LMFDB - Embedded newform 304.7.e.f.113.20

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			113.20
Character	$\chi$	$=$	304.113
Dual form			304.7.e.f.113.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+17.1584i q^{3} +166.964 q^{5} +600.271 q^{7} +434.588 q^{9} +O(q^{10})$	$q+17.1584i q^{3} +166.964 q^{5} +600.271 q^{7} +434.588 q^{9} -821.172 q^{11} +3128.21i q^{13} +2864.85i q^{15} +6605.76 q^{17} +(-2663.92 + 6320.55i) q^{19} +10299.7i q^{21} +3415.18 q^{23} +12252.1 q^{25} +19965.4i q^{27} +36396.6i q^{29} -38303.4i q^{31} -14090.0i q^{33} +100224. q^{35} -71402.1i q^{37} -53675.1 q^{39} -76636.1i q^{41} -128959. q^{43} +72560.7 q^{45} -100370. q^{47} +242677. q^{49} +113345. i q^{51} -151301. i q^{53} -137107. q^{55} +(-108451. - 45708.8i) q^{57} +86390.9i q^{59} +148346. q^{61} +260871. q^{63} +522299. i q^{65} +336548. i q^{67} +58599.2i q^{69} -68093.8i q^{71} +479983. q^{73} +210227. i q^{75} -492926. q^{77} -713891. i q^{79} -25759.6 q^{81} -350700. q^{83} +1.10293e6 q^{85} -624509. q^{87} +459383. i q^{89} +1.87777e6i q^{91} +657227. q^{93} +(-444781. + 1.05531e6i) q^{95} +348610. i q^{97} -356872. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			17.1584i			0.635498i	0.948175	+	0.317749i	$0.102927\pi$
$3$			17.1584i			0.635498i	−0.948175	+	0.317749i	$0.897073\pi$
$4$		0			0
$5$	166.964			1.33572			0.667858	−	0.744289i	$-0.267212\pi$
$5$	166.964			1.33572			0.667858	+	0.744289i	$0.267212\pi$
$6$		0			0
$7$	600.271			1.75006			0.875031	−	0.484066i	$-0.160841\pi$
$7$	600.271			1.75006			0.875031	+	0.484066i	$0.160841\pi$
$8$		0			0
$9$	434.588			0.596143
$10$		0			0
$11$	−821.172			−0.616959			−0.308479	−	0.951231i	$-0.599820\pi$
$11$	−821.172			−0.616959			−0.308479	+	0.951231i	$0.599820\pi$
$12$		0			0
$13$			3128.21i			1.42385i	0.702254	+	0.711927i	$0.252177\pi$
$13$			3128.21i			1.42385i	−0.702254	+	0.711927i	$0.747823\pi$
$14$		0			0
$15$			2864.85i			0.848844i
$16$		0			0
$17$	6605.76			1.34455			0.672274	−	0.740303i	$-0.265318\pi$
$17$	6605.76			1.34455			0.672274	+	0.740303i	$0.265318\pi$
$18$		0			0
$19$	−2663.92	+	6320.55i	−0.388384	+	0.921498i
$19$	−2663.92	+	6320.55i	−0.388384	+	0.921498i
$20$		0			0
$21$			10299.7i			1.11216i
$22$		0			0
$23$	3415.18			0.280692			0.140346	−	0.990103i	$-0.455179\pi$
$23$	3415.18			0.280692			0.140346	+	0.990103i	$0.455179\pi$
$24$		0			0
$25$	12252.1			0.784135
$26$		0			0
$27$			19965.4i			1.01434i
$28$		0			0
$29$			36396.6i			1.49234i	0.665757	+	0.746169i	$0.268109\pi$
$29$			36396.6i			1.49234i	−0.665757	+	0.746169i	$0.731891\pi$
$30$		0			0
$31$		−	38303.4i		−	1.28574i	−0.765976	−	0.642869i	$-0.777744\pi$
$31$		−	38303.4i		−	1.28574i	0.765976	−	0.642869i	$-0.222256\pi$
$32$		0			0
$33$		−	14090.0i		−	0.392076i
$34$		0			0
$35$	100224.			2.33758
$36$		0			0
$37$		−	71402.1i		−	1.40963i	−0.709390	−	0.704816i	$-0.751030\pi$
$37$		−	71402.1i		−	1.40963i	0.709390	−	0.704816i	$-0.248970\pi$
$38$		0			0
$39$	−53675.1			−0.904855
$40$		0			0
$41$		−	76636.1i		−	1.11194i	−0.831202	−	0.555970i	$-0.812347\pi$
$41$		−	76636.1i		−	1.11194i	0.831202	−	0.555970i	$-0.187653\pi$
$42$		0			0
$43$	−128959.			−1.62198			−0.810991	−	0.585059i	$-0.801071\pi$
$43$	−128959.			−1.62198			−0.810991	+	0.585059i	$0.801071\pi$
$44$		0			0
$45$	72560.7			0.796277
$46$		0			0
$47$	−100370.			−0.966737			−0.483369	−	0.875417i	$-0.660587\pi$
$47$	−100370.			−0.966737			−0.483369	+	0.875417i	$0.660587\pi$
$48$		0			0
$49$	242677.			2.06272
$50$		0			0
$51$			113345.i			0.854457i
$52$		0			0
$53$		−	151301.i		−	1.01628i	−0.861274	−	0.508140i	$-0.830333\pi$
$53$		−	151301.i		−	1.01628i	0.861274	−	0.508140i	$-0.169667\pi$
$54$		0			0
$55$	−137107.			−0.824081
$56$		0			0
$57$	−108451.	−	45708.8i	−0.585610	−	0.246817i
$58$		0			0
$59$			86390.9i			0.420641i	0.977632	+	0.210321i	$0.0674508\pi$
$59$			86390.9i			0.420641i	−0.977632	+	0.210321i	$0.932549\pi$
$60$		0			0
$61$	148346.			0.653561			0.326781	−	0.945100i	$-0.394036\pi$
$61$	148346.			0.653561			0.326781	+	0.945100i	$0.394036\pi$
$62$		0			0
$63$	260871.			1.04329
$64$		0			0
$65$			522299.i			1.90186i
$66$		0			0
$67$			336548.i			1.11898i	0.828837	+	0.559491i	$0.189003\pi$
$67$			336548.i			1.11898i	−0.828837	+	0.559491i	$0.810997\pi$
$68$		0			0
$69$			58599.2i			0.178379i
$70$		0			0
$71$		−	68093.8i		−	0.190253i	−0.995465	−	0.0951267i	$-0.969674\pi$
$71$		−	68093.8i		−	0.190253i	0.995465	−	0.0951267i	$-0.0303256\pi$
$72$		0			0
$73$	479983.			1.23383			0.616917	−	0.787028i	$-0.288381\pi$
$73$	479983.			1.23383			0.616917	+	0.787028i	$0.288381\pi$
$74$		0			0
$75$			210227.i			0.498316i
$76$		0			0
$77$	−492926.			−1.07972
$78$		0			0
$79$		−	713891.i		−	1.44794i	−0.689831	−	0.723970i	$-0.742315\pi$
$79$		−	713891.i		−	1.44794i	0.689831	−	0.723970i	$-0.257685\pi$
$80$		0			0
$81$	−25759.6			−0.0484712
$82$		0			0
$83$	−350700.			−0.613340			−0.306670	−	0.951816i	$-0.599215\pi$
$83$	−350700.			−0.613340			−0.306670	+	0.951816i	$0.599215\pi$
$84$		0			0
$85$	1.10293e6			1.79593
$86$		0			0
$87$	−624509.			−0.948377
$88$		0			0
$89$			459383.i			0.651636i	0.945432	+	0.325818i	$0.105640\pi$
$89$			459383.i			0.651636i	−0.945432	+	0.325818i	$0.894360\pi$
$90$		0			0
$91$			1.87777e6i			2.49183i
$92$		0			0
