218\pi$
$809$	5.89102e8			1.11262			0.556308	+	0.830976i	$0.312218\pi$
$810$		0			0
$811$			5.31055e8i			0.995582i	0.867297	+	0.497791i	$0.165855\pi$
$811$			5.31055e8i			0.995582i	−0.867297	+	0.497791i	$0.834145\pi$
$812$		0			0
$813$		−	6.66628e8i		−	1.24054i
$814$		0			0
$815$	2.55743e7			0.0472423
$816$		0			0
$817$	7.16207e8	−	9.24878e6i	1.31333	−	0.0169597i
$818$		0			0
$819$		−	7.57654e8i		−	1.37917i
$820$		0			0
$821$	9.12888e8			1.64964			0.824818	−	0.565399i	$-0.191278\pi$
$821$	9.12888e8			1.64964			0.824818	+	0.565399i	$0.191278\pi$
$822$		0			0
$823$	3.80355e7			0.0682322			0.0341161	−	0.999418i	$-0.489138\pi$
$823$	3.80355e7			0.0682322			0.0341161	+	0.999418i	$0.489138\pi$
$824$		0			0
$825$			1.19079e9i			2.12067i
$826$		0			0
$827$		−	6.09954e8i		−	1.07840i	−0.842177	−	0.539201i	$-0.818726\pi$
$827$		−	6.09954e8i		−	1.07840i	0.842177	−	0.539201i	$-0.181274\pi$
$828$		0			0
$829$		−	9.95748e8i		−	1.74778i	−0.486127	−	0.873888i	$-0.661591\pi$
$829$		−	9.95748e8i		−	1.74778i	0.486127	−	0.873888i	$-0.338409\pi$
$830$		0			0
$831$			7.45611e8i			1.29930i
$832$		0			0
$833$	−6.26795e6			−0.0108440
$834$		0			0
$835$		−	8.30308e6i		−	0.0142620i
$836$		0			0
$837$	−2.95469e7			−0.0503889
$838$		0			0
$839$			4.97180e8i			0.841836i	0.907099	+	0.420918i	$0.138292\pi$
$839$			4.97180e8i			0.841836i	−0.907099	+	0.420918i	$0.861708\pi$
$840$		0			0
$841$	−2.85131e8			−0.479354
$842$		0			0
$843$	7.16746e8			1.19642
$844$		0			0
$845$	−7.61069e7			−0.126140
$846$		0			0
$847$	−8.50991e8			−1.40047
$848$		0			0
$849$			7.47204e8i			1.22100i
$850$		0			0
$851$		−	5.52924e7i		−	0.0897174i
$852$		0			0
$853$	−1.13929e9			−1.83564			−0.917822	−	0.396993i	$-0.870054\pi$
$853$	−1.13929e9			−1.83564			−0.917822	+	0.396993i	$0.870054\pi$
$854$		0			0
$855$	−4.02303e7	+	519516.i	−0.0643657	+	0.000831190i
$856$		0			0
$857$			7.96845e8i			1.26599i	0.774154	+	0.632997i	$0.218175\pi$
$857$			7.96845e8i			1.26599i	−0.774154	+	0.632997i	$0.781825\pi$
$858$		0			0
$859$	6.96488e8			1.09884			0.549420	−	0.835547i	$-0.314849\pi$
$859$	6.96488e8			1.09884			0.549420	+	0.835547i	$0.314849\pi$
$860$		0			0
$861$	6.20627e8			0.972347
$862$		0			0
$863$			4.55068e8i			0.708018i	0.935242	+	0.354009i	$0.115182\pi$
$863$			4.55068e8i			0.708018i	−0.935242	+	0.354009i	$0.884818\pi$
$864$		0			0
$865$		−	8.65928e7i		−	0.133793i
$866$		0			0
$867$			8.53599e8i			1.30977i
$868$		0			0
$869$		−	8.69705e8i		−	1.32530i
$870$		0			0
$871$	−1.14337e9			−1.73034
$872$		0			0
$873$		−	5.88668e8i		−	0.884765i
$874$		0			0
$875$	9.68345e7			0.144546
$876$		0			0
$877$		−	1.12029e8i		−	0.166086i	−0.996546	−	0.0830430i	$-0.973536\pi$
$877$		−	1.12029e8i		−	0.166086i	0.996546	−	0.0830430i	$-0.0264639\pi$
$878$		0			0
$879$	−1.20986e9			−1.78142
$880$		0			0
$881$	6.27658e7			0.0917900			0.0458950	−	0.998946i	$-0.485386\pi$
$881$	6.27658e7			0.0917900			0.0458950	+	0.998946i	$0.485386\pi$