$93$	657227.			0.817083
$94$		0			0
$95$	−444781.	+	1.05531e6i	−0.518770	+	1.23086i
$96$		0			0
$97$			348610.i			0.381966i	0.981593	+	0.190983i	$0.0611676\pi$
$97$			348610.i			0.381966i	−0.981593	+	0.190983i	$0.938832\pi$
$98$		0			0
$99$	−356872.			−0.367796
$100$		0			0
$101$	−1.40939e6			−1.36794			−0.683969	−	0.729511i	$-0.739748\pi$
$101$	−1.40939e6			−1.36794			−0.683969	+	0.729511i	$0.739748\pi$
$102$		0			0
$103$			646428.i			0.591573i	0.955254	+	0.295786i	$0.0955817\pi$
$103$			646428.i			0.591573i	−0.955254	+	0.295786i	$0.904418\pi$
$104$		0			0
$105$			1.71969e6i			1.48553i
$106$		0			0
$107$		−	1.17898e6i		−	0.962397i	−0.876612	−	0.481199i	$-0.840202\pi$
$107$		−	1.17898e6i		−	0.962397i	0.876612	−	0.481199i	$-0.159798\pi$
$108$		0			0
$109$		−	1.67679e6i		−	1.29479i	−0.762155	−	0.647394i	$-0.775859\pi$
$109$		−	1.67679e6i		−	1.29479i	0.762155	−	0.647394i	$-0.224141\pi$
$110$		0			0
$111$	1.22515e6			0.895818
$112$		0			0
$113$			1.30764e6i			0.906260i	0.891444	+	0.453130i	$0.149693\pi$
$113$			1.30764e6i			0.906260i	−0.891444	+	0.453130i	$0.850307\pi$
$114$		0			0
$115$	570213.			0.374925
$116$		0			0
$117$			1.35948e6i			0.848820i
$118$		0			0
$119$	3.96525e6			2.35304
$120$		0			0
$121$	−1.09724e6			−0.619362
$122$		0			0
$123$	1.31495e6			0.706636
$124$		0			0
$125$	−563153.			−0.288334
$126$		0			0
$127$			896680.i			0.437750i	0.975753	+	0.218875i	$0.0702387\pi$
$127$			896680.i			0.437750i	−0.975753	+	0.218875i	$0.929761\pi$
$128$		0			0
$129$		−	2.21273e6i		−	1.03077i
$130$		0			0
$131$	1.06805e6			0.475090			0.237545	−	0.971377i	$-0.423657\pi$
$131$	1.06805e6			0.475090			0.237545	+	0.971377i	$0.423657\pi$
$132$		0			0
$133$	−1.59908e6	+	3.79405e6i	−0.679696	+	1.61268i
$134$		0			0
$135$			3.33350e6i			1.35488i
$136$		0			0
$137$	1.86193e6			0.724106			0.362053	−	0.932158i	$-0.382076\pi$
$137$	1.86193e6			0.724106			0.362053	+	0.932158i	$0.382076\pi$
$138$		0			0
$139$	−2.18110e6			−0.812139			−0.406070	−	0.913842i	$-0.633101\pi$
$139$	−2.18110e6			−0.812139			−0.406070	+	0.913842i	$0.633101\pi$
$140$		0			0
$141$		−	1.72218e6i		−	0.614359i
$142$		0			0
$143$		−	2.56880e6i		−	0.878459i
$144$		0			0
$145$			6.07694e6i			1.99334i
$146$		0			0
$147$			4.16395e6i			1.31085i
$148$		0			0
$149$	−4.07784e6			−1.23274			−0.616370	−	0.787457i	$-0.711397\pi$
$149$	−4.07784e6			−1.23274			−0.616370	+	0.787457i	$0.711397\pi$
$150$		0			0
$151$		−	3.21447e6i		−	0.933639i	−0.884353	−	0.466819i	$-0.845400\pi$
$151$		−	3.21447e6i		−	0.933639i	0.884353	−	0.466819i	$-0.154600\pi$
$152$		0			0
$153$	2.87078e6			0.801542
$154$		0			0
$155$		−	6.39531e6i		−	1.71738i
$156$		0			0
$157$	3.85657e6			0.996558			0.498279	−	0.867017i	$-0.333965\pi$
$157$	3.85657e6			0.996558			0.498279	+	0.867017i	$0.333965\pi$
$158$		0			0
$159$	2.59609e6			0.645844
$160$		0			0
$161$	2.05004e6			0.491229
$162$		0			0
$163$	4.21638e6			0.973593			0.486796	−	0.873516i	$-0.338165\pi$
$163$	4.21638e6			0.973593			0.486796	+	0.873516i	$0.338165\pi$
$164$		0			0
$165$		−	2.35253e6i		−	0.523702i
$166$		0			0
$167$			6.50328e6i			1.39631i	0.715945	+	0.698157i	$0.245996\pi$
$167$			6.50328e6i			1.39631i	−0.715945	+	0.698157i	$0.754004\pi$
$168$		0			0
$169$	−4.95886e6			−1.02736
$170$		0			0
$171$	−1.15771e6	+	2.74684e6i	−0.231532	+	0.549344i
$172$		0			0
$173$			590191.i			0.113987i	0.998375	+	0.0569934i	$0.0181514\pi$
$173$			590191.i			0.113987i	−0.998375	+	0.0569934i	$0.981849\pi$
$174$		0			0
$175$	7.35459e6			1.37228
$176$		0			0
$177$	−1.48233e6			−0.267317
$178$		0			0
$179$		−	9.70549e6i		−	1.69223i	−0.533004	−	0.846113i	$-0.678937\pi$
$179$		−	9.70549e6i		−	1.69223i	0.533004	−	0.846113i	$-0.321063\pi$
$180$		0			0
$181$			6.20826e6i			1.04697i	0.852035	+	0.523485i	$0.175368\pi$
$181$			6.20826e6i			1.04697i	−0.852035	+	0.523485i	$0.824632\pi$
$182$		0			0
$183$			2.54539e6i			0.415337i
$184$		0			0
$185$		−	1.19216e7i		−	1.88287i
$186$		0			0
$187$	−5.42447e6			−0.829530
$188$		0			0
$189$			1.19846e7i			1.77517i
$190$		0			0
$191$	−667378.			−0.0957793			−0.0478896	−	0.998853i	$-0.515250\pi$
$191$	−667378.			−0.0957793			−0.0478896	+	0.998853i	$0.515250\pi$
$192$		0			0
$193$			9.57318e6i			1.33163i	0.746116	+	0.665816i	$0.231916\pi$
$193$			9.57318e6i			1.33163i	−0.746116	+	0.665816i	$0.768084\pi$
$194$		0			0
$195$	−8.96183e6			−1.20863
$196$		0			0
$197$	1.30026e7			1.70071			0.850356	−	0.526208i	$-0.176387\pi$
$197$	1.30026e7			1.70071			0.850356	+	0.526208i	$0.176387\pi$
$198$		0			0
$199$	1.23078e7			1.56178			0.780892	−	0.624666i	$-0.214765\pi$
$199$	1.23078e7			1.56178			0.780892	+	0.624666i	$0.214765\pi$
$200$		0			0
$201$	−5.77464e6			−0.711110
$202$		0			0
$203$			2.18479e7i			2.61168i
$204$		0			0
$205$		−	1.27955e7i		−	1.48524i
$206$		0			0
$207$	1.48420e6			0.167333
$208$		0			0
$209$	2.18754e6	−	5.19026e6i	0.239617	−	0.568526i
$210$		0			0
$211$		−	3.75152e6i		−	0.399356i	−0.979862	−	0.199678i	$-0.936010\pi$
$211$		−	3.75152e6i		−	0.399356i	0.979862	−	0.199678i	$-0.0639896\pi$
$212$		0			0
$213$	1.16838e6			0.120906
$214$		0			0
$215$	−2.15315e7			−2.16651
$216$		0			0
$217$		−	2.29924e7i		−	2.25012i
$218$		0			0
$219$			8.23575e6i			0.784099i
$220$		0			0
$221$			2.06642e7i			1.91444i
$222$		0			0
$223$		−	7.50463e6i		−	0.676729i	−0.941015	−	0.338364i	$-0.890126\pi$
$223$		−	7.50463e6i		−	0.676729i	0.941015	−	0.338364i	$-0.109874\pi$
$224$		0			0
$225$	5.32462e6			0.467456
$226$		0			0
$227$		−	6.59536e6i		−	0.563846i	−0.959437	−	0.281923i	$-0.909028\pi$
$227$		−	6.59536e6i		−	0.563846i	0.959437	−	0.281923i	$-0.0909723\pi$
$228$		0			0
$229$	−1.84191e7			−1.53377			−0.766887	−	0.641782i	$-0.778196\pi$
$229$	−1.84191e7			−1.53377			−0.766887	+	0.641782i	$0.778196\pi$
$230$		0			0
$231$		−	8.45784e6i		−	0.686157i
$232$		0			0
$233$	−1.61221e7			−1.27454			−0.637270	−	0.770640i	$-0.719937\pi$
$233$	−1.61221e7			−1.27454			−0.637270	+	0.770640i	$0.719937\pi$
$234$		0			0
$235$	−1.67581e7			−1.29129
$236$		0			0
$237$	1.22493e7			0.920162
$238$		0			0
$239$	797649.			0.0584276			0.0292138	−	0.999573i	$-0.490700\pi$
$239$	797649.			0.0584276			0.0292138	+	0.999573i	$0.490700\pi$
$240$		0			0
$241$		−	1.60290e7i		−	1.14513i	−0.819858	−	0.572567i	$-0.805948\pi$
$241$		−	1.60290e7i		−	1.14513i	0.819858	−	0.572567i	$-0.194052\pi$
$242$		0			0
$243$			1.41127e7i			0.983542i
$244$		0			0
$245$	4.05184e7			2.75520
$246$		0			0
$247$	−1.97720e7	−	8.33330e6i	−1.31208	−	0.553002i
$248$		0			0
$249$		−	6.01746e6i		−	0.389776i
$250$		0			0
$251$	5.46408e6			0.345538			0.172769	−	0.984962i	$-0.444729\pi$
$251$	5.46408e6			0.345538			0.172769	+	0.984962i	$0.444729\pi$
$252$		0			0
$253$	−2.80445e6			−0.173175
$254$		0			0
$255$			1.89245e7i			1.14131i
$256$		0			0
$257$			361482.i			0.0212955i	0.999943	+	0.0106477i	$0.00338935\pi$
$257$			361482.i			0.0212955i	−0.999943	+	0.0106477i	$0.996611\pi$
$258$		0			0
$259$		−	4.28606e7i		−	2.46694i
$260$		0			0
$261$			1.58175e7i			0.889646i
$262$		0			0
$263$	−7.39160e6			−0.406323			−0.203161	−	0.979145i	$-0.565122\pi$
$263$	−7.39160e6			−0.406323			−0.203161	+	0.979145i	$0.565122\pi$
$264$		0			0
$265$		−	2.52618e7i		−	1.35746i
$266$		0			0
$267$	−7.88230e6			−0.414113
$268$		0			0
$269$		−	2.21850e7i		−	1.13973i	−0.821738	−	0.569866i	$-0.806995\pi$
$269$		−	2.21850e7i		−	1.13973i	0.821738	−	0.569866i	$-0.193005\pi$
$270$		0			0
$271$	−2.02916e7			−1.01955			−0.509775	−	0.860308i	$-0.670271\pi$
$271$	−2.02916e7			−1.01955			−0.509775	+	0.860308i	$0.670271\pi$
$272$		0			0
$273$	−3.22196e7			−1.58355
$274$		0			0
$275$	−1.00611e7			−0.483779
$276$		0			0
$277$	1.17205e7			0.551451			0.275726	−	0.961236i	$-0.411082\pi$
$277$	1.17205e7			0.551451			0.275726	+	0.961236i	$0.411082\pi$
$278$		0			0
$279$		−	1.66462e7i		−	0.766483i
$280$		0			0
$281$		−	1.68718e6i		−	0.0760401i	−0.999277	−	0.0380200i	$-0.987895\pi$
$281$		−	1.68718e6i		−	0.0760401i	0.999277	−	0.0380200i	$-0.0121051\pi$
$282$		0			0
$283$	3.81982e7			1.68533			0.842663	−	0.538442i	$-0.180987\pi$
$283$	3.81982e7			1.68533			0.842663	+	0.538442i	$0.180987\pi$
$284$		0			0
$285$	−1.81074e7	−	7.63174e6i	−0.782208	−	0.329677i
$286$		0			0
$287$		−	4.60024e7i		−	1.94597i
$288$		0			0
$289$	1.94985e7			0.807807
$290$		0			0
$291$	−5.98161e6			−0.242739
$292$		0			0
$293$			7.04988e6i			0.280272i	0.990132	+	0.140136i	$0.0447539\pi$
$293$			7.04988e6i			0.280272i	−0.990132	+	0.140136i	$0.955246\pi$
$294$		0			0
$295$			1.44242e7i			0.561857i
$296$		0			0
$297$		−	1.63950e7i		−	0.625809i
$298$		0			0
$299$			1.06834e7i			0.399664i
$300$		0			0
$301$	−7.74104e7			−2.83857
$302$		0			0
$303$		−	2.41829e7i		−	0.869322i
$304$		0			0
$305$	2.47685e7			0.872972
$306$		0			0
$307$		−	2.74067e7i		−	0.947199i	−0.880740	−	0.473599i	$-0.842954\pi$
$307$		−	2.74067e7i		−	0.947199i	0.880740	−	0.473599i	$-0.157046\pi$
$308$		0			0
$309$	−1.10917e7			−0.375943
$310$		0			0
$311$	4.85979e6			0.161561			0.0807805	−	0.996732i	$-0.474259\pi$
$311$	4.85979e6			0.161561			0.0807805	+	0.996732i	$0.474259\pi$
$312$		0			0
$313$	3.70710e7			1.20893			0.604465	−	0.796631i	$-0.293387\pi$
$313$	3.70710e7			1.20893			0.604465	+	0.796631i	$0.293387\pi$
$314$		0			0
$315$	4.35561e7			1.39353
$316$		0			0
$317$		−	3.48160e7i		−	1.09295i	−0.837475	−	0.546475i	$-0.815969\pi$
$317$		−	3.48160e7i		−	1.09295i	0.837475	−	0.546475i	$-0.184031\pi$
$318$		0			0
$319$		−	2.98879e7i		−	0.920711i
$320$		0			0
$321$	2.02294e7			0.611601
$322$		0			0
$323$	−1.75972e7	+	4.17521e7i	−0.522200	+	1.23900i
$324$		0			0
$325$			3.83271e7i			1.11649i
$326$		0			0
$327$	2.87711e7			0.822835
$328$		0			0
$329$	−6.02490e7			−1.69185
$330$		0			0
$331$			4.07867e7i			1.12469i	0.826901	+	0.562347i	$0.190102\pi$
$331$			4.07867e7i			1.12469i	−0.826901	+	0.562347i	$0.809898\pi$
$332$		0			0
$333$		−	3.10305e7i		−	0.840342i
$334$		0			0
$335$			5.61916e7i			1.49464i
$336$		0			0
$337$			2.96265e7i			0.774089i	0.922061	+	0.387045i	$0.126504\pi$
$337$			2.96265e7i			0.774089i	−0.922061	+	0.387045i	$0.873496\pi$
$338$		0			0
$339$	−2.24371e7			−0.575926
$340$		0			0
$341$			3.14537e7i			0.793247i
$342$		0			0
$343$	7.50506e7			1.85982
$344$		0			0
$345$			9.78397e6i			0.238264i
$346$		0			0
$347$	5.45176e7			1.30481			0.652407	−	0.757869i	$-0.273759\pi$
$347$	5.45176e7			1.30481			0.652407	+	0.757869i	$0.273759\pi$
$348$		0			0
$349$	4.85403e7			1.14190			0.570948	−	0.820986i	$-0.306576\pi$
$349$	4.85403e7			1.14190			0.570948	+	0.820986i	$0.306576\pi$
$350$		0			0