$882$		0			0
$883$	1.00439e9			1.45888			0.729438	−	0.684047i	$-0.239782\pi$
$883$	1.00439e9			1.45888			0.729438	+	0.684047i	$0.239782\pi$
$884$		0			0
$885$	1.28991e8			0.186093
$886$		0			0
$887$		−	1.40260e8i		−	0.200985i	−0.994938	−	0.100493i	$-0.967958\pi$
$887$		−	1.40260e8i		−	0.200985i	0.994938	−	0.100493i	$-0.0320419\pi$
$888$		0			0
$889$			1.23001e9i			1.75067i
$890$		0			0
$891$	−1.23469e9			−1.74552
$892$		0			0
$893$	−1.28611e9	+	1.66082e7i	−1.80602	+	0.0233221i
$894$		0			0
$895$			3.19926e7i			0.0446252i
$896$		0			0
$897$	−1.31318e9			−1.81948
$898$		0			0
$899$	−2.39785e8			−0.330023
$900$		0			0
$901$		−	5.58824e7i		−	0.0764013i
$902$		0			0
$903$			1.28436e9i			1.74431i
$904$		0			0
$905$		−	2.04392e7i		−	0.0275751i
$906$		0			0
$907$		−	8.39630e8i		−	1.12529i	−0.826697	−	0.562647i	$-0.809783\pi$
$907$		−	8.39630e8i		−	1.12529i	0.826697	−	0.562647i	$-0.190217\pi$
$908$		0			0
$909$	3.99333e8			0.531672
$910$		0			0
$911$		−	1.27025e8i		−	0.168009i	−0.996465	−	0.0840047i	$-0.973229\pi$
$911$		−	1.27025e8i		−	0.168009i	0.996465	−	0.0840047i	$-0.0267711\pi$
$912$		0			0
$913$	2.80752e8			0.368902
$914$		0			0
$915$		−	1.11984e8i		−	0.146182i
$916$		0			0
$917$	1.27163e9			1.64913
$918$		0			0
$919$	−1.02603e9			−1.32195			−0.660974	−	0.750408i	$-0.729857\pi$
$919$	−1.02603e9			−1.32195			−0.660974	+	0.750408i	$0.729857\pi$
$920$		0			0
$921$	−1.60102e9			−2.04936
$922$		0			0
$923$	1.71267e9			2.17805
$924$		0			0
$925$			8.69518e7i			0.109863i
$926$		0			0
$927$		−	2.29402e7i		−	0.0287977i
$928$		0			0
$929$	−5.76963e8			−0.719617			−0.359808	−	0.933026i	$-0.617158\pi$
$929$	−5.76963e8			−0.719617			−0.359808	+	0.933026i	$0.617158\pi$
$930$		0			0
$931$	−4.33973e7	+	560414.i	−0.0537791	+	0.000694480i
$932$		0			0
$933$			1.44088e9i			1.77412i
$934$		0			0
$935$	1.91793e7			0.0234638
$936$		0			0
$937$	−1.11930e9			−1.36059			−0.680296	−	0.732937i	$-0.738149\pi$
$937$	−1.11930e9			−1.36059			−0.680296	+	0.732937i	$0.738149\pi$
$938$		0			0
$939$		−	2.11650e9i		−	2.55636i
$940$		0			0
$941$			1.21766e9i			1.46137i	0.682717	+	0.730683i	$0.260798\pi$
$941$			1.21766e9i			1.46137i	−0.682717	+	0.730683i	$0.739202\pi$
$942$		0			0
$943$		−	4.98594e8i		−	0.594582i
$944$		0			0
$945$			1.13581e7i			0.0134589i
$946$		0			0
$947$	−1.34672e9			−1.58572			−0.792862	−	0.609401i	$-0.791410\pi$
$947$	−1.34672e9			−1.58572			−0.792862	+	0.609401i	$0.791410\pi$
$948$		0			0
$949$			1.54564e9i			1.80847i
$950$		0			0
$951$	9.54594e7			0.110988
$952$		0			0
$953$		−	1.09603e9i		−	1.26632i	−0.774019	−	0.633162i	$-0.781757\pi$
$953$		−	1.09603e9i		−	1.26632i	0.774019	−	0.633162i	$-0.218243\pi$
$954$		0			0
$955$	2.05087e6			0.00235466
$956$		0			0
$957$	−2.27333e9			−2.59374
$958$		0			0
$959$	1.06683e9			1.20960
$960$		0			0
$961$	8.22163e8			0.926377
$962$		0			0
$963$		−	6.67767e7i		−	0.0747732i
$964$		0			0
$965$			7.39296e7i			0.0822690i
$966$		0			0