$351$	−6.24557e7			−1.44428
$352$		0			0
$353$	9.45288e6			0.214902			0.107451	−	0.994210i	$-0.465731\pi$
$353$	9.45288e6			0.214902			0.107451	+	0.994210i	$0.465731\pi$
$354$		0			0
$355$		−	1.13692e7i		−	0.254124i
$356$		0			0
$357$			6.80375e7i			1.49535i
$358$		0			0
$359$	3.49669e7			0.755742			0.377871	−	0.925858i	$-0.376656\pi$
$359$	3.49669e7			0.755742			0.377871	+	0.925858i	$0.376656\pi$
$360$		0			0
$361$	−3.28529e7	−	3.36750e7i	−0.698316	−	0.715790i
$362$		0			0
$363$		−	1.88269e7i		−	0.393603i
$364$		0			0
$365$	8.01400e7			1.64805
$366$		0			0
$367$	−4.22417e6			−0.0854561			−0.0427280	−	0.999087i	$-0.513605\pi$
$367$	−4.22417e6			−0.0854561			−0.0427280	+	0.999087i	$0.513605\pi$
$368$		0			0
$369$		−	3.33051e7i		−	0.662875i
$370$		0			0
$371$		−	9.08216e7i		−	1.77855i
$372$		0			0
$373$		−	2.58083e7i		−	0.497317i	−0.968591	−	0.248659i	$-0.920010\pi$
$373$		−	2.58083e7i		−	0.497317i	0.968591	−	0.248659i	$-0.0799897\pi$
$374$		0			0
$375$		−	9.66283e6i		−	0.183236i
$376$		0			0
$377$	−1.13856e8			−2.12487
$378$		0			0
$379$		−	2.27060e7i		−	0.417083i	−0.978014	−	0.208541i	$-0.933128\pi$
$379$		−	2.27060e7i		−	0.417083i	0.978014	−	0.208541i	$-0.0668716\pi$
$380$		0			0
$381$	−1.53856e7			−0.278189
$382$		0			0
$383$			7.21914e7i			1.28496i	0.766303	+	0.642479i	$0.222094\pi$
$383$			7.21914e7i			1.28496i	−0.766303	+	0.642479i	$0.777906\pi$
$384$		0			0
$385$	−8.23011e7			−1.44219
$386$		0			0
$387$	−5.60440e7			−0.966933
$388$		0			0
$389$	−4.91629e7			−0.835196			−0.417598	−	0.908632i	$-0.637128\pi$
$389$	−4.91629e7			−0.835196			−0.417598	+	0.908632i	$0.637128\pi$
$390$		0			0
$391$	2.25599e7			0.377404
$392$		0			0
$393$			1.83260e7i			0.301918i
$394$		0			0
$395$		−	1.19194e8i		−	1.93404i
$396$		0			0
$397$	−6.82113e6			−0.109015			−0.0545073	−	0.998513i	$-0.517359\pi$
$397$	−6.82113e6			−0.109015			−0.0545073	+	0.998513i	$0.517359\pi$
$398$		0			0
$399$	−6.50999e7	−	2.74377e7i	−1.02485	−	0.431945i
$400$		0			0
$401$			2.97357e7i			0.461153i	0.973054	+	0.230576i	$0.0740611\pi$
$401$			2.97357e7i			0.461153i	−0.973054	+	0.230576i	$0.925939\pi$
$402$		0			0
$403$	1.19821e8			1.83070
$404$		0			0
$405$	−4.30093e6			−0.0647437
$406$		0			0
$407$			5.86334e7i			0.869685i
$408$		0			0
$409$		−	2.07315e7i		−	0.303013i	−0.988456	−	0.151506i	$-0.951588\pi$
$409$		−	2.07315e7i		−	0.303013i	0.988456	−	0.151506i	$-0.0484124\pi$
$410$		0			0
$411$			3.19478e7i			0.460168i
$412$		0			0
$413$			5.18580e7i			0.736149i
$414$		0			0
$415$	−5.85544e7			−0.819248
$416$		0			0
$417$		−	3.74242e7i		−	0.516113i
$418$		0			0
$419$	−1.25618e7			−0.170770			−0.0853850	−	0.996348i	$-0.527212\pi$
$419$	−1.25618e7			−0.170770			−0.0853850	+	0.996348i	$0.527212\pi$
$420$		0			0
$421$			3.09759e6i			0.0415124i	0.999785	+	0.0207562i	$0.00660738\pi$
$421$			3.09759e6i			0.0415124i	−0.999785	+	0.0207562i	$0.993393\pi$
$422$		0			0
$423$	−4.36194e7			−0.576313
$424$		0			0
$425$	8.09345e7			1.05431
$426$		0			0
$427$	8.90478e7			1.14377
$428$		0			0
$429$	4.40765e7			0.558259
$430$		0			0
$431$		−	8.09265e7i		−	1.01079i	−0.862889	−	0.505393i	$-0.831348\pi$
$431$		−	8.09265e7i		−	1.01079i	0.862889	−	0.505393i	$-0.168652\pi$
$432$		0			0
$433$			1.33096e7i			0.163946i	0.996635	+	0.0819728i	$0.0261221\pi$
$433$			1.33096e7i			0.163946i	−0.996635	+	0.0819728i	$0.973878\pi$
$434$		0			0
$435$	−1.04271e8			−1.26676
$436$		0			0
$437$	−9.09778e6	+	2.15858e7i	−0.109016	+	0.258657i
$438$		0			0
$439$			8.89142e7i			1.05094i	0.850813	+	0.525469i	$0.176110\pi$
$439$			8.89142e7i			1.05094i	−0.850813	+	0.525469i	$0.823890\pi$
$440$		0			0
$441$	1.05464e8			1.22967
$442$		0			0
$443$	−1.13457e8			−1.30503			−0.652514	−	0.757777i	$-0.726286\pi$
$443$	−1.13457e8			−1.30503			−0.652514	+	0.757777i	$0.726286\pi$
$444$		0			0
$445$			7.67007e7i			0.870401i
$446$		0			0
$447$		−	6.99694e7i		−	0.783403i
$448$		0			0
$449$		−	1.35437e8i		−	1.49623i	−0.663570	−	0.748114i	$-0.730960\pi$
$449$		−	1.35437e8i		−	1.49623i	0.663570	−	0.748114i	$-0.269040\pi$
$450$		0			0
$451$			6.29314e7i			0.686022i
$452$		0			0
$453$	5.51553e7			0.593325
$454$		0			0
$455$			3.13521e8i			3.32838i
$456$		0			0
$457$	−1.34065e8			−1.40464			−0.702321	−	0.711861i	$-0.747853\pi$
$457$	−1.34065e8			−1.40464			−0.702321	+	0.711861i	$0.747853\pi$
$458$		0			0
$459$			1.31886e8i			1.36383i
$460$		0			0
$461$	1.38677e8			1.41547			0.707736	−	0.706477i	$-0.249716\pi$
$461$	1.38677e8			1.41547			0.707736	+	0.706477i	$0.249716\pi$
$462$		0			0
$463$	−2.06366e7			−0.207919			−0.103960	−	0.994582i	$-0.533151\pi$
$463$	−2.06366e7			−0.207919			−0.103960	+	0.994582i	$0.533151\pi$
$464$		0			0
$465$	1.09733e8			1.09139
$466$		0			0
$467$	6.11181e7			0.600094			0.300047	−	0.953924i	$-0.402998\pi$
$467$	6.11181e7			0.600094			0.300047	+	0.953924i	$0.402998\pi$
$468$		0			0
$469$			2.02020e8i			1.95829i
$470$		0			0
$471$			6.61728e7i			0.633311i
$472$		0			0
$473$	1.05897e8			1.00070
$474$		0			0
$475$	−3.26387e7	+	7.74401e7i	−0.304545	+	0.722578i
$476$		0			0
$477$		−	6.57535e7i		−	0.605848i
$478$		0			0
$479$	6.56456e7			0.597309			0.298655	−	0.954361i	$-0.403462\pi$
$479$	6.56456e7			0.597309			0.298655	+	0.954361i	$0.403462\pi$
$480$		0			0
$481$	2.23360e8			2.00711
$482$		0			0
$483$			3.51754e7i			0.312175i
$484$		0			0
$485$			5.82055e7i			0.510198i
$486$		0			0
$487$			1.06958e8i			0.926031i	0.886350	+	0.463016i	$0.153233\pi$
$487$			1.06958e8i			0.926031i	−0.886350	+	0.463016i	$0.846767\pi$
$488$		0			0
$489$			7.23465e7i			0.618716i
$490$		0			0
$491$	5.43653e6			0.0459280			0.0229640	−	0.999736i	$-0.492690\pi$
$491$	5.43653e6			0.0459280			0.0229640	+	0.999736i	$0.492690\pi$
$492$		0			0
$493$			2.40427e8i			2.00652i
$494$		0			0
$495$	−5.95849e7			−0.491270
$496$		0			0
$497$		−	4.08748e7i		−	0.332955i
$498$		0			0
$499$	6.74644e7			0.542966			0.271483	−	0.962443i	$-0.412486\pi$
$499$	6.74644e7			0.542966			0.271483	+	0.962443i	$0.412486\pi$
$500$		0			0
$501$	−1.11586e8			−0.887354
$502$		0			0
$503$	1.16070e7			0.0912041			0.0456021	−	0.998960i	$-0.485479\pi$
$503$	1.16070e7			0.0912041			0.0456021	+	0.998960i	$0.485479\pi$
$504$		0			0
$505$	−2.35318e8			−1.82718
$506$		0			0
$507$		−	8.50863e7i		−	0.652884i
$508$		0			0
$509$		−	2.76714e7i		−	0.209835i	−0.994481	−	0.104917i	$-0.966542\pi$
$509$		−	2.76714e7i		−	0.209835i	0.994481	−	0.104917i	$-0.0334578\pi$
$510$		0			0
$511$	2.88120e8			2.15929
$512$		0			0
$513$	−1.26192e8	−	5.31862e7i	−0.934717	−	0.393955i
$514$		0			0
$515$			1.07930e8i			0.790173i
$516$		0			0
$517$	8.24207e7			0.596437
$518$		0			0
$519$	−1.01268e7			−0.0724383
$520$		0			0
$521$			4.77798e7i			0.337856i	0.985628	+	0.168928i	$0.0540305\pi$
$521$			4.77798e7i			0.337856i	−0.985628	+	0.168928i	$0.945970\pi$
$522$		0			0
$523$			1.55607e8i			1.08774i	0.839170	+	0.543869i	$0.183041\pi$
$523$			1.55607e8i			1.08774i	−0.839170	+	0.543869i	$0.816959\pi$
$524$		0			0
$525$			1.26193e8i			0.872084i
$526$		0			0
$527$		−	2.53023e8i		−	1.72873i
$528$		0			0
$529$	−1.36372e8			−0.921212
$530$		0			0
$531$			3.75445e7i			0.250762i
$532$		0			0
$533$	2.39733e8			1.58324
$534$		0			0
$535$		−	1.96847e8i		−	1.28549i
$536$		0			0
$537$	1.66531e8			1.07541
$538$		0			0
$539$	−1.99279e8			−1.27261
$540$		0			0
$541$	1.03400e8			0.653024			0.326512	−	0.945193i	$-0.394127\pi$
$541$	1.03400e8			0.653024			0.326512	+	0.945193i	$0.394127\pi$
$542$		0			0
$543$	−1.06524e8			−0.665347
$544$		0			0
$545$		−	2.79964e8i		−	1.72947i
$546$		0			0
$547$			6.69553e7i			0.409094i	0.978857	+	0.204547i	$0.0655722\pi$
$547$			6.69553e7i			0.409094i	−0.978857	+	0.204547i	$0.934428\pi$
$548$		0			0
$549$	6.44694e7			0.389616
$550$		0			0
$551$	−2.30047e8	−	9.69579e7i	−1.37519	−	0.579600i
$552$		0			0
$553$		−	4.28528e8i		−	2.53399i
$554$		0			0
$555$	2.04556e8			1.19656
$556$		0			0
$557$	1.29628e8			0.750125			0.375063	−	0.926999i	$-0.377621\pi$
$557$	1.29628e8			0.750125			0.375063	+	0.926999i	$0.377621\pi$
$558$		0			0
$559$		−	4.03410e8i		−	2.30946i
$560$		0			0
$561$		−	9.30754e7i		−	0.527165i
$562$		0			0
$563$		−	7.15428e7i		−	0.400904i	−0.979704	−	0.200452i	$-0.935759\pi$
$563$		−	7.15428e7i		−	0.400904i	0.979704	−	0.200452i	$-0.0642411\pi$
$564$		0			0
$565$			2.18329e8i			1.21051i
$566$		0			0
$567$	−1.54627e7			−0.0848277
$568$		0			0
$569$		−	3.64539e8i		−	1.97882i	−0.145134	−	0.989412i	$-0.546361\pi$
$569$		−	3.64539e8i		−	1.97882i	0.145134	−	0.989412i	$-0.453639\pi$
$570$		0			0
$571$	−1.18991e8			−0.639153			−0.319576	−	0.947560i	$-0.603541\pi$
$571$	−1.18991e8			−0.639153			−0.319576	+	0.947560i	$0.603541\pi$
$572$		0			0
$573$		−	1.14512e7i		−	0.0608675i
$574$		0			0
$575$	4.18431e7			0.220100
$576$		0			0
$577$	−1.21851e8			−0.634312			−0.317156	−	0.948373i	$-0.602728\pi$
$577$	−1.21851e8			−0.634312			−0.317156	+	0.948373i	$0.602728\pi$
$578$		0			0
$579$	−1.64261e8			−0.846249
$580$		0			0
$581$	−2.10515e8			−1.07338
$582$		0			0
$583$			1.24244e8i			0.627003i
$584$		0			0
$585$			2.26985e8i			1.13378i
$586$		0			0
$587$	−1.95435e8			−0.966248			−0.483124	−	0.875552i	$-0.660498\pi$
$587$	−1.95435e8			−0.966248			−0.483124	+	0.875552i	$0.660498\pi$
$588$		0			0
$589$	2.42099e8	+	1.02037e8i	1.18480	+	0.499360i
$590$		0			0
$591$			2.23104e8i			1.08080i
$592$		0			0
$593$	−3.49229e7			−0.167474			−0.0837369	−	0.996488i	$-0.526686\pi$
$593$	−3.49229e7			−0.167474			−0.0837369	+	0.996488i	$0.526686\pi$
$594$		0			0
$595$	6.62055e8			3.14299
$596$		0			0
$597$			2.11183e8i			0.992510i
$598$		0			0
$599$		−	3.74989e8i		−	1.74477i	−0.488821	−	0.872384i	$-0.662573\pi$
$599$		−	3.74989e8i		−	1.74477i	0.488821	−	0.872384i	$-0.337427\pi$
$600$		0			0
$601$			2.81030e8i			1.29458i	0.762243	+	0.647291i	$0.224098\pi$
$601$			2.81030e8i			1.29458i	−0.762243	+	0.647291i	$0.775902\pi$
$602$		0			0
$603$			1.46260e8i			0.667073i
$604$		0			0
$605$	−1.83200e8			−0.827291
$606$		0			0
$607$			8.16439e7i			0.365055i	0.983201	+	0.182527i	$0.0584278\pi$
$607$			8.16439e7i			0.365055i	−0.983201	+	0.182527i	$0.941572\pi$
$608$		0			0
$609$	−3.74875e8			−1.65972
$610$		0			0
$611$		−	3.13977e8i		−	1.37649i
$612$		0			0
$613$	−2.57661e8			−1.11858			−0.559289	−	0.828972i	$-0.688926\pi$
$613$	−2.57661e8			−1.11858			−0.559289	+	0.828972i	$0.688926\pi$
$614$		0			0
$615$	2.19551e8			0.943864
$616$		0			0
$617$	−2.37541e8			−1.01131			−0.505653	−	0.862737i	$-0.668748\pi$
$617$	−2.37541e8			−1.01131			−0.505653	+	0.862737i	$0.668748\pi$
$618$		0			0
$619$	−5.64614e7			−0.238056			−0.119028	−	0.992891i	$-0.537978\pi$