$967$	−1.36692e9			−1.51170			−0.755848	−	0.654748i	$-0.772775\pi$
$967$	−1.36692e9			−1.51170			−0.755848	+	0.654748i	$0.772775\pi$
$968$		0			0
$969$	−3.23402e6	−	2.50436e8i	−0.00355444	−	0.275249i
$970$		0			0
$971$			5.55859e8i			0.607165i	0.952805	+	0.303583i	$0.0981828\pi$
$971$			5.55859e8i			0.607165i	−0.952805	+	0.303583i	$0.901817\pi$
$972$		0			0
$973$	−6.99308e8			−0.759154
$974$		0			0
$975$	2.06508e9			2.22805
$976$		0			0
$977$		−	1.00569e9i		−	1.07840i	−0.842177	−	0.539202i	$-0.818726\pi$
$977$		−	1.00569e9i		−	1.07840i	0.842177	−	0.539202i	$-0.181274\pi$
$978$		0			0
$979$			1.34327e9i			1.43158i
$980$		0			0
$981$			9.41047e8i			0.996792i
$982$		0			0
$983$			5.73194e8i			0.603450i	0.953395	+	0.301725i	$0.0975624\pi$
$983$			5.73194e8i			0.603450i	−0.953395	+	0.301725i	$0.902438\pi$
$984$		0			0
$985$	−6.30190e7			−0.0659421
$986$		0			0
$987$		−	2.30635e9i		−	2.39869i
$988$		0			0
$989$	1.03182e9			1.06663
$990$		0			0
$991$			1.12059e9i			1.15140i	0.817660	+	0.575701i	$0.195271\pi$
$991$			1.12059e9i			1.15140i	−0.817660	+	0.575701i	$0.804729\pi$
$992$		0			0
$993$	−8.94097e8			−0.913139
$994$		0			0
$995$	−9.19323e7			−0.0933252
$996$		0			0
$997$	9.05063e8			0.913257			0.456629	−	0.889657i	$-0.349057\pi$
$997$	9.05063e8			0.913257			0.456629	+	0.889657i	$0.349057\pi$
$998$		0			0
$999$	−2.04548e7			−0.0205163

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.7.e.f.113.5		30
4.3	odd	2		152.7.e.a.113.26	yes	30
19.18	odd	2	inner	304.7.e.f.113.26		30
76.75	even	2		152.7.e.a.113.5	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.7.e.a.113.5	✓	30	76.75	even	2
152.7.e.a.113.26	yes	30	4.3	odd	2
304.7.e.f.113.5		30	1.1	even	1	trivial
304.7.e.f.113.26		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)

\(f(q)\)	\(=\)	\(q-36.8625i q^{3} +9.31317 q^{5} -333.649 q^{7} -629.840 q^{9} +O(q^{10})\)	\(q-36.8625i q^{3} +9.31317 q^{5} -333.649 q^{7} -629.840 q^{9} +2078.97 q^{11} -3605.38i q^{13} -343.306i q^{15} +990.575 q^{17} +(6858.43 - 88.5667i) q^{19} +12299.1i q^{21} +9880.74 q^{23} -15538.3 q^{25} -3655.26i q^{27} -29664.0i q^{29} -8083.37i q^{31} -76636.0i q^{33} -3107.33 q^{35} -5595.98i q^{37} -132903. q^{39} -50461.2i q^{41} +104427. q^{43} -5865.81 q^{45} -187522. q^{47} -6327.59 q^{49} -36515.0i q^{51} -56414.1i q^{53} +19361.8 q^{55} +(-3264.79 - 252818. i) q^{57} +375732. i q^{59} +326193. q^{61} +210145. q^{63} -33577.5i q^{65} -317128. i q^{67} -364228. i q^{69} +475031. i q^{71} -428704. q^{73} +572779. i q^{75} -693646. q^{77} -418334. i q^{79} -593896. q^{81} +135044. q^{83} +9225.40 q^{85} -1.09349e6 q^{87} +646123. i q^{89} +1.20293e6i q^{91} -297973. q^{93} +(63873.7 - 824.837i) q^{95} +934631. i q^{97} -1.30942e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 720 q^{7} - 8670 q^{9}+O(q^{10}) \) 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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