$619$	−5.64614e7			−0.238056			−0.119028	+	0.992891i	$0.537978\pi$
$620$		0			0
$621$			6.81853e7i			0.284719i
$622$		0			0
$623$			2.75755e8i			1.14040i
$624$		0			0
$625$	−2.85466e8			−1.16927
$626$		0			0
$627$	8.90568e7	+	3.75348e7i	0.361297	+	0.152276i
$628$		0			0
$629$		−	4.71665e8i		−	1.89532i
$630$		0			0
$631$	1.43549e8			0.571362			0.285681	−	0.958325i	$-0.407780\pi$
$631$	1.43549e8			0.571362			0.285681	+	0.958325i	$0.407780\pi$
$632$		0			0
$633$	6.43703e7			0.253790
$634$		0			0
$635$			1.49714e8i			0.584710i
$636$		0			0
$637$			7.59143e8i			2.93701i
$638$		0			0
$639$		−	2.95928e7i		−	0.113418i
$640$		0			0
$641$		−	1.41422e8i		−	0.536961i	−0.963285	−	0.268481i	$-0.913478\pi$
$641$		−	1.41422e8i		−	0.536961i	0.963285	−	0.268481i	$-0.0865215\pi$
$642$		0			0
$643$	−2.92646e8			−1.10080			−0.550401	−	0.834901i	$-0.685525\pi$
$643$	−2.92646e8			−1.10080			−0.550401	+	0.834901i	$0.685525\pi$
$644$		0			0
$645$		−	3.69448e8i		−	1.37681i
$646$		0			0
$647$	3.72347e8			1.37478			0.687392	−	0.726286i	$-0.258755\pi$
$647$	3.72347e8			1.37478			0.687392	+	0.726286i	$0.258755\pi$
$648$		0			0
$649$		−	7.09418e7i		−	0.259518i
$650$		0			0
$651$	3.94514e8			1.42995
$652$		0			0
$653$	−2.00301e8			−0.719355			−0.359678	−	0.933077i	$-0.617113\pi$
$653$	−2.00301e8			−0.719355			−0.359678	+	0.933077i	$0.617113\pi$
$654$		0			0
$655$	1.78326e8			0.634585
$656$		0			0
$657$	2.08595e8			0.735541
$658$		0			0
$659$		−	4.74776e8i		−	1.65895i	−0.558547	−	0.829473i	$-0.688641\pi$
$659$		−	4.74776e8i		−	1.65895i	0.558547	−	0.829473i	$-0.311359\pi$
$660$		0			0
$661$			3.42566e7i			0.118615i	0.998240	+	0.0593076i	$0.0188893\pi$
$661$			3.42566e7i			0.118615i	−0.998240	+	0.0593076i	$0.981111\pi$
$662$		0			0
$663$	−3.54565e8			−1.21662
$664$		0			0
$665$	−2.66989e8	+	6.33471e8i	−0.907880	+	2.15408i
$666$		0			0
$667$			1.24301e8i			0.418887i
$668$		0			0
$669$	1.28768e8			0.430060
$670$		0			0
$671$	−1.21818e8			−0.403220
$672$		0			0
$673$		−	5.41235e8i		−	1.77558i	−0.460246	−	0.887791i	$-0.652239\pi$
$673$		−	5.41235e8i		−	1.77558i	0.460246	−	0.887791i	$-0.347761\pi$
$674$		0			0
$675$			2.44618e8i			0.795383i
$676$		0			0
$677$		−	1.93475e8i		−	0.623532i	−0.950159	−	0.311766i	$-0.899080\pi$
$677$		−	1.93475e8i		−	0.623532i	0.950159	−	0.311766i	$-0.100920\pi$
$678$		0			0
$679$			2.09261e8i			0.668465i
$680$		0			0
$681$	1.13166e8			0.358323
$682$		0			0
$683$		−	1.50059e8i		−	0.470976i	−0.971877	−	0.235488i	$-0.924331\pi$
$683$		−	1.50059e8i		−	0.470976i	0.971877	−	0.235488i	$-0.0756688\pi$
$684$		0			0
$685$	3.10876e8			0.967199
$686$		0			0
$687$		−	3.16043e8i		−	0.974710i
$688$		0			0
$689$	4.73300e8			1.44703
$690$		0			0
$691$	−4.37586e8			−1.32626			−0.663131	−	0.748504i	$-0.730773\pi$
$691$	−4.37586e8			−1.32626			−0.663131	+	0.748504i	$0.730773\pi$
$692$		0			0
$693$	−2.14220e8			−0.643665
$694$		0			0
$695$	−3.64166e8			−1.08479
$696$		0			0
$697$		−	5.06239e8i		−	1.49506i
$698$		0			0
$699$		−	2.76630e8i		−	0.809968i
$700$		0			0
$701$	−1.35620e8			−0.393704			−0.196852	−	0.980433i	$-0.563072\pi$
$701$	−1.35620e8			−0.393704			−0.196852	+	0.980433i	$0.563072\pi$
$702$		0			0
$703$	4.51301e8	+	1.90210e8i	1.29897	+	0.547478i
$704$		0			0
$705$		−	2.87544e8i		−	0.820609i
$706$		0			0
$707$	−8.46016e8			−2.39398
$708$		0			0
$709$	−4.80711e8			−1.34879			−0.674396	−	0.738370i	$-0.735596\pi$
$709$	−4.80711e8			−1.34879			−0.674396	+	0.738370i	$0.735596\pi$
$710$		0			0
$711$		−	3.10248e8i		−	0.863179i
$712$		0			0
$713$		−	1.30813e8i		−	0.360896i
$714$		0			0
$715$		−	4.28897e8i		−	1.17337i
$716$		0			0
$717$			1.36864e7i			0.0371306i
$718$		0			0
$719$	1.71431e8			0.461215			0.230608	−	0.973047i	$-0.425929\pi$
$719$	1.71431e8			0.461215			0.230608	+	0.973047i	$0.425929\pi$
$720$		0			0
$721$			3.88032e8i			1.03529i
$722$		0			0
$723$	2.75033e8			0.727730
$724$		0			0
$725$			4.45935e8i			1.17019i
$726$		0			0
$727$	2.21362e8			0.576102			0.288051	−	0.957615i	$-0.406993\pi$
$727$	2.21362e8			0.576102			0.288051	+	0.957615i	$0.406993\pi$
$728$		0			0
$729$	−2.60931e8			−0.673510
$730$		0			0
$731$	−8.51872e8			−2.18083
$732$		0			0
$733$	2.30440e8			0.585120			0.292560	−	0.956247i	$-0.405493\pi$
$733$	2.30440e8			0.585120			0.292560	+	0.956247i	$0.405493\pi$
$734$		0			0
$735$			6.95232e8i			1.75093i
$736$		0			0
$737$		−	2.76364e8i		−	0.690366i
$738$		0			0
$739$	3.01741e8			0.747656			0.373828	−	0.927498i	$-0.378045\pi$
$739$	3.01741e8			0.747656			0.373828	+	0.927498i	$0.378045\pi$
$740$		0			0
$741$	1.42986e8	−	3.39256e8i	0.351431	−	0.833822i
$742$		0			0
$743$		−	1.69529e8i		−	0.413312i	−0.978414	−	0.206656i	$-0.933742\pi$
$743$		−	1.69529e8i		−	0.413312i	0.978414	−	0.206656i	$-0.0662581\pi$
$744$		0			0
$745$	−6.80854e8			−1.64659
$746$		0			0
$747$	−1.52410e8			−0.365638
$748$		0			0
$749$		−	7.07707e8i		−	1.68425i
$750$		0			0
$751$		−	4.49431e8i		−	1.06107i	−0.847664	−	0.530533i	$-0.821992\pi$
$751$		−	4.49431e8i		−	1.06107i	0.847664	−	0.530533i	$-0.178008\pi$
$752$		0			0
$753$			9.37550e7i			0.219588i
$754$		0			0
$755$		−	5.36703e8i		−	1.24708i
$756$		0			0
$757$	−2.84019e8			−0.654726			−0.327363	−	0.944899i	$-0.606160\pi$
$757$	−2.84019e8			−0.654726			−0.327363	+	0.944899i	$0.606160\pi$
$758$		0			0
$759$		−	4.81200e7i		−	0.110053i
$760$		0			0
$761$	1.13112e8			0.256659			0.128329	−	0.991732i	$-0.459039\pi$
$761$	1.13112e8			0.256659			0.128329	+	0.991732i	$0.459039\pi$
$762$		0			0
$763$		−	1.00653e9i		−	2.26596i
$764$		0			0
$765$	4.79319e8			1.07063
$766$		0			0
$767$	−2.70249e8			−0.598932
$768$		0			0
$769$	5.96029e6			0.0131066			0.00655328	−	0.999979i	$-0.497914\pi$
$769$	5.96029e6			0.0131066			0.00655328	+	0.999979i	$0.497914\pi$
$770$		0			0
$771$	−6.20247e6			−0.0135332
$772$		0			0
$773$		−	4.08609e8i		−	0.884646i	−0.896856	−	0.442323i	$-0.854154\pi$
$773$		−	4.08609e8i		−	0.884646i	0.896856	−	0.442323i	$-0.145846\pi$
$774$		0			0
$775$		−	4.69297e8i		−	1.00819i
$776$		0			0
$777$	7.35421e8			1.56774
$778$		0			0
$779$	4.84382e8	+	2.04153e8i	1.02465	+	0.431860i
$780$		0			0
$781$			5.59167e7i			0.117379i
$782$		0			0
$783$	−7.26671e8			−1.51375
$784$		0			0
$785$	6.43911e8			1.33112
$786$		0			0
$787$			6.96762e8i			1.42942i	0.699420	+	0.714711i	$0.253442\pi$
$787$			6.96762e8i			1.42942i	−0.699420	+	0.714711i	$0.746558\pi$
$788$		0			0
$789$		−	1.26828e8i		−	0.258217i
$790$		0			0
$791$			7.84939e8i			1.58601i
$792$		0			0
$793$			4.64057e8i			0.930575i
$794$		0			0
$795$	4.33454e8			0.862664
$796$		0			0
$797$		−	2.65404e8i		−	0.524243i	−0.965035	−	0.262121i	$-0.915578\pi$
$797$		−	2.65404e8i		−	0.524243i	0.965035	−	0.262121i	$-0.0844221\pi$
$798$		0			0
$799$	−6.63017e8			−1.29982
$800$		0			0
$801$			1.99643e8i			0.388468i
$802$		0			0
$803$	−3.94148e8			−0.761225
$804$		0			0
$805$	3.42283e8			0.656141
$806$		0			0
$807$	3.80660e8			0.724297
$808$		0			0
$809$	−7.88461e7			−0.148914			−0.0744569	−	0.997224i	$-0.523722\pi$
$809$	−7.88461e7			−0.148914			−0.0744569	+	0.997224i	$0.523722\pi$
$810$		0			0
$811$		−	5.18284e7i		−	0.0971640i	−0.998819	−	0.0485820i	$-0.984530\pi$
$811$		−	5.18284e7i		−	0.0971640i	0.998819	−	0.0485820i	$-0.0154702\pi$
$812$		0			0
$813$		−	3.48172e8i		−	0.647922i
$814$		0			0
$815$	7.03986e8			1.30044
$816$		0			0
$817$	3.43537e8	−	8.15092e8i	0.629952	−	1.49465i
$818$		0			0
$819$			8.16057e8i			1.48549i
$820$		0			0
$821$	−5.31582e8			−0.960597			−0.480298	−	0.877105i	$-0.659472\pi$
$821$	−5.31582e8			−0.960597			−0.480298	+	0.877105i	$0.659472\pi$
$822$		0			0
$823$	3.44247e8			0.617548			0.308774	−	0.951136i	$-0.400081\pi$
$823$	3.44247e8			0.617548			0.308774	+	0.951136i	$0.400081\pi$
$824$		0			0
$825$		−	1.72633e8i		−	0.307440i
$826$		0			0
$827$		−	4.60383e8i		−	0.813959i	−0.913437	−	0.406980i	$-0.866582\pi$
$827$		−	4.60383e8i		−	0.813959i	0.913437	−	0.406980i	$-0.133418\pi$
$828$		0			0
$829$			6.90551e8i			1.21208i	0.795433	+	0.606041i	$0.207243\pi$
$829$			6.90551e8i			1.21208i	−0.795433	+	0.606041i	$0.792757\pi$
$830$		0			0
$831$			2.01106e8i			0.350446i
$832$		0			0
$833$	1.60306e9			2.77342
$834$		0			0
$835$			1.08582e9i			1.86508i
$836$		0			0
$837$	7.64741e8			1.30418
$838$		0			0
$839$		−	1.66414e8i		−	0.281775i	−0.990026	−	0.140888i	$-0.955004\pi$
$839$		−	1.66414e8i		−	0.281775i	0.990026	−	0.140888i	$-0.0449957\pi$
$840$		0			0
$841$	−7.29891e8			−1.22707
$842$		0			0
$843$	2.89494e7			0.0483233
$844$		0			0
$845$	−8.27953e8			−1.37226
$846$		0			0
$847$	−6.58640e8			−1.08392
$848$		0			0
$849$			6.55422e8i			1.07102i
$850$		0			0
$851$		−	2.43851e8i		−	0.395672i
$852$		0			0
$853$	−5.07499e7			−0.0817689			−0.0408844	−	0.999164i	$-0.513018\pi$
$853$	−5.07499e7			−0.0817689			−0.0408844	+	0.999164i	$0.513018\pi$
$854$		0			0
$855$	−1.93296e8	+	4.58624e8i	−0.309261	+	0.733767i
$856$		0			0
$857$			6.23537e8i			0.990649i	0.868708	+	0.495325i	$0.164951\pi$
$857$			6.23537e8i			0.990649i	−0.868708	+	0.495325i	$0.835049\pi$
$858$		0			0
$859$	1.71151e8			0.270023			0.135011	−	0.990844i	$-0.456893\pi$
$859$	1.71151e8			0.270023			0.135011	+	0.990844i	$0.456893\pi$
$860$		0			0
$861$	7.89330e8			1.23666
$862$		0			0
$863$			1.11208e9i			1.73023i	0.501576	+	0.865114i	$0.332754\pi$
$863$			1.11208e9i			1.73023i	−0.501576	+	0.865114i	$0.667246\pi$
$864$		0			0
$865$			9.85409e7i			0.152254i
$866$		0			0
$867$			3.34564e8i			0.513359i
$868$		0			0
$869$			5.86227e8i			0.893319i
$870$		0			0
$871$	−1.05279e9			−1.59327
$872$		0			0
$873$			1.51502e8i			0.227706i
$874$		0			0
$875$	−3.38045e8			−0.504603
$876$		0			0
$877$		−	6.02801e8i		−	0.893665i	−0.894618	−	0.446833i	$-0.852552\pi$
$877$		−	6.02801e8i		−	0.893665i	0.894618	−	0.446833i	$-0.147448\pi$
$878$		0			0
$879$	−1.20965e8			−0.178112
$880$		0			0
$881$	−1.14813e9			−1.67905			−0.839527	−	0.543318i	$-0.817168\pi$
$881$	−1.14813e9			−1.67905			−0.839527	+	0.543318i	$0.817168\pi$
$882$		0			0
$883$	6.59311e8			0.957654			0.478827	−	0.877909i	$-0.341062\pi$
$883$	6.59311e8			0.957654			0.478827	+	0.877909i	$0.341062\pi$
$884$		0			0
$885$	−2.47497e8			−0.357059
$886$		0			0
$887$		−	7.63756e7i		−	0.109442i	−0.998502	−	0.0547210i	$-0.982573\pi$
$887$		−	7.63756e7i		−	0.109442i	0.998502	−	0.0547210i	$-0.0174269\pi$
$888$		0			0
$889$			5.38251e8i			0.766090i
$890$		0			0
$891$	2.11531e7			0.0299047
$892$		0			0
$893$	2.67377e8	−	6.34391e8i	0.375465	−	0.890846i
$894$		0			0
$895$		−	1.62047e9i		−	2.26033i
$896$		0			0
$897$	−1.83310e8			−0.253986
$898$		0			0
$899$	1.39411e9			1.91875
$900$		0			0
$901$		−	9.99457e8i		−	1.36644i
$902$		0			0
$903$		−	1.32824e9i		−	1.80390i
$904$		0			0
$905$			1.03656e9i			1.39845i
$906$		0			0
$907$		−	1.06677e8i		−	0.142971i	−0.997442	−	0.0714853i	$-0.977226\pi$
$907$		−	1.06677e8i		−	0.142971i	0.997442	−	0.0714853i	$-0.0227739\pi$
$908$		0			0
$909$	−6.12503e8			−0.815487
$910$		0			0
$911$			6.94052e8i			0.917988i	0.888439	+	0.458994i	$0.151790\pi$
$911$			6.94052e8i			0.917988i	−0.888439	+	0.458994i	$0.848210\pi$
$912$		0			0
$913$	2.87985e8			0.378406
$914$		0			0
$915$			4.24989e8i			0.554771i
$916$		0			0
$917$	6.41117e8			0.831437
$918$		0			0
$919$	6.81619e8			0.878203			0.439102	−	0.898437i	$-0.355297\pi$
$919$	6.81619e8			0.878203			0.439102	+	0.898437i	$0.355297\pi$
$920$		0			0
$921$	4.70256e8			0.601943
$922$		0			0
$923$	2.13011e8			0.270893
$924$		0			0
$925$		−	8.74826e8i		−	1.10534i
$926$		0			0
$927$			2.80930e8i			0.352662i
$928$		0			0
$929$	−9.14995e8			−1.14123			−0.570613	−	0.821219i	$-0.693294\pi$
$929$	−9.14995e8			−1.14123			−0.570613	+	0.821219i	$0.693294\pi$
$930$		0			0
$931$	−6.46473e8	+	1.53385e9i	−0.801127	+	1.90079i
$932$		0			0
$933$			8.33864e7i			0.102672i
$934$		0			0
$935$	−9.05693e8			−1.10802
$936$		0			0
$937$	9.69261e7			0.117821			0.0589104	−	0.998263i	$-0.481237\pi$
$937$	9.69261e7			0.117821			0.0589104	+	0.998263i	$0.481237\pi$
$938$		0			0
$939$			6.36081e8i			0.768273i
$940$		0			0
$941$			7.66604e8i			0.920031i	0.887911	+	0.460016i	$0.152156\pi$
$941$			7.66604e8i			0.920031i	−0.887911	+	0.460016i	$0.847844\pi$
$942$		0			0
$943$		−	2.61726e8i		−	0.312113i
$944$		0			0
$945$			2.00101e9i			2.37112i
$946$		0			0
$947$	7.76785e8			0.914641			0.457321	−	0.889302i	$-0.348809\pi$
$947$	7.76785e8			0.914641			0.457321	+	0.889302i	$0.348809\pi$
$948$		0			0
$949$			1.50148e9i			1.75680i
$950$		0			0
$951$	5.97387e8			0.694568
$952$		0			0
$953$		−	4.54292e8i		−	0.524876i	−0.964949	−	0.262438i	$-0.915474\pi$
$953$		−	4.54292e8i		−	0.524876i	0.964949	−	0.262438i	$-0.0845265\pi$
$954$		0			0
$955$	−1.11428e8			−0.127934
$956$		0			0
$957$	5.12830e8			0.585110
$958$		0			0
$959$	1.11766e9			1.26723
$960$		0			0
$961$	−5.79647e8			−0.653121
$962$		0			0
$963$		−	5.12370e8i		−	0.573726i
$964$		0			0
$965$			1.59838e9i			1.77868i
$966$		0			0
$967$	−8.05156e8			−0.890432			−0.445216	−	0.895423i	$-0.646873\pi$
$967$	−8.05156e8			−0.890432			−0.445216	+	0.895423i	$0.646873\pi$
$968$		0			0
$969$	−7.16400e8	−	3.01941e8i	−0.787380	−	0.331857i
$970$		0			0
$971$			2.84869e8i			0.311163i	0.987823	+	0.155581i	$0.0497251\pi$
$971$			2.84869e8i			0.311163i	−0.987823	+	0.155581i	$0.950275\pi$
$972$		0			0
$973$	−1.30925e9			−1.42129
$974$		0			0
$975$	−6.57633e8			−0.709529
$976$		0			0
$977$			4.70813e8i			0.504853i	0.967616	+	0.252427i	$0.0812286\pi$
$977$			4.70813e8i			0.504853i	−0.967616	+	0.252427i	$0.918771\pi$
$978$		0			0
$979$		−	3.77233e8i		−	0.402033i
$980$		0			0
$981$		−	7.28713e8i		−	0.771879i
$982$		0			0
$983$		−	1.42855e9i		−	1.50396i	−0.659187	−	0.751979i	$-0.729099\pi$
$983$		−	1.42855e9i		−	1.50396i	0.659187	−	0.751979i	$-0.270901\pi$
$984$		0			0
$985$	2.17097e9			2.27167
$986$		0			0
$987$		−	1.03378e9i		−	1.07517i
$988$		0			0
$989$	−4.40418e8			−0.455277
$990$		0			0
$991$		−	2.01089e7i		−	0.0206617i	−0.999947	−	0.0103309i	$-0.996712\pi$
$991$		−	2.01089e7i		−	0.0206617i	0.999947	−	0.0103309i	$-0.00328847\pi$
$992$		0			0
$993$	−6.99836e8			−0.714741
$994$		0			0
$995$	2.05496e9			2.08610
$996$		0			0
$997$	2.19010e8			0.220993			0.110497	−	0.993877i	$-0.464756\pi$
$997$	2.19010e8			0.220993			0.110497	+	0.993877i	$0.464756\pi$
$998$		0			0
$999$	1.42557e9			1.42985

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.7.e.f.113.20		30
4.3	odd	2		152.7.e.a.113.11	✓	30
19.18	odd	2	inner	304.7.e.f.113.11		30
76.75	even	2		152.7.e.a.113.20	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.7.e.a.113.11	✓	30	4.3	odd	2
152.7.e.a.113.20	yes	30	76.75	even	2
304.7.e.f.113.11		30	19.18	odd	2	inner
304.7.e.f.113.20		30	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+17.1584i q^{3} +166.964 q^{5} +600.271 q^{7} +434.588 q^{9} +O(q^{10})\)	\(q+17.1584i q^{3} +166.964 q^{5} +600.271 q^{7} +434.588 q^{9} -821.172 q^{11} +3128.21i q^{13} +2864.85i q^{15} +6605.76 q^{17} +(-2663.92 + 6320.55i) q^{19} +10299.7i q^{21} +3415.18 q^{23} +12252.1 q^{25} +19965.4i q^{27} +36396.6i q^{29} -38303.4i q^{31} -14090.0i q^{33} +100224. q^{35} -71402.1i q^{37} -53675.1 q^{39} -76636.1i q^{41} -128959. q^{43} +72560.7 q^{45} -100370. q^{47} +242677. q^{49} +113345. i q^{51} -151301. i q^{53} -137107. q^{55} +(-108451. - 45708.8i) q^{57} +86390.9i q^{59} +148346. q^{61} +260871. q^{63} +522299. i q^{65} +336548. i q^{67} +58599.2i q^{69} -68093.8i q^{71} +479983. q^{73} +210227. i q^{75} -492926. q^{77} -713891. i q^{79} -25759.6 q^{81} -350700. q^{83} +1.10293e6 q^{85} -624509. q^{87} +459383. i q^{89} +1.87777e6i q^{91} +657227. q^{93} +(-444781. + 1.05531e6i) q^{95} +348610. i q^{97} -356872. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 720 q^{7} - 8670 q^{9}+O(q^{10}) \) 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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